Cell migration is one of the most intriguing areas in cell biology, because it is crucial to many biological processes such as chemotaxis, inflammatory responses, wound healing and cancer metastasis. To model the migration of cells, one has to understand the subtle interplay between quite a few factors, including intracellular biochemical reactions, cell mechanics, as well as cell morphology. The phase field method has turned out to be a very good framework for constructing comprehensive models of cell motility, for it enables modelers to track the boundary of cells implicitly, thus keeping the computational cost quite affordable, even with plenty of model components in consideration. Previous modeling efforts with phase field models have captured many behaviors of individual "Mensenchymal" type cells in motion. In the first part of my thesis, I will introduce our work， which extends previous phase field models, by modeling small groups of interacting cells instead of single cells, and by modeling cells with very irregular shapes instead of cells with relatively stable shapes. For the cell-interaction scenario, we will explore how different polarity mechanisms may regulate the behavior of cell groups trapped in square or round micropatterns, and lead to persistent rotations. While in the case of irregular cell morphology, we will allow our phase field cells to undergo "Amoeboid" type of deformation by embedding some delicate chemical reaction advection diffusion systems into the cell membrane.

Equally important to understanding what's happening in moving cells, we must precisely measure the cellular forces onto the extra-cellular matrix (ECM). Recently, 3D traction force microscopy (TFM), with cells embedded in fibrous biopolymer networks, is increasingly popular in experiments. However, a nontrivial problem remains: how can one recover the cellular forces with the deformation of biopolymer networks. So, the second part of my thesis will focus on our work in exploring possible obstacles in the way of doing a precise force recovery. Our work is based on some network model of fibrous biopolymers in soft matter physics.