# A Hyperbolic Free Boundary Problem Modeling Tumor Growth

### Avner Friedman

Ohio State University, Columbus, USA### Shangbin Cui

Zhongshan University, Guangzhou, Guangdong, China

## 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 $\delta_0\leq R(t)\leq M$ for all $t\geq 0$, while $\lim_{t\to\infty}R(t)=\infty$ in the case $K_R=0$.

## Cite this article

Avner Friedman, Shangbin Cui, A Hyperbolic Free Boundary Problem Modeling Tumor Growth. Interfaces Free Bound. 5 (2003), no. 2, pp. 159–182

DOI 10.4171/IFB/76