Box-Behnken Design an Alternative for the Optimization

Analytica Chimica Acta 597 (2007) 179–186

Review

Box-Behnken design: An alternative for the optimization of analytical methods S.L.C. Ferreira a,∗ , R.E. Bruns b , H.S. Ferreira a , G.D. Matos a , J.M. David a , G.C. Brandão a , E.G.P. da Silva a , L.A. Portugal a , P.S. dos Reis c,a , A.S. Souza a , W.N.L. dos Santos c a

Universidade Federal da Bahia, Instituto de Qu´ımica, Campus Universitário de Ondina, Salvador, Bahia 40170-290, Brazil b Universidade Estadual de Campinas, Instituto de Qu´ımica, Campinas, S˜ ao Paulo 13084-971, Brazil c Universidade Do Estado da Bahia, Rua Silveira Martins, 2555, Cabula, Salvador-Bahia 41.195.001, Brazil Received 15 May 2007; received in revised form 1 July 2007; accepted 3 July 2007 Available online 23 July 2007

Abstract The present paper describes fundamentals, advantages and limitations of the Box-Behnken design (BBD) for the optimization of analytical methods. It establishes also a comparison between this design and composite central, three-level full factorial and Doehlert designs. A detailed study on factors and responses involved during the optimization of analytical systems is also presented. Functions developed for calculation of multiple responses are discussed, including the desirability function, which was proposed by Derringer and Suich in 1980. Concept and evaluation of robustness of analytical methods are also discussed. Finally, descriptions of applications of this technique for optimization of analytical methods are presented. © 2007 Elsevier B.V. All rights reserved. Keywords: Box-Behnken design; Multivariate optimization; Experimental design; Analytical methods; Desirability function; Robustness

Contents 1.

2.

3.

∗

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Application of multivariate techniques in analytical chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Factors and responses in multivariate optimization techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. The robustness of analytical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4. Box-Behnken design as a tool for multivariate optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Application of Box-Behnken designs (BBD) for optimization of analytical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Application of BBD for the optimization of the spectroanalytical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Application of BBD for the optimization of chromatographic methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Application of BBD for the optimization of capillary electrophoresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Application of BBD for the optimization of electroanalytical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Application of BBD for the optimization of sorption process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. Other applications of BBD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Corresponding author. Fax: +55 71 32355166. E-mail address: slcf@ufba.br (S.L.C. Ferreira).

0003-2670/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.aca.2007.07.011

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S.L.C. Ferreira et al. / Analytica Chimica Acta 597 (2007) 179–186

1. Introduction 1.1. Application of multivariate techniques in analytical chemistry In recent years, chemometric tools have been frequently applied to the optimization of analytical methods, considering their advantages such as a reduction in the number of experiments that need be executed resulting in lower reagent consumption and considerably less laboratory work. Furthermore these methods allow the development of mathematical models that permit assessment of the relevance as well as statistical significance of the factor effects being studied as well as evaluate the interaction effects between the factors. If there significant interaction effects between factors the optimal conditions indicated by the univariate studies will be different from the correct results of the multivariate optimization. The larger the interaction effects the greater the difference that will be found using univariate and multivariate optimization strategies. So the univariate procedure may fail since the effect of one variable can be dependent on the level of the others involved in the optimization. That is why multivariate optimization schemes involve designs for which the levels of all the variables are changed simultaneously. The first step of multivariate optimization is accomplished screening the factors studied (full factorial or fractional factorial design) in order to obtain the significant effects of the analytical system. After determining the significant factors, the optimum operation conditions are attained by using more complex experimental designs such as Doehlert matrix (DM), central composite designs (CCD) and three-level designs such as the Box-Behnken design (BBD) [1–3]. In analytical chemistry, multivariate techniques have been applied to the optimization of chemical factors during the development of analytical strategies involving pre-concentration systems using solid phase extraction [4–10] cloud point extraction [11–13], liquid–liquid extraction [14,15] and coprecipitation [16]; procedures for sample digestion [17–20]; sampling systems [21]; chromatographic methods [22–28]; capillary electrophoresis [29] methods employing flow injection analysis [30,31] and sequential injection analysis [32–35]; electroanalytical methods [36–39] and thermogravimetry [40]. Other applications include the optimization of instrumental parameters of equipment for analysis by graphite furnace atomic absorption spectrometry (GF AAS) [41,42], inductively coupled plasma optical emission spectrometry (ICP OES) [43,44] and inductively coupled plasma mass spectrometry (ICP-MS) [45]. Several review papers have been published on this subject [46,47]. 1.2. Factors and responses in multivariate optimization techniques During the multivariate optimization procedure, there are two types of variables: the responses and the factors. The responses are the dependent variables. Their values depend on the levels of the factors, which can be classified as qualitative or quantitative. In an optimization of a digestion process of lubricating

