In this research project we investigate the evolution of tumor growth relying on a non-linear 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 role of nutrient and drug application in the evolution of cancerous cells. We construct a numerical scheme to approximate solutions of the nonlinear system and establish its convergence by employing compactness methods in the spirit of P.L. Lions. 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 talk will present results obtained in collaboration with D. Donatelli and F. Weber.