肿瘤模型中时间延迟和血管生成的影响
The Impact of Time Delay and Angiogenesis in a Tumor Model
摘要：
我们考虑了一个自由边界肿瘤模型，在该模型中，加入了细胞位置信息，并考虑了血管生成和增殖过程中的时间延迟。假设肿瘤以与$α$成比例的速率吸引血管，并且参数$μ$与肿瘤的“侵袭性”成比例。本文首先证明了对所有正的$α$，$μ$，存在唯一的径向对称平稳解$\left(σ_{*}, p_{*}, R_{*}\right)$.然后，找到一个门限值$μ_\ast$，使得径向对称的静态解是线性稳定的，如果$μ<μ_\ast$和线性不稳定的，如果$μ>μ_\ast$。结果表明，血管生成参数$α$的增加会导致门限值$μ_\ast$的降低;增加时间延迟不会改变门限值$μ_\ast$，但会导致更大的静止肿瘤，肿瘤侵袭性参数$μ$越大，时间延迟对静止肿瘤大小的影响越大。
英文摘要：
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 $α$, and a parameter $μ$ is proportional to the `aggressiveness' of the tumor. In this paper, we first prove that there exists a unique radially symmetric stationary solution $\left(σ_{*}, p_{*}, R_{*}\right)$ for all positive $α$, $μ$. Then a threshold value $μ_\ast$ is found such that the radially symmetric stationary solution is linearly stable if $μ<μ_\ast$ and linearly unstable if $μ>μ_\ast$. Our results indicate that the increase of the angiogenesis parameter $α$ would result in the reduction of the threshold value $μ_\ast$; adding the time delay would not alter the threshold value $μ_\ast$, but result in a larger stationary tumor, and the larger the tumor aggressiveness parameter $μ$ is, the greater impact of time delay would have on the size of the stationary tumor.
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