A structure-preserving computational method in the simulation of the dynamics of cancer growth with radiotherapy

In this work, we consider a two-dimensional mathematical model that describes the growth dynamics of cancer when radiotherapy is considered. The mathematical model for the local density of the tumor is described by a parabolic partial differential equation with variable diffusion coefficient. The nonlinear reaction term considers both the logistic law of proliferation of tumor cells and the effect of a treatment against cancer. Suitable initial-boundary conditions are imposed on a bounded spatial domain, and a theorem on the existence and the uniqueness of solutions for the initial-boundary-value problem is proved. Motivated by this result, we design a finite-difference methodology to approximate the solutions of our mathematical model. The scheme is a linear method that is capable of preserving the positivity and the boundedness of the approximations. Some simulations are presented in order to illustrate the performance of the method. Among other conclusions, the numerical results show that the method is able to preserve the analytical features of the relevant solutions of the model.