A tumor growth model of Hele-Shaw type as a gradient flow

TL;DR

This paper models tumor growth using a Hele-Shaw type PDE as a gradient flow in a nonnegative measure space, establishing existence, uniqueness, and numerical schemes without regularity assumptions.

Contribution

It introduces a new gradient flow formulation for a tumor growth PDE using an optimal transport-growth metric, with proofs of existence, uniqueness, and a convergent numerical scheme.

Findings

Proved existence of solutions via minimizing movements.

Established uniqueness on convex domains using Evolutional Variational Inequality.

Developed a numerical scheme with proven convergence.

Abstract

In this paper, we characterize a degenerate PDE as the gradient flow in the space of nonnegative measures endowed with an optimal transport-growth metric. The PDE of concern, of Hele-Shaw type, was introduced by Perthame et. al. as a mechanical model for tumor growth and the metric was introduced recently in several articles as the analogue of the Wasserstein metric for nonnegative measures. We show existence of solutions using minimizing movements and show uniqueness of solutions on convex domains by proving the Evolutional Variational Inequality. Our analysis does not require any regularity assumption on the initial condition. We also derive a numerical scheme based on the discretization of the gradient flow and the idea of entropic regularization. We assess the convergence of the scheme on explicit solutions. In doing this analysis, we prove several new properties of the optimal transport-growth metric, which generally have a known counterpart for the Wasserstein metric.