Spatial Moran Models II. Tumor growth and progression

We study the accumulation of mutations in a spatial Moran model on a torus in Zd in which each cell gives birth at a rate equal to its fitness and replaces a neighbor at random with its offspring. Cells of type k have relative fitness (1 + s)k and mutate to type k + 1 at rate uk+1. When restricted to two cell types and no mutations, this model reduces to the biased voter model. We give a new result for the biased voter model that identifies the order of magnitude of the speed of propagation in the Bramson-Griffeath shape theorem, when s is small. However, our main focus is on σk, the time of birth of the first type k whose family line does not die out, and the growth of the number of type k cells, Zk(t). This investigation is a first step in understanding the spatial structure of the genetic heterogeneity of solid tumors.