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 < KR < ∞, KR = 0 and KR = ∞, where KR is the removal rate of dead cells. We also prove that in the cases 0 < KR < ∞ and KR = ∞ there exist positive numbers δ0 and M such that δ0 ≤ R(t) ≤ M for all t ≥ 0, while limt→∞R(t) = ∞in the case KR = 0.