Modeling Bio Electrical

f

VH

-V' . Jeq(r')dY' [r - r']

(2.18)

Exactly as Eq. (2.4) was rewritten as Eq. (2.9), we may rewrite Eq. (2.18) as
_1_ f Jeq(r') . V' (_1-) av' 4rrg [r - r'l e

VH

(2.19)

The Forward Problem of Electrocardiography

49

thereby identifying Jeq as an equivalent current-dipole density. Once again this interpretation only holds if the interstitial conductivity ge is homogeneous. Alternative formulations for Jeq based on Eqs. (2.13) and (2.14) are also possible. For the former, (2.20) and acts in an interstitial domain of homogeneous effective conductivity g.; for the latter, (2.21) and acts in an interstitial domain of homogeneous effective conductivity g. The above field equations were written assuming an infinite homogeneous extent for the bidomain myocardium. With the finite heart in the torso, this is evidently not the case. In particular, we have a bidomain-monodomain interface between heart and torso, and deciding which boundary conditions apply at this interface is not immediately evident. Three boundary conditions are needed. The first two are the continuity of the interstitial potential e at the epicardial surface to the torso potential <1>0 just outside the heart, and of the normal component of the total current that crosses over from the heart to the torso. In mathematical terms, these two conditions at the heart-torso interface may be expressed as

e

ae,

gi -

on

= <1>0

(2.22)

oe on

0<1>0

+ ge-- = 0'0-on

(2.23)

where the normal derivative ojon denotes the component of the gradient V· n along the outward normal n, and 0'0 denotes the torso conductivity just outside the interface. The third boundary condition generally used is that the intracellular current stops at the heart surface (Tung, 1978; Krassowska and Neu, 1994), so that we have, (2.24a) With this condition, only the normal component of the interstitial current in Eq. (2.23) crosses over into the torso. An alternative third boundary condition (Colli-Franzone et al., 1990) that has sometimes been employed is

ae,

g--

on

0<1>0

=0'0--

on

(2.24b)

This formulation is particularly convenient if Jeq is given by Eq. (2.21), since the interstitial medium in which this Jeq acts has conductivity g, and Eq. (2.24b) then simply expresses the continuity of the normal component of the current from this equivalent interstitial medium into the torso. Equations (2.16), (2.20) and (2.21) all identify Jeq as an equivalent current-dipole density per unit volume that exists wherever a spatial gradient of transmembrane or intracellular potential is present, for example, in the vicinity of a propagating excitation wavefront. An alternative equivalent surface current-dipole density that is placed on this excitation front

50

R. M. Gulrajani

FIGURE 2.3. Diagram illustrating the spatial distribution of the transmembrane potential as an excitation wavefront sweeps the myocardium along the direction indicated by the unit vector n. Reproduced, with permission, from Gulrajani (l998a).

can be derived using Eq. (2.16). Figure 2.3 shows such a propagating action potential wavefront of spatial extent d. Ahead of this front, the myocardial cells are at rest with Vm = V" where Vr denotes the resting transmembrane potential. Behind the front, the cells are depolarized with Vm = Vd, where Vd denotes the depolarized transmembrane potential. If we apply Eq. (2.16) to the transition region between the dashed lines, the total dipole moment p, associated with the wavefront of area A, is p = Jeq x Volume = -geq VVm x Ad ~

geq(Vd

-

Vr)An = geq VpAn

(2.25)

In Eq. (2.25), the gradient VVm has been approximated by (Vp/d)n, where Vp == Vd - Vr is the amplitude of the propagating action potential and the unit vector n denotes its direction of propagation. Equation (2.25) holds in the limit that the propagating wavefront is assumed infinitely thin and allows us to identify a surface dipole layer associated with the wavefront whose density is geq Vpn. Dipole orientations within the layer are everywhere normal to the wavefront, and, if the action potential amplitude is uniform everywhere, then so is the dipole density, whence the term uniform dipole layer used to describe this equivalent source.

2.2.2.2 Equations for an Anisotropic Bidomain-the Oblique Dipole Layer Equations (2.11) and (2.12) are rewritten for a homogeneous anisotropic bidomain by replacing the scalar effective conductivities gi and ge by their tensor equivalents G i and G e, respectively. We get,

v . (G i VJ

= I mv

V· (GeVe) = -Imv

(2.26) (2.27)

51

The Forward Problem of Electrocardiography

G; and G e are now the diagonal intracellular and inter stitial effect ive conductivity tensors, respectively, and can be written in 3 x 3 matrix form ,

(2.28)

where it is assumed that the cardiac fibers are oriented along the z axis. Although not strictly true (see Hooks et al., 2002), symmetry about this axis is generally assumed so that g ;x = g ;y and gex = gey' Note that G; V <1>; and G eVe in Eqs . (2.26) and (2.27), respectively, are vectors, being the product of a 3 x 3 tensor matrix with a 3 x 1 column matrix representation of the gradient vector. A major complication arises on account of the fiber rotation present in the real heart, which leads to an inhomogeneous myocardium. Under these circumstances, the conductivity tensors are diagonal only in a local coordinate system, characterized by the unit vectors e" e2, e3, where e3 is always oriented along the fiber direction. These local unit vectors may be expressed in terms of fixed global unit vectors ex, ey, e. , e.g., (2.29a) and reciprocally, the global unit vectors expressed in terms of the local unit vectors, e.g., (2.29b) Accordingly, a rotation matrix A, characterizing the transformation from local to glob al coordinates, may be written as

(2.30)

with the transposed matrix AT characterizing the reverse transformation from global to local coordinates. From the mathematical definition of a Cartesian tensor (Fung, 1977), it can be shown that the diagonal conductivity tensors G; and Ge in the local coordinate system transform to the symmetric conductivity tensors and G~ , respectively, in the global coordinate system that are given by the matrix products

G;

(2.31) Equations (2.26) and (2.27) continue to hold in global coordinates, but with the primed conductivity tensors G; and G~ replacing the unprimed tensors G; and G e . We have ,

V · (G;V; )

=

Imv

V . (G~ Ve) = - I m v

(2.32) (2.33)

52

R. M. Gulrajani

Note that the primed conductivity tensors vary from point to point in the myocardium with the change in fiber orientation. As for the isotropic case, the volume current density I m v may be eliminated from Eqs. (2.32) and (2.33), to yield (2.34) Upon adding V . (G;V

G;

G;

G;

G;.

(2.36)

+ n T G~ V

n T GoV
(2.37)

nTG;V
(2.38a)

n T G'V

(2.38b)

In Eqs. (2.37) and (2.38), each of the terms is a triple matrix product, involving a row matrix nT, a 3 x 3 conductivity matrix, and a gradient column matrix. That these Equations do express the continuity of the normal components of currents is evident if we note, for example, that the triple matrix product n T GoV
53

The Forward Problem of Electrocardiography

and giz = gil, where gil and gil are the transverse and longitudinal effective conductivities, then in a manner similar to that in which Eq. (2.25) was derived, we can show that the dipole moment p associated with a wavefront of area A is given by

where n I , nz and n3 are the components of the normal n to the wavefront in local coordinates. The above equation can be rewritten as (2.39) revealing that the oblique dipole consists of a component normal to the wavefront plus a second axial component along the fiber direction (Corbin and Scher, 1977; Colli-Franzone et al., 1982; 1983).

2.3 TORSO GEOMETRY REPRESENTATIONS Torso geometry may be represented either by a numerical discretization of its different interfaces, or by a full three-dimensional discretization of its entire volume. The choice is dictated by the solution methodology to be used (see Section 2.4). Thus in Figure 2.4, which shows a stylized torso, with interface discretization only the outer torso surface So, the lung surfaces Sj and Sz, the outer heart surface S3, and the blood-filled heart cavities S4 and Ss, are triangulated. The intervening regions are assumed of constant isotropic conductivity. If the anisotropic-conductivity skeletal muscle layer underlying the skin (not shown in Fig. 2.4) is to be included, it is first converted to an approximately-equivalent isotropic layer (McFee and Rush, 1968) prior to the triangulation of its interfaces. With volume discretization, the entire three-dimensional torso volume is modeled either in point-wise fashion as a collection of equispaced points, or in piece-wise fashion by a combination of tetrahedral and hexahedral (brick-shaped) elements. The major advantage of these volume representations is the ability to accurately model not only anisotropic conductivity regions, but also regions of continuously varying conductivity.

2.4 SOLUTION METHODOLOGIES FOR THE FORWARD PROBLEM Calculation of the torso potentials from heart source dipoles is done via one of two general approaches, namely surface methods or volume methods. As may be surmised, surface methods employ torso models with only the interfaces discretized, and obtain the potentials only on these interfaces. Volume methods, on the other hand, use full three-dimensional discretizations of the torso volume and obtain the potential everywhere. The number of potentials to be evaluated is accordingly much greater with volume methods, leading to large coefficient matrices. However, the potential at each point is expressed only in terms of its nearest neighbors, so that the matrices are sparse and may be inverted via sparse matrix solvers. With surface methods on the other hand, while the coefficient matrices are much smaller, the potential at any interface point is coupled to the potential at every other interface

54

R. M. Gulrajani

FIGURE 2.4. Torso with multiple regions of differing isotropic conductivity. See text for more details.

point, with the result that the coefficient matrix is fully populated and sparse matrix routines are inapplicable. Surface methods are also often termed "boundary-element methods" in the literature.

2.4.1 SURFACE METHODS 2.4.1.1 Solutions from Equivalent Dipoles Surface methods usually employ integral equations for the potentials to be calculated. These integral equations may be derived by applying Green's second identity to the torso geometry shown in Fig. 2.4, to obtain for the potential ¢k(r) at the observation point r on interface Sk (Barr et al., 1966),

(2.40)

The Forward Problem of Electrocardiography

55

where Jeq(r') denotes the equivalent dipole sources present in the heart myocardium VH, the torso interfaces S/ extend from 0 to N, (with internal and external conductivities (f/- and (f/+ , respectively), and dQrr' denotes the solid angle subtended at the observation point r by an element of the surface integral d S' at r' (dQ rr, = - I~~~; . tsd S'). The term involving the so-called "auto solid angle" (dQrr), representing the solid angle subtended at r by the surface element containing r, is excluded from the summation. The contribution of this auto solid angle term is 2rr if the surface around the observation point r is smooth, and has already been incorporated in the derivation of Eq. (2.40). The point to note is that the first term on the right-hand side in Eq. (2.40) is proportional to the infinite-medium potential due to the equivalent sources Jeq(r') (see Eq. 2.9). The second term explicitly represents the effect of the different torso interfaces. The simplest assumption with triangulated torso interfaces is to consider the potential as constant over each triangle face. With this constant potential assumption, the observation point r may be placed at the centroid of each triangle, and an equation such as Eq. (2.40) can be written as r is moved from triangle to triangle. The ensemble of NT equations, where NT is the total number of triangles, may be written in compact matrix form as (2.41 ) where is now an NT x I column matrix containing the triangle potentials, G is an NT x I matrix containing the first terms on the right ofEq. (2.40), and A is an NT x NT weighted solid angle matrix (with diagonal terms zero on account of the elimination of the auto solid angle term from the summation on the right-hand side in Eq. 2.40) that depends only on torso geometry and conductivities. Equation (2.41) can be rewritten as (I-A) = G , where 1 is an NT x NT identity matrix. A straightforward solution would entail inverting the coefficient matrix (I - A) to get = (I - A) - IG. Unfortunately, this is not possible as the matrix (I - A) is singular and does not possess an inverse. The singularity can be demonstrated mathematically (Lynn and Timlake , 1968), but it can be understood by realizing that potential measurements are never absolute but always with respect to a reference. Thus the potential is always indeterminate up to the constant value chosen for the reference potential, and mathematically the singularity of the coefficient matrix ensures that a unique solution for q, is not possible unless a choice for the reference potential is first made. In practice, the problem is circumvented by replacing A by a "deflated" matrix A * (the deflation procedure entails removing an eigenvalue x = 1 of A), so that we end up solving (I - A *)* = G. The coefficient matrix (I - A *) is no longer singular, and accordingly this equation can be inverted. Note, however, that * is not the original potential on account of the changed coefficient matrix. It can be shown that * = for triangles on the outer torso surface So, but for internal interfaces * and differ by a constant. It can also be shown that the deflation procedure sets the sum of the triangle potentials on So to zero, in effect establishing a zero reference for the potential equal to the mean of the triangle potentials on So. A more accurate procedure with triangulated torso interfaces is to assume a linear variation of the potential over each triangle interface, so as to better represent the varying interface potential. This leads to placing the observation point s r at triangle vertices rather than at triangle centroids. One immediate advantage is that for closed triangulated surfaces, the number of vertices is approximately half the number of triangles, so that the number of unknown potentials to be determined is also approximately half, leading to smaller

R. M. Gulrajani

56

matrix sizes . Alternatively, if the computer power is already present, we can use a finer triangulation doubling the number of triangle vertices and still keeping the number of unknown potentials the same . This vertex approach entails a slight modification of the governing integral equation (Eq. 2.40) as the auto solid angle is no longer 2][. This is because with an observation point r at a vertex, any small selected neighborhood 5E around r is no longer smooth, but subtends a difficult-to- calculate auto solid angle Q , s, at r . Equation (2.40) may be rewritten, but this time explicitly in terms of Q, s, . We get, [ak- (4][ - Q,sJ - a k+Q , s,lk(r) =

f

Jeq(r' ) . V' (Ir

~ rl) dV '

VH

(2.42)

As before, the summation on the right-hand side excludes the auto solid angle term, as this has already been moved over to the left-hand side. An equation such as Eq. (2.42) can be written as the observation point moves from vertex to vertex, and the ensemble of equations combined in matrix form . The terms on the left-hand side of this ensemble will eventually form the diagonal terms of the coefficient matrix that multiplies cPo Since the auto solid angle Q, s, is difficult to compute, and it only occurs in these diagonal terms, a common approximation is to set each diagonal term to be equal to the negative sum of the other terms in its row. This introduces a linear dependency between matrix columns, rendering the coefficient matrix singular, as mandated by the non-unique nature of the potential. Other approximations for Q,s, are discussed in Meijs et at. (1989), Heller (1990) and Wischmann et at. (1996) . As with the earlier centroid option, deflation is also needed with this vertex option for potential sites. An interesting variant of the traditional approach of expressing Jeq as the gradient of the transmembrane potential was derived by Geselowitz (1989) for the special situation of equal anisotropy, i.e., when the intracellular and interstitial conductivity tensors are proportional. In particular, for a heart with isotropic intracellular and interstitial conductivities (which is a degenerate case of equal anisotropy), Geselowitz showed that the equivalent dipole surface density could be repre sented as (2.43) where DH is a unit dipole perpendicular to all heart surfaces, epicardial as well as the endocardial surfaces adjacent to the blood masses (surfaces 5 3 ,54 and 55 in Fig. 2.4). Using this equivalent source representation simply converts the volume integral on the right hand side ofEq. (2.40) to a surface integral over these three heart surfaces. If we denote the union of these three surfaces by 5 H , we get

(2.44)

57

The Forward Problem of Electrocardiography

A similar conversion of the volume integral on the right-hand side of Eq. (2.42) also results when Eq. (2.43) is used as an equivalent source.

2.4.1.2 Solutions from Epicardial Potentials As mentioned in the Introduction, the forward problem can also be solved using the heart's epicardial potentials as the starting point (Barr et al., 1977). The governing equations may be determined by applying Green 's theorem to a torso model containing only the heart's epicardial surface S3 and the outer torso surface So (Fig. 2.4). By allowing the observation point r to first approach So, and then S3, the following two equations are obtained:

1 <1>B (r ) = - -

4n

f r1

,

,

- ,-V <1>E . DdS BE

If ,(I) , If ,(I) , f ,( I ) , If ,( I ) , + -If SE

+-

<1>EV

4n

~

1 <1> E(r) = - -

r BE

. DdS - -

4n

<1>BV

-,r BB

. DdS

(2.45)

<1>BV

- ,rEB

. DdS

(2.46)

~

1 , <1> E . DdS, - ,-V

4n

SE

4n

- ,-

rEE

<1>EV

~

- ,-

r EE

. DdS - -

4n

~

In Eqs . (2.45) and (2.46) , we have explicitly denoted epicardial and body surface potentials by the SUbscripts E and B, respectively, and the corresponding surfaces by SE(= S3) and SB(= So), respectively. The unit normals 0 to these surfaces are always outward. The scalar distance [r - r' 1is now denoted by r~E' etc., with the first subscript denoting the location of r and the second that of r' , and the prime on r~ E is used to simply reinforce the fact that the variable of integration is r'. Two sets of matrix equations result as the observation point r is moved from triangle to triangle on the body and heart surfaces, and these may be written in compact fashion as ABB~B +ABE~E +BBEr E =

(2.47)

AEB~B

(2.48)

+ AEE~E

0 + BEEr E = 0

In Eqs . (2.47) and (2.48), ~ B and ~ E are column matrices of body surface and epicardial potentials, T E is a column matrix of epicardial potential gradients, and the A and B coefficient matrices depend solely on integrations involving epicardial and body surface geometries. The first subscript on A (or B) indicates the surface on which the observation points are selected, and the second subscript whether the integration is over the epicardial or body surface. Equation (2.47) may be used to obtain an expression for r E , which when substituted in Eq. (2.48) yields: (2.49) The elements of T BE = [A BB - BB E(BEE)-IAEBr l[BBE(BEE)-IAEE - ABE] are the transfer coefficients relating the potential at a particular epicardial point to that at a particular

R. M. Gulrajani

58

torso surface point. As was the case for forward solutions using current dipoles , these points may be selected at triangle centroids, implying a potential that is constant over each triangle, or at triangle vertices with a potential that varies linearly over each triangle.

2.4.2 VOLUME METHODS 2.4.2.1 Finite-Difference Method The finite-difference method represents the torso volume by a three-dimensional array of regularly-spaced points or nodes that are connected to each other by intervening resistors, whose values are selected to best reflect the intervening resistance between the points. Kirchhoff's current law is written for each node, resulting in a large set of equations relating the potentials between adjacent nodes . In effect, the method represents a discrete approximation to the governing differential equation (Eq. 2.7), namely V . (aV
2.4.2.2 Finite-Element Method Here, the torso volume is represented by contiguous three-dimensional tetrahedra and/or hexahedra (brick-like elements). The finite-element method also solves Eq. (2.7), with I s v = - V . Jeq . The starting assumption is that the potential within each element can be approximated by r

(2.50)

(x ,y,z) = L,Bi(X ,y, Z)i i= l

where i denotes the potential at an element node, ,Bi is an appropriate interpolation polynomial (usually linear in x, y, and z, for tetrahedral elements), and r is the number of nodes for the element. Each ,Bi is equal to unity at node i, and is zero at all other element nodes. If we substitute Eq. (2.50) into Eq. (2.7), then on account of the approximation, we get V . (aV
+ Is v =

R

(2.51)

where a is the element conductivity and R denotes a residual. The technique of weighted residuals (Brebbia and Dominguez, 1992) is now invoked in an attempt to reduce R to zero , but in a "weak form" by reducing the set of weighted integrals below to zero:

f v

[V . (aV
+ Isv]WidV =

f v

RWidV

=0

i

= 1,2, ... . r

(2.52)

The Forward Problem of Electrocardiography

59

In Eq. (2.52), each Wi is a weighting polynomial, and the integration is over the element in question. Often Wi is set equal to the interpolating polynomial f3i, and we get the so-called Galerkin weighted-residual formulation, namely

f

[V' . (aV'¢)]f3i dV

v

+

f

i = 1,2, . . . . r

f3;Isv dV = 0

(2.53)

v

The first integral is now integrated by parts to obtain

-f S

f3i(aV'¢). odS +

f

(aV'¢)· V'f3i d V =

v

f

f3;Isv d V

i = 1,2, .... r

(2.54)

v

where the surface integral is over the bounding surfaces of the element and 0 is the unit outward normal. Consider, initially, internal volume elements that do not abut the outer torso surface So. The contribution of the surface integral in Eq. (2.54) will eventually be cancelled by similar terms from contiguous elements on account of the continuity of the normal component of the current, and because the f3i'S are selected such that the potential at a common interface is only determined by nodes on that interface. This last is done to ensure the continuity of the potential across the interface between elements. Both volume integrals in Eq. (2.54), however, need to be considered, with that on the right-hand side only contributing for elements where Isv is non-zero. Assuming tetrahedral elements and linear polynomials for the f3i, the set of r equations in Eq. (2.54) may be written in linear matrix fonn (2.55) where A (e) is an r x r coefficient matrix, ep(e) and F(e) are r x 1 column matrices, and the superscript (e) is used to denote that Eq. (2.55) holds for a particular element. The matrix ep(e)contains the element potentials to be computed, and F(e) is the matrix representation of the source terms and surface integrals of Eq. (2.54). The coefficient matrices from the different elements, internal as well as those that abut the outer torso surface, may be combined to result in a global matrix equation (2.56) where now A is m x m if there are m unknown potentials in the torso to be determined, and where ep and F are each m x 1. Surface integrals from contiguous elements will now cancel. For elements that abut the outer torso surface So, the surface integrals over the sides that form a part of So will remain uncancelled. These uncancelled surface integrals yield the set of surface integrals over So:

-f So

f3i(a V'¢) . tul S,

i = 1,2, . . . . m

(2.57)

60

R. M. Gulrajani

If the boundary conditions applicable on So are of the mixed form

=

a

(aV
(2.58a)

on SOl

= indS

on S02

(2.58b)

where a denotes the known potential on a portion SOl of So and in is the injected normal currentdensityoverthe remainingportion S02, then sincethe fJi reduceto zero over SOl there being no unknownpotentialsthere, Eq. (2.57) need only be integratedover S02. Substituting Eq. (2.58b) in Eq. (2.57), the uncancelled surface integrals become i = 1,2, . . . . m

(2.59)

Only the integralsfor i correspondingto surface nodes over S02 contribute. In buildingEq. (2.56), they form another source term due to the injectedcurrent, and add to the appropriate term fi in the matrix F. Thus, the so-called Neumann boundary condition of Eq. (2.58b) enters naturally into the finite-element formulation. The Dirichlet boundary condition of Eq. (2.58a), however, has to be introduced explicitly into the global matrix equation (Eq. 2.56). Thus if the potential at node k is a, then all matrix elements akl in row k are set equal to zero, except akk, which is set equal to unity; in addition fk is set equal to a' This also renders A non-singular. Since A is large, solutions to Eq. (2.56) may be obtained by iterative techniques, though many finite-element packages have direct solvers that exploit the sparse nature of A. Finally, the finite-element methodcan also be used to computebody surfacepotentials from epicardial potentials, in which case since the heart region is excluded, the volume integral on the right-hand side of Eq. (2.54) drops out. The uncancelled surface integrals of Eq. (2.57) are also zero since no current leaves the torso. Only the volume integral on the left-hand side of Eq. (2.54) remains, and leads to the global matrix equation A
f s

(a V <1» • DdS = -

f

IsvdV

(2.60)

v

Whereasthefinite-element methodsoughtto satisfyEq. (2.7),albeitin weakform,in a global manner over the entire torso volume, the finite-volume method aims to satisfy Eq. (2.60)

The Forward Problem of Electrocardiography

61

locally, over each torso element. These local elements are usually selected to be small cubic volumes or cells, over each of which Eq. (2.60) must be satisfied. In addition, continuity of the normal component of the current between cells must be ensured. Rosenfeld et al. (1996) describe the mechanics of solving Eq. (2.60), and of ensuring that the continuity condition is satisfied, by approximating the gradient V required in Eq. (2.60) by its integral definition (2.61)

The use of Eq. (2.61) leads to Eq. (2.60) being approximated over each cell by a linear equation involving the unknown potentials, at the center of the cell in question, and at the centers of its nearest neighbors. Eventually, when the equations from all the cells are combined, a linear matrix equation of the type A ep = F results, where the coefficient matrix A is again sparse. The non-unique nature of the potential is handled by applying the additional condition that Gauss' flux theorem should be satisfied over the entire volume. A better approximation than Eq. (2.61) for V can be obtained if, similar to the finite-element method, we start with an approximating equation such as Eq. (2.50) for the potential (Harrild and Henriquez, 1997).

2.4.3 COMBINATION METHODS Surface and volume methods may be combined such that the former is used where the volume conductor is isotropic and the latter where it is anisotropic. This aims to take advantage of the reduced computational load of surface methods, and at the same time be able to accurately represent anisotropic conductivities. Such a combination was illustrated by Stanley and Pilkington (1989), where torso potentials were computed from epicardial potentials in the presence of an anisotropic skeletal-muscle layer. A transfer-coefficient matrix between epicardial potentials and the inner surface of the skeletal muscle, using Eq. (2.49), was first determined from a boundary-element discretization ofthe torso up to this inner layer. The finite-element method was then used to represent the anisotropic skeletalmuscle and proceed from the potentials on this inner layer to the torso surface potentials. The methodology of a combination method employing higher-order interpolation, capable of matching the potential as well as potential gradients across the elements, has been described by Pullan (1996).

2.5 APPUCATIONS OF THE FORWARD PROBLEM Three categories of applications ofthe forward problem are described below. The first is its obvious use with computer heart models to calculate torso (and in some cases, epicardial) potentials. The second is to gauge the effects of torso conductivity inhomogeneities on electrocardiographic potentials. The final application is the reciprocal problem of obtaining the currents traversing the heart due to currents injected at the body surface. One further application of the forward problem that is only mentioned here in passing is its use in the

62

R. M. Gulrajani

inverse problem of electrocardiography. Almost all inverse solutions entail a prior forward problem calculation.

2.5.1 COMPUTER HEART MODELS Present-day computer heart models start by storing a realistic three-dimensional representation of the heart anatomy that, in tum, is properly positioned within a second realistic three-dimensional representation of the human torso. A full solution would then entail the solution of the two inhomogeneous-anisotropy bidomain equations (Eqs. 2.32 and 2.33) applicable in the heart region, Laplace's equation applicable in the torso region, plus the boundary conditions at the heart-torso interface and at the torso-air interface. Equations (2.32) and (2.33) also imply a description of I m v , the membrane currents of the cardiac cell. These should be obtained from one of the more recent models for cardiac membrane currents (e.g., Luo and Rudy, 1991). In essence, the solution to this mix should be the intracellular and interstitial potentials, <1>; and e, computed everywhere in the heart, as well as the torso potential <1>0, computed everywhere in the torso. The potentials <1>; and e then permit determination of the transmembrane potential Vm in the heart, and thus a description of the heart's excitation. In addition, the values of e at the heart surface will automatically yield the epicardial potential distribution. To date, on account of its complexity, only one group (Lines et al., 2003a) has achieved this complete solution for realistic heart and torso geometries. They show a single simulation of a transmembrane potential distribution in the heart along with a simultaneously-computed body surface distribution (Lines et al., 2003b). A second group (Buist and Pullan, 2002) has described two solution methodologies for the complete solution, but only illustrate their results with a two-dimensional model of a realistic-geometry heart slice placed inside a torso slice. Both these groups have pioneered the numerical techniques needed for the complete solution, but undoubtedly computer memory and speed limitations have prevented exploiting the techniques to the full. Earlier work by many other groups used a simplified methodology that entailed splitting the problem into two sub-problems. The first entails solving Eqs. (2.32) and (2.33) in the heart region, assuming that the heart is insulated from the torso, i.e., neither intracellular nor interstitial currents cross over into the torso. This would solve the sub-problem of determining the excitation of the heart. The second subproblem would then attempt to calculate torso surface potentials and/or epicardial potentials from the now-known excitation pattern of the heart, most commonly by using equivalent dipole representations for this excitation. Both sub-problems, as well as a review of the many simulations realized, are described briefly below.

2.5.1.1 Determining the Excitation Pattern ofthe Heart Historically, two categories of heart models may be defined, those that assume a fixed activation pattern for the heart, based on the activation isochrones first measured in the human heart by Durrer et al. (1970), and those that incorporate their own activation algorithm. The best example of the first type of heart model, with fixed activation isochrones corresponding to normal excitation, was the one developed by Miller and Geselowitz (1978). No solutions for the heart's excitation pattern are needed here; instead a pre-determined

63

The Forward Problem of Electrocardiography

form of the cardiac action potential is triggered at each model point at its corresponding isochrone time. This then serves to determine the spatial and temporal distribution of Vm . Clearly, heart models possessing an intrinsic activation algorithm are much more versatile and capable of simulating both normal and abnormal excitation. Early examples of such heart models were all of the so-called "cellular automaton" type. Here, as opposed to solving Eqs. (2.32) and (2.33), activation is determined by a set of rules that define the propagation velocities between model cells, the excitation and refractory states, and the form of the action potential. Such cellular automaton heart models originated with Okajima et al. (1968), and some of the better known ones are those of Solomon and Selvester (1971; 1973), Horacek and van Eck (1972), Lorange and Gulrajani (1993), and Werner et at. (2000). One interesting way to solve the excitation problem in more realistic fashion is to derive an approximating "eikonal" equation (Colli-Franzone and Guerri, 1993; Keener and Panfilov, 1995) for the activation wavefront from Eqs. (2.32) and (2.33). Solution of this simplified eikonal equation then yields just the activation time at a given myocardium point or, equivalently, the wavefront position at a given time. A pre-determined action potential is again triggered at each point corresponding to its activation time. This eikonal approach offers an approximate solution for the wavefronts, and was developed for want of the requisite computing power available at the time to solve the anisotropic bidomain equations. Another approach, also dictated by the lack of computing power, was that of Leon and Horacek (1991). These investigators, by assuming equal anisotropy, combined Eqs. (2.32) and (2.33) into a single equation for the transmembrane potential Vm . We rewrite these Equations below in slightly modified form

v . (G;'1J

f3lm - f3Istim V . (G~ '1e) = -131m

(2.62)

=

(2.63)

where 13 is the surface-to-volume ratio of the cardiac cells, and an intracellularly-injected stimulating current Istim (in Nm 2 ) has been introduced to permit excitation sites for the model. The surface-to-volume ratio permits the conversion of the current per unit volume Umv, expressed in Nm 3 ) to the current per unit membrane surface area Um, expressed in Nm 2 ) via the relation Imv = 131m. With equal anisotropy, i.e., = ~G~, and expressing the membrane current as the sum of the capacitive and ionic currents

G;

(2.64) Eqs. (2.62), (2.63) and (2.64) may be combined to result in a single governing reactiondiffusion equation: m -aV - - I

at -

[~ V · ( ) GI '1V f3C m -1 +~ e m

-

13

I

IOn

13]

-I . +1 + ~ sum

(2.65)

Leon and Horacek solved Eq. (2.65) for the subthreshold case, i.e. by assuming that the ionic current was passive and given by lion = G mVm, where G mis the constant resting membrane conductance per unit area. Once Vm reached a fixed threshold value, a pre-determined action potential waveform was triggered. Thus, the Leon- Horacek model was a hybrid, with

64

R. M. Gulrajani

correct subthreshold excitation, but with cellular automaton characteristics above threshold. A comprehensive review of most of these heart models, and of the electrocardiographic simulations realized with them, has been provided by Wei (1997). Later work focused on the solution of Eq. (2.65), but with both sub- as well as supra-threshold representations for lion. Simulations with a simple FitzHugh-Nagumo representation for lion have been described by Berenfeld and Abboud (1996) and by Panfilov (1997). Huiskamp (1998) described a particularly impressive study that solved Eq. (2.65) for an 800,000 point model of the dog ventricles, using modified Beeler-Reuter equations (Drouhard and Roberge, 1986) for lion' This dog model was developed by Hunter et al. (1992) and incorporated measured fiber directions at every point. Recent work by our group (Trudel et al., 2001) has employed a multi-processor computer to solve Eq. (2.65) for a 12-rnillion point high-resolution version of the earlier Lorange and Gulrajani (1993) human-heart model, with analytically-introduced fiber rotation, and with a Luo and Rudy (1991) membrane model for lion. The ventricular activation isochrones for normal excitation obtained in these simulations by Trudel et al. are shown in Figure 2.5. Current work by our group focuses on doing away with the equal anisotropy assumption and solving Eqs. (2.62) and (2.63) together, without their combination into the single Eq. (2.65). This is now feasible with our 12-rnillion point heart model, again due to a newer generation multi-processor computer now available to us. A second advantage of the solution of Eqs. (2.62) and (2.63) without their combination would be the automatic determination of the interstitial distribution

2.5.1.2 Calculating Torso and/or Epicardial Potentials Once the spatial and temporal distribution of Vm has been determined by one of the above-described excitation methodologies, subsequent calculation of the body surface potentials has usually assumed the myocardium to be isotropic. Accordingly, most investigators have used an equation such as Eq. (2.16), namely Jeq = -geq VVm , to compute an equivalent dipole density. This leads to individual dipoles at each model point, which are then combined vectorially for individual heart regions to realize a "multiple-dipole" heart model, e.g., the Miller-Geselowitz model used 23 such regional dipoles to represent activation of its ventricles. A surface methodology can then be used to compute the body surface potentials due to each regional dipole, which potentials are then added to get the final body surface potential. Thus, Figure 2.6 shows the normal 12-lead electrocardiogram corresponding to the activation isochrones of Fig. 2.5, computed from 58 regional dipoles of the Trudel et al. heart model, using the integral equation formulation of Eq. (2.40). Huiskamp (1998), on the other hand, used the variant equation Jeq = -gi VmDH (Eq. 2.43) described by Geselowitz (1989), to determine dipoles on the epicardial and endocardial surfaces of his dog heart model, prior to using these dipoles as sources to compute the potential on the torso surface. This approach may also permit the approximation of epicardial potentials by computing potentials on a surface just outside the real epicardium, as demonstrated by Simms and Geselowitz (1995). The disadvantage with this approach is that for an injured heart containing dead tissue, this dead tissue forms an additional interface on which equivalent dipoles need to be placed. The more traditional route of

40

30

msec

msec

FIGURE 2.5. Isochrones corresponding to normal activation of the ventricular heart model employed by Trudel et al. (200 I). A transverse section (top) and a longitudinal section (bottom) are depicted. The colors indicate the time of activation as per the color bars on the right. Isochrones start at 5 ms after ventricular activation and are spaced at 5 ms intervals. See the attached CD for color figure. © IEEE.

66

R. M. Gulrajani

O.5mV

aVR

'I

'"

aVL

aVF

/\

V1

2

V6

/\

FIGURE 2.6. The normal I2-lead EeG corresponding to theactivation isochrones of Figure2.5.Reproduced, with permission, from Trudel et al. (2001). © IEEE.

lumping dipoles computed by Eq. (2.16) into regional dipoles and then using Eq. (2.40) does not yield sufficiently accurate potential distributions just outside the epicardium, largely because of a loss of spatial resolution, due to the lumping, that shows up at close distances. However, as shown by Hren et al. (1998 ), reasonably-accurate approximations to the epicardial distribution can be obtained if the dipoles at the individual points are not combined but used individually in the computations. Hren et al. also used the oblique dipole model to compute their epicardial potentials. They also assumed that the oblique dipoles existed in a homogeneous, isotropic myocardium. Nevertheless, the computed epicardial potentials (Fig. 2.7) for paced stimulation at different intramural depth s in the right ventricular wall of Hren et al .'s heart model revealed both the characteristic one-minimum two-maxima pattern,

F

A

-33. 50/5.52

·15 .70/4.75

·32 .84/5.08

-8.24/4.96

-3 1.68/4.85

-6.12/4.93

-29.48/4.68

-4.36/5.00

-26.95/4.71

-1.93/5 .29

B

c

D

E

FIGURE 2.7. Simulated potential maps on a patch of epicardium 10 ms after the onset of activation, for pacing at different intramural depths in the right ventricular free wall. Pacing sites were 0.5 mm apart, progressing from the epicardium (panel A) to the endocardium (panel J). The epicardial projection of each pacing site is indicated by the black dot. Isopotentiallines are plotted for equal intervals, with no zero line; solid contours represent the positive and broken contours the negative values of the potential; the magnitudes of the minimum and maximum are given (in mV) at the bottom of each map. Note that the axis joining the two maxima rotates counterclockwise with increasing pacing depth following the transmural counterclockwise rotation of fibers from epicardium to endocardium. Figure reproduced, with permission, from Hren et al., 1998.

68

R. M. Gulrajani

as well as the rotation with depth of stimulation, observed experimentally by Taccardi et al . (1994). The use of the oblique dipole model, along with the approximation of a homogeneous isotropic myocardium, in order to compute torso surface potentials may be more contentious. In a simulation study involving a layered myocardium block placed inside a larger volume conductor block, Thivierge et al. (1997) showed that the effect of the axial dipole component on surface potentials is reduced both on account of its orientation along the high-conductivity fiber direction as well as due to a well-known effect described by Brody (1956), whereby dipoles oriented tangential to the high-conductivity blood masses are dimini shed in so far as torso surface potentials are concerned, whereas dipoles oriented radial to the blood masses are enhanced. In the heart, the fibers are in the main oriented tangential to the blood masses. During normal activation of the heart from endocardium to epicardium, the component of the oblique dipole normal to the wavefront ends up being radial to the blood masses and is therefore enhanced, whereas the axial component along the fibers is diminished both due to the Brody effect as well as due to its orientation along the high-conductivity direction of the myocardium. Furthermore, the axial dipole is small to begin with during normal activation. This is because the endocardial-to-epicardial spread of normal activation ensures that the largest component of the transmembrane potential gradient VVm lies in the endocardium-to-epicardium direction perpendicular to the fibers, and that the axial component of VVm parallel to the fibers is very small. Mathematically, this transla tes to a very small value for n 3 in Eq. (2.39), and hence a small axial component for p. All of these reasons may well explain why torso potential simulations that have ignored the axial component have often successfully reproduced clinical electrocardiograms (ECGs). If using the oblique dipole model for computing torso potentials is being contemplated, it may be necessary to include both the high-conductivity intraventricular blood masses as well as the myocardial anisotropy into the torso model so as to correctly reduce the effect of the axial component. If these inclusions are not possible, it may actually be preferable to use the uniform dipole model and an isotropic myocardium. On the other hand, for epicardial potentials, as the simulations by Hren et al. (1998) show, it is essential to include the axial component. It may also be essential to include this component for body surface computations during repolarization, which being less organized than depolarization, may have a larger axial component. Also abnormal stimulation of the ventricles at isolated sites will certainly lead to initial propagation along the fiber direction, and larger axial components of VVm • Intramural stimulation, such as that depicted in Fig. 2.7, therefore may warrant use of the oblique dipole model not only for epicardial potentials but also for torso surface potentials. All of the above computations of torso and/or epicardial potentials described above have been via surface methodologies. Fischer et al. (2000) recently described a mixed boundary element-finite element route to obtaining both epicard ial and torso potentials . The starting point was the transmembrane potential distribution Vm , calculated by Huiskamp (1998). The heart region of Huiskamp's model was then represented via finite elements , with the known Vm distribut ion acting as a source . Outside the heart, a boundary-element surface representation for the torso was employed. The governing equations were Eq. (2.35) in the heart, namely V · (G ' V e) = - V · (G ; V Vm), and Laplace's equation in the torso. Since Vm is known, only two boundary conditions are needed at the heart-torso interface.

69

The Forward Problem of Electrocardiography

The ones selected were Eqs. (2.36) and (2.37), the latter rewritten in terms of the known potential Vm, and the unknown potentials e and <1>0 being sought, namely, (2.66) Solution of this mixed finite-element and boundary-element problem yielded e and <1>0; the former then gives the epicardial potential distribution and the latter the torso surface potential distribution. Two important points to note are that due to Vm being used directly as the source and due to the finite-element methodology used for the heart, an accurate solution results for e and <1>0 without the need to draw on the oblique dipole interpretations used with surface methodologies. Yet the computed solution is the equivalent of a surface methodology that employs an oblique dipole in an inhomogeneous anisotropic heart, i.e., with both G; and G' varying tensor quantities. Fischer et al. went on to compare their results with those computed assuming G' to be an isotropic constant, i.e., equivalent to an oblique dipole but acting in an isotropic heart, and with those assuming both G; and G' isotropic constants, i.e., equivalent to a uniform dipole layer acting in an isotropic heart. Relative errors in the body surface potentials were approximately 34% for G' constant and 43% for both G; and G' constant, reflecting the importance of including myocardial anisotropy in the body surface computations. Qualitatively, however, the surface potential distributions were similar (Fig. 2.8). Interestingly, the solution with G' constant, i.e, equivalent to an oblique dipole in an isotropic heart, overestimated the potential variations, while the solution with both G; and G' constant, i.e., equivalent to a uniform dipole layer in an isotropic heart, underestimated the variations. This reinforces the finding by Thivierge et al. (1997) that the axial component of the oblique dipole does tend to get reduced by its orientation along the high-conductivity fiber direction. The high-conductivity blood masses were not included in the study by Fischer et al. so the Brody effect did not come into play in the potential distributions of Fig. 2.8. It would be interesting to see if the presence of blood masses

[mV)

-0 5

.,

.t 5

Ca)

(b )

(e)

FIGURE 2.8. Body surface potential maps on the anterior torso model 48 ms after the onset of activation: (a) oblique dipole layer model in an isotropic heart, (b) full anisotropic myocardium model, (c) uniform dipole layer model in an isotropic heart. Contours are plotted in steps of 0.5 mV. Field patterns are in qualitative agreement. but quantitative differences are large, even though the isotropic conductivities were chosen to realize the smallest difference. Figure reproduced, with permission, from Fischer et al., 2000.

70

R. M. Gulrajani

within the heart would bring the three torso surface distributions of Fig. 2.8 closer together. On the other hand, the computed epicardial distributions (not shown here) under the three conditions clearly revealed that the approximation of a uniform dipole layer in an isotropic heart was unable to reproduce, even in qualitative fashion, the correct epicardial potential distribution.

2.5.2 EFFECTS OF TORSO CONDUCTIVITY INHOMOGENEITIES The effects of torso conductivity inhomogeneities have long been of interest to researchers. Early interest was kindled simply by the desire to know the intrinsic effect of the major torso inhomogeneities such as the intracardiac blood masses, lungs, ribs, spine, skeletal muscle, and subcutaneous fat on the electrocardiographic potentials. A second motivation today is to gauge which of the above inhomogeneities has the largest perturbing effect on inverse solutions, and therefore needs to be carefully included in the torso volume conductor used in computing these inverse solutions. Many early forward problem studies used physical analogs of the torso, into which materials that mimicked the torso inhomogeneities could be inserted to see their effect on the surface potentials generated by a current dipole source placed within the analog. The most informative early studies were, however, analytical, employing spherical or cylindrical models of the heart and torso. We have already alluded to the Brody-effect which was deduced using the model of a spherical ventricular cavity of infinite conductivity that represented the blood masses, situated in an otherwise infinite homogeneous medium that represented the myocardium (Brody, 1956). The Brody model was extended by Rudy and coworkers (Rudy et aI., 1979; Rudy and Plonsey, 1980) who used an "eccentric-spheres" model in which the torso was represented by two systems of spheres. The inner system which mimicked the heart and blood-filled cavity was eccentric with respect to the outer system which represented lungs, skeletal muscle, and subcutaneous fat. Besides confirming the Brody-effect for radial dipoles, Rudy and coworkers described the effects of varying the conductivities of the lungs, skeletal muscle and fat regions. A review of their work has been written by Rudy (1987). The Brody effect has been revisited recently by van Oosterom and Plonsey (1991). They reiterate that, while generally correct, the magnitude of the Brody effect is dependent on the field point, and, if the medium is bounded, also on the location of the reference electrode. The Brody effect can also be negative under certain conditions, where the insertion of the blood masses causes the potential to shift from positive to negative. This corresponds to shifts of the zero isopotential line in the surface potential distribution. With the development of numerical techniques for computing the torso surface potentials, attention shifted to utilizing realistic-geometry torso models to study the effects of torso inhomogeneities. One such study was done by Gulrajani and Mailloux (1983) using a boundary-element torso model that comprised intraventricular blood masses, lungs and a skeletal muscle layer. This last was approximated as an isotropic layer of increased thickness as first suggested by McFee and Rush (1968). No subcutaneous fat layer was used, the approximated skeletal-muscle layer extending all the way to the surface. Equation (2.40) was used to compute the surface potentials due to the 23 individual current dipoles of the Miller-Geselowitz heart model as the inhomogeneities were introduced one-by-one into an otherwise homogeneous torso. Most of Gulrajani and Mailloux's findings were in

The Forward Problem of Electrocardiography

71

accordance with earlier work employing realistic torso models (Barnard et al., 1967; Selvester et aI., 1968; Horacek, 1971). Apart from qualitatively confirming the Brody effect, by activating the 23 Miller-Geselowitz dipoles in concert to generate normal activation Gulrajani and Mailloux could gauge the effects of the inhomogeneities on the normal ECG and the whole body surface potential map (BSPM). The major qualitative effects were restricted to a smoothing of notches in the ECG and of isopotentials in the BSPM due to, in descending order of importance, the blood masses, muscle layer and lungs. However, although qualitative pattern changes in the ECG and BSPM were limited to these smoothing effects, there were large quantitative changes in both, notably magnitude increases due to the blood masses and magnitude decreases due to the muscle layer. The latter is due to the increased distance of the torso surface from the dipoles on account of the increased effective thickness of the muscle layer, and the former is due to the Brody-effect enhancement on the predominantly radial orientation of the current dipoles associated with normal activation. The advent of powerful computers has seen more elaborate finite-element and finitedifference torso models being used for the study of inhomogeneity effects. One example is the finite-element model developed at the University of Utah (Johnson et al., 1992), which was constructed on the basis of magnetic resonance images of the torso, and incorporated lungs, an anisotropic skeletal-muscle layer, subcutaneous fat, as well as secondary inhomogeneities such as epicardial fatpads, blood-filled major arteries and veins, sternum, ribs, spine, and clavicles. This model has been used in a study by Klepfer et aI. (1997) on the effects of the inhomogeneities and anisotropies on a known, fixed, epicardial potential distribution. Klepfer etaI. estimated 11 to 15% changes in the BSPM due to addition or removal of either the lungs, anisotropic skeletal-muscle layer, or subcutaneous fat. No major BSPM pattern changes were noted. Klepfer et al. , however, may have lessened the impact of the inhomogeneities since the starting epicardial distribution was kept fixed and not "loaded" by the changing inhomogeneities. Changes in the amplitudes of epicardial potentials with changes in the torso conductivity have been reported by MacLeod et al. (1994) in an experimental study that used measured epicardial and surrogate torso potentials from a dog heart placed inside a human-shaped torso tank. Data from the Utah model was also used in a study by Bradley et at. (2000) that used boundary elements for the epicardial surface, lung surface and torso cavity, but finite elements for the anisotropic-conductivity skeletal muscle and subcutaneous fat layers. Bradley et at. used a single time-varying current dipole derived from a Frank vectorcardiographic signal to represent the excitation of the heart. This dipole was placed within the heart region and served as the source. Consequently, the calculated epicardial potentials are subject to loading effects as the inhomogeneities are removed or added. Again pattern changes in the BSPM were not noted, but up to 30% magnitude changes in the BSPM could be seen. Bradley et al. found that the effect of the subcutaneous fat to be more important than that of the skeletal muscle. This was in contrast to the earlier studies of Gulrajani and Mailloux (1983) and of Stanley and Pilkington (1986) who found that the effect of the skeletal muscle layer was important. However, in both these earlier studies the skeletal muscle layer extended all the way to the outer torso surface, there being no subcutaneous fat layer. It could well be that it is the layer that abuts the outer torso surface that needs to be correctly represented for more accurate torso potential magnitudes. On the other hand, Hyttinen et aI. (2000) in a study involving a finite-difference model of the torso, constructed on the basis of the US National Library of Medicine's Visible Human Man data, gauged the effect of a 10% increase in the conductivities of the

72

R. M. Gulrajani

individual inhomogeneities on torso surface potentials due to a heart dipole . They found that the effect of the heart muscle was largest, followed by the intracardiac blood , skeletal muscle, lungs and subcutaneous fat, in that order. This protocol of a 10% conductivity change, by standardizing the extent of the change, may be a better indication of the effect of a particular inhomogeneity, than adding or removing a homogeneity altogether. While the inhomogeneities mentioned above only affected potential magnitudes with little effect on BSPM patterns, Bradley et al. also found that the position and orientation of the heart in the torso made the most difference for both torso potential magnitudes and distributions. This is in accordance with recent work by Ramanathan and Rudy (2001 a) that used measured epicardial and surrogate torso potentials from a dog heart placed inside a human-shaped torso tank . They found that torso inhomogeneities have a minimal effect on torso patterns computed from the measured epicardial potentials, but that for a good match with measured torso potentials it was essential that heart and torso geometry be accurately represented. In an accompanying paper (2001b), they show that even a homogeneous torso can be used in the inverse computation of epicardial potential distributions, provided heart and torso geometry are correctly represented. This was also shown in the study by Hyttinen et al. (2000) cited earlier who, having access to heart geometries corresponding to both systole and diastole, found that the error in inversely-computed dipoles increased significantly if the wrong heart model was used in the inverse computations. Clearly the best way to judge the effect of torso inhomogeneities on surface potentials is to use a fully-coupled complete heart-torso solution. Buist and Pullan (2003) have done such a study but with their two-dimensional heart-torso slice mentioned earlier. They report that in none of their tested situations did the two-step equivalent dipole approach completely reproduce the fully-coupled results further supporting the above assertion. 2.5.3 DEFIBRILLATION

Defibrillation consists of applying a high-energy shock to the fibrillating heart , the idea being to simultaneously depolarize all the ventricular cells thereby halting the fibrillatory activity. Upon recovery from the shock, the sinus node often regains control of the heart and a normal heartbeat ensues. This is the "total extinction" hypothesis for defibrillation, first put forward by Wiggers (1940). Later, Zipes et al. (1975) proposed the "critical mass" hypothesis, whereby halting the fibrillatory activity in a certain critical mass of myocardium, thought to be greater than 75% of the total mass was sufficient for successful defibrillation. The remaining mass would then be incapable of sustaining the residual fibrillatory activity. More recently, Chen et al. (1986) suggested that even if a shock was strong enough to halt the fibrillatory activity everywhere or in a critical mass of myocardium, it could still reinitiate fibrillation upon removal, and therefore had to be somewhat larger. This hypothesis was based on two observations, first that fibrillation was only induced in dogs for shocks between a lower and an upper limit, and second that shocks above this upper limit of vulnerability never failed to defibrillate an already fibrillating heart, presumably because it never reinitiated fibrillation. Chen et al. 's hypothesis has come to be known as the "upper limit of vulnerability" hypothesis. Whichever of the above three hypotheses holds, it is clear that a certain minimum level of excitation is required at the heart for successful defibrillation which translates to a minimum value of the applied current density (or

73

The Forward Problem of Electrocardiography

voltage gradient) everywhere in the heart. The number used in modeling studies is between 12.5 - 35 mNcm 2 or 5-6 V/cm (Karlon et aI.,1994; Panescu et al., 1995; Min and Mehra, 1998). At the same time, it is necessary that the current density anywhere in the heart does not exceed approximately 500 mNcm2 , since at these densities tissue damage is likely to occur. Thus, it is important that the defibrillation electrodes ensure a reasonably uniform current distribution in the heart. For transthoracic defibrillation, the shock is applied via external paddle electrodes, but in internal defibrillation intraventricular catheters or epicardial patches are used for shock application. The finite-element method is the most direct way to calculate the current density in the heart due to the defibrillatory shock. The governing equation is obtained by replacing the surface integral in Eq. (2.54), by the uncancelled component of this integral over S02 that remains once Eq. (2.54) is applied to all volume elements. This uncancelled component is given by Eq. (2.59), so that we get,

f

(O'V
v

f

f3i lsvd V

V

+

f

i = 1,2, . . . . m

f3;Jn d S

(2.67)

~2

Note that the number of equations spans the number of unknown potentials in the entire volume conductor. Since during these current density calculations, the heart is treated as passive, the first volume integral on the right-hand side is zero. Moreover, during defibrillation it is incorrect to assume that the electrodes inject a uniform current density In' Thus, it is more appropriate to use a Dirichlet boundary condition, setting the potential at the nodes corresponding to the electrodes equal to the applied electrode voltage. This eliminates the surface integral in Eq. (2.67) also. Only the left-hand side remains in Eq. (2.67), which may then be reduced to global matrix form. Due to the complex torso models used, the matrices are large, and Ng et al. (1995) discuss the use of parallel computers in obtaining the solution. Once all unknown node potentials are calculated, the current density within the elements representing the heart is obtained from the gradient of the potential distribution <1>. By using a finite-element representation for the intraventricular catheter electrode, the finite-element approach can also be used for studying the heart current densities due to implantable defibrillators (Jorgenson et al., 1995). External or transthoracic defibrillation can also be studied with the boundary-element formulation (Claydon et al., 1988; Oostendorp and van Oosterom, 1991; Gale et al., 1994; Gale, 1995). While matrix sizes are smaller, the major disadvantage is that anisotropic conductivity variations, especially those in the myocardium, cannot be taken into account. The governing equation for the potential (r) anywhere in the heart is again obtained from an application of Green's second identity to the torso geometry ofFig. 2.4 and is (Oostendorp and van Oosterom, 1991):

41l'O'(r)(r)

~f (0'/- = L.J /=0

SI

a/+ )(r' )dQ rr,

f

+ -In(r') - , dS , So

[r - r

I

(2.68)

In Eq. (2.68), O'(r) is the conductivity at the observation point, and In(r') is the normal

74

R. M. Gulrajani

component of the injected current density at the defibrillation electrode. Oostendorp and van Oosterom discuss the numerical solution of Eq. (2.68) for the unknown potentials and current densities in terms of the known potentials at the defibrillation electrodes. Simulation studies of defibrillation have concentrated on determining the optimal positioning and size of defibrillation electrodes in order to ensure an adequate and approximately-uniform current density everywhere in the heart (Gale et aI., 1994; Gale, 1995; Camacho et al., 1995; Panescu et al., 1995; Schmidt and Johnson, 1995). More recent work has focused on determining the defibrillation threshold, namely the electrode voltage or energy needed for a given percentage of the myocardial mass (usually 95%) to attain a voltage gradient of at least 5 V/cm (Aguel et al., 1999; De Jongh et al., 1999; Eason et al., 1998; Kinst et al., 1997; Min and Mehra, 1998). This is in line with the critical mass hypothesis for defibrillation. One early study focused on the sensitivity of the current-density distribution in the heart to variations in skeletal muscle anisotropy (Karlon et al., 1994). It was found that in transthoracic defibrillation, the anisotropy made little difference to current flow patterns in the heart, but simply affected current magnitudes. On the other hand, the same study showed that other inhomogeneities such as the lungs, ribs and sternum affected both magnitudes and current patterns. Another study (Eason et aI., 1998) found that the voltage defibrillation threshold, for internal defibrillation between a right ventricular catheter electrode and the defibrillator can in the pectoral region of the left chest, differed by only 4.5% if the realistic fiber architecture in the heart model was replaced with an isotropic conductivity myocardium. It is being widely recognized today that defibrillation is an immensely complex phenomenon, and that the above simulation studies yield, at best, an estimate of the extracellular potential gradient and current density inside a monodomain passive heart. What really counts is the transmembrane potential distribution in the heart, and the response of the active cells in the heart to this transmembrane distribution. Theoretical analysis has revealed that even with extracellular stimulation of the passive bidomain heart, the unequal anisotropies of the intracellular and interstitial space result in contiguous regions of large transmembrane potential depolarization and hyperpolarization in the heart (Sepulveda et al., 1989). Local currents flowing between these regions have the ability to re-initiate fibrillation and negate the findings of studies that assume a monodomain heart. Experimental work and theoretical simulations both tend to support this re-initiation mechanism (Efimov et al., 2000; Skouibine et al., 2000). A second perturbing factor that affects real transmembrane voltage gradients in the heart is the effect of local conductivity discontinuities (e.g., gap junctions, fiber curvature, random clefts). Sobie et at. (1997) have shown how, using a "generalized activation function," the effects of both extracellular stimulation and of conductivity discontinuities can be taken into account in determining the transmembrane potential distribution. This generalized activation function suggests that more than the gradient of the extracellular potential, it is the second spatial derivative of the extracellular potential that determines the transmembrane potential change. Although Sobie et at. illustrated their approach in passive myocardium, it can just as easily be applied to active myocardium if sufficient computational resources are available. This represents the final, and most important step, in translating an applied extracellular potential to the actual response of the cardiac cells. Already, Skouibine et at. (2000) illustrate such active responses in simulations in a two-dimensional bidomain sheet of active myocardium subjected to a defibrillation shock.

The Forward Problem of Electrocardiography

75

2.6 FUTURE TRENDS This review of the forward problem of electrocardiography has presented its theoretical underpinnings, the solution methodologies employed, and finally its major applications. The greater accessibility of multiprocessor computers means that we shall likely see more accurate heart models capable of simulating more complex heart pathologies, with simultaneous computation of heart activation and torso potentials via a complete solution. Torso representations are also likely to be more accurate with better accounting of anisotropic conductivity, and with the effect of torso inhomogeneities gauged with a complete fully-coupled three-dimensional heart-torso solution. Finally, successful defibrillation is more than just a matter of knowing the extracellular potential or current density everywhere in the heart. Rather it is the interaction of this potential or current density with the cardiac cell, and the subsequent effect on the cellular action potentials that determines whether the defibrillation shock succeeds or fails. While the defibrillation simulations described above may be useful in electrode design, a better understanding of defibrillation necessitates simulations in which the heart is not treated simply as a passive conductor, but as an active bidomain. Already, Efimov et al. (2000) have described the transmembrane potentials generated in a three-dimensional realistic-geometry rabbit-heart model with varying fiber directions, following uniform electric-field application. Although they represented the myocardium with passive bidomain equations, it is only a matter of time before such whole-heart defibrillation simulations will be realized with an active membrane representation for the heart's ionic currents.

ACKNOWLEDGMENT Work supported by the Natural Sciences and Engineering Research Council of Canada.

REFERENCES Abboud, S., Eshel, Y, Levy, S., and Rosenfeld, M., 1994, Numerical calculation of the potential distribution due to dipole sources in a spherical model of the head, Comput. Biomed. Res. 27: 441-455. Aguel, E, Eason, J. c., Trayanova, N. A., Seikas, G., and Fishier, M. G., 1999, Impact of trans venous lead position on active-can ICD defibrillation: A computer simulation study, PACE 22 [Pt. II]: 158-164. Barnard, A. C. L., Duck, I. M., and Lynn, M. S., 1967, The application of electromagnetic theory to electrocardiology. II. Numerical solution of the integral equations, Biophys. J. 7: 463-491. Barr, R. c., Pilkington, T. c., Boineau, J. P., and Spach, M. S., 1966, Determining surface potentials from current dipoles with application to electrocardiography, IEEE Trans. Biomed. Eng. 13: 88-92. Barr, R. c., Ramsey, M., III, and Spach, M. S., 1977, Relating epicardial to body surface potential distributions by means of transfer coefficients based on geometry measurements, IEEE Trans. Biomed. Eng. 24: 1-11. Berenfe1d, 0., and Abboud, S., 1996, Simulation of cardiac activity and the ECG using a heart model with a reaction-diffusion action potential, Med. Eng. Phys.18: 615-625. Berne, R. M., and Levy, M. N., 1977, Cardiovascular Physiology, Mosby, St. Louis, chapter 4. Bradley, C. P., Pullan, A. J., and Hunter, P. J., 2000, Effects of material properties and geometry on electrocardiographic forward simulations, Ann. Biomed. Eng. 28: 721-741. Brebbia, C. A., and Dominguez, J., 1992, Boundary Elements. An Introductory Course, 2nd ed., WIT Press, Southampton, U.K., chapter 2.

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w.,

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WHOLE HEART MODELING AND COMPUTER SIMULATION DamingWei Graduate Department of Information System,The University of Aizu, Japan

3.1 INTRODUCTION Bioelectrical models of the heart are studied in three levels: the single-cell model, the cell-network (tissue ) model and the whole heart model (Wei, 1997). The single-cell model describes ionic current flow across myocardial cell membranes. The cell-network model describes ionic current flow between aggregates of myocardial cells in temporal and spatial domains. Details of these model s have been described in the previous chapters. The development and spread of ionic currents throughout the heart and body volume conductor result in electrical potentials that can be measured on the body surface, called electrocardiogram (ECG). A whole heart model describes three major mechanisms: the propagation of activation in the heart, the cardiac electrical sources, and the extracellular potentials within and on the body surface. Thi s kind of model is able to relate the body surface ECG waveforms to the action potential, conduction velocity of cardiac tissue and other electrophysiological properties of the heart and, thus, yield clinically comparable ECG waveforms. For this reason, the whole heart model offers a unique means to bridge the clinical applications with the single-cell or cell-network models . Conventionally, a whole heart model refers to a 3-dimentional (3D) heart-torso model that contains realistic geometry of the heart and torso. Some heart models contain both atria and ventricles to represent the entire heart , but some only contain a portion of the heart such as the ventricles, or just the left ventricle. A computer heart model is usually represented by a 3D voxel array. The voxel in each grid is called an element. As the array increases in size, the model exhibits higher resolution. In most of the existing whole heart models, the element is of an order of 1 mm in diameter. It is usually several thousand times larger than a true cardiac cell. In simulation studies, the element is usually treated as a cardiac cell, called a model cell in this chapter, but it is important to keep in mind that the element or model cell in a whole heart model is actually a lumped model of tissue. Tsuruga, Ikki-machi, Aizu-Wakamatsu City, Fukushima 965-8580, Japan, Tel. +81-242-37-2602, Fax. +81-24237-2728, [email protected]

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Whole heart modeling and computer simulation are typical topics associated with the electrocardiographic forward problem. Since the main theory and methodology for the forward problem have been described in detail in Chapter 2, this chapter will focus on how to construct a 3D heart-torso model, how a whole heart model can be applied to understanding mechanisms of bioelectric phenomena of the heart, and to relating the electrical activity of the heart to the surface ECG.

3.2 METHODOLOGY IN 3D WHOLE HEART MODELING The main issues in whole heart modeling shall include: heart-torso geometry modeling, specialized conduction systems, rotating fiber direction and myocardial anisotropy, action potential and propagation, cardiac electric sources, and body surface potential calculation. These topics are described in detail in the following sections.

3.2.1 HEART-TORSO GEOMETRY MODELING Descriptions of heart anatomy can be found in a vast number of textbooks. For heart modeling purposes, one should acquire some basic knowledge of the heart's anatomic features. First of all, we should have a quantitative concept of the heart's size. One point is that the left ventricle including the left free wall and the interventricular septum (wall thickness 9-11 mm) constitute the major mass of the heart. The right ventricular wall is about 1/3 the thickness of the left and, therefore, about 1/27 of the total myocardium in volume. The wall of the atria is even thinner. This means that the left ventricular myocardium provides the major contribution to the generation of ECG. To construct a heart model with a realistic shape, the first step is to map the heart geometry onto a matrix of a regular 3D grid. In the early stage of model development (Okajima et al., 1968), the heart geometry was constructed using traditional digitization techniques. With such a method, a heart is first embedded in a gelatin solution and then frozen. After that, the frozen heart is sliced and photographed, and then pictures of each slice are digitized and input into a computer. For construction of high-resolution heart models, it is necessary to use CT imaging technology. In the study of Rorange et al. (1993a), a human heart obtained from autopsy was first inflated with air at end-diastolic pressures and submerged in liquid nitrogen. The frozen heart was then CT-scanned in 1 mm intervals resulting in 132 slices of 512*512 image data. These data were processed to extract the heart edges and eventually to obtain a 3D matrix of about 250,000 points spaced 1 mm apart. Recently, several databases provide accurate and anatomically detailed images of the human body available for model studies. One of the most commonly used databases for this purpose is the Visible Human Project of the U.S. National Library of Medicine (http://www.nlm.nih.gov/research/visible/visibleJmman.html). It provides 1,871 section images of a human cadaver at 1 mm intervals. The original images are 1048 by 1216 pixels with 24-bit color. These data have been used in recently published models (Balasubramaniam et al., 1997; Kauppinen et al., 1999; Ramanathan and Ruddy 2001a). The main techniques to model the heart-torso shapes include image segmentation and element generation. To the author's knowledge, there is no full automation technology

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currently available for image segmentation for the purpose. In most studies, the segmentation is performed in a semi-automatic manner, which includes steps of digital filtering, image enhancement, region growing processing, and segment decision (Heinonen et al., 1996; Hsiao and Kao, 2000). After these procedures, the boundary contours of distinct anatomical regions are obtained. Based on these data, the anatomical regions are filled with discrete elements representing distinct cardiac cell types. The cell types usually include cells of the atria, ventricle and specialized conduction system. For an inhomogeneous torso model, the volume is usually divided into piecewise homogeneous regions that may include the heart with blood mass, the lung, the fat, and the bones, with distinct regions being assigned different values of electrical conductivity. Cell classification can only be performed manually with the current techniques. An important issue in whole heart modeling is spatial and temporal resolutions of the model. Qualitatively, the model's resolution mainly affects the propagation details, especially if rotating fiber directions are taken into account. For surface ECG calculations, the resolution of most existing heart models is sufficient, considering the fact that many studies reduce the total number of elemental dipoles to a few multiple dipoles before calculating the surface ECG potentials. A heart model of 1 mm spatial resolution has enabled the simulation of reentrant propagation such as spiral waves (Panfilov 1993, 1995). In a recent model of Hren and Horacek (1997, 1998), a high resolution 0.5 mm is adapted with 1.7 million cells. The heart model has enclosed more anatomic details including the myocardial wall, trabecular tissue, and papillary muscles.

