Analysis of a nonlinear free-boundary tumor model with three layers

TL;DR

This paper models the growth of spherically symmetric tumors with three distinct layers, analyzing free boundary evolution driven by nutrient supply and mass conservation, and establishing stability of stationary solutions.

Contribution

It introduces a nonlinear analysis method to handle free boundaries and discontinuous functions, revealing tumor evolution mechanisms and internal structure transformations.

Findings

Existence and uniqueness of radial stationary solutions proved.

Globally asymptotic stability towards dormant tumor states established.

Analyzed mutual relationships between free boundaries and tumor structure evolution.

Abstract

In this paper, we study a nonlinear free boundary problem modeling the growth of spherically symmetric tumors. The tumor consists of a central necrotic core, an intermediate annual quiescent-cell layer, and an outer proliferating-cell layer. The evolution of tumor layers and the movement of the tumor boundary are totally governed by external nutrient supply and conservation of mass. The three-layer structure generates three free boundaries with boundary conditions of different types. We develop a nonlinear analysis method to get over the great difficulty arising from free boundaries and the discontinuity of the nutrient-consumption rate function. By carefully studying the mutual relationships between the free boundaries, we reveal the evolutionary mechanism in tumor growth and the mutual transformation of its internal structures. The existence and uniqueness of the radial stationary solution is proved, and its globally asymptotic stability towards different dormant tumor states is established.