A Competition Model for Hypertumors

In a 2004 paper, John Nagy raised the possibility of the existence of a hypertumor—a focus of aggressively reproducing parenchyma cells that invade part or all of an existing tumor. The model in that study consists of a system of nonlinear ordinary differential equations that is used to find a set of conditions under which these hypertumors may exist. We have expanded that model by introducing spatial structure in an idealized spherical tumor with radial symmetry, whose evolution is described by a system of parabolic partial differential equations for the densities of two different phenotypes of parenchimal cells, as well as of vascular endothelial cells (VECs) and of the length of existing tumor microvessels. We use a free boundary condition to describe tumor growth. We assume one type of parenchimal  cells initially forms almost the entirety of the tumor while the other, much more aggressive, strain appears in a small circular region at the center of the tumor. The main goal of this research is to determine under what conditions a single strain of parenchimal cells will survive, when both strains may coexist and form a hypertumor, or both strains die out due to necrosis of the entire tumor mass. Some analytical and simulation results will be presented to help formulate a conjecture about the stability of the hypertumor and these theoretical predictions will then be compared with relevant oncological data.