3.2.2 INCLUSION OF SPECIALIZED CONDUCTION SYSTEM For studying bioelectrical phenomena, the heart can be simply thought of as an electric generator comprising a specialized conduction system (SCS) and excitable myocardial tissues, as diagrammed in Fig. 3.1. Understanding the SCS is important for whole heart modeling because it is one of the key issues that affect the propagation sequence of activation in normal and abnormal hearts. In the following paragraphs, preparative knowledge for the activation of the heart and the role of the SCS are briefly introduced while referring to Fig. 3.1. Detailed references can be found in other references (Guyton, 1986; Van Dam, 1989). For a normal heart, the electric pulse is spontaneously generated by the sinus node, the pacemaker (located in the upper right atrium as shown in Fig. 3.1). The pulse stimulates the neighboring atrial cells to activate them. The activating cells in tum stimulate their neighboring cells so that the excitation propagates throughout the atria. Although some studies suggest that there may exit a specialized conduction system in the atria, there is no definite evidence to support this assumption. As clearly illustrated in Fig. 3.1, there is no direct "electric connection" between the atria and the ventricles, except in some abnormal hearts such as those suffering from the WPW syndrome (described later in Section 3.4). For normal hearts, the only pathway for conduction of the excitation between the atria and the ventricles is the atrioventricular node (AV node) and a specialized conduction system consisting ofthe His bundle, the left and right bound branches, and the Puikinje network, as illustrated in Fig. 3.1. The most important feature of the specialized conduction system is the high conduction velocity in comparison to that of normal myocardial fiber. The typical conduction velocity in the His bundle and the main branches is 2 mS- I , while that in the ventricular myocardium is 0.5 mS- I . The

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FIGURE 3.1. A schematic diagram of the heart and the specialized conduction system.

distribution of the Purkinje network influences the heart excitation sequence the most, and therefore is very important to the model study. The Purkinje fibers penetrate the septal myocardium near the apex and are distributed in the subendocardium. In general, the Purkinje network is distributed on the apical half of the subendocardium and absent on the basal half. This configuration avoids outflow obstruction. In whole heart modeling, distributing the Purkinje network in the ventricular model greatly influences the excitation process of the heart model and the resulting body surface ECG . Because it is not possible to identify the Purkinje fiber from image data, modeling the Purkinje network is usually based on the excitation sequence of the ventricles observed by experiments. The most cited literature regarding human heart excitation is that of Durrer et al. (1970). The measured excitation isochrones in isolated human hearts are shown in Fig . 3.2. According to Durrer et aI., the early excited areas in the left ventricle were observed in three endocardial sides : high on the anterior paraseptal wall just below the attachment of the mitral valve, central on the left surface of the interventricular septum, and posterior paraseptal about one third of the distance from the apex to the base. Early excitation of the right ventricle was found near the insertion of the anterior papillary muscle. Septal activation was found to start at the middle third of the junction of the septum and posterior wall. The se data are useful for arranging the Purkinje fiber in a heart model and for evaluating the simulation results of the model. Inclusion of the SCS to a heart model is usually a time-consuming task in the construction of a model. The positions and distributions are usually adjusted repeatedly until

Whole Heart Modeling and Computer Simulation

o

5

8S

10 15 20 25 30 35 40 45 50 55 60 65 ms

FIGURE 3.2. Excitation sequence of the human heart. (Reproduced with permission from Durrer et aI., 1970).

simulated excitation isochrones fit the experimental data. In the adjustment, the distribution of the Purkinje network, the position of joint points connecting bundle branches to the Purkinje network, and the activation time arriving at these points have the most important effects on simulation results. In most whole heart models, the Purkinje network is represented by a one-layer sheet and the bundle branches are represented by cables (Aoki et al., 1987; Rorange et al., 1993). The sheet model of the Purkinje fiber network is supported by some experimental studies. In the model of Aoki et aI., the left bundle branch terminates at three points: the central region of the septum, the antero-basal region, and the postero-apical region of the left endocardium. The right bundle branch terminates at one point on the antero-apical region of the right endocardium. A more detailed model of the specialized system can be found in Al-Nashash and Lvov (1997), where the His-Purkinje electrogram is the target of simulation. In this study, a His-Purkinje model with a 3D curvature resembling the ventricular endocardial surface is built and the His-Purkinje system electrogram is simulated using the volume conductor theory.

3.2.3 INCORPORATING ROTATING FIBER DIRECTIONS Myocardial anisotropy is an important issue in studying cardiac phenomena. Nowadays, a whole heart model without inclusion of myocardial anisotropy would hardly be accepted. Inclusion of myocardial anisotropy in a heart model involves three aspects: anisotropic geometry of the myocardial muscle, anisotropic propagation, and anisotropic cardiac sources. Anisotropy geometry means that a heart model should have

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location-varying fiber directions for all discrete elements, based on the fact that the ventricular myocardium has a spiral structure with fiber orientations rotating from the epicardial surface to the endocardial surface in a total angle of 90 to 120 (Streeter et aI., 1969). This is known as rotational anisotropy. Anisotropic propagation requires a direction-dependent conduction velocity in controlling the excitation process of the model. The anisotropic cardiac source arises from the fact that both the intracellular and extracellular domains are anisotropic, and the anisotropic ratios are essentially different from point to point in myocardial muscle. There are several ways to incorporate rotating fiber directions in a heart model. The simplest way is called stylized representation, as used in Lorange et al. (1993). With this method, a family of nested ellipsoids of revolution extending from the endocardium to the epicardium is used, where the fiber angle varies 120 from the endocardial to the epicardial ellipsoids. The fiber direction at a model point is obtained by determining which ellipsoid passes through the point. A similar method is used in other models (Adam et al., 1987; Leon et aI., 1991a, b). The advantage of this method is the simplicity in calculating the point-by-point fiber direction. The disadvantage is that the fiber directions are too simple in comparison with actual heart anatomy. The most precise way to incorporate fiber direction is through microscopic determination of fiber orientation as reported in Panfilov et al. (1993). In this study, detailed data of the ventricular geometry and fiber orientation were microscopically measured on an intact canine heart to produce a finite element model (Nielson , 1991). These data were then mapped onto a regular 93*93 *93 grid with 1 mm of distance between the grid points . Compared to the stylized representation, this method provides more geometric details of the heart. Microscopic determination of fiber orientation is not always possible for model studies . Wei et al. (1989, 1995) proposed a discrete method to calculate point-by-point fiber direction. This method can generate intermediate precision between that of microscopic determination and stylized representation. This method uses the following assumption based on Streeter et al.: (1) the myocardial fibers of the ventricles have a layered structure, (2) all fiber orientations are parallel to each other in one layer but different from layer to layer, (3) the fiber orientations rotate counterclockwise over 90° to 1200 with increasing depth from the epicardium to the endocardium. Fig. 3.3 (a) shows a longitudinal cross-section of the heart model. The spatial configuration of the elements is illustrated in Fig. 3.3(b). To assign each element a fiber direction, the model is layered as shown in Fig. 3.3(c). The myocardial fibers in one layer are mathematically described as intersection curves by cutting the layer with a group of parallel planes, called fiber planes, as illustrated by Fig. 3.3(d) and (e). Obviously, the fiber planes in one layer have a common normal direction, called fiber plane direction (FPO). Following experimental data (Spaggiari , 1987), an FPO perpendicular to the geometric heart axis along the apex-to-base direction is assigned to the epicardial layer. Then , the FPO is rotated counterclockwise in the septal plane to determine FPOs for all layers in the epicardium-toendocardium sequence. The rotating angle for a layer I with respect to that of the FPO of epicardium is given by 0

0

0

a(l)

= I·

A

-

N

(3.1)

Whole HeartModeling and Computer Simulation

87

(a)

FIGURE 3.3. Heart mode l of Wei et al. (a) Longitudina l cross-sect ion of the heart model. P denotes the Purkinje fiber. (b) 3D configuration of mode l elements . (c) Model elements layered for solving point-by-point fiber directions. (d) Illustration of rotational fiber orientations in the layers. (e) Mathem atical relationships of fiber plane, fiber plane direction, and fiber plane rotatio n. FPo: fiber plane of the outermost layer; FPj: fiber plane of the i-th layer; FPDj: fiber plane direction of the i-th layer; ali): rotating angle of layer i; PS: septal plane (Reproduced with permission from Wei et al., 1995 ).

88

D.Wei

~

I11111111111111111'. ".

lnyei ] layer

.........

....... ... ··.............. .. , .. ... .... ....... "

•

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FIGURE 3.3. (cont.)

where N is the total number of layers of the model and A is the total rotating angle between the epicardial and endocardial layers, having a value between 90° and 120°. In implementation, the heart model is layered from the epicardium to the endocardium by applying simultaneous stimuli to all model elements on the outermost layer (the epicardium) of the model and solving the propagation sequence of "virtual excitation" for the entire model. The septum of the heart model is specially treated in the propagation process so as to make the septal fibers natural extensions of the left ventricular fibers. As a result,

Whole Heart Modeling and Computer Simulation

89

(c)

FIGURE 3.3. (cont.)

the sequence number of such a propagation process corresponds to the layer number of the model element. Then , the outer product of two unit vectors determines the local fiber direction at each element. One is the fiber plane direction of the layer to which the unit belongs, and the other is the normal direction of the layer at that element. If an element (i, i . k) belongs to layer l and the layer has a fiber plane direction of P(l) and a normal direction N (i, j , k) at (i, i . k), the local fiber direction is given by F(i , j, k) = P(l ) x N(i , i . k)

(3.2)

To confirm the rotating fiber directions of the heart model, the model is stimulated at the left ventricular wall at three depths : at the epicardial layer, the intramural layer, and the endocardial layer. The isochrones are shown in Fig. 3.4. The long axes ofthe isochrones show approximately 90° of total rotation from the epicardial (Fig. 3.4b), intramural (Fig. 3.4c), and endocardial (Fig. 3.4d) layers. For comparison, propagation in the isotropic model is shown in Fig. 3.4a. Determination of fiber direction can also be performed in terms of analytical methods. In the study of Hren and Horacek (1997), the epicardial and endocardial surfaces are analytically obtained with sampling data from CT images using surface harmonic expansion (Hren and Stroink, 1995). Then, the principal fiber direction at a point lying in a tangential plane at an element is determined so that the fiber direction rotates counterclockwise from the epicardial to the endocardial surface.

3.2.4 ACTION POTENTIALS AND ELECTROPHYSIOLOGIC PROPERTIES In simulation studies , the model cell mimics the actual cardiac cell to reconstruct the cardiac process . Like an actual cell , the model cell should own the action potentials and other electrophysiologic properties. These properties are the main input data for simulation studies .

( a)

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FIGURE3.4. Simulated isochrones created by applying stimuli at different depths in the left ventricle of the heart

model. The stimulu s is denoted by '+'. Time sequence is identified by the thickness of the circle and its center point: the thicker the circle and the center point, the more advanced the time seque nce. Each increase in thickness of the circle repre sents 6 ms, and that of the center point represent s 36 ms (circles with no center points represent the first 36 ms). (a) Epicardi al isochrones of the left ventricle with the isotropic propagation (b) Epicardial isochrones of the left ventricle with aniso tropic propagation (c) Intramurallayers beneath the left epicardium with aniso tropic propagation (d) Intramural isochrones 9 layers benea th the left epicardium with anisotropic propagation The isochrones from (b) through (d) show rotating fiber directions in the heart model. (Reproduced with permission from Wei et al., 1995).

91

Whole Heart Modeling and Computer Simulation

. . . . ......... .... .. .......... . .... .... ... ...... . . . . .... ....... . . . . ..... .. . .. ... .... . . . ..... ....... . . .... .... ... ......... ......... ........ .... .. .... ...... .. ...... ...... . . . . ...... . . .. . ...... ...... .

•• • • •• • • •. . . . • . • • . . . . . • . ; • • • • • • •• •• ~ t:"l ,

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(c)

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(d)

~ FIGURE 3.4. (cant.)

The ideal way to assign action potentials to the model cell is to use the single-cell model to generate action potential as a function of time and distances. However, since the calculation of action potentials with single-cell models is computationally demanding and the scale of a whole heart model is quite large, most existing whole heart models do not directly calculate the action potentials, but assign pre-defined action potentials to the model cells. For an electrophysiological simulation study, one should take into account at least the following basic nature of action potential in modeling. First, different kinds of cardiac cells have different action potentials. Second, at least for the ventricular cells, the action potential duration (APD) is location-dependent in the heart, longest on the endocardium and the base, and shortest on the epicedium and the apex (Harumi et al., 1964). Third, the APD

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is time-varying during premature excitation, depending on the restitution property with the coupling interval (Harumi et al., 1989a). In many studies, the cardiac cell is represented by finite state automaton. This is called a cellular automata (CA) model (Siregar et aI., 1996). In fact, the CA model was first used in an early study by Moe et al. (1964) to simulate atrial fibrillation . In this study, a discrete time step of 5 ms and a unit diameter of 4 mm were used. One time step was the propagation time in a fully recovered unit, corresponding to a normal conduction velocity of 80 cmls for the atrial tissue. The excitability of the units was represented by 5 states. The first state corresponded to the absolute refractory period, having a duration of R = K JC, where C was the preceding cycle length and K was a property of the unit, different from unit to unit. States 2, 3 and 4 together corresponded to the relative period. Dividing the relative period into different states incorporated the time-varying conduction velocity during the propagation process. From state 2 to state 4, the conduction velocity increased from 1/4 to 1/2 of the normal conduction velocity. In state 5, the conduction velocity returned to normal. State 5 lasted until the next excitation and corresponded to the recovery period. Because the model of Moe et al. was aimed at studying the propagation of activation, the waveform of the action potential was not needed . For models that need to calculate ECG potentials, the waveform of the action potential should be defined. A simple representation is the piecewise linear approximation as used in Siregar et al. (1996). In this model , the muscle cell is resented by four states : resting, depolarization, absolute refractory, and absolute refractory. The function of the action potential, Y, is expressed by (3.3) where e and i denote model cell and state, respectively, a and f3 define a line for the cell and state. As functions oftime, a and f3 are context-dependent so that the action potential is time-varying according to the prematurity of the incoming impulse. This model is capable of simulating typical types of arrhythmias. In assigning action potentials to the model cells, there is a technical difficulty due to the fact that a 3D heart model usually has a huge number of cells . If each model cell has to be accompanied with a dataset containing an action potential and other parameters, the memory space for modeling would be extremely large. One method for solving the memory problem used by Wei et al. (1995 ) is to assign the action potentials and electrophysiologic parameters to cell types, instead of individual cells. In this way, model cells are organized in limited cell types. Action potentials and other electrophysiologic properties are linked to cell types so that limited memory is required. There are sixteen cell types available in the model of Wei et al. Each of them is associated with a parameter table containing the action potential waveform and other electrophysiological parameters as listed in Table 3.1. The available cell types include normal cardiac cells like the atria, ventricles, and a specialized conduction system. Besides, any special cell types, such as ischemia, infarction, or ectopic beats , can be defined by parameter settings suitable to the study. The following paragraphs show how the action potential and other electrophysiological details are assigned to the model cells in Wei et al. ( 1990, 1995). See Table 3.1 for meanings and abbreviations for the parameters, and see Fig. 3.5 for the action potential definition. The basic parameter relative to action potential is action potent ial duration (APD ) in ms, taken as the sum of phases through 4. The parameters relative to time are given as a

°

TABLE 3.1. Electrophysiologic Parameters Parameter

Definition

Action Potential APD

TO Tl T2 T3 T4

VO VI

V2 V4 GRD DVT

DC Conduction: CVL CR

DC Automaticity: ICL PRT

DLY ACC BKP

action potential duration (ms) duration of action potential phase 0 (% of APD) duration of action potential phase 0 (% of APD) duration of action potential phase 0 (% of APD) duration of action potential phase 0 (% of APD) duration of action potential phase 0 (% of APD) potential of phase 0 (resting potential) (mv) maximum action potential (mv) potential of phase 2 (mv) potential after full recovery (mv) gradient of the APD distribution (mslIayer) deviation of APD for random distribution (mv) APD change to coupling interval (%) conduction velocity along the fiber axis (m1sec) anisotropic ratio (m1sec) conduction velocity after ARP (m1sec) intrinsic cycle length (ms) protected or non-protected (yes/no) maximum delay of the phase response (% of ICL) maximum acceleration of the phase response (% of ICL) break point position of the phase response (%)

Pacing: BCL

BN ICN

basic cycle length of pacing (ms) beat number of pacing for the simulation increment of cycle length per cycle (% of BCL)

T2

T3

V2

YO

4

FIGURE 3.5. Definition of action potential used in the simulation (Reproduced with permission from Wei et aI., 1995).

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percentage of the APD. Lines specified by time intervals and voltages are used to represent phases 0, I, 2 and 4 of the action potential waveform. Phase 3 is defined by a curve through interpolation (second order Lagrange interpolation) of sample data. The action potential waveform is linked to each cell type and used as a look-up table during simulation. To distribute the pre-defined action potential waveforms over the ventricles so that the APD lengthens from the epicardial to the endocardial and from the base to the apex, a parameter GRD (see Table 3.1) is defined to specify the gradient interval along the epicardial-to-endocardial and base-to-apex sequence. Thus, the value of APD for a cell at location (i, j, k) is given by

AP D(i,

i. k) =

AP o,

+ GRD· SQ(i, j, k)

(3.4)

where APDd is the defined value of APD for the cell type, and SQ(i, i. k) is a sequential number of the cell sorting along the epicardial-to-endocardial and base-to-apex directions. In this model, a positive GRD value of about 5 ms per sequence yields a normal T wave in the simulated ECG. Adjusting the parameter GRD in a simulation study is useful to simulate T wave abnormalities. To make the APD adaptive to the coupling interval so that it is dynamically modified during the simulation, a parameter called dynamic coefficient (DC) is defined as the ratio of the change in APD to the change in coupling interval. During simulation, the value of APD at any time, t, is dynamically modified by

AP D(t)

= AP D(t -

1) + DC . fj.C l(t)

(3.5)

where fj.C/ is the change in coupling interval at time t. In addition to the action potentials, many other electrophyosiologic properties are defined with parameters. These parameters are concerned with conduction velocity, pacing, and automaticity. The conduction velocity is made adaptive by assuming a piecewise linear function of the coupling interval. During the absolute refractory period, the conduction velocity is zero. After full refractory, the conduction velocity is assumed constant specified by a parameter, CVL (see Table 3.1). During the relative refractory period, the conduction velocity is a linear incremental function of the coupling interval. To define a cycle length and pacing times, anyone or more cells can be paced in the simulation. The pacing rate can be increased or decreased with the changing rate specified by a parameter INC, which is defined as a percentage increment to the basic cycle length. The parameters relative to automaticity define the electrotonic interaction between normal and ectopic pacemakers and is used to simulate cardiac arrhythmias (Wei, 1992).

3.2.5 PROPAGATION MODELS There are three types of propagation models used in whole heart modeling. These are propagations based on Huygens' principle, Hodgkin-Huxley (HH) formulism (Hodgkin and Huxley, 1952; Plonsey, 1987; Leon et al., 1991a, b; Beeler and Reuter, 1977; Luo and Rudy, 1991, 1994a, b), and the FitzHugh-Nagumo (FN) model (FitzHugh, 1961; Panfilov and Keener 1995; Berenfeld, 1996). The model using Huygens' principle provides the simplest way to simulate the propagation of the action potentials. It requires calculations to

Whole Heart Modeling and Computer Simulation

95

generate wavelets around activating cells at each discrete instant. Cells within the wavelets are to be activated at the next instant. Constructing the wavelets for each instance yields the excitation sequence of the heart model. The effectiveness and the details of this type of model depend on the way the action potential and other electrophysiologic properties are defined. If the definitions contain sufficient electrophysiologic details, the heart model is efficient enough to describe the dynamic process of the heart such as the cardiac arrhythmias. The propagation model based on HH formulism is theoretically most correct. It has the capability to respond to ionic current, membrane potentials, and other factors to provide detailed information that cannot be provided by the former. However, the computation complexity of such a model limits the application in large-scale models. While most published studies using HH formulism in simulating propagations are two-dimensional models, many researchers currently show increasing interest in applying the HH type propagation to 3D whole heart modeling. A recent work is discussed in Chapter 2. Propagation using the FitzHugh-Nagumo (FN) model is a compromise between the implementation complexity and the electrophysiologic details. The FN type cellular dynamics is an electrodynamic model that reconstructs the action potential by reaction state equations. Compared to HH type equations, numerical solutions for FN type propagation need much less computation time.

3.2.5.1 Propagation model ofHuygens' type As an example, the propagation of the Huygens' type is introduced with the model of Wei et al. In this model, the excitability of model cells is one of two basic types: conductive or non-conductive. Non-conductive activation is concerned with pacemaker cells. For these cells, the activation is obligatory whenever the firing time comes. Conductive activation is treated in three steps. The first step is to calculate the extent of propagation in one time step around each excited model unit by constructing an ellipsoidal wavelet based on local fiber direction as shown in Fig. 3.6. The long serniaxis of the wavelet is along the fiber direction, and the other two short serniaxes are along the two transversal directions. The lengths of the long and short serniaxes are RI

=

Vt(t)· T

(3.6)

and (3.7)

respectively, where T is the time step, Vt(t) is the longitudinal conduction velocity of the cell at time t, and k, is the conductivity ratio. The extent of propagation in one time step is described by (3.8)

where I, n, t are the distances along and across the fiber direction in the local coordinate system established at the excited element.

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(;70

(a)

Fiber direction

(b)

FIGURE 3.6. An illustration of the propagation wavelet.

Cells within the wavelet can be activated if they are excitable at that time. The excitability is checked by the principle of refractoriness as (3.9)

where Tpre is the starting time of the previous excitation of the model unit and TA R P (t) is the absolute refractory period at time t. If the model cell is recognized as excitable, the third step is to assign a conduction velocity to it for propagation at the next time step. Note that the conduction velocity in this model is time-varying, depending on the coupling interval. In this model, the automaticity of a pacemaker cell can be either protected or unprotected, depending on the parameter setting. For unprotected automaticity, stimuli from neighboring cells unconditionally activate the unit and reset it automatically after activation. For the protected automaticity, the model unit is not directly activated by surrounding stimuli, but its intrinsic cycle length is modulated based on a so-called phase response curve (PRe). Fig. 3.7(a) shows a PRC assigned to the cells of an ectopic pacemaker to simulate the ectopic firing of the ventricular parasystole . The biphasic line of the PRC was

Whole Heart Modeling and Computer Simulation

97

Delay (%)

SO 40 30

20

DLY

10 0 -10 -20 -30 -40

-SO (a)

(b)

FIGURE 3.7. (a) The phaseresponse curve assigned to cells of an ectopic pacemakerin simulatingectopic firing of ventricularparasystole (Reproduced with permission from Wei et aI., 1995). (b) SN: sinus node, the normal pacemaker; PVC: the ectopic pacemakercausing premature ventricular contraction. (c) From top to bottom are simulated ECGsof bigeminy, trigeminy, quadrigeminy and an operativeratio of the 5:I type.

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~.omv

O.2sec II

II

II (e)

FIGURE 3.7. (cont.)

used to approximate the experimental data (Jalife, 1976; Moe, 1977). By assigning proper values of intrinsic cycle length to sinus and ectopic pacemaker cells (Fig. 3.7(b», ECG waveforms of bigeminy, trigeminy, quadrigeminy, and an operative ratio 5: 1 type were simulated as shown in Fig. 3.7(c) . The simulated ECG waveforms are comparable with those in clinical findings. They are also in agreement with a theoretical model where the cardiac dynamics are represented by difference equations (Ikeda , 1983). The example demonstrates that a Huygens' type model is able to dynamically reproduce the electrophysiologic process of the heart when adaptive action potential and other properties are used in the simulation.

3.2.5.2 Propagation ofHodgkin-Huxley type The HH equation is expressed as (3.10) where L; is the transmembrane current density; Vm = <1>; - e is the transmembrane potential; em is the membrane capacitance; gk, gNa, g/ and Ei ; E Na , E/ represent the conductivity and Nemst potential of potassium, sodium and leakage, respectively. Originated from the HH model, the membrane ion kinetics has been modified based on new experimental findings in recent years, leading to a wide range of single-cell models such as the sinus cell model (Noble, 1989), the atrial cell model (Earm and Noble, 1990), the Purkinje fiber model (DiFrancesco and Noble, 1985), and the ventricular model (Beeler and Reuter, 1977; Luo and Rudy, 1991, 1994a, b). For simulating 2D or 3D propagations of activation in the ventricular myocardium, the Beeler and Reuter model (referred to as the BR model) and the Luo and Rudy (referred to as the LR model) are widely used. The BR and LR models are general mammalian ventricular cell models derived from experimental data and mathematically represented by HH type formulism. These models are expressed in physiological parameters so that they are well suited for simulation studies. Use of BR model takes a

99

Whole Heart Modeling and Computer Simulation

smallercomputationload than the LR modeldoes, but the LR modelhas incorporatedmore recent informationand provides more options such as the ability to chang the extracellular ion concentrations. To model HH type propagation, it is necessary to link the transmembrane current density, L; to an excitable tissue network arising from the electrical structure. For a onedimensionaltissue model, the propagation of the action potential is described (Plonsey and Barr, 1987)by 1 av, lm=-----

Lnair,

(3.11)

+ re ) az

where a is the radius of the fiber, r, and r; are the intracellularand extracellularresistances per unit length along the axial coordinate of z. The equation expanded to 20 and 3D with anisotropy can be foundin Plonseyand Barr(1987).Usually, the partialdifferentialequation is solved numerically with discrete steps in time and space domains. One difficulty in the implementation of HH type propagation is that the numerical solutionfor Vm(x, y, z, t) in a 3D model sometimestakes an impracticalcomputationload. This is the reason why early studies of HH type simulation used 20 tissue models (Virag et al., 1998). On the other hand, however, HH type propagation can relate surface ECG to cellularinformation whenusedin 3D wholeheartmodeling. Recently, someresearchgroups are trying to realize HH type propagation in 3D whole heart modeling in different ways. A Japanese group is establishing such a model using a supercomputer(Suzuki et aI., 2001). Gulrajani and coworkers have recently simulated the surface ECG successfully during the normal cardiac cycles using a 3D whole heart model employing the LR model using the parallel computing technique (see Chapter 2 for details). An alternative way to apply HH formulism to a 3D heart model is the combination of HH and Huygens' type models, as reported by Leon et al. (1991). In this study, when transmembrane potential is less than a potential threshold, the propagation in progress within the model cell is governed by

av = V . DVv at

Cm -

i;on(v) + i app

v<

Vth

(3.12)

where v is the transmembrane potential; i;on, the ionic current; iapp, the appliedcurrent; Cm, the membrane capacitance; and D, the conductivity tensor. When the transmembrane potential is greater than the potential threshold, the model reduces to a conventional Huygens' type and the propagation process is controlled by

V=!Ct,T ,C)

(3.13)

where! is a pre-assignedaction potentialwaveformfor a cell type r and its valuevarieswith time t, and the coupling interval C. In this manner, both subthreshold and suprathreshold phenomenawere simulated at a whole heart level.

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3.2.5.3 Propagation using Fitzllugh-Nagumo model

Taking the model of Berenfeld (1996) as an example, the action potential using the FitzHugh-Nagumo model is represented by state equations of V and U:

-dv = c(-U +bV dt dU

V 3)+ z

1

-dt = -c (-U +a + V)

(3.14) (3.15)

where a and b are constants, and parameter c is an adaptive function representing the reaction to the cellular mechanism of the depolarization and repolarization processes. As in this equation, the variable of state V is able to represent the property of membrane potential. It can be linked to the membrane current density by giving z the physiologic meaning of stimulus intensity, and thus obtain dv - = c( - U

dt

+ bV

3

- V ')

+ V' . (DV'V)

(3.16)

where D is the diffusion tensor dependent on fiber orientation. The term z = V' . (DV'V) behaves the same as with the transmembrane current term in the cable equation. Equations (3.16) and (3.14) constitute a so-called reaction-diffusion system . See details in Berenfeld (1996) , where a complete cycle of ECG is simulated with the reaction-diffusion system incorporating rotational anisotropy. 3.2.6 CARDIAC ELECTRIC SOURCES AND SURFACE ECG POTENTIALS

With any of the propagation strategies described above, we can estimate transmembrane potential Vm(x, y, z, t) at any voxel of the heart model. Therefore, we are able to determine the cardiac sources based on the spatial distribution of transmembrane potential at any instant of simulation (see Chapter 2 for details). It is usually convenient to calculate elemental sources at individual cells of the model, and then integrate them throughout the heart . From the cardiac sources, the ECG potentials on the surface of the torso model can be computed. Simulation of surface ECG potentials is one of the main purposes for whole heart modeling. The basis to determine the cardiac electric source is the bidomain theory. Solving the ECG potentials on the torso model is a typical volume conductor problem. Both of these topics are fully covered in Chapter 2. Therefore, the general principles and methodology will not be repeated in this chapter. Instead, we introduce the algorithms and implementations in our own model study as an example . We start from the Miller and Geselowitz (1978) formula that is based on bidomain theory: (3.17) where (J is the conductivity, ¢ is the membrane potential , and subscript i refers to the intracellular domain . It says that the current dipole density is proportional to the spatial

Whole Heart Modeling and Computer Simulation

101

gradient of intracellular potential distribution. In fact, the cardiac sources can be equivalently expressed in terms of either intracellular or extracellular membrane potentials depending on the form of effective interstitial conductivity used in simulation. Equation (3.17) can also be modified (Geselowitz, 1989) as (3.18) where
(3.20)

where a/ and at represent intracellular conductivity along and perpendicular to the fiber direction. Then D, is expressed in the global system as (3.21) where I, t1 and ti are unit vectors (column vectors) along each axis in the local coordinate system. If we further assume an equal conductivity ratio and let the ratio along and across the fiber axis be r, the conductivity tensor can be simplified as (3.22)

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where C= I

+ (r

- l)U T

(3.23)

In computing the surface ECG potentials , we (Aoki et al., 1987; Wei et al., 1995) developed an algorithm that transforms Poisson equation to Laplacian equation to simplify the volume conductor problem, and used the boundary element method (BEM) for solutions. The problem originated from the Poisson equation with respect to the surface potential ¢ is expressed as: (3.24) Since the surface potentials are results of the primary source owing to the current dipoles in the heart , and the secondary source owing to the boundary effect, we introduce an intermediate variable 1/J = ¢ - ¢o with the boundary condition of

a1/J

a¢o

-=-=-q

an

an

°

(3.25)

where ¢ o is the potentials in an infinite medium, and n represents normal direction to the body surface. Applying the BEM with respect to 1/J leads to (3.26) where G is the Green function of

G=

_

(3.27)

4nlr - r'l

where r is the distance from a dipole source and the integration is performed with respect to ; ' . Discretization of the integral equation (3.26) leads to linear equation s N

L hij1/J j

N

= - LgijqJ

j= ]

fori

=

1,2, . . . , IV

(3.28)

j =l

where hij and gij are coefficients depending on the torso geometry, and N is the number of elemental triangles which approximates the torso surface. To ensure a unique solution, an (N + 1)th equation N

'L aj1/Jj j=]

N

= - 'La j¢J

(3.29)

j=1

is added to (3.28), where aj is proportional to the area of an elemental triangle which has an indexj. This condition is used to define a potential reference so that the surface potential

Whole Heart Modeling and Computer Simulation

103

integral is zero. Rewrite the (N + 1) simultaneous equations of (3.28) and (3.29) in a matrix notation, we have

Hlp = _GQo - A

(3.30)

The final solution for surface potentials is obtained as (3.31) where (3.32) and (3.33)

3.3 COMPUTER SIMULATIONS AND APPliCATIONS If we look at the details of publications in the past, it is very clear that whole heart models and simulations have been extensively employed in research investigating cardiac mechanisms. Part of the significance of whole heart modeling is the capability of linking the undergoing pathology to the ECG features used in clinical applications. Typical examples are models of abnormal ST-T waves (Harurni, 1989b; Dube, 1996; Hyttinen et aI., 1997; Abildskov and Lux, 2000), myocardial infarction (StarttiSelvester et al., 1989, Zenda et aI., 2000), WPW syndrome (Lorange et aI., 1986; Wei, 1987, 1990), hypertrophy and cardiomyopathy (Harumi, 1989b; Wei et aI., 1999), reentrant and scroll waves in ventricular arrhythmias and fibrillations (Panfilov, 1993; Gray and Jalife, 1996; Okazaki et aI., 1998; Clayton et al., 2001a, b). Some new topics in the past five years are long QT syndrome (Clayton et al., 2001, Okazaki and Lux, 1999), T-wave altemans (Abildskov and Lux, 2000), and late potentials (Yamaki et aI., 1999). Furthermore, recent studies have extended model applications to the development of new instrumentation techniques. Examples are pace mapping (Xu et al., 1996, Hren and Horacek, 1997), epicardial mapping (Ramanathan and Rudy, 2001), Laplacian ECG mapping (Wu et aI., 1998; He and Wu, 1999; Wei and Mashima, 1999; Wei, 2001), artificial heart (Zhang et aI., 1999), and impedance CT (Kauppinen et al., 1999). In the following sections, some typical applications of the whole heart model are introduced. In each section, the undergoing pathology is briefly introduced before the description of the model study. For details of pathology, please refer to medical textbooks. Three major reference books used in the description are Goldman (1986), MacFarlane and Lawrie (1989), and de Luna (1993).

3.3.1 SIMULATION OF THE NORMAL ELECTROCARDIOGRAM A correct simulation of the normal ECG is usually the first step for whole heart modeling, before applying the model to special pathologic conditions. The normality of

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TABLE 3.2. Definition of l 2-lead Electrocard iogram Lead

Definit ion

Bipolar limb leads 1= EL - ER II = EF - ER III = EF - EL

I II III Augmented unipolar limb leads aV R a VL aVF Precord ial leads

aVR = (3j2)(ER - Ewe'> aVL = (3j2)(EL - Ewe,) aVF = (3j2)(EF - Ewe,)

Vj(I = 1, 2, .. . , 6) Ewo! is called Wilson Central Terminal. Ewct = (EI.+ ER+ EF)/3 . E denotes potential, subscripts L. R, and F denote left arm, right arm and left foot, respectively.See Fig. 3.8 for electrode positions of V;.

TABLE 3.3. Electrode Positions for Recording 12-lead Electrocardiogram Electrode

Position

Left arm (R) Right arm (L) Left foot (F) Right foot (G)

Left wrist Right wrist Left ankle Right ankle Right sternal margin, fourth intercostal space Left sternal margin, fourth intercostal space Midway betwee n V2 and V4 Left midclavi cular line, fifth intercostal space Left anterior axillary line, V4 level Left midaxi llary line, V4 and V5 level

VI V2 V3 V4 V5

V6

the simulation results can be evaluated with the excitation sequence of the heart model, the simulated vectorcardiogram (VCG) , l2-lead ECG, and body surface isopotential maps. The 12-lead ECG is the most popular lead system used in clinical practice. To evaluate models and simulations with different pathologies, the results are usually compared with clinical recorded l2-lead ECGs. Recording the l2-lead ECG requires 10 electrodes. Four electrodes are placed on limbs to record six limb leads and the other six are placed on the precordial chest wall to record the precordial leads. The definition of l2-lead ECG is summarized in Table 3.2, and the electrode positions for recording are shown in Fig. 3.8(a) and described in Table 3.3 (Horacek, 1989). In model studies, torso models usually do not include the limbs . The limb leads are moved to the closed positions on the torso . Because the potential difference on the limb is sufficiently small , this does not sign ificantly change the simulation results. Most torso models are represented by polygon meshes, and the nodal point s may not exactly overlap the electrode position. In this case, interpolation is usually needed to calculate either the electrode position s from positions of the surrounding nodal points, or the ECG potentials from the potentials on surrounding nodal points. A typical ECG waveform is illustrated in Fig. 3.8(b). The waves of the ECG are designated by Einthoven as P, Q, R, S, T, as shown in this figure . It is well known that the P

Whole Heart Modeling and Computer Simulation

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.

.

,; ' : " .r '

..·..·.\..'1 :

~

,. '

.,

•. '

".

.

V2-·. · · ....·-,f -. V3 -::

. .

'.

'. -,

(a)

QRS

Interval

R

l 1---+---1-+---+---+-----4STse~ent

8, ~

~

u

0.2 (b )

0.4

0.6

time ( 1mm=O,04s)

0,8

1.0

FIGURE 3.8. (a) Electrode positions for the precordial leads. (b) A typical electrocardiogram.

wave corre sponds to atrial depolarization, the QRS complex to ventricular depolarization, and the T wave to ventricular repolarization. The U wave is occasionally recorded after the T wave, and its mechanism remains unclear (di Bernardo and Murray, 2002). The criteria of normality for the ECG include time and amplitude standards. Clinically, the following parameters are evaluated for the normality of ECG: • • • • • •

Cardiac rhythm and heart rate PR interval and segment QRS interval QT interval Pattern and amplitude of P wave Pattern and amplitude of QRS complex

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• Pattern and amplitude of T wave • ST segment and T wave • Mean electrical axis (frontal plane) The limit of normal ECG can be found in MacFarlane and Lawrie (1989) and the textbook of de Luna (1993). The following are typical values: P wave interval = 100 ms, QRS interval = 90 ms, PR interval = 150 ms, and QT interval = 380 ms. Because most heart models do not yield absolute values for ECG potentials, the amplitudes can be evaluated by patterns and relative amplitudes. For example, the aVR wave should have an inversed waveform. The R wave should gradually increase from VI to V4 or V5 and fall from V5 or V6. In simulation studies, the use of VCG is very convenient in evaluating the simulation results. By summing the cardiac dipole sources in all model cells at each instance into a single dipole and plotting it in the frontal, horizontal and saggital planes, the VCG is obtained. It is theoretically close to the clinical VCG. From these figures, the mean electrical axises can easily be estimated. The normal P, QRS, and T axises should be between 0° to +90 30° to + 110°, and 0° to +90°. The body surface isopotential maps are also easily used for evaluation. The typical patterns for a normal heart can be found in many articles and textbooks (e.g., MacFarlane and Lawrie, 1989). Fig. 3.9 shows the simulated results of a normal heart mode by Wei at al. In Fig. 3.9(c), potential distributions on the torso during the QRS period are shown by isopotential countour maps. In most instances, the distributions show dipole fields with one positive maximum on the anterior chest and one negative potential minimum on the back at the early QRS and gradually reversed in the late QRS. In a short period of the middle QRS (time 183-189), we can find a multipole field represented by one potential maximum and two negative minimums. This is a typical pattern in the isopotenials maps representing the right ventricular breakthrough, as experimentally observed by Taccardi (1963). Simulation of a normal heart is generally a repetitive procedure to adjust the model until simulated excitation isochrones fit the experimental data, and the simulated vectorcardiogram and surface ECGs fall within the normal range. Generally, obtaining a normal P wave is not difficult. The excitation from the sinus node spreads leftward and downward does reproduce a normal P wave in the simulated ECG. Getting a normal QRS lead in the 12-lead ECG is more difficult. To keep normal waveforms in most leads on the surface of the torso model, including the normal time periods and normal amplitude relationships among the leads, it is important to correctly mount and adjust the specialized conduction system in the heart model. The simulation ofthe T wave requires a deep understanding of the T wave mechanismthe reason why positive T waves are measured in most of the 12-lead ECG. Suppose the action potential is uniform at all ventricular myocardium, the depolarization and repolarization would be along the same direction, and thus the T waves would have the opposite polarity as that of the QRS waves. But actually, positive T waves are observed in most leads. The mechanism can be interpreted with a theoretical model that assumes longer action potential duration in the sides of the endocardium and apex than toward the epicardium and the base (Harumi et al., 1964). In this way, the repolarization spreads along the opposite direction to the depolarization. As a result, the T wave polarity is the same as that of the QRS wave. The T wave model is consistent to the experimental measurement on the epicardium (Spack et al., 1977). A detailed description of a T wave mechanism can be found in Barr 0

, -

Whole Heart Modeling and Computer Simulation

avtvvt- r vt- vl v:k- v~ ~ I~ I~

( a)

Horizontal

107

av~ av~

frontal

sagittal

(b)

I(~6:~ II ~68 ~ II ~11 ~ II 8

~

e I [J~) °1 117 ms

pT~ 9~~ ~ ~ f'Pr8JlJ I~A'\ II ~ Mil (@) 01[ @) 01 195 ms

(c )

114 ~

198 ~

201 ~

204 ~

201

~

FIGURE 3.9. Simulation results of a normal heart model: (a) ECG ; (b) VCG; and (c) body surface isopotent ial maps. In (c), "+" and "-" show positions of potential maximum and minimum , respectively. Lines of light black represent contours of zero potential. The left area of six tenths correspond s to the anterior torso chest, and the right of four tenths corresponds to the back. Time is counted from the onset of the P wave. (Figures of a and bare reproduced with permiss ion from Wei et al., 1995).

(1989). The algorithm used in simulation to distribute the action potential in the 3D whole heart model is introducedin Section 3.2.4. 3.3.2 SIMULATION OF ST-T WAVES IN PATHOLOGIC CONDITIONS

Studyingthe pathologicchanges in the ST segmentand the T wavesare typicaland importantapplications of whole heart models. This is becausethe ST-T changes are associated with seriousheart diseases such as myocardial ischemia, hypertrophy, and cardiomyopathy.

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108

>'E

-

•

J.

TIme

1St

zoo (ms)

:&SO

FIGURE 3.10. Pre-defined action pote ntials assig ned to cells in normal and middle, moderate , and severely ischemic regions in the simulation of Dube et al. (Reproduced with permission. from Dube et al., 1996).

Myocardial ischemias arise from insufficient blood flow due to an occlusion of coronary arteries. In recent years , the percutaneous trasluminal coronary angioplasty (PTCA ) provides an opportunity to precisely confirm the relationship between the ECG features and the sides of the blood block by controlling the balloon inflation during the PTCA operation. Dube et al. (1996) simulated clinical body surface potential maps and ECGs using the PTCA protocol. Because the action potential change of the ischemic tissue is the direct cause of the ST segment changes in the ECG, the way of setting action potential in the heart model is a key to the simulation. Three transmural zones , middle, moderate and severe ischemia, were set to the heart model, located in the vicinity of the left anterior descending, left circumflex, and right coronary arteries . The action potentials, as shown in Fig. 3.10, were set to these regions, representing action potentials under middle, moderate and severe ischemias. The simulation produced ECG maps quantitatively similar to clinical maps. Fig. 3.11 shows another example that simulates "giant negative T waves" known as the main feature of the apical hypertrophic cardiomyopathy (Harumi, 1989b). The simulated ECG and VCG give surprisingly similar results to clinical findings. The results were obtained by modifying the APD gradient and the conductivity value for the pathologic zone. Unlike the ischemia, the heart with hypertrophic cardiomyopathy is impossible to make with experimental animals . In this sense , computer simulation is the only way of in vivo experimentation to study the unknown mechanism.

3.3.3 SIMULATION OF MYOCARDIAL INFARCTION Myocardial infarction is a typical concern in heart modeling. The example introduced in this section demon strates that model study is not only useful for understanding the mechanism, but also it is useful for helping to develop a diagnosis tool for clinical practice.

109

Whole Heart Modeling and Computer Simulation

Frontal r - - - r - - - - - , r - - -r --

--, Sagittal

FIGURE 3.11. Simulated ECG and VCG in the heart modelof apical myocardial cardiomyopathy (Reproduced with permission from Harumi et aI., 1989).

Myocardial infarction is caused by the occlusion of the coronary artery. The development of myocardial infarction is usually classified in three phases by ECG patterns (de Luna, 1993). The early phase of ischemia is characterized by T wave changes. The later phase of injury is characterized by ST segment changes. The final phase of necrosis is characterized by Q wave changes. The ST and T wave changes can be simulated with a whole heart model by changing the action potentials for the model cells. The action potentials during the reduced blood flow can be measured experimentally. The location and size of myocardial infarction is an important aspect in clinical diagnosis. In clinical practice, the locat ion and size of infarction are qualitatively interpreted with the theory of vectorcardiogram. With a whole heart model , StarttlSelvester et al. (1989) systematically simulated the infarcts due to three major coronary artery distributions, and expected Q waves were obtained in each case. They found that in any case the degree of QRS change was proportional to the degree of local infarction. Dividing the ventricles into four walls by 12 segments, they developed a quantitative method to estimate the location and size of the infarction. The result led to a practical tool known as the ECGNCG scoring system, where each point scored was set up to repre sent 3% of the left ventricular myocardium. The scoring system predicted a distribution of damage in the 12 left ventricular

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segments in good correlation with the average of planometric pathology found in the same subdivisions. They further developed a monogram based on the same simulation study, which relates the VCG changes to the infarct size. If the duration and magnitude of QRS deformity are measured before and after infarction, the infarct size can be simply found on the monogram. Details can be found in Startt/Selvester et al. (1989).

3.3.4 SIMULATION OF PACE MAPPING Pace mapping is a new technique that is used to speed up the procedure for localizing ectopic focus in catheterization (SippensGroenewegen et aI., 1993). In SippensGroenewegen et aI., body surface potential maps in patients with cardiac arrhythmias but with no evidence of structural heart disease were recorded during ectopic beats by catheter stimulation at different endocardial sites in the ventricles. Based on these data, they were able to classify the QRS integral map patterns with respect to the location of ectopic beats. The same procedure was reproduced by Xu et aI. (1996) with the heart model of Lorange et aI. In the simulation, 38 selected endocardial sites (25 on the left ventricle and 13 on the right ventricle) corresponding to SippensGroenewegen et aI. were paced to initialize the excitation process of the heart model. With a more detailed heart model (0.5 mm spatial resolution), Hren and Horacek (1997) generated a database of 155 QRS integral maps by pacing the epicardial surfaces in the left and right ventricles. This database would be useful in catheter pace mapping during treatment of ventricular arrhythmia. The simulation of pace mapping is a good example to show the clinical usefulness of whole heart models.

3.3.5 SPIRAL WAVES-A NEW HYPOTHESIS OF VENTRICULAR FIBRILLATION Reentrant excitation is recognized as a mechanism of life-threatening arrhythmias. Among several hypotheses to explain the reentrant excitation, spiral wave is the latest one and is getting more and more support from experimental studies (Gray, 1995, 1996). In addition to experimental studies, computer simulation of the spiral wave using the whole heart model with realistic geometry and anatomy is a useful tool because it is capable of producing comparable results with experiments. Fig. 3.12 shows an example of spiral waves simulated with a whole heart model (Gray and Jalife, 1996). The simulated isochrones and ECGs (right) are comparable with the measured results.

3.3.6 SIMULATION OF ANTIARRHYTHMIC DRUG EFFECT The example introduced in the following demonstrates how the drug effect is confirmed by a simulation study with a whole heart model (Wei et aI., 1992). The study was based on an experiment that investigated the relationship (Harumi, 1989a) between the restitution of premature action potential and the stimulation coupling interval for Purkinje fiber and ventricular muscle in dogs before and during the infusion of antiarrhythmic drugs. In the simulation, lines through linear regression as shown in Fig. 3.13(a) were obtained to approximate the experimental data. The slopes of these lines corresponding to a parameter called dynamic coefficient (DC) were input to the heart model. Note that, the figure shows different slopes for the ventricular muscle and the Purkinje fiber before and during the infusion

Whole Heart Modeling and Computer Simulation

111

FIGURE 3.12. Simulation of spiral wave.(Reproducedwith permission, from Gray and Ja1ife, 1996).

of antiarrhythmic drugs. In the simulation, ten successive extra-stimuli of 170 ms interval started at 300 ms after the first sinus pacing were applied to the epicardium of the ventricle. The simulated ECG shown on the top of Fig. 3.13(b) corresponds to the case where the APD changes follow the lines before the drug infusion. In this case, the stimulation caused two tachycardia-like waveforms followed by sustained VF. When APD changes follow the lines during the drug infusion, normal waveforms are restored after the stimulation, as shown by the waveform in the bottom of Fig. 3.13(b). Figure 3.13(c) shows a picture of animation developed for visualizing the propagation of excitation during the fibrillation. A number of propagation wavefronts developed due to a number of reentries can be seen in this picture. This simulation supported the assumption that different ratios of restitution in premature APD between the ventricular muscle and the Purkinje fiber may playa role in the induction of VF. It also demonstrated the antiarrhythmic drug effect in suppressing VF.

3.4 DISCUSSION Principles, methodology and applications in 3D whole heart modeling are described in this chapter. The significance of 3D whole heart models is that, as compared to experimental studies, computer simulations with whole heart models are always in vivo so as to provide information relating intracardiac events to the body surface electrocardiogram in different

Ventricular muscle Purkinje fiber

Ventricular muscl e Purkinje fiber

,

Q

~<::)

<::)

lI'\ N

lI'\ N

<::) <::)

<::) <::)

N

N

300

400

Coupling interval (rns)

(a)

500

300

400

Coupling interval (ms)

500

Lomv 0 2 Se C

(b)

FIGURE 3.13. (a) Approximation of experimental results describing restitutions of premature action potential to the stimulation coupling interval for Purkinje fiber and the ventricular muscle in dogs before (left) and during (right) the infusion of antiarrhythmic drug. (b) Simulated ECGs. The waveform on the top shows ventricular fibrillation induced by applying successive stimuli to the left ventricular wall. The waveform on the bottom shows that the fibrillation stops after the infusion of the antiarrhythmic drug. (c) An image of 3D animation showing excitation propagation during ventricular fibrillation.

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Whole Heart Modeling and Computer Simulation

pathologic processe s. Such information is sometimes difficult to obtain by experimental studies. Other advantages arise from the facts that the computer simulation is low-cost , fast and repeatable. As an experimental tool, a whole heart model can be used in laboratories for research purposes and used in classroom for computer-aided instruction in medical education. The main limitation of current whole heart models is the insufficiency of the electrophysiological details, due to the large scale of the whole heart and the computational capability related to the problem. In the present stage, the heart models in single-cell, cellnetwork and whole heart levels are still separately studied . More time is needed to realize and spread the 3D whole heart models of next-generation that can generate action potentials based on cellular mechanisms, that can simulate propagation based on HH type formulisrn , and reconstruct clinically comparable body surface electrocardiograms. This kind of model would be able to directly relate basic experimental findings to clinical applications. In a search of related abstracts on Medline in the U.S. National Library for the two decades till the end of 1980s, Malik and Camm (1991) found that both the absolute and relative numbers of publications in computer modeling and simulation have increased much faster than the total increase in cardiology as a whole . We have searched the same database for publications in the 1990s using the keywords "heart + model + simulation + computer." The result is shown in Fig . 3.14. Publication activity continues to increase. Nowadays, computer modeling and simulation have become more attractive to researchers than ever before . Because the biological system is so complex, no single experiment can synthesize the system as a whole . As the computation ability of computers becomes more and more powerful , computer modeling and simulation technologies are making it possible to study the biological system of humans as a whole. This is recently known as "Physiome" ("physi " means life, "orne" means whole , see http://www.physiome.orgl). Whole heart modeling is one of the common concerns in Physiome.

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i

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1990 1991 1992 1993 1994

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1996 1997 1998 1999 2000

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FIGURE 3.14. Publicati ons searched on MedPu b. (Keyword = "heart + model

+ simulation + computer").

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REFERENCES Adam , D., and Barta, E., 1987, The effect s of anisotro py on myocardial activation, in: Simulation and Cont rol of the Card iac System, Volume III (S. Sideman, and R. Beyar, eds. ), CRC, Boca Raton, Florida , pp. 13-27. Abildskov, J. A., and Lux, R. L., 2000, Mechan isms in T-wave alternans caused by intraventric ular block, J. Electro cardiol. 33 (4 ):3 11-3 19. Al-Nashash, H., and Lvov, 8. . 1997, Thr ee-dimensional model for the simulation of the HPS electrogram, Biomed. Mat er. Eng. 7(6 ):40 1-4 10. Aoki, M., Okamoto, Y., Musha, T., and Harum i, K., 1987, Th ree-dimensional simulation of the ventricular depolariza tion and repolari zation proce sses and body surface potentials: Normal heart and Bundle Branch Block, IEEE Trans . Biomed. Eng. 34 :454-462. Balasubramaniam, C; Go pakumaran , B., and Jagadeesh, J. M., 1997, Sim ulation of cardiac cond uction system in distributed computer environm ent, Biomed. Sci. Instrum. 33 :13-8 . Barr, R. C, 1989 , Gene sis of the elec troca rdiogram , in: Comp rehensive Electrocardiology (P. W. MacFarlane and T. D. V. Lawrie, eds. ), Pergamon Press, New York, pp. 129-151. Beeler, G. W., and Reuter, H., 1977, Reconstruction of the action potent ial of ventricular myocardial fibres, J. Physiol.268(1):I77-21O. Bercnfeld, 0 ., and Abboud, S., 1996, Simu lation of cardiac activity and the ECG using a heart model with a reaction-diffusion actio n potential, Med. Eng. Phys. 18:615-25. Clayto n, R. H., Bailey, A., Biktashev, V. N., and Holden, A. V., 200 1a, Re-entrant cardiac arrhy thm ias in com putational mode ls of long QT myocardium, 1. Theor. Bioi. 208 :2 15-25. Clayton, R. H., 2001 b, Computational mode ls of norma l and abnorma l action potentia l propagation in cardiac tissue: linking experi mental and clinical cardiology, Physi ol. Mea s. 22 : R 15- R34 . Cler c, L., 1976, Directional differences of impu lse spread in trabec ular muscle from mammalian heart . J. Physiol. 255 :335- 345. De Luna, A. B., 1993, Clin ical Electrocard iograph y: A Textbook , Futura Publ ishing Company Inc., New York. di Bern ardo, D., and Murray, A ., 2002, Origin on the electrocardiogram of Ll-waves and abnorma l U-wa ve inversion, Cardiova sc. Res. 531 :202- 208. DiFrancesco, D., and Nob le, D., 1985, A model of cardiac elec trical activity incorporating ionic pumps and concen tratio n change s, Phil. Trans. R. Soc. Lond. 8307:353- 98. Dubc, B., Gulrajani, R. M., Lorange, M., LeBl anc, A. R., Nasmi th, J., and Nadeau , R, A. , 1996, A com puter heart mode l incorporating anisotropic propaga tion. IV. Sim ulation of regional myoca rdial ischem ia, J. Electrocardiol. 29:91-103. Durrer, D., van Dam, R. T., Freud, G. E., Janse, M. J., Meijler, F. L. , and Arzbaec her, R. C; 1970, Th e total excita tion of the isolated human heart, Circ. 41 :899-912. Earrn, Y. E., and Noble, D., 1990, A mode l of the single atrial ce ll: betwee n calci um current and calcium release, Proc. Roy Soc. 240 :83- 96. FitzHugh, R., 196 1, Impu lses and phy siological states in theoret ical models of nerve memb rane . Biophys. J. 1:445-466. Gczelowitz, D., and Miller III, W. T., 1983, A bidomain model for anisotrop ic cardiac muscle, Ann. Biomed. Eng . 11:19 1-206. Gese lowitz, D. B., 1989, The ory and simulat ion of the Electrocardiogram, in: Comp rehensive Electrocardiology (P. W. MacFarlane and T. D. V. Lawrie, eds .), Pergamon Press, New York , pp. 18 1- 195. Goldman, M. J., 1986, Principles of Clinical Electroca rdiograph y, Lange Medical Pub lications, Los Altos, Califo rnia . Gray, R. A., Jalife, J., Panfilov, A. v.. Baxter, W. T., Cabo, c., Davidenko, J. M., and Pertsov, A. M., 1995, Mec hanism of cardiac fibrillation, Science 270 :1222- 1225. Gray, R. A., and Jalife, J., 1996, Spiral waves and the heart, lnternational J. of Bifurca tion and Chaos 6: 415-435. Gulrajani, R. M., and Mailloux, G. E., 1983, A simu lation study of the effects of torso inhomogeneities on electroca rdiographic poten tials, using realistic heart and torso models, Circ. Res. 52 :45- 56. Guyton, A. c., 1986, Textbook ofMed ical Physiology, W. 8. Saunders, London. Harumi. K., Burgress , M. J., and Abildskov, 1. A., 1964, A theoret ical mode l of the T wave, Circ. 34:657-668. Harumi, K., Tsutsumi, T., Sato, T.. and Seki ya, S., 1989a, Classification of antiarrhythmic drugs based on ventricular fibrillat ion threshold, Amer. J. Cardiol. 64 : IOJ-1 4J.

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4

HEART SURFACE ELECTROCARDIOGRAPHIC INVERSE SOLUTIONS Fred Greensite 1 Department of Radiological Services University of California, Irvine

4.1 INTRODUCTION In this chapter, we will review the problem of noninvasive and minimially invasive imaging of cardiac electrical function. We use the term "imaging" in the sense of methodology which seeks to spatially resolve distributed properties of cardiac muscle electrophysiology such as extracellular potential, or features of the action potential. Thus, we do not consider the problems of computing properties of an "equivalent" cardiac multipole, moving dipole(s), or any other source model that does not satisfy such criteria. We will further restrict ourselves to resolving such electrophysiological features on the epicardial or endocardial surfacesa reasonable restriction, since measurements currently accessed by invasive procedures are obtained on these surfaces, and also because the spatial dimension of the "source" domain then nominally matches the spatial dimension of the data domain. Thus, we will not consider the earliest distributed source model, representing intramural current density imaging (Barber and Fischman, 1961; Bellman et al., 1964), on which work continues (e.g., see (He and Wu, 2001), or the recent heart-excitation-model based 3D inverse imaging approach (Li and He, 2001) in Chapter 5 in this book). Following an historical perspective, we will discuss in some detail the inherent difficulties of this imaging problem (principally mathematical), and strategies developed to circumvent them. We will not attempt to comprehensively cite the voluminous work done on these formulations of the inverse electrocardiography problem. Excellent reviews for the period prior to 1990 exist (Gulrajani et al., 1989; Rudy and Messinger-Rapport, 1988), and more recent shorter reviews can be found in (MacLeod and Brooks, 1998; Gulrajani, 1998). I

Mailing Address: Fred. Greensite, Department of Radiological Sciences, University of California-Irvine Medical Center, Trailer 11, Route 140, 101 The City Drive South, Orange CA 92868 USA. E-mail: [email protected]. Telephone: (714) 456-7404, FAX: (714) 456-6380. 119

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Our principal objectives are to provide a meaningful presentation of the issues involved in this inverse problem, and to survey some work of the last several years.

4.1.1 THE RATIONALE FOR IMAGING CARDIAC ELECTRICAL FUNCTION Coordination of the heart's mechanical functioning is accomplished electrically. In fact, there is a tight coupling of mechanical and electrical function both at the microscopic cellular and macroscopic organ level. Global contraction of the heart proceeds from a sequence of local changes in cell membrane conductances to various ionic species (the opening and closing of voltage-gated species-specific membrane channels). These membrane conductance changes, resulting in transmembrane currents, generate local source currents which lead to Ohmic currents throughout the remainder of the body volume conductor. In this manner, the sequential cardiac muscle membrane changes, which characterize the heart's electrical functioning, are reflected in electrical potentials that can be measured at the body surface. The local changes in intracellular ionic concentrations, resulting from the transmembrane electrical current, trigger the local subcelluar mechanical events (the "sliding" of filaments over each other) which on a global scale summate to coordinated contraction of the muscle mass. Since the local mechanical events must be coordinated satisfactorily with each other to produce an effective organic contraction, a corollary is that a mechanical catastrophe can only be avoided if the inciting local electrical events are themselves coordinated properly on a global scale. Even when mechanical function appears adequate, the spatiotemporally distributed electrical functioning of the heart is of the greatest interest medically, since well characterized disturbances of cardiac electrical functioning are known to predispose to such mechanical catastrophes, and are known to be amenable to various interventions (pharmcological, catheter-based, or surgical). Evidently, it is useful to consider how such cardiac electrical distubances might be effectively diagnosed noninvasively. Although indications of electrophysiological dysfunction may be apparent from the unadorned clinical electrocardiogram (tracings of electrical potential at certain body surface locations), successful treatment often requires "imaging" of such disturbances-usually accomplished by laborious transvascular electrode catheter manipulations within a cardiac chamber. The prospect that such imaging might be effected in a less invasive and efficient manner, is the principal motivation for present efforts to develop a practical clinical technology for imaging cardiac electrical functioning.

4.1.2 A HISTORICAL PERSPECTIVE In our context, imaging could be looked at as a global depiction of local features. Thus, it can be understood in terms of microscopic and macroscopic cardiac electrophysiology.

Microscopic: Action Potential Sensory-neuro-muscular tissue is distinguished by its ability to generate and conduct action potentials. The latter represent cellular transmembrane potential evolution during the characteristic functioning of the tissue (figure 4.1). The concept was inherent in the work of Bernstein in the late 1800s, wherein functional neuronal cellular electrical activity was attributed to sequential membrane permeability changes. Muscle is particularly distinguished

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mV

-90

FIGURE 4.1. The action potential in heart Purkinje cells resembles a pulse (i.e., square wave). The upstroke (phase 0, "activation") corresponds to rapid intracellular influx of sodium. A limited fast repolarization (phase 1), and subsequent plateau (phase 2), correspond to an interval for which the location is refractory to stimulation. The downstroke (phase 3) is over an interval for which stimulation will lead to weakened activation. The horizontal axis can be either time (action potential duration is on the order of hundreds of milliseconds) or space (although simultaneous recording of transmembrane potential along a path in the heart muscle at a single time instant would usually only reveal the full action potential shape in the setting of re-entry type arrhythmias). The -90 millivolt baseline defines phase 4.

in that arrival of an action potential at a given location triggers a cascade of events leading to contraction. Wilson et al. (1933) expressed great insights into the significance of this transmembrane functioning in cardiac muscle, in a seminal work from the 1930's. All aspects of the action potential are potentially of interest. For example, phase 0 (signifying local activation) initiates the events leading to the local contraction (sliding of filiments). At the phase 0 time, the location "depolarizes" (transmembrane potential changes from roughly -90 millivolts to roughly zero millivolts). The amplitude of phase o has important implications as regards local muscle integrity. During Phase 1 (transient, limited, fast repolarization) and Phase 2 (plateau), the location cannot be further stimulated (absolute refractory period). Phase 3 (progression through to "repolarization") is a relative refractory period, during which attenuated responses to further stimulation are possible. The durations of the absolute and relative reftractory periods govern when the location will be susceptible to being triggered again (for the next heartbeat), and how strong that next contraction might be locally. During phase 4, the location is fully repolarized, i.e., maintains the baseline -90 millivolt transmembrane potential difference. Given the significance of these features, an obvious goal would be to noninvasively image (spatially resolve) the action potential at every location in the heart muscle (i.e., reconstruct the time series of the transmembrane potential at each location). However, articulation of such a goal has been long in coming.

Macroscopic: Electrocardiogram Recognition of the electrical functioning of muscle predates the nineteenth century (and possibly the eighteenth century) (Malmivuo and Plonsey, 1995). By the mid-nineteenth

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bn--_ _ I;

FIGURE 4.2. An illustration adapted from (Waller, 1889). The cardiac-generated potential field depicted is essentially dipolar in nature. [Adapted from figure 5, p. 186: Waller, A., 1889, On the electromotive changes connected with the beat of the mammalian heart, and of the human heart in particular, Phil. Trans. R. Soc. Lond. B. 180: 169-194. Used by permission.]

century, the epicardial potentials had been invasively accessed. Later in the century, Waller had the brilliant insight that the limbs could be viewed as electrode leads emanating from the heart-so that coupling them to an appropriate electrical apparatus should allow noninvasive assessment of the previously established cardiac electrical functioning. From his published illustrations, it appears that he conceived of modeling the heart as a current dipole source, begging the question of inversely "imaging" this source from his body surface potential measurements (Waller, 1889) (figure 4.2). Interestingly, Waller (1911) was not optimistic that such information would be of much use medically. However, nearly coincident with his cautionary remarks, Einthoven (1912) was engaged in the work of demonstrating the clinical efficacy of improved instrumentation to accomplish just such a goal. Indeed, early in the twentieth century Einthoven popularized (via the "Einthoven triangle") the concept of the cardiac dipole as (qualitatively) estimated from measurements at standardized electrode locations-the electrocardiogram. In ensuing decades, a number of investigators attempted to make such calculations more quantitative (e.g., (Wilson et al., 1947; Frank, 1954». In this context, Wilson's earlier work (Wilson et al., 1933) provided the conceptual link between the microscopic and macroscopic. In particular, it was observed that macroscopic cardiac activation wavefronts, arising from cellular membrane events, effect measurable remote potentials whose magnitude is roughly

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proportional to the solid angle subtended by the activation wavefront and the electrode location. However, the implication that cardiac electrical functioning might be remotely imaged had to await the diffusion of computer technology to the biomedical community in the early 1960s (Gerlernter and Swihart, 1964; Bellman et al., 1964; Barnard et al., 1967). 4.1.3 NOTATION AND CONVENTIONS The imaging formulas for cardiac electrophysiology derive from linear partial differential equations. When they are "discretized" for numerical solution (see Chapter 2 in this book), one encounters large scale algebraic manipulations. We will find the following vector and matrix conventions useful: We denote a matrix composed of zeros as 0, and an identity matrix as I. Superscript t applied to a matrix denotes the transpose, so when it is applied to a column vector it therefore denotes the corresponding row vector (and vice-versa). Thus, for column vectors a, b, the inner product will be written as either a . b or at b, while the outer product is written as either a ® b or ab' (the outer product of a and b is the matrix whose (i, j) entry is the product of the i-th component of a with the j-th component of b). A vector in three dimensions will be denoted by a bold face font (e.g., n will denote a unit normal to a surface, andj will denote a current density). Matrices will be denoted by capital letters, e.g., M. The i-th row of M will be denoted as M i : , and the j-th column will be denoted M: j • Random variables (and random vectors and random matrices) will sometimes be denoted with a superscripted bar, e.g., a. The probability (or probability density) of some event will be denoted p(.), so that p(alb) is the (conditional) probability of event a given that event b has been observed. As is customary in physics, elemental sources (e.g., electrical monopoles) are expressed via the Dirac delta function 8(x )-a convenient shorthand for the limiting notion of a function whose nonzero portion is localized to a tiny region, but whose integral over all space is unity (Jackson, 1975). We note that such entities are rigorously treated in the context of the theory of distributions (e.g., see (Keener, 1988) for details).