oil samples using focused-microwave assistance for determination of several metals employing ICP OES, the factors could be: type of acid mixture (qualitative factor), amount of acid mixture, power applied and digestion time (quantitative factors). In the same study, several responses could be evaluated: (a) residual acidity after digestion (this response is important considering the inconvenience of using very acidic solutions for quantification using ICP OES); (b) residual carbon (this parameter reflects directly the efficiency of the mineralization process of the organic matrix); (c) quantification of the metals using ICP OES (this response can be evaluated by recoveries for each metal and shows the efficiency of the digestion process, without considering the residual acidity and residual carbon). If one knows the natures of the relationships between the responses and the factors, i.e. the response surfaces, the optimal values of the factors can be determined. The optimization can be performed in two ways. Response surfaces can be determined for each response and these surfaces can be analyzed simultaneously. Or a model for a single composite function that takes into account all three responses can be determined to obtain a single response surface. The advantages of each approach are still being investigated. Another question that should be addressed is whether the different responses suffer similar effects on changing the factor levels. During the optimization of an analytical procedure involving a multielement technique (ICP OES, ICP-MS and chromatography) generally the composite response will be resultant of several single responses with similar effects. However, the digestion process described for quantification of metals in oil samples clearly exemplifies a situation where the individual responses have different effects. This aspect should be considered during the establishment of an appropriate optimization strategy. An increasingly popular form for treating multiple responses makes use of a desirability function D, which was proposed by Derringer and Suich in 1980 [48]. Individual response surfaces are determined for each response. Predicted values obtained from each response surface are transformed to a dimensionless scale di . The scale of the desirability function ranges between d = 0 (for an unacceptable response value) and d = 1 (for a completely desirable one). D is calculated combining the individual desirability values by applying the geometric mean: D = (d1 × d2 × . . . dm )1/m . An algorithm is then applied to the D function in order to determine the set of variable values that maximize it. This function has been frequently used during the optimization of analytical systems, which involve several responses. Garcia et al. optimized the chromatographic conditions for the determination of eight hormones employing gas chromatography with mass spectrometry detection. A desirability function was proposed for simultaneously optimizing the resolution and the peak width of the separation process [49]. Another paper used a desirability function involving the responses, size and coefficient of variation of the analytical signal, during the optimization of a flow injection system with electrochemical detection for the determination of hydroquinone in cosmetics [50]. Candioti et al. proposed a method for separation and determination of four active ingredients in


pharmaceutical preparations employing capillary electrophoresis. They used a desirability function to simultaneously optimize five responses: the three resolutions, the analysis time and the capillary current [51]. Ortiz et al. reported the use of a desirability function to optimize instrumental responses obtained in analysis involving electroanalytical methods. Two applications were discussed: (1) the simultaneous maximization of the peak current and minimization of its standard deviation for the determination of copper(II) by differential pulse anodic stripping voltammetry; and (2) the simultaneous maximization of the peak current and minimization of the blank signal for the determinations of nickel(II) and indomethacin by adsorptive stripping voltammetry. In all these cases, the experimental conditions for which the optima are found for each individual response are quite different so one is required to look for a compromise solution, that can be achieved using the desirability function [52]. A method was developed for the determination of pesticide multiresidues by matrix solid-phase dispersion and gas chromatography. The authors employed a desirability function to optimize simultaneously pesticide recoveries and matrix cleanup [53]. Concha-Herrera et al. [54] performed a chromatographic separation procedure for the determination of proteic primary amino acids. They used a desirability function involving resolution(s) between peaks (and) analysis time during the optimization step of method. Simultaneous optimization of the resolution and analysis time was achieved using Derringer’s desirability function for a method proposed for the determination of of phenyl thiohydantoin amino acids employing micellar liquid chromatography [55]. Other multi-response functions have been also proposed but these are not based on the Derringer–Suich desirability function [48]. These were established considering mainly the objective of the analytical system being optimized. Then, during the optimization step of an on-line pre-concentration system for the determination of copper by flame atomic absorption spectrometry, Ferreira et al. proposed a multi-response function, involving analytical signal (absorbance) and pre-concentration time. This multi-response function was called “sensitivity efficiency” and it has been defined as the analytical signal obtained for an online enrichment system for a pre-concentration time of 1 min [56]. This multi-response function was also used for the optimization of an on-line pre-concentration system proposed for the determination of lead using FAAS [57] and in another method performed for selective extraction and determination of catechol in water samples, using a polymeric sorbent based on molecular imprinting technology with subsequent determination by differential pulse voltammetry. The sensitivity efficiency was determined considering the electrochemical signal and the pre-concentration time [37]. During the optimization of a preconcentration procedure using cloud point extraction for the determination of six metal ions (cadmium, chromium, copper, manganese, nickel and lead) employing ICP OES, another multiple response function was found for obtaining of a simultaneous pre-concentration condition [58]. The behaviors of five of these ions as a function of varying experimental conditions are highly correlated and can all adequately be described by a first principal component whereas the nickel ion behav-

