The Asymptotic Behavior of a Probabilistic Model for Tumor Growth

Abstract

We discuss here the behavior for large time of a cancer model introduced by Williams and Bjerknes [5], Cells are of two types, normal and abnormal, and are located on the planar lattice, one at each site. With each cellular division, one daughter cell stays put; the other usurps the position of a neighbor, who disappears from the population. Abnormal cells are assumed to reproduce at a faster rate than normal cells; the cancer therefore has a tendency to spread. It is shown that, conditioned on its nonextinction, a tumor commencing from a single abnormal cell will have an asymptotically linear rate of radial growth; asymptotically, the tumor assumes a fixed shape.