4.2 THE BASIC MODEL AND SOURCE FORMULATIONS In this section, we will derive imaging equations relating body surface potentials to heart surface potentials. We do not present the analogous equations for magnetic field data, which can be derived in a similar manner from the equations of magnetostatics. As noted, description of muscle electrophysiology can be organized around the concept of an action potential, which reflects the opening and closing of ionic channels in the cell membrane-which in tum lead to local intracellular ionic environmental changes that trigger the local mechanical function, and the return to the resting state. By definition, an action potential is the transmembrane potential that occurs in sensory-neuromuscular tissue during its characteristic functioning. It encompasses the phenomena of self-propagation to adjacent locations, meaning that the form ofthe transmembrane potential at a fixed location as a function oftime is recapitulated in a tracing oftransmembrane potential at a fixed time as a function of location (along the path of activation). Propagation of the action potential coordinates local contraction over the full three-dimensional

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extent of the muscle, and implies its link to the global mechanical functioning of the tissue. Of course, the action potential is measured historically via microelectrodes inserted into cells. Our objective is to resolve features of the transmembrane potential remotely from the tissue. Thus , we must formulate the relationship between cardiac muscle action potential, and remote measurements (such as on the body surface) . This is conveniently accomplished via the so-called "bidornain" model , a concept first suggested by Schmitt (1969), and later given mathematical form by others (reviewed by Henriquez (1993)) . Although the model is discussed in Chapter 2, we re-present it here as we wish to derive our source formulations from "first principles", as well as provide a consistency of notation within the present chapter. At a level of resolution appropriate to this problem, it can be assumed that every body location consists of a small amount of intracellular space and a small amount of extracellular space. We can write (4.1)

where ¢i, ¢e, and ¢m, are intracellular potential, extracellular potential, and transmembrane potential, respectively. In the context of our remote sensing problem, it has been shown that capacitive, inductive, and electromagnetic propagative effects are "negligible" (the "quasistatic assumption") (Plonsey, 1969). Therefore, the intracellular or extracellular current densities at the bidomain point are each linearly dependent on the gradient of the intracellular or extracellular potential (Ohm' s Law). Thus, these respective current densities are given by j i = -GiV¢i

(4.2)

je = -G eV¢e,

(4.3)

where G, and G, represent conductivity in the intracellular and extracellular components of the bidomain point, and these conductivities are (in practical terms) independent of the potential (the medium in linear) (Plonsey, 1969). It is important to note that cardiac muscle tissue is "anisotropic", in that it is composed of (interconnected) muscle fibers whose intracellular conductivity along the fiber direction is much greater than the conductivity normal to the fiber direction. Due to the geometrical constraints imposed by the intervening fibers, the extracellular conductivity is also anisotropic. Therefore, G i and G e are tensors (in this case, symmetric 3 x 3 matrices) which thereby linearly map a gradient of potential (at a point) to a current density vector. If G i and G e were proportional to each other (implying that the principal axes, and the conductivity ratios for different pairs of principal axes, are the same for intracellular space as for extracelluar space), we could speak of there being "equal anisotropy". According to Eq. (4.2) and Eq. (4.3) , total current density j = j i + je at any of the bidomain "points" must satisfy (4.4)

The quasi-static assumption and charge conservation imply that any excess current (nonzero

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divergence) appearing in the extracellular component of a bidomain point must come from the intracellular component of the bidomain (via the cell membrane), and vice versa. That is, V' . j = O. Thus, Eq. (4.4) implies (4.5)

The divergences on both sides of the above equation express the location's role as a source of extracellular current (there is net transmembrane current if the divergence is nonzero). Transmembrane currents capable of influencing body surface potentials only occur in excitable tissue (i.e., sensory-neuro-muscular tissue)-the predominant one being the heart. Thus, outside the heart there is no propagation of action potentials, and no source current, since G i is zero there. Writing the extracellular potential e, as simply 4>, then using Eq. (4.1) we can rearrange Eq. (4.5) as - V' . [(G e

+ G i) V'4>] =

V' . [G i V' 4>m].

(4.6)

We have the further condition that no current leaves the body, so the component of current density normal to the body surface is zero. Thus, for y on the body surface and D y a unit normal to the body surface at y, we have D y . G e V'4>(y) = o. A uniformly zero boundary condition (such as this) is referred to as "homogeneous". Equation (4.6) is a partial differential equation, specifically, Poisson's equation. Let us imagine that we are given V' . [G i V' 4>m] (the "source"), and we wish to compute the resulting potential 4>-the so-called "Forward Problem" (see Chapter 2 of this book). One very important feature of the Poisson equation is its linearity. That is, if a solution to - V' . [(G e + G i) V'4>] = f satisfying the homogeneous boundary condition is known as 4>f' and a solution to - V' . [(G e + G i) V'4>] = g satisfying the homogeneous boundary condition is known as 4>g, then it is easy to verify that the solution to a solution to -V'. [(G e + GJV'4>] = c.] + C2g is given by cl4>f + c24>g (for Cl, C2 constants), and this solution also satisfies the homogeneous boundary condition. This means that if we know the solutions to Eq. (4.6) for a source localized to any single location in the heart, then to determine the solution for any more geometrically complex source within the heart volume V we only need to add up (integrate) the solutions that would be obtained for each single location comprising the complex source. Thus, consider Eq. (4.6) where its right-hand-side (the source) has unit strength when integrated over all space, but is zero everywhere except at x (i.e., the source is an electric monopole). Denoting the solution as 1/r(x, y) (where 1/r(x, y) as a function of y satisfies the homogeneous boundary condition), we have -V' . [(G e

+ GJV'1/r(x, y)]

= 8(x - y),

(4.7)

where the divergence and gradient operators are with respect to the field point y in threespace. Thus, 1/r(x, y) is the potential at point y in the body that would be induced by a unit strength source that was zero everywhere in the heart except at the location x (a unit strength source localized to a point is mathematically represented by a delta function). 1/r(x, y) is known as a Green's function. Since Poisson's equation is linear, we can now write the

126

F. Green site

solution to Eq. (4.6) for the geometrically complicated source (on its right-hand-side) as (4.8)

cPe(Y) = Iv 1fr(x , y )V . [Gi(x)VcPm(x)]dVx,

where V is the heart volume. Following Yamashita and Geselow itz (1985), integration by parts applied to Eq. (4.8) (i.e., application of the Divergence Theorem and the identity V . (fV g) = fV . Vg + V f . Vg ) leads to q,(y ) =

L

1fr(x , y) [G;VcPm(x ) , llx]dS x - Iv V1fr (x , y) . [G; VcPm (x )]dVx

= - Iv[G ;V1fr(X,y)] . VcPm(x)dVx

(4.9) (4.10)

where S is the surface surrounding the heart muscle volume V . Equation (4.10) follows firstly because the surface integral in Eq. (4.9) vanishes, i.e., G;VcPm ' n, is zero for x on the heart surface (G; VcPm is the source current, and therefore confined to the heart , so that it will have no component normal to the heart surface ). The integrand on the right-hand-side of Eq. (4.10) results becau se of the symmetry of G; (i.e., for vectors a, b and symmetric matrix C , a' Cb = b'Ca). A second integration by parts , now applied to Eq. (4.10), gives

Given that y is not in V (e.g., we typically consider y to be on the body surface ), the volume integral on the right-hand side ofEq. (4.11) is zero if G, is proportional to G e (equal anisotropy). This is because in that case we would have for some scalar a V · G;V1fr(x , y) = aV · (G;

+ Ge)V1fr(x, y) =

0,

(4. 12)

where the second equality ofEq. (4.12) follows from Eq. (4.7) since y on the body surface is external to the source domain V. In this case, the imaging equation (4.11) becomes cP (y) = - L[G;V1fr(X , y )] . ll xcPm(x )dSx.

(4.13)

Since 1fr(x , y ) satisfies Eq. (4.7), as a function of y it can be thought of as the field generated by a monopole at x. Thus, [G ;V1fr(x , y) ] . n, = V1fr (x , y) . [G;llx] can be thought of as the field generated by a current dipole at x pointing in the G;llx direction. This can be verified by introducing the second source -8(x ' - y) where x' is a point close to x with the line between x and x ' oriented as G;llx' This monopole of opposite polarity is associated with a second Green 's function -1fr(x ' , y ), so that the composite of monopole sources of opposite sign at x and x ' approach a dipole . The appropriate limiting procedure leads to a field determined by (G; n) . V1fr as in the integrand above.

Heart Surface Electrocardiographic Inverse Solutions

127

Thus, we have dervied a linear relationship between transmembrane potential at the cardiac surface S (endocardium plus epicardium), and measurable body surface potential. In the practical setting where the geometry is discretized, the function ¢ over body surface points is expressed as a column vector whose components are measured potentials at various electrode sites, while ¢m is a vector whose components are transmembrane potential at some set of locations on the heart surface, and -l[G;V1/J(X, y)]. nA·)dSx

(4.14)

is a matrix. The forward problem associated with the above transmembrane potential formulation requires knowledge of the anisotropic conductivity of the heart in construction of the operator Eq. (4.14). For example, solution of Eq. (4.7) for any given source point x requires knowledge of G; + G, throughout the body volume, including in the heart (where we have to know G; and G e as tensors). There is an additional complication in that the equal anisotropy assumption is not accurate, so one must also consider the second integral on the right-hand-side of Eq. (4.11). However, in the portion of the body external to the heart muscle we have that G; V¢; = 0, since G; is zero outside the heart. In that volume, we have from Eq. (4.5) that (4.15) i.e., Laplace's equation. The relevant volume is bounded by the epicardium and the body surface. Thus, the boundary conditions are divided into two parts: 1) the zero normal component of current density at the body surface, 2) the (unknown) epicardial potentials. Suppose we know the solution to this equation for the situation where the epicardial potential is identically zero except for having unit strength concentrated at location x-and call this solution kj(x, y). From the linearity of Eq. (4.15), we can then find the solution for any geometrically complex epicardial potential distribution by simply adding together such elemental solutions, just as was done for Poisson's equation. This again defines a linear relationship as (4.16) where S is now the epicardial surface and ¢ep; is the epicardial potential. Thus, we have the linear relationship between epicardial potentials and (measured) body surface potentials. The Green's function k, (x, y) is provided by solution of the forward problem (see Chapter 2 in this book). This formulation avoids having to consider the anisotropic myocardium in the construction of k, (x, y), since the volume under consideration for the partial differential equation does not include the heart volume. The last source formulation we will consider is that of the endocardial potentials. If we design a transvenous catheter such that its tip is embedded with many electrodes (the "probe"), pass it into a cardiac chamber, and register its location with respect to the endocardium (e.g., via ultrasound or electronic means), then we can consider the volume between the catheter probe and the endocardium. Laplace's equation (4.15) still holds for this volume (it is source-free). The boundary conditions are the zero component of current

128

F. Greensite

density normal to the probe surface, and the (unknown) endocardial potentials. Analogously, we can again use Green's functions to derive a linear relationship between the endocardial potentials and the probe electrode potentials, and we again have an equation of the form as above, i.e.,
i

k2(x , y)
(4.17)

That is, S is the endocardial surface,
(4.18)

where h and g are data and source vectors, respectively, and F is a (transfer) matrix.

4.3 HEARTSURFACE INVERSE PROBLEMS METHODOLOGY Some significant mathematical issues are involved in providing optimal solutions for the imaging equations of the last section. Firstly, one must deal with the concept of an "illposed problem" (intuitively, a problem whose nominal solution is unstable to small changes in data-e.g., unstable to small noise variations). Accordingly, it is necessary to introduce "regularization" formalisms. Since the problem is inextricably bound up with noise, we

129

Heart Surface Electrocardiographic Inverse Solutions

cavity G, a const ant

myocardium

"il . [G."il4>] = 0 (

_8<1»

an

~O

p,.obe

endocardium : epicardium: ¢'P'

"il·[G,"il¢] = 0

body G, com plicated

' - - - - - - - - - - - - - - - body su rfac e: ¢b. FIG URE 4.3. A diagram containing the elements used to define the endocardial versus epica rdial potential imaging problems . Epicardial potential imaging problem: The outer box contai ns the body. The outermost ellipse represents the epicardial surface, on which potentia l 1>el' i exists. Electrical potential satisfies Laplace's equation in the body volume external to the epicardium. The mixed boundary condit ions of the (unknown) epicardial potential 1>el' i , and absen t current component normal to the body surface, fully determine potential 1> in the body volume external to the heart. Green's second identity can be used to derive a linear dependence between potential at the body surface and potential at the epicardium. However, computation of this linear dependen ce (the forward problem) is dependent on knowledge of the extracellular conductivity Ge - which is very heterogeneous, e.g., due to the lungs, fat, and muscle (however, there is recent evidence that the impact of these inhomogeneities on an inverse solution may be small (Ramanathan and Rudy, 200 1» . Endocardial potential imaging problem: The innermost ellipse represents the surface of a catheter electrode probe, and the next ellipse represents the endocardial surface. The region between these two surfaces (the blood-filled lumen of a cardiac chamber) contains no current sources, so Laplaces equation holds in this volume. The boundary cond itions are the endocardial potential, and the absent component of current normal to the probe surface. Again, a linear relationship can be derived-this time between measured probe potentials, and the (unknown) endocardi al potential. The latter linear operator is much easier to compute than the corresponding operator for the epicardial potential problem, since the conductivity in the relevant volume is uniform (simply being the conductivity of blood), and the geometry is easily measured (i.e., it does not require CT or MRl ). However, the attendant advantages are tempered by the fact the the method is invasive.

emphasize the statistical approach, from which other methods (e.g., those of Tikhonov) can be interpreted as special cases . Secondly, one must be prepared to deal optimally with a time series of such problems-i-i.e., stochastic processes.

4.3.1 SOLUTION NONUNIQUENESS AND INSTABILITY As noted in the last section, we are required to deal with equations of the form h = F g, where the components of vector h are our noninvasive or minimally invasive measurements at various spatially acces sible locations, F is a tran sfer matrix (supplied by the forward

130

F. Greensite

problem solution), and g is the image of transmembrane, epicardial, or endocardial potential. Assuming that F has an inverse, one would presume that we could supply the image as g = F-1h.

To understand the naivity of such an approach, we can consider that our situation is similar to being given a highly blurred image of a scene (our electrodes are located remote from the sources, and all the sources potentially contribute to what is measured at each electrode). We have some knowledge of the "point spread function", and can accordingly attempt some "image enhancement" with the objective of "deconvolving" the source image from the point spread function. But a blurring operator attenuates the information responsible for higher resolution. Thus, the signal power of the high resolution information relative to the power of the low resolution information is much smaller in the blurred image than in the unblurred source image. At the same time, noise (not described by the blurring operator) is also present in the resulting blurred image. That is, the noise (being added in addition to the blurring) will not be blurred away (much of it can be thought of as being added after the blurring operation, e.g., due to electrode noise or imprecisions in computation of the transfer matrix F). Thus, it will be typical that the noise power exceeds the signal power in the high resolution subspaces of the data. If one naively applies the inverse of the blurring operator, the noise will continue to dominate the high resolution subspaces-thus assuring continued absence of identifiable high resolution features. Furthermore, since the blurring operator severely attenuates high resolution, its inverse must involve a marked amplification relevant to the high resolution subspaces-which then also markedly amplifies the noise in these subspaces, so that noise will dominate the entire solution (i.e., contribute most of the power to the resulting image). This means that a nonsense solution estimate will be obtained, also characterized by its instability to small changes in the data. Because of the dominance of noise in the high resolution subspaces, one must simply forgo restoration of the high resolution subspaces and be content with restoring those subspaces where the blurred signal outweighs the noise-unless physiologically meaningful and valid extrinsic constraints can be imposed. In other medical imaging modalities, one does not have this difficulty. In MR!, for example, the spins (the hydrogen protons in the body water or fat) are induced to produce a signal (picked up by an antenna) whose frequency reflects their spatial location. Thus, the amplitude ofthe signal at a particular frequency reflects the number of spins at corresponding locations. In fact, following selective excitation of the spins in a single slice of the body, the spins at a particular location in the slice are cleverly given two frequencies in different bands (one from their spatially-varying NMR gyroscopic precessional frequency resulting from a magnetic field gradient applied along one spatial direction, and another one via manipulations of their relative phase via application of magnetic field gradient pulses in the direction orthogonal to the first). The relationship of frequency to magnetic field strength (which varies in space due to the applied gradients) means that an image of the tissue can be obtained by applying a two-dimensional Fourier Transform to the antenna data, so that the magnitude of the resulting function is an image of spin density in the tissue slice. Unlike a blurring matrix, a (discrete) Fourier Transform does not attenuate information in any source subspace more or less than in any other subspace. Thus, inverting the effects of the operator does not involve any differential noise amplification.

Heart Surface Electrocardiographic Inverse Solutions

131

To intelligently approach our dilemma, one ultimately needs to make the above notions more quantitative. For this, it is useful to introduce the singular value decomposition (SVD) of a matrix. A matrix represents a linear transformation, mapping a domain vector to a range vector. For any linear transformation, it can be shown that there exists a particular orthogonal coordinate system in the domain space, and a particular orthogonal coordinate system in the range space, such that a vector pointing along a coordinate axis of the domain space, is mapped to a vector pointing along a coordinate axis of the range space, and whose magnitude is amplified by a nonnegative scalar depending only upon which domain axis it was pointing along. Since any vector in either space can be written as a linear combination of unit vectors in the above coordinate axis systems, the preceding wordy statement is equivalent to the assertion that any matrix F can be written as the SVD, F = USV I ,

(4.19)

where the columns of matrices V and U are the requisite orthogonal bases of the domain and range coordinate systems alluded to above, and S = diagts}, ... , sn) is a diagonal matrix whose diagonal entries (the singular values) are the amplification constants referred to above (they are arranged in order from largest to smallest). U and V are each orthogonal matrices, and S is referred to as the singular value matrix. Note that each singular value s, is associated with corresponding one-dimensional domain and range subspaces (the i-th columns of V and U). By convention, we will take U to be an (m x m) matrix, V to be an (n x n) matrix, so that S is a (m x n) matrix. We are now in a position to understand the severe mathematically determined difficulty of our problem. In any noninvasive imaging technique for cardiac electrophysiology, F is always severely ill-conditioned-because the field ¢ diminishes with distance from the source, and the field at a point has contributions from all sources (there is "blurring"). As a result, the ratio of its largest and smallest positive singular values is "large" (the value of the smallest positive singular value is "small" compared to the value of the largest singular value). That is, F is "ill-conditioned". In particular, the noise in the data in many of the singular subspaces (columns of U) dominates the signal in those subspaces. However, F- 1 = V S-I U I (assuming the inverse exists). A solution of the form F- 1h thereby entails application of l/si to the data component of h in the Uu subspace. If this is one of the many subspaces for which 1[s, is very large and in which the noise dominates the signal, we can appreciate that the noise is this subspace will be markedly amplified in the solution estimate (this will also imply that the solution estimate will be very unstable to small noise perturbations). Intuitively, we would thereby expect that it will be necessary to somehow attenuate the solution components associated with many (or most) of the subspacesmeaning that there will be a severe limit on the number of degrees of freedom in a meaningful estimate for g in Eq. (4.18) (essentially given by the number of singular values large enough not to attenuate signal components below the noise amplitude in the subspace defined by the corresponding column of U). Thus, much of the structure of an estimate for g must come in the form of a priori constraints-s-either by default (imposed as artifacts of the regularization procedure), or by design (constraints that truly reflect the class ofphysiologically meaningful

F. Greensite

132

solutions). Without such constraints, the solution estimate would be nonunique-since the addition of any vector in the supressed high resolution subspace to any solution estimate gives a new estimate that is also consistent with the accessed data. The field of Inverse Problems typically deals with situations where one is given data reflecting the effect of some operator on a "source" we would like to estimate, but where the inversion procedure (undoing the effect of the operator) is inherently unstable (e.g., highly noise amplifying), and (in practical terms) solution estimates are not unique. Such problems are loosely referred to as "ill-posed" (the latter term has a quite precise meaning in general Hilbert space settings, that we will not go into further). 4.3.2 LINEAR ESTIMATION AND REGULARIZATION

Taking noise vector v into account, Eq. (4.18) becomes h=Fg+v,

(4.20)

where it is required to estimate g given hand F. It might first occur to us that a useful estimate for g would be the one which maximizes p(hlg)-i.e., the choice that maximizes the conditional probability (density) that the measured h would occur given a particular candidate for g. This is known as the "maximum likelihood estimate". Assuming F has an inverse F- 1 , and the noise has zero mean, this is given by gml = F- 1h-because the likeliest value for the noise (its mean) is the zero vector (under conditions of zero mean Gaussian noise, the maximum likelihood solution coincides with the least-squares solution, i.e., if F- 1 does not exist, F- 1 is replaced by the pseudoin-verse). If F is ill-conditioned, we have seen that this is in general a very poor estimate for g, and is very unstable to noise variations. But one could alternatively take the estimate for g to be the choice which maximizes p(glh)-i.e., the choice that maximizes the chance that a particular g will be present given the observed noisy data h. This is referred to as the "maximum a posteriori" estimate, gmap. The relationship between the two conditional probability densities above is given by a version of Bayes Theorem: p(glh) = [p(h Ig)p(g)]/ p(h). gmap has the great advantage of being stable to noise variations-but the disadvantage that its calculation requires one to first supply nontrivial information concerning statistical properties of g (in fact, the entire "posterior" probability density p(g)).1f such is available and reliable, the methodology is referred to as "Bayesian". If one supplies the statistical properties as "drawn out of the air", or perhaps estimated from the given data h itself ("noninformative"), the methodology is referred to as "empirically" Bayesian. The general approach is also referred to as "Satatistical Regularization" (evidently, in discrete settings, the maximum likelihood estimate is equivalent to the Bayesian approach in a minimum information setting where every realization of g is considered equally likely to occur). All of this suggests that, once noise is introduced as in Eq. (4.20), it is useful to carefully consider statistical notions. We will endeavor now to make these a little more precise. Specifically, each component of noise vector v is a "realization" (outcome) of a "random variable", the composite of which defines random vector v. For our purposes, a random variable is an entity that associates a probability density value with every real number. From this, we can compute the probability that some realization of the random

Heart Surface Electrocardiographic Inverse Solutions

133

variable (outcome of a measurement) will yield a value falling in some given interval. Accordingly, the expectation tTl of some expression involving a random variable is the integral of the expression over all possible values of the random variable weighted by the probability density associated with each value (a zero mean random variable a is such that Era] = 0). Furthermore, a "Gaussian" random variable a has a Gaussian probability density (the familiar bell-shaped curve), and is fully characterized by its particular expectation £[a] and variance £[(a - £[a])2]. Similarly, a zero mean Gaussian random vector W is a column vector of zero mean jointly Gaussian random variables Wi, i.e., W = (WI, ... , wnY. A zero mean random vector is further characterized by its autocovariance matrix £[ww t ], which describes the dependence between all different pairs of components of W (note that the product of jointly Gaussian random variables is Gaussian). Similarly, the cross-covariance matrix of zero mean random vectors v, w is given by £[Dw t ] , and describes the mutual dependence of ii and W. Just as v is a realization of a random vector v, so too can g be considered to be an (unknown) realization of random vector g. For notational simplicity, in this subsection we will suppress the superscript" - ", and denote a random variable and its realization by the same symbol. However, we will resume the notational distinction in the next subsection. In approaching Eq. (4.20), a good objective is to find gopt such that £[lIg - gopt 11 2 ] is minimum (this being the "minimum-mean-square-error" estimate). If g and v are realizations of zero mean Gaussian random vectors, gopt is obtained via the Wiener filter. Under these conditions, the maximum a posteriori estimate gmap is equivalent to gopt. This linear estimation procedure develops as follows. A linear estimate of g is given by application of an "estimation matrix" Mest to data h, i.e., gest = Mesth. Ideally, we desire the solution estimate

(4.21) such that £[llg - Mopth 11 2 ] is minimum. Thus, it is sufficient to calculate M opt' The way to proceed follows from the "Orthogonality Principle", which asserts that gopt minimizes the mean-square-error when

(4.22) i.e., when the cross-covariance matrix of the "error of the estimate" (g - gest) and the "data vector" h is the zero matrix (so that the error and the data have no dependence). Intuitively, the Orthogonality Principle assures that every bit of useful information is extracted from the data h in making the solution estimate gopt. Substitution of Eq. (4.20) and Eq. (4.21) into Eq. (4.22) immediately gives

(4.23) Thus, assuming that g and v are independent (i.e., £[gv t] as

where Cg

== £[ggf]

and C;

== £[vv t]

= 0), Eq. (4.23) can be written

are the autocovariance matrices of signal g and

F. Greensite

134

noise v. Hence, (4.24) Thus, the optimal solution estimate for Eq. (4.20) is provided by Eq. (4.24) and Eq. (4.21)assuming we know the autocovariance matrices of signal and noise. It is interesting to express this estimation matrix M op I as a modification of F- 1 , assuming the latter exists. We have from Eq. (4.24) that

where Ci; = (FCgF I + C v ) is the autocovariance matrix of h (as is seen via Eq. (4.20)). Thus, M op I involves an initial preprocessing of the data h via (C h - CV)Ch1 (the classical Wiener filter (Papoulis, 1984)) followed by application of F- 1 (note that providing Ci; nominally requires knowledge of Cs- although one can attempt to estimate Ch using the given measurement h itself-essentially the problem of spectral estimation (Papoulis, 1984)). If Cv = I (i.e., if the noise is white), Eq. (4.24) becomes

a;

M op l =

(

2 -1)-1 rrs «;c; F, I

I

(4.26)

since (FIF+a;C;-I)CgF I = FI(FCgFI+a;l). If Cg=aiI, using the SVD, F= U SV I , we can write Eq. (4.26) as (4.27) An alternative applicable formulation for treating Eq. (4.20) is provided by the maximum likelihood method in concert with a deterministic constraint (such as that the signal power is equal to some a priori vaule E). Maximum likelihood then corresponds to minimizing IIFg - hll 2 subject to IIgl1 2 = E (assuming Gaussian noise). This also leads to a linear estimation matrix of similar form to the above. Ultimately, the distinction between the Bayesian and constrained maximum likelihood methods are that the latter treat only the noisy measurements in a statistical fashion, while the former also treats the underlying signal statistically. Thus, the Bayesian methods potentially allow (or require) introduction of a larger class of a priori information. A third alternative, Tikhonov regularization (Tikhonov, 1977), could be viewed as either a hybrid or an empirical Bayesian method. The linear estimation matrix supplied by it can typically be interpreted as resulting from the assumption of white noise and selection of a term proportional to Cg /a 2 in Eq. (4.26), where the proportionality scalar (the regularization parameter) is selected using either a deterministic or data-dependent constraint. Comparing the right-hand-side of Eq. (4.27) to F- 1 = V S-I U I (assuming the inverse exists), we can appreciate the violence that estimation theory does to the notion of a high resolution reconstruction of g. Reversal of the effects of F would require application of 1/s i to the data component in the U; subspace. Instead, for the regularized estimate, the factor sd(s; + y) is applied. As s, becomes small, this factor bears no resemblance to 1/ Si-SO there is no attempt at faithful reconstruction of components of the associated source subspaces. Another way of looking at the situation, is that a square-integrable

Heart Surface Electrocardiographic Inverse Solutions

135

(i.e., well-behaved) function (or image) must be such that its higher order Fourier coefficients tend to zero (we imagine the Fourier coefficients to be with respect to the SVD domain coordinate system of F, given by the columns of V in the discretized approximation). In the presence of white noise, whose Fourier coefficients therefore do not tend to zero, it is clear that higher order Fourier coefficients of data h are hopelessly noise-corrupted. The Wiener filter, and Tikhonov regularization, achieve stable results by removing any attempt at meaningful reconstruction of the high resolution components. There is only one way out of the dilemma of resolution loss. If physiological constraints exist which effectively reduce the dimension of the solution space to be commensurate with the number of useful data Fourier coefficients, one can anticipate that it will be possible to preserve spatial resolution. For example, for the inverse electroencephalography problem (where one wishes to image the brain sources of the scalp electrical potentials), it might be true that only a single focus is responsible for inciting an epileptic seizure, and that this focus can be modeled as a single current source dipole located at some unknown location in the brain. In that case, one is searching for an entity with six degrees of freedom (reflecting its location, orientation, and magnitude). High spatial resolution could conceivably be possible assuming there are six or more data Fourier coefficients (with respect to the transfer matrix SVD-derived coordinate system) that are not dominated by noise. At first blush, such an obvious constraint does not appear to be physiological in the heart, since the heart is not faithfully modeled as a single dipole. However, a deeper look at the geometry reveals that such constraints do in fact apply (in principle) for the "critical points" of ventricular activation-from which an activation map can be fashioned (see Section 4.6).

4.3.3 STOCHASTIC PROCESSES AND TIME SERIES OF INVERSE PROBLEMS The data available in our problem are distributed in time as well as space. Thus, Eq. (4.20) would be more appropriately written as hi

= Fig, + Vi,

(4.28)

i = 1,2, ... , n, where i indexes the time instants at which measurements are made (note that we leave open the possibility that the transfer matrix is time-varying, thus we write it as F;). Underlying Eq. (4.28) are the time series of random vectors, hi, gi, and vi-i.e., stochastic processes. The important additional feature of a stochastic process is that the i -th random vector may have correlations with the j-th random vector for j =1= i. However, a "state variable model" which embodies known correlations between gi and gj, for i =1= j, is not explicitly available in our problems. Given the lack of such explicit accurate constraints, it is usual to adopt a "minimum information" perspective. Though such an approach seems reasonable, and suggests that the equations ofEq. (4.28) might best be treated independently of each other (by simply applying the methods of the prior section to each one), the reality is more subtle. For convenience, let us define matrices H, G, N such that Hi = hi, G:i = gi, Ni = Vi-SO that Eq. (4.28) becomes Hi = Fi G:i

+ Ni'

(4.29)

F. Greensite

136

i = 1, ... , n. We then have the underlying random matrices as if, G,

N such

that their i-th columns are hi, gi, Vi, respectively. The usual assumption is that the entries of G are independent and identically distributed random variables (and similarly for the entries of N). This is equivalent to the statement that all row autocovariance matrices are proportional to the identity matrix (with the same proportionality constant), and all row cross-covariance matrices are the zero matrix (this is also equivalent to the statement that all column autocovariance matrices are proportional to the identity matrix, with the same proportionality constant, and all column cross-covariance matrices are the zero matrix). This minimum information assumption would imply that each member of equation sequence Eq. (4.28) can be treated independently of every other member of the sequence. However, if we leave open the possiblity that there are correlations between the different gi, the members of the equation sequence can no longer be considered necessarily independent-and an optimal processing of the data is subject to specification (or identification) of appropriate choices of the cross-covariance matrices of the columns of G. Thus, suppose we continue to assume that the row cross-covariance matrices of G are the zero matrix and the row autocovariance matrices of G are identical, but that the latter are not necessarily proportional to the identity matrix (thus, we will be rejecting the minimum information approach). This means that the column autocovariance matrices are proportional to the identity matrix-but we still have not specified the column cross-covariance matrices. Estimates for these will be derived from the data, i.e., empirically. This is actually not a radical thing to do, since even in the minimum information approach one typically derives the signal power (or signal-to-noise ratio) from the given data (thus, the minimum information approach is by no means "pure" in this respect). In fact, under the present conditions, there is a favored nontrivial choice of each cross-covariance matrix £[Gi 0 G:j]. For the purposes of linear estimation, Eq. (4.29) can be equivalently written in block matrix form as

(~:l) H: n

=

[~' .. ~....~ .. : ] (~l) + (~:1 ). 0

0

...

F;

G';

(4.30)

Nn

The Wiener filter (detailed in the last subsection) supplies an optimal estimate of the entries of G as given by

(4.31)

where diag(Fi ) is the block diagonal matrix on the right-hand-side of Eq. (4.30), and

c., = (£[G: i 0 C N = (£[N: i 0

G:j])i,j

s.»;

(4.32)

(4.33)

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Heart Surface Electrocardiographic Inverse Solutions

i.e., Cc is the block matrix whose (i, j) block entry is the cross-covariance matrix of ifi with etc. Thus, C c is the (large) autocovariance matrix ofthe random vector consisting of the entries of G. Equation (4.31) is simply the composite of Eq. (4.21) and Eq. (4.24) applied to Eq. (4.30). In the "Standard Method" (the minimum information approach), one takes E[G: i 0 G: j ] to be the zero matrix when i #- j. In the "New Method", for i #- j, one takes

c.;

(4.34) assuming the trace of F! F, is not zero. It can be shown that the mean-square-error in the resulting estimate of signal autocovariance matrix C c is smaller than the estimate used in the Standard Method (Greensite, 2002). For the case of white noise, and assuming the F, are identical (i.e., F, = F, for all i), it can also be shown that the New Method reduces to the following procedure: Instead of individually treating the equations

Hi = FG:i + Ni,

(4.35)

i = 1, ... , n, we instead individually treat the equations

HX: i = FGX: i

+ NX: i ,

(4.36)

i = 1, ... , n, where the columns of n x n matrix X are the eigenvectors of HI H. Denoting the solution to the i-th equation ofEq. (4.36) as (GX):i (the i-th column of a matrix (GX)), we take the solution estimate for G to be (4.37) The method generalizes to the case where there are nontrivial spatial correlations, and also to the case where a priori constraints are available regarding time correlations. However, if nonwhite characteristics of the noise are known, the route of Eq. (4.37) is unavailable, and one is left with the computationally complex method resulting directly from Eqs. (4.30)-(4.34) (Greensite, 2002). Underlying the New Method is the recognition of a fundamental asymmetry regarding H = F G + N. That is, the signal G undergoes a spatial transformation, but does not undergo a time transformation. For an equation of the form h = F g + v, where we consider h, g, v to be spatial vectors, it is quite reasonable in a filtering context to impose a signal (g) autocovariance matrix proportional to the identity matrix. This would imply an autocovariance matrix for the noiseless portion of h as given by F I F -implying nontrivial filtering of noisy h (see the second equality in (4.25)). However, for an equation of the form h = h o + v, where the h and v are considered time series, it is quite unreasonable to set the signal (h o) autocovariance matrix proportional to the identity, since the resulting (Wiener) filter is no filter at all (assuming white noise). Thus, since F is a spatial transformation, while G is spatiotemporal, one cannot simply impose the minimum information condition that the entries of G are independent and identically distributed-assuming that one wishes to

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effectively filter in the time domain. In the setting of "minimum constraints" as opposed to "minimum information", the New Method is a means of performing spatiotemporal filtering in a manner dictated by the broken-symmetry of the problem , and the desire to minimize mean-square-error in the utilized signal autocovariance matrix .

4.4 EPICARDIAL POTENTIAL IMAGING The source formulations for the inverse problem of electrocardiography have included those of a single moving dipole (Gabor and Nelson , 1954), two moving dipoles (Gulrajani et al., 1984), dipole arrays (Lynn et al., 1967; Barber and Fischman, 1961; Bellman et al., 1964; He and Wu, 2001), multipole expansion coefficients (Geselowitz, 1967), and a heart excitation model (Li and He, 2001). But in a very influential letter to the editor, Zablow (1966) asserted the need to reconstruct an actual anatomically-based entity that was already being accessed invasively-so that artifacts of the source model might be minimized, and the result could be thought of as representing some sort of verifiable physiological truth. In essence, he noted that a linear relationship existed between the epicardial potentials and measurable potentials at the body surface, and suggested the former as the source formulation to be reconstructed. This was particularly attractive, since physiologists were already engaged in measuring the epicardial potentials invasively, and these were deemed useful. Over the next several decades many investigators pursued the objective of epicardial potential imaging. The many proposed refinements of technique can be divided into those pertaining to • • • • • • • • •

Statistical regularization, Tikhonov regularization, Truncated SVD regularization, Constrained least squares regularization, Nonlinear regularization methods , Augmented source formulation , Different methods for selecting regularization parameters, Preprocessing the data, Introduction of spatiotemporal constraints.

Before embarking on a discussion of these refinements, we observe that there is no consensus regarding which method s are the most worthy of employment-and we do not attempt such value judgements here . A comprehensive approach to this question is itself a sizable objective that has not yet been achieved. Ultimately, the difficulty is in the experimental setup required-i.e., the need for (ideally ) simultaneous collection of body surface and epicardial data , with coincident anatomical imaging and electrode registration, in a series of animals and human subjects with a variety of pathological conditions (Nash et al., 2000).

4.4.1 STATISTICAL REGULARIZATION In what was apparently the first serious treatment of the epicardial potential source formulation , Martin and Pilkington (1972) reported on the dismal prospects of "unconstrained"

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inverse epicardial potenti al imaging , identifying implications of the problem ill-po sedness discus sed in Section 4.3. They subsequently applied the Weiner filter, and reported more encouraging results in followup simul ations (Martin et aI., 1975). This approach requires estimates of both the signal and noise autocovariance matrices. While the noise might be considered white (ignoring inaccuracies in the forward problem construct F), a choice for the signal autocovariance matrix is less obvious. They proposed two ways of choosing one. The first was based on estimating the spatial autocovariance from time ensembles of epicardial potential maps supplied from a representative set of activation sequences. The second was a Monte Carlo method, whereby each epicardial location was given some a priori probability of being activated at any given time, and epicardial maps were then generated by random numbers assigned to each location-thus leading to a computation for the signal autocovariance matrix . Following the innovat ions of Barr et al. (1977) on the forward problem, Barr and Spach (1978) reported on inverse calculation of epicardial potential in twelve dogs with chronically implanted epicardial electrodes. In applying the Wiener filter, they simply opted to take the signal autocovariance matrix as proportional to the identity (i.e., as random variables , the epicardial potential s at all locations on the epicardium were presumed to be independent and identically distributed). They concluded that some features of the epicardial potential distribution through time can be imaged, particularly in dogs for which detailed geometry measurements.were available (postmortem). Recently, van Oosterom (1999 ) has re-examined the statistical regularization approach, concluding that impressive improvements in accuracy (compared with other regularization methods ) are possible if a nontrivial accurate signal covariance matrix is available. He suggested that the signal autocovariance matrix could be based on prior estimation of the activation sequence via other techniques (e.g., as in Section 4.6).

4.4.2 TIKHONOV REGULARIZATION AND ITS MODIFICATIONS Despite the presence of noise v in Eq. (4.20), we are still tempted to view our problem as one of applying a kind of "inverse" of F . Indeed, given that the magnitude of v is "small", we are even tempted to pretend that we are dealing with h = F g . Our prior discussion has surely revealed that one cannot expect to apply the actual inverse of F to h becau se of instability problems, but it is also useful to specifically address the (typical) setting where F doesn 't even have an inverse (which will always be the case if F is not square). Firstly, it could be that h is not even in the range of F. In that case , we might (naively) choose the solution estimate g est that minimizes (4.38) over all point s g in m-space (the expres sion Eq. (4.38) is known as the "residual" or "discrepancy"). But a second problem could be that the residual may not have a unique minimizer (as occurs when the dimension of h is smaller than the dimension of g ). One could then ask for the estimate gest that is of minimum norm among all the minimizers of the residual. From convexity arguments, it can be shown that the minimizer of the residual Eq. (4.38) of smallest norm is unique . In fact, it can be shown that the minimum-norm

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least-squares solution for gin h = Fg is given by

where Ft is the "pseudoinverse" of F. For the SVD of F as in Eq. (4.19), the pseudoinverse is given by

where st is the diagonal matrix whose i -th diagonal entry 1/s, if s, -=I- 0, and zero otherwise. Intuitively, it is easy to see why this works: The null space of Ft is the subspace orthogonal to the range of F. Thus, Ft h does not burden the estimate with any component that doesn't contribute to fitting the data. Otherwise, Ft simply undoes the attenuation s, that components of g experience when F is applied. However, the above is simply a fix for the situation where F does not have an inverse. From Section 4.3, we know that, even if F has an inverse, we are faced with solution estimate instability if F is ill-conditioned-because of the subspaces corresponding to small positive singular values. This consideration obviously will still hold for the pseudoinversebased solution. We have already encountered the Wiener filter regularization approach, which requires a priori estimates of at least the form of the signal and noise autocovariance matrices. Colli-Franzone et at. (1985) introduced the Tikhonov regularization method to the epicardial potential imaging problem, ostensibly avoiding the problem of providing estimates for the signal autocovariance and noise autocovariance matrices. This approach skirts the usual notions of stochastic processes, and instead begins with the desire to minimize Eq. (4.38). But instead of simply searching for a solution estimate gest which minimizes the residual (which would lead to an estimate that is exceedingly noise sensitive and unstable), in the Tikhonov approach one searches for the solution estimate that minimizes (4.39) where R is some matrix. Thus, one seeks an estimate for which the residual II Fgest - h f is "small", while at the same time some other property of the estimate, measured by II Rg est [[2, is also small. For example, R might be the identity, in which case one is looking for an estimate with small residual as well as a small norm (unstable nonphysiological solutions will tend to have large norms). Alternatively, R could be such that II Rg est 11 2 reflects the first or second spatial derivative of the solution estimate-so that the regularized estimate would have to be relatively "smooth". These approaches require selection of a "regularization parameter" y, which regulates how strong an influence the co-minimized second property has in determining the solution estimate. In the Tikhonov approach, the regularized estimate is given by (4.40)

obtained by setting to zero the derivative of Eq. (4.39) with respect to g (a gradient), assembling the simultaneous equations into a matrix equation, and solving for g (note

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that Eq. (4.39) is a function of the variable g-a point in m-space; thus, its gradient is a vector in m-space). In the sense of Section 4.3, Eq. (4.40) evidently describes a linear estimation method-the estimation matrix being the expression in brackets on the righthand-side of Eq. (4.40). Many alternative regularization operators R can be used. With "zero-order Tikhonov", the estimation matrix results from the choice R = I-so the technique corresponds to statistical regularization under the assumption that the signal and noise covariance matrices are both proportional to the identity matrix, with the regularization parameter presumptively being the inverse of the square of the signal-to-noise ratio (i.e., compare Eq. (4.40) to the first equality in Eq. (4.27)). First-order and second-order Tikhonov regularization correspond to a choice of R derived from the gradient and Laplacian operators. Thinking of these in the context of statistical regularization (compare Eq. (4.40) to Eq. (4.26)), these symmetric higher-order Tikhonov reguarization operators correspond to a signal autocovariance matrix that is a smoothed version of the sharp "ridge" represented by the identity matrix (with the assumption of white noise). That is, there is now a nonzero covariance between spatially proximate locations-instead of these being taken to be independent (as with a signal autocovariance matrix proportional to I). Although Colli-Franzone et at. (1985) suggested that first-order Tikhonov regularization was more accurate in in vitro experiments, Messenger-Rapport and Rudy (1988) found no significant differences in the results obtained with zero-order, first-order, or second-order Tikhonov regularization. When expressed in terms of an SVD, F = U sv', the zero-order Tikhonov solution estimate is given by (4.41) which employs a linear estimation matrix comparable to that in Eq. (4.27). A so-called "regional regularization" scheme was suggested by Oster and Rudy (1997), whereby the solution is given as

where D is a diagonal matrix whose diagonal elements take on a few different values-in effect, the diagonal values of D represent multiple regularization parameters. Again, this can be interpretted as an attempt to supply the signal autocovariance matrix. The "spatial regularization" method (Velipasaoglu et al., 2000) selects R in a manner inspired by the fact that the noisy data fails to satisfy the Discrete Picard Condition (Hansen, 1992; Throne and Olsen, 2000). The latter condition asserts that a stable solution requires that the squares of the Fourier coefficients of the data with respect to the eigenvectors of F P, should on average decay faster than the eigenvalues of F Ft. A means of modifying the data to conform to this Condition is the basis of this approach. 4.4.3 TRUNCATION SCHEMES As noted earlier, the minimum-norm least-squares solution for gin h = Fg is given by gest = Ft h = Y sto' h, where st is the diagonal matrix whose i-th diagonal entry I/si if s, =J 0, and zero otherwise. With this in mind, one could consider a regularized solution

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estimate given by (4.42) where st is the diagonal matrix whose i -th diagonal entry is 1/ s, if s, > f and 0 otherwise. This regularization method is known as Truncated SVD, or TSVD regularization (Hansen, 1992). The value f functions as a regularization parameter. TSVD performance is usually very similar to that of zero-order Tikhonov regularization. The truncation idea is also incorporated in the "Generalized Eigensystems" approach of Throne and Olson (Throne and Olsen, 1994), which is relevant to a finite element discretization of the body (rather than a boundary element discretization). Instead of truncating a solution expanded in the singular vectors of the transfer matrix F (as in Eq. (4.42)), they consider a set of generalized eigenvectors defined over the entire finite element mesh, having the properties that each generalized eigenvector satisfies the boundary conditions on the forward problem, as well as Laplace's equation within the volume, and the "subvectors" consi sting of the components on the epicardial surface are orthogonal. One then constructs a linear combination of the generalized eigenvectors such that the body surface potential data h is fitted by the components that correspond to locations on the body surface . The components corresponding to the epicardial potential locations then take on values determined by this linear combination-which would be the presumed inverse solution desired . However, this will nominally lead to an unstable noise-dominated solution . Therefore, one truncates the linear combination, using only the generalized eigenvectors associated with the largest generalized eigenvalues-thus achieving a stable solution estimate . Instead of expanding (and truncating) the solution series in terms of the eigenvectors of F F' as with a TSVD (a sequence which most efficiently represents the effects of F for a given (truncated) number of terms), one is truncating a series derived from a set of field vectors that most efficiently pack the power of the field over the entire body volume in a given (truncated) set of components. Truncation is also employed in the "local regularization" scheme of Johnson and MacLeod (Johnson, 2001). In this approach, it is recognized that F is expressed in terms of the "inverses" of three different submatrices, when a finite element discretization of the forward problem is employed. Since these matrices have much different condition number, the implication is that they should each be receive different degrees of regularization (e.g., individualized SVD truncation).

4.4.4 SPECIFIC CONSTRAINTS IN REGULARIZATION The Tikhonov type formulation Eq. (4.39) also suggests a way in which other types of constraints could be formulated . For example, suppose one has knowledge that the solution shares some features with a preliminary estimate grough ' One could then suggest the minimization of (4.43) The explicit expression for g esl is then obtained by setting the gradient of the above expression to 0, and solving for g. This is known as the Twomey method (Oster and Rudy, 1992).

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Iakovidis and Gulrajani (1992) introduced a method whereby a deliberately overregularized estimate (to be used to estimate the location of the epicardial zero-potential line, but otherwise having too few interesting features ) was used to constrain the solution for what would otherwise be an under-regularized estimate (i.e., the second regularization parameter would have been too small if the constraint had not been present, in the sense that the estimate obtained would have been unstable and "noisy").

4.4.5 NONLINEAR REGULARIZATION METHODOLOGY Overwhelmingly, linear estimation methods have been applied to obtain regularized (i.e.•noise-stable) estimates of linear formulations of the inverse electrocardiography problems. However, nonlinear (e.g., information theoretic) methods have also been applied (a particularly recent example is provided in (He et al., 2000)). We will not describe such approaches here.

4.4.6 AN AUGMENTED SOURCE FORMULATION If one treats Eq. (4.5) via a Green's function approach along the lines of what was done with Eq. (4.6) (i.e., performing two integrations-by-parts analogous to Eq. (4.9) and Eq. (4.11)), one obtains an expression relating body surface potentials to the composite of epicardial potentials and the normal component of epicardial current density (as in (Greensite, 2001, p. 151-152),

Noting that such a formulation occurs as an intermediate step in the development of the forward problem method expounded by Barr et al. (1977) , Horacek and Clements (1997) investigated solutions obtained where both the epicardial potential and epicardial normal current density are inversely computed. They suggested that this problem might be slightly better posed than the traditional epicardial potential formulation, and they also investigated refinements in the regularization technique.

4.4.7 DIFFERENT METHODS FOR REGULARIZATION PARAMETER SELECTION The Tikhonov estimate Eq. (4.40) requires selection of a value for the regularization parameter. Some guidance in this regard is provided by the "Discrepancy Principle" (Hansen, 1992). Here the reasonable assumption is made that the discrepancy (or residual) IIF g - h 11 2 is not zero-because of the noise present. Rather, the discrepancy (for the true solution g) is most likely to be the noise power. Thus, it makes sense to look for a solution estimate which produces this discrepancy. So, consider the constrained minimization problem: "minimize IIgll 2 such that I Fg - hll 2 = f" . Again , IIgl1 2 and II Fg - hll 2 are each functions of g (which varies over points in m-space). We know from Calculus that minimization of the first function subject to the constraint on the second function implies (under rather general conditions) that the gradients of the two functions at the constrained minimum gm in lie on the same ray-i.e., the gradients are proportional at gmi n' Another

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way of saying this is that there exists a scalar y such that the gradient of Eq. (4.39) is zero at gmin (in this case R = 1). As we know, the requisite gmin is given by the right-hand-side of Eq. (4.40), where the regularization parameter y was to be determined. But now, the regularization parameter is simply given as that which produces a Tikhonov solution estimate gmin satisfying the discrepancy expression [IFgmin - h 11 2 = E, where E is the noise power. However, the error in the data (noise power E) is not known (being a composite of electrode noise and modeling errors in F). There are actually several other methods for regularization parameter selection. In the so-called "Lcurve method" (Hansen, 1992), one computes a log-log plot of the first term in Eq. (4.39) versus the second term in Eq. (4.39) (the residual versus the solution estimate seminorm). The solution estimate is chosen as the one corresponding to the "comer" of the above L-shaped graph (a balance between small seminorm and small discrepancy). It should be noted, however, that a comer on the L-curve does not always exist. The Composite Residual and Smoothing Operator (CRESO) method (Colli et al., 1985) chooses the smallest positive value of the regularization parameter for which the second derivative of the first term in Eq. (4.39) with respect to the regularization parameter equals the second derivative of the second term in Eq. (4.39) with respect to the regularization parameter. The cross-validation method (Whaba, 1977) is another important regularization parameter selection method, though it has not been prominently applied in the context of the inverse electrocardiography problem.

4.4.8 THE BODY SURFACE LAPLACIAN APPROACH As we have previously noted (see equation (4.25)), statistical regularization can be viewed as the composite of a particular "preprocessing step" followed by application of the standard inverse (if such exists). This suggests the possible application of other preprocessing steps. He and Wu (1997) proposed preprocessing the data to be in the form of body surface Laplacian measurements, and solving the inverse problem using this processed data as input. The resulting transfer matrix is of a different nature (evidently, it results from left multiplying the usual transfer matrix by the same matrix that is used to preprocess the body surface data), in that the surface Laplacian of potential at a body surface location is significantly influenced by many fewer source locations than potential itself (i.e., there is less "blurring"). As a trade-off, the effect of the source on the "data" falls of more rapidly (with the fourth power of the distance from the source, rather than the second power of the distance, as with body surface potential), and there is a theoretical noise amplification in numerical differentiation of the body surface potential. Ignoring the curvature of the body surface, the body surface Laplacian of body surface potential h with respect to an (x, y) body surface coordinate system is given by

Ultimately, we can write this as L[h], where L is the Laplacian-a linear operator. After the problem is discretized for numerical treatment, L[·] is simply a matrix. Thus, h = Fg + v

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becomes Lh = (L F) g

+ L v,

Our data is now Lh, and we now need to "invert" (L F) to estimate epicardial potential g. The regularization tools remain the same as before. L is a differential operator-and thus can be thought of as akin to a high pass filter. With respect to a Fourier expansion of h , application of L amplifies the high frequency terms. In particular, the high frequency components of the noise are greatly amplified (in the nondiscretized setting, the noise is "unboundedly" amplified). However, the latter is in principle taken care of by the fact that LF is more "singular" than F -meaning that it's "inverse" will be smoother (i.e., application of the inverse is more stable, and tends to smooth noise). If one has an expectation of somehow reducing the L v contribution to data Lh prior to application of the "inverse", one might expect greater stability and fidelity in the estimate of g than that obtainable with the direct treatment of h = Fg + v. Such an expectation could be reasonable with use of Laplacian electrodes (He and Cohen, 1992), though it has not been established that these can be accurately designed in practice. In the absence of such, investigations have proceeded with direct application of L to measured data h . The approach seems to have potential in identifying and spatially distinguishing cardiac sources close to the body surface electrodes (Johnston , 1997; He and Wu, 1997). 4.4.9 SPATIOTEMPORAL REGULARIZATION

Ultimately, one is faced with a time series of problems (4.44) i = 1, 2, ..., where the subscript i now refers to the source and data at the i-th time point in the cardiac cycle. Oster and Rudy (1992) suggested using preliminary (e.g., zero-order Tikhonov) estimates at time points i - I and/or i + 1 to constrain the regularization at the i-th time step. This was done using a Twomey regularization formalism via equat ion Eq. (4.43). On the other hand, Eq. (4.44) evidently describes a stochastic process, which begs the question of applying Kalman filter theory. This requires that a stochastic model be applied, defining the presumed interdependence of the epicardial potentials between different times (Joly et al., 1993)-is itself a not entirely trivial problem. Temporal and spatial constraints can also be joined by the the method of Brooks et al. which employs two or more regularization parameters in a traditional constrained minimization format (Brooks et al., 1999). Thus , Eq. (4.44) is written as

(4.45)

One now writes a functional to be minimized, consisting of the residual (for this augmented

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problem), a spatial regularizing operator (e.g., expressing the sum of the norms of the solution estimates at each time point), and a temporal regularizing operator (e.g., the magnitude of the discretized "time derivative" of the solution estimates over all the time points)where the latter two operators are given their own regularization parameters. The solution estimate is ultimately expressed as

where diag(F) is the block matrix on the right-hand-side ofEq. (4.45), B is the discretized version of a temporal differential operator, and Yl, Yz are the two regularization parameters. The "admissible solution" approach of Ahmed et al. (1998), posits that any solution satisfying a sufficiently robust composite of constraints is deemed satisfactory, and such constraints can include those related to time. The solution algorithm requires the constraints to be convex, e.g., the "ball" of vectors g satisfying IIg liZ < C is an example of a convex set (any line joining any two members of the set consists only of members lying in the set). The need for regularization parameters is replaced by the need for bounds defining the required convexity. Finally, the approach of Greensite (1998; 2002) (described in Section 4.3.3) effectively replaces the original sequence of (nonindependent) Eq. (4.35) (or Eq. (4.44)) with a smaller number of mutually independent equations Eq. (4.36)-without the imposition of any extrinsic temporal constraints (or temporal regularization parameter). The method derives from the recognition that there is something intrinsically wrong with the assumption that the entries of G are realizations of independent and identically distributed random variables. Indeed, this symmetry condition is broken once one poses the problem described by Eq. (4.44). Given the assumption that the rows of G are independent and have identical autocovariance matrices, the solution mechanism uses a more accurate signal autocovariance matrix estimate than the other methods (in a mean-square-error sense).

4.4.10 RECENT IN VITRO AND IN VIVO WORK There has been a significant amount of in vitro work done with the Utah torso tank. The experimental setup employs a heart suspended in a torso shaped electrolytic tank, perfused by an anesthetized dog external to the tank (Oster et al., 1997). Electrodes are present on the outer margin of the tank, and also in proximity to the epicardium. Such work has addressed inverse reconstruction of epicardial potentials and activation in the setting of sinus rhythm, pacing, and arrhythmias (Burnes et al., 2001), and has also been used to assess the impact of torso inhomogeneities (Ramanathan and Rudy, 2001). Perhaps the most impressive demonstration of the potential promise of the epicardial potential imaging formulation is in the quasi-in vivo work where densely sampled epicardial potential data was accessed from infarcting dogs, and used in simulations with numerical human torsos to test the fidelity of the zero-order Tikhonov inversion under realistic conditions of modeling and electronic noise (figure 4.4) (Burnes et al., 2000). This approach has also been applied to demonstrate the feasibility of reconstructing repolarization properties of interest (e.g., increased dispersion of repolarization (Ghanem et al., 2001)). On the other hand, earlier results of a different group using invasive data from patients undergoing arrhythmia surgery, suggested that the usual epicardial potential regularization

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FIG URE 4.4. Maps of epicardial potential at two different times during ventricular activation. in the study of (Burnes et al., 2000). The invasively measured epicardial potential data (from dogs) is used to forward generate body surface potentials on a numerical human torso surface. Geometri cal and "electronic" noise is added to these body surface potentials , and inversely reconstructed epicardial potential maps are computed. [From: Burnes, J. E., Taccardi , B.• Macleod, R. S., and Rudy, Y, 2000, Noninvasive ECG imaging of elec trophysiologcially abnorma l substrates in infarcted hearts, a model study, Circulation. 101: 533- 540. Used by permission.]

methodology was able to usefully image epicardial potential during the QRS inteval only in its initial portions (Shahidi et ai., 1994). Finally, a report by Penney et ai. (2000) (extending work by MacLeod et al. (1995)) identified local changes in inversely computed epicardial electrograms in patients whose data was accessed during coronary catheterization, preceed ing and following angioplasty balloon catheter inflation. In the eighteen study patients, the predicted region of ischemia following balloon inflation correlated with the expected region of perfusion deficit based on the vessel occluded.

4.5 ENDOCARDIAL POTENTIAL IMAGING Interventional cardiologists employ transvenous catheter procedures to treat arrhythmogenic foci and aberrant conduction pathways. Such treatment first requires mapping the endocardial potential. This initial invasive imaging is cumbersome, tedious, and lengthy. Typically, a roving probing trasvenous electrode catheter is brought into contact with many endocardial location s over the course of many heartbeats, and a depiction of an endocardial activation map is thereby inferred (an improved technology along these lines is given by Gepstein et ai. (1996)). The number of sites accessed is limited , and there is no accounting for beat-to-beat activation variability when reconstructing the maps from the many beats. Recently introduced expandible basket electrode arrays (Schmitt et ai., 1999) have their own problems related to limited numbers of electrodes, the need to contact (and perhap s

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irritate) the endocardium, and the possibility of difficulties in collapsing the basket at the end of the acquisition. These problems can be potentially addressed by the use of a transvenous catheter whose tip is studded with multiple electrodes, and which is placed somewhere in the midst of a cardiac chamber (without contacting the endocardium). Once the catheter location relative to the endocardium is registered, it becomes theoretically possible to inversely compute the endocardial potentials from a single heartbeat-indeed, to follow dynamic isopotential maps within a single beat, as well as beat-to-beat changes in activation maps. For this inverse problem, the volume is bounded by the endocardial surface and the multielectrode probe surface. Laplace's equation holds in this volume, and the boundary conditions are the (unknown) endocardial potentials, and the zero normal current density at the multi-electrode probe surface. As in Section 4.2, a linear relationship is derived between the endocardial potentials and the catheter electrode potentials. Notwithstanding the inconvenience of the required cardiac catheterization, there are two very significant advantages of this formulation over the technique of imaging the epicardial potentials from the body surface. First, the electrodes are relatively close to all portions of the surface to be imaged (e.g., as opposed to the distance between body surface electrodes and the posterior wall of the heart). Second, the relevant volume is composed only of blood in the lumen of the cardiac chamber. Therefore, the geometric modeling required for estimation of the transfer matrix is vastly less, and the uncertainties in the values of key components of the model (i.e., tissue conductivities) are markedly diminished (the blood has uniform isotropic conductivity). The initial proposal and work on a multielectrode noncontact array, placed in a cardiac chamber for purposes of accessing endocardial potentials, was due to Taccardi et al. (1987). In the past few years there has been much significant work reported on successors to this idea. For example, in experiments on dogs, Khoury et al. (1998) used a 128 electrode catheter, inserted via a purse string suture in the left ventricular apex, and showed that faithful renditions of endocardial activation, both with paced and spontaneous beats, was possible by solving the inverse problem. Ischemic zones were also well defined. A spiral catheter design has also been investigated (Jia et al., 2000). An impressive series of experiments has been performed with a competing system, developed by Endocardial Solutions, Inc. In addition to a 64-electrode 7.5 ml inflatable balloon catheter, a second transvacular catheter is passed and dragged along the endocardium. As it is dragged, a several kHz signal is passed between it and the electrode catheter, localizing its position with respect to the electrode catheter. In this way, a rendition of the endocardium with respect to the electrode catheter is produced. Following construction of a "virtual endocardium" via a convex hull algorithm applied to the above anatomical data, the inverse problem is then solved, generating several thousand "virtual electrograms" on the virtual endocardium (figure 4.5 and figure 4.6). The literature on this subject is growing rapidly, and we site only a few of examples. Overall, very impressive utility and fidelity is being established. For example, a report by Schilling et al. (2000) describes the classification of atrial fibrillation in humans in terms of numbers of independent reentrant wavefronts identified. A report by Strickberger et al. (2000) describes the successful ablation of fifteen instances of ventricular tachycardial guided by this catheter system. A recent report by Paul et al. (2001) describes the utility of the system in directing catheter ablative therapy in subjects with atrial arrhythmias refractory to pharmacologic therapy.

Heart Surface Electrocardiographic Inverse Solutions

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r

ECGI ~~ C '\,.-"'""-..rVvr-_'\_\J,(~---...,.. N/~ R -'-----~~/\.../"'v'\~ FIG URE 4.5. Surface ECG from lead I (ECG I), endocardial electrogram via a contact electrode (C), and inversely reconstructed electrogram using input from a noncontact multielectrod e probe in the atrium (R), with C and R from the same location, in three patient s with atrial fibrillation, in the study of (Schilling et al., 2000, [From: Schillin g, R. J., Kadish, A. H., Peters, N. S., Goldberger, J., Wyn Davies, D., 2000, Endocard ial mappin g of atrial fibrillation in the human right atrium using a non-contac t catheter, European Heart Journal. 21: 550-564. Used by permission of the publisher, WB Saunders .]

4.6 IMAGING FEATURES OF THE ACTION POTENTIAL 4.6.1 MYOCARDIAL ACTIVATION IMAGING Epicardial and endocardial potential imaging addresses the need to reconstruct something that is currently accessed invasively, and is thus of evident interest. However, such potentials are not themselves a clinical endpoint. Ultimately, clinicians are interested in the action potential---or at least , feature s of the action potential. The most important features of the action potential are activation time (time of arrival of phase zero at every location, the aggragate of which globally describe conduction disturbances), phase zero amplitude (reflecting ischemia), and action potential duration (reflecting refractory periods, potentially associated with propensity for re-entrant arrhythmias). The marker for activation in an electrogram (a tracing of epicardial or endocardial potential at a given cardiac site) is the "intrinsic deflection"-defined as the steepest downward deflection of the electrogram. Recall that the source is the gradient of transmembrane potential (e.g., Eq. (4.10» , and that during cardiac activation this is usually appreciably nonzero only at the locus of points undergoing action potential phase O. This locus is approximately a surface (the interface between depolarized and nondepolarized muscle). Electrically, this behaves approximately as a propagating surface of dipole moment density (a double layer). There is a discontinuity of potential as the double layer is crossed. Ideally, as an extracellular location is passed over by the activation wavefront, there will then be a sharp downward deflection in the extracellular (electrogram) potential-the intrinsic deflection. However,

150

F. G reensi te

FIGURE 4.6. Time sequential views of a portion of the "virtual endocardium" depiction of the atria in the study of (Paul et al., 2001), showing isopotential maps at six successive times. Spreading endocardial activation wavefronts can be appreciated (e.g., two of these collide in E and F) during an atrial reentrant tachycardia. See the attached CD for color figure. [From: Paul, T., Windhagen-Mahnen , B., Kriebel, T., Bertram, H., Kaulitz, R.. Korte, T., Niehaus, M., and Tebbenjohanns, 1., 200 I, Atrial Reentrant Tachycardia After Surgery for Congenital Heart Disease Endocardial Mapping and Radiofrequency Catheter Ablation Using a Novel, Noncontact Mapping System, Circulation. 103: 2266-227 1. Used by permission.]

the reality is that it is not infrequent that there is more than one reasonable candidate for the intrinsic deflection within a given location's electrogram. Furthermore, the intrinsic deflection is often rather lengthy, so the selection of a single activation time within the intrinsic deflection is to some extent arbitrary (Ideker et al., 1989; Paul et al., 1990). The activation time is presumably the inflection point of the deflection (which itself is poorly defined in the noisy setting). To a large extent, these problems are inherent in the source formulation : The epicardial (or endocardial) potential at a location actually reflects contributions from electrical activity at all surrounding locations , when in fact we desire to resolve results of the membrane function at a single location-i.e., the local action potential. In this section we examine work done on imaging the myocardial activation feature of the action potential , rather than the epicardial potential. Enthusiasm for immedi ately attacki ng the problem of recon structing the transmembrane potential m(x), or its gradient, is tempered by recogniti on of a dimen sionality problem : our measurements are confined to a surface (of the body), while the source V . (G i Vm) permiates a volume (the heart). Inherentl y, we are faced with a "projection" of the three

151

Heart Surface Electrocardiographic Inverse Solutions

dimensional source on the two dimensional body volume (a further exacerbation of the already described ill-posedness of the problem). However, building on the work of Wilson et at. (1933), Frank (1954) noted that the source during the QRS inteval was roughly a double layer (i.e., a surface), which tends to mitigate the above dimensionality problem. While Frank was interested in quantifying the inaccuracy of the single moving dipole model of the heart via forward computations (rather than imaging the double layer), two decades later Dotti (1974) made an interesting observation: Neglecting anisotropic conductivity of the heart, assuming uniform action potential amplitude, and recognizing the fact that the gradient of transmembrane potential propagates as a dipolar wavefront (double layer), he noted that the source surface at any time is electrically equivalent (as regards points external to the heart) to a double layer consisting of the portions of the endocardium and epicardium already depolarized (figure 4.7). This is a consequence of the well-known fact from electrostatics that a closed uniform double layer in an isotropic medium generates no external potential. This means that one can derive a relationship between the body surface potential at a given time, and the locus of points on the cardiac surface that have been activated. Thus, the dimensionality problem resolves, and the surface of interest is actually fixed. Dotti presented a very small scale two dimensional simulation illustrating this concept. Similar observations were made independently by Salu (1978) a few years later. However, the concept can be said to have been formally introduced in a more complete engineering context by Cuppen and van Oosteroom in the early 1980s. They presented the imaging equation as ¢(y, t)

=

i

(4.46)

A(x, y)H(t - r(x»dS,

where S is the composite of the endocardial and epicardial surfaces, and the action potential (during the QRS interval) is modeled using the Heaviside function H(t) (zero for t < 0, unity for t > 0). Thus, the action potential (figure 4.1) is taken to be the step function a(x)

+ b(x)H(t -

r(x».

(4.47)

The action potential amplitude b(x) is assumed to be constant over the ventricles, and is subsumed into the transfer function A(x, y). The offset a(x) is also assumed constant, and thus has no effect since A(x, y)dS = 0 (i.e., a uniform closed double layer generates no external potential). Note that in the absence of reentrant arrhythmias there is no repolarization during the QRS interval, so the action potential can then be modeled as a step function in that interval. Thus, Eq. (4.46) is fully consistent with Eq. (4.13) (dervied from the bidomain). Using Eq. (4.46), one wishes to determine rex), the time that point x on the surface surrounding the heart undergoes action potential phase zero. Note that the equation is nonlinear. Equation Eq. (4.46) achieves a superficially satisfying form upon integration over the activation QRS interval,

Is

f QRS

¢(y, t)dt = [ A(x, y)f

1s

QRS

H(t - r(x»dtdSx = -

[ A(x, y)r(x)dSx .

