Numerical methods are developed for the study of cell migration and chemotaxis with specific focus on reaction-diffusion based models on moving domains. This thesis focuses building on the previous model of cell migration and chemotaxis introduced by Neilson et al . It does this by introducing new and discussing existing numerical methods in a cell migration and chemotaxis context. A new approach to the solution of forced mean curvature flow is introduced in the form of a moving mesh partial differential equation (MMPDE). This MMPDE is split into two partial differential algebraic equations: one for the normal velocity and one for the tangential velocity. We examine curves moving with a prescribed normal velocity and this decoupling of the velocity components allows mesh adaption to be considered along the tangential component by means of a mesh adaption monitor function. This new method is not restricted to cell migration models and could be used in other contexts where mesh adaption of an evolving curve domain would be desired. A second-order conservative ALE-FEM scheme is also introduced for the solutions of reaction-diffusion equations on an evolving curve boundary and is subsequently used to derive the concentrations of various chemicals which lay on the model of the cell's membrane. The cell migration model is also extended to higher dimensions using a second ALE-FEM scheme which couples the solutions of reactions taking place on different domains together by means of a flux boundary condition and a two-dimensional mesh movement strategy which complements the new one-dimensional MMPDE approach for curves. This scheme is used to model diffusion of ligand molecules in the region exterior to the moving cell. Simulations are presented of this new two-dimensional model indicating the effect of cell movement and receptor-ligand binding dynamics on the cell micro-environment and research is currently underway to use this model to investigate biological theories [84, 86].
|Date of Award||20 Apr 2018|
- University Of Strathclyde
|Sponsors||EPSRC (Engineering and Physical Sciences Research Council)|
|Supervisor||John MacKenzie (Supervisor) & Steven Webb (Supervisor)|