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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. [ABSTRACT FROM AUTHOR]

Mechanical models for tumor growth have been used extensively in recent years for the analysis of medical observations and for the prediction of cancer evolution based on image analysis. This work deals with the numerical approximation of a mechanical model for tumor growth and the analysis of its dynamics. The system under investigation is given by a multi-phase flow model: The densities of the different cells are governed by a transport equation for the evolution of tumor cells, whereas the velocity field is given by a Brinkman regularization of the classical Darcy’s law. An efficient finite difference scheme is proposed and shown to converge to a weak solution of the system. Our approach relies on convergence and compactness arguments in the spirit of Lions [P.-L. Lions, Mathematical topics in fluid mechanics. Vol. 2. Vol. 10 of Oxford Lecture Series Math. Appl. The Clarendon Press, Oxford University Press, New York (1998)]. [ABSTRACT FROM AUTHOR]