TUGAS AKHIR M6 MATEMATIKA PEMODELAN MATEMATIKA DAN METODE NUMERIK

TUGAS AKHIR M6 MATEMATIKA PEMODELAN MATEMATIKA DAN METODE NUMERIK Disusun Untuk Memenuhi sebagian Persyaratan Ketercapaian Tugas Akhir M6 PPG dalam Jabatan Angkatan 2 Tahun 2019

Oleh: Muhammad Darmawan Dewanto, S.Pd 19040318010315

PPG DALAM JABATAN ANGKATAN 2 PROGRAM PENDIDIKAN MATEMATIKA FAKULTAS KEGURUAN DAN ILMU PENDIDIKAN UNIVERSITAS NEGERI YOGYAKARTA 2019

1. Carilah sebuah artikel jurnal imternasional (3 tahun terakhir) yang menggunakan pemodelan matematika, Buatlah resume artikel tersebut dengan menyebutkan langkahlangkah pemodelan sesuai yang telah Anda pelajari, 2. Lingkungan sekitar dapat menjadi inspirasi dalam mendesain soal matematika, termasuk lingkungan sekolah, a. Dengan mengacu pada kriteria yang telah dibahas pada modul 6,2, buatlah sebuah soal bertipe pemodelan matematika sederhana untuk pembelajaran matematika di sekolah, b. Dengan mengikuti model siklus pemodelan matematika yang telah dibahas dalam modul, selesaikan soal yang telah didesain pada poin a, c. Masing-masing siswa mungkin akan memberikan jawaban yang bermacammacam dan perlu diprediksi sebelum menggunakan soal tersebut dalam proses pembelajaran, Oleh karena itu, menyelesaikan

soal

tersebut,

berikan beberapa alternatif lain cara gunakan

juga

software

matematis

memungkinkan, 3. Nilai Viskositas air dapat ditentukan dengan menggunakan tabel berikut ini: T(ºC)

(10-3 Ns/m2)

0

1,792

10

1,308

30

0,801

50

0,549

70

0,406

90

0,317

100

0,284

Perkirakan harga viskositas air  pada temperatur 40o menggunakan polinom Newton,

jika

1. Judul artikel jurnal internasional (artikel jurnal terlampir): Mathematical Modeling of the Effects of Tumor Heterogeneity on the Efficiency of Radiation Treatment Schedule Author: Farinaz Forouzannia1 · Heiko Enderling2, Mohammad Kohandel1 Bull Math Biol (2018) 80:283–293 https://doi.org/10.1007/s11538-017-0371-5 Received: 17 November 2016 / Accepted: 22 November 2017 / Published online: 7 December 2017 © Society for Mathematical Biology 2017 Hasil Resume: Penelitian ini berfokus pada radioterapi menggunakan energi dosis tinggi untuk membasmi dan mengendalikan sel kanker tumor. Berbagai jadwal perawatan telah dikembangkan dan dicoba dalam uji klinis, namun masih ada kendala signifikan untuk meningkatkan fraksinasi radioterapi. Metode penelitian ini mengembangkan model matematika dua kompartemen untuk menganalisis efek radiasi terapi pada dua subpopulasi fenotipik yang berbeda dari radioresisten dan sel kanker adiosensitif. Pada kanker payudara, populasi ini telah diidentifikasi oleh, masing-masing, CD44highCD24low (CD +; biomarker positif) dan CD44lowCD24high (CD−; biomarker negatif), yang juga merupakan penanda batang kanker. Lebih lanjut, penelitian ini menggunakan parameter estimasi. Simulasi stokastik dan deterministik telah dibandingkan dengan dua set percobaan data untuk mendapatkan nilai yang sesuai untuk parameter model. Studi eksperimental pada sel pemicu kanker payudara (mis., CSC) dan uji pembentukan mammosphere (MFA) data mengkalibrasi fraksi CSCs biomarker positif. Setelah dilakukan pemodelan, dan menggunakan parameter estimasi, dan diilustrasikan dengan gambar, hasil penelitian menunjukkan bahwa sel-sel tumor resisten dan sensitif dianggap memiliki radiosensitivitas yang berbeda. Dengan demikian, total populasi sel pada waktu t diberikan oleh N (t) = NS (t) + NP (t) (selanjutnya, NS ≡ NS + NRS dan NP ≡ NP + NRP kecuali dinyatakan lain); populasi sel setelah terpapar pengobatan selama periode waktu τ. Kesimpulan pada penelitian ini adalah model matematika dua kompartemen telah dikembangkan untuk menilai efek heterogenitas tumor dan fraksinasi radioterapi pada respon pengobatan. Simulasi model menunjukkan bahwa radioterapi dapat mengubah dan meningkatkan heterogenitas tumor fraksi sel resisten. Studi selanjutnya dapat lebih meningkatkan kompleksitas biologis dengan mempertimbangkan peningkatan pembaruan diri populasi resisten sebagai respons

terhadap radiasi (Gao et al. 2013). Jika dosis radiasi total tidak cukup untuk memberantasnya tumor, pengayaan dalam sel induk kanker dapat menyebabkan kekambuhan dan kekambuhan tumor. Karena itu, jika kontrol total tumor tidak dapat dicapai terapi yang optimal harus seimbang berkurangnya beban tumor dan pencegahan pertumbuhan subpopulasi yang paling resisten. 2. Lingkungan sekitar dapat menjadi inspirasi dalam mendesain soal matematika, termasuk lingkungan sekolah, a. Soal pemodelan matematika sederhana “Sebuah kapal terlihat dari atas mercusuar yang tingginya 40 meter. Ujung depan kapal teramati dengan sudut depresi 45°, sedangkan ujung belakangnya 30°. Jika kapal bergerak dengan arah menuju mercusuar, panjang kapal tersebut adalah... meter.” b. Pada kasus soal pada poin (a), berdasarkan Modul 6 Kegiatan Belajar 2, permasalahan pada soal tersebut dinamakan dengan permasalahan dengan “situasi nyata”. Langkah selanjutnya adalah siswa diharapkan mampu menganalisis

