广义Cahn-Hilliard方程在生物学中的应用
A generalized Cahn-Hilliard equation for biological applications
摘要：
最近，我们考虑了一个随机离散模型，它描述了正面的细胞侵入伤口\cite{KSS}。在该模型中，细胞可以移动、增殖并经历细胞-细胞粘附。在这项工作中，我们专注于一个连续描述这种现象的广义Cahn-Hilliard方程（GCH）与增殖项。在离散模型中，有两个有趣的制度。对于亚临界粘附，有传播的“拉”前，类似于那些的Fisher-Kolmogorov方程。前速度选择的问题进行检查，我们的理论预测是在一个很好的协议与GCH方程的数值解。对于超临界粘附，有一个非平凡的瞬态行为，其中密度分布表现出第二峰。为了分析这一制度，我们研究了无增殖的Cahn-Hilliard方程的弛豫动力学。我们发现，弛豫过程表现出自相似行为。对于我们分析的不同区域，连续模型和离散模型的结果彼此吻合得很好。
英文摘要：
Recently we considered a stochastic discrete model which describes fronts of cells invading a wound \cite{KSS}. In the model cells can move, proliferate, and experience cell-cell adhesion. In this work we focus on a continuum description of this phenomenon by means of a generalized Cahn-Hilliard equation (GCH) with a proliferation term. As in the discrete model, there are two interesting regimes. For subcritical adhesion, there are propagating "pulled" fronts, similarly to those of Fisher-Kolmogorov equation. The problem of front velocity selection is examined, and our theoretical predictions are in a good agreement with a numerical solution of the GCH equation. For supercritical adhesion, there is a nontrivial transient behavior, where density profile exhibits a secondary peak. To analyze this regime, we investigated relaxation dynamics for the Cahn-Hilliard equation without proliferation. We found that the relaxation process exhibits self-similar behavior. The results of continuum and discrete models are in a good agreement with each other for the different regimes we analyzed.
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