1s

(4.48)

152

F. Greensite

B

c FIGURE 4.7. An electrical double layer in an infinite homogene ous volume conduct or of infinite extent generates potential at a point proportional to the solid angie subtended by the point and the double layer. The above diagram depicts how ventricular intramural depolarization wavefronts generate potential equivalent to that generated by "virtual" double layers on the epicardium and/or endoca rdium. [From: van Oosterom, A., 1987, Comp uting the depolar ization sequence at the ventricular surface from body surface potentials, in: Pediatric and Fundam ental Electro cardiography , (1. Liebman, R. Plonsey, and Y. Rudy, eds.), Martinu s Nijhoff, Zoetermeer, The Netherlands, pp. 75-89. Used by permissi on.]

Apparently, the imaging of myocardial activation is also a linear problem. But attempts to solve Eq. (4.48) soon run up against the problem that the computed activation times are entirely unrealistic-because regularization schemes typically favor solution estimates with lower norm (even with higher order Tikhonov regularization). Thus, the computed

Heart Surface Electrocardiographic Inverse Solutions

153

QRS interval becomes highly contracted. Furthermore, there is the impressionthat one has been wasteful of the temporally resolved (dynamical) information inherent in Eq. (4.46), by integrating it all away in Eq. (4.48). This is unacceptable in an already very ill-posed problem.Huiskampand van Oosterom(1988)addressedthis objectionablefeature by using the regularized solution to Eq. (4.48) as a seed for a quasi-Newton routine for solving a regularized version of the full nonlinear expression Eq. (4.46). As with the basic Newton procedure from Calculus, which extracts the root of a nonlinear function nearest the seed, the quasi-Newton procedure applied here is a means of finding a root (i.e., the appropriate r(x» for a regularizedversionof¢(y, t) A(x, y)H(t - r(x»dS = O.However,aswith the basic Newton method, one is dealing with an intrisically local procedure that does not perform a global optimization. The solution estimate obtained is highly influencedby the initial seed from Eq. (4.48).On the other hand, there is no reason why a global optimization routine such as simulated annealing, could not be used (in fact, this is proposed in a very recent paperon activation time and actionpotentialamplitudeimaging(Ohyu et al., 2002». However, a furtherproblemis that Eq. (4.46) is validonly under the assumptionthat cardiac muscle has isotropicconductivity, or satisfies equal anisotropy. A differentapproachwas taken by Greensite(1994; 1995).The general idea is that the myocardial surface activation function r(x), like any (nominally differentiable) function, is greatly characterized by its relative extrema-e.g., its relative maxima and minima. Predominantly, these are the epicardial breakthrough points and activation sinks of the transmuraldepolarization wavefront. Indeed, since t (x) is definedover a compact domain (the heart surface), and has a finite range (the QRS interval), knowledge of these "critical points" reduces the space of admissible solutions to that of a compact set of functions. The problem of reconstructing the rest of r(x) from Eq. (4.48) is nominally a well-posed problem.In simpleterms,if the relativemaximaand minimaof r (x) are known,the problem of determining the rest of r(x) becomes simply a matter of optimized interpolation-for which the constraints embodied by Eq. (4.46) should be sufficient. An efficientmeans for computing the critical points was given in the Critical Point Theorem (Greensite, 1995). Consider the "data operator"

Is

¢[.] =

1

¢(y, t)(·)dy.

QRS

In the practical setting, ¢ is a space-time matrix, each of whose rows is the body surface potential time series (ECG) at a particular electrode location. The Critical Point Theorem states that x' is a critical point of r(x) if and only if A(x l , y) is in the space spanned by the eigenfunctions of ¢¢t. In fact, the Theorem holds even in the case of an anisotropic myocardium. Complications ensue once noise is added to the formulation, but an efficient algorithm employing these ideas in a noisy context was proposed in (Huiskamp and Greensite, 1997). Oostendorp et at. at the University of Nijmegen/University of Helsinki have produced work evaluating the latter approach both in vitro (Oostendorp et aI., 1997) and in vivo (Oostendorp and Pesola, 1998) (validation in hearts removed at the time of cardiac transplantation, figure 4.8). Work on invasive validation of these latter ideas has also recently been undertaken by a group at the Technical University of Graz, (Tilg et aI., 1999; Modre et aI., 2001a,

154

F. Greensite

FIGURE 4.8. One of a series of four hearts, removed at transplantation, in the study of (Oostendorp and Pesola, 1998). The two upper images of the anterior and posterior ventricular epicardium show the activation maps obtained at the time of surgery (prior to cardiac transplantation) via application of an epicardial electrode sock (epicardial electrode locations indicated by circles). The lower two images show the corresponding preoperative activation map, inversely computed from body surface potential electrode data. [From: Oostendorp, T., and Pesola, K., 1998, Non-invasive determination of the activation time sequence of the heart: validation by comparison with invasive human data, Computers in Cardiology. 25:313-316. Copyright IEEE. Used by permission].

Wach et ai., 2001; Tilg et ai., 2001; Modre et ai., 2001b), and a group at the University of Auckland/University of Oxford (Pullan et ai., 2001).

4.6.2 IMAGING OTHER FEATURES OF THE ACTION POTENTIAL Let tl be a time during the TP interval of the ECG-i.e., a time during which all ventricular locations are in action potential phase 4 (fully repolarized). Let t: be a time during the ST interval of the ECG-i.e., a time during which all ventricular locations are in phase 2 (fully depolarized). From Eq. (4.13) and the action potential model Eq. (4.47), ¢(y, t2) - ¢(y, t1)

= L[G iV1/r(X, y)]. nA¢m(x, t2) = L[GiV1/r(X, y)]. nxb(x)dS.

¢m(x, tl»dS

(4.49)

Note that if b(x) is a constant, both sides of the above equation will be zero (e.g., a closed uniform double layer generates no external potential). Indeed, the body surface potential

155

Heart Surface Electrocardiographic Inverse Solutions

during the TP and ST segments have the same value in healthy subjects. However, in the case of cardiac ischemia, the action potential amplitude is spatially varying. In that setting, one can imagine solving the above integral equation to obtain the spatially-varying action potential amplitude-up to a spatial constant (the null space of the operator is the space of constant functions). Since the phase 0 amplitude in healthy myocytes is already known to be approximately 90 millivolts, one can then (in principle) image the action potential amplitude h(x) fully. Reflecting on the approach of Cuppen and van Oosterom (1984), Geselowitz (1985) noted that it would be possible to image the area under the action potential (i.e., the integral with respect to the baseline of the action potential) by simply extending the time interval of integration in Eq. (4.48) to be the interval (encompassing the time period of activation and repolarization). Thus, {

JQRST

(y, t)dt = -

{

([G;V1/f(x, y)]. Dxm(X, t)dSdt

JQRST Js

=- ([G;V1/f(X,y)].Dxb(X) (

Js

= -l[G;V1/f(X,

JQRST

H(t-r(x))dtdS

-n DxfL(X)dS,

where fL(x) is the area under the action potential at x. Now one can use knowledge of h(x) and fL(X) to create an image of action potential duration as fL(X)jb(x). Thus, one might anticipate imaging action potential attributes such as action potential amplitude and action potential duration, in addition to phase 0 time (activation imaging). Apparently, this joining of the prior two paragraphs has not been investigated, and the practicality of such manipulations is speculative.

4.7 DISCUSSION Among the many engineering challenges posed by the imaging problem treated in this chapter, the necessity of a proper mathematical understanding of the computational difficulties (and their optimal treatment) has some pre-eminence. In this regard, there are lively controversies regarding which is the favored source formulation to be imaged (epicardial/endocardial potentials versus action potential features), the possible role of preprocessing the raw signals (e.g., Laplacian electrocardiography), the reductionist role in activation imaging (e.g., the Critical Point Theorem), and the desirability of integrating the temporal data from a stochastic processes standpoint. Recent history has shown that there is surely room for improvement in algorithmic technique. Methodological refinements continue to be proposed by many different groups. At the same time, the biophysical understanding, technical apparatus, and mathematical methodology, are clearly already in place to create images of extracellular potential and action potential features on the epicardial and endocardial surfaces. The principal question is whether the resulting images are either too blurred to be of much use, or are otherwise unreliable and misleading (e.g., due to the inherent ill-posedness of the problem, lack

156

F. Greensite

of sufficiently powerful mitigating constraints, or insufficiently accurate forward problem solutions due to uncertainties in knowledge of body tissue conductivities and anisotropies). Thus, image validation is presently a major question of interest in this field. Such validation is fairly well advanced in the case of the minimally invasive techniques of imaging endocardial potential via trans vascular catheter probe electrode arrays. However, for the epicardial imaging approaches, it is particularly difficult to address the validation question adequately, because validation ideally requires the simultaneous acquisition of epicardial signals, body surface signals, and anatomical body imaging (via CT or MRI). That is, the body imaging should be conducted closed chest (otherwise, the body surface signals would be subject to an unrealistic transfer matrix), despite the simultaneous need for "gold standard" invasively obtained epicardial potentials for the validation. Nevertheless, the latter validation goal is being aggressively pursued by a number of groups worldwide. It is likely that the field is maturing to the extent that the next several years will see clarification of the true potential and promise of the methodologies discussed in this chapter. Accordingly, the era of Noninvasive Imaging of Cardiac Electrophysiology (NICE) could soon be at hand.

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Papoulis, A., 1984, Probability, Random Variables, and Stochastic Processes, McGraw-Hill, New York. Paul, T, Moak, J. P., Morris, C., and Garson, A., 1990, Epicardial mapping: how to measure local activation, PACE. 12:285-292. Paul, T, Windhagen-Mahnert, B., Kriebel, T., Bertram, H., Kaulitz, R, Korte, T, Niehaus, M., and Tebbenjohanns, J., 2001, Atrial Reentrant Tachycardia After Surgery for Congenital Heart Disease Endocardial Mapping and Radiofrequency Catheter Ablation Using a Novel, Noncontact Mapping System, Circulation. 103:2266-2271. Penney, C. 1., Clements, J. C., and Horacek, B. M., 2000, Non-invasive imaging of epicardial electrograms during controlled myocardial ischemia, Computers in Cardiology 2000.27:103-106. Plonsey, R., 1969, Bioelectric Phenomena, McGraw-Hill, New York. Pullan, A. J., Cheng, L.K., Nash, M.P., Bradley, c.P., Paterson, DJ., 2001, Noninvasive electrical imaging of the heart: theory and model development. Ann. Biomed. Eng. 29:817-836. Ramanathan, C., and Rudy, Y, 2001, Electrocardiographic Imaging: II. Effect of torso inhomogeneities on noninvasive reconstruction of epicardial potentials, electrograms, and isochrones. J. Cardiovasc. Electrophysiol. 12:242-252. Reese, T, Weisskoff, R., Smith, R, Rosen, B., Dinsmore, R, and Wedeen, v., 1995, Imaging myocardial fiber architecture in vivo with magnetic resonance, Magnetic Resonance in Medicine. 34:786-791. Rudy, Y, and Messinger-Rapport, B. J., 1988, The inverse problem in electrocardiography: solutions in terms of epicardial potentials, Crit. Rev. Biomed. Eng. 16:215-268. Salu, Y, 1978, Relating the multipole moments of the heart to activated parts of the epicardium and endocardium, Ann. Biomed. Eng., 6:492-505. Schilling, R. J., Kadish, A. H., Peters, N. S., Goldberger, J., Wyn Davies, D., 2000, Endocardial mapping of atrial fibrillation in the human right atrium using a non-contact catheter, European Heart Journal. 21: 550-564. Schmitt, C; Zrenner, B., Schneider, M., Karch, M., Ndrepepa, G., Deisenhofer, I., Weyerbrock, S., Schreieck, J., and Schoemig, A., 1999, Clinical experience with a novel multielectrode basket catheter in right atrial tachycardias, Circulation. 99:2414-2422. Schmitt, O. H., 1969, Biological information processing using the concept of interpenetrating domains, in Information Processing in the Nervous System, (Leibovic, K. N. ed.), Spinger-Verlag, New York. Shahidi, A. v., Savard, P., and Nadeau, R., 1994, Forward and inverse problems of electrocardiography: modeling and recovery of epicardial potentials in humans, IEEE Trans. Biomed. Eng. 41:249-256. Strickberger, S. A., Knight, B. P., Michaud, G. E, Pelosi, E, and Morady, E, 2000, Mapping and ablation of ventricular tachycardia guided by virtual electrograms using a noncontact, computerized mapping system. 1. Am. Col. Cardiol. 35:414-421. Taccardi, B., Arisi, G., Macchi, E., Baruffi, S., and Spaggiari, S., 1987, A new intracavitary probe for detecting the site of origin of ectopic ventricular beats during one cardiac cycle, Circulation. 75: 272-281. Throne, R, and Olsen, L., 1994, A generalized eigensystem approach to the inverse problem of electrocardiography, IEEE Trans. Biomed. Eng. 41:592-600. Throne, R. D., Olsen, L. G., 2000, A comparision of spatial regularization with zero and first order Tikhonov regularization for the inverse problem of electrocardiography, Computers in Cardiology. 27:493--496. Tikhonov, A., and Arsenin, v., 1977, Solutions of Ill-Posed Problems, John Wiley and Sons, New York. Tilg, B., Wach, E, SippensGroenwegen, A., Fischer, G., Modre, R., Roithinger, E Mlynash, M., Reddyuu, G., Roberts, T., Lesh, M., and Steiner, P., 1999, Closed-chest validation of source imaging from human ECG and MCG mapping data, in: Proceedings of the 21st Annual International Conference of the IEEE EMBS, October 19991 First Joint BMESIEMBS Conference, IEEE Press. Tilg, B., Fischer, G., Modre, R., Hanser, E, Messnarz, B., Wach, P., Pachinger, 0., Hintringer, E, Berger, T., Abou-Harb, M., Schoke, M., Kremser, C., and Roithinger, E, 2001, Feasibility of activation time imaging within the human atria and ventricles in the catheter laboratory, Biomedizinishe Technik 46:213-215. Tuch, D. S., Wedeen, V. J., Dale, A. M., and Belliveau, J. 1997, Conductivity maps of white matter fibertracts using magnetic resonance diffusion tensor imaging, Proc. Third int. conf. On Fundamental Mapping of the Human Brain, Neuroimage. 5:s44. Twomey, S., 1963, On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature, J. ACM, 10:97-101. Ueno, S., and Iriguchi, N., 1998, Impedance magnetic resonance imaging: a method for imaging of impedance distribution based on magnetic resonance imaging, J. Appl. Phys. 83:6450-6452.

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van Oosterom, A., 1987, Computing the depolarization sequence at the ventricular surface from body surface potentials, in: Pediatric and Fundamental Electrocardiography, (1. Liebman, R. Plonsey, and Y. Rudy, eds.), Martinus Nijhoff, Zoetermeer, The Netherlands, pp. 75-89. van Oosterom, A., 1999, The use of the spatial covariance in computing pericardial potentials. IEEE Trans. Biomed. Eng. 46:778-787. Velipasaoglu, E. P., Sun, H., Zhang, E, Berrier, K. L., and Khoury, D. S., 2000, Spatial regulariation of the electrocardiographic inverse problem and its application to endocardial mapping, IEEE Trans. Biomed. Eng. 47:327-337. Wach, P., Modre, R., Tilg, B., Fischer, G., 2001, An iterative linearized optimization technique for non-linear ill-posed problems applied to cardiac activation time imaging, COMPEL 20:676-688. Waller, A., 1889, On the electromotive changes connected with the beat of the mammalian heart, and of the human heart in particular, Phil. Trans. R. Soc. Lond. B. 180: 169-194. Waller, A., 1911, quoted in Cooper 1. K., 1987, Electrocardiography 100 years ago: origins, pioneers, and contributors, NEJM. 315:461-464. Wahba, G., 1977, Practical approximated solutions to linear operator equations when the data are noisy, SIAM 1. Numer. Anal. 14:651-667. Wilson, EN., Macleod, A. G., and Barker, P. S., 1933, The distribution of the action currents produced by heart muscle and other excitable tissues immersed in extensive conducting media, 1. Gen. Physiol. 16:423-456. Wilson, EN., Johnston, E D., and Kossmann, C. E., 1947, The substitution of the tetrahedron for the Einthoven triangle. Am. Heart J., 33:594--603. Yamashita, Y., and Geselowitz, D., 1985, Source-field relationships for cardiac generators on the heart surface based on their transfer coefficients, IEEE Trans. Biomed. Eng. BME-32:964--970. Zablow, L., 1966, An equivalent cardiac generator which preserves topolgraphy, Biophys. 1. 6:535-536.

5

THREE-DIMENSIONAL ELECTROCARDIOGRAPHIC TOMOGRAPHIC IMAGING Bin He* University of Illinois at Chicago

5.1 INTRODUCTION Cardiac electrical activity is distributed over the three dimensional (3D) myocardium. It is of significance to noninvasively image distributed cardiac electrical activity throughout the 3D volume of the myocardium. Such knowledge of the source distribution would play an important role in our effort to relate the electrocardiographic inverse solutions with regional cardiac activity. Historically, attempts to noninvasively obtain spatial information regarding cardiac electrical activity started from body surface potential mapping by using a larger number of recording leads covering the entire surface of the body (Taccardi, 1962). From such measurements, instantaneous equipotential contour maps on the body surface have been obtained and shown to provide additional information when compared to a conventional electrocardiogram (See Flowers & Horan, 1995 for review). Since body surface potential maps (BSPMs) are manifestation of cardiac electrical sources on the body surface, efforts have been made to solve the electrocardiography inverse problem-to seek the generators of BSPMs. Equivalent dipole solutions have been investigated with the aim of extracting useful information regarding cardiac electrical activity. Such efforts included (l) Moving Dipole Solutions (Mirvis et al., 1977; Savard et al., 1980; Okamoto et al., 1983; Gulrajani et al., 1984), in which one or more current dipoles are estimated at the location(s) that best describe the body surface recorded electrocardiograms; and (2) Fixed Dipoles

* Present address for correspondence: University of Minnesota, Department of Biomedical Engineering, 7-105 BSBE, 312 Church Street, Minneapolis, MN 55455 E-mail: [email protected]

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Solutions (Barber & Fischman, 1961; Bellman et aI., 1964; He & Wu, 2001), in which an array of dipoles are arranged at fixed locations where their moments are determined by minimizing the difference between the model-generated and the measured body surface electrocardiograms. It has been demonstrated that the single moving dipole solution can provide a good representation of well-localized cardiac electrical activity (Savard et aI., 1980). Efforts have also been made to estimate two moving dipole solutions, although technical challenges exist when the number of equivalent dipoles increases from one to two (Okamoto et aI., 1983; Gulrajani et al., 1984). Due to the ill-posedness of the inverse problem, currently there is no well established method to estimate three or more moving dipoles. In addition to equivalent dipole approach, equivalent multipole models have also been investigated (Geselowitz 1960; Hlavin & Plonsey, 1963; Pilkington & Morrow, 1982) during the early stage of electrocardiography inverse solutions, in an attempt to obtain equivalent 3D information on cardiac electrical activity. The limitation of the multipole approach, however, is its inability of localizing cardiac electrical activity. In the past decade, most research on the electrocardiography inverse problem has been carried out in the line of heart surface inverse solutions. As reviewed in Chapter 4, these research efforts are mainly related to epicardial potential inverse solutions or heart surface activation imaging. Three dimensional electrocardiographic tomographic imaging has received much attention since 2000. He and Wu reported their effort on electrocardiographic tomography in their presentation at the World Congress on Medical Physics and Biomedical Engineering held in Chicago in 2000. In this work, He & Wu demonstrated the feasibility in a computer simulation study to image the 3D distribution of cardiac dipole source distribution from noninvasive body surface electrograms by using the Laplacian weighted minimum norm approach (He & Wu, 2000, 2001). In subsequent work, He and coworkers developed a heart-model based 3D activation imaging approach (He & Li, 2002; He et al., 2002), and a 3D transmembrane potential (TMP) imaging approach (He et al., 2003), which were introduced in a presentation at the 4th International Conference of Bioelectromagnetism in 2002 (He & Li, 2002). Ohyu et al. (2002) have developed an approach to estimate the activation time and approximate amplitude of the TMP from magnetocardiograms using the Wiener estimation technique. Skipa et al. presented their effort to estimate transmembrane potentials from body surface electrocardiograms at the 4th International Conference on Bioelectromagnetism (2002). In this Chapter, we review the principles and methods of performing 3D electrocardiographic tomographic imaging, with a focus on introducing the recently developed distributed 3D electrocardiographic tomographic imaging techniques.

5.2 THREE-DIMENSIONAL MYOCARDIAL DIPOLE SOURCE IMAGING 5.2.1 EQUIVALENT MOVING DIPOLE MODEL Equivalent dipole inverse solutions were among the earliest efforts of obtaining electrocardiography inverse solutions. Early efforts were made on moving dipole inverse solutions

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where one or two equivalent dipoles were used to represent cardiac electrical activity in the sense that the dipole-generated body surface potential maps (BSPMs) matches the measured BSPMs well (Mirvis et aI., 1977; Savard et al., 1980; Okamoto et al., 1983; Gulrajani et aI., 1984). In the moving dipole model, the locations of the equivalent dipoles vary from time to time which provide information on the centers of gravity of electrical activity within the heart. Such location information offers an important capability for moving dipole solutions to localize the regions of myocardial tissues which are most responsible for the measured BSPMs. A limitation of this approach, however, is that the inverse solution is sensitive to measurement noise and thus limiting the number of moving dipoles that can be reliably estimated from the measured BSPMs. For this reason, the moving dipole inverse solution may be useful in localizing a focal cardiac source during the initial phase ofcardiac activation for a single activity. For general cardiac activation, the moving dipole inverse solution fails to represent the complex cardiac electrical activity. A detailed review on an equivalent moving dipole solution can be found in reference (Gulrajani et al., 1984).

5.2.2 EQUIVALENT DIPOLE DISTRIBUTION MODEL As early as the 1960's, Barber and Fischman suggested the possibility of modeling cardiac electrical activity using an array of current dipoles located at fixed locations within the myocardium (Barber & Fischman 1961; Bellman et aI., 1964). In this model, the dipoles are not moving butfixed over time, while its moments remain variable. Yet this model did not receive much attention in the field of electrocardiography inverse problem in the past three decades, partially due to the dominance of the epicardial potential (Barr et aI., 1977; Frazone et aI., 1978; Shahidi et aI., 1994; Throne & Olson, 1994, 1997; Johnston & Gulrajani, 1997; Oster et aI., 1997; He & Wu, 1997; Greensite & Huiskamp, 1998; Burnes et al., 2000) and the heart-surface activation time (Cuppen and van Oosterom 1984; Huiskamp & Greensite 1997; Greensite 2001; Modre et al., 2001; Pullan et al., 2001) inverse solutions developed during the same period. Recently, the fixed dipole array model has been expanded into a volume distribution of current dipoles for the purpose of tomographic dipole source imaging (He & Wu, 2000, 2001). In this work, He & Wu modeled cardiac electrical sources by means of a large number of current dipoles over the 3D volume of the ventricles. Each of the dipoles was located at a particular position, representing the local electrical activity, while the moments of the dipoles were varied over time. The magnitude function of the regional dipoles provided a spatial distribution of current strength with the 3D myocardial volume. Estimation of this current dipole moment provides a means of imaging the spatial distribution of current sources.

5.2.3 INVERSE ESTIMATION OF 3D DIPOLE DISTRIBUTION The key hypothesis under the 3D dipole distribution imaging is based on the assumption that the electrical sources located at a small region of myocardial tissue are coherent and

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can be approximated by a current dipole. By assigning one such current dipole to each "small" region of the myocardium, the following mathematical model, which relates the current dipole distribution inside the myocardium to the body surface ECG measurements, can be obtained:

V=AX

(5.1)

where V is the vector consisting of m body surface-recorded ECG signals, X is the unknown vector consisting of the moments of the current dipoles, which are located at n sites covering the entire myocardial volume, and A is the transfer matrix. The measurement at each electrode sensor is produced by a linear combination of all dipole components, with columns in A serve as weighting factors. By solving (5.1), one obtains an estimation of 3D current dipole source distribution corresponding to each measured BSPM. Since the number of measurement electrodes is always far less than the dimension of the unknown dipole source vector X, this problem is an underdetermined inverse problem and a proper regularization strategy is necessary for obtaining a reasonable solution to (5.1). The minimum norm (MN) solution is one of the feasible solutions (Hamalainen & Ilmoniemi, 1984) (5.2) where (*)+ denotes the Moore-Penrose inverse. As the minimum norm solution is intrinsically biased towards the superficial position, the weighted minimum norm solution (Jeffs et aI., 1987) and the Laplacian weighted minimum norm solution (LWMN) (Pascual-Marqui et aI., 1994) has been proposed to solve the linear inverse problem. He & Wu investigated 3D electrocardiography dipole source imaging using the principles of LWMN (He & Wu, 2000,2001). LWMN utilizes a weighting operator LW, where L is a Laplacian operator and W is a diagonal 3n by 3n matrix with Wii = I[ Ai II and Ai is the z-th column of the transfer matrix A. Assuming the weighting factor is nonsingular, then (5.3)

and the LWMN solution of (5.1) becomes (5.4)

When using LWMN the resulting solution tends to be over-smoothed due to the constraints of minimizing the Laplacian of the signal. For well-focused cardiac sources, such as the sites of origins of cardiac arrhythmias, a recursive weighting strategy, which was previously developed for improving the performance of MN MEG imaging (Gorodnitsky et aI., 1995), has been used to search for focal sources in the heart from initial LWMN estimates. This algorithm recursively enhances the values of some of the initial solution elements, while decreasing the rest of the elements until they become zero. In the end, only a small number of winning elements remain non-zero, yielding the desired type of localized energy distribution of the solution.

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5.2.4 NUMERICAL EXAMPLE OF 3D MYOCARDIAL DIPOLE SOURCE IMAGING Computer simulation results for 3D myocardial dipole source imaging have recently been reported (He & Wu, 2001). A 3D heart-torso inhomogeneous volume conductor model (Wu et aI., 1999) was used in the simulation. Considering the low conductivity of the lungs , the conductivity ratio of torso to lungs was set to 1:0.2; and the conductivity for myocardial muscle was assumed to be the same as the torso (Mulmivuo & Plonsey, 1995; Gulrajani , 1998). The ventricles were divided into an equi-distant lattice structure of 1,124 nodes with a resolution of 6.7 mm. A regional current dipole was assigned on each of the nodes, resulting in 1,124 regional dipoles. Fig. 5.1 illustrates an example of3D cardiac dipole source imaging . Two dipole sources , oriented from the waist towards the neck, were used to approxiate two localized cardiac sources, which were located close to the endocardium at the right ventricle and the epicardium at the left ventricle (Fig. 5.l(a)). Gaussian white noise of 5% was added to the body surface potentials calculated from assumed cardiac dipole sources to simulate noisecontamin ated body surface ECG measurements. The inverse imaging algorithm described in Section 5.2.3 was used to attempt to reconstruct the source distribut ion within the myocardium , without a priori knowledge of the number of primary current dipoles. The LWMN solution is illustrated in Fig . 5.1(b), where the red and yellow color s illustrate the strength of the equivalent dipole source distribution throughout the ventricle s. Fig. 5.1(b) shows that the LWMN solution reached maxima in both the right ventricle and the left ventricle, overlying with the locations of the source dipoles. The LWMN solution showed a stronger source distribution over the left ventricle probably because the dipole in the left ventricle is located closer to the chest when compared with the dipole located in the right ventricle. In addition, Fig . 5.1(b) shows that there is another major area of activity appearing over the posterior ventricular wall in the LWMN solution . The recursively weighted LWMN solution is illustrated in Fig. 5.l (c) where after 20 iterations the source strength distribution is well focused at two locations. One of the source localization results was consistent with the "true" dipole at the right ventricle , while the other was located at the left ventricle, but shifted about 1 em towards the direction of the endocardium from the "true" diople position . Previous studies have shown that the equivalent dipole solution suffers from existing experimental noise if the number of the moving dipoles increases to two or more (Okamoto et al., 1983). In a clinical setting, however, it is necessary to localize and image sites of origins of arrhythmias even without knowing in advance how many dipoles should be used. Hence , there is a need to develop a technique, which can localize and image sites of origins of cardiac arrhythmias without a priori constraints on the number of equivalent moving dipoles (such as one dipole ). The LWMN approach which He & Wu (2000, 2001) have applied to cardiac dipole source imaging , on the other hand, does not attempt to make assumptions on the number of focal cardiac sources. Some of the a priori inform ation being taken into account in the 3D cardiac dipole source imaging approach is that the myocardial electrical activation is smooth over a reasonably small region. With such constraints , the estimated inverse solution provide s a smoothed distribution of current density over a large area of myocardium (Fig. 5.1(b)). For focal sources, as illustrated in Fig. 5.1(a ), additional strategy

(b)

(a)

(c)

Recursive So lution

FIGURE 5.1. A numeri cal example of 3D cardiac dipole source imaging. (a) Two dipole sources (red dots), oriented from the waist toward s the neck, were used to approxiate two localized cardiac sources , located close to the endocardium at the right ventricl e and the epicardium at the left ventricle . Gaussian white noise of 5% was added to the body surface potentials to simulate noise-contaminated body surface ECG measurem ents. (b) The LWMN solution shows smoo thed distrubution of current den sity distributi on within the myocardium. where the red and yellow colors illustrate the strength of the equivalent dipole source distribution. (c) The recursively weighted LWMN solution shows well focused source strength distribution which correspond well with the two "true" dipole sources as shown in (a). See the attached CD for color figure. (from He & Wu, 200 1 with permission) @ IEEE

LWMN Solution

Simu lated Sources

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such as recursive focusing is needed to obtain the inverse solution, in which it is assumed that the sites of origins of cardiac arrhythmias are localized over small regions inside the myocardium. With this additional constraint, the inverse dipole source distribution shows localized distribution of current density close to the original "true" dipole sources. Note, however, that the LWMN inverse solution with a recursive weighting strategy still shows certain shift towards the "interior" of the myocardium from the "true" solution positions (Fig. 5.1). Both the LWMN algorithm and the recursive weighting algorithm may contribute to such "shift." Although the work reported by He & Wu suggests the promise of imaging cardiac electrical activity using LWMN approach, a systematic study should be conducted to evaluate the reconstruction results for a number of source configurations including sources located in various regions of the heart with various orientations.

5.3 THREE-DIMENSIONAL MYOCARDIAL ACTIVATION IMAGING Myocardial activation imaging received much attention in recent years, in which local activation time over the heart surface is estimated from BSPMs (Cuppen and van Oosterom 1984, Huiskamp and Greensite 1997, Greensite 2001, Modre et al., 2001, Pullan et al 2001). As reviewed in Chapter 4, this approach is based on the bidomain theory, which allows direct linking of the heart surface activation time with body surface potentials under the assumption of the electrical isotropy (or "equal anisotropy") within the myocardium (Greensite 2001). Recently, the concept of myocardial activation imaging has been extended from 2D heart surface to 3D myocardial volume (He et al., 2002; He & Li, 2002; Ohyu et al., 2002). In these approaches, the activation time throughout the 3D myocardium is estimated from body surface electrograms by means of a heart-excitation-model (He et al., 2002; He & Li, 2002) or a Wiener inverse filter (Ohyu et al., 2002). In this Section, the heart-model-based 3D activation imaging approach (He et al., 2002) is presented.

5.3.1 OUTLlNE OF THE HEART-MODEL BASED 3D ACTlVATION TIME IMAGING APPROACH

The 3D distribution of activation time throughout the ventricles has been estimated with the aid of a heart-model based approach, in which the a priori knowledge on cardiac electrophysiology is embedded. Fig. 5.2 illustrates a schematic diagram of this approach. A realistic geometry computer heart-torso model is used to represent the relationship between 3D activation sequences within the myocardium with BSPMs. The a priori knowledge on cardiac electrophysiology and the detailed anatomic information on the heart and torso are embedded in this heart-torso model. The 3D myocardial activation sequence is estimated as the parameter of the heart-model and obtained by means of a nonlinear estimation procedure.

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FIGURE 5.2. Schematic diagram of 3D electrocardiography tomographic imaging. See attached CD for color figure (From He et aI., Phys Med & BioI, 2002 with permission)

A preliminary classification system (PCS) is employed to determine the cardiac status based on the a priori knowledge of the cardiac electrophysiology and the measured BSPM by means of an artificial neural network (ANN) (Li & He, 2001). The output of the ANN based PCS provides the initial estimate of heart model parameters to be used later in a nonlinear optimization system. Using these initial parameters as the result of PCS, the optimization system then minimizes objective functions that assess the dissimilarity between the measured and heart-torso-model-calculated BSPMs. The heart model parameters corresponding with the calculated BSPM are employed to produce 3D myocardial activation sequence if the measured BSPM and the heart-torso-model-calculated BSPM matches well. Before the objective functions satisfy the given convergent criteria, the heart model parameters are adjusted with the aid of the optimization algorithms and the optimization procedure proceeds.

5.3.2 COMPUTER HEART EXCITATION MODEL Numerous efforts have been made to develop computer heart models that can simulate cardiac electrophysiological processes as well as the relationship between cardiac activities and BSPMs (See Chapter 2 for review). Although the more detailed information incorporated the better, the cellular automaton heart excitation model (Aoki et al., 1987; Lu et al.,

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1993) has been used in the 3D activation time imaging research due to its capability of simulating cardiac activation, BSPMs, and computational efficiency. In this work (He et aI., 2002), we used a cellular automaton ventricle model that was constructed as a 3D array of approximately 42,000 myocardial cell units with a spatial resolution of 1.5 mm. The ventricles consisted of 50 layers with inter-layer distance of being 1.5 mm, and were divided into 53 myocardial segments. Each segment is comprised of approximately the same number of myocardial cell units. The action potential of each of heart units was already determined according to the cardiac action potential experimentally observed and stored in the action potential data file. From the epicardium to the endocardium, the refractory period of the action potential of cardiac cellular units gradually increased for the T-wave simulation. The primary current dipole sources are proportional to the gradient of the transmembrane potentials at adjacent cardiac units (Miller & Geselowitz, 1978). The anisotropic propagation of excitation in the ventricular myocardium was incorporated into this heart model (He et aI., 2003) in order to obtain more accurate simulation of the body surface ECG and myocardial activation sequence (Nenonen et al., 1991, Lorange and Gulrajani 1993; Wei et al., 1995; Franzone et al., 1998; Huiskamp, 1998; Fischer et aI., 2000). Ventricular myocardium was divided into different layers with thickness of 1.5 mm from epicardium to endocardium. The myocardial fiber orientations were rotated counterclockwise over 1200 from the outermost layer (epicardium, -60°) to the innermost layer (endocardium, +60°) (Streeter et aI., 1969) with identical increment between the consecutive layers. All units on a myocardial layer of ventricles from epicardial layer to endocardial layer had identical fiber orientation. For each myocardial unit, a fiber direction vector, which is located on its local tangential plane, was determined by its fiber angle. The fiber orientations of all myocardial units of ventricles were determined, and put in the realistically shaped inhomogeneous torso model for calculating the body surface ECG. Excitation conduction velocity of myocardial units was set to 0.6 m/s and 0.2 m/s along the longitudinal and transverse fiber direction, respectively. Electrical conductivity of myocardial units was set to 1.5 mS/cm along the longitudinal fiber direction and 0.5 mS/cm along the transverse fiber direction (Nenonen et aI., 1991). Fig. 5.3 shows the realistic geometry inhomogeneous heart-torso model (a), an example of simulated sinus rhythm (b), and an example of paced activity (c). Fig. 5.3(b) shows the activation sequence corresponding to sinus rhythm (left), and an example of the anterior BSPM and a chest ECG lead simulated during sinus rhythm (right). Fig. 5.3(c) shows an example of the simulated BSPM on anterior chest (middle-bottom) at 30 ms following pacing the anterior wall of the ventricle, and the ventricular excitation sequence over the epicardium (middle-top) corresponding to the pacing site (left), and a chest ECG lead (right). The pacing site is shown on the left.

5.3.3 PRELIMINARY CLASSIFICATION SYSTEM There are a large number of parameters associated with the cellular automaton heart model that need to be determined in order to estimate the activation sequence. A Preliminary Classification System (PCS) is used to approximately classify, from BSPMs, cardiac status with the a priori knowledge on cardiac electrophysiology and the mapping relationship between cardiac activation and resulting BSPMs. An ANN has been used to serve as a PCS. In this implementation (Li & He, 2001), a three-layer feed-forward ANN was used, with the

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Sin

Rhythm

Pa in

l

~ ( a)

( c)

FI G URE 5.3. Illustration of computer heart-torso modeling and simulation. (a) Realistic geometry heart torso model. (b) Simulation of sinus rhythm: Left panel-Intracardiac activation sequence over three slices within the ventricles; Right panel-An example of simulated anterior BSPM and chest ECG lead. (c) Simulation of epicardial pacing: Left panel- Heart model and a pacing site at middle anterior epicardium of ventricle. Middle panels- Anterior view of simulated epicardial isochrone (top), and an example of the BSPM over the anterior chest following pacing (bottom). Righ panel-A simulated chest ECG lead. See attached CD for color figure. (Modified from He et aI., Phys Med & BioI, 2002 with permission)

number of neurons in the input layer being set to the number of body surface electrod es, and the number of neurons in the output layer being set to the number of myocardial segments being studied. Gaussian white noise (GWN) was added to the BSPMs to simulate noisecont aminat ed body surface ECG measurements . The BSPM maps during 25 to 50 ms after initial activation were used as inputs to train the ANN.

5.3.4 NONLINEAR OPTIMIZATION SYSTEM The heart model parameters associated with myocardial activation sequence are estimated by minimizing dissimilarity between the measured and heart-torso-model-generated (referred to as "simulated" below) BSPMs. The dissimilarity between the measured and simulated BSPMs is described by appropri ate characteristic parameters extracted from the BSPMs. The following three objective functions (Li & He, 200 1) have been used to reflect the dissimilarity between the measured and simulated BSPMs for paced ventricular activation: (a) Ecd x), which was constru cted with the average correlation coefficient (CC) between the measured and simulated BSPM s from instant T 1 to instant T2 of the cardiac excitation after detection of initial activation, is defined as:

L [l Tz

EccCx) =

I=T,

CCms(x , t ))/ (T 2

-

T J)

(5.5)

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where CCms(X, t) is the CC between the measured and simulated BSPMs at instant t. x is a parameter vector of the spatial location of initial activation in the computer heart-excitation model. (b) Eminp(x), which was constructed with the deviation of the positions of minima of the measured and simulated BSPMs from instant T] to instant T2 , is defined as: T2

Eminp(X) =

L II P~n(O - r:», 0[1

(5.6)

t=T,

where P~n(O and P~in(x, t) represent the positions of the minima in the measured and simulated BSPMs at instant t, respectively. The definition of x is the same as that in Eq. (5.5). (c) ENPL(x), which was constructed with the relative error of the number of body surface recording leads, at which the potentials are less than a certain negative threshold, in the measured and simulated BSPMs from instant T 1 to instant T2 , is defined as: (5.7)

where L%(t) = L~~l U(¢T - ¢(t, i» and L~(x, t) = L;:;] U(¢T - ¢(x, t, i), are the numbers of recording leads, at which the potentials are less than a given threshold ¢T( < 0), in the measured and simulated BSPMs at instant t, respectively. ¢(t, i) and ¢(x, t, i) are the ith-lead measured and simulated potentials at instant t, respectively. u(e) is the unit-step function, which gives a unity output if the potential at a lead is less than the pre-set threshold. NL is the number of body surface recording leads. The definition of x is the same as that in Eq. (5.5). Combining the above objective functions, the mathematical model of the optimization for this heart-model-based electrocardiographic imaging can be represented as the following minimization problem: min(Ecdx» = E~c' Eminp(X) < XEX

£minp,

ENPL(x) < £NPL

(5.8)

where X is the probable value region of the parameters in the computer heart-excitationmodel. x is a vector of heart model parameters. E~c is the optimal value of the objective function Ecdx). £minp and £NPL are the allowable errors of the objective function Eminp(x) and ENPL(x), respectively. Eq. (5.8) was solved by means of the Simplex Method.

5.3.5 COMPUTER SIMULATION The feasibility of 3D myocardial activation imaging has been suggested in a computer simulation study (He et al., 2002), and presented in this section. In this simulation study, pacing protocols were used to simulate paced cardiac activation. By setting pacing sites in different myocardial regions of the heart excitation model, sequential pace maps were obtained by solving the forward problem using the heart-torso computer model. Two pacing protocols, single-site pacing and dual-site pacing, were used to evaluate the performance of

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the 3D myocardial activation imaging approach. Gaussian white noise (GWN) of 10 !.LV was added to the BSPMs at each time instant after the onset of pacing, to simulate the noisecontaminated body surface potential measurements. The maximum value of the BSPM during the QRS complex was set to 3 m V. The performance of activation time imaging was tested by single-site pacing in 24 different sites throughout the ventricles. The CC and RE between the vector of simulated activation time and the vector of estimated activation time were calculated for each of the 24 pacing sites. The vector of activation times consists of the activation time of each voxel within the ventricles. Averaged over all 24 sites, the RE and CC between the "true" and estimated activation times are 0.07 ± 0.03 and 0.9989 ± 0.0008, respectively, suggesting the high degree of fidelity of the inverse estimation of activation time in the ventricles. Fig. 5.4 shows two typical simulation examples. The top rows show the simulated "true" activation sequence, and the bottom rows show the inversely estimated activation sequence. Each row shows the activation sequence in 5 longitudinal sections «b)'""-'(f)) and I transverse section of ventricles (a). Five horizontal lines in the transverse section

True

(A) Estimated (8)

(b)

(c)

(d)

(b)

(c)

(d)

(e)

(I)

True

(B) Estimated (8)

(e)

(I)

FIGUR E 5.4. Two examples of activation time imaging results during dual-site ventricular pacing. The activation sequence within the ventricles was inversely estimated from the BSPM with !Of!. V Gaussian white noise being added. First row shows the simulated "true" activation sequence , and second row the activation sequence corresp onding to the inversely estimated result. Each row shows the isochrone in 5 longitudinal sections «b)- (f» and I transverse section of the ventricles (a). Five horizontal black lines in the transverse section of the ventricles from top to bottom respectively indicate the positions of 5 longitudinal sections from (b) to (f). The unit of color bar is millisecond. The distance between both neighboring sections is 4.5 mm. (A) One pacing site at septum endoc ardium of left ventricle, and another at intramural of left-anterior wall (marked by yellow dots). (B) Both pacing sites at left-posterior intramural adjacent to endocardium (marked by yellow dots). See attached CD for color figure. (From He et aI., Phys Med & Bioi, 2002 with permission)

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TABLE 5.1. Effects of heart and torso geometry uncertainty on the activation time imaging Pacing Region

BA

BRW

BLW

BS

BP

Mean

NM LSX RSX FSY BSY NTM+ 10% NTM - 10%

0.08 0.08 0.06 0.08 0.08 0.09 0.09

0.03 0.03 0.03 0.03 0.06 0.11 0.10

0.06 0.06 0.06 0.07 0.07 0.08 0.06

0.04 0.10 0.09 0.10 0.12 0.05 0.10

0.05 0.06 0.07 0.06 0.05 0.07 0.07

0.05 0.06 0.06 0.05 0.07 0.08 0.08

Note: NM: LSXlRSX: FSYIBSY: NTM + IO%/NMT - 10%:

± SD

± 0.02 ± 0.03 ± 0.02 ± 0.02 ± 0.03 ± 0.02 ± 0.02

Normal Model. Left/Right shift along the x-direction. Front/Back shift along the y-direction. 10% enlargement and reduction of the normal torso model.

of ventricles from top to bottom respectively indicate the positions of the 5 longitudinal sections from (b) to (f). In panel (A), one pacing site is located at the left ventricular septal endocardium, and another pacing site is located at the left anterior intramural wall. In panel (B), both pacing sites are located at left-posterior intramural adjacent to endocardium. In both cases, 10 f.L V GWN was added to the BSPMs to simulate noise-contaminated body surface ECG recordings. Fig. 5.4 suggests that, for dual site pacing at two separate locations (A) or adjacent locations (B), the 3D myocardial activation imaging can reconstruct well the activation sequence, although the estimated activation sequence showed little delayed activation as compared with the "true" activation sequence. Effects of heart-torso geometry uncertainties were tested by selecting five pacing sites, in five different regions adjacent to the AV-ring (BA: basal-anterior; BRW: basal-right-wall; BP: basal-posterior; BLW: basal-left-wall; BS: basal-septum). The modified (enlarge or reduce by 10%) torso models or position-shifted heart models (in 4 directions) were used in the forward BSPM simulation, and 10 f.L V GWN was added to the simulated BSPMs. By using the modified heart-torso models in the forward simulations while the standard model in the inverse calculation, the effect of inter-subject geometry variation was initially evaluated. Table 5.1 shows the RE between the simulated "true" ventricular activation sequence and the estimated activation sequence following a single-site pacing. NM refers to normal case, in which only measurement noise is introduced without geometry uncertainty. Note that the heart-torso geometry uncertainty showed little effect on the activation sequence estimation as determined by the RE measure. For example, 2% increase in RE was obtained for the backward shift of the heart along the y-direction (BSY), as compared with the NM case. The estimation errors associated with the 10% enlarged or reduced torso models have averaged REof8%. Effects of conduction velocity of ventricular activation in the heart model were assessed by varying the conduction velocity in the forward heart model. The BSPMs were simulated by using this altered ventricle model and noise was added to simulate noise-contaminated BSPM measurements. The standard heart model, in which the average conduction velocity was used, was then used to estimate the inverse solutions. Fig. 5.5 shows a typical simulation example, with the same format as in Fig. 5.4. GWN of 10 f.L V was added to the BSPMs

174

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Troe

(a)

(0)

(c)

(d)

(e)

(0

FIGURE 5.5. An example of activation time imaging results with variation in the conduction velocity. Same format of display as in Fig. 5.5. The top rows show the simulate d "true " activation sequence with altered condu ction velocity in the forward heart model, and the bottom rows show the inversely estimated activation sequence. The results correspond to 10% increase in con duction velocity following a single-site pacing. See attached CD for color figure. (From He et aI., Phys Mcd & Bioi, 2002 with permission)

to simulate noise-contaminated body surface ECG recordings. The top row shows the simulated "true" activation sequence with altered conduction velocity in the forward heart model, and the bottom row shows the inversely estimated activation sequence, following a single-site pacing when the conduction velocity of the forward heart model was increased by 10%. When the average conduction velocity is different in the forward heart model when compared with that in the heart model used in the inverse procedure, the estimated activation time at specific region s within the ventricles differs from the original activation time distribution in the forward solution. In particular, the early activation moved down toward the apex direction. Nevertheless, the overall distributions of the activation time are not affected substantially. The RE and CC between the "true" and estimated activation sequences within the ventricles are 0.0916/0.106 and 0.996/0 .998, respectively, corre sponding to 5%110% increa se in the conduction velocity.

5.3.6 DISCUSSION In this Section, a new approach for noninvasive 3D cardiac activation time imaging by means of a heart-excitation-model is reviewed. This approach is based on the observation that a priori information regarding cardiac electrophysiology should be incorporated into the cardiac inverse solutions in order to obtain useful information on the 3D cardiac activation from the two-dimensional electrical measurements over the body surface . In this approach, the a priori information on cardiac electrophysiology is incorporated into the heart-excitation-model, which is not an equivalent physical source model but an equivalent physiological source model. By linking this physiological source model with body surface ECG measurements, physiological parameters of interest are estimated from body surface ECG recordings. A unique feature of such an approach is that the rich knowledge we have gained in the forward whole heart modeling (See Chapters 2 and 3) can be directly applied to the 3D cardiac imaging. Furthermore, the anisotropic nature of myocardial propagation can also be incorporated in the 3D myocardial activation imaging , as shown in this Section. Such a priori electrophysiological information serves as constraints when solving the inverse problem, leading to robust 3D inverse solutions.

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Although a cellular-automaton heart-excitation-model has been used for the 3D activation time imaging, it is anticipated that more sophisticated 3D heart-excitation-models (e.g. Gulrajani et al., 2001) can be realized in 3D activation time imaging in the future, providing needed spatial resolution for more complicated cardiac activation sequences. With rapid advancement in computer technology, this goal seems to be more feasible than expected. In parallel to the heart-excitation-model based 3D activation imaging approach, Ohyu et al. (2002) has reported another 3D activation imaging approach in which the action potentials are approximated by a step function with the initial activation being varied from site to site within isotropic ventricles. The distribution of activation time was then connected to the BSPMs and estimated by means of Wiener inverse filter. Both this activation imaging approach (Ohyu et al., 2002) and the dipole source distribution imaging approach (He & Wu, 2001) reviewed in Section 5.2 use linear systems connecting the inverse solution parameters directly with the BSPMs. It would be of interest to evaluate the heart-model-based activation imaging reported by He et al. (2002), with linear system based activation imaging reported by Ohyu et al. (2002).

5.4 THREE-DIMENSIONAL MYOCARDIAL TRANSMEMBRANE POTENTIAL IMAGING In association with activation time, the transmembrane potential (TMP) reflects important electrophysiological properties on the local myocardial tissues. The TMP has been estimated over the heart surface from magnetocardiograms (Wach et al., 1997). Recently, efforts have been extended from the heart surface to the 3D myocardium. He and coworkers reported (2002, 2003) their effort to estimate TMP distribution within the 3D ventricles from body surface electrocardiograms by means of a heart-model based imaging approach. Ohyu et al. has developed an approach to estimate the activation time and approximate amplitude of the TMP from magnetocardiograms using Wiener inverse filter (2002). Skipa et al. reported their initial results on estimation of TMP distribution within the heart from BSPMs (2002). In this Section, we present the heart-mode-based 3D TMP imaging approach and its applications to imaging TMP distributions associated with paced ventricular activity, and acute myocardial infarction. The whole procedure of the heart-model-based TMP imaging approach may also be illustrated in the schematic diagram in Fig. 5.2, except that the reconstructed 3D cardiac sources are spatio-temporal distribution of TMP. Similar to the 3D heart-model based activation imaging, the following procedures are used. A realistic geometry 3D heart-torso-model is constructed based on the knowledge of cardiac electrophysiology and geometric measurements via CTIMRI. The anisotropic nature of myocardium can be incorporated into this computer heart model. The BSPMs are linked with the 3D TMP distribution by means of the heart-torso-model. To reduce the dimensionality of the parameter space, a preliminary classification system (PCS) is employed to classify cardiac status based on the a priori knowledge of cardiac electrophysiology and the BSPM, by means of an ANN. The output of the PCS provides the initial estimate of heart model parameters which are employed later in a nonlinear optimization system. The nonlinear optimization system then minimizes the objective functions that assess the dissimilarity between the "measured" and model-generated BSPMs.

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If the "measured" BSPM and the heart-torso-model-generated BSPM match well, the 3D distribution of transmembrane potentials is determined from the heart model parameters corresponding with the resulting BSPM. If the results do not match, the heart model parameters are adjusted with the aid of the optimization algorithms and the optimization procedure proceeds until the objective functions satisfy the given convergent criteria. When the procedure converges, the TMP distribution throughout the 3D myocardium is determined. The feasibility of imaging 3D TMP distribution has been suggested in computer simulation in paced activities (He et al., 2003). The performance of the above TMP imaging approach was tested by single-site pacing in 24 different sites throughout the ventricles. GWN of 10 J.L V was added to the BSPMs, and GWN of 10 mm was added to the body surface electrode positions, to simulate noise-contaminated body surface ECG recordings. Fig. 5.6 shows the TMP amplitude distributions (a-b) of ventricular depolarization following a single-site pacing at the septum in 5 longitudinal sections within the ventricles (c). The TMP distribution in each longitudinal section at 8 typical instances (from 6 ms to 48 ms with a time step of 6ms) after the onset of pacing is shown in one row. The 5 longitudinal sections within the ventricles arc illustrated in Fig. 5.6(c) by 5 horizontal black lines in the

Layel'-3

•

(a)

6ms

12ms

18ms

24ms

3Jms

35ms

42ms

48ms

FI GURE 5.6. An exam ple of TMP imaging results during single-site ventricular pacing. (a) and (b) illustrate the forward and inverse solution of the TMP distributions in 5 longitudinal sections within the ventricles, duri ng ventricu lar depo larization following a single-site pacing at the septum . The TMP distri bution in each longitud inal section at 8 typical instances (from 6 ms to 48 ms with a time step of 6 ms) after the onset of pacing is shown in one row. The 5 longitudinal sections within the ventricles are illustrated in (c) by 5 horizontal black lines in the transverse section of the ventricles from top to bottom indicating their positions. The gray regions in the longitudina l sections indicate the resting cell units. The Max and Min of color bars correspond to the maximum and minimum values of the TMP amplitude during the first 60 ms from the onset of activation. See attac hed CD for color figure.

Three-Dimensional Electrocardiographic Tomographic Imaging

La

177

3

uyu.1S (b)

6nu

12ms

JSms

(c)

FIGURE 5.6. (cont.)

transverse section of the ventricles from top to bottom indicating their positions. The gray regions in the longitudinal sections indicate the resting cell units . The Max and Min of color bars correspond to the maximum and minimum values of the TMP amplitude during the first 60 ms from the onset of activation . Fig . 5.6 suggests that the inverse TMP distribution captures well the overall spatio-temporal patterns of the forward TMP distribution following a single site pacing at the septum, but with a slight shift of the area of initial activation towards the base (as observed in Layer 3 in the inverse TMP distribution). Averaged over 24 sites for single-pacing, the RE and CC between the "true" and estimated TMP distributions are 0.1266 ± 0.0326 and 0.9915 ± 0.0041 , respectively, indicating that the 3D TMP imaging approach can reconstruct well the TMP distributions within the ventricles corresponding to a well-localized ventricular activati on.

B.He

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(a)

(b)

FIGURE 5.7. A numerical example of myocardial infarction imaging. (a) Green shows preset acute myocardial infarction. (b) Red shows estimated infarcted area over the same layer in the heart model. See attached CD for color figure.

The effects of heart-torso geometry uncertainties on the performance of the 3D TMP imaging approach has also been evaluated with torso size, heart position uncertainties being considered. The 3D TMP imaging approach was found to be robust for up to 10% torso size variation and up to 10 mm heart position shift. See reference (He et al., 2003) for detailed descriptions of the results for TMP imaging of paced activities. The 3D TMP imaging approach has also been applied to image acute myocardial infarction (MI) (Li & He, in press). Fig. 5.7 illustrates an example of myocardial infarction imaging. In this case, GWN of5 f..L V was added to the BSPMs to simulate noise-contaminated body surface ECG recordings. Fig. 5.7(a) shows the "true" MI that is marked by green color, and Fig. 5.7(b) shows the inversely estimated MI that is marked by red color. A "true" MI is located in the middle left wall (MLW) of ventricle and distributes in 7 myocardial layers (not shown here). The estimated MI is located close to the "true" MI site, and has similar shape with the "true" MI, except that some small MI areas occurred in layers adjacent to the 7 "true" MI layers (not shown here). Fig. 5.7 suggests the feasibility of applying the 3D TMP imaging approach to imaging the spatial location and extent of the acute MI in the ventricles .

5.5 DISCUSSION In this Chapter, we reviewed the recent progres s in the development of noninvasive 3D electrocardiography tomographic imaging approaches, including 3D dipole source distribution imaging , 3D activation imaging , and 3D transmembrane potential imaging. These activities may be classified as two general approaches in terms of methodology. One is to solve the system equations connecting electrophysiological characteristics (such as current density, activation time and TMP) to BSPMs. This approach involves solving the system equations using inverse techniques such as weighted minimum norm (He & Wu, 2000,200 I ; Skipa et al., 2002) and Weiner technique (Ohyu et al., 2002). Another approach is to solve the electrocardiography tomographic imaging problem indirectly, by means of a heart-model. In this heart-model based approach, we have developed a localization approach to localize the site of origin of activation from body surface ECG recordings (Li & He, 2001), an activation imaging approach to image the activation time distribution (He et al., 2002) , and an TMP imaging approach to image distribution of transmembrane potential s

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throughout the ventricles from BSPMs (He et al., 2003). The 3D TMP imaging approach has been applied to image dynamic spatiotemporal patterns of activation induced by pacing (He et al., 2003), and to image and localize the site and size of acute myocardial infarction (Li & He, in press). The heart-model based approaches are based on our observation that a priori information regarding the distributed cardiac electrophysiological process should be incorporated into the cardiac inverse solutions in order to obtain useful information on the distributed 3D cardiac electrical activity from the two-dimensional BSPMs. In the present approach, the a priori information on cardiac electrophysiology is incorporated into the distributed heart-model, which is not an equivalent physical source model but an electrophysiological source model, in which knowledge of electrophysiology and pathophysiology is imbedded. The distributed electrophysiological process within the heart is represented by cellular automata, on each of which the site of origin of activation, activation time, or transmembrane potential are determined based on the knowledge of cardiac electrophysiology. In such approaches, since substantial electrophysiology a priori information is incorporated into the inverse solutions, more accurate inverse solutions are anticipated as compared with other approaches without taking this information into account. Such electrophysiology a priori information not only includes more accurate forward solution at each time point, but also a more realistic time-varying dynamics as set by the heart electrophysiology model. Therefore, it is not surprising that good matches between "true" cardiac electrical activity and estimated inverse solutions are obtained by means of the heart model based approaches. On the other hand, the system equation approach has the benefit that there is no need to limit the search space for heart model parameters, as currently being practiced in the heartmodel based approaches. The inverse solutions are obtained directly by solving the system equations that link the electrophysiological properties with BSPMs via biophysical relationships. It would be of interest to compare the performance of these two approaches for 3D electrocardiography tomographic imaging. The inverse problem of electrocardiography has been solved by means of equivalent point sources (dipole localization), distributed two-dimensional heart surface imaging methods (epicardial potential imaging, and heart surface activation imaging), and 3D distributed source imaging approaches. While the 3D distributed source imaging, as reviewed in this chapter, represents an important advancement in the field of electrocardiography inverse problem, all 3D electrocardiography tomographic imaging approaches have only been evaluated, up to date, in computer simulations. It is of ultimate importance and significance to experimentally validate the 3D distributed source imaging approaches, in order to establish electrocardiography tomographic imaging as a useful means for imaging noninvasive three dimensional distribution of cardiac electrical activity, for aiding clinical diagnosis and management of a variety of cardiac diseases, and for guiding radio-frequency catheter ablative interventions.

ACKNOWLEDGEMENT The author wishes to thank his postdoctoral associates and graduate students, Dr. Guanglin Li, Dr. Dongsheng Wu, and Xin Zhang, with whom this work was conducted. This work was supported in part by NSF BES-0201939, a grant from the American Heart Association #0140132N, and NSF CAREER Award BES-9875344.

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6

BODY SURFACE LAPLACIAN MAPPING OF BIOELECTRIC SOURCES Bin He* and lie Lian Department of Bioengineering, University of Illinois at Chicago

6.1 INTRODUCTION 6.1.1 HIGH-RESOLUTION ECG AND EEG Targeting two of the most life-critical organs, the heart and brain, the electrocardiogram (ECG) and the electroencephalogram (EEG) are the two important bioelectric recordings to study the cardiac and neural activity. Conventional ECG and EEG have many advantages. First, they are noninvasive measurement. Second, they are very convenient for application and have relatively low cost. More importantly, they have unsurpassed millisecond-scale temporal resolution, which is essential for revealing rapid change of dynamic patterns of heart and brain activities. However, the major limitation of the conventional ECG and EEG is their relatively low spatial resolution as compared to some other imaging modalities, such as the computed tomography (CT) or the magnetic resonance imaging (MRI). One reason contributing to the low spatial resolution is the limited spatial sampling. Conventional EEG uses the standard international 10-20 system, which has about 20 electrodes over the scalp, with corresponding inter-electrode distance of about 6 em (Nunez et al., 1994). For the ECG measurement, the most commonly used configuration in a clinical setting is the 12-lead ECG. Despite its great success in many clinical applications, it has a major limitation in that it contains very little spatial information, and doctors have to infer the cardiac status mainly based on temporal analysis of the ECG waveforms. Therefore, one way to enhance the spatial resolution of ECG and EEG is to increase the spatial sampling, by using larger number of surface electrodes in ECG and EEG measurement. *Address all correspondence to: Bin He, Ph.D. University of Minnesota, Department of Biomedical Engineering, 7-105 BSBE, 312 Church Street, Minneapolis, MN 55455. E-mail: [email protected]

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However, even with very high-density spatial sampling, the spatial resolution of the EEG and ECG is still limited, because of the volume conduction effect. In other words, the electrical signals will get smeared as they pass through the media between the bioelectric sources and the body surface sensors. For the brain, it's the head volume conductor, particularly the skull layer, which has low conductivity (Nunez, 1981, 1995). For the heart, it's the torso volume conductor, including the effects of lungs, the ribs and other tissues (Mirvis et al., 1977; Spach et al., 1977; Rudy & Plonsey, 1980). Therefore, advanced techniques are desired in order to compensate for the volume conduction effect and enhance the spatial resolution of the ECG and EEG. As reviewed in Chapters 4 and 5 with applications to the heart, one of such methods is to solve the so-called inverse problem, which attempts to estimate the bioelectric sources from the body surface potential measurements. Another method is the surface Laplacian, which will be thoroughly discussed in this chapter.

6.1.2 BIOPHYSICAL BACKGROUND OF THE SURFACE LAPLACIAN The concept of the Laplacian originated centuries ago, and the Laplacian operator has been widely used in digital image processing as a spatial enhancement method. Similarly, the Laplacian technique can also be used for high-resolution bioelectric mapping. By definition, the surface Laplacian (SL) is defined as the 2nd order spatial derivative of the surface potential. Due to its intrinsic spatial high-pass filtering characteristics, the SL can reduce the volume conduction effect by enhancing the high-frequency spatial components, therefore can achieve higher spatial resolution than the surface potentials (Figure 6-1). Consider the non-orthogonal curvilinear coordinate system on a general surface Q, u = x, v = y, and z = f (u, v), where f(u, v) is a continuous function whose 2nd order partial derivatives exist. If V(u, v) is the analytical surface potential function (whose 2nd order partial derivatives exist) on Q, the SL of V (u, v) can be written in tensorial formulation (Courant & Hilbert, 1966; Babiloni et al., 1996): Vs2 V

1 =.y'g

{aau ["fi (av av)] + ava ["fi (av av)]} gIla-;; + g12a;;g21a-;; + g22a;;-

(6-1)

where the components of the metric tensor are given by (Babiloni et al., 1996): (6-2a)

(6-2b)

af af au av

---

gl2 = g21 =

g22

=

g

1+(~r g

(6-2c)

(6-2d)

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FIGURE 6-1. Schematic illustration of the SL as a spatial enhancement method. The cardiac activity located at the anterior apex (black circle) is sensed by potential measurement over the larger area on the chest (light grey) but by Laplacian measurement over the smaller area on the chest (dark grey). (From Tsai et aI., Electromagnetics, 2001 with permission)

For the plane model where z = j(u, v) = 0, u = x and v = y, the SL is reduced to: (6-3)

For the sphere model, assume by (Perrin et al., 1987a):

z=

.Jl - u 2 - v 2 ,

U

= x and v = y, the SL is then given

(6-4) The Laplacian electrogram (we refer electrogram to either ECG or EEG when the heart or the brain is concerned) shall be defined as the negative SL of the surface potential electrogram (He, 1999; He & Wu, 1999), to facilitate the interpretation of the Laplacian maps in comparison to the potential maps. As stated in equation (6-3), assuming a planar surface in the vicinity of the observation point, a reasonable approximation of the local area of the body surface would be the tangential plane at the point of interest, over which a local Cartesian coordinate system (x, y, z) can be considered. Assuming z to be normal to the tangential plane, the Laplacian ECGIEEG at the observation point becomes (He, 1999; He & Wu, 1999):

where J denotes the current density and Jeq is an equivalent current source (He & Cohen, 1992a, 1995; He, 1997, 1998a, 1999; He & Wu, 1999). Unlike the ECG and EEG inverse problems, the SL approach does not attempt to locate the bioelectric sources inside the heart and brain. Instead, the Laplacian ECGIEEG

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can be viewed as a two-dimensional (2D) projection of the three-dimensional (3D) bioelectric source onto the body surface. Therefore, as shown in equation (6-5), the Laplacian ECG/EEG can be interpreted as an equivalent current source density on the body surface, which has the similar physical units as the primary bioelectric source density. On the other hand, compared to the ECG and EEG inverse approaches, the SL approach does not require exact knowledge about the conductivity distribution inside the torso and head volume conductors and has unique advantage of reference-independence as compared with the potential measurement.

6.2 SURFACE LAPLACIAN ESTIMATION TECHNIQUES 6.2.1 LOCAL LAPLACIAN ESTIMATES If the body surface in the vicinity of the measurement point can be approximately represented by a planar surface, the SL can be calculated using equation (6-3). In practice, the second order derivatives can be approximated by means of finite difference (Hjoth, 1975). Consider a grid of unipolar electrodes with equal inter-electrode distance b on the body surface (Figure 6-2A), the regular Laplacian electrogram at each non-boundary electrode (the cross-hatched circle) can be estimated by using the regular finite difference representation (Hjoth, 1975; He & Cohen, 1992a; Wu et al., 1999; Lian et al., 2002):

LR(i, j)

~~

{V(i, j) -

~ [V(i -

1, j) + V(i

+ 1, j) + v«, j

- 1) + V(i, j

+ I)]} (6-6)

where V(i, j) and LR(i, j) represent the potential and the regular Laplacian electrogram at the electrode (i, j), respectively. For each non-boundary electrode, equation (6-6) uses the potential measurement at five electrodes (the cross-hatched circle and its four neighboring open circles in Figure 6-2A) to estimate the Laplacian electrogram at the center electrode (the cross- hatched circle). Similarly, the Laplacian electrogram at electrode (i, j) can also be estimated from the potential recorded from this electrode and those recorded from its other four neighboring electrodes in the diagonal direction (neighboring black circles surrounding the cross-hatched circle in Figure 6-2A). Denote the distance from the center electrode to its diagonal neighboring electrodes as d (for uniform grid, d = .fib), the diagonal Laplacian electrogram can also be estimated by (Wu et al., 1999; Lian et al., 2002):

LD(i, j)

~

:2

{V(i, j) -

~[V(i -

1, j - I) + V(i - 1, j

+ 1)

+V(i+l,j-l)+V(i+l,j+I)]} where L D(i, j) represents the diagonal Laplacian electrogram at the electrode (i, j).

(6-7)

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••••••••••••••• •••••••••••••••

,.

•• 0 ••••••••••••

• { •, •

• 0 0 ••••••••••• •• 0 ••••••••••••

••••••••••••••• ••••••••••••••• ••••••••••••••• ••••••••••••••• ••••••••••••••• A

....... .