permasalahan

tersebut,

kemudian

siswa

dapat

membuat

“pemodelan nyata” sebagai dasar untuk menyusun dalam bentuk “pemodelan matematis”. Berdasarkan pemodelan yang sudah disusun, kemudian diharapkan siswa dapat menyelesaikan persoalan dengan ”perhitungan matematis”. Berikut adalah solusi penyelesaian dari permasalahan yang disajikan.

Diketahui: ∠𝐸𝐷𝐵 = 45° , ∠𝐸𝐷𝐶 = 30°, dan tinggi mercusuar = 40 m Sehingga: ∠𝐴𝐷𝐵 = 45° ∠𝐴𝐷𝐶 = 60° Diperoleh: 𝐴𝐵 tan 45° = 𝐴𝐷 𝐴𝐵

1= 40 40 = 𝐴𝐵 𝐴𝐶 dan tan 60° = 𝐴𝐷 √3 =

“Perhitungan matematis”

𝐴𝐶 40

40√3 = 𝐴𝐶 Jadi, panjang kapal adalah (BC) = AC-AB = 40√3 - 40 = 40(√3 − 1) meter c. Alternatif yang dimungkinkan siswa dalam menjawab. 𝐴𝐶

tan 60° = 𝐴𝐷 √3 =

𝐴𝐶 40

40√3 = 𝐴𝐶 dan tan 45° = 𝐴𝐵

𝐴𝐵 𝐴𝐷

1= 40 40 = 𝐴𝐵 Jadi, panjang kapal adalah (BC) = AC-AB = 40√3 - 40 = 40(√3 − 1) meter 3. Perkiraan harga viskositas air  pada temperatur 40o menggunakan polinom Newton sebagai berikut. Solusi ST-1 𝑓(𝑥1 , 𝑥0 ) =

𝑓(𝑥1 ) − 𝑓(𝑥0 ) 1,308 − 1,792 = = −0,04840 𝑥1 − 𝑥0 10 − 0

𝑓(𝑥2 , 𝑥1 ) =

𝑓(𝑥2 ) − 𝑓(𝑥1 ) 0,801 − 1,308 = = −0,02535 𝑥2 − 𝑥1 30 − 10

𝑓(𝑥3 , 𝑥2 ) =

𝑓(𝑥3 ) − 𝑓(𝑥2 ) 0,549 − 0,801 = = −0,01260 𝑥3 − 𝑥2 50 − 30

𝑓(𝑥4 , 𝑥3 ) =

𝑓(𝑥4 ) − 𝑓(𝑥3 ) 0,406 − 0,549 = = −0,00715 𝑥4 − 𝑥3 70 − 50

𝑓(𝑥5 , 𝑥4 ) =

𝑓(𝑥5 ) − 𝑓(𝑥4 ) 0,317 − 0,406 = = −0,00445 𝑥5 − 𝑥4 90 − 70

𝑓(𝑥6 , 𝑥5 ) =

𝑓(𝑥6 ) − 𝑓(𝑥5 ) 0,284 − 0,317 = = −0,00330 𝑥6 − 𝑥5 100 − 90

Iterasi

x 0

f(x) 1,792

10

1,308

30

0,801

50

0,549

70

0,406

90

0,317

100

0,284

0 1 2 3 4 5 6

ST-1 0,04840 0,02535 0,01260 0,00715 0,00445 0,00330

Solusi ST-2 𝑓(𝑥2 , 𝑥1 , 𝑥0 ) =

𝑓(𝑥2 , 𝑥1 ) − 𝑓(𝑥1 , 𝑥0 ) −0,02535 − (−0,04840) = = 7,68 𝑥 10−4 𝑥2 − 𝑥0 30 − 0

𝑓(𝑥3 , 𝑥2 , 𝑥1 ) =

𝑓(𝑥3 , 𝑥2 ) − 𝑓(𝑥2 , 𝑥1 ) −0,01260 − (−0,02535) = = 3,19 𝑥 10−4 𝑥3 − 𝑥1 50 − 10

𝑓(𝑥4 , 𝑥3 , 𝑥2 ) =

𝑓(𝑥4 , 𝑥3 ) − 𝑓(𝑥3 , 𝑥2 ) −0,00715 − (−0,01260) = = 1,36 𝑥 10−4 𝑥4 − 𝑥2 70 − 30

𝑓(𝑥5 , 𝑥4 , 𝑥3 ) =

𝑓(𝑥5 , 𝑥4 ) − 𝑓(𝑥4 , 𝑥3 ) −0,00445 − (−0,00715) = = 6,75 𝑥 10−5 𝑥5 − 𝑥3 90 − 50

𝑓(𝑥6 , 𝑥5 , 𝑥4 ) =

𝑓(𝑥6 , 𝑥5 ) − 𝑓(𝑥5 , 𝑥4 ) −0,00330 − (−0,00445) = = 3,83 𝑥 10−5 𝑥6 − 𝑥4 100 − 70

Iterasi

X 0

f(x) 1,792

10

1,308

30

0,801

50

0,549

70

0,406

90

0,317

100

0,284

0 1 2 3 4 5 6

ST-1 0,04840 0,02535 0,01260 0,00715 0,00445 0,00330

ST-2 7,68E04 3,19E04 1,36E04 6,75E05 3,83E05

Solusi ST-3 𝑓(𝑥3, 𝑥2 , 𝑥1 , 𝑥0 ) =

𝑓(𝑥3 , 𝑥2 , 𝑥1 ) − 𝑓(𝑥2 , 𝑥1 , 𝑥0 ) 3,19 𝑥 10−4 − (7,68 𝑥 10−4 ) = 𝑥3 − 𝑥0 50 − 0

