Qualitative analysis of a time-delayed free boundary problem for tumor growth with angiogenesis and Gibbs-Thomson relation

Shihe Xu; Junde Wu; Shihe Xu; Junde Wu

doi:10.3934/mbe.2019372

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 6: 7433-7446. doi: 10.3934/mbe.2019372

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Research article

Qualitative analysis of a time-delayed free boundary problem for tumor growth with angiogenesis and Gibbs-Thomson relation

Shihe Xu ^{1
,
,},
Junde Wu ²

1.
School of Mathematics and Statistics, Zhaoqing University, Zhaoqing, Guangdong 526061, China
2.
Department of Mathematics, Soochow University, Suzhou, Jiangsu 215006, China

Received: 02 May 2019 Accepted: 30 July 2019 Published: 14 August 2019

In this paper we consider a time-delayed mathematical model describing tumor growth with angiogenesis and Gibbs-Thomson relation. In the model there are two unknown functions: One is $\sigma(r, t)$ which is the nutrient concentration at time $t$ and radius $r$, and the other one is $R(t)$ which is the outer tumor radius at time $t$. Since $R(t)$ is unknown and varies with time, this problem has a free boundary. Assume $\alpha(t)$ is the rate at which the tumor attracts blood vessels and the Gibbs-Thomson relation is considered for the concentration of nutrient at outer boundary of the tumor, so that on the outer boundary, the condition $ \dfrac{\partial \sigma}{\partial r}+\alpha(t)\left(\sigma-N(t)\right) = 0, ~~r = R(t) $ holds, where $N(t) = \bar{\sigma}\left(1-\dfrac{\gamma}{R(t)}\right)H(R(t))$ is derived from Gibbs-Thomson relation. $H(\cdot)$ is smooth on $(0, \infty)$ satisfying $H(x) = 0$ if $x\leq \gamma$, $H(x) = 1$ if $x\geq 2\gamma$ and $0\leq H'(x)\leq 2/\gamma$ for all $x\geq 0$. In the case where $\alpha$ is a constant, the existence of steady-state solutions is discussed and the stability of the steady-state solutions is proved. In another case where $\alpha$ depends on time, we show that $R(t)$ will be also bounded if $\alpha(t)$ is bounded and some sufficient conditions for the disappearance of tumors are given.
- tumor growth,
- time delay,
- free boundary problem,
- Gibbs-Thomson relation,
- stability
Citation: Shihe Xu, Junde Wu. Qualitative analysis of a time-delayed free boundary problem for tumor growth with angiogenesis and Gibbs-Thomson relation[J]. Mathematical Biosciences and Engineering, 2019, 16(6): 7433-7446. doi: 10.3934/mbe.2019372

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Abstract

In this paper we consider a time-delayed mathematical model describing tumor growth with angiogenesis and Gibbs-Thomson relation. In the model there are two unknown functions: One is $\sigma(r, t)$ which is the nutrient concentration at time $t$ and radius $r$, and the other one is $R(t)$ which is the outer tumor radius at time $t$. Since $R(t)$ is unknown and varies with time, this problem has a free boundary. Assume $\alpha(t)$ is the rate at which the tumor attracts blood vessels and the Gibbs-Thomson relation is considered for the concentration of nutrient at outer boundary of the tumor, so that on the outer boundary, the condition $ \dfrac{\partial \sigma}{\partial r}+\alpha(t)\left(\sigma-N(t)\right) = 0, ~~r = R(t) $ holds, where $N(t) = \bar{\sigma}\left(1-\dfrac{\gamma}{R(t)}\right)H(R(t))$ is derived from Gibbs-Thomson relation. $H(\cdot)$ is smooth on $(0, \infty)$ satisfying $H(x) = 0$ if $x\leq \gamma$, $H(x) = 1$ if $x\geq 2\gamma$ and $0\leq H'(x)\leq 2/\gamma$ for all $x\geq 0$. In the case where $\alpha$ is a constant, the existence of steady-state solutions is discussed and the stability of the steady-state solutions is proved. In another case where $\alpha$ depends on time, we show that $R(t)$ will be also bounded if $\alpha(t)$ is bounded and some sufficient conditions for the disappearance of tumors are given.

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