Incompressible limit of a mechanical model of tumour growth with viscosity.
Various models of tumour growth are available in the literature. The first type describe the evolution of the cell number density when considered as a continuous visco-elastic material with growth. The second type describe the tumour as a set, and rules for the free boundary are given related to the classical Hele-Shaw model of fluid dynamics. Following previous papers where the material is described by a purely elastic material, or when active cell motion is included, we make the link between the two types of description considering the 'stiff pressure law' limit. Even though viscosity is a regularizing effect, new mathematical difficulties arise in the visco-elastic case because estimates on the pressure field are weaker and do not immediately imply compactness. For instance, travelling wave solutions and numerical simulations show that the pressure is discontinuous in space, which is not the case for an elastic material.
Sorbonne Universités, UPMC University of Paris 06, Paris, France CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, Paris 75005, France INRIA-Paris-Rocquencourt, EPC MAMBA, Domaine de Voluceau, BP105, Le Chesnay Cedex 78153, France benoit.perthame@upmc.fr.
Benoît
B
Perthame
Sorbonne Universités, UPMC University of Paris 06, Paris, France CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, Paris 75005, France INRIA-Paris-Rocquencourt, EPC MAMBA, Domaine de Voluceau, BP105, Le Chesnay Cedex 78153, France.
Nicolas
N
Vauchelet
D000818:Animals; D002452:Cell Count; D049109:Cell Proliferation; D019245:Compressive Strength; D003198:Computer Simulation; D004548:Elasticity; D006801:Humans; D040542:Mechanotransduction, Cellular; D008954:Models, Biological; D009369:Neoplasms / Q000473:pathology* / Q000503:physiopathology*; D011312:Pressure; D012212:Rheology / Q000379:methods; D014783:Viscosity
Hele-Shaw equation; free boundary problems; porous media; tumour growth; visco-elastic media