Derivation of a Hele-Shaw type system from a cell model with active motion

Abstract : 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 some kind of \lq mushy region' prepares the tumor invasion.
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Benoît Perthame, Fernando Quirós, Min Tang, Nicolas Vauchelet. Derivation of a Hele-Shaw type system from a cell model with active motion. Interfaces and Free Boundaries, European Mathematical Society, 2014, 14 (4), pp.489-508. ⟨10.4171/IFB/327⟩. ⟨hal-00906168⟩

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