Incompressible limit of mechanical model of tumor growth with viscosity

TL;DR

This paper investigates the mathematical behavior of tumor growth models incorporating viscosity, focusing on the incompressible limit and revealing new challenges such as pressure discontinuities not seen in elastic models.

Contribution

It establishes the link between elastic and visco-elastic tumor models in the incompressible limit, highlighting the mathematical difficulties introduced by viscosity.

Findings

Viscosity introduces weaker pressure estimates.

Pressure can be discontinuous in space in the visco-elastic case.

Mathematical challenges arise in proving compactness due to viscosity.

Abstract

Various models of tumor growth are available in the litterature. A first class describes the evolution of the cell number density when considered as a continuous visco-elastic material with growth. A second class, describes the tumor as a set and rules for the free boundary are given related to the classical Hele-Shaw model of fluid dynamics. Following the lines of 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 levels 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 imply immediately compactness. For instance, traveling wave solutions and numerical simulations show that the pressure may be discontinous in space which is not the case for the elastic case.