= −8,99 𝑥 10−6 𝑓(𝑥4 , 𝑥3 , 𝑥2 ) − 𝑓(𝑥3 , 𝑥2 , 𝑥1 ) 1,36 𝑥 10−4 − 3,19 𝑥 10−4 𝑓(𝑥4, 𝑥3 , 𝑥2 , 𝑥1 ) = = 𝑥4 − 𝑥1 70 − 10 = −3,04 𝑥 10−6 𝑓(𝑥5 , 𝑥4 , 𝑥3 ) − 𝑓(𝑥4 , 𝑥3 , 𝑥2 ) 6,75 𝑥 10−5 − 1,36 𝑥 10−4 𝑓(𝑥5, 𝑥4 , 𝑥3 , 𝑥2 ) = = 𝑥5 − 𝑥2 90 − 30 = −1,15 𝑥 10−6 𝑓(𝑥6 , 𝑥5 , 𝑥4 ) − 𝑓(𝑥5 , 𝑥4 , 𝑥3 ) 3,83 𝑥 10−5 − 6,75 𝑥 10−5 𝑓(𝑥6, 𝑥5 , 𝑥4 , 𝑥3 ) = = 𝑥6 − 𝑥3 100 − 50 = −5,83 𝑥 10−7 Iterasi

X 0

f(x) 1,792

10

1,308

30

0,801

50

0,549

70

0,406

90

0,317

100

0,284

0 1 2 3 4 5 6

ST-1 0,04840 0,02535 0,01260 0,00715 0,00445 0,00330

ST-2 7,68E04 3,19E04 1,36E04 6,75E05 3,83E05

ST-3 -8,99E06 -3,04E06 -1,15E06 -5,83E07

Solusi ST-4 𝑓(𝑥4, 𝑥3, 𝑥2 , 𝑥1 , 𝑥0 ) =

𝑓(𝑥4 , 𝑥3 , 𝑥2 , 𝑥1 ) − 𝑓(𝑥3, 𝑥2 , 𝑥1 , 𝑥0 ) 𝑥4 − 𝑥0

−3,04 𝑥 10−6 − (−8,99 𝑥 10−6 ) = = 8,50 𝑥 10−8 70 − 0 𝑓(𝑥5, 𝑥4, 𝑥3, 𝑥2 , 𝑥1 ) =

𝑓(𝑥5 , 𝑥4 , 𝑥3 , 𝑥2 ) − 𝑓(𝑥4, 𝑥3, 𝑥2 , 𝑥1 ) 𝑥5 − 𝑥1

−1,15 𝑥 10−6 − (−3,04 𝑥 10−6 ) = = 2,37 𝑥 10−8 90 − 10 𝑓(𝑥6, 𝑥5, 𝑥4, 𝑥3, 𝑥2 ) = =

