A Hyperbolic Free Boundary Problem Modeling Tumor Growth

Abstract

In this paper we study a free boundary problem modeling the growth of tumors with three cell populations: proliferating cells, quiescent cells and dead cells. The densities of these cells satisfy a system of nonlinear first order hyperbolic equations in the tumor, with tumor surface as a free boundary. The nutrient concentration satisfies a diffusion equation, and the free boundary r = R(t) satisfies an integro-differential equation. We consider the radially symmetric case of this free boundary problem, and prove that it has a unique global solution for all the three cases 0 < K R < , K R = 0 and K R = , where K R is the removal rate of dead cells. We also prove that in the cases 0 < K R < and K R = there exist positive numbers 0 and M such that 0 R(t) M for all t 0, while lim t R(t) = in the case K R = 0.