An elliptic-hyperbolic free boundary problem modelling cancer therapy

In this paper we study a free boundary problem modelling the growth of an avascular tumour with drug application. The tumour consists of two cell populations: live cells and dead cells. The densities of these cells satisfy a system of nonlinear first order hyperbolic equations. The tumour surface is a moving boundary, which satisfies an integro-differential equation. The nutrient concentration and the drug concentration satisfy nonlinear diffusion equations. The nutrient drives the growth of the tumour, whereas the drug is capable of killing cells with Michaelis-Menten kinetics. We prove that this free boundary problem has a unique global solution. Furthermore, we investigate the combined effects of a drug and a nutrient on an avascular tumour growth. We prove that the tumour shrinks to a necrotic core with radius R-s > 0 and that the global solution converges to a trivial steady-state solution under some natural assumptions on the model parameters. We also prove that an untreated tumour shrinks to a dead core or continually grows to an infinite size, which depends on the different parameter conditions.