𝑓(𝑥6 , 𝑥5 , 𝑥4 , 𝑥3 ) − 𝑓(𝑥5, 𝑥4, 𝑥3, 𝑥2 ) 𝑥6 − 𝑥2

−5,83 𝑥 10−7 − (−1,15 𝑥 10−6 ) = 8,04 𝑥 10−9 100 − 30

Iterasi

x 0

f(x) 1,792

10

1,308

30

0,801

50

0,549

70

0,406

90

0,317

100

0,284

0 1 2 3 4 5 6

ST-1 0,04840 0,02535 0,01260 0,00715 0,00445 0,00330

ST-2 7,68E04 3,19E04 1,36E04 6,75E05 3,83E05

ST-3 -8,99E06 -3,04E06 -1,15E06 -5,83E07

ST-4 8,50E08 2,37E08 8,04E09

Solusi ST-5 𝑓(𝑥5, 𝑥4, 𝑥3, 𝑥2 , 𝑥1 , 𝑥0 ) = =

2,37 𝑥 10−8 − 8,50 𝑥 10−8 = −6,81 𝑥 10−10 90 − 0

𝑓(𝑥6, 𝑥5, 𝑥4, 𝑥3, 𝑥2 , 𝑥1 ) = = Iterasi

𝑓(𝑥5 , 𝑥4 , 𝑥3 , 𝑥2 , 𝑥1 ) − 𝑓(𝑥4, 𝑥3, 𝑥2 , 𝑥1 , 𝑥0 ) 𝑥5 − 𝑥0

𝑓(𝑥6 , 𝑥5 , 𝑥4 , 𝑥3 , 𝑥2 ) − 𝑓(𝑥5 , 𝑥4, 𝑥3, 𝑥2 , 𝑥1 ) 𝑥6 − 𝑥1

8,04 𝑥 10−9 − 2,37 𝑥 10−8 = −1,74 𝑥 10−10 100 − 10

x 0

f(x) 1,792

10

1,308

30

0,801

50

0,549

70

0,406

90

0,317

100

0,284

0 1 2 3 4 5 6

ST-1 0,04840 0,02535 0,01260 0,00715 0,00445 0,00330

ST-2 7,68E04 3,19E04 1,36E04 6,75E05 3,83E05

ST-3 -8,99E06 -3,04E06 -1,15E06 -5,83E07

ST-4 8,50E08 2,37E08 8,04E09

Solusi ST-6 𝑓(𝑥6, 𝑥5, 𝑥4, 𝑥3, 𝑥2 , 𝑥1 , 𝑥0 ) = =

𝑓(𝑥6, 𝑥5 , 𝑥4 , 𝑥3 , 𝑥2 , 𝑥1 ) − 𝑓(𝑥5, 𝑥4, 𝑥3, 𝑥2 , 𝑥1 , 𝑥0 ) 𝑥6 − 𝑥0

−1,74 𝑥 10−10 − 6,81 𝑥 10−10 = 5,07 𝑥 10−12 100 − 0

ST-5 -6,81E10 -1,74E10

Itera si

x

f(x)

ST-1

0

1,79 2

0,0484 0 0,0253 5 0,0126 0 0,0071 5 0,0044 5 0,0033 0

0 10

1,30 8

1 30

0,80 1

2 50

0,54 9

3 70

0,40 6

4 90

0,31 7

10 0

0,28 4

5 6

ST-2 7,68E04 3,19E04 1,36E04 6,75E05

ST-3 8,99E06 3,04E06 1,15E06 5,83E07

ST-4 8,50E08 2,37E08

ST-5 6,81E10 1,74E10

ST-6 5,07E12

8,04E09

3,83E05

Dengan demikian persamaan polinomnya adalah: 𝑝6 (𝑥) = 𝑎0 + 𝑎1 (𝑥 − 𝑥0 )+𝑎2 (𝑥 − 𝑥0 )(𝑥 − 𝑥1 ) + 𝑎3 (𝑥 − 𝑥0 )(𝑥 − 𝑥1 )(𝑥 − 𝑥2 ) + 𝑎4 (𝑥 − 𝑥0 )(𝑥 − 𝑥1 )(𝑥 − 𝑥2 )(𝑥 − 𝑥3 ) + 𝑎5 (𝑥 − 𝑥0 )(𝑥 − 𝑥1 )(𝑥 − 𝑥2 )(𝑥 − 𝑥3 )(𝑥 − 𝑥4 )(𝑥 − 𝑥4 ) + 𝑎6 (𝑥 − 𝑥0 )(𝑥 − 𝑥1 )(𝑥 − 𝑥2 )(𝑥 − 𝑥3 )(𝑥 − 𝑥4 ) + (𝑥 − 𝑥5 ) Untuk x = 40 maka 𝑝6 (40) = 1,792 − 0,04840(40 − 0) + 7,68 𝑥 10−4 (40 − 0)(40 − 10) − 8,99 𝑥 10−6 (40 − 0)(40 − 10)(4 − 30) + 8,50 𝑥 10−8 (40 − 0)(40 − 10)(40 − 30)(40 − 50) − 6,81 𝑥 10−10 (40 − 0)(40 − 10)(40 − 30)(40 − 50)(40 − 70) + 5,07 𝑥 10−12 (40 − 0)(40 − 10)(40 − 30)(40 − 50)(40 − 70)(40 − 90) 𝑝6 (40) = 1,79200 − 1,93600 + 0,92200 − 0,10790 − 0,01020 − 0,00245 − 0,00091 𝑝6 (40) = 0,657 Maka perkirakan harga viskositas air  pada temperatur 40o adalah 0,657 (10-3 Ns/m2)

Bull Math Biol (2018) 80:283–293 https://doi.org/10.1007/s11538-017-0371-5 ORIGINAL ARTICLE

Mathematical Modeling of the Effects of Tumor Heterogeneity on the Efficiency of Radiation Treatment Schedule Farinaz Forouzannia1 · Heiko Enderling2 · Mohammad Kohandel1

Received: 17 November 2016 / Accepted: 22 November 2017 / Published online: 7 December 2017 © Society for Mathematical Biology 2017

Abstract Radiotherapy uses high doses of energy to eradicate cancer cells and control tumors. Various treatment schedules have been developed and tried in clinical trials, yet significant obstacles remain to improving the radiotherapy fractionation. Genetic and non-genetic cellular diversity within tumors can lead to different radiosensitivity among cancer cells that can affect radiation treatment outcome. We propose a minimal mathematical model to study the effect of tumor heterogeneity and repair in different radiation treatment schedules. We perform stochastic and deterministic simulations to estimate model parameters using available experimental data. Our results suggest that gross tumor volume reduction is insufficient to control the disease if a fraction of radioresistant cells survives therapy. If cure cannot be achieved, protocols should balance volume reduction with minimal selection for radioresistant cells. We show that the most efficient treatment schedule is dependent on biology and model parameter values and, therefore, emphasize the need for careful tumor-specific model calibration before clinically actionable conclusions can be drawn.

Electronic supplementary material The online version of this article (https://doi.org/10.1007/s11538017-0371-5) contains supplementary material, which is available to authorized users.

