Analysis of a nonlinear free-boundary tumor model with three layers
Junde Wu, Hao Xu, Yuehong Zhuang
TL;DR
The paper analyzes a nonlinear free boundary model for three-layer tumor growth under radial symmetry, featuring a necrotic core, a quiescent shell, and a proliferating outer region. Using an inside-out shooting approach across the three layers and nonlinear nutrient consumption, it establishes existence and uniqueness of radial stationary solutions and identifies two critical nutrient thresholds $\sigma^*$ and $\sigma_*$ that determine whether the dormant tumor adopts three-layer, two-layer, or one-layer structures. It then proves global well-posedness and reduces the dynamics to $\dfrac{dR}{dt} = RF(R,\bar{\sigma})$, showing exponential convergence to the corresponding stationary states and detailing structural transitions among six possible tumor states as $\bar{\sigma}$ and the initial radius vary. The results reveal how external nutrient supply drives tumor architecture, offering a potential mechanism to influence tumor structure in practical settings, and connect to observations of spheroid layering in experiments. Regularity and monotonicity analyses underpin the robustness of the long-time outcomes.
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.