# The Impact of Time Delay in a Tumor Model

## TL;DR

This paper analyzes how time delay influences the stability and size of tumors in a mathematical model, revealing that increased delay can enlarge tumors and that stability depends on tumor aggressiveness.

## Contribution

It introduces a tumor growth model with time delay, proving stability conditions and showing how delay affects tumor size and stability thresholds.

## Key findings

- Existence of a unique stable stationary tumor solution for any tumor aggressiveness.
- Stability depends on a critical aggressiveness parameter, with instability for higher values.
- Time delay increases tumor size and has a greater impact with higher tumor aggressiveness.

## Abstract

In this paper we consider a free boundary tumor growth model with a time delay in cell proliferation and study how time delay affects the stability and the size of the tumor. The model is a coupled system of an elliptic equation, a parabolic equation and an ordinary differential equation. It incorporates the cell location under the presence of time delay, with the tumor boundary as a free boundary. A parameter $\mu$ in the model is proportional to the "aggressiveness" of the tumor. It is proved that there exists a unique classical radially symmetric stationary solution $(\sigma_*, p_*, R_*)$ which is stable for any $\mu > 0$ with respect to all radially symmetric perturbations (c.f. \cite{delay1}). However, under non-radially symmetric perturbations, we prove that there exists a critical value $\mu_*$, such that if $\mu<\mu_*$ then the stationary solution $(\sigma_*, p_*, R_*)$ is linearly stable; whereas if $\mu>\mu_*$ the stationary solution is unstable. It is actually unrealistic to expect the problem to be stable for large tumor aggressiveness parameter, therefore our result is more reasonable. Furthermore, we established that adding the time delay in the model would result in a larger stationary tumor, and if the tumor aggressiveness parameter is larger, then the time delay would have a greater impact on the size of the tumor.