# American Institute of Mathematical Sciences

2013, 10(3): 761-775. doi: 10.3934/mbe.2013.10.761

## Spatial stochastic models of cancer: Fitness, migration, invasion

 1 Department of Mathematics, University of California Irvine, Irvine CA 92697, United States

Received  July 2012 Revised  November 2012 Published  April 2013

Cancer progression is driven by genetic and epigenetic events giving rise to heterogeneity of cell phenotypes, and by selection forces that shape the changing composition of tumors. The selection forces are dynamic and depend on many factors. The cells favored by selection are said to be more fit'' than others. They tend to leave more viable offspring and spread through the population. What cellular characteristics make certain cells more fit than others? What combinations of the mutant characteristics and background'' characteristics make the mutant cells win the evolutionary competition? In this review we concentrate on two phenotypic characteristics of cells: their reproductive potential and their motility. We show that migration has a direct positive impact on the ability of a single mutant cell to invade a pre-existing colony. Thus, a decrease in the reproductive potential can be compensated by an increase in cell migration. We further demonstrate that the neutral ridges (the set of all types with the invasion probability equal to that of the host cells) remain invariant under the increase of system size (for large system sizes), thus making the invasion probability a universal characteristic of the cells' selection status. We list very general conditions under which the optimal phenotype is just one single strategy (thus leading to a nearly-homogeneous type invading the colony), or a large set of strategies that differ by their reproductive potentials and migration characteristics, but have a nearly-equal fitness. In the latter case the evolutionary competition will result in a highly heterogeneous population.
Citation: Natalia L. Komarova. Spatial stochastic models of cancer: Fitness, migration, invasion. Mathematical Biosciences & Engineering, 2013, 10 (3) : 761-775. doi: 10.3934/mbe.2013.10.761
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