Analysis and Simulations on a Model for the Evolution of Tumors Under the Influence of Nutrient and Drug Application

We investigate the growth of tumors using a nonlinear model of partial differential equations which incorporates mechanical laws for tissue compression combined with rules for nutrients availability and drug application. Rigorous analysis and simulations are presented which show the effect of nutrient and drug applications on the progression of the tumor. We construct a convergent finite difference scheme to approximate solutions of the nonlinear system of partial differential equations. Extensive numerical tests show that solutions exhibit a necrotic core when the nutrient level falls below a critical level in accordance with medical observations. The same numerical experiment is performed in the case of drug application for the purpose of comparison. Depending on the balance between nutrient and drug both shrinkage and growth of tumors can occur. The role of inhomogeneous boundary conditions, vascularization, and anisotropies in the development of tumor shape irregularities are discussed.