'

.

• •• •I

.-.--

./

••

B

FIGURE 6-2. Schematic illustration of the local Laplacian estimates. (A) Regular or diagonal 5-point local Laplacian estimation. (B) Circular finite difference local Laplacian estimation.

A more general form of the finite difference reprentation of the SL utilizes the potential information from more local electrodes to realize the circular Laplacian electrode (He & Cohen, 1992a). As illustrated in Figure 6-2B, to estimate the Laplacian electrogram at the center electrode, the unipolar potential data are obtained from this electrode as well as from n electrodes located along a small circle (with radius r) surronding it, and the finite difference representation of the Laplacian electrogram is given by (Le et aI., 1994; Wei et al., 1995; He, 1997, 1998a, 1999; Wei & Mashima, 1999; Wei, 2001):

4 ( Vo u, ~ 2"

r

n ) -1 LVi

n

(6-8)

i=!

where Vo and La represent the potential and circular Laplacian electrogram at the center electrode, respectively, and V;(i = 1, 2, ... , n) represents the potential at one of the surrounding electrodes. Another local Laplacian estimate uses bipolar concentric electrode that consists of two parts: a conductive disk at the center and a surrounding conductive ring (Fattorusso et al., 1949; He & Cohen, 1992a). In the bipolar approach, the Laplacian electrogram may be estimated as (He & Cohen, 1992a, 1995; He, 1997):

Lo

~ i2 (Vo __1_ 1. Vdl) r

2nr

r

(6-9)

where the integral is taken around a circle of radius r. In addition, some other local-based Laplacian algorithms were proposed in order to achieve more acurate numerical estimates, for instance, by local modeling of the scalp and the potential distribution (Le et aI., 1994), or by means of the local polynomial fitting (Wang & Begleiter, 1999).

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6.2.2 GLOBAL LAPLACIAN ESTIMATES From equations (6-1) and (6-2), it can be seen that both the analytical models of the potential distribution V (u, v ) and the surface geometry f eu, v) are required to calculate the SL. However, in real applications, the only data available are the limited number of potential recording s and body surface geometri c coordinate samplings. Thus, V (u , v) and f eu, v ) must be interpolated or approximated , and the most widely used interpolation scheme is the spline interpolation. Among the investigations on the spline SL, of noteworthy is the spherical spline SL (Perrin et al., 1987a,b, 1989), the ellipsoidal spline SL (Law et al., 1993), and the realistic geometry spline SL (Babiloni et al., 1996, 1998; He et al., 2001, 2002 ; Zhao & He, 2001). Considering the non-planar shape of the body surface , a global SL estimation using spline technique will be more accurate than the local-based SL estimates. Furthermore, as a major advantage over the local-based SL estimation, the spline SL has been shown to provide a more robust characteristic against noise (Perrin et al., 1987a; Law et aI., 1993; Babiloni et al., 1996; He et al., 2001, 2002). In the following, a recently developed realistic geometry spline Laplacian estimation algorithm is presented (He et al., 2001, 2002). Estimation of the parameters associated with the spline Laplacian is formulated by seeking the general inverse of a transfer matrix. The number of spline parameters, which need to be determined through regularization, is reduced to one in the present approach , thus enabling easy implementation of the realistic geometry spline Laplacian estimator.

6.2.2.1 Spline interpolation ofthe surface geometry Given body surface sampling point s (Xi, Yi, z.) . i = 1, ... , M, where M is the number of sampling points for coordinate measurement, the mathematical model of the body surface geometry Z = f (x , y) can be described by 2D thin plate spline (Harder & Desmarais, 1972; Perrin et al., 1987a,b; Babiloni et al., 1996; He et al., 2001, 2002): M ~

M ~

i=1

i=l

2(m- l)

z = f (x , Y ) = ~ Pi Km - I + Qm - l = ~ p;d;

m- I

d

( 2 2) ~ ~ d-k k log d +W + ~ ~ qdk x Y i

d=Ok=O

(6-10)

where m (spline order) is set to 2 (Perrin et aI., 1987a,b; Babiloni et aI., 1996; He et aI., Qm - I are basis function and osculating function , respectively, and w is a constant which accounts for effective radius of the recording sensor (Harder & Desmarais, 1972; Perrin et al., 1987a). The coefficients Pi and qdk are the solutions of following matrix equation s (Duchon, 1976; Perrin et aI., 1987a,b; He et al., 2002):

2001,2002 ), d; = (x - Xi)2 + (y - Yi)2, Km- I and

KP +EQ=Z

(6-11a)

ETp =0

(6- 11b)

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Body Surface Laplacian Mapping of Bioelectric Sources

where P, Q, and Z are the vectors containing Pi , qdk. and Zi respectively, the matrices K and

E are composed of elements of basis function and sampling coordinates, respectively. 6.2.2.2 Spline interpolation ofthe surface potential distribution Similarly, given body surface potential recordings Vi at positions ( Xi, Yi, z. ), i = 1, 2, . . . , N, where N is the number of recording electrodes, the body surface potential distribution over the 3D space at an arbitrary point (x, y, z) can be modeled by the 3D spline (Babiloni et al., 1996, 1998; He et aI., 2001 , 2002): N

V (x,y, z )

