Sains Malaysiana 49(11)(2020): 2833-2846

http://dx.doi.org/10.17576/jsm-2020-4911-22

 

Stability Analysis of Radiotherapy Cancer Treatment Model with Fractional Derivative

(Analisis Kestabilan Model Rawatan Radioterapi Kanser dengan Terbitan Pecahan)

 

MUSILIU FOLARIN FARAYOLA, SHARIDAN SHAFIE* & FUAADA MOHD SIAM

 

Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 Johor Bahru, Johor Darul Takzim, Malaysia

 

Received: 27 January 2020/Accepted: 9 June 2020

 

ABSTRACT

This paper presents the condition for uniqueness, the stability analysis, and the bifurcation analysis of a mathematical model that simulates a radiotherapy cancer treatment process. The presented model was the previous cancer treatment model integrated with the Caputo fractional derivative and the Linear-Quadratic with the repopulation model. The metric space analysis was used to establish the conditions for the presence of unique fixed points for the model, which indicated the presence of unique solutions. After establishing uniqueness, the model was used to simulate the fractionated treatment process of six cancer patients treated with radiotherapy. The simulations of the cancer treatment process were done in MATLAB with numerical and radiation parameters. The numerical parameters were obtained from previous literature and the radiation parameters were obtained from reported clinical data. The solutions of the simulations represented the final volumes of tumors and normal cells. Subsequently, the initial values of the model were varied with 200 different values for each patient and the corresponding solutions were recorded. The continuity of the solutions was used to investigate the stability of the solutions with respect to initial values. In addition, the value of the Caputo fractional derivative was chosen as the bifurcation parameter. This parameter was varied with 500 different values to determine the bifurcation values. It was concluded that the solutions are unique and stable, hence the model is well-posed. Therefore, it can be used to simulate a cancer treatment process as well as to predict outcomes of other radiation protocols. 

 

Keywords: Caputo fractional derivative; Linear-Quadratic; radiotherapy

 

ABSTRAK

Kertas ini membentangkan syarat keunikan, analisis kestabilan, dan analisis bifurkasi terhadap model matematik bagi simulasi proses rawatan kanser radioterapi. Model yang digunakan adalah model rawatan kanser terdahulu yang disepadukan dengan terbitan pecahan Caputo dan Kuadratik-Linear dengan model populasi semula. Analisis ruang metrik digunakan untuk menentukan syarat-syarat kehadiran titik tetap unik untuk model, yang menunjukkan kehadiran penyelesaian unik. Setelah keunikan ditentukan, model ini digunakan bagi simulasi proses rawatan berbahagi terhadap enam pesakit kanser yang dirawat dengan radioterapi. Simulasi proses rawatan kanser dilakukan dalam MATLAB dengan parameter berangka dan parameter radiasi. Parameter berangka diperoleh daripada kajian sebelumnya dan parameter radiasi diperoleh daripada data klinikal yang dilaporkan. Penyelesaian simulasi mewakili isi padu tumor terakhir dan sel normal. Selanjutnya, nilai-nilai awal model telah dipelbagaikan dengan 200 nilai yang berbeza untuk setiap pesakit dan penyelesaian yang berpadanan direkodkan. Keselanjaran penyelesaian telah digunakan untuk mengkaji kestabilan penyelesaian terhadap nilai-nilai awal. Selain itu, nilai terbitan pecahan Caputo dipilih sebagai parameter bifurkasi. Parameter ini dipelbagaikan dengan 500 nilai yang berbeza untuk menentukan nilai-nilai bifurkasi. Didapati bahawa, penyelesaian adalah unik dan stabil, maka model adalah teraju rapi. Oleh itu, model boleh digunakan bagi simulasi proses rawatan kanser serta meramalkan hasil keputusan protokol radiasi yang lain.

 

Kata kunci: Kuadratik-Linear; radioterapi; terbitan pecahan Caputo

 

References

Abuasad, S. & Hashim, I. 2018. Homotopy decomposition method for solving higher-order time- fractional diffusion equation via modified beta derivative. Sains Malaysiana 47(11): 2899-2905.

Awadalla, M., Yameni, Y. & Abuassba, K. 2019. A new fractional model for the cancer treatment by radiotherapy using the Hadamard fractional derivative. Online Mathematics 1(2): 14-18.

Barnett, G.C., West, C.M.L., Dunning, A.M., Elliott, R.M., Coles, C.E., Pharoah, P.D.P. & Burnet, N.G. 2009. Normal tissue reactions to radiotherapy: Towards tailoring treatment dose by genotype. Nature Reviews Cancer 9(2): 134-142.

Belfatto, A., Riboldi, M., Ciardo, D., Cattani, F., Cecconi, A., Lazzari, R., Jereczek-Fossa, B.A., Orecchia, R., Baroni, G. & Cerveri, P. 2016. Kinetic models for predicting cervical cancer response to radiation therapy on individual basis using tumor regression measured in vivo with volumetric imaging. Technology in Cancer Research & Treatment 15(1): 146-158.

Belostotski, G. & Freedman, H.I. 2005. A control theory model for cancer treatment by radiotherapy. International Journal of Pure and Applied Mathematics 25: 447-480.

