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Mathematical models describing the spatial spreading and invasion ofpopulations of biological cells are often developed in a continuum modellingframework using reaction-diffusion equations. While continuum models based onlinear diffusion are routinely employed and known to capture key experimentalobservations, linear diffusion fails to predict well-defined sharp fronts thatare often observed experimentally. This observation has motivated the use ofnonlinear degenerate diffusion, however these nonlinear models and theassociated parameters lack a clear biological motivation and interpretation.Here we take a different approach by developing a stochastic discretelattice-based model incorporating biologically-inspired mechanisms and thenderiving the reaction-diffusion continuum limit. Inspired by experimentalobservations, agents in the simulation deposit extracellular material, that wecall a substrate, locally onto the lattice, and the motility of agents is takento be proportional to the substrate density. Discrete simulations that mimic atwo--dimensional circular barrier assay illustrate how the discrete modelsupports both smooth and sharp-fronted density profiles depending on the rateof substrate deposition. Coarse-graining the discrete model leads to a novelpartial differential equation (PDE) model whose solution accuratelyapproximates averaged data from the discrete model. The new discrete model andPDE approximation provides a simple, biologically motivated framework formodelling the spreading, growth and invasion of cell populations withwell-defined sharp fronts

Discrete and continuous mathematical models of sharp-fronted collective cell migration and invasion