Derivation of a Hele-Shaw type system from a cell model with active motion
نویسندگان
چکیده
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 ‘mushy region’ prepares the tumor invasion. Key-words: Tumor growth; Hele-Shaw equation; porous medium equation; free boundary problems. Mathematics Subject Classification 35K55; 35B25; 76D27; 92C50.
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