Analysis of a mathematical model of the effect of inhibitors on the growth of tumors

In this paper, we study a model of tumor growth in the presence of inhibitors. The tumor is assumed to be spherically symmetric and its boundary is an unknown function r=R(t). Within the tumor the concentration of nutrient and the concentration of inhibitor (drug) satisfy a system of reaction–diffusion equations. The important parameters are Λ0 (which depends on the tumorʹs parameters when no inhibitors are present), γ which depends only on the specific properties of the inhibitor, and β̄ which is the (normalized) external concentration of the inhibitor. In this paper, we give precise conditions under which there exist one dormant tumor, two dormant tumors, or none. We then prove that in the first case, the dormant tumor is globally asymptotically stable, and in the second case, if the radii of the dormant tumors are denoted by Rs−,Rs+ with Rs−<Rs+, then the smaller one is asymptotically stable, so that limt→∞R(t)=Rs−, provided the initial radius R0 is smaller than Rs+; if however R0>Rs+ then the initial tumor in general grows unboundedly in time. The above analysis suggests an effective strategy for treatment of tumors.