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It has been shown experimentally that contact interactions may influence themigration of cancer cells. Previous works have modelized this thanks tostochastic, discrete models (cellular automata) at the cell level. However, forthe study of the growth of real-size tumors with several millions of cells, itis best to use a macroscopic model having the form of a partial differentialequation (PDE) for the density of cells. The difficulty is to predict theeffect, at the macroscopic scale, of contact interactions that take place atthe microscopic scale. To address this we use a multiscale approach: startingfrom a very simple, yet experimentally validated, microscopic model ofmigration with contact interactions, we derive a macroscopic model. We showthat a diffusion equation arises, as is often postulated in the field of gliomamodeling, but it is nonlinear because of the interactions. We give the explicitdependence of diffusivity on the cell density and on a parameter governingcell-cell interactions. We discuss in details the conditions of validity of theapproximations used in the derivation and we compare analytic results from ourPDE to numerical simulations and to some in vitro experiments. We notice thatthe family of microscopic models we started from includes as special cases somekinetically constrained models that were introduced for the study of thephysics of glasses, supercooled liquids and jamming systems.

Modeling tumor cell migration: from microscopic to macroscopic