B B

Farinaz Forouzannia [email protected] Mohammad Kohandel [email protected] Heiko Enderling [email protected]

1

Department of Applied Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada

2

Department of Integrated Mathematical Oncology, Department of Radiation Oncology, H. Lee Moffitt Cancer Center and Research Institute, Tampa, FL 33647, USA

123

284

F. Forouzannia et al.

Keywords Cancer stem cell · Fractionation · Tumor control

1 Introduction Radiation is commonly used in cancer treatment, either as monotherapy or as combination treatment with surgery and chemotherapy. Radiation is a DNA damaging agent, and radiation as cancer therapy benefits from cancer cells being less efficient in repairing radiation-induced damage than normal cells. Total radiation dose is divided into small frequent fractions to provide temporal windows for normal tissue recovery. Treatment schedules (fractionation) are predominantly based on evolved empirical knowledge and wisdom, but greatly constrained by logistical considerations. Recent developments include hypo- and hyperfractionation for various cancer types, that is delivery of either larger doses temporally further separated or smaller doses more frequently. Despite many technical improvements in the efficiency of radiotherapy, many tumors are refractory to irradiation. Various clinical and biological factors explain such complications, including DNA damage repair (Hall and Giaccia 2006; Mathews et al. 2013), prevalence of hypoxia, and tumor heterogeneity and plasticity. Recently, the presence of cancer stem cells and a tumor hierarchy has been discussed as source of intratumoral heterogeneity and therapy response (Marjanovic et al. 2013; Shackleton et al. 2009). The cancer stem cell hypothesis proposes that the small subpopulation of so-called cancer stem cells (CSCs) is critically important for the initiation and maintenance of the tumor. These CSCs are able to self-renew indefinitely and undergo symmetric and asymmetric divisions to retrospectively increase the CSC population and produce progenitor cells that will make up the bulk of the tumor (Reya et al. 2001). Recent evidence suggests plasticity between non-CSC and CSC states (Gupta et al. 2011), due to genetic or microenvironmental perturbations. CSCs have been shown to utilize superior radiation-induced DNA damage repair mechanisms to prevent cell death (Bao et al. 2006). After radiation exposure, cells with damaged DNA attempt different pathways of repair, and the repair time is likely dependent on the delivered radiation dose (Lagadec et al. 2010; Sarcar et al. 2011). The conventional radiotherapy protocol for most tumors delivers a total dose of 50– 70 Gy in 2 Gy fractions on each weekday, with no treatment given on weekends. To reduce toxicity and increase efficiency, alternative treatments have been considered, including a hyperfractionated protocol with 1 Gy per fraction twice a day; an accelerated regimen of 1.2 Gy per fraction twice daily; and hypofractionation with 5 Gy twice-a-week fractions. Here, we develop a minimal mathematical model to study the effect of tumor heterogeneity and repair in tumors exposed to theses different radiation treatment schedules. Several mathematical models have been developed to simulate the effects of radiotherapy. Most models utilize the so-called linear quadratic (LQ) model and its various extensions (Hall and Giaccia 2006). In the original LQ model, cell survival probability S after acute doses of radiation d can be estimated with S(d) = exp(−αd − βd 2 ),

123

(1)

Mathematical Modeling of the Effects of Tumor Heterogeneity…

285

where α (Gy−1 ) and β (Gy−2 ) are tissue-specific radiosensitivity parameters that are usually derived from fitting the LQ model to clonogenic survival data (Hall and Giaccia 2006). More recently, mathematical frameworks have been combined with experimental data to investigate the different responses to clinically available radiation protocols (Dhawan et al. 2014; Dionysiou et al. 2004; Enderling et al. 2009; Stamatakos et al. 2014). Recently, Leder et al. (2014) proposed an optimized radiation dosing schedule for PDGF-driven glioblastoma. The model, however, is dependent on a large number of parameters and, with limited biological data, some parameters are far from biological realism. In particular, tissue-specific radiosenstivity parameters α and β are derived such that the derived ratio of α/β = 865, 789 Gy is five orders of magnitude larger than frequently derived α/β = 3 − 10 Gy (Leder et al. 2014). Nevertheless, the model-predicted optimal dose fractionation showed prolonged survival in subsequent mouse experiments, emphasizing that the currently applied standard-of-care radiation fractionation may not yield optimal outcomes. Mathematical models may help decipher the complex biology underlying cancer cell response to irradiation, with the ultimate aim to improve clinical application of radiotherapy. Herein we propose a simple mathematical model of breast cancer cell dynamics under fractionated radiation exposure. The model includes phenotypic cell heterogeneity and plasticity, as well as radiation-induced cell cycle arrest, which may play a pivotal role in analyzing radiation protocols with multiple doses per day. The effect of different model parameters and repair mechanisms on heterogeneity are studied for different clinically feasible radiotherapy treatments.

2 Method We developed a two-compartment mathematical model to analyze the effect of radiation therapy on the two phenotypically distinct subpopulations of radioresistant and radiosensitive cancer cells. In breast cancer, these populations have been identified by, respectively, CD44highCD24low (C D + ; biomarker positive) and CD44lowCD24high (C D − ; biomarker negative), which are also markers of cancer stemness (Al-Hajj et al. 2003; Fillmore and Kuperwasser 2008). Both subpopulations are capable of selfrenewal, albeit with lower rates for biomarker negative CD− cells that also feature higher death rates. We discuss the balance of self-renewal and cell death as the net population growth rate, which does not affect the behavior of the system. As a visualization of phenotypic plasticity, cells can switch from one phenotype to the other (Marjanovic et al. 2013). After exposure to radiation, cells in each compartment are forced into cell cycle arrest to attempt repair from radiation-induced DNA damage. Biomarker positive cells have been shown to have better repair mechanisms (Bao et al. 2006; Baumann et al. 2008) and, thus, a larger fraction of growth-arrested biomarker positive CD+ (calculated by the LQ model with phenotype-specific α S and β S parameters) returns into the viable population after successful repair. Figure 1 shows a schematic diagram of the proposed model, and model parameters are summarized in Table 1. We denote by N S , N R S , N P , and N R P the population of resistant cells, resistant repairing cells, sensitive cells, and sensitive repairing cells, respectively. The model can be mathematically represented by the following system of equations

