JournalsifbVol. 5 , No. 2DOI 10.4171/ifb/76

A Hyperbolic Free Boundary Problem Modeling Tumor Growth

  • Avner Friedman

    Ohio State University, Columbus, USA
  • Shangbin Cui

    Zhongshan University, Guangzhou, Guangdong, China
A Hyperbolic Free Boundary Problem Modeling Tumor Growth cover

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)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<KR<0<K_R<\infty, KR=0K_R=0 and KR=K_R=\infty, where KRK_R is the removal rate of dead cells. We also prove that in the cases 0<KR<0<K_R<\infty and KR=K_R=\infty there exist positive numbers δ0\delta_0 and MM such that δ0R(t)M\delta_0\leq R(t)\leq M for all t0t\geq 0, while limtR(t)=\lim_{t\to\infty}R(t)=\infty in the case KR=0K_R=0.