A mathematical and computational framework for modelling epithelial cell morphodynamics

Mathematical and computational modelling provides a useful framework within which to investigate the organisation of biological tissues. As experimental biologists generate increasingly detailed descriptions of cellular behaviour, models that consider cells as discrete entities have become a common tool to study how cell-level processes affect collective dynamics, form and function at the tissue level. To date, however, models incorporating detailed biophysical descriptions of cell shape dynamics have gained little traction among the modelling community. Few model implementations are publicly available, and there often remains no comprehensive account of their method of solution, computational implementation, or analysis of parameter scaling, hindering our ability to utilise such models in practice. In addition, the quality of software underpinning such models (and academic research more widely) is coming under increasing scrutiny, with a growing recognition of the need for correct, reliable and sustainable research software tools. This thesis aims to address these needs for one such cell-based modelling approach, the immersed boundary method. We develop and analyse the immersed boundary method and provide an efficient, open source implementation for simulating cell populations. The implementation is undertaken within Chaste, an open source C++ library that allows one to easily change constitutive assumptions, and is designed and built with software best practices in mind. We explore such best practices and refine and add to the infrastructure and functionality of Chaste. We then carefully compare the immersed boundary method with the vertex model, a competing cell-based modelling approach for epithelial tissues, clearly elucidating relative strengths and weaknesses of the immersed boundary method. This furthers our understanding of the circumstances under which the immersed boundary method and similar cell-based approaches are an appropriate computational tool. Finally, to demonstrate the efficacy of the immersed boundary method, we investigate the mechanics underpinning a novel form of epithelial bending, advancing our understanding of the formation of placodes, structures of epithelial thickening in the cranial region of developing embryos. Together, the contributions in this thesis advance the use of reproducible software development approaches in cell-based modelling.