= ~ti ~

H

m-I

+

R

m-I

N

= ~tiri ~

i=1

(2m- 3)/2

m- I d k ~~~

+ ~~~rdkgX d-k Yk- gZ g

(6-12 )

d= O k=O g= O

i=1

where m (spline order) is set to 3 (Law et aI., 1993; Babiloni et aI., 1996, 1998; He et aI., 2001, 2002), rl = (x - Xi)2 + (y - Yi)2 + (z - Zi)2, Hm - l and R m-l are basis and osculating functions, respectively, and the coefficients t i and r dkg can be determined by solving the matrix equations (Law et aI., 1993; Babiloni et aI., 1996; He et al., 2002): (6-13a) (6-13b ) where T, R, and V are the vectors containing t., r dkg, and V i , respectively, the matrices H and F are composed of elements of basis function and electrode coordinates, respectively.

6.2.2.3 Determination ofthe spline parameters In an attempt to overcome the ill-poseness of the systems, approximation instead of interpolation of the surface geometry and potential distribution are used by introducing correction terms in equations (6-11a) and (6-13a), which are respectively changed to (Babiloni et aI., 1996, 1998; He et aI., 2001, 2002):

+ wI)P + EQ = Z ( H + AI)T + F R = V

(K

(6-14a) (6-14b)

where 1 is the identity matrix, parameters wand A. are used to improve the numerical stability of the systems. The optimal values of these two parameters need be determined separately by either "tuning procedure" or other regularization techniques (Babiloni et al., 1996, 1998). Instead of searching the optimal parameters in two dimensions, the above equations can be reformulated by combining equations (6-11a,b) and (6-13a,b) into one linear system equation (He et aI., 2001 , 2002):

AX=B

(6-15)

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where

A-

[ff,

E

0 0 0

0

0

x= [P B = [Z

Q T 0

V

0 0

~] Rf

(6-16a)

H

FT

(6-16b)

O]T

(6-16c)

Then the problem becomes seeking the solution of equation (6-15). Applying the concept of the general inverse, we have (He et al., 2001, 2002): (6-17) where A# is the pseudo-inverse of A. Matrix A is ill-posed, thus regularization methods must be used to improve the stability of the system. Notably, after reformulating the matrix equations into one unified linear system in equation (6-15), only one single regularization parameter needs to be determined when seeking the general inverse A #. Therefore, the present method not only can significantly reduce the computation effort and improve the efficiency and stability of the spline SL algorithm, but also can be combined with many regularization techniques which have been extensively studied to determine the parameter in seeking the general inverse A# (for details on the regularization techniques, see Chapter 4). 6.2.3 SURFACE LAPLACIAN BASED INVERSE PROBLEM

The SL-based ECG or EEG inverse problem has also been explored to achieve highresolution heart or brain electric source imaging. One of the approaches is to estimate the epicardial potentials from the body surface Laplacian ECG (He, 1994; Wu et al., 1995, 1998; He & Wu, 1997, 1999; Johnston, 1997; Throne & Olson, 2000), or estimate the cortical potentials from the scalp Laplacian EEG (He, 1998; Babiloni et al., 2000; Bradshaw & Wikswo, 2001). As illustrated in Figure 6-3, if Vis an isotropic homogeneous volume conductor surrounded by an outer surface S 1 and an inner surface Sz, and there is no current source existing within V, the potential on the inner surface can be related to the potential or Laplacians on the outer surface. Applying Green's second identity to the volume V results in (Barr et al., 1977): U("'l r

= -.L ~

If

51

U .

dQ - -.L ~

If

52

u . dQ - -.L ~

If

1.r . 2E..dS ~

(6-18)

52

where u(r'l-the electrical potential at the observation point r* dQ-the solid angle of an infinitesimal surface element ds as seen from r* -the first derivative of potential u with respect to the outward normal to dS aau r;

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Body Surface Laplacian Mapping of Bioelectric Sources

Sl

FIGURE 6-3. Schematic illustration of the arbitrarily shaped volume conductor.

By discretizing the surfaces Sl and S2 into triangular elements, and taking the limit of observation point approaching the surface element on St and S 2 , respectively, from the inside of V, the following matrix equations can be obtained: ~

PIIU, ~

P21U j

-

+ P'2U2 + G I2r2 = 0 + P22U2 + G22r2 = 0 ~

(6-19) (6-20)

where U k is the vector consisting of the electrical potentials at every surface element on Si , and r k is the vector consisting of the normal derivatives of the electrical potentials at every triangle element on Sk but just inside of VI. P lI , P 12, P21 , P22, G12, and G22 are coefficient matrices (Barr et aI., 1977). Solving equations (6-19) and (6-20) leads to the followingequation that relates the inner surface potential U 2 to the outer surface potential U,: (6-2 1)

where Tl2 = (P lI - Gl 2G221 P2d-l(G 12G221 P22 - P12 ). The surface Laplacian of the potential at the position r *at the outer surface Sl can be written as follows (Wu et aI., 1998; He & Wu, 1999): L s(r4* ) = -

1 4rr

If

2

(a

Q U • d a,;T

Ir*) +

1 4rr

SI

If

2

U •

I )+ 1

Q r* d (aan2

S2

4rr

If

2

I)

au · ( an2( a ,:) 1 r* dS ar.

S2

(6-22) where n is the normal direction of the surface Sl at r*. Similarly, by discretizing the surfaces Sl and S2 into triangular elements, the following matrix equation can be obtained: ........

....

--->.

......

Ls=AU,+BU 2+Cr 2

(6-23)

192

B. He and

J. Lian

~

where L, is the vector consisting of the surface Laplacians at every surface element on S" and A, B, and C are coefficient matrices (Wu et aI., 1998; He & Wu, 1999). From equations (6-19) , (6-:?l) , and (6-23 ), we can relate the inner surface potential U 2 to the outer surface Laplacian L, by transfer matrix H: (6-24) where H = A · T l2 + B - C · G 22' . ( P2, . Tl2 + P22) The potential-based inverse problem seeks the inner surface potentials from the outer surface potentials by solving equation (6-2 1): (6-25) On the other hand, the SL-based inverse problem seeks the inner surface potentials based on solving equation (6-24 ): (6-26) where # denotes the general inverse of the transfer matrix. In addition, the hybrid potential-Laplacian-based inverse solution can also be solved by minimizing the error function (He & Wu, 1999; Throne & Olson, 2000): (6-27) where ex is a weighting coefficient. The resulting inverse solution is given by (He & Wu, 1999): (6-28) Equation (6-28) suggests that the inner surface potentials can be estimated from both the outer surface Laplacians and out surface potentials.

6.3 SURFACE LAPLACIAN IMAGING OF HEART ELECTRICAL ACTIVITY 6.3.1 HIGH-RESOLUTION LAPLACIAN ECG MAPPING By applying the SL technique to the potential ECG , body surface Laplacian mapping (BSLM) was first proposed by He and Cohen (l992a,b). Theoretical and experimental studies have been carried out, demonstrating the unique feature of BSLM in effectively reducing the torso volume conduction effect and enhancing the capability of localizing and mapping multiple simultaneously active myocardial electrical events (He & Cohen,

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1992a,b; He et aI., 1993, 1995, 1997,2002; Oostendorp & van Oosterom, 1996; Umetani et aI., 1998; Wei & Harasawa, 1999; Wu et aI., 1999; Tsai el aI., 2001; Besio et al., 2001 ; Wei et aI., 2001 ; Li et aI., 2002 ).

6.3.2 PERFORMANCE EVALUATION OF THE SPLINE LAPLACIAN ECG Through human experiments and computer simulations, we have systematically evaluated the signal to noise ratio of the Laplacian ECG , during ventricular depolarization and repolarization, and demonstrated the feasibility of recording the Laplacian ECG, using the 5-point local SL estimator (Wu et aI., 1999; Lian et aI., 2001 ,2002). Further improvement of the Laplacian ECG estimation may be achieved by using the global-based spline Laplacian technique. In this section, we present the performance evaluation of the 3D spline SL algorithm (see Section 6.2.2). Computer simulations were conducted using both a spherical model and a realistic geometry heart-torso model, and comparison studies were also made with the 5-point local SL estimator (He et aI., 2002). Given the torso surface geometry coordinates and the potential measurement, the Laplacian ECG was estimated by using the realistic geometry spline SL algorithm as detailed in Section 6.2.2. The linear inverse problem in equation (6-17) was solved by using the truncated singular value decomp osition (TSVD) (Shim & Cho , 1981), and the truncation parameter was determined by means of the discrepancy principle (Morozov, 1984).

6.3.2.1 Effects of noise We first evaluated the effects of noise on the SL estimation, by approximating the torso volume conductor as a homogeneous single-layer unit-radius sphere model , with normalized interior conductivity of 1.0. A radial dipole or a tangential dipole was used to represent a localized cardiac electrical source. Three eccentricities (0.5, 0.6, 0.7) were used to assess the effect of the source depth on the SL estimation. The Gaussian white noise (GWN) of different noise levels (5%, 7%, 10%) was added to the dipole-generated surface potentials sampled from 129 surface electrodes, simulating noise-contaminated potential ECG measurement. Two cases of geometry noise were also considered (2% geometry noise plus 10% potential noise, and 5% geometry noise plus 5% potential noise). For each noise level, ten trials of noise were generated and simul ations were conducted. The correlation coefficient (cq values between the estimated SL and the analytical SL for all ten trials were averaged and shown in Table 6-1. The SL was estimated by three different methods : (1) 5-point local SL (5PL), (2) two-parameter spline SL (2SL), and (3) the recently developed one-parameter spline SL (1SL). The two-parameter spline SL was estimated optimally by using the tuning procedure (Babiloni et aI., 1996, 1998) to find the optimal values of (J) and A in equation (6-14) and the one-parameter spline SL was estimated optimally by searching the optimal truncation parameter in TSVD procedure (He et aI., 2002). The optimal parameters correspond to the maximum CC between the analytical SL and the estimated SL. The quant ity w in equation (6-10) was set to 0.16 in spline SL estimation.

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TABLE 6-1. The CC values between the analytical and estimated SL under different levels of noise in one-sphere model GWN I-RD, r=0.5 I-RD, r = 0.6 I-RD, r= 0.7 I-TD, r = 0.5 I-TD, r = 0.6 I-TD, r = 0.7

5%PN 5PL 2SL ISL

7%PN 5PL 2SL ISL

10%PN 5PL 2SL ISL

2% GN+IO%PN 5%GN+5%PN 5PL 2SL ISL 5PL 2SL ISL

0.79 0.90 0.96 0.71 0.84 0.94

0.73 0.85 0.93 0.62 0.78 0.91

0.63 0.77 0.89 0.50 0.67 0.85

0.58 0.58 0.76 0.65 0.60 0.71

0.98 0.97 0.96 0.98 0.95 0.96

0.96 0.98 0.96 0.96 0.98 0.96

0.97 0.96 0.95 0.97 0.95 0.95

0.95 0.97 0.95 0.95 0.95 0.95

0.96 0.95 0.93 0.96 0.94 0.91

0.94 0.95 0.92 0.94 0.95 0.93

0.93 0.91 0.89 0.90 0.92 0.88

0.92 0.93 0.90 0.91 0.92 0.88

0.72 0.70 0.80 0.67 0.66 0.79

0.91 0.91 0.92 0.88 0.92 0.91

0.90 0.92 0.93 0.90 0.92 0.92

Note: RO-radial dipole, TO-tangential dipole, PN-potential noise, GN-geometry noise.

Three findings are obvious from Table 6-1. First, for all three different SL estimators, the higher the noise level, the smaller the Cc. Second, for all the cases studied, the spline SL has superior performance (higher CC) than the 5-point local SL, while the two-parameter spline SL and one-parameter spline SL have similar performance. Third, the 5-point local SL has the best performance for superficial sources and under low potential noise level, but its performance degrades dramatically as the source moves to deeper position or under higher noise levels. On the other hand, the spline SL generally has good performance over a broader source depths (from 0.5 to 0.7), and shows more robust characteristics against the noise in potential measurement. Specifically, for the one-parameter spline SL estimator, the CC values for all cases studied are greater than 0.92 under 10% potential noise, and equal or greater than 0.90 under 5% potential noise plus 5% geometry noise.

6.3.2.2 Effects ofnumber ofrecording electrodes Table 6-2 shows the effects of number of recording electrodes on the SL estimation. One or multiple dipoles with varying orientations were placed in the spherical conductor model (see note under Table 6-2 for dipole configurations), and 5% GWN was added to the dipoles-generated potentials. The CC values between the analytical SL and the estimated SL with different electrode numbers and different dipole configurations are shown in Table 6-2. Similarly, three different SL estimators were evaluated and compared. The quantity w in equation (6-10) was set to 0.16, 0.18, 0.20, and 0.24 corresponding to 129,96,64, and TABLE 6-2. The CC values between the analytical and estimated SL corresponding to different electrode numbers and dipole configurations in one-sphere model Electrode Number

129 5PL 2SL ISL

96 5PL 2SL ISL

64 5PL 2SL ISL

32 5PL 2SL ISL

Config. A Config. B Config. C Config. D Config.E

0.84 0.90 0.77 0.87 0.74

0.78 0.92 0.82 0.88 0.75

0.88 0.92 0.80 0.88 0.77

0.72 0.85 0.59 0.49 0.47

0.95 0.97 0.96 0.96 0.94

0.98 0.98 0.99 0.98 0.98

0.97 0.97 0.94 0.97 0.89

0.97 0.98 0.97 0.95 0.94

0.96 0.97 0.94 0.97 0.92

0.97 0.97 0.96 0.90 0.92

0.95 0.96 0.92 0.73 0.67

0.84 0.96 0.88 0.71 0.83

Note: Configurations A: I-TO at r= 0.6; B: I-RO at r = 0.6; C: two +z-direction dipoles at (±0.3, 0.0, 0.5); 0: one -l-x-direction dipole at (0.0, 0.0, 0.7) and two -l-z-direction dipoles at (0.0, ±OA, 0.5); E: 4-RO at r = 0,6, each one is ttl] with respect to the z-axis. RO-radial dipole, TO-tangential dipole.

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32 electrodes, respectively. The spline SL was estimated optimally by seeking the optimal regularization parameter(s). Table 6-2 clearly indicates the correlation between the goodness of SL estimation and the number of surface electrodes. In general, the more electrodes being used, the higher CC of the SL estimation. The CC values drop significantly when 32 electrodes are used, which is consistent with the fact that a minimum sampling in the space domain is needed to restore the spatial frequency spectrum. Again, Table 6-2 indicates that two-parameter spline SL and one-parameter spline SL have comparable performance, and are more robust against measurement noise than the 5-point local SL estimation.

6.3.2.3 Effects ofregularization In simulations, the optimal SL can be estimated by means of a priori information of the analytical SL, i.e., by seeking the optimal parameter that maximizes the CC between the analytical SL and the estimated SL. In real applications, the SL can also be estimated without using the a priori information of the analytical SL, for example, by using the discrepancy principle (Morozov, 1984). In Table 6-3, the upper rows show the CC between the analytical SL and the optimal estimated SL. The lower rows show the CC between the analytical SL and the estimated SL obtained by means of the discrepancy principle, without the a priori information on the analytical SL. In this simulation, multiple dipoles with varying orientations were placed in the spherical conductor model (see note under Table 6-3 dipole configurations), and 5% GWN was added to the analytical surface potentials sampled at 129 recording electrodes. The quantity w in equation (6-17) was set to 0.16 in spline SL estimation. Table 6-3 indicates that the regularization results always have lower CC than the optimal results (by definition). However, the results obtained via regularization (by using the discrepancy principle in this case) are comparable to the optimal SL estimates. Out of four source configurations, the CC values for configurations B and D are almost similar for these two types of results. For Configuration C, the CC of the regularization is smaller than the optimal result by 1%. For configuration A, the CC of the regularization result is smaller than the optimal result by less than 3%. However, the absolute CC is 96% or above, suggesting the feasibility of the estimation of the SL through regularization. Figure 6-4 depicts one typical example of the normalized surface potential maps and the SL maps corresponding to source Configuration C in Table 6-3. In this figure, (A) is the noise-contaminated surface potential map, (B) is the analytical Laplacian ECG map, (C) is the optimal spline Laplacian ECG map estimated by means of a priori information, (D) is TABLE6-3. Comparison of the optimalestimated spline SL and the spline SL estimated by using the discrepancy principle in one-sphere model Dipole Configuration

A

B

C

D

Optimal spline SL Regularized spline SL

0.99 0.96

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0.98 0.97

0.98 0.98

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the spline Laplacian ECG map estimated by means of the discrepancy principle, and (E) is the Laplacian ECG map estimated by the 5-point SL estimator. Figure 6-4 indicates that, from the viewpoint of imaging and mapping, the regularization spline SL estimate is almost identical to the optimal spline SL estimate, and similar to the analytical SL result, for the case studied. This is consistent with the high CC values obtained (Table 6-3) between the analytical SL, the optimal spline SL, and the regularization spline SL estimates. Also noted, the 5-point local SL is more sensitive to the measurement noise as compared to the spline SL, especially at the border regions.

6.3.2.4 Simulation in a realistic geometry heart-torso model The performance of the present 3D spline SL estimator was further examined using a realistic geometry heart-torso model, where cardiac electric activity was simulated by pacing one or two sites in the ventricles of the heart model (He et al., 2002) (Figure 6-5A). The potential ECG induced by ventricular pacing was simulated by means of the boundary element method (Aoki et al., 1987). The GWN was added to the simulated surface potentials to simulate noise-contaminated potential BCG measurement. The Laplacian ECG was estimated from the noise-contaminated potential ECG using the 3D spline SL algorithm, and comparison was also made with the conventional 5-point local SL estimation. Figure 6-5B depicts the simulation results when simultaneously pacing two sites (site #1 at the free wall of right ventricle, site #2 at the ventricular anterior) in the ventricular base, and the single site pacing results corresponding to these two sites are shown in Figure 6-5C and Figure 6-5D, respectively. In these figures, (i) shows the activation sequence inside the ventricles induced by the pacing. (ii) shows the 5% GWN contaminated body surface potential map over the anterior chest immediately following the pacing. (iii) and (iv) respectively show the estimated body surface Laplacian ECG maps over the anterior chest, by using the 5-point local SL estimator and the one-parameter spline SL estimator. Note that in Figure 6-5B, the estimated Laplacian ECG maps provide multiple and more localized areas of activity overlying the two pacing sites, whereas the body surface potential map does not reveal the spatial details on this source multiplicity, due to the smearing effect of the torso volume conductor. Also noted, that the spline SL estimate is less noisy and can separate the two areas of activity more efficiently than the 5-point local SL estimate. In Figures 6-5C-D, both the potential ECG and Laplacian ECG maps corresponding to single site pacing reveal one pair of negative/positive activity, while the Laplacian ECG maps

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provide much more localized spatial pattern. The two pairs of negative/positive activities revealed in Figure 6-5B correspond well to the activities observed in Figures 6-5C-D. Consistently, the 5-point local SL estimates are noisier than the spline SL estimates.

6.3.2.5 Spline Laplacian ECG mapping in Humans Applying the spline Laplacian algorithm we have developed , body surface Laplacian mapping has been explored in a group of healthy male subjects during ventricular and atrial depolarization. Ninety-five channel body surface potential ECG was recorded simultaneously over the anterolateral chest in the subjects. The Laplacian ECG was estimated from

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the recorded potentials during QRS complex and the P-wave by means of the one-parameter spline SL estimator. For all subjects, more spatial details were observed in the SL ECG maps as compared with the potential ECG maps, with spline SL more robust against noise than the 5-point SL (Li et al., 2003). Figure 6-6 shows one example of the SL ECG map over the anterolateral chest of a healthy male subject around the peak of R-wave (Figure 6-6A). Figure 6-6B shows the potential map, which shows a pair of positivity and negativity over the anterolateral chest. The corresponding spline BSLM map is shown in Figure 6-6C, illustrating a localized negative activity, N2, located over the central chest, a positive activity P2 slightly shifted toward the left lateral chest with respect to the position of N2, another positive activity to the left of P2, and another negative activity N3 appeared in left-superior area. Figure 6-6D shows the SL ECG map estimated using the 5-point local SL estimator. Note that the local 5-point SL estimate (Figure 6-6D) shows more focused activities as compared with the potential map (Figure 6-6B), but failed to reveal the spatial details as illustrated in the spline SL map (Figure 6-6C). The negative and positive activities observed in the group of human subjects have been related to the epicardial events (Li et al., 2003). Figure 6-7 shows an example of spline SL mapping in a healthy human subject during atrial depolarization (Lian et al., 2002b). Compared with the diffused potential map (Figure 6-7B), the corresponding spline BSLM map (Figure 6-7C) clearly shows two major positive activities, PI and P2, representing the local maxima on the right and left anterior chest,

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respectively. Correspondingly, three associated negative activities can also be observed and denoted as Nl , N2, and N3, representing the local minima on right, middle and left chest separated by PI and P2, respectively. Data analysis andcomputer simulation studies suggest that the positivities PI and P2 may correspond to the activation wavefronts in the right and left atria, respectively (Lian et al., 2002b). Compared to the smooth pattems of the BSPMs, more spatial details are revealed in the BSLM maps during ventricular and atrial activation, which could be correlated with the underlying multiple myocardial activation wavefronts. 6.3.3 SURFACE LAPLACIAN BASED EPICARDIAL INVERSE PROBLEM

The feasibility on solving the ECG inverse problem by means of the Laplacian ECG (He, 1994; Wu et aI., 1995, 1998; He & Wu, 1997, 1999; Johnston, 1997), and a hybrid approach using both potential ECG and Laplacian ECG for epicaridial inverse problem (He & Wu, 1999;Throne & Olson, 2000) have also been explored (see Section 6.2.3). As an example, Figure 6-8 shows the simulation results for testing the feasibility of the Laplacian ECG based epicardial potential inverse solution in a realistically shaped heart-torso model. Current dipoles located inside the anterior myocardium pointing from endocardium to the epicardium were used to simulate anterior sources. To simulate the noise-contaminated experimental recordings, up to 20%GWN wasadded to the heart-model generated body surface ECG signals before the epicardial potentials were reconstructed. Figure 6-8 plots the relative error (RE) and CC between the epicardial potentials calculated from two anteriordipoles using the boundary element method, and the epicardial potentials

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reconstructed from the noise-contaminated potential ECG and surface Laplacian ECG over the whole torso. Note that the Laplacian ECG based epicardial inverse solutions always provide smaller RE as compared with the potential ECG based epicardial inverse solutions for the same noise level. For a larger noise level of 20% in the Laplacian ECG, the Laplacian ECG based epicardial inverse solutions still show a comparable performance as compared with the potential ECG based inverse solutions at a lower noise level of about 3%. Due to increased noise level in Laplacian ECG as compared with the potential ECG, the real merits of the Laplacian ECG based inverse solution would depend on how accurate one may record or estimate the Laplacian ECG in an experimental setting. The data comparing the epicardial inverse solutions obtained from the body surface potentials with those obtained from the body surface Laplacians, which are estimated from the noise-contaminated potentials, are currently lacking in the literature.

6.4 SURFACE LAPLACIAN IMAGING OF BRAIN ELECTRICAL ACTIVITY 6.4.1 HIGH-RESOLUTION LAPLACIAN EEG MAPPING As a spatial enhancement method, the SL technique has been applied for many years to high-resolution EEG mapping. The SL has been considered an estimate of the local current density flowing perpendicular to the skull into the scalp, thus it has also been termed current source density or scalp current density (Perrin et aI., 1987a,b; Nunez et aI., 1994). In addition, the relationship between the SL and the cortical potentials has also been investigated (Nunez et aI., 1994; Srinivasan et aI., 1996). Since Hjorth's early exploration on scalp Laplacian EEG (Hjorth, 1975), a number of efforts has been made to develop reliable and easy-to-use SL techniques. Of noteworthy is the development of spherical spline SL (Perrin et aI., 1987a,b), ellipsoidal spline SL (Law et aI., 1993), and the realistic geometry spline SL (Babi1oni et al., 1996, 1998; He, 1999; Zhao & He, 2001; He et al., 2001).

6.4.2 PERFORMANCE EVALUATION OF THE SPLINE LAPLACIAN EEG In this section, we present the performance evaluation of the realistic geometry spline Laplacian estimation algorithm in high-resolution EEG mapping (He et al., 2001). The evaluation was conducted by computer simulations using both a 3-concentric-sphere head model (Rush & Driscoll, 1969) and a realistic geometry head model. In addition, we examined the performance of the spline SL algorithm in high-resolution mapping of neural sources using experimental visual evoked potential (VEP) data. The realistic geometry spline SL estimation algorithm is detailed in Section 6.2.2. The linear inverse problem in equation (6-17) was solved by the TSVD (Shim & Cho, 1981), and the truncation parameter was determined by means of the discrepancy principle (Morozov, 1984).

6.4.2.1 Effects ofnoise The analytical SL was used to evaluate the numerical accuracy and reliability of the spline SL estimation from the scalp potentials. The accuracy of the spline SL estimator was evaluated by the CC and RE between the estimated and analytic SL distributions.

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Figure 6-9 shows an example of the simulation results for the effects of noise in the 3-concentric-sphere head model. A radial dipole (Rad) or a tangential dipole (Tag) located at an eccentricity of 0.6 was used to represent a well-localized areas of brain electrical activity. The scalp potential and the scalp SL at 129 electrodes were calculated analytically (Perrin et al., 1987a). Noise of up to 25% was added to the analytical potentials to simulate noise-contaminated scalp potential measurements. For each noise level, ten trials of GWN were generated and simulation conducted . The RE and the CC between the estimated SL and the analytical SL for all ten trials were averaged and displayed in Figure 6-9. The SL was estimated by minimizing the RE between the analytical SL and the SL estimate obtained by solving equation (6-17) for a regularization parameter. The parameter w in equation (6-10) was set to 0.16. Figure 6-9 indicates that the higher the noise level the larger the RE (or smaller the CC). The CC value was greater than 96% for the radial dipole , and greater than 91% for the tangential dipole, for up to 25% noise level. Figure 6-9 suggests that the one-parameter realistic geometry SL estimator is robust against the additive white noise in the scalp potential measurements.

6.4.2.2 Effects ofnumber ofrecording electrodes Table 6-4 shows an examp le of the simulation results with different number of surface electrodes in the 3-concentric-sphere head model. The RE and CC between the analytical SL and the estimated spline SL with different electrode number and different dipole configurations are shown in Table 6-4. One or multiple dipoles with unity strength were placed in the spherical conductor model (see note under Table 6-4 for detailed description of the parameters of the dipole configurations). The GWN of specified noise level was added to the scalp potentials to simulate noise contaminated EEG measurements. The parameter w in equation (6-10) was set to 0.16,0.18,0.20 and 0.24 corresponding to 129,96,64 and 32 electrodes, respectively. The SL was estimated by minimizing the RE between the analytic SL and the SL estimate. Table 6-4 clearly indicates the correlation between the goodne ss of the SL estimation and the number of surface electrodes. In general , the more electrodes used, the higher

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TABLE 6-4. The RE and CC values between analytical and estimated SL corresponding to different electrode numbers and dipole configurations in 3-sphere model Electrode Number Configuration A Configuration B Configuration C Configuration D Configuration E

RE CC RE CC RE CC RE CC RE CC

129

96

64

32

0.25 0.97 0.30 0.95 0.37 0.93 0.28 0.97 0.20 0.98

0.27 0.96 0.31 0.95 0.40 0.92 0.33 0.94 0.22 0.98

0.29 0.96 0.40 0.92 0.46 0.89 0.36 0.93 0.28 0.96

0.32 0.95 0.51 0.86 0.51 0.86 0.64 0.77 0.42 0.91

Note: Configurations A: I-RO at r = 0.5, with 15% GWN; B: I-TO at r = 0.7, with 5% GWN; C: I dipole at (0.0, 0.1,0.75) pointing to +x direction, another dipole at (0.0, -0.1,0.6) pointing to +z direction, with 5%GWN; 0: 2-RO at r = 0.7, and I-TO at r = 0.6, each has an angle of rr/6 with respect to z-axis, with 5% GWN; E: 2-RO at x-axis and 2-TO at y-axis, all at r = 0.65, and each has an angle of rr/6 with respect to z-axis, with 10%. RO: radial dipole. TO: tangential dipole.

CC (or lower RE) of the SL estimation. This phenomenon is consistent with the fact that a minimum sampling in the space domain is needed to restore the spatial frequency spectrum. As can be seen from Table 6-4, the CC for the SL estimation is 92% or larger for all cases when 96 or more electrodes were used. The CC for Configuration C was about 89% when 64 electrodes were used and 86% when 32 electrodes were used. This relatively low CC values, as compared with other configurations may be explained by the large eccentricity of the dipoles in the Configuration C. The closer the dipole is located to the scalp, the sharper the spatial distribution of the scalp Laplacian. Thus higher spatial sampling rate is desired. This phenomenon is further observed when only 32 electrodes are used, the CC values dropped lower than 90% for Configurations B, C and D. Of interesting is the low CC value for Configuration D when 32 electrodes were used. The CC dropped from 93%, when 64 electrodes were used, to 77% when 32 electrodes were used. Table 6-4 suggests that a high-density electrode array of 96 or more is desirable for scalp spline SL mapping.

6.4.2.3 Effects ofregularization Table 6-5 shows examples of the simulation results comparing the SL estimate obtained by means of a priori information of the analytical SL (denoted below as the optimal SL estimate), and the SL estimate without using the a priori information of the analytical SL. In this simulation, the discrepancy principle (Morozov, 1984) was used to determine the truncation parameter of the TSVD procedure. In Table 6-5, the upper rows show the RE and CC between the analytical SL and the optimal SL estimate. The lower rows show the RE and CC between the analytical SL and the estimated SL obtained by means of the discrepancy principle. In this simulation, 129 electrodes were uniformly distributed over the upper hemisphere, the parameter w in equation (6-10) was set to 0.16, and GWN of varying noise level was added to the analytical scalp potentials to simulate noise-contaminated potential measurements.

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TABLE6-5. Comparison of the optimal estimated spline SL and the spline SL estimated by using the discrepancy principle in 3-sphere model Dipole Configuration RE CC RE CC

Optimal TSVD Estimated TSVD

A

B

C

D

0.36 0.95 0.38 0.94

0.34 0.94 0.40 0.93

0.20 0.98 0.25 0.97

0.29 0.98 0.44 0.91

Note: ConfigurationsA: 2-RD at r = 0.70, each has an angle of '!f/8 with respect to z-axis, with 5% GWN; B: 2-RD at r = 0.6, and I-TD at r = 0.7, each has an angle of '!f/6 with respect to z-axis, with 10% GWN; C: 2-RD

and 2-TD, all at r = 0.65, and each has an angle of n 16with respect to z-axis, with 10% GWN; D: 4-RD at r = 0.75, each has an angle of '!f17 with respect to z-axis, with 5% GWN. RD: radial dipole. TD: tangential dipole.

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Table6-5 indicatesthat the regularizationresults are always worse than the optimal SL estimates (by definition), since the optimal SL estimates are obtained by minimizingthe RE betweenthe analyticaland estimatedSL. However, Table6-5 showsthat the results obtained via regularization (by using the discrepancy principle in this case) are comparable to the optimal SL estimates.Out of four source configurations, the CC for Configurations A, Band C are almost similar for these two types of results. For Configuration D, the CC of the regularization is smaller than the optimal SL estimate by about 6.5%. Howeverthe absolute CC is above 91%, suggesting the feasibility of the estimation of the SL through regularization. Figure6-10depictstwoexamplesof the normalizedpotentialand LaplacianEEGdistributionscorrespondingto Configuration A and Configuration B in Table6-5. For each source configuration, the first panel shows is the noise-contaminatedscalp potential map, the second panel showsthe analytical splineLaplacianEEG map, the third panel showsthe optimal

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estimated Laplacian EEG map by means of a priori information, and the last panel shows the regularization estimated Laplacian EEG map by means of the discrepancy principle.

6.4.2.4 Simulation in a realistic geometry head model The computer simulation was also conducted by using a realistic geometry head model built from one healthy subject (male, 34 years old) who was later involved in the VEP experiment. The CT images of the subjects were obtained, and the BEM models of the scalp, the skull, and the brain surfaces of the subject were constructed (Figure 6-11). Two artificial dipoles were used to simulate two simultaneously active brain electrical sources. One dipole source is located in right medial temporal lobe with orientation tangential to the cortical surface, and another dipole is located in right inferior frontal lobe with radial orientation. Figure 6-12 shows the result of spline Laplacian imaging of the simulated dipole sources in realistic geometry head model. The scalp potentials generated by the two artificial dipoles were contaminated with 5% GWN to simulate the measurement noise, and show a blurred dipole pattern of distribution with frontal positivity and posterior negativity. The spline Laplacian EEG map, however, effectively reduces the blurring effect caused by the head volume conductor, and clearly reveals two localized activities corresponding to the underlying dipole sources.

6.4.2.5 Surface Laplacian imaging of visual evoked potential activity Besides simulations, human VEP experiments were carried out to examine the performance of the spline SL estimator. The same above subject who gave written informed consent was studied in accordance with a protocol approved by the UIC/IRB. Visual stimuli were generated by the STIM system (Neuro Scan Labs, VA). The 96-channel VEP signals referenced to right earlobe were amplified with a gain of 500 and band pass filtered from 1 Hz to 200 Hz by Synamps (Neuro Scan Labs, VA), and were acquired at a sampling rate of

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FIGURE 6-12. Spline Laplacian mapping of the simulated dipole sources in a realistic geometry head model. (A) Scalp potential map. (B) Estimated spline Laplacian EEG map. See the attached CD for color figure. (From He & Lian, Crit Rev BME, 2002 with permission from Begel! House)

1,000 Hz by using SCAN 4.1 software (Neuro Scan Labs, VA). The electrodes' locations were measured using Polhemus Fastrack (Polhemus Inc. , Vermont). Full or half visual field pattern reversal checkerboards (black and white) with reversal interval of 0.5 sec served as visual stimuli and 300 reversals were recorded to obtain averaged VEP signals . The display had a total viewing angle of 14.3 0 by ILl 0 , and the checksize was set to be 175' by 135' expressed in arc minutes . The SL was estimated at the peak of the P100 component. Figure 6-13 shows the recorded scalp potential maps and the estimated spline Laplacian EEG maps at the PIOO peak time point of the pattern reversal VEP recorded from 96

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electrodes over the scalp. As shown in Figure 6-13A , the scalp potential map elicited by the full visual field stimuli is characterized by the strong but diffused activity that distributed symmetrically over the occipital area. The estimated spline Laplacian EEG map (Figure 6-13B), on the other hand, greatly improves the spatial resolution and clearly reveals two dipole-like sources located in the visual cortices in both hemispheres. Notably, the distribution of the positivity and negativity on each part of the scalp suggests orientation of cortical current sources , pointing toward the contralateral hemisphere. As shown in Figure 6-13C, in response to the left visual field stimuli, a dominant positive potential component was elicited with a widespread distribution on the left scalp . However, the estimated spline Laplacian EEG map (Figure 6-13D) shows a dominant dipole-like current source located in the right visual cortex. Similarly, the positivity-negativity distribution on the right scalp in the spline Laplacian EEG map suggests orientation of cortical current sources, pointing toward the contralateral hemisphere.

6.4.3 SURFACE LAPLACIAN BASED CORTICAL IMAGING Using the procedure as detailed in Section 6.2.3, the application of using the scalp SL to reconstruct the cortical potentials has also been explored (He, 1998b). Figure 6-14 shows

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FIGURE 6-14. An example of cortical imaging from scalp Laplacian EEG. (a) The "ture" cortical potential distribution generated by four radial dipoles located at eccentricity of 0.8 in the 3-sphere head model. (b)-(d) cortical potential distributions reconstructed from scalp Laplacian EEG with (b) 10%, (c) 30%, and (d) 50% Gaussion white noise, respectively. See the attached CD for color figure. (From He, IEEE-EMB, 1998 with permission) © IEEE

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FIGURE 6-15. Cortical imaging of the SEP activity over a realistic geometry head model in a human subject, using standard and the SL pre-filtered WMN estimate. See text for details. See the attached CD for Color figure. (From Babiloni et aI., MBEC, 2000 with permission)

an example of the simulation study based on the 3-concentric-sphere head model (Rush & Driscoll, 1969). Figure 6-14(a) shows the "true" cortical potential distribution generated by four radial dipoles. Figures 6-14(b)-(d) show the inverse cortical potential distributions estimated from the scalp SL with 10%, 30%, and 50% GWN added to the scalp SL, respectively. Notice that the higher the noise level in the scalp SL, the higher the background noise level in the estimated cortical potentials. However, reconstruction of the multiple extrema of the cortical potential distribution is quite robust, suggesting the feasibility and unique feature of the cortical potential imaging from the scalp SL. In a separate study, by investigating the spatial filter characteristics of the source-Laplacian relationship, Bradshaw and Wikswo (2001) demonstrated that dramatic improvement is evident in the SL-based inverse solution, as compared with inverse reconstruction from the raw data. In another approach, the SL pre-filtered EEG data was used as input for the weighted minimum norm (WMN) linear inverse estimate of the cortical current sources, in order to remove subcortically originated EEG potentials from the scalp potential distribution (Babiloni et aI., 2000). As an example, Figure 6-15 shows the application of this technique to the cortical imaging of the somatosensory evoked potentials (SEPs) over the realistic geometry head model of one subject. For all the SEP components being examined (P20N20, P22, N30-P30), the cortical imaging inverse solutions have enhanced spatial resolution than the scalp potential maps. Moreover, with respect to the WMN estimate, the SL prefiltered WMN estimate presented enhanced spatial information content, in that the potential maxima over the cortical surface were sharper and more localized.

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6.5 DISCUSSION The SL, as demonstrated by many investigators, enjoys enhanced spatial resolution and sensitivity to regional bioelectrical activity located close to the surface recording electrodes, and has unique advantage of reference independence. Conventionally, the local-based SL operators have been used to estimate the Laplacian ECG or Laplacian EEG, by approximating planar surface at the recording electrode. More accurate estimation can be achieved by taking into account the realistic geometry of the body surface using the spline interpolation scheme. On the other hand, due to the high-pass spatial filtering characteristics of the local SL operator, amplification of the noise associated with the potential measurements is unavoidable. Spatial low-pass filters, such as the Gaussian filter (Le et al., 1994) and Wiener filter (He, 1998a), have been shown to be useful in improving the signal-to-noise ratio of the SL. The spline SL, on the other hand, has been shown to provide an intrinsic spatial-low-pass filtering in addition to its spatial-high-pass filtering characteristics (Nunez et al., 1994; Srinivasan et al., 1998). Estimation of the spline SL from potentials does not require the information on the conductivity distribution inside the volume conductor. On the other hand, the spline SL estimation techniques do need a mathematical model describing the geometry of the surface over which the SL is to be estimated. The spline SL has been estimated over a spherical, ellipsoidal, and a realistic geometry surface (Perrin et al., 1987a,b, 1989; Law et al., 1993; Babiloni et aI., 1996, 1998; He, 1999; Zhao & He, 2001; He et al., 2001, 2002). Furthermore, the recently developed 3D spline SL algorithm (see Section 6.2.2) eliminates the need of determining two spline parameters, and provides a rational determination of the spline parameter through regularization process (He et al., 2001,2002). Such new approach provides comparable computational accuracy and stability, while substantially reduces the computational burden of optimizing two independent regularization parameters, as required in the previously reported approaches. The performance of the present realistic geometry spline SL estimator has been evaluated through a series of computer simulations. Based on the one-sphere homogeneous volume conductor model, the simulation results demonstrate that the performance of the one-parameter spline SL algorithm is comparable with that of the traditional two-parameter spline SL algorithm (Tables 6-1-6-3), but with much greater computational efficiency. The simulation study also demonstrates that the spline SL estimators are more robust against additive noise in both potential and geometry measurements as compared to the 5-point local SL estimator (Table 6-1), and consistent results are found for different numbers of recording electrodes (Tables 6-2). An interesting finding is that the 5-point local SL has good performance only for shallow sources and under low noise level, while the spline SL has good performance over a broader source depths and under variant noise levels (Table 6-1). This can be explained by the high-pass spatial filter property of the 5-point local SL estimator, versus the band-pass spatial filter property of the spline SL estimator (Nunez et al., 1994; Srinivasan et al., 1998). Table 6-3 and Figure 6-4 further suggest that the SLcan be estimated by using the well established regularization techniques, such as the discrepancy principle (Morozov, 1984), without a priori information of the "true" SL. In addition, based on the 3-sphere inhomogeneous volume conductor model, consistent evidence is shown that the spline SL estimation algorithm is robust against noise in potential measurements (Figure 6-9), provides consistent performance for different number of recording electrodes (Table 6-4), and can be estimated by means of the discrepancy principle (Table 6-5).

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The application of the 3D spline SL estimator to the realistic geometry volume conductor models further suggests the potential use of the realistic geometry spline Laplacian ECGIEEG estimation. Figure 6-5 indicates that the Laplacian ECG maps provide a much-localized projection onto the body surface in the areas directly overlying the heart, and are especially useful in identifying and characterizing the source multiplicity as compared with the potential ECG maps. Similarly, Figure 6-12 indicates that the Laplacian EEG map can effectively reduces the smoothing effect of the head volume conductor, and clearly localizes the underlying brain electrical sources. The human YEP experiments further suggest the usefulness of the Laplacian EEG mapping. The full visual field stimuli elicited symmetrical potential distribution about the midline of the occipital scalp. The estimated Laplacian EEG map showed much more localized dipolar current sources in both visual cortices, with dipolar orientations pointing toward the respective opposite hemisphere. It is widely accepted that the half visual field stimuli activates the visual cortex on the contralateral hemisphere of the brain. But paradoxically, the left visual field stimuli elicited stronger positive potential distribution over the ipsilateral side of the scalp, which might be misinterpreted as left visual cortex activation. However, by using the 3D spline SL method, the estimated Laplacian EEG map clearly indicated that the right visual cortex was activated, and these results are consistent with previous reports (Barrett et aI., 1976; Blumhardt et aI., 1977; Towle et aI., 1995). In summary, the SL, as a spatial enhancement method, can enhance the high-frequency spatial components of the surface EEG and ECG. The 3D spline SL algorithm can take into consideration of the realistic geometry of the body surface, and is applicable to both brain and heart electrical source imaging. Only one spline parameter needs to be determined through regularization procedure in this spline SL algorithm, thus enabling easy implementation of the spline SL in an arbitrarily shaped surface of a volume conductor. Both computer simulations and preliminary human experiments have demonstrated the excellent performance of the 3D spline SL in high-resolution ECG and EEG mapping, suggesting it may become an alternative for noninvasive mapping of heart and brain electrical activity.

ACKNOWLEDGEMENT The authors would like to thank their colleagues Dr. G. Li and Dr. D. Wu for useful discussions. This work was supported in part by a grant from American Heart Association #0140132N, NSF CAREER Award BES-9875344, and NSF BES-0201939.

REFERENCES Aoki, M., Okamoto, Y, Musha, T., and Harurni, K.: Three-dimensional simulation of the ventricular depolarization and repolarization processes and body surface potentials: normal heart and bundle branch block. IEEE Trans. Biomed. Eng., 34: 45~62, 1987. Babiloni, E, Babiloni, C, Carducci, E, Fattorini, 1., Onorati, P., and Urbano. A.: Spline Laplacian estimate of EEG potentials over a realistic magnetic resonance-constructed scalp surface model. Electroenceph. din. Neurophysiol., 98: 363-373,1996. Babiloni, E, Carducci, E, Babiloni, C, and Urbano, A.: Improved realistic Laplacian estimate of highly-sampled EEG potentials by regularization techniques. Electroenceph. Clin. Neurophysiol., 106: 336-343, 1998. Babiloni, E, Babiloni, C, Locche, 1., Cincotti, E, Rossini, P.M., and Carducci, E: High-resolution electroencephalogram: source estimates of Laplacian-transformed somatosensory-evoked potentials using a realistic

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subject head model constructed from magnetic resonance images. Med. Bioi. Eng. Comput., 38: 512-519, 2000. Barr, R.C., Ramsey, M. III, and Spach, M.S.: Relating epicardial to body surface potential distributions by means of transfer coefficients based on geometry measurements. IEEE Trans. Biomed. Eng., 24: I-II, 1977. Barrett, G., Blumhardt, L., Halliday, A.M., Halliday, E., and Kriss, A.: A paradox in the lateralisation of the visual evoked response. Nature, 261: 253-255,1976. Besio, WG; Lu, C.C., and Tarjan, P.P.: A feasibility study for body surface cardiac propagation maps of humans from Laplacian moments of activation. Electromagnetics, 21: 621-632, 2001. Blumhardt, L.D., Barrett, G., and Halliday, A.M.: The asymmetrical visual evoked potential to pattern reversal in one half field and its significance for the analysis of visual field defects. British J. Ophthalmology, 61: 454-461, 1977. Bradshaw, L.A., and Wikswo, J.PJr.: Spatial filter approach for evaluation of the surface Laplacian of the electroencephalogram and magnetoencephalogram. Ann. Biomed. Eng., 29: 202-213, 2001. Courant, R. and Hilbert, D.: Methods of mathematical physics. New York: Interscience. 1966. Duchon, J.: Interpolation des fonctions de deux variables suivant Ie principe de la flexion des plaques minces. R.A.I.R.O. Anal. Num., 10: 5-12,1976. Fattorusso, v., Thaon, M., Tilmant, 1.: Contribution of I'etude de I'electrocardiogramme precordial. Acta Cardiologica., 4: 464-487,1949. Harder, R. and Desmarais, R.: Interpolation using surface spline. 1. Aircraft, 9: 189-191, 1972. He, B. and Cohen, RJ.: Body surface Laplacian ECG mapping. IEEE Trans. Biomed. Eng., 39: 1179-1191, I992a. He, B. and Cohen, RJ.: Body surface Laplacian mapping of cardiac electrical activity. Am. J. Cardiol., 70: 1617-1620, 1992b. He, B., Kirby, D., Mullen, T., and Cohen, RJ.: Body surface Laplacian mapping of cardiac excitation in intact pigs. Pacing Clin. Electrophysiol., 16: 1017-1026, 1993. He, B.: On the Laplacian inverse electrocardiography. Proc. Ann. Int. Conf. IEEE Eng. Med. BioI. Soc., 145-146, 1994. He, B., Chernyak, Y, and Cohen, RJ.: An equivalent body surface charge model representing three dimensional bioelectrical activity. IEEE Trans. Biomed. Eng., 42: 637-646, 1995. He, B. and Cohen, RJ.: Body surface Laplacian ECG mapping-A review. Crit. Rev. Biomed. Eng., 23: 475-510, 1995. He, B.: Principles and applications ofthe Laplacian electrocardiogram. IEEE Eng. Med. BioI. Mag., 16: 133-138, 1997. He, B. and Wu, D.: A bioelectric inverse imaging technique based on surface Laplacians. IEEE Trans. Biomed. Eng., 44: 529-538, 1997. He, B., Yu, X., Wu, D., and Mehdi, N.: Body surface Laplacian mapping of bioelectrical activity. Methods In! Med., 36: 326-328, 1997. He, B.: Theory and applications of body-surface Laplacian ECG mapping. IEEE Eng. Med. BioI. Mag., 17: 102-109,1998a. He, B.: High resolution source imaging of brain electrical activity. IEEE Eng. Med. BioI. Mag., 17: 123-129, I998b. He, B.: Brain electrical source imaging: Scalp Laplacian mapping and cortical imaging. Crit. Rev. Biomed. Eng., 27: 149-188, 1999. He, B. and Wu, D.: Laplacian electrocardiography. Crit. Rev. Biomed. Eng., 27: 285-338,1999. He, B., Lian, 1., and Li, G.: High-resolution EEG: a new realistic geometry spline Laplacian estimation technique. Clin. Neurophysiol., 112: 845-852,2001. He, B., Li, G., and Lian, J.: A spline Laplacian ECG estimator in a realistic geometry volume conductor. IEEE Trans. Biomed. Eng., 49: 110-117, 2002. He, B., Lian, J.: Spatio-ternporal Functional Neuroimaging of Brain Electric Activity. Critical Review ofBiomedical Engineering, 30: 283-306, 2002. Hjorth, B.: An on-line transformation of EEG scalp potentials into orthogonal source derivations. Electroenceph. Clin. Neurophysiol., 39: 526-530, 1975. Johnston, P.R.: The Laplacian inverse problem of electrocardiography: an eccentric spheres study. IEEE Trans. Biomed. Eng., 44: 539-48,1997.

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Law, S.K., Nunez, P.L., and Wijesinghe, R.S.: High-resolution EEG using spline generated surface on spherical and ellipsoidal surfaces. IEEE Trans. Biomed. Eng., 40: 145-153, 1993. Le, J., Menon, V, and Gevins, A.: Local estimate of surface Laplacian derivation on a realistically shaped scalp surface and its performance on noisy data. Electroenceph. Clin. Neurophysiol., 92: 433-441, 1994. Lian, 1., Srinivasan, S., Tsai, H., and He, B.: Comments on "Is accurate recording of the ECG surface Laplacian feasible?" IEEE Trans. Biomed. Eng., 48: 610-613, 2001. Lian, J., Srinivasan, S., Tsai, H., Wu, D., and He, B.: On the estimation of noise level and signal to noise ratio of Laplacian ECG during ventricular depolarization and repolarization. Pacing Clin. Electrophysiol., 25(10): 1474-1487,2002. Lian, J., Li, G., Cheng, J., Avitall, B., and He, B.: Body surface Laplacian mapping of atrial depolarization in healthy human subjects. Med. Bio!. Eng. Comput., 40(6): 650-659, 2002b. Li, G., Lian, J., He, B.: On the Spatial Resolution of Body Surface Potential and Laplacian Pace Mapping. Pacing and Clinical Electrophysiology, 25: 420-429, 2002. Li, G., Lian, J., Salla, P., Cheng, J., Ramachandra, 1., Shah, P., Avitall, B., and He, B.: Body surface Laplacian electrogram of ventricular depolarization in normal human subjects. J. Cardiovasc. Electrophysiol., 140): 16-27,2003. Mirvis, D.M., Keller, EW., Ideker, RE., Cox, r.w, Zettergren, D.G., and Dowdie, RJ.: Values and limitations of surface isopotential mapping techniques in the detection and localization of multiple discrete epicardial events. 1. Electrocardiol., 10: 347-358,1977. Morozov, VA.: Methods for solving incorrectly posed problems. Berlin: Springer-Verlag, 1984. Nunez, PL.: Electric field ofthe brain. London: Oxford University Press, 1981. Nunez P.L.: Neocortical dynamics and human EEG rhythms. New York: Oxford University Press, 1995. Nunez, P.L., Silibertein, RB., Cdush, PJ., Wijesinghe, R.S., Westdrop, A.E, and Srinivasan, R.: A theoretical and experimental study of high resolution EEG based on surface Laplacian and cortical imaging. Electroenceph. Clin. Neurophysiol., 90: 40-57,1994. Oostendorp, T.E and van Oosterom, A.: The surface Laplacian of the potential: theory and application. IEEE Trans. Biomed. Eng., 43: 394-403, 1996. Perrin, E, Bertrand, 0., and Pernier, J.: Scalp current density mapping: value and estimation from potential data. IEEE Trans. Biomed. Eng., 34: 283-288, 1987a. Perrin, E, Pernier, J., Bertrand, 0., Giard, M.H., and Echallier, J.E: Mapping of scalp potentials by surface spline interpolation. Electroenceph. Clin. Neurophysiol., 66: 75-81, 1987b. Perrin, E, Pernier, J., Bertrand, 0., and Echallier, J.E: Spherical splines for scalp potential and current density mapping. Electroenceph. Clin. Neurophysiol., 72: 184-187, 1989. Rudy, Y and Plonsey, R.: A comparison of volume conductor and source geometry effects on body surface and epicardial potentials. eire. Res., 46: 283-291,1980. Rush, S. and Driscoll, D.A.: EEG electrode sensitivity-an application of reciprocity. IEEE Trans. Biomed. Eng., 16: 15-22, 1969. Shim, YS. and Cho, Z.H.: SVD pseudoinversion image reconstruction. IEEE Trans. Acoust. Speech. Processing, 29: 904-909,1981. Spach, M.S., Barr, R.C., Lanning, C.E, and Tucek, P.c.: Origin of body surface QRS and T-wave potentials from epicardial potential distributions in the intact chimpanzee. Circulation, 55: 268-278, 1977. Srinivasan, R., Nunez, P.L., Tucker, D.M., Silberstein, RB., Cadusch, PJ.: Spatial sampling and filtering of EEG with spline Laplacian to estimate cortical potentia!. Brain Topography, 8(4): 355-366, 1996. Srinivasan, R, Nunez, P.L. and Silberstein, R.B.: Spatial filtering and neocortical dynamics: estimates of EEG coherence. IEEE Trans. Biomed. Eng., 45: 814-826,1998. Throne, R.D. and Olson, L.G.: Fusion of body surface potential and body surface Laplacian signals for electrocardiographic imaging. IEEE Trans. Biomed. Eng., 47: 452-462, 2000. Towle, VL., Cakmur, R., Cao, Y, Brigell, M., and Parmeggiani, L. Locating VEP equivalent dipoles in magnetic resonance images. Int. J. Neurosci., 80: 105-116, 1995a. Tsai, H., Ceccoli, H., Avitall, B., and He, B.: Body surface Laplacian mapping of anterior myocardial infarction in man. Electromagnetics, 21: 607-619, 2001. Umetani, K., Okamoto, Y, Mashima, S., Ono, K., Hosaka, H., and He, B.: Body Surface Laplacian mapping in patients with left or right ventricular bundle branch block. Pacing Clin. Electrophysiol., 21: 2043-2054, 1998.

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7

NEUROMAGNETIC SOURCE RECONSTRUCTION AND INVERSE MODELING Kensuke Sekihara' and Srikantan S. Nagarajarr' I Departmentof

ElectronicSystems and Engineering, Tokyo Metropolitan Institute of Technology, Asahigaoka 6-6, Hino, Tokyo 191-0065, Japan 2Department of Radiology, University of California, San Francisco, 513 Parnassus Avenue, S362 San Francisco, CA 94143, USA

7.1 INTRODUCTION The human brain ha s approximately 1010 neurons in its cerebral cortex. Their electrophy siological activity generates weak but measurable magnetic fields outside the scalp. Magnetoencephalography (MEG) is a method which measure s these neuromagnetic fields to obtain information about these neu ral activitie s (Hamalainen et al., 1993 ; Roberts et al., 1998; Lewine et al., 1995). Among the various kind s of funct ional neuroimaging methods, such a neuro-electromagnetic approach has a major advantage in that it can provide fine time resolution of millisecond order. Therefore, the goal of neuromagnetic imaging is to visualize neural activit ies with such fine time resolution and to provide functional information about brain dynamics. To attain this goal, one technical hurdle must be overcome. That is, an efficient method to reconstruct the spatio-temporal neural activities from neuromagnetic measurements needs to be developed. Toward this goal, a number of algorithms for reconstructing spatio-temporal source activities have been investigated (Baillet et al., 2001) . Th is chapter deals with this neuromagnetic reconstruction problem . However, we do not provide a general review of variou s algorithms for this reconstruction problem. Instead, we describe a particular class of source reconstruction techniques referred to as the spatial filter, which allows the spat io-temporal recon struction of neural activities without assuming any kind of source model. Furthermore, among the spatial filter techniques, we focus on adapti ve spatial filter techniques. The se techniques were origi nally developed in the field s of array signal processing , including radar, sonar, and seismic exploration , and have been

Corresponding author: Kensuke Sekihara Ph.D., Tokyo Metropolitan Institute of Technology, Asahigaoka 6-6, Hino, Tokyo 191-0065, Japan, Tel:8 1-42-585-8642, Fax:8 1-42-585-8642, E-mail: ksekiha@cc .tmit.ac.jp 213

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widely used in such fields (van Veen et al., 1988). Nonetheless, the adaptive spatial filter techniques are relatively less acknowledged in the MEG/EEG community. In this chapter, we also formulate reconstruction techniques based on linear leastsquares methods (Hamalainen and Ilmoniemi, 1984) as the non-adaptive spatial filter. This formulation enables us to compare the least-squares-based techniques with the adaptive spatial filter techniques on a common, unified basis. Actually, we compare these two types of techniques by using the same figure of merit called the resolution kernel, and show that the adaptive techniques can provide much higher spatial resolution than the least-squares-based methods. The organization of this chapter is as follows: Following a brief review of the neuromagnetometer hardware in Section 7.2, we describe the forward modeling and some basic properties of MEG signals in Section 7.3. Section 7.4 presents the formulation of the linear least-squares-based methods as the non-adaptive spatial filter. Section 7.5 describes the adaptive spatial filter techniques. In Section 7.6, we present a quantitative comparison between the adaptive and non-adaptive methods using the resolution kernel criterion. Section 7.7 presents a series of numerical experiments on the adaptive spatial filter performance. In Section 7.8, we demonstrate the effectiveness of the adaptive spatial filter techniques by applying them to two sets of MEG data.

7.2 BRIEF SUMMARY OF NEUROMAGNETOMETER HARDWARE It is generally believed that the neuromagnetic field is generated by the post-synaptic ionic current in the pyramidal cells of cortical layer III. Here, neuronal cells are organized into so-called columnar structure, and the synchronous activity of these cells results in superimposed magnetic fields strong enough to be measured outside the human head. The average intensity of this neuromagnetic field, however, is around a few hundred femtoTesla (fT)*. To measure such an extremely weak magnetic field, a neuromagnetometer uses a special device called a super-conducting quantum interference device (SQUID) (Clarke, 1994; Drung et al., 1991). This device is so sensitive that it can in principle measure a single quantum of magnetic flux. When measuring such a weak neuromagnetic field, a major problem arises from the background environmental magnetic noise. Background magnetic noise is generated by electronic appliances such as computers, power lines, cars, and elevators. Such noise is common at a site where the neuromagnetometer is installed, and its average intensity is usually five to six orders of magnitude greater than the neuromagnetic field. One obvious way to reduce it is to use a magnetically-shielded room. However, a typical medium-quality shielded room, most commonly used in MEG measurements, can reduce the background noise by up to only three orders of magnitude. To further reduce the background noise, neuromagnetometers are usually equipped with a special type of detector configuration called a gradiometer (Hamalainen et al., 1993; Lewine et al., 1995). The first-order gradiometer consists of two coils of exactly the same area; they are connected in series, but wound in opposite directions. Therefore, the gradiometer cancels the electric current induced by the background noise fields because the * One IT is equivalent to I x 10- 15 T.

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sources of such noise fields are generally far from the gradiometer and induce nearly the same amount of electric current in both coils. The gradiometer can achieve two to three orders of magnitude reduction in the background noise, and can remove the influence of the residual noise field within the magnetically-shielded room . The reduction performance, however, depends on the manufacturing precision of the two coils. Aside from the gradiometer, several other methods of removing the external noise have been investigated (Vrba and Robinson , 2001; Adachi et ai., 200 1). Many of these methods use extra sensors that measure only the noise fields. (Such sensors are usually located apart from the sensor array which measures the neuromagnetic fields.) Quasi-real-time electronics then perform the on-line subtraction between the outputs from the extra sensors and from the regular sensors. A method that does not require additional sensor channels has also been developed. This method applies a technique called signal-space projection, and can be implemented completely as a post-processing procedure (Parkkonen et ai., 1999). These external noise cancellation methods make it possible to use a magnetometer as a sensor coil instead of the gradiometer. They also permit the measurement of neuromagnetic fields outside a magnetically-shielded room. The most remarkable advance in neuromagnetometer hardware over the last ten years has been the rapid increase in the number of sensors. Since neuromagnetometers with a 37-channel sensor array became commercially available in the late 80s (Lewine et al., 1995), the number of sensors in commercially available neuromagnetometers has constantly increased . The latest neuromagnetometers are equipped with 200-300 sensor channels, with whole-head coverage of the sensor array.

7.3 FORWARD MODELING 7.3.1 DEFINITIONS Let us define the magnetic field measured by the rnth detector coil at time t as bm(t), and a column vector bet) = [b,(t) , b2(t) , . .. , bM(t)]T as a set of measured data where M is the total number of detector coils and the superscript T indicates the matrix transpose . A spatial location is represented by a three-dimensional vector r : r = (x, y , z), The sourcecurrent density at r and time t is defined as a three-dimensional column vector s(r, t) . The magnitude of the source current is denoted as s(r, t) (= js(r, t)[), and the orientation of the source is defined as a three-dimensional column vector "'l(r, t) = s(r, t)/s(r, t) = [1JxCr, t), 1Jy(r, t), 1Jz(r, t)] T, whose ~ component (where ~ equals x, y, or z in this chapter), is equal to the cosine of the angle between the direction of the source moment and the ~ direction. Let us define l!n(r) as the rnth sensor output induced by the unit-magnitude source located at r and directed in the ~ direction. The column vector I I;(r) is defined as II;(r) = [fr(r) , l~(r) , . . . , It(r)] T . Then , we define a matrix which represents the sensitivity of the whole sensor array at r as L(r) = [IX(r ), per ), F(r)]. The rnth row of L(r ), I m(r) = [f~(r ), l~(r), l~(r )], represents the sensitivity at r of the rnth sensor. Then, using the superposition law, the relationship between b(t) and s (r , t ) is expressed as bet) =

f

L(r)s(r , t)dr

+ net)·

(7.1)

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Here, n(t) is the noise vector at t. The sensor sensitivity pattern, represented by the matrix L(r), is customarily called the sensor lead field (Hamalainen et al., 1993; Sarvas, 1987), and this matrix is called the lead-field matrix. We define, for later use, the lead-field vector in the source-moment direction as l(r), which is obtained by using l(r) = L(r)'Y/(r).

7.3.2 ESTIMATION OF THE SENSOR LEAD FlEW The problem of the source reconstruction is the problem of obtaining the best estimate of s(r, t) from the array measurement b(t). It is thus apparent that, to solve this reconstruction problem, we need to have a reasonable estimate of the lead-field matrix. In this subsection, we describe how we can obtain the lead field using the spherically symmetric homogeneous conductor model, which is most commonly used in estimating the MEG sensor lead field. Also, we briefly mention a realistically-shaped volume-conductor model, which can generally provide more accurate lead field estimates, particularly for non-superficial brain regions. Estimation of the lead-field is called the forward problem, because this is equivalent to estimating the magnetic field from a point source located at a known location; this problem stands in contrast to the inverse problem in which the source configuration is estimated from a known magnetic field distribution. Let us define the electric potential as V, the magnetic field as B, and the electric current density as j. The source current s(r) defined in Section 7.3.1 is called the primary current, (alternatively called the impressed current), which is directly generated from the neural activities. There is another type of electric current called the return current or volume current. It results from the electric field in the conducting medium, and it is not directly caused by the neural activities. Defining the conductivity as p, the return current is expressed as - p V V where - V V is equal to the electric field. Thus, the total electric current j is expressed as j(r) = s(r) - pVV.

(7.2)

The relationship between the total current j and the resultant magnetic field B is given by the Biot-Savart law, /.Lo B(r) = -

4n

f itr': .

x r - r' ,dr,I

Ir - r'l·

(7.3)

where /.Lo is the magnetic permeability of the free space. In order to derive the analytical expression for the relationship between the primary source current and the magnetic field, we first consider the case in which a whole space is filled with a conductor with constant conductivity p. In this case, it is easy to show that the following relationship holds: /.Lo Bo(r)=-

4n

fs(r)x i r -r' ,dr. I

Ir - r'l'

(7.4)

Here the magnetic field is denoted as Bo for later convenience. Note that Eq. (7.4) is similar to Eq. (7.3). The only difference is that the total current density j(r) is replaced by the primary current density s(r).

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We then proceed to deriving a formula for the magnetic field outside a sphericallysymmetric homogeneous conductor, B. To do so, we make use of the fact that the radial component of the magnetic field B is equal to the radial component of B o in Eq. (7.4) (Cuffin and Cohen, 1977; Sarvas, 1987), i.e.,

B . I, = B o . I"

(7.5)

where I, is the unit vector in the radial direction defined as I, = r Ilrl. (Note that we set the coordinate origin at the sphere origin.) The relationship V x B = 0 holds outside the volume conductor, because there is no electric current. Thus, B can be expressed in terms of the magnetic scalar potential U (r), B = -J),oVU(r).

(7.6)

This potential function is derived from

1

00

U(r) = - 1

~

0

B(r

+ TI,)· l,dT =

1

00

-1

~

0

Bo(r

+ TI,)· f .d»,

(7.7)

where we use the relationship in Eq. (7.5). By substituting Eq. (7.4) intoEq. (7.7), we finally obtain 1 U(r) = - -

4;rr

f

s(r') x r' . r , dr A '

(7.8)

where A=

Ir -r'I(lr -r/lr + Irl2 -r' -r).

The formula for B is then obtained by substituting Eq. (7.8) into Eq. (7.6), i.e., B(r) = J),o

4;rr

f ~[AS(r/) A

x r' - (s(r /) x r' . r)VA]dr',

(7.9)

where

VA = [

Ir-r

Irl

/2 1

+

(r-r')·r Ir -

r'l

+21r -r/l +2Irl]r -

[

Ir -r/l

+21rl +

(r-rl).r] Ir - r/l

r',

To obtain the component of the lead-field matrix, l~(ro), we first calculate B(r m ) (where r m is the mth sensor location) by using Eq. (7.9) with s(r ') = ft;o(r ' - ro), where I ~ is the unit vector in the ~ direction. When the sensor coil is a magnetometer coil, (which measures only the magnetic field component normal to the sensor coil), l~(ro) is calculated from l~(ro) = B(r m) . I,::il where I'::il is a unit vector expressing the normal direction of the mth sensor coil. When the sensor coil is a first-order axial gradiometer with a baseline This l~(ro) of D, l~(ro) is calculated from l~(ro) = B(r m) . I,::il - B(r m + D I,::il) .

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218

represents the sensitivity of the mth sensor to the primary current density located at r o and directed in the ~ direction. One of the important properties of the lead field obtained using the sphericallysymmetric homogeneous conductor model is that if s(r o) and r o are parallel, i.e., if the primary current source is oriented in the radial direction, no magnetic fields are generated outside the spherical conductor from such a radial source. Also , we can see that when r o approaches the center of the sphere, l~ (r 0) becomes zero, and no magnetic field is generated outside the conductor from a source at the origin. The spherically-symmetric homogeneous conductor is generally satisfactory in explaining the measured magnetic field when only superficial sources exist , i.e., when all sources are located relati vely close to the sensor array. This is because the curvature of the upper half of the brain is well approximated by a sphere. However, for sources located in lower regions of the brain, the model becomes inaccurate because the curvature of the lower brain regions significantly differs from a sphere. Such errors caused by misfits of the model may be reduced by using a spheroidally-symmetric conductor model (Cuffin and Cohen , 1977) or an eccentric sphere conductor model (Cuffin, 1991). More fundamental improvements can be obtained by using realistically-shaped volume-conductor models. Such conductor models can be constructed by first extracting the brain boundary surface from the subject's 3D MRI. We denote this surface ~. We then use the following Geselowitz formula (Ge selowitz, 1970; Sarvas , 1987) to calculate magnetic fields outside the volume conductor:

J.lo B(r ) = Bo(r ) - p 4rr

l ' :E

V Cr )JzJr') x r - r''3 d S, Ir - r I

(7.10 )

where the integral on the right-hand side indicates the surface integral over ~; r' represents a point on ~, and! :E(r' ) is a unit vector perpendicular to ~ at r ' . Here , we assume that the conductivity within the brain is uniform, and denote it p. The second term on the right-hand side of Eq. (7.10) represents the influence from the volume current, and to calculate this term, we need to know VCr) on ~ , which is obtained by solving (Sarvas, 1987)

p

p

-V(r)=Vo(r ) - 2 4rr

1 ' :E

,

V(r)!:E(r) ·

r -r' 3d S,

Ir - r 'l -

(7.11 )

where Vo(r )

= -1

4rr

/ s(r ' ). r -r' :J dr ,, Ir - r 'I '

(7.12)

and rand r' are points on the boundary surface. Because the brain boundaries are irregular, the magnetic field B (r ) can only be obtained numerically using the boundary element method (BEM ). In this calculation, we first estimate the electric potential V (r) on the brain boundary surface ~ by numerically solving Eq. (7. 11). We then calculate magnetic fields outside the brain using Eq. (7.10). The details of these numerical calculations are out of the scope of this chapter, and they are found in (Barnard et al., 1967; Hamalainen and Sarva s, 1989). The numerical method

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mentioned so far assumes uniform conductivity within the brain boundary, and is called single-compartment BEM. It is usually used in estimating MEG sensor lead fields (Fuchs et al., 1998). It can be extended to the multiple-compartment BEM and such models are usually used for estimating the EEG sensor lead fields. The BEM-based realistically-shaped volume conductor models generally provide significant improvements in the accuracy of the forward calculation particularly for deep sources (Fuchs et al., 1998; Cuffin, 1996), although they are computationally expensive. Improvements in the computational efficiency of the BEM have been reported (Bradley et al., 2001; van't Ent et al., 2001).

7.3.3 LOW-RANK SIGNALS AND THEIR PROPERTIES Let us consider specific cases where the primary current consists of localized discrete sources. The number of sources is denoted Q, and we assume that Q is less than the number of sensors M. The locations of these sources are denotedr , .r-, ... ,r Q. The source-moment distribution is then expressed as Q

s(r, t) = LSD(rq, t)8(r - r q),

(7.13)

q=]

where s D(rq) = f s(r)dr and this integral extends the small region around r q where the qth source is confined. This type of localized source is called the equivalent current dipole with moment s D. Here, s D is called the moment because it has a dimension of current x distance. The basis underlying the equivalent current dipole is physiologically plausible (Okada et al., 1987), and sources of neuromagnetic fields are often modeled with the current dipoles. Since there is little advantage to explicitly differentiating the current density s(rq v t) and the current moment sD(r q, t), for simplicity we keep the same notation s(r q, t) to express the current moment. Then, the Q-dimensional source magnitude vector is defined as vet) = [s(r], t), s(r2, t), ... , s(rQ' t)]T. We define a 3Q x Q matrix that expresses the orientations of all Q sources as Wet) such that

lJi"(t)=

[

TJ(r~ , t)

o

o

o

The composite lead-field matrix for the entire set of Q sources is defined as (7.14) Then, substituting Eq. (7.13) into Eq. (7.1), we have the discrete form of the basic relationship between bet) and vet) such that

bet) = [LclJi"(t)]v(t)

+ net).

(7.15)

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Let us define the measurement covariance matrix as R b ; i.e., R b = (b(t)b T (t»), where (.) indicates the ensemble average. (This ensemble average is usually replaced by the time average over a certain time window.) Let us also define the covariance matrix of the sourcemoment activity as R s ; i.e., R, = (!li(t)v(t)v T (t)I]i"(t)T). Then, using Eq. (7.15), we get the relationship between the measurement covariance matrix and the source-activity covariance matrix such that (7.16) where the noise in the measured data is assumed to be white Gaussian noise with a variance of (J2, and I is the M x M identity matrix. Let us define the kth eigenvalue and eigenvector of R b as Ak and ei, respectively. Let us assume, for simplicity, that all sources have fixed orientations. Then, unless some source activities are perfectly correlated with each other, the rank of R, is equal to the number of sources Q. Therefore, according to Eq. (7.16), Rb has Q eigenvalues greater than (J2 and M - Q eigenvalues that are equal to (J2. The signal whose covariance matrix has such properties is referred to as the low-rank signal (Paulraj et al., 1993; Sekihara et al., 2000). Let us define the matrices E s and EN as E s = [er, ... , eQ] and EN = [eQ+]' ... ' eM]. The column span of E s is the maximum-likelihood estimate of the signal subspace of R b , and the span of EN is that of the noise subspace (Scharf, 1991). In the low-rank signals, the measurement covariance matrix R b can be decomposed into its signal and noise subspace components; i.e., (7.17) Here, we define the matrices As and AN as

As = diag[AI, ... , AQ]

and

AN = diag[AQ+l, ... , AM],

(7.18)

where diagl- ..] indicates a diagonal matrix whose diagonal elements are equal to the entries in the brackets. The most important property of the low-rank signal is that, at source locations, the lead-field matrix is orthogonal to the noise subspace of Rb. This can be understood by first considering that (7.19) Since L; is a full column-rank matrix, and we assume that R, is a full rank matrix, the above equation gives (7.20) This implies that the lead-field matrices at the true source locations are orthogonal to any noise level eigenvector; that is, they are orthogonal to the noise subspace (Schmidt, 1981),

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Neuromagnetic Source Reconstruction and Inverse Modeling

i.e., (7.21) Since the equation above holds, the relationship 1T (r q)E N = 0 also holds. These orthogonality relationships are the basis of the eigenspace-projection adaptive beamformer described in Section 7.5.2, as well as the basis of the well-known MUSIC algorithm (Schmidt, 1986; Schmidt, 1981; Mosher et al., 1992).

7.4 SPATIAL FILTER FORMULATION AND NON-ADAPTIVE SPATIAL FILTER TECHNIQUES 7.4.1 SPATIAL FILTER FORMULATION

Spatial filter techniques estimate the source current density (or the source moment) by applying a linear filter to the measured data. Because the source is a three-dimensional vector quantity, there are two ways to implement the spatial filter approach in the neuromagnetic source reconstruction. One is the scalar spatial filter and the other is the vector spatial filter. In the scalar approach , we use a single set weights that characterizes the properties of the spatial filter, and define the set of the filter weights as a column vector w (r , 7]) = [WI (r, 7]) , w2(r , 7]) , . . . , wM(r, 7])]T . Here the weight vector depends both on the location r and the source orientation 7]. This weight vector w(r , 7]) should only pass the signal from a source with a particular location r and an orientation 7]. The weight vector rejects not only the signals from other locations but also the signal from the location r if the orientation of the source at r differs from 7]. Then, the magnitude of the source moment is estimated using a simple linear operation ,

L wm(r , 7])bm(t ), M

S{r, t ) = w T (r, 7])b(t ) =

(7.22)

m=1

where the estimate of the source magnitude is denoted s (r , t ). When using the scalar-type beamformer in Eq. (7.22), we need to first determine the beamformer orientation 7] to estimate the source activity at a specific location r . However, this 7] is generally unknown, although several techniques have been developed to obtain the optimum estimate of the source orientation (Sekihara and Scholz , 1996; Mosher et al., 1992). The vector spatial filter uses a weight matrix W(r ) that contains three weight vectors wAr), w y(r ), and w z(r ), which respectively estimate the x, y , and z components of the source moment. That is, the source current vector is estimated from S(r , t )

= W T(r )b(t ) = [w Ar ), w y(r), w z(r )fb(t),

(7.23)

where s (r , t) is the estimate of the source current vector. The vector spatial filter estimate the source orientation as well as the source magnitude. The application of a spatial filter weight artificially focuses the sensitivity of a sensor array on a specific location r , and this location r is a controllable parameter. Therefore, in

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K. Sekihara and S. S. Nagarajan

a post-processing procedure, we can scan the focused region over a region of intere st to recon struct the entire source distribution.

7.4.2 RESOLUTION KERNEL The major problem with spatial filter techniques is how to derive a weight vector with desirable properties. To develop such weight vectors, we need a criterion which characterizes how appropriately the weig ht has been designed. The resolution kernel can play this role. Comb ining Eqs. (7.1) and (7.23), we obtain the relationship, s (r, t ) =

!

W T (r )L(r ' )s(r ' , t )dr' =

!

lR(r, r ' )s(r ' , t )dr' ,

(7.24)

where, lR(r,r ')

=

WT(r)L(r ') .

(7.25)

This lR(r, r ') is called the resolution kernel, which expres ses the relationship between the original and estimated source distributions (Menendez et al, 1997; Menendez et al., 1996; Lutkenhoner and Menendez, 1997). Therefore, the resoluti on kernel can provide a measure of the appropriateness of the filter weight. In other words, the weight must be chosen so that the resolution kernel has a desirable shape, which generally satisfies the three propert ies: (i) peak at the source location, (ii) a small main-lobe width, and (iii) a low side-lobe ampl itude . The most important property among them is that the kernel should be peaked at the source locat ion. Only in this case, the recon structed source distribution can be interpreted as a smoothed version of the true source distribution . However, if this condition is not met, the recon structed source distribution should contain systematic bias and may be totally different from the true source distributi on. The kernel should also have a small main-lob e width, so that the recon struction results have high spatial resolution. When the kernel has a lower side-lobe amplitude, the results have less systematic noise and artifacts.

7.4.3 NON-ADAPTIVE SPATIAL FILTER

Minimum norm spatialfilter There are generally two types of spatial filter techniques. One is a non-adaptive method in which the filter weight is independent of the measurements. The other is an adaptive method in which the filter weight depends on the measurements. The primary intere st of this chapter is the application of the adaptive spatial filter techniqu e to the neuromagnetic source reconstruct ion. However, before proceeding to describe the adaptive spatial filter, we briefly describe the non-adaptive spatial filter in order to clarify the difference between these two types of spatial filter methods. The best-known non-adaptive spatial filter is the minimum-norm estimate (Hamalainen and Ilmoniemi , 1984; Hamalainen and Ilmoniemi , 1994; Wang et al., 1992; Graumann, 1991). The filter weight can be obtained by the following minimiz ation: min! IIlR(r, r' ) - 8(r - r ') 1I 2dr' ,

(7.26)

Neuromagnetic Source Reconstruction and Inverse Modeling

223

where b(r) is the three-dimensional delta function. By making the resolution kernel close to the delta-function, the weight is obtained and it is expressed as

The estimated current density is then expressed as (7.28) The matrix G is often referred to as the gram matrix. The p and q element of G is given by calculating the overlap between the lead fields of the pth and qth sensors,

o.. =

f Ip(r)l~(r)dr.

(7.29)

Unfortunately, in biomagnetic instruments, the overlaps between the adjacent sensor lead fields are very large, as depicted in Fig. 7.1(a). As a result, G p •q has a more-or-less similar value for various pairs of p and q. Consequently, the matrix G is generally very poorly conditioned. This fact greatly affects the performance of this non-adaptive spatial filter method because it requires calculation of the inverse of G, a process which is very erroneous if G is nearly singular. This gram matrix G is usually numerically calculated by introducing pixel grids throughout the source space. Let us denote the locations of the pixel grid points r 1, r 2, ... , r N, and the composite lead-field matrix for the entire pixel grids L N:

Then, the matrix G is calculated from (7.30)

(a)

(b)

FIGURE 7.1. Schematic views of the sensor lead field. (a) Biomagnetic instrument and (b) X-ray computed tomography.

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K. Sekihara and S. S. Nagarajan

However, to avoid the numerical instability when inverting it, the following regularized version is usually used: (7.31) where y is the regularization parameter. The final solution is expressed as (7.32) The regularization, however, inevitably introduces considerable amounts of smearing into the reconstruction results. Besides, the solution obtained using the minimum-norm spatial filter suffers a geometric bias in that the current estimates are forced to be closer to the sensor array than their actual locations. It should, however, emphasized that the poor performance of the minimum-norm spatial filter is not because the method itself has a serious defect but because a mismatch exists between the method and the biomagnetic instruments. This can be understood by considering a situation for other imaging modalities such as the X-ray computed tomography (CT). As is shown in Fig. 7.1(b), the overlaps between the lead fields of different sensors are very small for the X-ray CT. As a result, the matrix G is close to the identity matrix and the non-adaptive spatial filter method works quite well. Indeed, the minimum-norm spatial filter technique is considered identical to the filtered-backprojection algorithm (Herman, 1980) used for image reconstruction from projections in commercial X-ray CT systems. In the next subsection, we briefly describe investigations into ways of improving the performance of the minimum-norm-based spatial filter.

Least-squares-based interpretation ofthe minimum-norm methods The minimum-norm spatial filter is commonly derived by minimizing the leastsquares-based cost function. Actually, this least-squares-based interpretation is much more popular than the spatial-filter-based interpretation described in Section 7.4.3. Namely, the solution in Eq. (7.32) minimizes the cost function, (7.33) where S N is a source vector whose elements consist of the current estimate at the pixel points, i.e.'sN = [S(rj, t), ... ,S(rN, t)]T. In Eq. (7.33), the first term on the right-hand side is the least-squares error term and the second term is the total sum of the current norm. Therefore, the optimum solution minimizes the total current norm as well as the least-squares error. This is why the method is often referred to as the minimum-norm estimate. The trick to improving the performance of the minimum-norm method is to use a more general form of the cost function, expressed as (7.34) where iJ! represents some kind of weighting applied to the solution vector SN, and Y represents the weighting applied to the residual of the least-squares term. The solution

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Neuromagnetic Source Reconstruction and Inverse Modeling

derived by minimizing this cost function is expressed as (7.35)

In this solution, the gram matrix becomes G= LNP-I L~ + yy-I. The inclusion of the matrices P and Y gives a greater degree of freedom in regularizing G, and by choosing appropriate forms for these matrices, the numerical instability can be improved without introducing unwanted side effects such as image blur. In general, the matrix P is derived from a desired property of the solution. One widely used example is the minimum weighted-norm constraint in which we use P = p L whose non-diagonal elements are zero, and diagonal terms are given by pL

_

3k+l,3k+l -

1 IlfY(rd[[2'

and

pL

3k+2,3k+2

=

1

IJlZ(r k) 11 2

(7 36) •

This weight pL can reduce the geometric bias of the minimum-norm solution to some extent, compensating for the variation in the lead-field norm. Low resolution electromagnetic tomography (LORETA) (Pascual-Marqui and Michel, 1994; Wagner et al., 1996) is another popular application of this particular type of P. It seeks the maximally smooth solution by using P = pLpR, where pR is the Laplacian smoothing matrix. Bayesian-type estimation methods determine P based on prior knowledge of the neural current distribution (Schmidt et al., 1999; Baillet and Gamero, 1997). Determination of P by fMRI has been proposed (Liu et al., 1998; Dale et al., 2000). The matrix Y is generally determined from the noise properties. When the measurements contain non-white noise and we know the noise covariance matrix, Y is usually set to the inverse of the noise covariance matrix. The determination of the optimum forms for the matrices p and r has been an active research topic, and many investigations have been performed in this direction. However, we will not digress into the details of these investigations. Instead, in the following section we describe different approaches known as the adaptive spatial filter, which does not use the gram matrix of the lead field.

7.4.4 NOISE GAIN AND WEIGHT NORMALIZATION The spatial filter weights determine the gain for the noise in the reconstructed results. In the scalar spatial filter techniques, the output noise power due to the noise input, Pn, is given by (7.37) where R; is the noise covariance matrix. When the noise is uncorrelated white Gaussian noise, the output noise power is equal to (7.38) where 0'2 = (n(t)n T (t)) is the power of the input noise. Therefore, the norm of the filter weight vector IIw(r)[!2 is called the noise power gain or the white noise gain. In vector

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K. Sekihara and S. S. Nagarajan

spatial filter techniques, the output noise power is expressed as

Pn = tr{W T (r)(n(t)nT (t)) W T (r)} = tr{W T (r)R n W T (r)}.

(7.39)

When the input noise is uncorrelated white Gaussian noise, this reduces to (7.40) Here, the square sum of the norm of the filter weights tr{WT(r)W(r)} = IlwxCr)/I2 + Ilwy(r)f + /Iw z(r)11 2is the noise power gain. A minimum-norm spatial filter with weight normalization has also been proposed (Dale et al., 2000) The output of this spatial filter is expressed as ~ s(r, t) =

WT(r)b(t) T

tr{W (r)W(r)}

.

(7.41)

Because the weight norm is the noise gain, the output of this spatial filter is interpreted as being equal to the SNR of the minimum-norm filter outputs.

7.5 ADAPTIVE SPATIAL FILTER TECHNIQUES 7.5.1 SCALAR MINIMUM- VARIANCE-BASED BEAMFORMER TECHNIQUES The adaptive spatial filter techniques use a weight vector that depends on measurement. The best-known adaptive spatial filter technique is probably the minimum-variance beamformer. The term "beamformer" has been customarily used in the signal-processing community with the same meaning as "spatial filter". In this method, the spatial filter weights are obtained by solving the constrained optimization problem, subject to

(7.42)

and consequently we get (7.43) where I(r) is defined as I(r) = L(r )17. The idea behind the above optimization is that the filter weight is designed to minimize the total output signal power while maintaining the signal from the pointing location r. Therefore, ideally, this weight only passes the signal from a source at the location r with the orientation 17, and suppresses the signals from sources at other locations or orientations. One difficulty arises when applying it to actual MEGfEEG source localization problems. That is, when we use the spherically symmetric homogeneous conductor model to calculate I(r), the beamformer output has erroneously large values near the center of the sphere. This is because III (r) II becomes very small when r approaches the center of the sphere.

Neuromagnetic Source Reconstruction and Inverse Modeling

227

A variant of the minimum-variance beamformer, proposed by Borgiotti and Kaplan (Borgiotti and Kaplan, 1979), uses the optimization, subject to

(7.44)

The resultant weight vector is expressed as (7.45)

Because w T (r )w(r) represents the noise power gain, the output of the above bearnformer directly corresponds to the power of the source activity normalized by the power of the output noise. This Borgiotti-Kaplan beamformer is known to provide a spatial resolution higher than that of the minimum-variance beamformer (Borgiotti and Kaplan, 1979). Moreover, it can easily be seen that the output of the beamformer in Eq. (7.45) does not depend on III(r) II. Thus, III (r) 11- related artifacts are avoided. Another more serious problem with the adaptive beamformer techniques described so far is that they are very sensitive to errors in the forward modeling or errors in estimating the data covariance matrix. Since such errors are nearly inevitable in neuromagnetic measurements, these techniques generally provide noisy spatio-temporal reconstruction results, as demonstrated in Section 7.7. One technique has been developed to overcome such poor performance (Cox et al., 1987; Carlson, 1988). The technique, referred to as diagonal loading, uses the regularized inverse of the measurement covariance matrix, instead of its direct matrix inverse. Although this technique has been applied to the MEG source localization problem (Robinson and Vrba, 1999; Gross and Ioannides, 1999; Gross et al., 2001), it is known that the regularization leads to a trade-off between the spatial resolution and the SNR of the beamformer output.

7.5.2 EXTENSION TO EIGENSPACE-PROJECTION BEAMFORMER We here describe the eigenspace-projection beamformer (van Veen, 1988; Feldman and Griffiths, 1991), which is tolerant of the above-mentioned errors and provides improved output SNR without sacrificing the spatial resolution in practical low-rank signal situations. Using Eqs. (7.43) and (7.17), and defining a = l/[IT(r)Rbl/(r)], we rewrite the weight vector for the minimum-variance beamformer as (7.46) where

In Eq. (7.46), the second term on the right-hand side, arNI(r), should ideally be equal to zero because the lead-field matrix I (r) is orthogonal to EN at the source locations as indicated by Eq. (7.21). Various factors, however, prevent this term from being zero, and a

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non-zero aTNL(r) seriouslydegradesSNR as explainedin the next section.Therefore,the eigenspace-based beamformeruses only the first term of Eq. (7.46) to calculate its weight vector w(r); i.e., _ w (r )

= aTs/(r ) =

T

T s/(r ) 1

I (r )RJ; I (r )

•

(7.47)

Note that w(r) is equal to the projection of w(r) onto the signal subspace of Rb • Namely, the following relationship holds (Feldman and Griffiths, 1991; Yu and Yeh, 1995): (7.48)

w(r) = EsEJ w(r).

Therefore, the extension to an eigenspace-projection beamformeris attained by projecting the weight vectors onto the signal subspaceof the measurement covariance matrix. 7.5.3 COMPARISON BETWEEN MINIMUM- VARIANCE AND EIGENSPACE BEAMFORMER TECHNIQUES

Althoughthe minimum-variancebeamformerideallyhas exactlythe sameSNR as that of theeigenspace-based beamformer, theSNRof theeigenspacebeamformeris significantly higher in practical applications. The reason for this high SNR can be understoodas follows. Let us assume that a single source with a moment magnitudeequal to set) exists at r. We assume that the estimated lead field, l (r ), is slightly different from the true lead field I (r) . The estimate of set),s et), is derived bys(t ) = w T(r )b(t ) = w T(r)/(r)s(t ) and the average power of the estimated source momentFir ), p." is expressed as (7.49)

where P, is the average power of set) defined by P, = (s(t) 2). For the minimum-variance beamformer, this Ps is expressed as ~

r,

2

[i T (r )R -1 I T]2 (r) = a 2 Ps[1~ T (r )rsl T (r)] 2, b

= a Ps I

(7.50)

where we use the orthogonality relationship IT(r)E N = O. For the eigenspace-projection beamformer, it is also expressed as

r. = a

~

2

~T

T

2

Ps[1 (r )TsI (r) ] .

(7.51)

The average noise power Pn is obtained using Eqs. (7.38) and (7.46) and, for the minimum-variance beamformer, it is expressed as (7.52)

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Neuromagnetic Source Reconstruction and Inverse Modeling

For the eigenspace-projection beamformer, Pn is expressed as (7.53 )

Thus, the output SNR of the minimum-variance beamformer, SNR(MV), is expressed as (Chang and Yeh, 1992; Chang and Yeh, 1993)

(7.54)

The SNR for the eigenspace-based beamformer, SNR(ES). is thus

(7.55)

The only difference between Eqs. (7.54) and (7.55 ) is the presence of the second term I T (r )r~I(r ) in the denominatorof the right-hand side ofEq. (7.54). It is readily apparent thatSNR(MV) and SN~ES ) are equalif we can use an accuratenoise subspaceestimateandan T accuratelead-field vector, because the termi (r )r~I(r) isexactlyequal to zero in thiscase. It is, however, generally difficultto attain the relationship, I T (r)r~I(r) = O. One obvious reasonfor this difficultyis that whencalculatingr ~ inpractice,pstead of usi'¥ R b , the ~am­ pIe covariancematrix Rb must be used; R b is calculated from R b = 1/ k L k=l b(tk )b T (tk) where K is the number of time points. Anotherfactorthat is specifictoMEGandcausesl (r ) r~1 (r ) to havea non-zerovalue is that it is almost impossible to use a perfectly accurate lead-field vector. This is because the conductivity distribution in the brain is usually approximated by using some kind of conductor model-such as the spherically homogeneous conductor model-to calculate the lead field matrix. Although this error may be reduced to a certain extent by using a realistic head model, the error cannot be perfectly avoided. Let us define the overall error in estimating I(r) as e. Assuming that 11/(r) [1 2»11 e /12, we can rewriteEq. (7.54) as ~T

~

(7.56)

Note that, in the denominatorof the right-hand side of this equation, the norm of the matrix e T r~e has an order of magnitudeproportionalto /I e 11 2 /A~, where AN representsone of the noise-level eigenvalues of Ri : The eigenvalue AN is usually significantly smaller than the signal-level eigenvalues. Therefore, Equation (7.56 ) indicates that even when the error " e " is very small, the term e T r ~g may not be negligiblysmallcompared to the firstterm in the denominator. Thus, in practice the eigenspace-projection beamformerattainsan SNR significantly higher than that of the minimum-variance beamformer.

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7.5.4 VECTOR-TYPE ADAPTIVE SPATIAL FILTER The scalar beamfonner techniques described in the preceding subsections require determination of the beamformer orientation "l to calculate l(r). We, here, describe the extension to vector-type adaptive spatial filter techniques, which does not require the predetermination of "l. Problem of virtual source correlation

