The growth and dynamics of epithelial cell sheets and self-assembly of cells into tissues play a central role in morphogenesis, tissue renewal and tumour growth. Alongside experimental approaches, mathematical models offer a useful tool for investigating these processes. I use a variety of modelling approaches to address questions in this area, from compartment models to individual cell-based and multiscale models. Below I summarise some of my major research strands to date.

Tissue self-renewal and carcinogenesis in the intestinal epithelium

In normal adult epithelial tissues, cell production, migration and death are tightly regulated by subcellular and long-range signalling, the disruption of which by sporadic mutants is the cause of over 80% of cancers. Modelling can help provide insight into these processes. A particularly important epithelium is that of the intestine, which is folded to form invaginations called crypts and (in the small intestine) protrusions called villi.

Using mouse models of colorectal cancer for parameterization, I have developed a multiscale model of a crypt [1]. This couples a stochastic description of cell proliferation with an individual-based model of cell movement, comprising a large nonlinear ODE system. Taking account of the crypt's curved geometry, I simulated the process of monoclonal conversion, where the progeny of one cell takes over the entire crypt, facilitating the spread of mutations [7]. Comparison with a classical Moran model highlighted the significant effect of the crypt's spatial structure on monoclonal conversion times. I have since extended this model to study the disruption of crypt dynamics due to altered proliferation and adhesion [9]. Varying the mutant 'phenotype' revealed a strong dependence of probability of mutant monoclonal conversion on dysregulation of cell adhesion, and provided strong evidence for the 'bottom-up' theory of crypt invasion. My model also suggested that the 'wiggles' in the width of a clonal ribbon represents a temporal record of that clone's stem cell population dynamics. This observation has been applied to human histological data, enabling us to infer, for the first time, the quantitative dynamics of human intestinal stem cells [20].

Tissue size control in the developing embryo

In the embryo, a 'symphony' of signals regulates the growth of epithelial tissues to ensure the correct overall size, while the coordinated movement of individual cells allows epithelial tissues to fold from flat sheets into complex three-dimensional structures. We lack a quantitative understanding of how cells incorporate external inputs from their surroundings to execute these developmental processes. This has implications for tissue engineering, cancer treatment and regenerative therapies. Models help provide insight into these processes through the connection of multiple levels of description, from gene regulation to pattern formation and tissue deformation.

Guided by molecular and live-imaging data in the fruit fly Drosophila from my experimental collaborators, I have created a computational modelling framework for developing epithelia based on the movement of cell junctions or vertices in a flat sheet [16]. In such vertex models [17], a first-order stochastic differential equation describes the overdamped movement of each vertex in response to nonlinear forces due to interfacial tension and other processes. Rules for cell rearrangements, proliferation and death are specified according to hypothesised mechanisms relating morphogen (diffusible chemical signal) levels and mechanical feedback to apoptosis (programmed cell death) and cell dynamics. Applications of this work include the role of cell competition in tissue size control [20].

Chaste: a consistent computational framework for multiscale modelling of cell populations

There is a variety of individual-based and multiscale modelling frameworks for the spatiotemporal dynamics of cell populations. Which model is best suited to a given problem remains an open question. It is often impossible to distinguish between differences in model behaviour that are due to underlying assumptions, and those due to differences in the in silico implementation of the model.

To address this issue I am a lead developer in the cell-based arm of Chaste, an extensible simulation framework [2, 10]. To support my research, I have created open source implementations of modelling approaches ranging from cellular automata and the cellular Potts model to off-lattice vertex models and subcellular element models. Chaste allows for the comparison of competing mathematical modelling paradigms within a consistent computational framework, so that one can be sure that any differences in the results are due to differences in the mathematics, rather than differences in their instantiation and numerical solution in the software implementation.