A free boundary problem for a coupled system of elliptic, hyperbolic, and Stokes equations modeling tumor growth

Abstract

We consider a tumor model with three populations of cells: proliferating, quiescent, and necrotic. Cells may change from one type to another at a rate which depends on the nutrient concentration. We assume that the tumor tissue is a fluid subject to the Stokes equation with sources determined by the proliferation rate of the proliferating cells. The boundary of the tumor is a free boundary held together by cell-to-cell adhesiveness of intensity $\gamma$. Thus, on the free boundary the stress tensor $T$ and the mean curvature $\kappa$ are related by $T\vec{n}=-\gamma \kappa \vec{n}$ where $\vec{n}$ is the outward normal. We prove that the coupled system of PDEs for the densities of the three types of cells, the nutrient concentration, and the fluid velocity and pressure have a unique smooth solution, with a smooth free boundary, for a small time interval.