A naive way of extending to the vector beamformer is to simply use the scalar beamformer weight vector obtained with "l = f ~ to estimate the source in the ~ direction (where ~ = x, y,orz).Letustrytoestimate~(t)byusing~(t) = wT(r, f~)b(t), where uif (r , f~) is obtained using Eq. (7.43). The use of such weight vectors, however, generally gives erroneous results, and the cause of this estimation failure can be explained as follows. Let us assume that a single source with its orientation equal to "l = [1]x, 1]y, 1];f exists at r. Its activation time course is assumed to be s(t). Then, we can express the measured magnetic field as bet) = 1]xs(t)r(r) + 1] ys(t)fY(r) + 1] zs(t)[Z(r). This can be interpreted as showing that the magnetic field is generated from three perfectly correlated sources located at the same location r , with moments equal to 1]xs(t)fx' 1] ys(t)f yo and 1] zs(t)f z. Let us, for example, consider the case of estimating the x component of the source moment. The estimated moment, s,,(t), is expressed as s,,(t)

=

w]' (r )b(t)

=

[1]x w T(r, fx)r (r)

+ 1]vw T(r, f,)fY (r) + 1]; w T(r, fz)F(r )]s(t). (7.57)

Since the weight w T(r, fx) is obtained by imposing the constraint w T(r, fx)r(r) = 1, we have (7.58)

In this equation, there is no guarantee that the relationships w T (r , fx)fY (r) = 0 and w T(r , f x)F(r) = 0 hold. Instead, w T tr , f x)fY(r) and w T(r , f x)[Z(r) generally have fairly large negative values, resulting in considerable errors. A vector-extended minimum-variance beamformer

The above analysis also suggests how we can avoid such errors. Equation (7.57) indicates that the weight should be derived with the multiple constraints, (7.59)

w]'r(r) = 1,

That is, we impose the null constraints on the directions orthogonal to the one to be estimated. We here omit the notation (r) for the weight expression unless this omission causes ambiguity. Similarly, to derive Wy and lL!:, the following constraints should be imposed, w;r(r) = 0, wIrer)

= 0,

T

=

1,

and

w yT[Z(r)

wzTfY(r) = 0,

and

w;[Z(r)

W

y fY(r)

= 0, = 1.

(7.60) (7.61)

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Neuromagnetic Source Reconstruction and Inverse Modeling

The minimum-variance beamformer with such multiple linear constraints, referred to as the linearly constrained minimum-variance beamformer (Frost, 1972), is known to have the following solution: (7.62) It is clear from the discussion above that, when estimating one of the three orthogonal components of the source moment, we need to suppress the other two components. By doing this, we can avoid the errors caused by the perfectly correlated virtual sources, so the beamformer can detect the source moment projected in three orthogonal directions. Note that the set of weight vectors in Eq. (7.62) has been previously reported (van Dronge1en et al., 1996; van Veen et al., 1997; Spencer et al., 1992). In these reports, however, the necessity for imposing the null constraints was not fully explained.

Vector-extended Borgiotti-Kaplan beamformer The extension of the Borgiotti-Kaplan beamformer in Eq. (7.45) is performed in the similar manner. The weight vectors are obtained by using the following constrained minimizations, . TR mmw x bWx Wx

subject to

. w TR bWy subject to mm y w,

min WzT Rbw z W;::

,

subject to

WxT Wx = 1,

w;fY(r) = 0,

and

w;"fZ(r) = 0,

(7.63)

w vTx 1 (r) = 0,

w yT w y = 1,

and

w yTIZ(r) = 0 ,

(7.64)

w/"r(r) = 0,

w!fY(r) = 0, ,.

and

T Wz Wz = 1. (7.65)

We first derive the expression for w .r- Let us introduce a scalar constant ~ such that w;r(r) = ~ where ~ can be determined from the relationship w; Wx = 1. Then, the constrained optimization problem in Eq. (7.63) becomes

(7.66)

The solution of this optimization problem is known to have the form (7.67) Then, we have T 2 T Wx Wx = ~ f x

nf x'

where

n=

l l. T [L (r)R{;1 L(r)r L T(r)R{;2L(r)[L T (r)R/;1 L(r)r

(7.68)

232

K. Sekihara and S. S. Nagarajan

l/jI; nIx

w;

x

Thus , we get ( = from the relationship W = 1. Using exactly the same derivation, the weights wyand W z can be derived, and a set of the weights is expressed as l

w ~=

RbI L(r )[L T (r)R b l L(r )r I~

jI[nh

.

(7.69)

Extension to eigenspace-projection vector beamformer The extension to the eigenspace-projection vector beamformer is attained by using (7.70) The projection onto the signal subspace , however, cannot preserve the null constraints imposed on the orthogonal components. This can be understood by considering, for example , the case of wx' The null constraints in this case should be w.~fY (r ) = 0 and w~F (r ) = O. However, let us consider

w;JY(r) = (EsEJ wxfJY(r) = w;EsE JF (r),

w.~fZ(r)

= (E sE J wxfZZ(r) = w; EsE JfZ(r).

(7.71)

Because P(r) and fZ(r) are not necessarily in the signal subspace, we generally have EsEJP(r ) i= P(r) and EsEJZZ(r) i= fZ(r) , and therefore w~P(r) i= 0 and w~ZZ(r) i= O. It can, however, be shown that the eigenspace-projection beamformer in Eq. (7.70) can still detect the three orthogonal components of the source moment even though the null constraints are not preserved (Sekihara et aI., 2001).

7.6 NUMERICAL EXPERIMENTS: RESOLUTION KERNEL COMPARISON BETWEEN ADAPTIVE AND NON-ADAPTIVE SPATIAL FILTERS 7.6.1 RESOLUTION KERNEL FOR THE MINIMUM-NORM SPATIAL FILTER We compare the resolution kernels for the minimum-norm and the minimum-variance spatial filter techniques. These two methods are typical and basic spatial filter techniques in each category. In these numerical experiments, we use the coil configuration of the 148channel Magnes 2500™ neuromagnetometer (4D Neuroimaging, San Diego). The sensor coils are arranged on a helmet-shaped surface whose sensor locations are shown in Fig. 7.2. The coordinate origin is chosen as the center of the sensor array. The z direction is defined as the direction perpendicular to the plane of the detector coil located at this center. The x direction is defined as that from the posterior to the anterior, and the y direction is defined as that from the left to the right hemisphere. The values of the spatial coordinates (x, y , z)

233

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are expressed in centimeters. The coordinate system is also shown in Fig. 7.2. The origin of the spherically symmetric homogeneous conductor was set to (0, 0, -11). To plot the resolution kernel, we assume a vertical plane x = 0 located below the center of the sensor array (Fig. 7.2). The power ofthe resolution kernel II lR 11 2 was calculated using Eqs. (7.25), (7.27), and (7.31) for the minimum-norm method. A point source was assumed to exist at (0,0, -6), i.e., r' in Eq. (7.25) was set to (0,0, -6). The kernel was plotted within a region defined as -5 :s y :s 5 and -9 :s z :s -Ion the vertical plane of x = O. The resulting resolution kernels are shown in Fig. 7.3. Here, the results in Fig.7.3(a) show the kernel obtained from the original minimum-norm method. It is well known that the original minimum-norm method suffers from a strong geometric bias toward the sensors. The results in Fig. 7.3(a) confirm this fact. The kernels of the minimum-norm method with the lead-field normalization are shown in Fig.7 .3(b) and (c). Here, cpL in Eq. (7.36) was used and the regularization parameter y was set at O.OOOlA l for (b) and O.OOlA l t for (c). These results show that the lead-field normalization significantly improves the performance of the minimum-norm method. However, the resolution is still significantly low, particularly in the depth direction. Moreover, the peak of the kernel is located a few centimeters shallower than the assumed location; the depth difference depends on the choice of the regularization parameter. The results in Fig.7.3(d) show the kernel from the minimum-norm method with the normalized weight (Eq. (7.41)). The main lobe is significantly sharper than those in the lead-field normalization cases of (b) and (c). However, the peak is located 2cm deeper than its original position.

t This A1 is the largesteigenvalueof the gram matrix L J; LN.

234

K. Sekihara and S. S. Nagarajan -I

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( em)

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(e m )

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c

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·7 -8 . <)

(c)

y (e m)

(d)

FIGURE 7.3. Results of plotting the resolution kernels for minimum-norm-based spatial filter techniques. (a) Results for the original minimum-norm method. (b) Results for the minimum-nom, method with the normalized lead field. The regularization parameter y was set to O.OOO])cj. (c) Results for the minimum-norm method with the normalized lead field. The regularization parameter y was set to 0.00 lA I. (d) Results for the minimum-norm method with weight normalization. The parameter y was set to 0.000 I AI for the results in (a) and (d). Here, Al is the largest eigenvalue of the gram matrix L~LN.

7.6.2 RESOLUTION KERNEL FOR THE MINIMUM- VARIANCE ADAPTIVE SPATIAL FILTER Next, the resolution kernel for the minimum-variance vector beamformer technique was plotted. The kernel was calculated using Eqs. (7.25) and (7.62). Here, the covariance matrix was calculated from R b = (a 21+ PJ(rl)[T (r l)) wherer ' is setto the sourcelocation equal to (0,0, -6). The calculated resolution kernels for the four SNR values of p"ja 2 are shown in Fig. 7.4. First of all, the kernel is peaked exactly at the target source location (0,0, -6). The kernel's sharpness depends on the SNR value. This is one characteristic of the adaptive spatial filter methods. We encounter the SNR between 8 and 2 in most actual measurements. In such an SNR range, the kernel sharpness obtained with the minimumvariance spatial filter is significantly higher than that with the non-adaptive minimum-norm method. These results clearly demonstrate that the minimum-variance filter can provide more accurate reconstruction results with significantly higher spatial resolution than the minimum-norm spatial filter.

J

235

Neuromagnetic Source Reconstruction and Inverse Modeling -I

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0

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(b)

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y(em)

-8 -4

(c)

-2

0 y(em )

4

-9

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(d )

-2

0

2

4

y (em)

FIGURE 7.4. Results ofplotting the resolution kernels for the minimum-variance adaptive spatial filter technique. The plots are for SNR values of (a) 8, (b) 4, (c) 2, and (d) I.

7.7 NUMERICAL EXPERIMENTS: EVALUATION OF ADAPTIVE BEAMFORMER PERFORMANCE 7.7.1 DATA GENERATION AND RECONSTRUCTION CONDITION We next conducted a series of numerical experiments to test the performance of the adaptive beamformer technique. Here, we assume the 37-sensor array of Magnes" neuromagnetometer in which the sensor coils are arranged in a uniform, concentric array on a spherical surface with a radius of 12.2 em. The sensors are configured as first-order axial gradiometers with a baseline of 5 em. Three signal sources were assumed to exist on a plane defined as x = O. The source configuration is schematically shown in Fig. 7.5(a) and their time courses are shown Fig. 7.5(b). The magnetic field was generated at a l-ms interval from 0 to 400 ms. Gaussian noise was added to the generated magnetic field, and the SNR, defined as the ratio of the Frobenius norm of the signal-magnetic-field data matrix to that of the noise matrix, was set to 16. The generated magnetic field is shown also in Fig.7.5(b). The covariance matrix R b was calculated using the data obtained with this time window between 0 and 400 ms. We express the source-moment vector using the two tangential components ((), ¢), and the radial component is assumed to be zero. The reconstruction was

236

K. Sekiha ra and S. S. Nagaraj an

37-channel sensor array

-5

three sources

..--

E

u

';:;' -10

-15

5 sphere for the forward modeling

(a)

FIGURE 7.5. (a) The coordinate system and the source configuration used in the numerical experiments in Section 7.7. The cross section atx = is shown. The square shows the reconstruction region for the experimental results shown in Figs. 7.6-7.9. (b) Time courses of the three sources assumed in the numerical experiments. Time courses from the first to the third sources are shown from the top to the third row. respectively. The three vertical broken lines indicate the time instants 220, 268, and 300 ms, at which the source-moment magnitude is displayed. The bottom row shows the generated magnetic field.

°

performed by using sq,(r, t )

= wI (r)b(t)

and so(r, t)

= w[ (r)b( t) .

(7.72)

The reconstruction region was defined as the area between -4 :::: y :::: 4 and -8 :::: z :::: - 3 on the plane x = 0, and the reconstruction interval was 1 mm in the y and z directions. Once sq,(r, t) and so(r , t) were obtained, cp, an angle repre senting the mean source direction in the ¢ - plane was calculated using

e

cp = arctan (

(so(t ?) ) (Sq, (t )2)

~ = -an:tan (

(;,(1)') ) (sq,(t)2)

if

(so(t )) > 0 (sq,(t)) - ,

if

(so(t)) < 0 (sq, (t )) ,

(7.73)

where (.) indicates the average over the time window with which R b was calculated. Then , the time course expressed in the mean source direction , slI(r , t ), and that in its orthogonal

237

Neuromagnetic Source Reconstruction and Inverse Modeling

220

268 300

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:;

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0

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c

100

200

300

400

late ncy (ms)

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2

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E 0 <:J

'S «: "E 100

0 (b )

200 latency (ms)

300

400

FIGURE 7.5. (cont .)

direction, S1-(r , t) were given by slI(r, t )

=s¢(r, t ) cos q; + so(r, t ) sin q; ,

h(r, t ) = so(r, t ) cos q; - ;;P(r, t) sin q;.

(7.74)

In the following experiments, we used slI(r, t ) and h(r. t) when displaying the time course of a source activity. To display the results of the spatio-temporal reconstruction, three time points at 220, 268, and 300 ms were selected . The amplitude of the second source happened to be zero at 220 ms, and all the sources had non-zero amplitudes at 268 ms, while only the second

238

K. Sekihara and S. S. Nagarajan

source had a non-zero amplitude at 300 ms. The snapshots of the source magnitude distribution [S(r , t )1 = reconstruction

s ~ (r , t ) + s~ (r , t ) at these three time points, and the time averaged

J (s (r , t )2) are presented in the following experiments.

7.7.2 RESULTS FROM MINIMUM-VARIANCE VECTOR BEAMFORMER

The results of the spatio-temporal reconstruction obtained using the minimum-variance vector beamformer in Eq. (7.62) are shown in Fig. 7.6(a). The estimated time courses at the pixels nearest to the three source locations are shown in Fig. 7.6(b). These results show that the reconstruction at each instant in time was fairly noisy : the snapshot at 220 ms showed some influence from the second source, and the snapshot at 300 ms contained the activities of the first and third sources. The time-averaged reconstruction, however, resolved three active sources. In Fig. 7.6(b), only slI(r, t) shows the source activity time courses, but S.L (r , t) contains no significant activities. This confirms the fact that the source orientation is fixed during the observation period. We next tested the minimum-variance beamformer with the regularized inverse (Rb + y I)-I instead of R"b 1. The regularization parameter was set to 0.003).. It . The results in Fig. 7.7(a) show that a considerable amount of blur was introduced. The estimated time courses are shown in Fig. 7.7(b). This figure shows that the SNR of the beamformer output increased considerably in this case, although each time course shows some influence from neighboring sources. The results demonstrate that the regularization leads to a trade-off between the spatial resolution and the SNR of the beamform er output. 7.7.3 RESULTS FROM THE VECTOR-EXTENDED BORGIOTT/-KAPLAN BEAMFORMER

The reconstruction results from weight vectors obtained using Eq. (7.69) are shown in Fig. 7.8. The weight is equivalent to the vector-extended Borgiotti-Kaplan (B-K) beamformer without the eigen space projection. Comparison between the time-averaged reconstruction in Fig. 7.6(a) and in Fig . 7.8(a) confirms that the Borgiotti-Kaplan-type beamformer has a spatial resolution much higher than the minimum-variance beamformer. The spatio-temporal reconstruction, however, is very noisy in both cases. 7.7.4 RESULTS FROM THE EIGENSPACE PROJECTED VECTOR-EXTENDED BORGIOTTI- KAPLAN BEAMFORMER

We then applied the eigensp ace-projected B-K beamformer obtained using Eqs. (7.69) and (7.70) to the same computer-generated data set. The reconstructed source distribution s are shown in Fig. 7.9(a), and the estimated time courses are shown in Fig. 7.9(b). Comparison between Figs. 7.8 and 7.9 confirms that the eigenspace projection can improve the SNR with almost no sacrifice in spatial resolution. Comparing the results in Fig. 7.9 with the minimum-variance results in Fig. 7.6, we can clearly see that the eigenspace-projected B-K beamformer technique significantly improved both spatial resolution and output SNR. :j: This AI is the largest eigenvalue of Rh.

239

Neuromagnetic Source Reconstruction and InverseModeling

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E. ... 6

8 -I

Q ,

6

8

2

2

(a)

~ ~

:-

.~

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100 ( 0)

200

300

-100

latency (m s)

FIGURE 7.6. (a) Results of the spatio-temporal reconstruction obtained using the minimum-variance-based vector beamformer in Eq. (7.62). The upper-left, upper-right, and lower-left maps show the snapshots of the source-moment magnitude at 220 ms, 268 ms, and 300 ms, respectively. The lower-right map shows the timeaveraged reconstruction. (b) Estimated time courses from the first to the third sources are shown from the top to the bottom, respectively. The two time courses in each panel correspond to slI(r, t) and h(r, I). The three vertical broken lines indicate the time instants at 220, 268, and 300 ms.

240

K. Sekihara and S. S. Nagarajan

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(a)

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c:; > u

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100 (b)

200 latency (ms)

300

400

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Neuromagnetic SourceReconstruction and Inverse Modeling

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242

K. Sekihara and S. S. Nagarajan

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Neuromagnetic Source Reconstruction and Inverse Modeling

243

z

FIGURE 7.10. The x, y, and z coordin ates used to express the reconstructi on results in Section 7.8. The coordinate origin is defined as the midpoint between the left and right pre-auricular points. The axis directed away from the origin toward the left pre-auricular point is defined as the +y axis, and that from the origin to the nasion as the +x axis. The +z axis is defined as the axis perpendicular to both these axes and is directed from the origin to the vertex.

7.8 APPLICATION OF ADAPTIVE SPATIAL FILTER TECHNIQUE TO MEG DATA This section describes the application of the adaptive spatial filter technique to actual MEG data. The MEG data sets were collected using the 37-channel Magnes" neuromagnetometer. The first data set is an auditory and somatosensory combined response, which contains two major source activities. We show that the adaptive spatial filter technique can reconstruct these two sources and retrieve their time courses. The second data set is the somatosensory response with very high SNR achieved by averaging 10000 trials. With this data set, we show that the adaptive technique can separate cortical activities only 0.7-cm apart . Throughout this section, we use the head coordinates shown in Fig. 7.10 to express the reconstruction results.

7.8.1 APPLICATION TO AUDITORY-SOMATOSENSORY COMBINED RESPONSE The evoked response was measured by simultaneously presenting an auditory stimulus and a somatosensory stimulus to a male subject. The auditory stimulus was a 200 ms puretone pulse with 1 kHz frequency presented to the subject's right ear, and the somatosensory stimulus was a 30 ms tactile pulse delivered to the distal segment of the right index finger. These two stimuli started at the same time. The sensor array was placed above the subject's left hemisphere with the position adjusted to optimally record the Nlm auditory evoked field. A total of 256 epochs were measured, and the response averaged over all the epochs is shown in the upper part of Fig . 7.11. The adaptive vector beamformer technique was applied to localize sources from this data set. The covariance matrix R b was calculated with the time window between 0 ms and 300 ms. We calculated, by using Eqs. (7.69) and (7.70), the eigenspace-projected Borgiotti-Kaplan weight matrix containing the two weight vectors [w , we], and estimated the source magnitude vector Flr, t ) using Eq. (7.72). The signal subspace dimension Q was

244

K. Sekihara and S. S. Nagarajan

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350

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5

5

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94ms

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FIGURE 7.11. Results of the spatio-temporal reconstruction from the auditory-somatosensory combined response shown in the upper trace of this figure. The auditory-somatosensory combined response was measured by simultaneously applying an auditory stimulus and a somatosensory stimulus. A total of256 epochs were averaged. The contour maps show reconstructed source magnitude distributions at three different latencies (65, 138, and 194 ms). The reconstruction grid spacing was set to 5 mm. The maximum-intensity projections onto the axial (left column), coronal (middle column), and sagittal (right column) directions are shown. The letters Land R indicate the left and right hemispheres. The circles depicting a human head show the projections of the sphere used for the forward modeling.

set to two because the eigenvalue spectrum of R b showed two distinctly large eigenvalues. The maximum-intensity projections of the reconstructed moment magnitude Us(r, t) 11 2 onto the axial, coronal, and sagittal planes are shown in Fig. 7.11. The source magnitude at three latencies, (65, 138, and 194 ms), is shown in this figure. The source magnitude map at 138 ms contains a source activity presumably in the primary somatosensory cortex. The source magnitude map at 194 ms shows a source activity in the primary auditory cortex. The map at 65 ms contains both of these activities. The time courses of points in the primary somatosensory and auditory cortices are shown in Figs. 7.l2(a) and (b), respectively. The coordinates of these cortices were determined from the maximum points in the source magnitude maps at 138 ms and 194 ms. In Fig. 7.12(a) the P50 peak, which is known to represent the activity of the primary

245

Neuromagnetic Source Reconstruction and Inverse Modeling

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somatosensory cortex, is observed at a latency of about 50 ms. In Fig. 7.12(b), the auditory Nlm peak is observed at a latency of about 100 ms.

7.8.2 APPLICATION TO SOMATOSENSORY RESPONSE: HIGH-RESOLUTION IMAGING EXPERIMENTS Electrical stimuli with 0.2 ms duration were delivered to the right posterior tibial nerve at the ankle with a repetition rate of 4 Hz. The MEG recordings were taken from the vertex centering at Cz of the intemationall0-20 system. An epoch of 60 ms duration was digitized at a 4000 Hz sampling frequency and 10000 epochs were averaged. The upper part of Fig. 7.13 shows the MEG signals, recorded over the foot somatosensory region in the left hemisphere. The eigenspace-projected Borgiotti-Kaplan beamformer was applied to this MEG recording. The covariance matrix Rb was calculated with a time window between 20 and 45 ms containing 100 time samples. The maximum-intensity projections of the reconstructed source magnitude IIs(r, t) 11 2 onto the axial, coronal, and sagittal planes are shown in Fig. 7.13. The source magnitude maps revealed initial activation in the anterior part of the S1 foot area at 33.1 ms, followed by co-activation of the posterior part of S 1 cortex at 36.2 ms. The posterior activation became dominant at 37.2 ms and the initial anterior activation completely disappeared at 39.1 ms. Fig. 7.14 shows the source magnitude map at 36.9 ms overlaid, with proper thresholding, onto the subject's MRI. Here, the anterior source was probably in area 3b and the posterior source was in an area near the marginal sulcus. The

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(em)

FIGURE 7.13. Results of the spatio-temporal reconstruction from the somatosensory response shown in the upper trace of this figure. The somatosensory response was measured using the right posterior tibial nerve stimulation. The contour maps show reconstructed source magnitude distributions at four different latencies. The reconstruction grid spacing was set to I mm. The maximum-intensity projections onto the axial (left column), coronal (middle column), and sagittal (right column) directions are shown. The letters Land R indicate the left and right hemispheres. The circles depicting a human head show the projections of the sphere used for the forward modeling.

separation of the two sources was approximately 7 mm, demonstrating the high-resolution imaging capability of the adaptive spatial filter techniques. Details of this investigation have been reported (Hashimoto et al., 2001a), and the results of applying the adaptive beamformer technique to the response from the median nerve stimulation have also been reported (Hashimoto et al., 200Ib).

ACKNOWLEDGMENTS The author would like to thank Dr. D. Poeppel, Dr. A. Marantz, and Dr. T. Roberts for providing the auditory data. We are also grateful to Dr. I. Hashimoto, and Dr. K. Sakuma for

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FIGURE 7.14. The source magnitude reconstruction results at a latency of 36.9 ms. The source magnitude map was properly thresholded and overlaid onto the sagittal cross section of the subject's MRI. The colors represent the relative intensity of the source magnitude; the relationship between the colors and relative intensities is indicated by the color bar. The anterior source was probably in area 3b and the posterior source was in an area near the marginal sulcus. The separation of the two sources was approximately 7 mm in this case. See the attached CD for color figure.

providing the somatosensory data and for useful discussion regarding the interpretation of the reconstructed results. This work has been supported by Grants-in-Aid from the Kayamori Foundation of Informational Science Advancement; Grants-in-Aid from the Suzuki Foundation; and Grants-in-Aid from the Ministry of Education, Science, Culture and Sports in Japan (C13680948). This work has also been supported by the Whitaker Foundation, and by National Institute of Health. (P4IRRI2553-03 and ROI-DC004855-0IAI).

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8 MULTIMODAL IMAGING FROM NEUROELECTROMAGNETIC AND FUNCTIONAL MAGNETIC

RESONANCE RECORDINGS Fabio Babiloni and Febo Cincotti

Dipartimento di Fisiologia Umana e Farmacologia, Universita di Roma "La Sapienza", Roma, Italy

8.1 INTRODUCTION Human neocortical processes involve temporal and spatial scales spanning several orders of magnitude, from the rapidly shifting somatosensory processes characterized by a temporal scale of milliseconds and a spatial scales of few square millimeters to the memory processes, involving time periods of seconds and spatial scale of square centimeters. Information about the brain activity can be obtained by measuring different physical variables arising from the brain processes, such as the increase in consumption of oxygen by the neural tissues or a variation of the electric potential over the scalp surface. All these variables are connected in direct or indirect way to the neural ongoing processes, and each variable has its own spatial and temporal resolution. The different neuroimaging techniques are then confined to the spatiotemporal resolution offered by the monitored variables. For instance, it is known from physiology that the temporal resolution of the hemodynamic deoxyhemoglobin increase/decrease lies in the range of 1-2 seconds, while its spatial resolution is generally observable with the current imaging techniques at few mm scale. Today, no neuroimaging method allows a spatial resolution on a mm scale and a temporal resolution on a msec scale. Hence, it is of interest to study the possibility to integrate the information offered by the different physiological variables in a unique mathematical context. This operation is called the "multimodal integration" of variable X and Y, when the X variable has typically particular appealing spatial resolution property (mm scale) and the Y variable has particular attractive temporal properties (on a ms scale). Nevertheless, the issue of several temporal and spatial domains is Corresponding author: Dr. Fabio Babiloni, Dipartimento di Fisiologia Umana e Farmacologia, Universita di Roma "La Sapienza", P.le A. Mow 5, 00185 Roma, Italy, Tel: +39-06-49910317, Fax: +39-06-49910917, Email: fabio.babiloniteuniromal.it

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critical in the study of the brain functions, since different properties could become observable, depending on the spatio-temporal scales at which the brain processes are measured. Electroencephalography (EEG) and magnetoencephalography (MEG) are two interesting techniques that present a high temporal resolution, on the millisecond scale, adequate to follow brain activity. Unlikely, both techniques have a relatively modest spatial resolution, beyond the centimeter. In spite ofa lack of spatial resolution, neural sources can be localized from EEG or MEG data by making a priori hypotheses on their number and extension. A more detailed description of the techniques involved in the high resolution EEG and MEG recordings and imaging can be found in Chapter 7 and Chapter 8. Here, we briefly recall that the so called high resolution EEG methods included: (i) subject's multi-compartment head model (scalp, skull, dura mater, cortex) constructed from magnetic resonance images, the sampling of EEG potentials from 64-128 electrodes; (ii) the computation of the surface Laplacian from the scalp potential recordings and/or the use of multi-dipole source model for characterizing the neural active sources. However, the spatial resolution of the EEG/MEG techniques is fundamentally limited by the inter-sensor distances and by the fundamental laws of electromagnetism (Nunez, 1981). On the other hand, the use of apriori information from other neuroimaging techniques like functional magnetic resonance imaging (fMRI) with high spatial resolution could improve the localization of sources from EEG/MEG data. This chapter deals with the multimodal integration of electrical, magnetic and hemodynamic data to locate neural sources responsible for the recorded EEG/MEG activity. The rationale of the multimodal approach based on fMRI, MEG and EEG data to locate brain activity is that neural activity generating EEG potentials or MEG fields increases glucose and oxygen demands (Magistretti et aI., 1999). This results in an increase in the local hemodynamic response that can be measured by fMRI (Grinvald et al., 1986; Puce et al., 1997). On the whole, such a correlation between electrical and hemodynamic concomitants provides the basis for a spatial correspondence between fMRI responses and EEG/MEG source activity. The chapter is organized as follows: first, a brief introduction on the principles at the basis of the fMRI recordings will be presented; then a recall of the principal techniques used for EEG and MEG for locating neural sources will be presented, with special emphasis on cortical imaging and linear distributed solutions. This last technique will be employed to show both the mathematical principle and the practical applications of the multimodal integration of EEG, MEG and fMRI for the localization of sources responsible for intentional movements.

8.2 GENERALITIES ON FUNCTIONAL MAGNETIC RESONANCE IMAGING A brain imaging method, known as fMRI, has gained favor among neuroscientists over the last few years. Functional MRI reflects oxygen consumption and, as oxygen consumption is tied to processing or neural activation, can give a map of functional activity. When neurons fire, they consume oxygen and this causes the local oxygen levels to briefly decrease and then actually increase above the resting level as nearby capillaries dilate to let more oxygenated blood flow into the active area. The most used acquisition paradigm is the so-called Blood Oxygen Level Dependence (BOLD), in which the fMRI scanner works by imaging blood oxygenation. The BOLD paradigm relies on the brain mechanisms, which overcompensate for oxygen usage (activation causes an influx of oxygenated blood in excess of that

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A

B

FI GUR E 8.1. Physiologic principle at the base of the generation of fMRI signals . A) Neurons increase their firing rates, increasing also the oxygen consumption. B) Hemodynamic response in a second scale increases the diameter of the vessel close to the activated neurons. The induced increase in blood flow overco mes the need for oxyge n supply. As a consequence, the percentage of deoxyhemoglobin in the blood flow decreases in the vesse l with respect to the figure A). See the attached CD for color figure.

used and therefore the local oxyhemoglobinconcentration increases). Oxygen is carried to the brain in the hemoglobin molecules of blood red cells. Fig. 8.1 shows the physiologic principle at the base of the generation of tMRI signals. In this figure it is shown how the hemodynamic responses elicited by an increased neuronal activity (A) produces a decrease of the deoxyhemoglobin content of the blood flow in the same neuronal district after few seconds (B). The magnetic properties of hemoglobin differ when it is saturated with oxygen compared to when it has given up oxygen. Technically, deoxygenated hemoglobin is "paramagnetic" and therefore has a short T2 relaxation time. As the ratio of oxygenated to deoxygenated hemoglobin increases, so does the signal recorded by the MRI. Deoxyhemoglobin increases the rate of depolarization of hydrogen nuclei creating the MR signal thus decreases the intensity of the T2 image. The bottom line is that the intensity of images increases with the increase of brain activation.The problem is that at the standard intensity used for the static magnetic field (1.5 Tesla) this increase is small (usually less than 2%) and easily obscured by noise and different artifacts. By increasing the static field of the fMRI scanner, the signal to noise ratio increases to more convenient values. Static field values of 3 Tesla are now commonly used for research on humans, while recently fMRI scanner at 7 Tesla was employed to map hemodynamic responses in the human brain (Bonmassar et al., 2001). At such high field value, the possibility to detect the initial increase of deoxyhemoglobin (the initial "dip") increase. The interest in the detection of the dip is based on the fact that this hemodynamic response happens on timescale of 500 ms (as revealedby hemodynamic optic measures; Malonek and Grinvald, 1996), compared to 1-2 seconds needed for the response of the vascular system to the oxygen demand. Furthermore, in the latter case the response has a temporal extension well beyond the activation occurred (10 seconds). As a last point, the spatial distribution of the initial dip (as described by using the optical dyes;

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Malonek and Grinvald, 1996) is sharper than those related to the vascular response of the oxygenated hemoglobin. Recentl y, with high field strength MR scanners at 7 or even 9.4 Tesla (on animals), a resolution down to cortic al column level has been achieved (Kim et al., 2000). However, at the standard field intensity commonly used in fMRI studies (1.5 or 3 Tesla), the identification of such initial transient increase of deoxyhemoglobin is controversial. Compared to positron emitted tomography (PET) or single photon emitted tomography (SPECT), fMRI does not require the injection of radio-labeled substances, and its images have a higher resolution (reviewed in Rosen et al., 1998). PET, however, is still the most informati ve technique for directly imaging metabolic processes and neurotransmitter turnover.

8.2.1 BLOCK-DESIGN AND EVENT-RELATED fMRI Though dynamic fMRI experiments were early recognized to be fundamentally different from previous hemodynamically based functional imaging methods (like, for instance, Positron Emitted Tomography ; PET), early studies in fMRI typically used experimental paradigms that could have been easily performed by using previous nuclear technologies. Spec ifically, most experiments were performed by using extended period s of "on" versus "off" activations, in a way called block designs paradigm . Such paradigm had been used in dozens of functional studies of sensory and higher cortic al function using PET and single photon emission computed tomography (SPECT) for more than a decade. Nevertheless, although such block designs are a necessity when imaging hemodynamics by using techn iques that requir e a quasi-equilibrium physiological state for periods up to I min, they were clearl y not requi red for fMRI experiments, where activity was detectable within seco nds from stimulus onset. Movement away from block designs was gradual and aided by a number of studies exploring fMRI signal responses to brief stimulus events (as long as 2 seco nds or less; Blamire et al., 1992). A detectable signal change in fMRI was shown to be produ ced by 2 s or shorter stimulus (Blamire et al., 1992; Bandett ini, 1993). More over, it was also shown that visual stimulation as brief as 34 msec in duration could elicit small, but clea rly detectable, signal changes (Savoy et al., 1995). All these data suggest that tMRI is sensitive to transient phenomena and can provide at least some degree of quantitative information on the underlying neuronal behavior. Together, these results thus suggest that it should be possible to interpret transient fMRI signal changes in ways directly analogous to electrophysiologic evoked potenti als. A first step in this direction was made by Dale and Buckner (reviewed in Rosen et al., 1998) who showed that visual stimuli lateralized to one hemifield could be detected within intermixed trial paradigms. By using methods similar to those applied in the field of evoked response potential research, the trials were selectively averaged to reveal the predicted pattern of contralateral visual cortex activation. Taken together with the above observations, these collective data demonstrate convincingly that fMRI is capable of detecting changes related to single-tas k events and brief epoch s of stimulation. Hence, the paradigm in which the fMRI information was collected on a trial-by-trial basis is called "event-related fMRI" .

8.3 INVERSE TECHNIQUES The ultimate goal of any EEG, MEG and fMRI recordin gs is to produce information about the brain activity of a subject during a particular sensorimotor or cogniti ve task.

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The mathematical procedures that allow us to recover information about the activity of the neural sources from the non-invasive EEG/MEG recordings are called inverse techniques. The inverse techniques have been systematically treated and reviewed, with application to ECG (Chapter 4), and MEG (Chapter 7). Mathematical models for the head as volume conductor and for the neural sources are employed by linear and non-linear minimization procedures to localize putative sources of EEG data. Several studies have indicated the adequacy of the equivalent current dipole as a model for the cortical sources (Nunez, 1981, 1995), while the importance of realistic geometry head volume conductor models for the localization of cortical activity has been stressed more recently (Gevins, 1989; Gevins et aI., 1991, 1999, Nunez, 1995). Results of previous intracranial EEG studies have led support to the idea that high resolution EEG techniques (including head/source models and proper regularization inverse procedures) might model with an acceptable approximation the strengths and extension of cortical sources of surface EEG data, at least in certain conditions (Le and Gevins, 1993; Gevins et aI., 1994; He et aI., 2002). We briefly present a survey on the principal inverse techniques, with a particular emphasis on the so-called "distributed" solutions, which we will use to demonstrate the multimodal integration of EEG/MEG and fMRI.

8.3.1 ACQUISITION OF VOLUME CONDUCTOR GEOMETRY A key point of high-resolution EEG and MEG technologies is the availability of the accurate model of the head as a volume conductor by using anatomical MRIs. These images are obtained by using the MRI facilities largely available in all research and clinical institutions worldwide. Reference landmarks such as nasion, inion, vertex, and preauricular points may be labeled using vitamin E pills as markers. Tl-weighted MR images are typically used since they present maximal contrast between the structures of interest. Contouring algorithms allow the segmentation of the principal tissues (scalp, skull, dura mater) from the MR images (Dale et aI., 1999). Separate surfaces of scalp, skull, dura mater and cortical envelopes are extracted for each experimental subject, yielding a closed triangulated mesh. This procedure produces an initial description of the anatomical structure that uses several hundred thousands points--quite too much for subsequent mathematical procedures. These structures are thus down-sampled and triangulated to produce scalp, skull and dura mater geometrical models with about 1000-1300 triangles for each surface. These triangulations were found adequate to model the spatial structures of these head tissues. A different number of triangles are used in the modeling of the cortical surface, since its envelope is more convoluted than the scalp, skull and dura mater structures. A number of triangles variable from 5000 to 6000 may be used to model the cortical envelope for the purpose of following the spatial shape of the cerebral cortex. In order to allow coregistration with other geometrical information, the coordinates of the triangulated structures are referred to an orthogonal coordinate system (x, y, z) based on the positions of nasion and pre-auricular points extracted from the MR images. For instance, the midpoint of the line connecting the pre-auricular points can be set as the origin of the coordinate system and, with the y axis going through the right pre-auricular point, the x axis lying on the plane determined by nasion and pre-auricular points (directed anteriorly) and the z axis normal to this plane (directed upward). Once the model of scalp surface has been generated, the integration of the electrodes' positions are accomplished by using the information about the sensor locations

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FIGURE 8.2. Realistic MRI-constructed head of a human subject. Electrode positions (128) are shown on the MRI-constructed scalp surface and on the underlying cortex surface.

produced by the 3-D digitizer. The sensors positions on the scalp model are determined by using a non-linear fitting technique. Fig. 8.2 shows the results of the integration between the EEG scalp electrodes position and a realistic head model.

8.3.2 DIPOLE LOCALIZATION TECHNIQUES Dipole localization techniques produced estimates of the position and moment of one or several equivalent current dipoles localized in a head model from the non-invasive EEG and/or MEG recordings. From the position of the localized current dipoles in the head model, inferences about the neural sources in the real brain are inferred. So far, two approaches to dipole localization have become popular in neuroscience, and both rely on the solution of non-linear minimization algorithms. The first approach is the so-called "moving dipole" method (Cohen et al., 1990). Dipoles are found at a succession of discrete times, with no a priori assumption about the relation of the localized dipoles at different time instants. In general, it is difficult to locate more than two dipoles for each potential/magnetic field recorded due to the numerical instability of the inverse procedure. However, also with this limitation, this procedure is very popular and allows locating sources mainly in the primary sensory cortical areas. Fig. 8.3 shows the localization of a current dipole (the red arrow) indicating the restricted cortical areas responsible for the generation of the characteristic magnetic field distribution recorded by the magnetic sensors 20 ms after a stimulus delivery at right wrist in humans.

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FIGURE 8.3. Localization of the equivalent current dipole (the red arrow) indicating the restricted cortical areas responsible for the generation of the characteristic magnetic field distribution occurring over the magnetic sensors 20 ms after a stimulus delivered at the right wrist of a subject (N201P20). The position of the dipole is integrated into a realistic head model built by segmenting sequential magnetic resonance images of the subject. See the attached CD for color figure.

The position of the localized dipole was then integrated into a realistic head model built according to the procedures described above. The second approach to the dipole localization combines both the spatial and temporal properties of scalp potentials/fields, to increase the ratio of available data to degrees of freedom for the minimization procedures. This results in an increase of the number of dipoles that may be reliably localized from EEG and MEG recordings. Different constraints are applied to find the best inverse solutions, for example setting the position of the dipoles and estimating the time/series of the dipole moments, or determining the orientation of the dipoles and setting their positions. This last approach is called multiple source analysis (MSA; Scherg and von Cramon, 1984; Ebersole 1997, 1999; Scherg et aI., 1999).

8.3.3 CORTICAL IMAGING The possibility to model the complex head geometry with the finite element technique allowed Alan Gevins and colleagues to derive a method, they called deblurring, that estimates potential distribution on the dura mater surface by using non invasive EEG recordings (Le and Gevins, 1993; Gevins et aI., 1994). This method still uses non-linear minimization techniques but did not use any explicit model of the neural sources. In fact, by just applying Poisson's equation, Gevins and co-workers were able to move back from the scalp potential distribution to the dura mater potential distribution. This method was also validated by

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i

~· I 3) Generated cortical f potentials from the estimated dipole strenghts

I) EEG signa s

2) Estimated cortical strenghts from EEG siznals

FIGUR E 8.4. Possible representation of the corti cal imaging technique. The acquired EEG scalp potentials ( I) are used to estimate the cortical dipole strengths at the dipo le layer level, here represented with the realistic cortical surface . The estimated cortical dipole strength s (2) are then used to generate the potential distribution over a dura mater surface (3) using the basic laws of electromagnetism. Such distribution can be generated at any other modeled head structure.

using epicortical recording s, and the deblurred dura mater potential distributions showed a clear improvement with respect to the examination of the raw potential distribution s over the scalp. It is worth of note that the mathematical model supporting the deblurring method is not suitable to accommodate fMRI or PET inform ation. In fact, mathematical framework that allow integration between electromagnetic and metabolic modalities, require the sources of currents in the brain to be explicitly modeled. Another technique useful to recover improved images of cortical distributions from EEG scalp recording s is known as cortical imaging . In this technique, an explicit model of the neural sources, i.e. the current dipole, is used. In general, a layer of current dipoles simulates the cortical surface, and the retrieved dipole strengths are then used to generate potential distributions over a surface of the head model simulating the dura mater. Fig. 8.4 shows the idea at the base of the cortic al imaging technique. A three- shell (scalp, inner and outer surface of the skull) realistic head volume conductor is represented, together with the cortical dipole layer. It has been proven that even the use of homogeneous spherical volume conductor for the head and a realistic cortical surface for the dipole layer provided more focused and detailed information than the raw scalp potential s (Sidman et al., 1992, Srebro et aI., 1993, Srebro and Oguz , 1997). However, it must be noted that the condu ctivity ratio between skull and scalp is far than 1 as assumed in homogeneous models. The value adopted for such ratio by all the researchers in the field in these last 30 years is 1:80 (Rush and Driscoll, 1968), or even I :15 as stated more recently (Ooste ndorp et al., 2000) . According to this observation, several researchers (He et aI., 1996; Babiloni et aI., 1997; He, 1999; He et al., 1999) developed cortical imaging techniques that took into account the inhomogeneity of the head as volume conductor by using realistic head models and boundary element mathem atics. By regularization, dura mater potentials obtaine d both

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from simulation and real recordings presented improved spatial characteristics with respect to the use of raw scalp potentials. It is worth of note that the mathematical framework of the cortical imaging technique allows in principle the integration of tMRI priors. In fact, since the cortical imaging method is a linear inverse technique (He, 1999), all the multimodal integration we will present in the following paragraph for the EEG/MEG and tMRI data for the distributed linear inverse solution can be in theory applied also for the cortical imaging. Beside to the use of Green's second identity, another approach to the imaging of the cortical potential distribution from non-invasive recordings was made by using the spherical harmonic functions (Nunez et al., 1994; Edlinger et al., 1998). However, the mathematical framework developed for the estimation of cortical potentials do not allow easily the integration of tMRI information. In fact, since this method recovers the "deblurred" cortical potential distribution but not the cortical current strengths, it is difficult to integrate the information about the activation of patches of cortical tissues obtained by tMRI.

8.3.4 DISTRIBUTED LINEAR INVERSE ESTIMATION As seen before, when the EEG activity is mainly generated by circumscribed cortical sources (i.e. short-latency evoked potentials/magnetic fields), the location and strength of these sources can be reliably estimated by the dipole localization technique (Scherg et al., 1984, Salmelin et al., 1995). In contrast, when EEG activity is generated by extended cortical sources (i.e. event-related potentials/magnetic fields), the underlying cortical sources can be described by using a distributed source model with spherical or realistic head models (Grave de Peralta et al., 1997; Pascual-Marqui, 1995; Dale and Sereno, 1993). With this approach, typically thousands of equivalent current dipoles covering the cortical surface modeled and located at the triangle center were used, and their strength was estimated by using linear and non linear inverse procedures (Dale and Sereno, 1993; Uutela et al., 1999). Taking into account the measurement noise n, supposed to be normally distributed, an estimate of the dipole source configuration that generated a measured potential b can be obtained by solving the linear system:

Ax-l n

e

b

(8.1)

where A is a m x n matrix with number of rows equal to the number of sensors and number of columns equal to the number of modeled sources. We denote with A j the potential distribution over the m sensors due to each unitary j-th cortical dipole. The collection of all the m-dimensional vectors A j , (j = 1, ... , n) describes how each dipole generates the potential distribution over the head model, and this collection is called the lead field matrix A This is a strongly under-determined linear system, in which the number of unknowns, dimension of the vector x, is greater than the number of measurements b of about one order of magnitude. In this case from the linear algebra we know that infinite solutions for the x dipole strength vector are available, explaining in the same way the data vector b. Furthermore, the linear system is ill-conditioned as results of the substantial equivalence of several columns of the electromagnetic lead field matrix A In fact, we know that each column of the lead field matrix arose from the potential distribution generated by the dipolar sources that are located in similar positions and have orientations along the cortical model used. Regularization of the inverse problem consists in attenuating the oscillatory modes

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generated by vectors that are associated with the smallest singular values of the lead field matrix A, introducing supplementary and apriori information on the sources to be estimated. In the following, we characterize with the term "source space" the vector space in which the "best" current strength solution x will be found. "Data space" is the vector space in which the vector b of the measured data is considered. The electrical lead field matrix A and the data vector b must be referenced consistently. Before we proceed to the derivation of a possible solution for the problem drawn in (8.1) we recall few definitions of algebra useful for the following. A more complete introduction to the theory of vector spaces is out of the scope of this chapter, and the interested readers could refer to related textbooks (Spiegel, 1978; Rao and Mitra, 1977). In a vector space provided with a definition of a inner product (-,.), it is possible to associate a value or modulus to a vector b by using the notation (b, b) = [b]. The notion of length of a vector can be generalized even in a vector space in which the space axes are not orthogonal. Any symmetric positive definite matrix M is said a metric for the vector space furnished with the inner product (.,.) and the squared modulus of a vector b in a space equipped with the norm M is described by

(8.2) With these recalls in mind, we now face the problem to derive a general solution of the problem described in Eq. 8.1 under the assumption of the existence of two distinct metrics Nand M for the source and the data space, respectively. Since the system is undetermined, infinite solutions exist. However, we are looking for a particular vector solution that has the following properties: 1) it has the minimum residual in fitting the data vector b under the norm M in the data space 2) it has the minimum strength in the source space under the norm N. To take into account these properties, we have to solve the problem utilizing the Lagrange multiplier A and minimizing the following functional that express the desired properties for the sources x (Tikhonov and Arsenin, 1977; Dale and Sereno, 1993; Menke, 1989; Grave de Peralta and Gonzalez Andino, 1998; Liu, 2000) :

e

=

(IIAx - bll~ +)..1 IIxlI~)

(8.3)

The solution of the variational problem depends on adequacy of the data and source space metrics. Under the hypothesis ofM and N positive definite , the solution of Eq. 3 is given by taking the derivatives of the functional and setting it to zero. After few straightforward computations the solution is

(8.4) where G is called the pseudoinverse matrix, or the inverse operator, that maps the measured Note that the requirements of positive definite matrices for data b onto the source space the metric Nand M allow to consider their inverses . Last equation stated that the inverse operator G depends on the matrices M and N that describe the norm of the measurements and the source space, respectively. The metric M, characterizing the idea of closeness in the data space, can be particularized by taking into account the sensors noise level by using the Mahalanobis distance (Grave de Peralta and Gonzalez Andino, 1998). If no a priori information is available for the solution of linear inverse problem, the matrices M

e.

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and N are set to the identity, and the minimum norm estimation is obtained (Hamalainen and Ilmoniemi, 1984). However, it was recognized that in this particular application the solutions obtained with the minimum norm constraints are biased toward those sources that are located nearest to the sensors. In fact, there is a dependence of the distance on the law of potential (and magnetic field) generation and this dependence tends to increase the activity of the more superficial sources while depresses the activity of the sources far from the sensors. The solution to this bias was obtained by taking into account a compensation factor for each dipole that equalizes the "visibility" of the dipole from the sensors. Such technique, called column norm normalization by Lawson and Hanson in 1974, was used in the linear inverse problem by Pascual-Marqui, 1985 and then adopted largely by the scientists in this field. With the column norm normalization the inverse of the resulting source metric is (8.5) in which (N- I)ii is the i-th element of the inverse of the diagonal matrix N and II Ai II is the L2 norm of the i-th column of the lead field matrix A. In this way, dipoles close to the sensors, and hence with a large II Ai II, will be depressed in the solution of the inverse problem, since their activations are not convenient from the point of view of the functional cost. The use of this definition of matrix N in the source estimation is known as weighted minimum norm solution (Pascual-Marqui, 1995; Grave de Peralta et aI., 1997). The described mathematical framework is able to accommodate the information coming from EEG, MEG and tMRI data, as we will demonstrate in the following paragraphs.

8.4 MULTIMODAL INTEGRATION OF EEG, MEG AND FMRI DATA Before we describe how it can be possible to implement methods that fuse data from all modalities, some remarks are necessary about the neural sources that mayor may not be retrieved by multimodal EEG-MEG-tMRI integration. In the following paragraphs, we will present possible techniques for multimodal integration of EEG, MEG and tMRI data by using a particularization of the metrics of the data and the source space, in the context of the distributed linear inverse problem. In particular, we will show that the metric of the data space M can be characterized to take into account the EEG and MEG data. Furthermore, we will demonstrate how the source metric N can be particularized by taking into account the information from the hemodynamic responses of the brain voxels.

8.4.1 VISIBLE AND INVISIBLE SOURCES Any neuroimaging technique has its own visible and invisible sources. The visible sources for a particular neuroimaging technique are those neuronal pools whose spatiotemporal activity can be at least in part detected. In contrast, invisible sources are those neural assemblies that produce a pattern of the spatio-temporal activity not detectable by the analyzed neuroimaging technique. In the case ofEEG (or MEG) technique, it is clear that the visible sources are generally located at the cortical level, since the cortical assemblies are close to the recording sensors, and the morphology of the cortical layers allows the

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generation of open (rather than closed) electromagnetic fields. On the other hand, it is often poorly understood that the invisible sources for the EEG (or MEG) are all those cortical assemblies that do not fire synchronously together. In fact, in a dipole layer composed by M coherent sources and N incoherent ones, the potentials due to individual coherent sources are combined by linear superposition, while the combination of the incoherent sources is only due to statistical fluctuations. The ratio between the contributions of coherent to incoherent source can be expressed by M 1v'N(Nunez, 1995). Hence, if N is very large, say about 10 million of incoherent neurons that fire continuously, and M is a small percentage of such neurons (say 1%; about 100,000 neurons) that instead fire synchronously, we obtain that the potential measured at the scalp level will be determined by 1051M , with a net result of about 30. Hence, only 1% of the active sources produce a potential larger than the other 99% by a factor of 30 just because of the synchronicity property. This means that a cortical patch may generate an EEG signal with no modification of its metabolic consumption, simply by increasing the firing coherence of a small percentage of neurons. As a consequence of that, neuroimaging techniques based on imaging of the metabolic/hemodynamic request of the neural assemblies may detect no activity change with respect to the baseline condition. However, there are other situations in which the visible sources for metabolic techniques such as fMRI and PET can be invisible for EEG or MEG techniques. Stellate cells are neurons present in the human cerebral cortex, and represent 15% of the neural population of the neocortex (Braitenberg and Schuz, 1991). These cells occupy a spherical volume within the cortex, thus generating essentially a closed-field electromagnetic pattern. Such a field cannot be recorded at the scalp level by electrical or magnetic sensors, although the actual firing rate of such stellate neurons is rather high with respect to the other cortical neurons. This means that these neuronal populations present high metabolism requirements that can be detected by the fMRI or PET techniques, while at the same time they are "invisible sources" for the EEG and MEG techniques. Other example of invisible sources for the EEG and MEG techniques are represented by the neural assemblies located at the thalamic level, since they are also arranged in such a way to produce closed electromagnetic field, while having high metabolic requirements.

8.4.2 EXPERIMENTAL DESIGN AND CO-REGISTRATION ISSUES 8.4.2.a Experimental design Experimental setups that take into account both the electrical and the hemodynamic responses as dependent variables have to be designed with particular attention. There are two main important classes of setups that can be considered in a study of this type, depending whether simultaneous EEG and fMRI measurements or just separate EEG/MEG and fMRI recordings are scheduled. In the first case, many issues related to the co-registration of the head can be easily overcome. However, in both cases the differences between the hemodynamic and electric behavior have to be taken into account. In fact, consideration about the signal to noise ratio (SNR) can limit the use of similar paradigms for EEG and fMRI recordings. For instance, EEG/MEG response to very brief stimuli (such as Somatosensory Evoked Potentials; SEPs, i.e. short electrical shock) can be recorded with a high SNR, while the hemodynamic responses decrease its SNR by decreasing the stimulation length. Furthermore, it has also been demonstrated that while EEG amplitudes decrease

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by increasing the stimulation rate, the opposite it is true for the hemodynamic response amplitudes (Wikstrom et al., 1996; Kampe et al., 2000). Experimental design for either separate or simultaneous collection of electrophysiological and hemodynamic variables can be easier when event-related fMRI technique is used, in contrast to the block-design fMRI. In these experimental paradigms, the availability of the time behavior of the hemodynamic response can be useful to design similar stimulation setup for both modalities.

S.4.2.b Co-registration In the multimodal integration ofEEGIMEG data with the fMRI, a common geometrical framework has to be derived in order to locate appropriately the voxels whose EEG responses is high and voxels whose hemodynamic response is increased/decreased during the task performance. The issue of deriving a common geometrical framework for the data obtained by different imaging modalities is called the "co-registration" problem (van den Elsen et al., 1993). Several techniques can be used to produce an optimal match between the realistic head reconstruction obtained in the high resolution EEGIMEG by the MRIs of the experimental subject and the fMRI image coordinates. The first body of techniques is based on the presence of landmarks on the both images used for the co-registration. Corresponding landmarks have to be determined in both modalities (Fuchs et al., 1995). A second body of techniques is based instead on the matching of surfaces belong to the same head structure, as obtained by the different image modalities. In these techniques a prerequisite is the segmentation of the structures whose surfaces have to be matched (Wagner and Fuchs, 2001). With the volume-based registration technique no additional information as landmarks or surface detection is necessary (Wells et al., 1997). In the case in which the multimodal EEG and fMRI is performed simultaneously, the setup of a common geometrical framework becomes simpler. In this case registration can be performed based on a scanner coordinate system. As additional advantage, simultaneous measurement of EEG and fMRI also allows an accurate co-registration of the electrode positions, a problem that in the other cases have to be solved by using non-linear minimization techniques.

8.4.3 INTEGRATION OF EEG AND MEG DATA As mentioned before, electroencephalography (EEG) and magneto-encephalography (MEG) are useful tools for the study of brain dynamics and functional cortical connectivity due to their high temporal resolution (in the range of milliseconds). While the EEG reflects the activity of neural generators oriented both tangentially or radially with respect to the surfaces of electrodes, the MEG is more sensitive to cortical generators oriented tangentially to the surfaces of sensors. However, the recorded EEG is a distorted copy of the cortical potential distribution due to the poor conductivity of the skull, while the MEG is insensitive to the different head tissues conductivities. In this framework an important question arises, namely the importance of using one (EEG, MEG) or both modalities (EEG and MEG) for increasing the accuracy of the estimated neural activity. Simulation studies aimed at integrating data from MEG and EEG sensors with phantoms demonstrated an improvement of spatial accuracy of the reconstruction methods when MEG and EEG data are fused together (Phillips et al., 1997, Baillet et al., 1997, 1999). These simulation studies suggest the possibility to practically integrate data from both EEG and MEG modalities in the solution

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of some neurophysiological problems also in the case of distributed source problem. It was also demonstrated that the use of combined EEG and MEG data increase the stability and the accuracy of the source activity estimate from primary sensory cortical areas of man with respect to using modalities separately (Stock et aI., 1987; Fuchs et aI., 1998). In this context, the question if the use of combined EEG and MEG measurements lead to a better estimate of the distributed cortical activity with respect to the use of EEG or MEG separately, has been recently addressed (Babiloni et aI., 2001). Here, we would like to expand in the following paragraph concepts about the possibility to integrate EEG and MEG data in the context of distributed linear inverse solutions. At a first look, the attempt to integrate the EEG and MEG data in order to increase the quality of the source reconstruction fails when we consider that the units of the potential and magnetic field differ. How can one fuse such data together? In order to combine the different measures of electric and magnetic data, both have to be converted to a common basis. This conversion was performed by normalizing the measured signals to their individual noise amplitudes, yielding unit-free measures for both electric and magnetic modalities. Such normalization procedure was accomplished by using the covariance matrix of the electric and magnetic noise as metric in the data space for the solution of the linear inverse problem. The estimation of the noise covariance matrices requires the recording of several single sweeps of EEG and MEG data, and the possibility to determine a segment of the recorded data in which no task-related activity is present. Then, on all the sweeps recorded and for the time period of interest, the maximum likelihood estimate for the covariance matrices of the electrical N, and magnetic Nm noise matrices have to be computed. With the use of these matrices we can produce the block covariance matrix of the electromagnetic measurement, by posing S = [ N, N m ] , i.e horizontally adjoining the two matrices. The forward solution specifying the potential scalp field due to an arbitrary dipole source configuration is solved on the basis of the linear system (8.6)

where (i) E is the electric lead field matrix obtained by the boundary element technique for the realistic MRI-constructed head model; (ii) B is the magnetic lead field matrix obtained for the same head model; (iii) x is the array of the unknown cortical dipole strengths; (iv) v is the array of the recorded potential values; and (v) m is the array of magnetic field values. The lead field matrix E and the array v were referenced consistently. In order to scale EEG and MEG, the rows of the lead field matrix E and B were first normalized by the norm of rows (Phillips et aI., 1997). This scaling was equally applied on the electrical and magnetic measurements arrays, v and m. As noted before, the inverse operator G is expressed in terms of the matrices M and N that regulates the metric in the measurement and source space, respectively. Here, M is now equal to the inverse of the covariance matrix S of the noise of the normalized EEG and MEG sensors, while N is the matrix that regulates how each EEG or MEG sensor is influenced by dipoles located at different depths of the source model. The covariance matrix S was derived from the normalized EEG (v) and MEG (m) data by maximum likelihood estimation as described before. The matrix N is a diagonal matrix in which the i-th element is equal to the norm of the i-th column of the normalized lead field matrix A.

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RIGHT MOVEMENT EEG

-t

.{l.5

-Hl.5

+1

0 -Hl.5 T1ME<-ec)

+1

0 TIME~

ME G

-t

-ns

FIGURE 8.5. Disposition of the electric (up) and magnetic (bottom) sensors for the recording of EEG and MEG data related to unilateral voluntary fingermovements (separate recording sessions). Averaged MEG and EEG time series (waveforms) recorded from two selected magnetic (Ml and M2) and electric (El and E2) sensors are shown on the right of the figure. These sensors overlay the primary sensorimotor cortex contralateral to the movement. See the attached CD for color figure.

In the following , the above methodology is applied to the EEG and MEG data related to the preparation and the execution of the voluntary movement of the right index finger. Fig. 8.5 presents the disposition of the electric (up) and magnetic (bottom) sensors for the recording of EEG and MEG data (separate recording sessions). Averaged MEG and EEG time series (waveforms) recorded from two selected magnetic (M I and M2) and electric (E I and E2) sensors in a normal healthy subject executing the movement are shown on the right of Fig. 8.5. These sensors overlay the primary sensorimotor cortex contralateral to the movement, which is known to be active both during preparation and execution of the movement. Fig. 8.6 shows linear inverse estimates from EEG , MEG , and combined EEGMEG data recorded from the subjec t about 110 ms after the onset ofEMG activity associated with self-paced right finger movement. Raw EEG and MEG distributions present large and distant negative and positive maxima preponderant in the side contralateral to the movement, the electric field being tilted of 90· with respect to the magnetic field. In contrast, linear inverse estimates were characterized by circumscribed zone of negativity and positivity in both sides. Linear inverse estimate of EEG data (Movement-Related Response 1; MRR 1) shows negative maxima in the mesial-frontal and contralater al frontal areas , and a zone of minor negativity in the ipsilateral frontal area. In addition, there is a reversed parietal

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FIGURE 8.6. Amplitude gray scale 3-D maps showing linear inverse estimates of electroencephalographic (EEG), magnetoencephalographic (MEG), and combined EEG-MEG data recorded (128-50 channels, respectively) from a subject about 110 ms after the onset of the electromyographic response accompanying a voluntary brisk right middle finger extension. Percent gray scale is normalized with reference to the maximum amplitude calculated for each map. Maximum negativity (-100%) is coded in white and maximum positivity (+ 100%) in black.

positivity in both sides. Compared to this distribution, linear inverse estimate of MEG data (Movement Evoked Field l; MEF 1) present closer bilateral frontal negativity and parietal positivity as well as no negativity in the mesial-frontal area. The linear inverse estimate of the combined EEG-MEG data show a high spatial resolution content integrating features of linear inverse estimate of EEG and MEG data considered separately. This result does not depend on the increase of the total number of sensors (EEG electrodes + MEG coils) since it was also reproducible by comparing purely EEG data and data obtained substituting some of the EEG channels with MEG channels (thus preserving the total number of channels in the two data-sets, Babiloni et al., 2001). It is worth of note that the presented results are obtained with separate EEG and MEG recordings. An important source of variance in the linear inverse source analysis of combined EEG- MEG data might be caused by the non-simultaneous recording of these data sets, when attentional, learning and emotional variables are unpaired across the recording blocks. Regarding the experiments presented above, it must be stressed that movement-related potentials/fields are very stable across experimental sessions performed in different days. Furthermore, the participant subjects were preliminary trained to stabilize a simple motor

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performance in which learning, attentional attentive and emotional concomitants seem to be negligible. Finally, the possibility of combining MEG and EEG data recorded in different experimental sessions offers the opportunity of scientific and clinical cooperation between centers that own only one of the imaging systems. In contrast, the use of simultaneous multimodal EEG-MEG recordings is nowadays possible only in very few laboratories. In conclusion, the present results for the multimodal integration of EEG and MEG suggest that the linear inverse source estimates of the combined EEG-MEG data improve with respect to those of EEG or MEG data considered separately. The methodological approach includes MR-based realistic head and cortical source modeling, high spatial sampling of real EEG and MEG data, and regularized linear inverse estimate mathematics (weighted minimum norm source estimate). The application of this technology supports the hypothesis that in humans the preparation and execution of one of the simple unilateral volitional digit acts is subserved by a distributed cortical system including SMA and contralaterally preponderant Ml and S1.

8.4.4 FUNCTIONAL HEMODYNAMIC COUPLING AND INVERSE ESTIMATION OF SOURCE ACTIVITY Here we describe how it is possible to take into account the information from the hemodynamic coupling between the cortical areas in the estimation of the time-varying source strengths by solving the electromagnetic inverse problem.

8.4.4.a Multimodal integration ofEEG/MEG and fMRI data with dipole localization techniques As described previously, the dipole localization techniques locate neural activity by using few current equivalent dipoles from non-invasive EEG and/or MEG recordings. The use of spatial information from hemodynamic methods as a constraint to the electromagnetic inverse problem necessitates the assumption that the brain areas that appear active with different methods are to some extent the same. Dipole localization techniques could use the tMRI information essentially in two way: 1) by setting the foci of the tMRI hotspots in the brain as initial locations of the equivalent current dipoles localization procedure. Since the minimization procedure is non linear, it is hence dependent by the initial position of the current dipole(s); 2) by setting the position of the current dipoles at the tMRI hotspots, allowing to the localization procedure to rotate freely only the direction of the dipoles to fit the EEG or MEG data. Methods I) and 2) can also be combined. In order to take into account possible electrical sources that were not detected as active fMRI spots, some current dipoles that are not constrained to fMRI spots' positions may be added to the source model. Recently, a simulation study suggested that EEG dipole fits could benefit from fMRI constraints (Wagner and Fuchs, 2001) . A reconstructed dipole in the vicinity of each fMRI hotspot yields the corresponding source time course . However, spatially unconstrained dipoles are then necessary to account for remaining simulated EEG activity, to appropriately locate also sources that are invisible to the fMRI hotspots . Moving from the simulation to the application studies, the spatial correspondence between EEG/MEG and tMRI has been mostly investigated in the context of motor and somatosensory evoked activity. In the localization of the primary sensory and motor areas using equivalent current dipole

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modeling, typical spatial differences between the dipole location and the center of tMRI activation have been between 10 to 16 millimeters (Sanders et al., 1996; Beistener et al., 1997). In light of the current limited knowledge, the localization results seem to agree reasonably well when locating responses from primary sensory areas. Extending comparison to more complex cortical networks has indicated mostly converging activation patterns (Ahlfors et al., 1999; Korvenoja et al., 1999). In the last study of Korvenoja and co-workers, five subjects participated in both a somatosensory evoked field (SEFs) recording with a Neuromag 122 and a tMRI recording. The goal was to maximize the response amplitude in both imaging modalities, in order to achieve the best possible signal-to-noise ratio. The SEFs were first modeled independently of tMRI results. Thereafter a multi-dipole model was constructed by placing equivalent current dipoles (ECDs) to tMRI activation centroids. If MEG data indicated activity in some of the eight source areas but no tMRI activation was seen, then ECD location from independent MEG data analysis was used. The eight dipoles were spatially fixed, but were allowed to change their orientation and amplitude to explain the data (rotating dipoles). The ECD orientations were remarkably stable over the whole analysis period (0 to 400 ms post stimulus). The time-courses of activation in the model were found to agree with data, which have been obtained with invasive electrophysiological methods (Allison et al., 1989; 1996). In another study performed by the same research group, the temporal dynamics of visual motion areas was analyzed by combining tMRI and MEG data (Ahlfors et al., 1999). Also in that case, results indicated that while complete overlap of activation patterns determined independently from MEG and tMRI did not exist on individual level, the activation patterns did converge at the group level. In a study on the sources of movement-related magnetic fields (Baillet et al., 2001), the locations of the contralateral and SMA sources found with MEG dipole localizations were found rather close to the maximum of the closest tMRI clusters (12 mm and 4 mm, respectively). Cortical remapping of the focal parametric source model was performed and the cortical clusters corresponding to the contralateral and central sources indicated a good match in location with the tMRI regions. Other studies have combined the analysis of hemodynamic and electrophysiological data that were collected separately, by using both the Positron emission tomography (PET) and EEG (Heinze et al., 1994; Snyder et al., 1995; Heinze et al., 1998) or the tMRI and EEGIMEG (Belliveau 1993; Morioka et al., 1995; George et al., 1995; Menon et al., 1997; Opitz et al., 1999). Example of integration of hemodynamic or metabolic techniques and invasive cortical recordings can be found in the study of Luck and colleagues with tMRI and EEG invasive recordings (Luck, 1999) or in that of Lamusuo and co-workers, that integrates PET, MEG and invasive recordings (Lamusuo et aI., 1999). Simultaneous acquisition of EEG and tMRI data is necessary when the activity of interest cannot be easily reproduced, as it happens in the case of the epilepsy studies. In this case the epileptiform activity between seizures could be produced by different cortical generators. Simultaneous EEG/tMRI recordings (Huang-Hellinger et al., 1995; Warach et aI., 1996; Seeck et aI., 1998; Krakow et al., 1999) have been used to measure such activity, by hypothesizing that epileptiform discharges are likely to produce neural activity measurable by tMRI (Ives et aI., 1993). These studies recorded interleaved EEG and tMRI to monitor for the presence of interictal activity without, however, localizing the EEG activity. In a recent study of comparison of spike-triggered tMRI activation hotspots and EEG dipole model in six epileptic patients, an average of 3 em of mismatch between the tMRI hotspots and EEG localized current dipoles was found (Lemieux et al., 2001). In this

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FIGURE 8.7. Comparisons between foci of interictal epileptic activity as obtained by MEG dipole localization technique, over threshold fMRI clusters and site of brain lesion in a patient evaluated before surgery. See the attached CD for color figure.

case the authors concluded that the combination of EEG and tMRI techniques offers the possibility of advancing the study of the generators of epileptiform electrical activity. Such results are in line with those found by the group of Romani and co-workers, in which they found some centimeters of displacement between tMRI hotspots, localized dipoles on the base of MEG recordings and brain l~sions before surgical operation in epileptic patients. Fig. 8.7 presents such displacements between the various hotspots and the brain lesion. There are occasions where disagreement in spatial activation patterns could exist for the integration of EEG/MEG and tMRI data. It is not clear, for example, whether very short-lasting synchronous firing, which can be detected in EEG and MEG, will produce a detectable hemodynamic change. Event-related synchronization and resynchronization are phenomena that possibly remain undetected by observing hemodynamic changes. For example, in the study by Ahlfors et al. (1999) MEG indicated activity over the frontal cortex bilaterally while tMRI did not demonstrate any activity in similar areas. This is a typical example of the co-existence of visible and invisible sources in a same behavioral task for the MEG and tMRI techniques. Taken together, the results above indicate that while it may not be possible to simply restrict the source model solutions to areas where tMRI shows activation, it still seems to be a valuable aid in the validation of the source model. Converging lines of evidence from multiple methods will increase the likelihood of correct solution. The ultimate way to validate the inverse solution would be the invasive recordings. However, it is worth of note that

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fMRI priors used in conjunction with the dipole localization techniques require a significant manual intervention, since one must decide in which regions of the fMRI activation a dipole is to be "placed". This could provide at least a difficulty in the replication of findings between different researchers, since the dependence of the results by experimenter's choice. In this respect, it is of interest to note that the combined fMRI, EEG and MEG procedure that utilizes the mathematical framework of distributed linear inverse solutions do not requires such "manual" intervention. Such procedure will be presented in the following paragraphs.

8.4.4.b Multimodal integration ofEEG/MEG and fMRI data with distributed model by using diagonal source metric Here, we present two characterizations of the source metric N that can provide the basis for the inclusion of the information about the statistical hemodynamic activation of i-th cortical voxel into the linear inverse estimation of the cortical source activity. In the fMRI analysis, several methods to quantify the brain hemodynamic response to a particular task have been developed. However, in the following we analyze the case in which a particular fMRI quantification technique has been used, called Percent Change (PC) technique. This measure quantifies the percentage increase of the fMRI signal during the task performance with respect the rest state (Kim et aI., 1993). The visualization of the voxels' distribution in the brain space that is statistically increased during the task condition with respect to the rest is called the PC map. The difference between the mean rest- and movement-related signal intensity is generally calculated voxel-by-voxeI. The rest-related fMRI signal intensity is obtained by averaging the pre-movement and recovery fMRI. Bonferroni-corrected Student's t-test is also used to minimize alpha inflation effects due to multiple statistical voxel-by-voxel comparisons (Type I error; p < 0.05). The introduction of fMRI priors into the linear inverse estimation produces a bias in the estimation of the current density strength of the modeled cortical dipoles. Statistically significantly activated fMRI voxels, which are returned by the percentage change approach (Kim et aI., 1993), are weighted to account for the EEG measured potentials. In fact, a reasonable hypothesis is that there is a positive correlation between local electric or magnetic activity and local hemodynamic response over time. This correlation can be expressed as a decrease of the cost in the functional of Eq. 8.3 for the sources Xj in which fMRI activation can be observed. This increases the probability for those particular sources Xj to be present in the solution of the electromagnetic problem. Such thoughts can be formalized by particularizing the source metric N, to take into account the information coming from the fMRI. The inverse of the resulting metric is then proposed as follows (Babiloni et aI., 2000): (8.7)

in which (N- 1)ii and IIA i II has the same meaning described above. g(Ui) is a function of the statistically significant percentage increase of the fMRI signal assigned to the i-th dipole of the modeled source space. This function is expressed as g(uJ 2=1+(K-1)

U· 1,

maxto.)

K2:1,

Ui2:0

(8.8)

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where a ; is the percentage increase of the tMRI signal during the task state for the i-th voxel and the factor K tunes tMRI constraints in the source space. Fixing K = 1 let us disregard tMRI priors , thus returning to a purely electrical solution; a value for K » 1 allows only the source s associated with tMRI active voxels to participate in the solution . It was shown that a value for K in the order of 10 (90% of constraints for the tMRI information) is useful to avoid mislocalization due to over constrained solutions (Liu et aI., 1998; Dale et aI., 2000 ; Liu, 2000). In the following the estimation of the cortical activity obtained with this metric will be denoted as diag-tMRI, since the previous definition of the source metric N results in a matrix in which the off-diagonal elements are zero.

8.4.4.c Multimodal integration of EEG/MEG and fMRI data with distributed model by using full source metric In the previous paragraphs, we observed that incorporating a priori information for each cortical voxel about geometrical orientation with respect to the cortical surface and about hemodynamic response we obtain an estimate of the cortical activity that improves the reconstruction generated without such constraints. However, it must be noted that all the formulations presented in literature on the integration of EEG, MEG with tMRI did not take into account informati on about the functional coupling of the neural sources. In fact, these formulation s only use the information about the presence or absence of a particular source located at the voxel level in the set of those whose hemodynamic responses have been elicited by the considered task. However, the theoretical possib ility to include this source of information in the linear inverse problem was already mentioned in previous articles (Dale and Sereno , 1993; Liu et aI., 1998; Dale et aI., 2000; Liu, 2000). In this paragraph we present an extension of the linear inverse problem aimed to taking also into account information about the functional coupling of the cortical sources, as provided experimentally by the hemodynamic response s returned by the event-related tMRI. In particular, we estimate the hemodynamic correlation of the neural sources by using the cross-correlation technique on the hemodynamic waveforms obtained during the performance of the task under the tMRI scanner. These correlation values are then used as additional a priori constraints in the solution of the electromagnetic linear inverse problem together with the cortical orientation constraints and the presence of statistically significant activation of the hemodynamic response. We take advantage of the off-diagonal elements of the matrix N to insert the information about the functional coupling of the cortical sources. In particular we set the generic (i, j) entry of the inverse of matrix N as in the following (8.9) where IIA.; II and g(a ;) have the same meaning described above and cor rij is the degree of functional coupling between source i and source j during the particular task analyzed . Information on coupling is revealed by the correlation of their hemodynamic responses obtained by the event-related tMRI data. In the following the estimation of the cortical activity obtained with this metric will be denoted as corr-tMRI. It is of interest that in the case of uncorrelated sources tcorn , == 0, i i= j; corn, == 1), the corr-tMRI formulation leads back to the diag-tMRI one. Fig . 8.8 summarizes the different approaches pursued here

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in order to insert the hemodynamic constraints in the solution of the linear inverse problem for the estimation of the cortical sources of the recorded EEG in a unique mathematical formulation .

8.4.4.d Application ofthe multimodal EEG-fMRI integration techniques to the estimation ofsources ofself-paced movements In this section we will provide a practical example of the application of multimodal integration techniques of EEG, MEG and fMRI (as theoretically described in the previous sections) to the problem of detection of neural sources subserving unilateral self-paced movements in humans. The high resolution EEG recordings (128 scalp electrodes) were performed on normal healthy subjects by using the facilities available at the laboratory of the Department of Human Physiology, University of Rome "La Sapienza". Realistic head models were used, each one provided with a cortical surface reconstruction tessellated with 3,000 current dipoles . Separate block design and event-related fMRI recordings of the same subject s were performed by using the facilities of the Istituto Tecnologie Avanzate Biomediche (ITAB) of Chiety, Italy, leaded by prof. Gian Luca Romani. Distributed linear inverse solutions by using hemodynamic constraints were obtained according to the methodology presented above . An example of multimodal integration between EEG and fMRI related to a simple voluntary movement task by using only the hemodynamic information relative to the strength

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FIGURE 8.9. Amplitude gray scale 3-D maps showing linear inverse estimates from high resolution electroencephalographic (HREEG) and combined functional magnetic resonance image (fMRI)-HREEG data computed from a subject about 20 ms after the onset of the electromyographic activity associated with self-paced right middle finger movements (motor potential peak, MPp). Percent gray scale of HREEG and combined fMRI-HREEG data is normalized with reference to the maximum amplitude calculated for each map. Maximum negativity ( -I 00% ) is coded in white and maximum positivity (+ I00%) in black.

of fMRI data (according to Eq. 8.7) is provided in Fig. 8.9. This figure shows amplitude gray scale maps of linear source inverse estimates from EEG and combined fMRI-EEG data, computed about 20 ms after the onset of the electromyographic response to voluntary right finger movements (Motor Potential peak; MPp). fMRI data indicate maximum activated voxels clustered in bilateral primary motor (Ml), primary somatosensory (Sl), and supplementary motor (SMA) areas, the fMRI signal intensity being much more higher on the contralateral (left) side. The linear inverse estimate of neural activity for the HREEG and combined fMRI-HREEG data were mapped over the cortical compartment of a realistic MRI-constructed subject's head model. The MPp map presents maximum responses in the contralateral Ml and Sl and in the modeled SMA. With respect to the HREEG solutions (left), the fMRI-HREEG solutions present more circumscribed Ml, S1, and SMA responses. In addition, the contralateral Ml and Sl responses have similar intensity and are spatially dissociated. An example of the multimodal integration between EEG and fMRI data by using both block and event related experimental designs is depicted in Fig. 8.10. In this figure, the upper row illustrates the topographic map of readiness potential distribution recorded at the scalp about 200 ms before a right middle finger extension for another subject analyzed. Note the extension of the maximum of the negative scalp potential distribution, roughly overlying

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Right finger movement Readiness Potential

tMRI Response

Linear Inverse solutions diag-tMRI

no-tMRI

-100%

corr-tMRI

+100%

FIGURE 8.10. Top left: scalp potential distribution recorded about 200 ms before the movement onset (128 recording channels) in a separate session. This distribution is representative of the so-called readiness potential. Percent color scale in which maximum negativity is coded in red and maximum positivity is coded in black. Top right: cortical tMRI response related to the movement. Only the tMRI voxels close to the cortical surface contribute to source weighting. The intensity of yellow codes the percentage of the increase of the tMRI response. Bottom row: Cortical distributions of the current density estimated with a linear inverse approach from the readiness potential shown in the top row. Linear inverse estimates are obtained with no tMRI constraints (left, no-tMRI) and two kinds oftMRI constraints, one based on the strengths of the cortical tMRI responses (center, diag-tMRI) and the other on the correlation between tMRI responsive cortical areas (right, corr-tMRI). Percent color scale: maximum negativity is coded in red and maximum positivity is coded in black. See the attached CD for color figure.

frontal and centro-parietal areas contralateral to the movement. The percent values of the tMRI response during the movement in a separate experimental session are also illustrated. The maximum values of the tMRI responses are located in the voxels roughly corresponding to the primary somatosensory and motor areas (hand representation) contralateral to the movement. In fact, during the self-paced unilateral finger extension, somatosensory reafference inputs from finger joints as well as cutaneous nerves are directed to the primary somatosensory area, while centrifugal commands from the primary motor area are directed

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toward the spinal cord via the pyramidal system. The lower row of Fig. 8.10 illustrates the cortical distribution of the current density estimated with linear inverse approach from the potentialdistributionof the upperrow. Linearinverseprocedureused no-tMRI constraint as well as two types offMRI constraints, i.e. one based on block-design (diag-fMRI, Eq. 8.7) and the other on event-related design (corr-fMRI, Eq. 8.9). The cortical distributions are represented on the realistic subject's head volume conductor model. Linear inverse solutions obtained with the fMRI priors (diag- and corr-fMRI)present more localized spots of activations with respect to thoseobtainedwith the no fMRI priors.Remarkably, the spots of activation are localizedin the hand region of the primary somatosensory (post-central) and motor (pre-central) areas contralateralto the movement. In addition, spots of minor activation wereobservedin the fronto-central medial areas (includingsupplementary motor area) and in the primary somatosensory and motor areas of the ipsilateral hemisphere. Similar results wereobtainedin the other main componentsof the movement-related potentials(i.e. motor potentials and movement-evoked potential).

8.5 DISCUSSION This chapterrevieweda mathematical framework for the integration ofEEG, MEGand fMRIdata. Besides, advantages and drawbacks relatedto the localizationtechniquesin conjunction with fMRI and PET data have also been reviewed. In general,there is a rather large consensus about the need and utility of the multimodal integration of metabolic, hemodynamicand neuroelectrical data. Resultsreviewedin literatureas wellas those presentedhere suggesta real improvement in the spatialdetailsof the estimatedneuralsourcesby performing multimodal integration. However, whilefor themultimodalintegration ofEEG andMEG data a precise electromagnetic theory exists, a clear mathematical and physiologiclink betweenmetabolic demandsand firing ratesof the neuronsis still lacking.It is out of doubtthat whenthislinkis furtherclarified, the modelingof the interaction betweenhemodynamic and neural firing rate can be further refined. This will lead us to a more proper characterization of the issues of visible and invisiblesource that at the moment represent the major concern about the applicability of the multimodal integration techniques (Nunez et aI., 2000). The results for the multimodal integration of EEG/MEG and fMRI presented in this chapter are in line with those regarding the coupling between cortical electrical activity and hemodynamic measure as indicated by a direct comparison of maps obtained using voltage-sensitive dyes (which reflect depolarization of neuronal membranes in superficial corticallayers) and mapsderivedfromintrinsicopticalsignals(whichreflectchangesin light absorptiondue to changes in blood volumeand oxygenconsumption, Shohamet aI., 1999). Furthermore, previous studies on animals have also shown a strong correlation between local field potentials,spikingactivity, and voltage-sensitive dye signals(Arieli et aI., 1996). Finally, studies in humans comparing the localization of functional activity by invasive electrical recordings and fMRI have provided evidence of a correlation between the local electrophysiological and hemodynamic responses(Puceet aI., 1997). It is worth of note that recentlya study aimed at investigating this link has been produced(Logothetis et aI., 2001). In this study, intracortical recordingsof neural signals and simultaneousfMRI signals were acquired in monkeys. The comparisons were made between the local field potentials, the multi-unit spiking activity and BOLD signals in the visual cortex. The study supports the

