Studying the interplay of spatio-temporal interactions and evolutionary dynamics during cancer cell invasion

Genome instability and mutations as well as the activation of invasion are defining characteristics of cancer. However, in most mathematical models only one of the two aspects is studied at a time, neglecting the complex interplay between the spatio-temporal interactions and evolutionary dynamics. To fill this gap, we here propose a mathematical model of individual cells that migrate, proliferate, die, and pass on their properties to their offspring with small variations.In particular, we assume that the set of individual properties results in a phenomenological fitness of each cell influencing its proliferation rate.In computer simulations, we show that the interplay of evolution and spatio-temporal dynamics leads to a propagating wave of invading cells, where the wave speed increases over time and clones of higher fitness appear preferably at the wave front.We use a mean-field approach to show that the system can be approximated by a PDE that is similar to the KPP-Fisher equation.We also show that the increase in average fitness over time is proportional to the variance in fitness in the population, in agreement with Fisher's fundamental theorem of natural selection.