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286


Positive biomarker cells CD44high/CD24low (CD+)

ρS

Repairing cells (CD+)

δS f Dead cells (CD+)

g(d)(1− e(−αS d−βS d ) ) 2

ρSP

ρ PS

g(d)e(−αS d−βS d

2

)

δP f

ρP

Dead cells (CD-)

g(d)e(−α P d−βP d

Negative biomarker cells CD44low/CD24high (CD-)

2

)

g(d)(1− e(−αP d−βP d ) ) 2

Repairing cells (CD-)

Fig. 1 (Color figure online) Schematic diagram of the model

Table 1 Description of the model parameters Parameters

Description

ρS

Net proliferation rate of C D + cells

ρS P

Rate of switching of C D + cells to C D − cells

ρP

Net proliferation rate of C D − cells

ρP S

Rate of switching of C D − cells to C D + cells

δS f

Rate at which C D + cells go for repair

δP f

Rate at which C D − cells go for repair

2 g(d)(1 − e(−α S d−β S d ) )

Rate at which C D + cells that are in repair die

2 g(d)(1 − e(−α P d−β P d ) )

Rate at which C D − cells that are in repair die

2 g(d)e(−α S d−β S d )

Rate at which C D + cells that are in repair become active again

2 g(d)e(−α P d−β P d )

Rate at which C D − cells that are in repair become active again

dN S dt dNRS dt dN P dt dNRP dt

123

= ρ S N S + ρ P S N P + g(d)e(−α S d−β S d ) NRS − ρSP N S − δ S f N S , 2

= δ S f N S − g(d)NRS , = ρ P N P + ρSP N S + g(d)e = δ P f N P − g(d)NRP .

(−α P d−β P d 2 )

(2) NRP − ρPS N P − δ P f N P ,


287

Table 2 Fractionated irradition effect on CSC population and mammosphere formation capacity (Lagadec et al. 2010) C D24low/−/ C D44high Dose

% of CSCs

Sphere forming capacity

2

6.54 (± 1.95)

13.49 (± 1.32)

2–2

8.04 (± 1.47)

10.76 (± 0.96)

2–2–2–2

8.56 (± 1.21)

11.85 (± 1.81)

Overall average

7.71

12.03

Cells acquire on average one DNA double strand break after exposure to 1 Gy of radiation. Therefore, we assume that each cell will enter cell cycle arrest and attempt repair after each radiation fraction, but no new damage arises in the interval between radiation treatments. Hence, f = 1 at discrete times when radiation is given, and f = 0 otherwise. Dependent on radiosensitivity parameters αi and βi with i ∈ {S, P}, cells will either die due to radiation-induced DNA damage with probability 1 − Si (d) or return to the viable non-repairing population S or P with probability Si (d) at dosedependent rate g(d). We assume the function g(d) to be of the order of inverse square of the dose (Lagadec et al. 2010; Sarcar et al. 2011), such that cells irradiated with a dose of 1 Gy spend on average 1 h attempting repair, and 4 h after exposure to 2 Gy.

2.1 Parameter estimation Stochastic and deterministic simulations have been compared to two sets of experimental data to derive suitable values for model parameters. The experimental study on breast cancer initiating cells (i.e., CSCs) and mammosphere formation assay (MFA) data calibrates the fraction of biomarker positive CSCs (Lagadec et al. 2010). In this study, the breast cancer cell line is irradiated with a single dose or daily doses of 2 Gy. After 48 hours, single cells are seeded to form spheres for 20 days. The fraction of CSCs and mammosphere formation capacity are reported in Table 2 (Lagadec et al. 2010). As the MFA experiment was initiated from a single cell (Lagadec et al. 2010), stochastic effects are important. We apply the Gillespie algorithm to compare model sphere forming capacity predictions with the experimental data in Table 2. Since running the Gillespie algorithm is computationally expensive, it is only used to fit the parameters of the model when δs = δ p = g = 0. We vary model parameters without repair to obtain the best fit to the experimental data. At the same time, we use the deterministic Eq. 2 to compare the theoretical results of average CSC fraction to experimental data. The estimated model parameters are summarized in Table 3, alongside the fraction of CSCs and sphere forming capacity using those values, which show good agreement with the experimental results in Table 2. Of note is that the reported parameter value combinations are not unique and, thus, we will perform a sensitivity analysis to investigate the impact of each parameter on model outcome.

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288


Table 3 Estimated model parameters when δ S = δ P = g = 0 Parameters

ρS

ρSP

ρP

ρPS

% of CSCs

Sphere forming capacity

Values

0.2

0.7

0.1

0.05

7.6

12.7

The values of fraction of CSCs and sphere forming capacity that are evaluated based on the estimated parameters values are also reported. The unit of all parameters is 1/day 100 Fitted curve Experimental data

Survival fraction

10−1

10−2

10−3

10−4

0

1

2

3

4

5

6

7

8

9

10

Radiation dose (Gy) Fig. 2 (Color figure online) Fitting the modified linear quadratic model to the experimental data of (Phillips et al. 2006). The black points are the extracted experimental data. The solid curve is the model result using the estimated radiosensitivity parameters α S = 0.14 Gy−1 , β S = 0.048 Gy−2 , α P = 0.41 Gy−1 and β P = 0.17 Gy−2