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link between the local field potentials and BOLD mechanism, which is at the base of the procedure of the multimodal integration of EEGIMEG with fMRI described above. This may suggest that the local fMRI responses can be reliably used to bias the estimation of the electrical activity in the regions showing a prominent hemodynamicresponse. It may be argued that combined EEG-fMRI responses could be less reliable for the modeling of cortical activation in the case of a spatial mismatch between electrical and hemodynamicresponses. However, previousstudies have suggestedthat by using the fMRI data as a partial constraint in the liner inverse procedure, it is possible to obtain accurate sourceestimatesof electricalactivityevenin the presenceof some spatialmismatchbetween the generators ofEEG data and the fMRI signals (Liu et al., 1998; Liu, 2000). Furthermore, it is questionable whether the level of bias for the hemodynamic constraints in the linear inverse estimation can be the same with the diag-fMRI and corr-fMRI approaches. This issue seems to deserve a specific simulation study, using the literature indexes capable of assessing the quality of the linear inversesolutions(PascualMarqui, 1995;Gravede Peralta et al., 1996; Grave de Peralta and Gonzalez Andino, 1998, Babiloni et al., 2001). The multimodalintegrationof fMRI, MEG and EEG data constitutesan unsurpassable non-invasive technologyfor the analysis of human higher brain functions at a high temporal and a good spatial resolution.

ACKNOWLEDGMENTS The Authors express their gratitude to the following colleagues that have participated in the researches described above: dr. Claudio Babiloni, dr. Filippo Carducci, prof. Gian Luca Romani, dr. Cosimo Del Gratta, dr. Vittorio Pizzella, prof. Paolo Rossini.

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Luck S. J. 1999. Direct and indirect integration of event-related potentials, functional magnetic resonance images, and single-unit recordings. Hum. Brain Map. 8:115-201. Magistretti, PJ., Pellerin, L., Rothman, D.L., and Shulman, RG., 1999, Energy on demand, Science 283(5401):496-7. Malonek D., Grinvald A., 1996, Interactions between electrical activity and cortical microcirculation revealed by imaging spectroscopy: implications for functional brain mapping. Science, 272(5261 ):551-4. Menke W. Geophysical Data Analysis: Discrete Inverse Theory. San Diego, CA Academic Press, 1989. Menon, Y, Ford, J.M., Lim, KO., Glover, G.H., and Pfefferbaum, A, 1997, Combined Event-Related fMRI and EEG Evidence For Temporal-Parietal Cortex Activation During Target Detection, NeuroReport 8: 3029-37. Morioka, T., Mizushima, A, Yamamoto, T., Tobimatsu, S., Matsumoto, S., Hasuo, K., Fujii, K., and Fukui, M., 1995, Functional mapping of the sensorimotor cortex: combined use of magnetoencephalography, functional MRI, and motor evoked potentials, Neuroradiology 37:526-30. Nunez, EL., Silberstein, R., 2000, On the relationship of synaptic activity to macroscopic measurements: does co-registration of EEG with fMRI make sense? Brain Topogr. 13(2):79-96. Nunez, E Electric fields of the brain. Oxford University Press, New York, 1981. Nunez, P. L., 1995, Neocortical dynamics and human EEG rhythms, Oxford University Press, New York. Opitz, B., Mecklinger, A., Von Cramon, D.Y., and Kruggel, E, 1999, Combining electrophysiological and hemodynamic measures of the auditory oddball. Psychophysiology 36: 142-7. Oostendorp, T.E, Delbeke, J., Stegeman, D.E, 2000, The conductivity of the human skull: results of in vivo and in vitro measurements. IEEE Trans Biomed Eng. 47(11): 1487-92. Pascual-Marqui, RD. (1995) Reply to comments by Hamalainen, Ilmoniemi and Nunez. In ISBET Newsletter N.6, December 1995. Ed: W. Skrandies., 16-28. Phillips, J.W., Leahy, R., and Mosher, J.e., 1997, MEG-based imaging of focal neuronal current sources, IEEE Trans. Med. Imag., vol. 16., n.3, pp. 338-348. Puce, A., Allison, T., Spencer, S.S., Spencer, D.D., and McCarthy, G., 1997, Comparison of cortical activation evoked by faces measured by intracranial field potentials and functional MRI: two case studies, Hum Brain Mapp 5(4):298-305. Rao, e.R, and Mitra, S.K., Generalized inverse of matrices and its applications. Wiley, New York, 1977. Rosen, B., Buckner, R., and Dale, A., 1998, Event-related fMRI: past, present and future. PNAS, 95:773-780. Rush S., and Driscoll, D.A., 1968, Current distribution in the brain from surface electrodes, Anesthesia Analgesia, 47:717-23. Salmelin, R., Forss, N., Knuutila, J., and Hari, R., 1995, Bilateral activation of the human somatomotor cortex by distal hand movements, Electroenceph Clin NeurophysioI95:444-52. Sanders, J.A, Lewine, J.D., Orrison, W.w., 1996, Comparison of primary motor localization using functional magnetic resonance imaging and magnetoencephalography. Human Brain Mapping 4:47-57. Savoy, R.L., Bandettini, P.A., O'Craven, KM., Kwong, K.K, Davis, T.L., Baker, J.R., Weisskoff, R.M., and Rosen, B.R., 1995, Proc. Soc. Magn. Reson. Med. Third Sci. Meeting Exhib. 2:450. Scherg, M., von Cramon, D., and Elton, M., 1984, Brain-stem auditory-evoked potentials in post-comatose patients after severe closed head trauma, J NeuroI231(1):1-5. Scherg, M., Bast T., and Berg, P., 1999, Multiple source analysis of interictal spikes: goals, requirements, and clinical value. Journal of Clinical Neurophysiology, 16:214-224. Seeck, M., Lazeyras, E, Michel, CM; Blamke, 0., Gericke, e.A., Ives, J., Delavelle, J., Golay, X., Haenggeli, e.A., De Tribolet, N., and Landis, T., 1998, Non-invasive epileptic focus localization using EEG-triggered functional MRI and electromagnetic tomography, Electroenceph. and Clin. Neurophysiol. 106:508-12. Shoham, D., Glaser, D.E., Arieli, A., Kenet, T., Wijnbergen, C; Toledo, Y., Hildesheim, R., and Grinvald, A., 1999, Imaging cortical dynamics at high spatial and temporal resolution with novel blue voltage-sensitive dyes, Neuron 24:791-802. Sidman, R., Vincent, D., Smith, D., and Lu, L., 1992, Experimental tests of the cortical imaging techniqueapplications to the response to median nerve stimulation and the localization of epileptiform discharges, IEEE Trans. Biomed. Eng. 39:437-444. Spiegel, M. Theory and problems of vector analysis and an introduction to tensor analysis. Me Graw Hill, New York, 1978. Srebro, R., Oguz, RM., Hughlett, K, and Purdy, P.D., 1993, Estimating regional brain activity from evoked potential field on the scalp, IEEE Trans. Biomed. Eng.; 40:509-516.

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Srebro, R, and Oguz, RM., 1997, Estimating cortical activity from VEPS with the shrinking ellipsoid inverse, Electroenceph. & din. Neurophysi.; 102:343-355. Snyder, A. Z., Abdullaev, Y.G., Posner, M. I., and Raichle, M. E., 1995, Scalp electrical potentials reflect regional cerebral blood flow responses during processing of written words, Proc.Natl. Acad. Sci. USA. 92: 1689-93. Stok, CJ., Meijs, J.W, and Peters MJ., 1987, Inverse solutions based on MEG and EEG applied to volume conductor analysis. Phys Med Bioi 32:99-104. Tikhonov, A.N., and Arsenin, Y.Y., Solutions of ill-posedproblems. Washington D.C., Winston, 1977 Uutela, K., Hamalainen, M., and Somersalo, E., 1999, Visualization of magnetoencephalographic data using minimum current estimates, Neuroimage, 10(2): 173-80. van den Elsen, P.A., Pol, EJ., Viergever M., 1993, Medical image matching - A review with classification, IEEE Engineeringin Medicine and Biology, 12:26-39. Wagner, M., and Fuchs, M. 2001, Integration of Functional MR!, Structural MR!, EEG, and MEG, International Journal of Bioelectromagnetism, 1(3). Warach, S., Ives, J.R., Schlaug, G., Patel, M.R, Darby, D.G., Thangaraj, Y., Edelman, R.R., and Schomer, D.L., 1996, EEG-triggered echo-planar functional MRI in epilepsy. Neurology 47:89-93. Wells WM., Viola P., Atsumi H., Nakajima S., Kikinis R., 1997, Multi-modal volume registration by maximization of mutual information, Medical Image Analysis 1:35-51. Wikstrom H., Huttunen J., Korvenoja A., Virtanen J., Salonen 0., Aronen H., Ilmoniemi RJ. 1996, Effects of interstimulus interval on somatosensory evoked magnetic fields (SEFs): a hypothesis concerning SEF generation at the primary sensorimotor cortex. Electroencephalography and Clinical Neurophysiology 100(6):479-87.

9

THE ELECTRICAL CONDUCTIVITY OF LIVING TISSUE: A PARAMETER IN THE BIOELECTRICAL INVERSE PROBLEM Maria J. Peters, Jeroen G. Stinstra, and Ibolya Leveles Faculty of AppliedPhysics, Low Temperature Division, University of Twente

9.1 INTRODUCTION Electrically active cells within the human body generate currents in the tissues surrounding these cells. These currents are called volume currents. The volume currents in turn give rise to potential differences between electrodes attached to the body. When these electrodes are attached to the torso, electrical potential differences generated by the heart are recorded. The recording of these electrical potential differences as a function of time is called an electrocardiogram (ECG). ECG measurements can be used to compute the generators within the heart. This is called the solution of the ECG inverse problem. This solution may be of interest for diagnostic purposes. For instance, it can be used to localize an extra conducting pathway between atria and ventricles. This pathway can then subsequently be removed by radio-frequent ablation through a catheter. When the active cells are situated within the brain and the electrodes are attached to the scalp, the recording of the potential difference measured between two electrodes as a function of time is called an electroencephalogram (EEG). The EEG inverse problem can, for example, be used to localize an epileptic focus as part of the presurgical evaluation. The frequencies involved in electrocardiograms and electroencephalograms are in the range of I-1000Hz. Therefore, the Maxwell equations can be used in a quasi-static approximation, implicating that capacitive and inductive effects and wave phenomena are ignored as argued by Plonsey and Heppner (1967). To solve the inverse problem a model is needed of the source and the surrounding tissues, i.e. the volume conductor. Customarily, the source is modeled by a current dipole or a current dipole layer and the volume conductor is described by a compartment model, Corresponding author: Prof. Dr. M. J. Peters, Faculty of Applied Physics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands, Tel. 31534893138, Fax 31534891099, E-mail m.j.peterstetn.utwente.nl

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where all compartments are considered to be homogeneous. The head may have a scalp, a skull, a cerebrospinal fluid and a brain compartment. The torso compartment model may include the ventricular cavities, the lungs and the surrounding homogeneous medium. The shape may be a rough approximation of the real geometry, or the surfaces of the various compartments may have a realistic shape that is obtained from magnetic resonance images. An electrical conductivity is assigned to each compartment. If the inverse solution is used to localize the sources of the measured potentials, then only the ratio between the conductivities assigned to the various compartments is of importance. If the inverse solution is also used to estimate the strengths of the sources, then the absolute values of these conductivities are of importance. In general, the conductivities of the various human tissues are among other things dependent on the blood content and temperature, they are a function of the frequency and strength of the applied current, they show an inter-individual variability, and they are inhomogeneous and anisotropic (Robillard and Poussart, 1977; Rosell et aZ., 1988; Law, 1993). Moreover, the conductivity may be dependent on the health of the subject, for instance, edema will change the conductivity, so does the presence of scar tissue or tumors. The conductivity is called inhomogeneous when the conductivity differs from place to place. The conductivity is called anisotropic when the conductivity is different in different directions. For low current densities, the current density is linear with the applied electric field, in other words the law of Ohm is valid in this case. The averaged Ohmic conductivity that is assigned to a compartment is called the effective conductivity. The effective conductivity of an inhomogeneous tissue is the conductivity of a hypothetical homogeneous medium, which mimics the potential distribution that is found outside the inhomogeneous tissue. For instance, in case of EEG, the effective conductivities assigned to the various tissues in the head have to give approximately the same potential distribution at the scalp as the real inhomogeneous tissues . The problem addressed in the present chapter is: Which value should be assigned to a certain compartment?

9.1.1 SCOPE OF THIS CHAPTER In order to estimate the effective conductivity of a certain tissue, two approaches are possible. First, the conductivity of a tissue can be measured in vitro or in vivo by applying a potential difference by means of a set electrodes and measuring the resulting current. A second approach is applying knowledge of the chemical composition and biological morphology of the tissue in order to compute the effective conductivity. As measured values of the effective conductivities of tissue reported in the literature vary widely, the second approach may be of help to restrict the uncertainties in the conductivities involved in the volume conduction problem . In section 9.1.2, the concept that the conductivity is not a straightforward property of the material is discussed. The effective conductivity from an experimental point of view will be discussed in section 9.1.3. In section 9.2 the tissue is modeled as a suspension ofcells in an aqueous surrounding. If only one type of cells is present and the suspension is a dilute one, the effective conductivity is expressed by Maxwell's mixture equation. This equation will be discussed in section 9.2.3. If more than one type of cell is present and the cells are densely packed or clustered, the effective conductivity is expressed by Archie's law. This will be the subject of section 9.2.4. The applicability of Maxwell's mixture equation and Archie's law will be illustrated for various tissues, such as blood, fat, liver, and skeletal muscles .

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The Electrical Conductivity of Living Tissue

In section 9.3, it is taken into account that most tissues have a layered structure, each layer having a different conductivity. The effective conductivity of an entire compartment will be discussed in section 9.4. The conductivity of a composite medium, like human tissue, cannot have any value but will be within certain limits. section 9.5 of this chapter is dedicated to these limits.

9.1.2 AMBIGUITY OF THE EFFECTIVE CONDUCTIVITY Electrical currents in the human body are due to the movements of ions. As a result of an electric force acting on an ion that is dissolved in fluid, the ion will develop a mean drift velocity between collisions. Despite the fact that the average spacing between cells may be no more than 20 nm, the mean free path of an ion in the extracellular space is only about 0.01 nm. This represents the distance between collisions with other molecules. Almost all these collisions take place with water. An ion rarely encounters a cell membrane and behaves most of the time as though it were in a continuum. The drift velocity in Ohmic conductors is directly proportional to the electric field. The current density reads: (9.1)

where (T is the conductivity and E the electric field. The electrical conductivity cannot be determined unambiguously, because it depends on the direction of the currents, the extent of the current source, and the positions of the measuring electrodes. This is illustrated in Fig. 9.1 where two different situations are compared. In the first situation, a uniform current is applied between two parallel flat electrodes. In the second situation the current is generated by a point source. The expressions for the conductivity are different for the two situations showing that the effective conductivity is not merely dependent on the material, but also on the configuration given by the orientation and location of the source and the measurement points. If a medium is piece-wise homogeneous and isotropic then the proper boundary conditions are that the normal component of the current density is continuous and the normal

p

so urce co nductivity between flat electrodes a=VA / IL

co nductivity of a poi nt so urce a=V 4rcr / I

FIGURE 9.1. This figure illustrates that the conductivity measured in a homogeneous medium of infinite extent depends on the electrodes used.

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M. J. Peters, J. G. Stinstra, and I. Leveles

component of the dielectric displacement makes a step that is equal to the surface charge density. Hence, all interfaces (including the outerfaces of the cells) will have a continuous distribution of surface charge representing genuine accumulations of charge. If the material is non-uniform or if the conductivity is anisotropic, we get accumulations of charge within the material as well as on the interfaces. The microscopic electrical conductivity is the conductivity that characterizes a part of tissue that is comparable in size with the dimensions of the cells. The macroscopic effective conductivity characterizes a part of the tissue that is large compared to the dimensions of the cells. Several levels of inhomogeneities can be distinguished. We will restrict ourselves to three levels of inhomogeneities, the microscopic level with typical dimensions of microns, the millimeter level and the macroscopic level, e.g. compartments with dimensions of several centimeters. This is illustrated in Fig. 9.2. In the microscopic point of view, forms and dimensions of cells and the interstitial fluid are taken into account. Near a cell, the electrical field may change in direction or amplitude due to the charge density on the surface and the presence of counterions near the surface. For macroscopic purposes one has to consider the field averaged over regions large enough to contain many thousands of cells or fibers so that microscopic fluctuations are smoothed over, the 'graininess' of the material is blurred by distance. At a millimeter level, the layered structure or columnar structure of an organ is taken into account. For instance, the skin is composed of three layers, namely the epidermis (the outer non-sensitive and non-vascular layer of the skin that overlies the dermis), the dermis and subcutis. The epidermis is composed of stratum corneum, stratum lucidum, stratum granulosum and stratum germinativum. These layers differ in composition and morphology and consequently in conductivity. The conductivity of the various layers is averaged and the material acts as a continuum, the averaged conductivity being the effective conductivity.

9.1.3 MEASURING THE EFFECTIVE CONDUCTIVITY The uncertainty in measured conductivity values is high, because the measurements are very complicated. The reasons why they are so complicated will be shortly discussed in this section. Usually, the Ohmic low-frequency conductivity of a piece of tissue is determined from the current-voltage relation using a two- or four points method. The currents used for the measurements have to be low (about lmA) in order not to trigger an activation of cells. The field near one cell is very much influenced by the presence of the cell. In order to have an effective conductivity thousands of cells have to be present within the piece of tissue measured. Therefore the effective conductivity has to be measured with electrodes with dimensions of millimeters, these electrodes have to be millimeters apart. The sources in the brain and heart of EEG and ECG are due to at least 105 cells that are active in synchrony else the EEG or ECG would not be measurable. Consequently, the condition that the source has dimensions of millimeters is met if brain or heart activity is used for conductivity measurements. If the measurements are carried outin vitro (i.e. outside the body), the accuracy may be low because tissue properties change after death. The conductivity will initially drop after circulatory arrest due to emptying blood vessels and drainage of fluids. For instance, it was found that the conductivity of frog muscle at 10Hz decreased a factor two after 2.5 hours of death and that of chicken muscle decreased 70 percent in the first 60 minutes (Zheng et al.,

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285

FIGURE 9.2. Illustration of the three levels of inhomogeneities discussed in this chapter. at the left) the head that is usually modeled by three compartments; in the middle) the layered gray matter; at the right) a suspension of cells.

1984). However, the conductivity of dog muscle increased immediately after death (Zheng

et al., 1984). This may be caused by a change in the osmotic pressure causing some cells to swell and burst. Gielen (1983) measured in vivo the low-frequency conductivity of the muscles of a rabbit that were prepared free , the blood supply was unimpaired. He found after finishing his experiments that the outermost muscle layer was damaged. The cross sections of the fibers were much larger and rounder than normal. This damage was related to an increase in conductivity. This phenomenon was blamed to osmotic proces ses. The increase

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will grow in time (Schwan, 1985), because cell membranes after death allow currents to pass more easily. In other words, all in vitro conductivity measurements should be completed within a very limited time. If the measurements are carried out in vivo, commonly animal tissue instead of human tissue is used . However, it is not clear whether the animal tissue has the same electrical properties as human tissue although sometimes they seem to be comparable. Moreo ver, the in vivo measurements depend on the surrounding tissues. When electrod es are implanted in living tissue, the currents applied by these electrodes will not be confined to the tissue that is between the electrodes, but will spread out through all surrounding tissues. Hence, it is difficult to estimate which part of the current will flow through the tissue of interest. Some investigators try to avoid this problem by measuring the conductivity with two electrodes spaced closely together. However, this raises the question wheth er on such a scale the inhomogeneities in the structure of the tissue do not disable the measurement of an effective macroscopic conductivity because the electric field within the tissue near cells has a complicated pattern. If one happe ns to be near a cell, the field may be small or point in a totally different direction. Another problem experienced is a relatively large extra capacity between electrode and tissue at low frequencies. Moreover, the electrode-el ectrolyte interface can produce large errors that depend on the pressure between electrode and organ tissue. Thence, it is not surprising that the measured low-frequen cy conductivities reported in the literature vary over a wide range. As example some values found for the low-frequency conductivity of skeletal muscle tissue at 37°C are given in table 9. 1. No attempt to give a complete overview has been made. The conductivity in muscle tissue is anisotropic, the condu ctivity along the fibers is higher than the condu ctivity perpendicular to the fibers a l . Five degrees misalignment from true parallel or perpendicul ar orientation durin g the measurement would result in an 18 percent overestimate of al and a 0.4 underestimate of a h (Epstein and Forster, 1983). Consequentl y, misalignment errors will be smallest in a h and the anisotropy factor will be easily undere stimated. Out of theoretical studies, there might develop insight into the nature of volume conduction that would permit a proper choice from the values that are reported .

a"

TABLE 9.1. Som e values fo und in the literature of the co nductivity of skeletal muscle at 37°C in the frequency range of 0-1000 Hz species cow rabbi t dog dog frog(2I CC ) dog rabbit monkey dog rabbit rat

ah(S/m )

a\(S/m)

0.41 0.8 0.67 0.43 0.09 0.70 0.75 0.8 1 0.52 0.5 0.5

0. 15 0.06 0.04 0.2 1 0.05 0.06 0.04 0.06 0.08 0.08 0.Q7

anisotropy factor ahla[ 2.7 13 17 2.0 1.8 II 17 13 6.5 6.3 6.1

referenc e Burger and van Dongen (1961) Rush et al. ( 1963) Burger and van Milaan ( 1943) Hart et al. ( 1999) Zhe ng et al. (1984)

Epstein and Foster (1983) Gielen et at. (1984) McCrae and Esrick (1993)

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The Electrical Conductivity of Living Tissue

TABLE9.2. Examples of the temperature dependence of tissues material

conductivity (S/m)

temperature e C)

bile (cow-pig)

1.66 1.28 1.54 2.04 3.33 2.56

37 20

0.42 0.66 0.08 0.20

36.7 44.5 (after 30 minutes) 36.6 44.5 (after 30 minutes)

amniotic fluid (sheep) Urine (cow-pig) skeletal muscle (rat) parallel perpendicular

25 37.5 37 20

9.1.4 TEMPERATURE DEPENDENCE The electrical condu ction in living tissue depend s on the temperature. Me Rae and Esrick (1993) heated freshly excised rat skeletal muscle from 39.5 to 50°C. Abo ve 44°C healthy tissue will be damaged irreversibly. Initially. a rapid increase of the low-frequenc y conductivity (with about a factor two) was observed followed by a much slower increase during which the low-frequency conducti vity graduall y approached the high-frequency values. The initial rapid change was associated with microscopically observed fiber roundin g and shrinkage in the radial direction and increa sing edema. The subsequent slow change was associated with disintegration of the tissue. After the cell membranes are destroyed. the conductivity is no longer frequency dependent and the low-frequ ency conductivity has the same value previously measured at high frequencies. The temperature dependency of the conductivity of tissue is also caused by the temperature-dependent regulation of the vessel diameter (vasodilatation). In other words. the blood supply is temperature dependent. The temperature dependen ce of the conductivity is illustrated in Fig. 9.3 and table 9.2.

9.1.5 FREQUENCY DEPENDENCE The low-frequency conductivity of most tissues like heart muscle . skin. liver. lung. fat. and uterus is not strongly dependent on frequency. although measurements show that the conductivity increases with frequency. also at low frequencie s. Commonly. this phenomenon is not taken into account and the conductivity is presumed to be frequency independent for frequencies lower than 1000Hz (e.g., Schwan and Foster. 1980). Nevertheless. the conductivity does increase with the frequency regardless the kind of tissue. Nicholson (1965) measured the conductivity of cerebral white matter of a cat. He found an enhancement of the conductivity in the direction normal to the fibers from 0.11Sim to 0.13S/m. when the frequency changed from 20 to 200Hz. No enhancement was found for the conductivity parallel to the fibers. Comparison between the experim ental data presented by Gabriel et al. (1996a ) . and corresponding data previously reported in the literature show good agreement (Stuchly and Stuchly, 1980). The conductivity of the gray matter as measured by Gabriel et al. (1996b ) in the frequency range from 10Hz to 20GHz is given in Fig. 9.4.

M. J. Peters, J. G. Stinstra,and I. Leveles

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The Electrical Conductivity of Living Tissue

289

9.1.5.1 Impact ofthe frequency dependence on the EEG The frequency dependence of the conductivity may influence the EEG. In order to estimate the impact on EEG, the effects of a frequency-dependent conductivity were simulated. The volume conductor model consisted of three concentric spheres representing the brain, skull and scalp, with radii of 87, 92 and 100mm. This model is often used to describe the head as a volume conductor. The source was modeled as a current dipole. Simulations were carried out for three current dipoles, i.e. a central dipole, and a radial and a tangential dipole at a radius of 80mm. As observation point, a point at the surface was taken where the potential, found for the lowest frequency, was maximal. The transfer function was calculated. The transfer function is the relationship between the strength of the current dipole and the potential in a point at the outermost surface as a function of the frequency. To enable a comparison between the different cases, all transfer functions were normalized. If there is no frequency dependence, the transfer function would have the value one for all frequencies. Two cases were studied. First, the gray matter conductivity was taken from Fig. 9.4 in which the conductivity increases by a factor four; the scalp conductivity was 0.33S/m and that of the skull 0.0042S/m. Second, the conductivity of all compartments increased 20% in the interval from 1Hz to 100Hz. At 1Hz the conductivity of the brain and scalp was 0.33S/m and that of the skull 0.0042S/m. The results of these simulations are given in Fig. 9.5. As shown the volume conductor acts as a low-pass filter. The potential may drop by approximately a factor two when the frequency is increased from 1 to 100Hz. The transfer function depends on the depth of the source.

9.2 MODELS OF HUMAN TISSUE Tissues are composed of cells. The interstitial space between the cells contains fluid. So, the effective conductivity of a tissue depends on the conductivity of the cells, the volume fraction occupied by the cells, and the conductivity of the extracellular medium.

9.2.1 COMPOSITES OF HUMAN TISSUE This section starts with a brief description of tissue at a cellular level. Next, the conductivity of a cell and that of the extracellular fluid will be discussed.

9.2.1.1 Cells All human cells stem from the round-shaped fertilized egg cell. There is no typical cell shape. Cells come in all shapes: cubes (cells lining sweat ducts), spheres (white blood cells of the immune system), Bismarck doughnuts (red blood cells), columnar cells, balloon-like cells (cells lining the urinary bladder), needle shaped ellipsoids or rods (skeletal muscle cells) and pancakes (cells on the surface of the skin) as illustrated in Fig. 9.6. Cells vary also considerably in size, and function. For instance, the diameter of a red blood cell is 7.5f.1m, the diameter of a human egg cell is 140fJ..m, a smooth muscle cell has a length of 20 to 500f.1m, while a skeletal muscle cell may have a length of 30cm. All cells

290

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0.1 0

10

20

30

40

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80

90

100

FIG UR E 9.5. Transfer functions for EEG for three dipoles using a three-sphere model of the head. Two cases of frequency-dependent behaviour are considered. In the first case the conductivity of the brain compartment is chose n according to Fig. 9.4. In the second case the conductiv ity of all three compartments increases linearly.

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blood cells skin cells

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Fat cells •

291

The Electrical Conductivity of LivingTissue

TABLE9.3. Values of a cell as foundin theliterature(adaptedfromKotnik et al., 1997) cell values

lower limit

higher limit

extracellular medium conductivity a e membrane conductivity am cytoplasmicconductivity a i cell radius R Average membranethickness: t = 5 nm

5.0 x IO-4Sm - 1 1.0 x IO-8Sm - 1 2.0 x IO- 2Sm- 1 1 11 m

2.0 Sm- I 1.2 x IO- 6Sm- 1 1.0 Sm- I 100 11m

are surrounded by a membrane 5 to lOnm thick, visible only with the electron microscope. The electrical properties of the cell components are different. The highest and lowest values reported in the literature for the electrical properties of biological cells are given in table 9.3. The intracellular fluid accounts for about 70 percent of the inner cell volume. The membranes maintain the integrity of the cell, breaking down after death. Membranes are highly selective permeability barriers and consist mainly of lipids and proteins. Lipid bilayer membranes have a very low permeability for ions and most polar molecules. An exception is water that easily crosses such membranes, and creates a balance in the whole organism. An ion such as Na+ crosses membranes very slowly because the removal of its shell of water molecules is highly unfavorable energetically. Membranes contain specific channels and pumps that regulate the molecular and ionic composition of the intercellular compartment. A nerve impulse, or action potential is mediated by transient changes in Na+ and K+ permeability. An action potential is generated when the membrane potential is depolarized beyond a critical threshold value. When the transmembrane potential is not exceeding the threshold value, the relationship between the potential and the current is approximately linear, so it obeys Ohm's law. For low frequencies, this is the case when the current density is smaller than 0.5J.LNcm2 •

9.2.1.2 Volume fraction occupied by cells One can analyse blood and determine the hematocrit content (the volume fraction occupied by the red blood cells) by measuring the electrical conductivity of blood. The electrical conductivity is a function of the hematocrit. This function is given in Fig. 9.7.

9.2.1.3 The extracellularfluid Mammals have a water content of 65 to 70 percent. In the early fetal period of humans, approximately 95 percent of the fetus is water. The proportion of total body weight that is water decreases throughout the fetal period to reach 75 percent at term. The water content of various tissues of adults is given in table 9.4. Biological tissues are inhomogeneous materials with discrete domains: the extracellular and the intracellular space. Although the cells are of microscopic size, still they are much larger than the ions in the extracellular space so that the extracellular fluid can be considered as a continuum. The extracellular and intracellular fluids are electrically neutral, however, the ion concentrations are quite different in each. Thus, the electrical properties of the intracellular fluid and the extracellular fluid are different.

M. J. Peters, J. G. Stinstra, and I. Leveles

292

TABLE 9.4. Water content of various organs (Pehtig and Kell, 1987; Foster and Schwan, 1986)

X

tissue type

volume fraction of water

gray matter (brain) white matter (brain) skeletal muscle Fat Liver Spleen

0.84 0.74 0.795 0.09 0.795 0.795

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.. ... . ..

, .... . ..

~ ~

's

13 :::J

8

"C

c:

8 6 .4

2l....-----l.-----1 - L -_ _----L 10 20 30 40 50 percentage hematocrit

....L..-_ _---I

60

70

FIGURE 9.7. The conductiv ity of blood as a function of the percentage of red blood cells (i.e. the hematocri t). The dashed line is the measured curve , the solid line is calculated using Maxwell's mixture equation, the line with points and dashes is calculated using Archie's law.

9.2.2 CONDUCTIVITIES OF COMPOSITES OF HUMAN TISSUE A composite cell includes a nucleus, cytoplasm, and a cell membrane. The nucleus is enclosed by a thin cell membrane. The cytoplasm is a mass of fluid that surrounds the nucleus and is encircled by the plasma membrane. Within the cytoplasm are specialized structures called cytoplasmic organelles. In other words, the structure of a cell is such that the conductivity cannot be expected to be homogeneous. The effective conductivity is the

293

The Electrical Conductivity of Living Tissue

E

FIGURE 9.8. Model used to calculate the effective conductivity of a spherical cell.

conductivity ascribed to an entire cell. In the next section this effective conductivity is estimated using a simple model for the cell.

9.2.2.1 Effectiveconductivity ofa sphericalcell The effective conductivity of a cell at low frequencies and for low current densities will be estimated for a spherical model of a cell. In the model, a sphere of radius rj and conductivity (Tj is surrounded by a thin concentric shell of thickness t and conductivity (Tm representing the membrane. The cell is suspended in a medium with conductivity (Te. The model is depicted in Fig. 9.8. Typically, tis Snm for a cell and its radius R = r, + tis 10/-Lm, thence t « R. The law of conservation of charge in quasistatic approximation reads:

(9.2)

J

Inserting = (T E and Laplace's equation

E=

- VV yields for a homogeneous region where a is constant

(9.3)

We assume that a homogeneous electric field E is applied along the z-axis. The electrical potential outside the two-layered particle can be calculated by solving Laplace's equation with the proper boundary conditions. The boundary conditions are that the potential and the normal component of the current density are continuous across the boundary. On the other hand we can calculate the potential outside a homogeneous sphere of conductivity (Teff suspended in a medium of conductivity (Te. The effective conductivity of the cell is the conductivity of a uniform sphere that gives the same electrical potential outside the cell as the two-layered one. Equating the two solutions for the potential and neglecting the higher order terms of tIR in both the denominator and numerator as t « R, yields

M. J. Peters, J. G. Stinstra, and I. Leveles

294

I

I

I

I

., I

,

\

,

..

FIGURE 9.9. Thecurrentdensity around a sphericalcell.The applied fieldwas initially uniform. The intracellular current density is too small to be depicted.

(Takashima, 1989)

(9.4)

Inserting the values given in table 9.3 shows that cells can be described as non-conducting particles because the effective conductivity of a cell is about 10- 5 times that of the surrounding fluid. In good approximation, at low frequencies the currents flow around the cells rather than through them as shown in Fig. 9.9. Thence, at low frequencies, the conductivity is dominated by the conductivity of the extracellular space .

9.2.2.2 Effective conductivity ofa cylindrical cell Similar calculations can be carried out to deter mine the effective conductivity of a cylindrical cell. We assume that a current is applied perpendicular to the axis. The effective conductivity reads

(9.5)

leading to the conclusion that in this case the cylindrical cell like the spherical one can be considered as a non-conducting particle. However, if the field is applied parallel to the axis it is a different situation. The cell has a resistance for a current parallel with the axis. The resistance of an element of material

295

The Electrical Conductivity of Living Tissue

of length L and cross section A is

R = L/(aA)

(9.6)

The cylindrical shell of thickness t, that represents the membrane, is connected in parallel with the inner cylinder of radius (r - t). The part of the membrane at the top and that at the bottom of the cylinder are connected in series with the intracellular fluid of the cell. So the resistance reads

1

Reff

Jl'

(r - t)2

L - 2t

2t -aj- +am -

(9.7)

where the effective conductivity is defined as the conductivity of an element of homogeneous material with the same dimensions, that has the same resistance as the inhomogeneous element. In good approximation expression (9.7) leads to the expression (9.8) Expression (9.8) is only true when the cells are intact, i.e. the membranes at the bottom and top of the cylinder are not damaged.

9.2.2.3 Conductivity ofextracellularfluid Since the intracellular space is not playing a role, as the effective conductivity of a cell is in general zero for low frequencies, the intracellular fluid is not considered here further. Materials with the highest conductivity are fluids with a low concentration of cells like urine, amniotic fluid, cerebrospinal fluid and plasma, which will be discussed below. Using the ionic concentrations of the different cations and anions in the extracellular fluid, the conductivity can be approached by (9.9) where Ai are the ionic conductivities at 37°C and Cj are the molar concentrations of the different ions. In table 9.5 the ionic conductivities of various ions at a temperature of 37°C are given as well as the ionic concentrations in the interstitial fluid, the cerebrospinal fluid and the blood plasma. Using these values, the conductivity of these fluids can be estimated. In table 9.6 the computed values are compared with the values cited in literature. From these values it becomes apparent that the computed values are about 15 to 25 percent higher than the measured values. This overestimation may be explained by the presence of proteins in the actual solution and the presence of counterions surrounding its cells. The extracellular concentration of ions such as K+ undergoes frequent small fluctuations, particularly after meals or bouts of exercise. An exception is the brain; if the brain

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TABLE9.5. The ionic concentration mmoIll of differentfluids encountered in the body and the specific conductivity of each ion in a dilute solution at 37°C.The valueswereobtainedfromAseyev(1998). "Thesevalueswereonly available for 25°C and have been correctedby 2 percent/tC Ion

blood plasma

interstitial fluid

cerebrospinal fluid

Ai 1O-4S m2 jmol

Na+ K+ Mg 2+ Ca2+

142 4 2 5 102 26 2 17.0 6.0

145 4 I 4 116 29 2 0.0 6.7

153 2 3 3 123

63.9 95.7 142.4 154.9 95.4 55* 296*

ClHC0 3 PO~protein Other

TABLE9.6. Calculated and measured values of the conductivity of body fluids at 37°C. The measured values are taken from literature: "Geddes and Baker (1967), "Schwan and Takashima (1993), cBaumann et al. (1997) conductivity of body fluids (at 37°C in S/m)

blood plasma

interstitial fluid

cerebrospinal fluid

computed measured

2.08 1.58"

2.22 2.0 b

2.12 1.79c

were exposed to such fluctuations the result might be uncontrolled nervous activity, because K+ ions influence the threshold for the firing of nerve cells. The conductivity of electrolytes is dependent on the temperature. The conductivity of the amniotic fluid was measured by De Luca et al. (1996) at a temperature of 20°C. The mayor contributors to the conductivity are the Nat and CI- ions. The ionic conductivity of the former increases by 2.1 percent per °C and that of the latter by 1.9 percent per "C. An overall increase of 17 x 2 = 34 percent for the amniotic fluid heated from 20 to 37°C is thus expected. The same rate of increase was also obtained by Baumann et al. (1997), who measured the conductivity of cerebrospinal fluid at 25 and 37°C.

9.2.3 MAXWELL'S MIXTURE EQUATION Spherical particles suspended in a solvent are considered to be a relevant model of biological tissues. Maxwell (1891) derived an equation for the effective conductivity of dilute suspensions in aqueous media of spherical particles (that do not have a permanent electric moment). The derivation of this equation is given in short below. Next, Maxwell's equation is extended for ellipsoidal particles, as many biological cells are better described by ellipsoids instead of spheres. The applicability of Maxwell's equation will be illustrated for blood.

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The Electrical Conductivity of Living Tissue

FIGURE 9.10. A spherical particle of conductivity an original uniform field E.

surrounded by a solvent with conductivity a e placed in

apart

9.2.3.1 A dilute solution ofspheres First, let us consider a spherical particle of radius R and conductivity apart embedded in a homogeneous solvent with conductivity a; when a uniform electric field Eo is applied. An expression for the potential outside the sphere is found by solving the Laplace equation with the proper boundary conditions using spherical co-ordinates. The origin of the co-ordinates (r, rp) is at the center of the sphere . This yields for the potential V outside the sphere of radius R:

e,

ae -

V(r, e) = - ( 1 + 2ae

apart

+ apart

R r

3

3

)

Eorcose

(9.10)

Within the sphere, the field is parallel and uniform. Outside the sphere, the field is the original field plus the field of a dipole at the center of the sphere (see Fig. 9.10). Second, the suspension is modeled by N small spherical particles of radius R and conductivity apart that are surrounded by a large spherical boundary of radius R' as depicted in Fig. 9.11. Two assumptions are made. First, the volume fraction occupied by the cells is assumed to be low, the average distance between them being larger than their dimensions and as a consequence the spheres do not influence each other although they act as dipoles . The potential is the sum of the potential due to N small homogeneously distributed particles, yielding:

ae -

V(r, e) = - ( 1 + N , 2ae -r- apart apart

3

R r

3

)

Eorcose.

(9.11)

Instead of the microscopic point of view, we can look at the sphere from a macroscopic point of view. We have a spherical shaped medium consisting of an aqueous solution of spherical particles. The effective conductivity of this sphere is per definition the conductivity that gives a potential outside of the sphere that is expressed by (9.10). Hence, the potential at

M. J. Peters, J. G. Stinstra, and I. Leveles

298

p

FIGURE 9.11. The model used for the derivation of the Maxwell mixture equation.

a distance r > R' from the center of the sphere describing the solution reads: 3

ae - aeff R/ ) V(r, B) = - ( 1 + - 3 EorcosB 2ae + aeff r

(9.12)

where aeff is the effective conductivity of the large sphere containing N particles. Formulas (9.11) and (9.12) are equivalent expressions for the potential outside the sphere with radius R'. Noting that N R 3 / R'3 = P is the volume fraction occupied by the particles that are suspended in the large spherical boundary we obtain by equating expressions (9.11) and (9.12) the so-called Maxwell's mixture equation: (9.13)

As argued before at low frequencies and low current densities, the currents will in general be only in the extracellular space, in that case one can insert apart = O. A spherical model is a good approximation for many colloidal particles, including biological cells. However, biological cells often have a complex geometry. Many cells are better described by ellipsoids; a, b and c are the semi-axes of the ellipsoid. Maxwell's mixture equation has been derived for ellipsoidal particles that are orientated in parallel by Sillars (1937). When p is very low and the field is applied along the a-axis, it reads: _ -,-p_(a-,-part_-_a_e_)_] aeff = ae [ 1 + a« + (apart - ae)L a

(9.14)

where La is the depolarization factor of the ellipsoid in the direction of the a-axis, (Boyle, 1985). abc

La =

['JO

2 10

ds (a2

+

s)J(a2

+ s)(b 2 + s)(c 2 + s);

La + L,

+ L, =

1

(9.15)

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The Electrical Conductivity of Living Tissue

0 .9 0 .8 0.7 0.6

~ 0 .5 0.4 0.3 0.2 0 .1 0

0

2

3

4

5

6

7

alb FIGURE 9.12. The depolarization factor in the directionof the a-axisfor an ellipsoidof revolution withsemi-axes a, b, and c, where b = c.

There are three cases of considerably interest for us: spheres, needle-shaped ellipsoids and disk-like particles. For a needle-shaped ellipsoid or a particle with the shape of a long cylinder with a = b « c, La, L, and L, tend to 1/2, 1/2 and 0, respectively. For a sphere with a = b = c, the depolarization factor La = 1/3. For a disk-shaped ellipsoid with a = b » c, La, t., and t, tend to 0, 0 and 1 respectively (Fricke, 1953). In Fig. 9.12 the depolarization factor La is depicted for a spheroid with semi-axes a, b = c. Maxwell's mixture equation has been experimentally tested by Cole et al. (1969) for variously shaped objects. The particles could be spheres, cubes, or cylinders arranged in a cubic or hexagonal array or randomly distributed plates. They found that the Maxwell equation was surprisingly accurate within one percent at concentrations from 30 percent or less to 90 percent.

9.2.3.1.a. Blood To illustrate the applicability of Maxwell's mixture equation, the conductivity of blood is discussed. Blood of normal subjects consists predominantly of red blood cells (erythrocytes) in plasma. The mature erythrocyte is a cell surrounded by a deformable membrane well adapted to the need to transverse narrow capillaries. The red cells are biconcave disks, each with a diameter of about Sum, a thickness of 2fJ..m at its edge, and a volume of about 94fJ..m3 . In normal adults, the red cells occupy on the average about 48 percent of the volume of blood of males and about 42 percent in the blood of females. The percentage of the volume of blood made up by erythrocytes is defined as the hematocrit. Erythrocytes are essentially

M. J. Peters, J. G. Stinstra, and I. Leveles

300

TABLE 9.7. Parametervalues usedtoestimate the conductivity perpendicular to the fibers and that parallel to the fibers. ("Kobayashi and Yonemura, 1967) parameter effective intracellular conductivity extracellular conductivity membrane conductivity volume fraction occupied by fibers fiber length membrane thickness

symbol

minimum

maximum

aj

0.55 S/m 2.0 S/m 1.0 X 1O-8S/m

0.80 S/m 2.4 S/m 1.2 X 1O- 6 S/m

L

0.85 5 mm

0.9" 30cm

t

0.1 lim

I lim

a. am

p

non-conducting in fields with frequencies up to 100kHz (Trautman and Newbower, 1983). A typical value for the low-frequency conductivity of human blood at body temperature and normal hematocrit is in the range 0.43-Q.76S/m (Geddes and Baker, 1967). The orientation of the ellipsoids with respect to the electric field depends on the blood flow. There is a periodic noise component in continuous measurements of conductivity, which is presumed to result from the cyclic reorientations due to the flow. The magnitude of the change with flow is of the order of three percent of the baseline conductivity. Since the shape of red blood cells of most species is non-spherical and is subject to changes in aggregation and orientation with flow, variations in the effective conductivity are understandable. Geddes and Sadler (1973) measured the conductivity for human , canine, bovine and equine blood at 25kHz and 37°C, having a hematocrit range extending from 0 to 70 percent. For human blood they found an exponential fit between the measured data and an exponential curve for p = 53.2eo.022H with a correlation coefficient of 0.99. Blood can be modeled as a dilute solution of non-conducting spherical particles within a conducting medium . In Fig. 9.7 both the experimental measurements and the results using Maxwell 's mixture equation for spherical particles are displayed. From this figure one can conclude that the description of Maxwell is rather accurate. With increa sing concentration of red blood cells (p is larger) the effective conductivity is reduced. The relation ship between p and the conductivity can be used to estimate the hematocrit content (Ulgen and Sezdi, 1998).

9.2.4 ARCHIE'S LAW Measurements of the conductivity of the cortex do not fit so well with the theory of Maxwell, they fit better with Archie's law (Archie, 1942; Nicholson and Rice, 1986). Moreover, Maxwell's theory can only be applied if there is only one type of cell present and most human tissues are constituted of various types of cells . Archie's law that will be discussed in the next section is applicable if there are various types of densely -packed non-conducting cells suspended in a conducting solution. To derive Archie's law the following assumptions are made . a) Tissue is modeled by cells of various shapes that are suspended in an aqueous conducting medium; b) The effective conductivity is calculated for a region containing thousands of particles; c) The particles are assumed to be homog eneou sly distributed; d) Currents are Ohmic and are only present in the extracellular space; e) The solvent is homogeneous and isotropic ; f) An electric field is applied which is initially uniform; g) The particles do not have a permanent electric dipole moment.

The Electrical Conductivity of Living Tissue

301

9.2.4.1 Ellipsoidal particles withthe same orientation The derivation of an equation for the effective conductivity for higher values of p is given by Hanai (1960). It starts with equation (9.14). Subsequently, the number of particles is increased. Due to this increase, the effective conductivity aeff increases with a fraction ~aeff' In the next stage the previous mixture acts as a host, and so on. An expression for the effective conductivity of a densely packed suspension of ellipsoidal particles is obtained by integration yielding: aeff -

apart

----'----ae -

apart

L'

ae

(

-

aeff

)

=l-p

(9.16)

Inserting in this equation apart = 0 yields Archie's law: (9.17) where a e is the conductivity of the fluid surrounding the non-conducting cells, p is the volume fraction occupied by the cells, and m is the so-called cementation factor that depends on the shape and orientation of the particles, but not on their sizes. According to Archie's law, the effective conductivity is proportional to the extracellular conductivity o.: The value of m depends on the shape of the cells. For instance, its value is 3/2 for spherical particles, m is 2 for long cylinders with the axis perpendicular to the external field, and it is 1 for cylinders with the axis parallel to the field. Archie's law also holds for ellipsoidal particles. If the field is applied along the a-axis of the ellipsoids and the ellipsoids have the same orientation, (9.18)

9.2.4.1.a. Fat Since fat tissue consists of about 9 percent water (Foster and Schwan, 1989) and the interior of the cells is almost completely filled with fat, the interstitial fluid occupies about 9 percent of the volume (upper limit). Histology shows that fat cells are sphericalshaped particles. According to equation (9.17) an upper limit for the conductivity is 1.9 x (0.09)3/2 ~ 0.05 S/m.

9.2.4.1.b. Skeletal muscles A skeletal muscle fiber represents a single cell of muscle. Each skeletal muscle fiber is a thin elongated cylinder with rounded ends that are attached to connective tissues associated with the muscle. Just beneath its cell membrane, the cytoplasm of the fiber contains many small nuclei and mitochondria. The cytoplasm also contains numerous threadlike myofibrils that lie parallel to the cylinder axis. The diameter may vary within muscles, between muscles in the same animals, and between species. Muscle fibers increase in diameter from birth to maturity and also in response to exercise. The length of a fiber may vary between millimeters and several tens of centimeters. The amount of connective tissue relative to muscle fibers is much greater in some muscles than in others and may range from 3 to 30 percent. Connective tissue is composed of collagen fibers, reticular fibers, elastic fibers and several varieties of cells, such as fat cells. There is an increase in elastic tissue with aging. Apart from the

302

M. J. Peters, J. G. Stinstra, and I. Leveles

fibers in the tongue, the fibers have no branches. In long muscles , as in the longest muscle in the human body the sartorius (52cm long) , the fibers are arranged in parallel. Covering the surface of each muscle fiber is a thin membrane of about O.I/-Lm thickness. The model used to describe muscle tissue consists of homogeneously distributed cylinders. So far as connective tissue, blood vessels, and nerve tissue cannot be described by parallel cylinders, their influence will be neglected. The effective conductivity of one fiber when the field is applied parallel to the axis is according to formula (9.9) dependent on the length ofthe fiber. Inserting the values ofTable 9.7 in formula (9.8) yields 0.25 x 1O-5S/m (short fiber) < apart < 15 x 1O- 2S/m (long fiber). When the fiber is damaged (forinstance, Mc Rae and Esrick (1993) trimmed the fibers) then aj may be as high as the effective intracellular conductivity. Currents parallel to the fibers will be in both domains, the intracellular and extracellular space are connected in parallel. The effective conductivity of the muscle tissue will be given by (1 - p) aextracell + paeff. When the current is applied perpendicular to the fibers, the cell can be described as a non-conducting cylinder. The effective conductivity perpendicular to the fibers at according to equation (9.17) will be within the limits 2.0 x (0.1)2 S aI,eff ::: 2.4 x (0.15)2. Thus the effective conductivity perpendicular to the fibers will be in the range of 0.02-0.05S/m.

9.2.4.1.c. Cardiac tissue Cardiac cells can be described as cylinders with a diameter of about 15/-Lm and a length of about 100/-Lm. The currents perpendicular to the fibers will circumvent the fibers. The effective conductivity in the direction perpendicular to the fibers can be calculated using expression (9.17) with m = 2. The volume fraction occupied by the fibers reported by Clerc ( 1976) is p = 0.7. The conductivity of the extracellular space is assumed to be that of Ringer (p = 69Qcm) yielding a conductivity of I.4S/m. So the conductivity perpendicular to the fibers is estimated to be 1.4 x (0.3)2 = O.13S/m. Rush et ai. (1963 ) measured the conductivity of cardiac tissue and found for the conductivity perpendicular to the axis 0.18S/m. Currents parallel to the fibers will be present both in the extracellular and in the intracellular space. Cardiac cells are joined at their ends by intercalated disks and each cell is connected to its neighbors by gap junctions passing through these disks. Cardiac muscle fibers can be modeled by cylinders that are interconnected by junctions as depicted in Fig. 9.13. All cylinders are homogeneously distributed and are arranged in parallel. The current is applied parallel to the axes, such that the volume between the electrodes used to apply the current contains 104 fibers or more. Each cylinder has a length of lOO/-Lm and a radius of 7.5/-Lm. All parameters are chosen conform the parameters chosen by Plonsey and Barr ( 1986) from literature. These values were measured in mammalian ventricular tissue at a temperature of about 20°e. The resistance of a junction is Rj = 0.85 x lO6Q . The resistivity of intracellular fluid is 282Qcm (Chapman and Frye, 1978). Consequently, the resistance of a cylinder with a length of lO-2cm and a diameter of 15/-Lm of intracellular fluid is

The junction and the intracellular space are connected in series , so the resistance of a single

303

The Electrical Conductivity of Living Tissue

Electrical scheme ,,

a e=2 S/~ ,, , Current

p

=0.7

1.p =0.3 Cross section

FIGURE 9.13. Model of cardiac tissue.

cylindrical cell is Rcell = R, + Rj = (0.85

+ 1.60) x 106 Q .

A homogeneous cylinder with the same dimensions will have the same resistance Rcell, when an effective conductivity aeff is ascribed to the entire cylinder. aeff = Ij(RcellO),

where

1= 10-4m, O = n(7.5 x 10- 6)2, yielding aeff = 0.23Sjm.

The two composites of the cardiac tissue, namely the intra- and extracellular medium are connected in parallel. Thus, the effective conductivity along the fibers for cardiac tissue is (1 - p)aextracell

+ paeff = 0.3

x 1.4 + 0.7 x 0.23

= 0.58Sjm.

Rush et al. (1963) found for the conductivity parallel to the axis of cardiac fibers a value of O.4S/m.

9.2.4.2 Randomly orientatedellipsoidal particles The architecture of human tissue may be such that fibers and cells are oriented along each other. For instance, skeletal muscle fibers are more or less parallel. However, the assumption that the cells or fibers have the same orientation is often not plausible. For example, the red blood cells in blood outside of the body can roughly be described by oblate

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304

4 r--

--r--

-

r--

--r--

-

..----

---,--

-

....---

---"T-

-----,

3 .5

3 E

2.5

FIGURE 9.14. The cementat ion factor for a solution of randomly orientated spheroids with semi-axes a. b. and c, where b = c.

spheroids (ellipsoids with a < b = c) that are randomly oriented . Boned and Peyrelasse (1983) derived an expression for a solution of randomly orientated ellipsoidal particle s. Every orientation of the particle has the same probability and therefore the Hanai procedure is performed such that infinitesimal amounts of particles are added such that 1/3 of the ellipsoids have their a-axes in the direction of the electric field, 1/3 their b-axes and 1/3 their c-axes. For non-conducting particles again Archie's law is found with a cementation factor

(9.19)

The low-frequency conductivity of suspensions of ellipsoidal particles and that of doubletshaped (budding yeast cells) or biconcave particles (erythrocytes) differs only a few percent as can be concluded from simulation studies using as numerical method the boundary element method (Sekine, 2000). In Fig. 9.14 the cementation factor is depicted for an ellipsoid of revolution with semiaxes a, b and c, where b = c. As can be seen m is minimal for La = 1/3, the depolarization factor for a sphere, in that case CTeff is maximal.

9.2.4.2.a. Blood Archie's law can be applicated for blood. Fitting the measured data for human blood of Geddes and Sadler (1973) to Archie's law leads to a fit for m = 1.46 (see Fig. 9.7). For blood of different animals, a slightly different cementation factor is found .

305

The Electrical Conductivity of Living Tissue

9.2.4.3 Cells of different shape

Human tissue can often be described by particle s of different shapes that are immersed in an aqueou s medium. The cortex, for instance , has pyramid al cells and glial cells . The sizes of the particle s do not play a role in the theory derived. A different shaped ellipsoidal particle is one that has a different ratio of the radii along the three axes. In order to obtain an expression for a suspension of particles of different shapes, we applied the Hanai procedure by adding successively infinitesimal numbers of particle s in proportion to their relative volume fractions , leading again to Archie 's law

.h WIt

n

m

=" L...,Pi mji= 1

p

and

I( I)

m, = - ,C , - 3 L...,1-LJ=a

(9.20)

J

where n is the number of types of non-conducting ellipsoids, Pi the volume fraction of a particle of type i, p the sum of all volume fractions occupied by particles and Lj the depolarization factor in the direction of one of the axes (Peters et al., 2001).

9.2.4.3.a. Gray matte r As an example , the effective conductivity of the superficial cortex is estimated, using parameters found in the literature. The principal neuron types in superficial cortex are pyramidal cells in layer II and stellate cells in layer III. Axons and dendrites from deeper layers and axons from subcortical layers occupy a substantial portion of the tissue. The glial cells of superficial cortex are mostly star-shaped cells, i.e. astrocytes and a lesser fraction of oligondendrocytes. The latter resemble astrocytes, but are smaller and have fewer proce sses. A branching astrocyte process has a length of about 40 micron s and an average diameter of about one micron. The shape of the glial cell body in vitro is an intermediate between a flat disk and a sphere. The pyramid al cells have a cell body and dendrites. The cell body has a roughly pyramidal shape. The branching properties of dendrite s of neurons show a mean branching angle of about 60 degrees. The diameter of motoneurons is reported as 7 to 14f.lm for the proximal segment with decreasing values of about 5, 3, and 2f.lm for succe ssive branches. The fact that solutes of high molecular weight readily pass between cerebrospinal fluid (CSF) and interstitial fluid supports the supposition that no significant concentration gradients exist between the two compartments in the steady state. Hence the CSF reflects the ionic composition of the interstitial fluid and the conductivity of the CSF will be identical to that of the interstitial fluid i.e.1.8S/m (Baum ann et al., 1997). The gray matter is composed of glial cells that are modeled as spheres, occupying a volume fraction of 38 percent and pyramidal cells that are modeled as cylinders, occupying a volume fraction of 46 percent (Havstad, 1967). Both neurons and glial cells are described by non-conducting particles, i.e. a particle = O. The effective conductivity is computed using equation (9.20) yielding a cff = 0.097S/m. A measurement that used a uniform current to determine the conductivity of gray matter has been carried out by Ludt and Hermann (1973), who reported a value of 0.10S/m for the rabbit. However, the measurements were carried out in vitro 15 minutes after death and at room temperature. As found by van Harreveld and Ochs (1956), the conductivity drops after circulatory arrest by 30 to 35 percent due to the emptying of blood vessels and drainage of fluid. Thus Ludt and Hermann's

M. J. Peters, J. G. Stinstra, and I. Leveles

306

measurements indicate that the effective conductivity of cortical tissue in vivo will be about 0.15S/m. Archie's law led to a value of about 0.10S/m. However, the values used for the calculation may be not optimal. The volume fraction occupied by the extracellular fluid is in the range 17 to 28 percent (van Harreveld et aI., 1965; Nicholson and Rice, 1986). The interstitial space and the vascular volume together constitute the extracellular space. The vascular space has a volume of about 1 to 3 percent. So the value of (l - p) in practice will be somewhat higher than the value handled by us in the computation of the effective conductivity.

9.2.4.4 Clustered cells When spherical particles are clustered in a chain or in a close-packed lattice, the conductivity is such that these clusters behave as randomly orientated ellipsoids and thus Archie's law will be obeyed (Grandqvist and Hunderi, 1978).

9.2.4.4.a. Blood Blood with a high hematocrit content may aggregate. Pftitzner (1987) measured the conductivity of blood samples containing red blood cells of varying diameter between 100Hz and 100kHz. The conductivity was essentially independent of the diameter of the cells and the frequency. He found that at a hematocrit of60 percent or more that the dielectric constant shows a distinct decrease. This was explained by the aggregation of blood cells. An aggregation in a single cell chain, for instance, would lead to effective depolarization factors La = 0.13; Lb = L, = 0.435, leading according relation (9.19) to m = 1.6. In such case, the effective conductivity at 80 percent hematocrit would practically be the same as without aggregation (2 percent difference).

9.2.4.4.b. Liver The liver consists of different types of cells. These can be divided in hepatocytes and non-hepatocytes (Raicu et al., 1998a ) . But since the volume fraction of non-hepatocytes (0.06) is relatively small to that of the hepatocytes (0.72) a model is used consisting of one cell-type. The hepatic cells are clustered forming plates of one layer in thickness (see Fig. 9.15). The close-packed clusters of cells are assumed to act as oblate spheroids with axes

FIGURE 9.15. A schematic drawing of a part of a lobule of the liver.

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The Electrical Conductivity of Living Tissue

Epiderm is

Dermis

Fat

FIGURE 9.16. A schematic drawing of the layered human skin.

a, b, c, where a = b » c. Choosing the effective depolarization factors La, Lb , and L, to be 0.1,0.1 and O.S, respectively and applying Archie's law with p = 0.72 and as = 0.65S/m (the conductivity of blood) results in O'eff = 0.03S/m. This finding is in accordance with findings of Gabriel et al. (1996 a ) who reported values in the range of 0.02-o.03S/m. Choosing La, Lb , and L, to be 0.2, 0.2 and 0.6 leads to an effective conductivity of O'eff = O.OSS/m, which is more in accordance with the values given by Geddes and Baker (1967) and Raicu et al. (199S b ) .

9.3 LAYERED STRUCTURES Many tissues are organized in layers, such as the gastrointestinal tract, the retina, or the gray matter. An example of a layered structure is shown in Fig. 9.16, where the scalp is depicted. At an interface current lines will change direction. Corning from a medium with higher conductivity to a medium with lower conductivity, current lines will bend in the direction of the normal. Coming from a medium with lower conductivity to a medium with higher conductivity, current lines will bend from the normal. As a consequence, currents tend to cross the skull in a direction perpendicular to its surface. In other words , the layers of the skull are approximately traversed in series. Currents in the scalp tend to be parallel to its surface and the layers of the scalp can be approximated by a parallel connection of resistances. The simulations discussed in sections 3a and 3b show that this is indeed the case .

9.3.1 THE SCALP The outer layer of the scalp is the epidermis of about 0.2mm thickness. This layer is nonsensitive and non-vascular and overlies the dermis. The epidermis is a poorly conductive

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layer; this is ascribed to the dead nature of one of the layers within the epidermis, the stratum corneum. The conductivity of the epidermis is approximately 0.026S/m (Sernrov et al., 1997). The dermis is connective tissue with a thickness of 2rnrn and its conductivity is estimated to be 0.22S/m (Yamanoto and Yamanoto, 1976). The basis of the dermis is a supporting matrix with a remarkable capacity for holding water. The dermis has a very rich blood supply. At the side of the head, subcutaneous fat is found of about 3mm thickness. Sernrov et al. (1997) used a conductivity value of 0.08S/m for this layer. To assess the effective conductivity of the scalp, a simulation is performed with a spherical volume conductor. Two models are used. The first model consists of five shells representing the brain, skull, fat layer (3mm), dermis (2rnrn) and the epidermis (0.2rnrn). The conductivities of the three layers are mentioned above. The radii and conductivity of the brain and skull are 78 and 83mm, and 0.33 and 0.0042S/m, respectively. The other model has 3 shells: the brain, skull and the skin. The radii and conductivities of the brain and skull are equal to those of the first model. The thickness of the skin layer is chosen to be the sum of the thickness of the fat layer, dermis and epidermis. The effective conductivity of the entire skin layer is calculated (assuming that the currents are crossing the scalp such that the layers are approximately connected in parallel) by means of (9.21) where d, and a i denote the thickness and conductivity of layer i. This yields a value of 0.13S/m for the effective conductivity of the skin. A current dipole on the z-axis is used as source. The potential is calculated at the outer sphere by means of an analytical expression (Burik, 1999). The potential using the three-shell model is compared with the potential using the other five-shell model. The differences are expressed by the relative difference measure (RDM) defined as

RDM=

(9.22)

where Vi is the calculated value of the potential for the three-shell model, Vc,i is the calculated value of the potential for the five-shell model and N is the number of points were the values are calculated, The calculations are repeated varying the eccentricities and orientations of the dipole, and the ratio of the thickness of the three layers of the skin. Some results are shown in Fig. 9.17. It can be seen that equation (9.21) can be used to calculate the effective conductivity of the scalp when the thickness of the various layers is known. If these thicknesses are not known then a = 0.13S/m will be an appropriate choice for the effective conductivity of the scalp.

9.3.2 THE SKULL Another example of a layered structure is the skull that can also be subdivided into three layers. The upper and lower layers are structures made out of bone, which are bad conductors. The middle layer is a relatively good conductor, because it is spongy and

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The Electrical Conductivity of Living Tissue

6 ,95E-03

- - radial dipole .. . .. annential

:i!:

6,80E-03

c

~

6,65E-Q3

6,50E-Q3 -+--...,....----r-- ----,-- ---,----,- - ,----.,--...,.......- -' 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 o.s

eccentricity FIGURE 9.17. The RDM between the scalp potential calculated using an effective conductivity and the one calculated for a skin composed of 3 layers, namely the epidermis, the dermis and the fat.

contains blood. The total thickness varies across the skull between 4 and 11mm, the mean is around 6mm. The skull has a conductivity in the direction perpendicular to the surface of a factor 10 smaller than in the direction parallel to the surface. Usually, a value of 0.0042S/m is assigned to the conductivity of the entire skull, although recent measurements of the conductivity of the skull using bone that was temporarily removed during epilepsy surgery led to values that were a factor ten higher (Hoekema et al., 2001). Simulations are performed with a spherical volume conductor. The volume conductor consisting of the brain, skull and scalp is modelled by either a three-shell model in which an effective conductivity is assigned to the skull or a five-shell model. The radii and conductivity of the brain and scalp are 78 and 89.2 mm, and 0.33 and O.l3S/m, respectively. The skull is described by three layers of conductivity of 0.0029,0.029 and 0.0029S/m, respectively. These values are chosen such that for a mean skull thickness of 6mm consisting of layers of 2mm each, the effective conductivity is 0.0042S/m. The other model has 3 shells: the brain, skull and the scalp. The radii and conductivities of the brain and scalp are equal to those of the first model. The thickness of the skull is varied between 2 and 10mm. The ratio between the thicknesses of the three layers describing the skull is varied as well. A current dipole is taken as the source, whose position and orientation are varied. As the conductivity of the skull is much lower than that of the surrounding tissues one may expect that the currents will cross the skull perpendicularly. This implies that the three layers of the skull are crossed in series. If the effective conductivity is given by

(9.23) then the RDM is well below 2 percent. Hence, we may conclude that the effective

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310

conductivity of the skull is approximated quite well by expression (9.23). When realisticallyshaped models are used in the inverse solution, both the thickness and the conductivity should vary.

9.3.3 A LAYER OF SKELETAL MUSCLE In the ECG inverse problem the muscle layer may be considered as one of the compartments. Some simulation studies suggest that this anisotropic layer is the only thoracic inhomogeneity with a significant effect on the relationship between epicardial and torso potentials (Stanley et al., 1991). The muscles underneath the skin of the torso are essentially directed parallel to the body surface but otherwise almost uniformly distributed over all angles(Rush, 1967). This implies that the conductivity parallel to the body surface differs from that perpendicular to it. Half of the fibers that are parallel to the body surface are perpendicular to the other half. The conductivity parallel to the body surface will be about am = (at + ah) /2. The conductivity perpendicular to the body surface is al. We choose a co-ordinate system such that ax = a y = am and a z = aj. Expression (9.2) reads in this case: (9.24) If the muscle layer is modeled as layer of infinite extent with thickness t, then the proper boundary conditions are that at the surface the normal component of the current density is continuous and the potential is continuous. Equation (9.24) can be solved by a co-ordinate transformation from (x,y,z) to (x' ,y' ,z'), where x' = x;y' = y

and

z' =

(9.25)

zJam/al

The scale transformation is such that the currents and the potential are chosen to be invariant. In other words they are the same at corresponding points of the primed and unprimed system. The components of the current density being a current divided by a surface transform as: and

'1

•

Jz = Jz

(9.26)

The electric field being the gradient of the potential transforms as: (9.27) As a consequence, in the primed system:

=0

n2lv, = v

and

J7 1

~a aE~ = yUmUj

(9.28)

Hence, after transformation we have to solve Laplace's equation with the proper boundary condition (i.e. the z-component of the current density is continuous), which is invariant under the co-ordinate transformation. The primed system is isotropic with conductivity

The Electrical Conductivity of Living Tissue

311

-Jamal. In other words, the potential of the anisotropic muscle layer is equivalent to the potential in a homogeneous isotropic medium with an effective conductivity -Jamal, if the thickness is enhanced with a factor -Jam/a,. Inserting aj = 0.05S/m, ah = 0.5S/m for skeletal muscle and taking a layer thickness t = lcm yields for the effective conductivity of the muscle layer aeff ;:::, O.IIS/m and an effective thickness of the muscle layer of 5cm. Stanley et at. (1986) showed that the agreement between calculated and measured torso potentials significantly improved if the anisotropic nature of the muscle layer was taken into account. Their results were based on a canine study.

9.4 COMPARTMENTS Customarily, the volume conductor is described by a compartment model. Each compartment is considered to be homogeneous. The number of compartments may vary. To solve the inverse problem for EEG, the head is often described by three compartments, the brain, skull and scalp. Usually, the conductivity assigned to the brain and to the scalp compartment are identical. The geometry of these compartments may be a sphere covered by two spherical shells or the compartments may have a realistic shape. Especially in the case that a three-sphere model of the head is used with standard radii of the spheres, the interfaces between the compartments will not coincide with the actual interfaces between the different tissues. Consequently, a compartment will consist of more than one type of tissue. For instance, the muscles that are present in the occipital and orbito-frontal areas of the scalp may be partly assigned to the scalp compartment and partly to the skull compartment. The cerebrospinal fluid may be assigned mainly to the brain compartment, but also partly to the skull compartment. Consequently, the effective conductivities do not represent the actual conductivities of brain matter, skull bone or skin. The effective conductivities are those values that minimize the differences between the actual EEG and the calculated EEG. The effective conductivity of a compartment can be estimated in three ways. First, implanted electrodes can be used to act as a source. Measurements of the potential or magnetic field distribution generated by the source can be used to estimate the source. The actual source can be compared with the calculated one. By varying the effective conductivities used in the models the differences between the actual source and the calculated one can be minimized. The conductivities that give the smallest differences are taken as the effective ones. Second, the effective conductivities can be determined by fitting evoked magnetic field measurements with those of the electrical potential in case that the potentials and magnetic fields are due to the same source. The third method is based on impedance tomography.

9.4.1 USING IMPLANTED ELECTRODES Sometimes electrodes are inserted in the brain through trephine holes during presurgical evaluation of epileptic patients (Veelen et al., 1990). Normally, these electrodes are used for the measurement of the potential, especially during long term seizure monitoring. However, when a current is applied to a pair of electrodes, it will act as an artificial source. Its location and orientation is known from X-ray or magnetic resonance images. The artificial source can be localized from potential measurements on the scalp. The true and calculated

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M. J. Peters, J. G. Stinstra, and I. Leveles

location should coincide. This can be established by varying parameters of the model used in the inverse solution, such as the ratio between the conductivities in a three-compartment model of the head. Homma et al. (1994) used a realistically shaped model and found that the inverse solution was the best when the ratio between the conductivities was 1 : 1/80 : 1.

9.4.2 COMBINING MEASUREMENTS OF THE POTENTIAL AND THE MAGNETIC FlEW Cohen and Cuffin (1983) measured magnetic fields and electrical potentials that were evoked by the same stimulus. Both types of measurements were used to localize a dipole within a standard three-sphere model of the head. The locations coincided when using the ratio 1 : 1/80 : 1 for the equivalent conductivity of the brain, the skull, and the scalp compartment. Goncalves et at. (2001) repeated these measurements in four subjects. They reported values for the conductivity ratio between scalp and skull that varied between 43 and 85. Part of the observed variability may be ascribed to errors in the volume conductor model, numerical errors, or errors in the measurement. The effect of these errors on the potential distribution will differ from that on the magnetic field distribution. Part will be due to the differences between the heads of the four individuals. In each head the distribution of inhomogeneities within compartments will be different.

9.4.3 ESTIMATION OF THE EQUIVALENT CONDUCTIVITY USING IMPEDANCE TOMOGRAPHY When a current is applied to the scalp surface, a potential distribution develops across the head. The relation between the applied currents and the resulting potential depends on the conductivity distribution within the head. The distribution of the internal conductivity can be estimated from measurements of this scalp potential. This type of research is known as electrical impedance tomography. Goncalves et al. (2001) used electrical impedance tomography to estimate the ratio between the effective conductivity of the skull and the brain compartment for five subjects under the condition that the brain has the same effective conductivity as the scalp. They used both a spherical and a realistically-shaped three-compartment model. For the spherical model they found for the ratio of the brain and scalp conductivities values that varied between 31 and 124. They ascribed the observed variability to geometrical errors. Differences between the actual head geometry and the threecompartment model are compensated by adjusting the values of the electrical conductivity of the compartments. These errors will be much smaller in case realistically-shaped models are used. And indeed for their realistically-shaped models the spread is decreased. In the latter case the ratio of the effective conductivity of brain and skull varied between 17 and 65. They suggest that electrical impedance tomography should be a part of the EEG inverse problem in order to take the individual differences in effective conductivities into account. Oostendorp et al. (2000) used this method to estimate the effective conductivity of the skull and scalp compartment for two subjects. Fitting their potential measurements to the potentials computed by means of the boundary element method yielded a skull conductivity of O.013S/m and a brain conductivity of 0.20S/m. The value found for the ratio of brain-to-skull conductivity was 15 : 1.

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The Electrical Conductivity of Living Tissue

Impedance tomography can also be used to estimate the effective conductivities used in a compartment model of the torso. Eynboglu et al. (1994) used this method to estimate the effective conductivities of the heart, lungs and body in a dog. However, the accuracy of the estimated effective conductivities was only about 40 percent, because the method is not sensitive for changes in conductivities in the various compartments.

9.5 UPPER AND LOWER BOUNDS In most cases, it is very difficult to actually calculate the effective conductivity of a composite material because oflack of detailed information on the micro-geometric structure. Based on information available, lower and upper bounds for the effective conductivity can be given. The range between these bounds decreases with increasing knowledge about the constituents of the composite medium. The conductivity of a composite medium cannot be higher than that of the best conducting phase and cannot not be lower than the worst conducting phase. Thus, for a material consisting of n phases, denoted by the index i = 1,2,3, ... , n, with (Jj increasing with i, the limits are (9.29) Stricter bounds are found when apart from the conductivities also the volume fractions Pi of all phases are known. These bounds read (Hashin and Shtrikman, 1962) (9.30) The upper bound is, for instance, attained for a two-phase composite medium consisting of a suspension of needle-shaped particles with the applied field in the direction of their principal axes. The lower bound is, for instance, attained for a suspension of thin disks that are stacked and the field is applied perpendicular to the disks. In case it is known that the distribution of the constituents is homogeneous and the medium is isotropic, more rigorous limits are applicable (Hashin and Shtrikman, 1962). In this case, the upper and lower bound for a two-phase material is given by: (J\

P2 + --;---=-----

---+~ (J2 -

(J\

3(J]

<

-

(Jeff

<

-

(J2

p] + --;-----

---+~

(J\ -

(J2

(9.31)

3(J2

If the composite constitutes of homogeneously distributed and randomly orientated particles within a conducting medium, and the particles are non-conducting spheroids (L, = Lc ) , the upper and lower bounds can be obtained from Archie's law. The depolarization factors are restrained to La + L, + L, = 1 and La > 0, L, > 0, L, > O. According to expression (9.19), the cementation factor m in Archie's law reads in this case:

(9.32)

M. J. Peters, J. G. Stinstra, and I. Leveles

314

The maximum effective conductivity is found when

-a ~ff = O"solv(l a~

yielding an upper bound for La = L,

m am p) In(l - p) -

a~

=0

(9.33)

= L, = 1/3

O"crr .:::: O"solv (l - p) 3/2

(9.34)

In other words , the upper bound is attained when the particles are spheres. Apparently, such a suspension has the least impact on the flow of currents through the material. The lower bound O"cff = 0 is attained when the particles are thin disks (La = 1, L, = L, = 0). Often the ratio between the axes of the spheroidal particles is not known, but it is known that a > b = c. In that case the lower bound is found for cylinders, where La = 0, L, = L, = 1/2, leading to O"solv (l - P)5/3 .:::: O"cff

(9.35)

In summary, for elongated, homogeneously distributed and randomly orientated nonconducting spheroids in suspen sion the effective conductivity is limited by the following bounds O"solv(l - P)5/ 3 .:::: O"cff .:::: O"solv (l _ P)3/2

(9.36)

In Fig. 9.18 the various upper and lower bounds are plotted for a two-phase composite med ium consisting of a non-conducting phase embedded in a conducting medium as a function of the volume fraction occupied by the non-conducting phase.

9.5.1 WHITE MATTER White matter appears white becau se it contains fiber groups that possess myelin sheaths that consist of layers of membranes. These membranes, composed largely of a lipoprotein called myelin, have a higher proportion of lipid than other surface membranes. The oligodendrocytes (i.e. star-shaped cells) are arranged in rows parallel to the myelinated nerve fibers, with long processes running in the same direction. The white matter can be modeled as a suspension of elongated particles that may have any direct ion. The extracellular space measured by van Harreveld and Ochs (1956) in the cerebellum of mice varied between 18.1 and 25.5 percent with a mean of 23.6 percent. The conductivity of the interstitial fluid is that of the cerebrospinal fluid, i.e.l .8S/m. Thus according to equation (9.36), the effective conductivity will be between the limits O.lOS/m < O"cff < 0.23S/m.

9.5.2 THE FETUS In order to simulate the fetal electrocardiogram, a single compartment may describ e the fetu s. As no measured values for the conductivity of the human fetus are available, its conductivity is estimated. In order to be able to use the theory presented above, the

315

The Electrical Conductivity of Living Tissue

x 02 1 - - - - - ,- - -

- -..-....,...----r-----,------, 1a

0.8

:f O .7[ ~

"0

0.6

c:

80.5 Ql

>

'U 0.4 ~

Ql

0.3 0.2 0.1 1b,2b,3b,4b

0.2

0.4 0.6 volumefraction particles(p)

0.8

FIGURE 9.18. Upper and lower bounds for a material consisting of two composites where one phase is nonconducting as a function of the volume fraction occupied by the non-conducting phase. The conductivity of the conducting phase is oz. (I a) Upper bound if the only information that is available is the value of (f 2 ; the lower bound (l b) coincides with the x-axis. (2a) upper bound and (2b) lower bound if the only information that is available is the value of (f2 and the volume fraction p. (3a) upper bound, (3b) lower bound if the phases are homogeneously distributed and (f 2 and p are known. (4a) upper bound and (4b) lower bound if the non-conducting phase consists of spheroidal, homogeneously distributed particles and the values of (f2 and p are known. (Sa) upper bound and (5b) lower bound if the non-conducting phase consists of elongated particles that are homogeneously distributed and randomly orientated and the values of oz and p are known.

fetus is assumed to be a homogeneous conductor. It is assumed that the cells in the fetus are homogeneously distributed and randomly orientated and have a shape somewhere between a sphere and a cylinder. Looking at the histology of the fetus most tissues consist of elongated spheroids or spheres. Disc-like cells are less commonly encountered. Based on these assumptions the conductiv ity of the fetus can be estimated to be between the limits oAI - p)5/3 :s afetus :s ae (1 - p)3/2. The volume fraction of the extracellular space at the end of gestation is about 40 percent of the total body volume (Brace, 1998; Costarino and Brans, 1998). The extracellular space include s besides the interstitial fluid the fluids in the body cavities like the cerebrospinal fluid and the blood plasma . The blood plasma is about 18 percent of the total extracellular water content. As the fetus is considered as one single entity, there is no objection in taking all the extracellular fluid in account in estimating the conductivity, as all contribute to the conductivity. In compari son the extracellular fluid fraction in an adult is about 20 percent. Thence, the fetus is at least a factor two more conducting than the maternal abdomen. Assuming the conductivity of the extracellular space in both fetus and adult to be comparable at a value of 2S/m, equation (9.36) predicts a range

316

M.

J. Peters, J. G. Stinstra, and I. Leveles

of 1.9 X (004)5/3 ~ OAIS/m::::: afetus ::::: 1.9 X (0.6)3/2 ~ 0.88S/m. Assuming that the volume fraction will be somewhere between 40 and 60 percent and will approach 40 percent at the end of gestation, a value of O.SS/m is a reasonable choice for the conductivity of the fetus in the third trimester of pregnancy. This value is used for the solution of the inverse problem for fetal ECG and leads to reasonable results (Stinstra, 2001).

9.6 DISCUSSION The accuracy of measurements will be limited because the measurements are very complicated. The accuracy of the computation is limited because the cells vary in shape, they are not homogeneously distributed, blood supply plays a role, etcetera. Since the model used to describe a tissue in this chapter is a simplification the results are only an approximation. However, the results are useful in clarifying the relation between the conductivity and the structure of the tissue. The results can be used to predict the effects of changes due to, for instance, temperature, illnesses or age. Anyhow, it makes no sense to use values of the effective conductivity that suggest an accuracy higher than ten percent by giving the values with too many digits. The effective electrical conductivity is a macroscopic parameter that represents the electrical conductivity of the tissue averaged in space over many cells. Many of the tissues in the body such as lung, liver, fat, and blood have cells structures that macroscopically show no preferred direction. Even the heart, which is muscular, has its muscle strands wound in such a complicated fashion that, overall, no preferred direction can be readily discerned. Baynham and Knisley (1999) measured the effective epicardial resistance of rabbit ventricles and found that in contrast to isolated fibers the ventricular epicardium exhibits an isotropic effective resistance due to transmural rotation of fibers. Only skeletal muscle cells have a definite preferred direction when many cells are averaged (Rush et al., 1984). Most cells have an elongated shape. Thence, the bounds given in section 9.5 can be used to estimate the effective conductivity. These bounds are not so far apart, so they will help to restrict the uncertainties in the effective conductivity to be used in the bioelectrical inverse problem. An exception form long skeletal muscle and heart tissue, as the conductivity parallel to the fibers will take place both in the extracellular and the intracellular space.

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Brace, R.A., 1998, Fluid distribution in the fetus and neonate, in: Fetal and neonata l Physiology. (R. A. Polin, and W.w. Fox, eds.), Saunders Comp., Philadelphia, pp. 1703- 1713. Burger, H. c., and Dongen, R. van, 196 1, Specific electric resistance of body tissues, Phys. Med. Biol., 5: 431-437. Burger, H. C., and Milaan, J. B. van. 1943, Measurement of the specific resistance of the human body to direct current, Act. Med. Scand., 114:585-607. Burik, M. J. van, 1999, Physical aspects of EEG , PhD thesis, University of Twente, the Netherlands. Chapman, R.A., and Frye, C.H., 1978. An analysis of the cable properties of frog ventricular myocardium, J. Physiol., 283:263-283. Clerc, L., 1976, Directional differences of impulse spread in trabecular muscle from mammali an heart, Ibid. 255:335- 346. Cohen, D., and Cuffin, B.N., 1983, Demonstration of useful differences between magnetoencephalogram and electroencephalogram, Electroen. din. Neuro., 56:38-51. Cole, K. S., Li, C., and Bak, A. E. 1969. Electrical analogues for tissues. Exp. Neurol., 24:459-473. Costarino, A.T.. and Brans, Y. w., 1998, Fetal and neonatal body fluid composi tion with reference to growth and development, in: Fetal and neonatal Physiology, (R. A. Polin, and W.w. Fox, eds.), Saunders Comp ., Philadelphia, pp. 1713-1721. De Luca, E, Cametti, C, Zimatore, G., Maraviglia, B., and Pachi, A., 1996, Use of low-frequency electrical impedance measurements to determine phospholipid content in amniotic fluid, Phys. Med. BioI., 41:18631869. Epstein. B. R., and Foster, K. R., 1983, Anisotropy in the dielect ric properties of skeletal muscle, Med. BioI. Eng. Comput., 21:51-55. Eyuboglu , B. M., Pilkington, T. C; and Wolf, P. D., 1994, Estimation of tissue resistiv ities from multiple-electrode measurements, Phys. Med. BioI. 39: 1-17 . Foster, K. R., and Schwan, H. P., 1989, Dielectric properties of issues and biological materials: a critical review, Crit. Rev. Biomed. Eng., 17:25- 104. Foster, K. R., and Schwan, H. P., 1986, Dielectric permittivity and electrical conductivity of biological materials, in: Handbook of Biological Effects of Electromagnetic Fields, (C. Polk, and E. Postow, eds.), CRC Press, Inc., Boca Raton, pp. 27. Fricke, H., 1953, The Maxwel l-Wagner dispersion in a suspension of ellipsoids , J. Phys. Chem., 57 :934-937. Gabriel, S., Lau, R. w. , and Gabriel, C., 1996 a, The dielectric properties of tissue: II. Measurements in the frequency range 10Hz to 20GHz, Phys Med BioI., 41:225 1- 2269. Gabriel, S., Lau, R. w. , and Gabriel, C., 1996b , The dielectric properties of biological tissues: III. Parametric models for the dielectric spectrum of tissues, Phys. Med. Bioi., 41:2271-2293. Geddes, L. A., and Baker, L. E., 1967, The specific resistance of biological material-A compendium of data for the biomedical engineer and physiologist, Med. BioI. Eng., 5:27 1- 293. Geddes, L. A., and Sadler, C, 1973, The specific resistance of blood at body temperature, Med. BioI. Eng., 11:336-339. Gcrsing, E., 1998, Monitoring temperature induced changes in tissue during hyperthermia by impedance methods, Proc. of the X.ICEBI, Universitat Politecnica de Cataluya. Gielcn, E, 1983, Electrical conductivity and histological structure of skeletal muscle. PhD Thesis, University of Twente, the Netherlands. Gielen, E L. H., Wallinga-de Jonge, w., and Boon, K. L. , 1984, Electrical conductivity of skeletal tissue: experimental results from different muscles in vivo, Med. BioI. Eng. Comput ., 22:569-577 . Goncalves, S., Munck, J. C. de, Heethaar, R. M., Lopes da Silva E H., and Dij k, B. W. van, 2000, The application of electrical impedance tomography to reduce systematic errors in the EEG inverse problem-a simulation study, Physiol. Meas., 21:379-393. Grandqvist, C. G., and Hunderi, 0 ., 1978, Conductivity of inhomogeneous materials: effective medium theory with dipole-dipole interaction, Phys. Rev. B, 18:1554-1561. Hanai, T., 1960, Theory of the dielectric dispersion due to the interfacial polarization and its application to emulsions, Kolloid-Z., 171:23- 3 1. Harreveld, A. van, Crowell, J., Malhtotra S.A., 1965, A study of extracellular space in central nervous tissue by freeze-substitution, J. Cell Bioi., 25:117-1 37. Harreveld, A. van, and Ochs, S., 1956, Cerebral impedance charges after circulatory arrest, Am. J. Physiol., 187:203-207.

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INDEX

Action potential 2, 89, 120 Activation time imaging 167 Adaptive spatial filter 226 Anisotropic bidoma in 50 Archie's law 300 Beamformer 226, 230, 231 Bidomain model 124 Bidomain myocardi um 46 Biot-Savart law 216 Block-design fMRI 254 Body surface isopotential map 107 Body surface Laplacian 144, 183 Body surface Laplacian mapping 192 Body surface potential map 126 BOLD 252 BSLM 192 BSPM 126 Cable theory 17 Cardiac action potential 7, 8, 9 Cardiac arrhythmia 27 Cardiac tissue 302 Cell networks 23 Cells 289 Computer heart model 62 Conductivity tensor 5 1 Co-registration 262, 263 Cortical imaging 257 Current dipole density 46 DAD 14,25 Defibrillation 72 Dipole distrib ution imaging 164 Dipole distribution 163, 164 Dipole localization 256, 267 Dipole source imaging 163, 165 Dipole source 44

Drug integration 35 EAD 14,25 ECG 183 EEG 183,263 Effective conductivity 47, 283 Electrical conductivity 28 1 Electrocardiographic tomographic imaging 16 1, 168,175 Endocardial potential imaging 129, 147 Endocardial potential 127, 128 Epicardial potential imaging 129, 138 Epicardial potential 57 Equivalent conductivity 312 Equivalent current density 49 Equivalent dipole 54 Equivalent moving dipole 163 Event-related fMRI 254 Extracellular electrogra m 20 Extracellular fluid 29 1, 295 Fat 301 Fiber orientation 52, 86 Finite difference laplacia n 186 Finite difference method 58 Finite element method 58 Finite volume method 60 FitzHugh-Nagumo model 100 fMRI 252 Forward problem 43, 53 Functional magnetic resona nce imaging 252 Genetic integration 35 Global Laplacia n estimate 188 Gradiometer 2 15 Gray matter 305 Green's function 54, 125

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