181

ior is quite different and is described by a second principal component. 1.3. The robustness of analytical methods In validation studies multivariate optimization techniques are also used for determination of robustness, which is defined as the capacity of an analytical method to reproduce results when the procedure is performed under small changes in the nominal values of the experimental factors established in the optimization step [59,60,45,61–63]. 1.4. Box-Behnken design as a tool for multivariate optimization Box-Behnken designs (BBD) [64] are a class of rotatable or nearly rotatable second-order designs based on three-level incomplete factorial designs. For three factors its graphical representation can be seen in two forms: 1a. A cube that consists of the central point and the middle points of the edges, as can be observed in Fig. 1a.

Fig. 1. (a) the cube for BBD and three interlocking 22 factorial design (b).

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Table 1 Coded factor levels for a Box-Behnken design of a three-variable system Experiment

x1

x2

x3

1 2 3 4 5 6 7 8 9 10 11 12 C C C

−1 1 −1 1 −1 1 −1 1 0 0 0 0 0 0 0 0

−1 −1 1 1 0 0 0 0 −1 1 −1 1 0 0 0 0

0 0 0 0 −1 −1 1 1 −1 −1 1 1 0 0 0 0

1b. A figure of three interlocking 22 factorial designs and a central point, as shown in Fig. 1b. The number of experiments (N) required for the development of BBD is defined as N = 2k(k − 1) + C0 , (where k is number of factors and Co is the number of central points). For comparison, the number of experiments for a central composite design is N = 2k + 2k + C0 . Tables 1 and 2 contain the coded values of the factor levels for BBD on three, four and five factors, respectively. A comparison between the BBD and other response surface designs (central composite, Doehlert matrix and three-level full factorial design) has demonstrated that the BBD and Doehlert matrix are slightly more efficient than the central composite design but much more efficient than the three-level full factorial designs where the efficiency of one experimental design

is defined as the number of coefficients in the estimated model divided by the number of experiments. Table 3 establishes a comparison among the efficiencies of the BBD and other response surface designs for the quadratic model. This Table demonstrates also that the three-level full factorial designs are costly when the factor number is higher than 2. Another advantage of the BBD is that it does not contain combinations for which all factors are simultaneously at their highest or lowest levels. So these designs are useful in avoiding experiments performed under extreme conditions, for which unsatisfactory results might occur. Conversely, they are not indicated for situations in which we would like to know the responses at the extremes, that is, at the vertices of the cube. BBD for four and five factors can be arranged in orthogonal blocks, as shown in Table 2. In this table, each (±1, ±1) combination within a row represents a full 22 design. Dashed lines separate the different blocks. Because of block orthogonality, the second-order model can be augmented to include block effects without affecting the parameter estimates, that is, the effects themselves are orthogonal to the block effects. This orthogonal blocking is a desirable property when the experiments have to be arranged in blocks and the block effects are likely to be large. 2. Application of Box-Behnken designs (BBD) for optimization of analytical systems 2.1. Application of BBD for the optimization of the spectroanalytical method Korn and de Oliveira used BBD during the optimization of a sequential injection analysis method proposed for the determination of sulphate in ethanol automotive fuel employing molecular absorption spectrophotometry (MAS) [65]. Araucaria angustifolia (named pinhão) wastes were tested as solid

Table 2 Coded factor levels for Box-Behnken designs for optimizations involving four and five factors


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Table 3 Comparison of efficiency of central composite design (CCD), Box-Behnken design (BBD) and Doehlert design (DM) Factors (k)

