  It has been shown experimentally that contact interactions may influence the
migration of cancer cells. Previous works have modelized this thanks to
stochastic, discrete models (cellular automata) at the cell level. However, for
the study of the growth of real-size tumors with several millions of cells, it
is best to use a macroscopic model having the form of a partial differential
equation (PDE) for the density of cells. The difficulty is to predict the
effect, at the macroscopic scale, of contact interactions that take place at
the microscopic scale. To address this we use a multiscale approach: starting
from a very simple, yet experimentally validated, microscopic model of
migration with contact interactions, we derive a macroscopic model. We show
that a diffusion equation arises, as is often postulated in the field of glioma
modeling, but it is nonlinear because of the interactions. We give the explicit
dependence of diffusivity on the cell density and on a parameter governing
cell-cell interactions. We discuss in details the conditions of validity of the
approximations used in the derivation and we compare analytic results from our
PDE to numerical simulations and to some in vitro experiments. We notice that
the family of microscopic models we started from includes as special cases some
kinetically constrained models that were introduced for the study of the
physics of glasses, supercooled liquids and jamming systems.
