Analysis of a Mathematical Model of the Growth of Necrotic Tumors

AbstractIn this paper we study a model of necrotic tumor growth. The tumor comprises necrotic cells which occupy a radially symmetric core and life proliferating cells which occupy a radially symmetric shell adjacent to the core. The proliferating cells receive nutrients through diffusion from the outer boundary as well as by means of blood flow through a network of capillary vessels. The mathematical model describes the evolution of the nutrient concentration σ between the boundary of the necrotic core r=ρ(t) and the outer boundary of the tumor r=R(t); within the core itself the concentration is a constant σ=σnec, a level under which life cells cannot be sustained. Both of the surfaces r=ρ(t) and r=R(t) are free boundaries, which are unknown in advance. Under some assumptions on the parameters, we prove that (i) there exists a stationary solution with radii r=ρs, r=Rs; (ii) for any initial data near the stationary solution, the time dependent model has a unique solution σ(r,t) with free boundaries r=ρ(t), r=R(t); and (iii) ρ(t)→ρs and R(t)→Rs as t→∞