2 3 4 5 6 7 8

Number of coefficients (p)

6 10 15 21 28 36 45

Number of experiments (f)

Efficiency (p/f)

CCD

DM

BBD

CCD

DM

BBD

9 15 25 43 77 143 273

7 13 21 31 43 57 73

– 13 25 41 61 85 113

0.67 0.67 0.60 0.49 0.36 0.25 0.16

0.86 0.77 0.71 0.68 0.65 0.63 0.62

– 0.77 0.60 0.61 0.46 0.42 0.40

phase for extraction of chromium VI. BBD was used for optimization of the experimental factors [66]. Chemical factors of a method proposed for the quantification of amikacin in pharmaceutical formulations were optimized also using BBD [67]. Otero-Rey et al. used BBD for optimization of several experimental parameters on arsenic and selenium leaching from coal fly ash samples and their determination using hydride generation coupled with atomic fluorescence spectrometry (HG AFS) [68]. Ferreira et al. proposed two on-line pre-concentration systems for the determination of cadmium and lead in drinking water by FAAS employing knotted reactor and 1-(2-pyridylazo)-2naphthol (PAN) as complexing reagent. In both methods, the optimization step was performed using BBD [69,70]. Zougagh et al. developed an on-line system for the pre-concentration and determination of lead in water using ICP OES. The solid phase extraction process was optimized using BBD [71]. BosqueSendra et al. [72] described the advantages of the BBD as response surface methodology for obtaining second order models in full detail. They used this design for re-optimization of the pararosaniline classical method for the determination of formaldehyde employing MAS. A comparison with the classical method [73], which was optimized using a univariate strategy, revealed that the re-optimized procedure using BBD has a sensitivity almost twice as large as the univariate result. BBD was used also for the optimization of the factors of a direct-current plasma system. The factors involved were horizontal position, vertical position, nebulizer pressure and electrode sleeve pressure. Three responses (precision, drift and sensitivity) were evaluated. All the optimization was carried out by quantification of the copper signal. An evaluation of the performance of the equipment

using the optimized conditions was performed for six other elements [74]. Table 4 summarizes applications of BBD for the optimization of chemical systems involving spectroanalytical techniques. 2.2. Application of BBD for the optimization of chromatographic methods Carasek and coworkers optimized a microextraction process for the determination of 2,4,6-trichloroanisole and 2,4,6-tribromoanisole in wine samples employing BBD [75]. Pyrzynska and coworkers used BBD for optimization of the derivatization reaction established during the development of a method proposed for the determination of aliphatic aldehydes by HPLC [76]. BBD was used for optimization of a procedure using microwave-assisted extraction proposed for the determination of persistent organochlorine pesticides in sediment using GC–MS [77]. McKenon and coworkers used BBD for optimization of method involving supercritical fluid extraction for the determination of fatty acid composition of castor seeds using GC-FID [78]. The separation process performed for the determination of captopril in pharmaceutical tablets using HPLC was optimized also using BBD [79]. Petz and coworkers proposed a CG–MS method for the determination of aminoglycoside antibiotics. BBD was used for the optimization of the derivatization reaction [80]. Walters and Qiu employed BBD for the optimization of the separation process of hydroxamates (arginine, leucine, threonine, histidine and Tryptophan) using paper chromatography [81]. Table 5 presents applications of BBD for optimization of chromatographic methods.

Table 4 Application of BBD for the optimization of spectroanalytical methods Analyte

Sample

Analytical technique

Optimized parameters

References

Sulphate Chromium VI Amikacin Arsenic and selenium Cadmium Lead Lead Formaldehyde

Ethanol automotive fuel Water Pharmaceutical formulations Coal fly ash Drinking water Drinking water Water –

SIA/MAS MAS Chemiluminescence HG AFS KR-FAAS KR-FAAS SPE-ICP OES MAS

Instrumental factors Chemical factors Chemical factors Chemical factors Chemical factors Chemical factors Chemical factors Chemical factors

[65] [66] [67] [68] [69] [70] [71] [72]

Sequential injection analysis (SIA); molecular absorption spectrophotometry (MAS); knotted reactor (KR); hydride generation coupled with atomic fluorescence spectrometry (HG AFS); solid phase extraction (SPE); inductively coupled plasma optical emission spectrometry (ICP OES).

