Modeling the migration of cancer cells: from microscopic to macroscopic models
by
Christophe Deroulers
Laboratory IMNC, Campus d'Orsay, 91406 Orsay Cedex, France
Coauthors: Mathilde BADOUAL, Marine AUBERT, Basile GRAMMATICOS

It is well known that the migration of cancer cells plays a key role in the development of some brain tumors, such as gliomas. Because cancer cells invade tissues far from the tumor center, the tumors have no sharp boundary. A surgeon cannot remove all cancer cells, and cancer reccurs invariably.

Therefore, it is crucial to take cell migration into account in the modelling of glioma. This is easy in a cellular automaton-like model, where some stochastic rules tell how the individual cells move, duplicate and die, and some of us used such a model to reproduce in vitro experiments of cancer cells migration and to show that cells interact while migrating [1]. However, a cellular automaton is not so convenient as partial differential equations (PDEs) that are often used to model the density of cancer cells in the brain. Especially, it is not suited for the study of real-size tumors with several millions of cells. Therefore, it would be nice to derive some PDE that takes the migration and interaction of cancer cells into account.

We give one analytic technique to go from the definition of a microscopic model (a cellular automaton) to a macroscopic model (a PDE). We apply this technique to the situation of [1] and we show that our PDE reproduces both simulation and in vitro experimental results [2]. We notice that the PDE we obtain is, because of the interaction of cancer cells, a nonlinear diffusion equation (belonging to the family of the porous media equations), whereas it is often postulated that diffusion of cancer cells is linear. Interestingly, we notice that our model is closely related to some kinetically constrained models introduced for the study of the physics of glasses, supercooled liquids and jamming systems.

We also give some results on the effect of taking into account the shape of cancer cells, which are not point-like as assumed in [2] but elongated, and on dealing with statistical correlations of the position of cancer cells.

[1] M. Aubert, M. Badoual, S. Fereol, C. Christov and B. Grammaticos, A cellular automaton model for the migration of glioma cells, Phys. Biol. vol. 3 p. 93 (2006).

[2] C. Deroulers, M. Badoual, M. Aubert and B. Grammaticos, Modelling tumour cell migration: from microscopic to macroscopic, submitted (2008).

Date received: May 15, 2008