A free boundary problem for a coupled system of elliptic, hyperbolic, and Stokes equations modeling tumor growth
Avner Friedman · Interfaces and Free Boundaries Mathematical Analysis Computation and Applications · 2006
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 γ .Thus, on the free boundary the stress tensor T and the mean curvature κ are related by T n = -γ κ n where 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.