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Table 5 Application of BBD for the optimization of chromatographic methods Analyte

Sample

Chromatographic technique

Optimized process

References

2-4-6 TCA, 2-4-6 TBA Aliphatic aldehydes Organochlorine pesticides Fatty acid composition Captopril Aminoglycoside antibiotics Aminoacids hydroxamates

Wine – Sediments Castor oil Pharmaceutical tablets –

GC-ECD HPLC GC–MS GC-FID HPLC GC–MS PC

Extraction step Derivatization reaction Extraction step Extraction step Separation step Derivatization reaction Separation step

[75] [76] [77] [78] [79] [80] [81]

2,4,6-trichloroanisole (2-4-6TCA); 2,4,6-tribromoanisole (2-4-6TBA); gas chromatography and electron-capture detection (GC-ECD); high performance liquid chromatography (HPLC); gas chromatography–mass spectrometry (GC–MS); gas chromatography with flame ionisation detection GC-FID; paper chromatography (PC).

2.3. Application of BBD for the optimization of capillary electrophoresis

the optimization of the sorption process of verofix red using a biopolymer [92].

Gong and coworkers compared response surfaces based on complementary three-level Box-Behnken, face-centered central composite and full factorial designs during the optimization of procedures employing capillary electrophoresis (CE) for the determination of tamsulosin enantiomers [82] and also ascorbic acid and isoascorbic acid [83]. These authors concluded that the optimizations using the three designs furnished similar and efficient results. A procedure for neuropeptide separation employing CE was optimized using BBD [84]. Ragonese et al. performed a separation procedure for the determination of ethambutol hydrochloride in pharmaceutical formulations using CE. The optimization step was carried out using BBD [85]. Hows et al. proposed a method for separation and determination of sulphonamides, dihydrofolate reductase inhibitors and beta-lactam antibiotics also employing CE. The experimental factors were optimized using BBD [86].

2.6. Other applications of BBD

2.4. Application of BBD for the optimization of electroanalytical methods BBD has not been used for the optimization of electroanalytical methods. Only a procedure employing adsorption stripping voltammetry for the determination of nalidxic acid and 7-hydroximethylanalidxic acid in urine samples was established using this design [87]. 2.5. Application of BBD for the optimization of sorption process

Matthews et al. used BBD for the optimization of an enzymatic procedure for the determination of arsenic in aqueous solutions [93]. Silva and coworkers developed a study in order to detect the most important factors that effect the formation of the four trihalomethanes (THM) (chloroform, bromodichloromethane, chlorodibromomethane and bromoform) in water disinfection processes using chlorine. BBD was used during the optimization step [94]. Petz and Lamar developed a receptor protein microplate assay for the detection and determination of penicillins and cephalosporins with intact beta-lactam in milk, bovine and porcine muscle juice, honey and egg samples. The optimization step was performed using BBD [95]. Wu and coworkers developed a photoelectrocatalytic oxidation system using a Ti/TiO2 electrode for the degradation of fulvic acid (FA). The optimization step was carried out using BBD [96]. BBD was employed for the optimization of an electrochemical process using reticulated vitreous carbon-supported-onpolyaniline cathodes for the reduction of hexavalent chromium of industrial wastewater samples [97]. Rajkumar et al. investigated the electrochemical oxidation process of phenol using a Ti/TiO2 –RuO2 –IrO2 anode. The experimental factors were optimized using BBD [98]. 3. Conclusions

Kannan et al. proposed the use of straw carbon for the adsorption of the copper(II), cadmium(II) and nickel(II) metal ions. The optimization step was carried out using BBD [88]. Madaria et al. applied carbon aerogel for electrolytic removal of mercury from aqueous solutions. Experimental factors were optimized employing BBD [89]. Activated carbon immobilized with Pseudomonas putida was evaluated as solid phase for extraction of phenol. The optimization step was performed using BBD [90]. The optimization of the adsorption process of Rhodamine 6G from water using both chitosan and activated carbon was performed also using BBD [91]. BBD was used for

The Box-Behnken is a good design for response surface methodology because it permits: (i) estimation of the parameters of the quadratic model; (ii) building of sequential designs; (iii) detection of lack of fit of the model; and (iv) use of blocks. A comparison between the Box-Behnken design and other response surface designs (central composite, Doehlert matrix and three-level full factorial design) has demonstrated that the Box-Behnken design and Doehlert matrix are slightly more efficient than the central composite design but much more efficient than the three-level full factorial designs.


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Box-Behnken Design an Alternative for the Optimization

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