Qualitative analysis of a time-delayed free boundary problem for tumor growth with angiogenesis and Gibbs-Thomson relation
| Autor: | Jun de Wu, Shi He Xu |
|---|---|
| Rok vydání: | 2019 |
| Předmět: |
Time Factors
Quantitative Biology::Tissues and Organs Boundary (topology) Apoptosis 02 engineering and technology Models Biological Combinatorics Qualitative analysis Neoplasms 0502 economics and business 0202 electrical engineering electronic engineering information engineering Free boundary problem Humans Tumor growth Computer Simulation Cell Proliferation Physics Models Statistical Neovascularization Pathologic Applied Mathematics 05 social sciences General Medicine Radius Computational Mathematics Time delayed Modeling and Simulation Bounded function 020201 artificial intelligence & image processing General Agricultural and Biological Sciences Constant (mathematics) 050203 business & management Algorithms |
| Zdroj: | Mathematical biosciences and engineering : MBE. 16(6) |
| ISSN: | 1551-0018 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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