The Impact of Time Delay and Angiogenesis in a Tumor Model

TL;DR

This paper analyzes a tumor growth model incorporating angiogenesis and time delays, establishing conditions for stability of stationary solutions and how parameters influence tumor size and stability.

Contribution

It proves the existence and uniqueness of radially symmetric stationary solutions and identifies a threshold for stability based on tumor aggressiveness.

Findings

Increased angiogenesis parameter reduces the stability threshold.

Time delay enlarges the stationary tumor size without changing the stability threshold.

Higher tumor aggressiveness amplifies the impact of time delay on tumor size.

Abstract

We consider a free boundary tumor model under the presence of angiogenesis and time delays in the process of proliferation, in which the cell location is incorporated. It is assumed that the tumor attracts blood vessels at a rate proportional to $\alpha$, and a parameter $\mu$ is proportional to the `aggressiveness' of the tumor. In this paper, we first prove that there exists a unique radially symmetric stationary solution $\left(\sigma_{*}, p_{*}, R_{*}\right)$ for all positive $\alpha$, $\mu$. Then a threshold value $\mu_\ast$ is found such that the radially symmetric stationary solution is linearly stable if $\mu<\mu_\ast$ and linearly unstable if $\mu>\mu_\ast$. Our results indicate that the increase of the angiogenesis parameter $\alpha$ would result in the reduction of the threshold value $\mu_\ast$; adding the time delay would not alter the threshold value $\mu_\ast$, but result in a larger stationary tumor, and the larger the tumor aggressiveness parameter $\mu$ is, the greater impact of time delay would have on the size of the stationary tumor.