Herein, resistant and sensitive tumor cells are considered to have different radiosensitivities. Thus, the total population of cells at time t is given by N (t) = N S (t)+ N P (t) (hereafter, N S ≡ N S + NRS and N P ≡ N P + NRP unless stated otherwise); the population of cells after exposure to treatment for the period of time τ can be calculated as N S (t) (−α S d−β S d 2 ) N P (t) (−α P d−β P d 2 ) N (t + τ ) = e e + . N (t) N (t) N (t)

(3)

Assuming that the fraction of CSCs is at its steady-state value before the radiation, (t) = 0.076 (and and using the experimental data of (Lagadec et al. 2010), we set NNS(t) N P (t) N (t)

= 0.924). Then, the modified linear quadratic model (Eq. 3) is used to fit model results to the experimental data of (Phillips et al. 2006), which yields α S = 0.14 Gy−1 , β S = 0.048 Gy−2 (α S /β S =2.9 Gy), α P = 0.41 Gy−1 and β P = 0.17 Gy−2 (α S /β S =2.4 Gy) (Fig. 2). Since at each radiation fraction the majority of damaged cells undergo repair mechanisms, we assume that 90% of cells will be arrested (Withers 1992). However, sensitivity analysis shows that reducing this fraction as low as to 40% does not qualitatively change the results (see Figure S1 and Figure S2 in supplementary materials). The list of all model parameters and their estimated values are reported in Table 4.

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289

Table 4 Model parameter values Parameters

Value (unit)

Reference

ρS

0.2 (day−1 )

Using experimental data (Lagadec et al. 2010)

ρS P

0.7 (day−1 )


ρP

0.1 (day−1 )


ρP S

0.05 (day−1 )


δS = δ P αS

310 (day−1 )

Assuming 90% of cancer cells undergo repair

0.14 (Gy−1 )

Using experimental data (Phillips et al. 2006)

αP

0.41 (Gy−1 )

Using experimental data(Phillips et al. 2006)

βS

0.048 (Gy−2 )


βP

0.17 (Gy−2 )


f

1 or 0

1: radiation, 0: no radiation

3 Results We consider different clinical radiotherapy treatment protocols for one week including standard of care (SoC; daily doses of 2 Gy), hyperfractionated (HR; two daily doses of 1 Gy), accelerated hyperfractionation (AC; two daily doses of 1.2 Gy), and hypofractionated (HO; twice-a-week doses of 5 Gy). Additionally, we simulate the recently suggested optimal protocol for PDGF-driven glioblastoma by Leder (Leder et al. 2014) (Optimum-1, OP; see Table 5). All protocols deliver a total dose of D = 10 Gy per week, except accelerated hyperfractionated with a total dose of D = 12 Gy. However, the accelerated hyperfractionated protocol has the same biologically effective dose (BED) as SoC (see Table 6). BED is used to describe the biological effect of dose fractionation and is defined as d − ln(S F(d)) = D 1+ , (4) BED = α α/β where S F(d) is the LQ model derived single dose d-dependent survival fraction with radiobiological parameters α and β (compare Eq. 1). Due to the linear quadratic relationship of dose and survival, total dose can be increased when smaller doses are given in each fraction (Fowler 1989). The model introduced in Sect. 2 considered two subgroups of cancer cells (resistant cells and sensitive cells) with different radiosensitivities. Thus, following the survival fraction of cancer cells in Eq. 3, the BED is given by BED =

− ln(m S FS + (1 − m)S FP ) , mα S + (1 − m)α P

(5)

where S FS and S FP are survival fractions for resistant cells and sensitive cells, respectively. The constant m represents the proportion of resistant cells in the tumor prior to irradiation. Table 6 shows the BED for the standard of care (SoC), hyperfrac-

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290


Table 5 Radiotherapy schedules for one week of treatment. Different colors are used for corresponding colors in the figures Schedule

Day 1

Day 2

Day 3

Day 4

Day 5

Standard of care (SoC)

2

2

2

2

2

Hyperfractionated (HR)

2×1

2×1

2×1

2×1

2×1

Optimum-1 (OP)

3×1

1

2×1

1

3×1

Hypofractionated (HO)

5

–

–

–

5

Accelerated hyperfractionated (AC)

2 × 1.2

2 × 1.2

2 × 1.2

2 × 1.2

2 × 1.2

Table 6 Biological effective dose for hyperfractionation, standard of care, and accelerated hyperfractionated protocols BED

Schedules HR (d = 1, n = 10)

SoC (d = 2, n = 5)

AC (d = 1.2 n = 10)

BED S (m = 1)

13.5

17.1

17.1

BED P (m = 0)

14.1

18.3

17.9

BED S P (m = 0.076)

11.1

12.7

12.7

tionated (HR), and accelerated hyperfractionated (AC) protocols with different initial fractions of resistant cells: tumors containing only resistant cells (m = 1), tumors containing only sensitive cells (m = 0), and heterogeneous tumors with a small subpopulation of resistant cells (m = 0.076) as estimated. The BED is almost identical for standard-of-care and accelerated hyperfractionation, but significantly smaller for hyperfractionation. Figure 3 shows the number of cancer cells N S + N P and fraction of resistant cells N S /(N S + N P ) for all considered radiation schedules (compare Table 5). Simulations show that protocols with larger number of fractions leads to more cell kill, with accelerated hyperfractionation yielding the smallest number of cells after one week of therapy. However, the fraction of stem cells is largest compared to the other radiotherapy protocols. Hypofractionation with smallest overall cell kill leads also to least competitive release of the most resistant stem cell subpopulation (Enderling et al. 2009). For the chosen parameter combinations (Table 4), accelerated hyperfractionation and SoC schedules yield the lowest number of cancer cells after one week of treatment (Fig. 3). Sensitivity analysis showed that the results are robust to changes in the parameter values (Table 7), with the exception that decreasing α P and β P by 50% produces hypofractionated and accelerated hyperfractionated as best protocols (Fig. 4). Heretofore we assumed that 90% of cells undergo arrest to attempt repair, and the parameter values for δ S and δ P were chosen large enough to satisfy this assumption. However, decreasing the values for these parameters so that less than 40% of cells attempt repair suggests accelerated hyperfractionated as the best schedule (see Figure

