A two species hyperbolic-parabolic model of tissue growth

TL;DR

This paper proves the existence of weak solutions for a complex two-species tissue growth model involving hyperbolic and parabolic equations, addressing mathematical challenges like discontinuities and vacuum singularities in higher dimensions.

Contribution

It introduces a novel proof strategy focusing on pressure gradient compactness, extending results to higher dimensions and handling singularities, improving upon previous methods.

Findings

Established existence of weak solutions for the model

Extended analysis to higher spatial dimensions

Developed new estimates and handled vacuum singularities

Abstract

Models of tissue growth are now well established, in particular in relation to their applications to cancer. They describe the dynamics of cells subject to motion resulting from a pressure gradient generated by the death and birth of cells, itself controlled primarily by pressure through contact inhibition. In the compressible regime we consider, when pressure results from the cell densities and when two different populations of cells are considered, a specific difficulty arises from the hyperbolic character of the equation for each cell density, and to the parabolic aspect of the equation for the total cell density. For that reason, few a priori estimates are available and discontinuities may occur. Therefore the existence of solutions is a difficult problem. Here, we establish the existence of weak solutions to the model with two cell populations which react similarly to the pressure in terms of their motion but undergo different growth/death rates. In opposition to the method used in the recent paper of J. A. Carrillo, S. Fagioli, F. Santambrogio, and M.Schmidtchen, Splitting schemes & segregation in reaction-(cross-)diffusion systems, our strategy is to ignore compactness on the cell densities and to prove strong compactness on the pressure gradient. We improve known results in two directions; we obtain new estimates, we treat higher dimension than 1 and we deal with singularities resulting from vacuum.