Tumor growth with a necrotic core as an obstacle problem in pressure

TL;DR

This paper models tumor growth with a necrotic core using an obstacle problem framework, providing analytical insights into the solution structure, phase transitions, and traveling wave solutions with external densities.

Contribution

It introduces a semi-analytical solution approach and proves the existence of traveling wave solutions with non-zero outer densities in a tumor growth model.

Findings

Quantitative analysis of necrotic core development phases

Long-term behavior characterized by traveling wave solutions

Existence of solutions with external densities below a threshold

Abstract

Motivated by the incompressible limit of a cell density model, we propose a free boundary tumor growth model where the pressure satisfies an obstacle problem on an evolving domain $\Omega(t)$, and the coincidence set $\Lambda(t)$ captures the emerging necrotic core. We contribute to the analytical characterization of the solution structure in the following two aspects. By deriving a semi-analytical solution and studying its dynamical behavior, we obtain quantitative transitional properties of the solution separating phases in the development of necrotic cores and establish its long time limit with the traveling wave solutions. Also, we prove the existence of traveling wave solutions incorporating non-zero outer densities outside the tumor bulk, provided that the size of the outer density is below a threshold.