123


Total number of cells

Fraction of resistant cells

Approximately 90% of cells go for repair

106

SoC HR OP HO AC

105 104 3

10

102

0

1

2

3

4

5

6

7

0.4

291


0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

1

2

Days

3

4

5

Days

Fig. 3 (Color figure online) Number of cancer cells N S + N P and the fraction of resistant cells N S /(N S + N P ) for radiotherapy protocols reported in Table 5 Table 7 Sensitivity analysis for different parameters of the model αS , βS

αP , βP

ρS

Default

0.14, 0.05

0.41, 0.17

0.2

α S , β S (+ 50%)

0.21, 0.07

0.41, 0.17

0.2

Parameters

ρP

ρPS

ρSP

0.1

0.044

0.73

AC

SoC

0.1

0.044

0.73

AC

SoC

1st best

2st best

α S , β S (− 50%)

0.07, 0.04

0.41, 0.17

0.2

0.1

0.044

0.73

AC

SoC

α P , β P (+ 50%)

0.14, 0.05

0.61, 0.25

0.2

0.1

0.044

0.73

AC

SoC

α P , β P (− 50%)

0.14, 0.05

0.21, 0.08

0.2

0.1

0.044

0.73

HO

AC

ρ S (+ 50%)

0.14, 0.05

0.41, 0.17

0.3

0.1

0.044

0.73

AC

SoC

ρ S (− 50%)

0.14, 0.05

0.41, 0.17

0.1

0.1

0.044

0.73

AC

SoC

0.14, 0.05

0.41, 0.17

0.2 0.15

0.044

0.73

AC

SoC

0.14, 0.05

0.41, 0.17

0.2 0.05

0.044

0.73

AC

SoC

ρ P S (+ 50%)

0.14, 0.05

0.41, 0.17

0.2

0.1

0.066

0.73

AC

SoC

ρ P S (− 50%)

0.14, 0.05

0.41, 0.17

0.2

0.1

0.022

0.73

AC

SoC

ρ S P (+ 50%)

0.14, 0.05

0.41, 0.17

0.2

0.1

0.044

1.09

AC

SoC

ρ S P (− 50%)

0.14, 0.05

0.41, 0.17

0.2

0.1

0.044

0.36

AC

SoC


Total number of cells

106

SoC HR OP HO AC

105

104

3

10

0

1

2

3

4

Days

5

6

7

Fraction of resistant cells

ρ P (+ 50%) ρ P (− 50%)

0.11


0.1 0.09 0.08 0.07 0.06 0

1

2

3

4

5

Days

Fig. 4 (Color figure online) Number of cancer cells N S + N P and the fraction of resistant cells N S /(N S + N P ) when α P and β P are changed to α P − 0.5α P and β P − 0.5β P for radiotherapy protocols reported in Table 7

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292


S1 and Figure S2 in supplementary materials). Furthermore, the model considers that cells leave arrest with rate g(d), which is assumed to be on the order of 1/dose2 . If g(d) was proportional to 1/dose the number of cancer cells at the end of the course of radiation therapy decreases significantly; however, the accelerated hyperfractionated schedule remains the best treatment protocol followed by SoC. Moreover, if g(d) = 1, the order of the best treatment regimens remains (see Figures S3, S4, and S5 in supplementary materials).

4 Conclusion In this paper, a two-compartment mathematical model has been developed to assess the effect of tumor heterogeneity and radiotherapy fractionation on treatment response. Model simulations suggest that radiotherapy can alter tumor heterogeneity and elevates the fraction of resistant cells. Future studies may further increase biological complexity by considering increased self-renewal of the resistant population in response to radiation (Gao et al. 2013). If the total radiation dose is insufficient to eradicate the tumor, enrichment in cancer stem cells may lead to tumor relapse and recurrence. Therefore, if total tumor control cannot be achieved optimal therapies should balance decreases in tumor burden and prevention of outgrowth of the most resistant subpopulation. Interestingly, none of our simulations suggested standard-of-care fractionation as the best therapeutic approach, further emphasizing the need to prospectively evaluate alternative fractionation protocols in the clinic. Furthermore, our model calibrated for breast cancer was unable to confirm the optimum treatment schedule for PDGFdriven glioblastoma (as in Leder et al. 2014). This suggests that treatment optimization may be highly tumor biology, mathematical model, and model parameter dependent, and utmost importance must be paid to identifying underlying biological mechanisms. Hence, a general optimal radiation schedule as suggested in (Conforti et al. 2008; Leder et al. 2014; Wein et al. 2000) may not be feasible and designing an efficient protocol may be required for each type of cancer, and even each individual patient. At the moment, the study is hypothesis generating, and we sincerely hope that the presented results are exciting and encouraging for experimentalists and clinicians to test the presented model predictions. Acknowledgements Financial support by the Natural Sciences and Engineering Research Council of Canada (NSERC) (MK) is gratefully acknowledged.

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TUGAS AKHIR M6 MATEMATIKA PEMODELAN MATEMATIKA DAN METODE NUMERIK

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