Numerical simulations and analysis for mathematical model of avascular tumor growth using Gompertz growth rate function

Abstract

Tumor growth models have proved as an important tool to produce an engineering background for cancer therapy either by designing therapeutic procedures combined with control engineering or by using the models for simulations and evaluation of treatment procedures. Mathematical modeling has been critical for the description of tumor growth, which is a highly complex process, as a finely chiseled tumor growth model always outlines the measurements and the physiological processes of the tumors. Therefore, a mathematical model involving partial differential equations for the growth of tumor have been studied, modified and developed in this paper. It is based on the deterministic model of an avascular tumor growth which is framed in a system of nonlinear coupled PDEs, describing the proliferating, quiescent, necrotic, and surrounding cells densities, accompanied by a supply of the nutrients. These equations are explained numerically by the Finite Difference Method. Furthermore, the simulations are carried out by introducing the Gompertz growth rate function for mitosis rate function g(c) of proliferating cells in the model. The discretization forms a system of coupled nonlinear difference equations which are solved at each iteration. The algorithm is implemented sequentially in MATLAB 2018a and numerical results suggest that the employed model is an authentic tool for analyzing the dynamics of tumor.