Numerical simulation of tumor growth and cell migration in 1d and 2d
AbstractNowadays, cancer has become a worldwide public health issue; therefore, it is essential to know the basic biological dynamics that govern the carcinogenesis in order to develop new techniques of treatment and prevention of this disease. Our objective was to build computational models in 1d and 2d that would allow us to study the interactions of a tumor with its surrounding tissue. In this paper, we studied the mathematical model of tumor progression and metastasis analyzed by Kolev and Zubik-Kowal. This mathematical model consists of a system of four equations representing the temporal evolution of cancer cell density, extracellular matrix (ECM) density, concentration of matrix metalloproteinases (MMPs), and concentration of endogenous tissue inhibitors of metalloproteinases (TIMPs). To solve the mathematical model, we implemented a numerical solution in 1d and 2d through the finite element method (MEF); additionally, for the temporal discretization of the model, we used the finite difference method (MDF). Our computational model was encoded in the programming language FreeFem++. The simulations results show the proliferation and subsequent migration of cancer cells, the relation of the tumor with the increase of MMPs, and the relation of MMPs with the ECM and TIMPs. Finally, with our algorithm we get accurate solutions in 1d and 2d, which are highly scalable to solutions in 3d for complex geometries that allow us to extend our computational model to the simulation of specific types of colon, cervix, or breast cancer.