Asymptotic behavior of a free boundary problem for the growth of multi-layer tumors in necrotic phase

TL;DR

This paper analyzes a free boundary model for multi-layer tumor growth in necrotic phase, establishing conditions for stability of stationary solutions based on cell adhesiveness.

Contribution

It introduces a mathematical model with two free boundaries for tumor growth and proves the stability criteria for stationary solutions depending on cell adhesiveness.

Findings

Existence of a unique flat stationary solution.

Stability of the solution depends on a critical cell adhesiveness parameter.

Stationary solution is stable if cell adhesiveness exceeds a threshold.

Abstract

In this paper we study a free boundary problem for the growth of multi-layer tumors in necrotic phase. The tumor region is strip-like and divided into necrotic region and proliferating region with two free boundaries. The upper free boundary is tumor surface and governed by a Stefan condition. The lower free boundary is the interface separating necrotic region from proliferating region, its evolution is implicit and intrinsically governed by an obstacle problem. We prove that the problem has a unique flat stationary solution, and there exists a positive constant $\gamma_*$, such that the flat stationary solution is asymptotically stable for cell-to-cell adhesiveness $\gamma>\gamma_*$, and unstable for $0<\gamma<\gamma_*$.