Lipschitz stability for Bayesian inference in porous medium tissue growth models

TL;DR

This paper proves Lipschitz stability of solutions in a porous medium tissue growth model with respect to the pressure-density relation parameter, facilitating Bayesian parameter estimation from experimental data.

Contribution

It establishes Lipschitz continuity of solutions relative to the diffusion exponent, advancing the mathematical foundation for Bayesian inverse methods in tissue growth modeling.

Findings

Lipschitz continuity of solutions w.r.t. the diffusion parameter in L1 norm.

Provides a mathematical basis for Bayesian inverse problem application.

Enhances parameter estimation techniques in biological tissue models.

Abstract

We consider a macroscopic model for the dynamics of living tissues incorporating pressure-driven dispersal and pressure-modulated proliferation. Given a power-law constitutive relation between the pressure and cell density, the model can be written as a porous medium equation with a growth term. We prove Lipschitz continuity of the mild solutions of the model with respect to the diffusion parameter (the exponent $\gamma$ in the pressure-density law) in the $L_1$ norm. While of independent analytical interest, our motivation for this result is to provide a vital step towards using Bayesian inverse problem methodology for parameter estimation based on experimental data -- such stability estimates are indispensable for applying sampling algorithms which rely on the gradient of the likelihood function.