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Integropartial differential equation models for cellcell adhesion and its application

 Author / Creator
 Buttenschoen, Andreas

In both health and disease, cells interact with one another through cellular adhesions. Normal development, wound healing, and metastasis all depend on these interactions. These phenomena are commonly studied using continuum models (partial differential equations). However, a mathematical description of cell adhesion in such tissue models had remained a challenge until 2006, when Armstrong et al. proposed the use of an integropartial differential equation (iPDE) model. The initial success of the model was the replication of the cellsorting experiments of Steinberg. Since then, this approach has proven popular in applications to embryogenesis, wound healing, and cancer cell invasions. In this thesis, I present a first systematic derivation of nonlocal (iPDE) models for adhesive and chemotactic motion. For this purpose, I develop a framework by which nonlocal models can be derived from a spacejump process. I show how the notions of cell motility and cell polarization can be naturally included. The significance of such a derivation is that, it allows me to take celllevel properties such as cellsize, cell protrusion length or adhesion molecule densities into account. Thus this derivation validates these popular nonlocal models. I show that particular choices of these properties yield the original Armstrong cellcell adhesion model. Finally, I extend the cell adhesion model to include volume exclusion, and complex cell adhesion molecule kinetics. In Chapter 3, I present a first in depth analytical study of the steadystates of the nonlocal cell adhesion model derived in this thesis. The importance of steady states is that these are the patterns observed in nature and tissues (e.g. cellsorting experiments). As a prerequisite to the subsequent analysis, I present an indepth study of the properties of the nonlocal cell adhesion operator. I present results on its continuity, compactness, and spectral properties. I then combine bifurcation techniques pioneered by Rabinowitz, equivariant bifurcation theory, and the properties of the equationâs solution, to obtain the existence of an unbounded bifurcation branch of nonhomogeneous solutions. Using the equationâs symmetries, I further classify the solution branches by the derivativeâs number of zeros (i.e., number of extrema remains fixed). Significantly, this parallels the classifications of solutions of nonlinear SturmLiouville problems and equivariant nonlinear elliptic equations. I identify the bifurcation type as pitchfork, and show that integration kernel in the nonlocal term determines whether an immediate switch in the solutionâs stability takes places. Finally, using numerical techniques, I verify my analysis, and demonstrate the existence of rotating waves of steady states. In the final chapter, I extend the nonlocal cell adhesion model to a bounded domain with noflux boundary conditions. In the past, boundary conditions for nonlocal equations were avoided, because their construction is subtle. I encounter three challenges: (1) ensure the nonlocal operator is welldefined near the boundary, (2) ensure that the nonlocal operator satisfies the noflux boundary conditions, and (3) the constructed nonlocal operators are nonunique in the boundary region. I address the first challenge by introducing spatially dependent integration limits. While more complicated, the new nonlocal operators share many of their mathematical properties with the periodic nonlocal operator. However, the spatial dependence of the integration limits complicates differentiation. Indeed, the theory of distributions must be used, to compute the nonlocal operatorâs weak derivative. Using the modelâs derivation, I ensure that the constructed operators satisfy the noflux boundary conditions. Finally, I classify the constructed nonlocal operators, by comparing their action to the periodic nonlocal adhesion operator, into either repellent, neutral, or attractive. It is however, an open problem to match these newly constructed nonlocal operators to cell behaviour near physical boundaries.

 Graduation date
 Spring 2018

 Type of Item
 Thesis

 Degree
 Doctor of Philosophy

 License
 This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for noncommercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.