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 $0<K_R<\infty$, $K_R=0$ and $K_R=\infty$, where $K_R$ is the removal rate of dead cells. We also prove that in the cases $0<K_R<\infty$ and $K_R=\infty$ there exist positive numbers ${\delta }_0$ and $M$ such that ${\delta }_0\le R(t)\le M$ for all $t\ge 0$, while ${\lim }_{t\to \infty }R(t)=\infty$ in the case $K_R=0$.