Free boundary regularity for tumor growth with nutrients and diffusion

TL;DR

This paper analyzes the regularity of tumor boundaries in a growth model driven by nutrients, introducing a novel Hamilton-Jacobi-Bellman interpretation to prove boundary smoothness and stability.

Contribution

It introduces a new HJB framework for tumor growth models, establishing boundary regularity and nondegeneracy results beyond topological changes.

Findings

The hitting time T is proven to be Hölder continuous.

Tumor boundary is regular in spacetime except on a small Hausdorff dimension set.

The tumor patch boundary is locally $C^{1,eta}$ in space-time on regular points.

Abstract

In this paper, we study a tumor growth model where the growth is driven by nutrient availability and the tumor expands according to Darcy's law with a mechanical pressure resulting from the incompressibility of the cells. Our focus is on the free boundary regularity of the tumor patch that holds beyond topological changes. A crucial element in our analysis is establishing the regularity of the hitting time T, which records the first time the tumor patch reaches a given point. We achieve this by introducing a novel Hamilton-Jacobi-Bellman (HJB) interpretation of the pressure, which is of independent interest. The HJB structure is obtained by viewing the model as a limit of the Porous Media Equation (PME) and building upon a new variant of the AB estimate. Using the HJB structure, we establish a new Hopf-Lax type formula for the pressure variable. Combined with barrier arguments, the formula allows us to show that T is C^{\alpha}, where \alpha depends only on the dimension, which translates into a mild nondegeneracy of the tumor patch evolution. Building on this and obstacle problem theory, we show that the tumor patch boundary is regular in spacetime except on a set of Hausdorff dimension at most $d-\alpha$. On the set of regular points, we further show that the tumor patch is locally $C^{1,\alpha}$ in space-time. This conclusively establishes that instabilities in the boundary evolution do not amplify arbitrarily high frequencies.