Simulation of a mathematical model of tumoral growth using finite differences

Abstract

The work presents a study of the non-linear mathematical model of tumor growth, proposed by Kolev and Zubik-Kowal (2011). The model is described by a system composed of four partial differential equations that represent the evolution of the density of cancer cells, density of the extracellular matrix (ECM), concentration of matrix-degrading enzyme (MDE) and concentration of tissue metalloproteinase inhibitors. For numerical simulations, the finite difference method is used, in which the temporal terms of the equations are discretized using a two-stage method. In spatial terms, finite central differences are used. A study of numerical convergence for the proposed scheme is presented, using analytical solutions manufactured in a rectangular geometry. Finally, simulations of the tumor growth model are performed, using a non-regular mesh that represents the geometry of a female breast. To simulate the model in non-regular geometry, the technique used is to approximate the contour of the physical domain by mesh segments. The simulations showed that the model has important characteristics of the interactions between tumor cells and the surrounding tissue.