Analysis of a nonlinear necrotic tumor model with angiogenesis and a periodic supply of external nutrients

TL;DR

This paper models necrotic tumor growth with angiogenesis under periodic nutrient supply, analyzing conditions for tumor extinction or persistence, and establishing existence and stability of periodic solutions.

Contribution

It introduces a nonlinear free boundary tumor model with periodic external nutrients and characterizes tumor outcomes based on nutrient supply averages.

Findings

Tumors vanish if the average of S(φ(t)) is nonpositive.

Existence of a unique positive periodic solution when the average is positive.

Tumors converge to this periodic solution under certain conditions.

Abstract

In this paper, we consider a free boundary problem modeling the growth of spherically symmetric necrotic tumors with angiogenesis and a $\omega$-periodic supply $\phi(t)$ of external nutrients. In the model, the consumption rate of the nutrient and the proliferation rate of tumor cells $S(\sigma)$ are both general nonlinear functions. The well-posedness and asymptotic behavior of solutions are studied. We show that if the average of $S(\phi(t))$ is nonpositive, then all evolutionary tumors will finally vanish; the converse is also ture. If instead the average of $S(\phi(t))$ is positive, then there exists a unique positive periodic solution and all other evolutionary tumors will converge to this periodic state.