A note on the numerical approach for the reaction-diffusion problem to model the density of the tumor growth dynamics

In this article, we numerically solve an equation modeling the evolution of the density of glioma in the brain - the most malignant form of brain tumor quantified in terms of net rates of proliferation and invasion. We employ a non-linear heterogeneous diffusion logistic density model. This model assumes that glioma cell invasion throughout the brain is a reaction-diffusion process and that the coefficient of diffusion can vary according to the gray and white matter composition of the brain at that location. The analysis provided in this article demonstrates that using the correct finite difference scheme can overcome the stability issues caused by the discontinuities of the diffusion coefficient. We also observe that at the steady-state these discontinuities vanish. To visualize and investigate numerically the behavior of the evolution of tumor concentration of the glioma, we calculated and plotted the number of tumor cells, the average mean radial distance, and the speed of the tumor cells along with charting the effects of net dispersal rate and net proliferation rate terms versus time for different center position values of Gaussian initial profile for each zone (gray and white matter tissues). We have proposed two numerical methods, the implicit backward Euler and the averaging in time and forward differences in space (the Crank-Nicolson scheme), both in combination with Newton's method for solving the governing equations. These methods are compared in terms of their performance in varying time-step and mesh-discretization. The Crank-Nicolson implicit method is shown to be the better choice to solve the equation.