Summary: Modeling tumor cell migration: From microscopic to macroscopic models
Christophe Deroulers,* Marine Aubert,
and Basil Grammaticos§
IMNC, Universités Paris VII-Paris XI-CNRS, UMR 8165, Bâtiment 104, 91406 Orsay Cedex, France
Received 19 December 2008; revised manuscript received 13 February 2009; published 25 March 2009
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 million 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 experimen-
tally 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 detail 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.