Analysis and computation of some tumor growth models with nutrient: From cell density models to free boundary dynamics
Analysis and computation of some tumor growth models with nutrient: From cell density models to free boundary dynamics
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DOI：
10.3934/dcdsb.2018297
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发表时间：
2018-02
期刊：
影响因子：
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通讯作者：
Jian‐Guo Liu;M. Tang;Li Wang;Zhennan Zhou
中科院分区：
文献类型：
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作者：
Jian‐Guo Liu;M. Tang;Li Wang;Zhennan Zhou
在本文中，我们研究了肿瘤生长方程以及营养成分的各种模型，包括体外模型和体内模型。在细胞密度水平上，肿瘤密度的空间可获得性由达西定律通过压力$p(N)=n^{\γ}$来控制。对于有限元，我们证明了一些关于肿瘤生长模型的先验估计，如营养密度的有界性，以及肿瘤密度的非负性和增长估计。细胞密度模型形式上收敛于Hele-Shaw流动模型，它决定了肿瘤组织在不可压缩极限下的自由边界动力学。我们推导了Hele-Shaw流动模型的几个解析解，作为肿瘤前沿传播几何运动的基准解。最后，我们将守恒正性的数值格式应用于胞体密度模型，数值结果验证了胞体密度模型与自由边界动力学模型之间的联系。
In this paper, we study the tumor growth equation along with various models for the nutrient component, including the \emph{in vitro} model and the \emph{in vivo} model. At the cell density level, the spatial availability of the tumor density $n$ is governed by the Darcy law via the pressure $p(n)=n^{\gamma}$. For finite $\gamma$, we prove some a priori estimates of the tumor growth model, such as boundedness of the nutrient density, and non-negativity and growth estimate of the tumor density. As $\gamma \rightarrow \infty$, the cell density models formally converge to Hele-Shaw flow models, which determine the free boundary dynamics of the tumor tissue in the incompressible limit. We derive several analytical solutions to the Hele-Shaw flow models, which serve as benchmark solutions to the geometric motion of tumor front propagation. Finally, we apply a conservative and positivity preserving numerical scheme to the cell density models, with numerical results verifying the link between cell density models and the free boundary dynamical models.