Analysis of a nonlinear free boundary problem modeling the radial growth of two-layer tumors
Junde Wu, Hao Xu, Yuehong Zhuang
TL;DR
This work analyzes a nonlinear, radially symmetric free boundary model for two-layer tumor growth with a proliferating outer shell and a quiescent core. The authors decompose nutrient diffusion into inner and outer regions, solve via a shooting method, and reduce the stationary problem to a root condition $F(R)=0$, establishing a sharp threshold $\sigma^*$: stationary solutions with a quiescent core exist if and only if $\bar{σ}>\sigma^*$. They prove global well-posedness and asymptotic stability of the stationary states, characterize three long-time regimes (extinction, proliferating-only, and proliferating-quiescent with core), and reveal a deep connection to necrotic-core models by showing convergence as the quiescent-cells consumption $h(\σ)$ vanishes. The results also demonstrate how external nutrient supply can control tumor morphology and unify two-layer quiescent-core and necrotic-core descriptions under a common framework.
Abstract
In this paper we study a nonlinear free boundary problem on the radial growth of a two-layer solid tumor with a quiescent core. The tumor surface and its inner interface separating the proliferating cells and the quiescent cells are both free boundaries. By deeply analyzing their relationship and employing the maximum principle, we show this problem is globally well-posed and prove the existence of a unique positive threshold $σ^*$ such that the problem admits a unique stationary solution with a quiescent core if and only if the externally supplied nutrient $\barσ> σ^*$. The stationary solution is globally asymptotically stable. The formation of the quiescent core and its interesting connection with the necrotic core are also given.