Theory and applications of motions of anisotropic interfacial networks

考虑到受体积保留约束的各向异性表面能梯度流动，提出了一种基于水平集的界面网络各向异性运动实现方法。首先，我们考虑了一种带有惩罚技术的矢量型Merriman-Bence-Osher （MBO）阈值方案的变分方法，该方案通过平滑技术实现了保持体积的晶体运动，但需要一个特殊的方案来满足正确的连接角。在感觉上皮细胞模式形成的激励下，我们考虑了由细胞-细胞粘附驱动的各向异性运动。由于三细胞连接的复杂性，我们随后开发了一种具有局部拍卖动态的mbo型卷积方案，该方案实现了正确的细胞接触角，并具有与水平集表示的良好兼容性以及坚实的数学背景，包括稳定性和收敛性。在这里，细胞-细胞连接使用水平集隐式表示，后续动态由阈值确定。我们解决了与细胞生物学相关的几个方面，例如，我们的定位步骤可以防止细胞在重排过程中发生非自然分裂。我们的方法成功地再现了感觉系统中细胞模式的发展，包括细胞分选、完全吞噬和细胞内化。采用C语言和Matlab软件进行仿真。研究结果已提交出版，并在本地和海外会议上发表，并受到好评。

Considering the gradient flow of anisotropic surface energy constrained by volume preservation, we developed a level set-based method for realizing anisotropic motion of an interface network. First, we considered a variational approach to a vector-type Merriman-Bence-Osher (MBO) thresholding scheme with penalization technique, which realized volume-preserving crystalline motion with a smoothing technique but required an ad hoc scheme to satisfy the correct junction angles. Motivated by cellular pattern formations in sensory epithelium, we considered anisotropic motions driven by cell-cell adhesion. Due to the complexity of tricellular junctions, we then developed an MBO-type convolution scheme with localized auction dynamics which realizes correct cell contact angles and features good compatibility with level set representation and a solid mathematical background including stability and convergence. Here, cell-cell junctions are implicitly represented using level sets and succeeding dynamics are numerically determined by thresholding. We addressed several aspects pertinent to cell biology, e.g., our localization step prevents cells from unnatural splitting during rearrangement. Our method successfully reproduced development of cellular patterns in sensory systems, including cell sorting, total engulfment, and cell internalization. These simulations were implemented using C programming language and Matlab software. Results have been submitted for publication, as well as, presented and well received in local and overseas conferences.