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

Diserahkan: 27 Januari 2020/Diterima: 9 Jun 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

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*Pengarang untuk surat-menyurat; email: sharidan@utm.my