On a Nonlinear Model for Tumor Growth in a Cellular Medium

Abstract

We investigate the dynamics of a nonlinear model for tumor growth within a cellular medium. In this setting the “tumor” is viewed as a multiphase flow consisting of cancerous cells in either proliferating phase or quiescent phase and a collection of cells accounting for the “waste” and/or dead cells in the presence of a nutrient. Here, the tumor is thought of as a growing continuum $$\Omega $$ with boundary $$\partial \Omega $$ both of which evolve in time. In particular, the evolution of the boundary $$\partial \Omega $$ is prescibed by a given velocity $${{{\varvec{V}}}.}$$ The key characteristic of the present model is that the total density of cancerous cells is allowed to vary, which is often the case within cellular media. We refer the reader to the articles (Enault in Mathematical study of models of tumor growth, 2010; Li and Lowengrub in J Theor Biol, 343:79–91, 2014) where compressible type tumor growth models are investigated. Global-in-time weak solutions are obtained using an approach based on penalization of the boundary behavior, diffusion, viscosity and pressure in the weak formulation, as well as convergence and compactness arguments in the spirit of Lions (Mathematical topics in fluid dynamics. Compressible models, 1998) [see also Donatelli and Trivisa (J Math Fluid Mech 16: 787–803, 2004), Feireisl (Dynamics of viscous compressible fluids, 2014)].