Qualitative analysis of a time-delayed free boundary problem for tumor growth with Gibbs-Thomson relation in the presence of inhibitors

Huiyan Peng; Xuemei Wei; Huiyan Peng; Xuemei Wei

doi:10.3934/math.20231140

AIMS Mathematics

2023, Volume 8, Issue 9: 22354-22370. doi: 10.3934/math.20231140

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Research article

Qualitative analysis of a time-delayed free boundary problem for tumor growth with Gibbs-Thomson relation in the presence of inhibitors

Huiyan Peng ,
Xuemei Wei ^,

Mathematics and Statistics, Guangdong University of Technology, Guangzhou 510520, China

Received: 17 May 2023 Revised: 18 June 2023 Accepted: 25 June 2023 Published: 13 July 2023
MSC : 35K57, 35Q92, 35R35

In this paper, we study a time-delayed free boundary of tumor growth with Gibbs-Thomson relation in the presence of inhibitors. The model consists of two reaction diffusion equations and an ordinary differential equation. The reaction diffusion equations describe the nutrient and inhibitor diffusion within tumors and take into account the Gibbs-Thomson relation at the outer boundary of the tumor. The tumor radius evolution is described by the ordinary differential equation. It is assumed that the regulatory apoptosis process takes longer than the natural apoptosis and proliferation processes. We first show the existence and uniqueness of the solution to the model. Next, we further demonstrate the existence of the stationary solutions and the asymptotic behavior of the stationary solutions when the blood vessel density is a constant. Finally, we further demonstrate the existence of the stationary solutions and the asymptotic behavior of the stationary solutions when the blood vessel density is bounded. The result implies that, under certain conditions, the tumor will probably become dormant or will finally disappear. The conclusions are illustrated by numerical computations.
- time delay,
- tumor growth,
- free boundary problem,
- Gibbs-Thomson relation,
- stability
Citation: Huiyan Peng, Xuemei Wei. Qualitative analysis of a time-delayed free boundary problem for tumor growth with Gibbs-Thomson relation in the presence of inhibitors[J]. AIMS Mathematics, 2023, 8(9): 22354-22370. doi: 10.3934/math.20231140

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Abstract

In this paper, we study a time-delayed free boundary of tumor growth with Gibbs-Thomson relation in the presence of inhibitors. The model consists of two reaction diffusion equations and an ordinary differential equation. The reaction diffusion equations describe the nutrient and inhibitor diffusion within tumors and take into account the Gibbs-Thomson relation at the outer boundary of the tumor. The tumor radius evolution is described by the ordinary differential equation. It is assumed that the regulatory apoptosis process takes longer than the natural apoptosis and proliferation processes. We first show the existence and uniqueness of the solution to the model. Next, we further demonstrate the existence of the stationary solutions and the asymptotic behavior of the stationary solutions when the blood vessel density is a constant. Finally, we further demonstrate the existence of the stationary solutions and the asymptotic behavior of the stationary solutions when the blood vessel density is bounded. The result implies that, under certain conditions, the tumor will probably become dormant or will finally disappear. The conclusions are illustrated by numerical computations.

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