We formulate a Hele–Shaw type free boundary problem for a tumor growing under the combined effects of pressure forces, cell multiplication and active motion, the latter being the novelty of the present paper. This new ingredient is considered here as a standard diffusion process. The free boundary model is derived from a description at the cell level using the asymptotic of a stiff pressure limit.
Compared to the case when active motion is neglected, the pressure satisfies the same complementarity Hele-Shaw type formula. However, the cell density is smoother (Lipschitz continuous), while there is a deep change in the free boundary velocity, which is no longer given by the gradient of the pressure, because a region, with limited population but diffusing with long range, can prepare the tumor invasion.
Cite this article
Benoît Perthame, Fernando Quirós, Min Tang, Nicolas Vauchelet, Derivation of a Hele–Shaw type system from a cell model with active motion. Interfaces Free Bound. 16 (2014), no. 4, pp. 489–508DOI 10.4171/IFB/327