Analysis and Derivation of Continuum 3D Blebbing Model with Simulations

In this thesis we attempt to give a strong mathematical basis to cellular blebbing models. After outlining the biology and literature surrounding these phenomena we derive a continuum model for a 3D cell. This derivation results in a fourth order parabolic equation with nonlinear second and lower order terms. Using an operator splitting we obtain a system of second order equations. These can be approximated using only linear finite elements. The model generalizes to an abstract formulation. In this general semi-discrete scheme (continuous in time and discrete in space) we can show weak well posedness and show a priori error estimates. This includes formulating a finite element approximation, proving stability bounds then showing the convergence of the discrete to its limiting case. We formulate a fully discrete scheme and include a stability proof which allows us to implement the problem in Dune-fem. We can then showcase the implementation by showing the convergence rate of the error on two different surfaces. Finally, we round off the thesis by showing some more biologically focused examples. Here we use our implementation in two cases and look at the effects of parameters on potential bleb formulations. These use 3D surfaces with no restriction that they be of a certain shape. We also showcase the adaptability of our implementation by applying it to cell image data. We explain how this can be run using Dune-fempy software which offers a lower barrier for entry into creating and editing the scheme. This potentially gives a bridge to test how successful our model or similar models are in modelling blebs