Stochastic differential equation modelling of cancer cell migration and tissue invasion

Invasion of the surrounding tissue is a key aspect of cancer growth and spread involving a coordinated effort between cell migration and matrix degradation, and has been the subject of mathematical modelling for almost 30 years. In this current paper we address a long-standing question in the field of cancer cell migration modelling. Namely, identify the migratory pattern and spread of individual cancer cells, or small clusters of cancer cells, when the macroscopic evolution of the cancer cell colony is dictated by a specific partial differential equation (PDE). We show that the usual heuristic understanding of the diffusion and advection terms of the PDE being one-to-one responsible for the random and biased motion of the solitary cancer cells, respectively, is not precise. On the contrary, we show that the drift term of the correct stochastic differential equation scheme that dictates the individual cancer cell migration, should account also for the divergence of the diffusion of the PDE. We support our claims with a number of numerical experiments and computational simulations.

Automated summary

This paper addresses the question of identifying the migratory pattern and spread of individual cancer cells or small clusters of cancer cells when the macroscopic evolution of the cancer cell colony is determined by a specific partial differential equation (PDE). The authors demonstrate that the traditional understanding of the diffusion and advection terms of the PDE as responsible for random and biased motion of solitary cancer cells, respectively, is imprecise. Instead, they show that the drift term of the correct stochastic differential equation scheme should also take into account the divergence of the PDE diffusion. Numerical experiments and computational simulations are used to support their claims.