JournalsifbVol. 8 , No. 2DOI 10.4171/ifb/142

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

  • Avner Friedman

    Ohio State University, Columbus, USA
A free boundary problem for a coupled system of elliptic, hyperbolic, and Stokes equations modeling tumor growth cover

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 γ\gamma. Thus, on the free boundary the stress tensor TT and the mean curvature κ\kappa are related by Tn=γκnT\vec n=-\gamma\kappa\vec n where n\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.