Benzekry, S., Lamont, C., Beheshti, A., Tracz, A., Ebos, J.M.L., Hlatky, L. & Hahnfeldt, P. 2014. Classical mathematical models for description and prediction of experimental tumor growth. PloS Computational Biology 10(8): e1003800.

Bertuzzi, A., Bruni, C., Papa, F. & Sinisgalli, C. 2013. Optimal solution for a cancer radiotherapy problem. Journal of Mathematical Biology 66(1-2): 311-349.

Borisut, P., Khammahawong, K. & Kumam, P. 2018. Fixed point theory approach to existence of solutions with differential equations. Differential Equations 1: 1-34.

Caputo, M. & Fabrizio, M. 2015. A new definition of fractional derivative without singular kernel. Progress in Fractional Differentiation and Applications 1(2): 1-13.

Diethelm, K. 2003. Efficient solution of multi-term fractional differential equations using P(EC)mE methods. Computing 71(4): 305-319.

Diethelm, K. & Freed, A.D. 1998. The FracPECE subroutine for the numerical solution of differential equations of fractional order. Forschung und wissenschaftliches Rechnen 1999: 57-71.

Diethelm, K., Ford, N.J. & Freed, A.D. 2004. Detailed error analysis for a fractional Adams method. Numerical Algorithms 36(1): 31-52.

Dokuyucu, A.M., Celik, E., Bulut, H. & Mehmet Baskonus, H. 2018. Cancer treatment model with the Caputo-Fabrizio fractional derivative. The European Physical Journal Plus 133(3): 92.

Du, M., Wang, Z. & Hu, H. 2013. Measuring memory with the order of fractional derivative. Scientific Reports 3: 3431.

Emami, B. 2013. Tolerance of normal tissue to therapeutic radiation. Reports of Radiotherapy and Oncology 1(1): 35-48.

Farayola, M.F., Shafie, S., Mohd Siam, F. & Khan, I. 2020a. Mathematical modeling of radiotherapy cancer treatment using Caputo fractional derivative. Computer Methods and Programs in Biomedicine 188: 105306.

Farayola, M.F., Shafie, S., Mohd Siam, F. & Khan, I. 2020b. Numerical simulation of normal and cancer cells’ populations with fractional derivative under radiotherapy. Computer Methods and Programs in Biomedicine 187: 105202.

Fowler, J.F. 2006. Development of radiobiology for oncology - A personal view. Physics in Medicine and Biology. 51(13): R263-R286.

Freedman, H.I. & Belostotski, G. 2009. Perturbed models for cancer treatment by radiotherapy. Differential Equations and Dynamical Systems 17(1-2): 115-133.

Garrappa, R. 2010. On linear stability of predictor-corrector algorithms for fractional differential equations. International Journal of Computer Mathematics 87(10): 2281-2290.

Hairer, E., Lubich, Ch. & Schlichte, M. 1985. Fast numerical solution of nonlinear Volterra convolution equations. SIAM Journal on Science and Statistical Computing 6(3): 532-541.

Jones, B. 1999. Mathematical models of tumour and normal tissue response. Acta Oncologica 38(7): 883-893.

Khalid, A.F.B., Tan, J.J. & Yong, Y.K. 2018. Malaysian Tualang honey and its potential anti-cancer properties: A review. Sains Malaysiana 47(11): 2705-2711.

Lee, S.P., Leu, M.Y., Smathers, J.B., McBride, W.H., Parker, R.G. & Withers, H.R. 1995. Biologically effective dose distribution based on the linear quadratic model and its clinical relevance. International Journal of Radiation Oncology, Biology, Physics 33(2): 375-389.

Liu, Z. & Yang, C. 2014. A mathematical model of cancer treatment by radiotherapy. Computational and Mathematical Methods in Medicine 2014: Article ID. 172923.

Lokman, N., Ab. Hamid, S.A. & Bachok, N. 2017. Survival study and prognostic factors of ovarian cancer registered in a teaching hospital in Malaysia. Sains Malaysiana 46(4): 559-565.

Losada, J. & Nieto, J.J. 2015. Properties of a new fractional derivative without singular kernel. Progress in Fractional Differentiation and Applications 1(2): 87-92.

Nawrocki, S. & Zubik-Kowal, B. 2015. Clinical study and numerical simulation of brain cancer dynamics under radiotherapy. Communications in Nonlinear Science and Numerical Simulation 22(1-3): 564-573.

O’Rourke, S.F.C., McAneney, H. & Hillen, T. 2009. Linear quadratic and tumour control probability modelling in external beam radiotherapy. Journal of Mathematical Biology 58(4-5): 799-817.

Rashid, H., Mohd Siam, F. & Maan, N. 2018. Parameter estimation for a model of ionizing radiation effects on targeted cells using genetic algorithm and pattern search method. Matematika 34(3): 1-13.

Wheeler, N. 1997. Construction and physical application of the fractional calculus. Reeds College Physics Seminar. Reeds College Physics Seminar.

 

*Corresponding author; email: sharidan@utm.my

 

 

 

 

previous