Spatial quantification and mathematical modelling of tissue development

Abstract

In this thesis, we study biological tissue development, during which cells organise themselves into structures which perform a specific function. Understanding how particular types of mechanisms lead to the emergence of various cell patterns in tissues is the main motivation of this research. Quantifying the tissue patterns is a first step towards understanding which mechanisms are at work in particular experiments. For this purpose, we develop pair-correlation functions (PCFs) which quantify how a spatial distribution of cells deviates from complete spatial randomness over specified directions. We evaluate the usefulness of PCFs for studying the three-dimensional organisation of cells in tumour spheroids and show that the PCFs robustly reveal information about their spatial structure. In particular, we demonstrate that the boundary that separates the necrotic and viable zones in the tumour spheroids can be detected using the PCF with a high degree of accuracy. We then turn to development of mathematical models to investigate the types of patterns that can arise from simple hypothesised interactions between cells. We begin in Chapter 3 by developing an on-lattice agent-based model (ABM) to investigate tumour spheroid growth using two different culture methods: suspension culture, and culture within a microgel. Our results suggest that stratifying the seeded cells into multiple layers and also reducing cell death are the key effects of the microgel that enable it to produce more uniformly-sized spheroids. In Chapter 4, we extend the ABM to study systems with two interacting species. A huge variety of aggregation patterns can arise in these systems, depending upon the underlying attractive-repulsive mechanisms. More specifically, we show that the run-and chase mechanism can produce a striped pattern, similar to that observed on the skin of zebrafish. Finally, we develop a non-local continuous model, approximating the mean behaviour of the ABM. This provides a connection between the cell-level and population-level models of tissue development. A linear stability analysis of the continuous model allows us to investigate parameter regimes that produce striped patterns. Importantly, we also point out the disparities that may arise between the behaviours of the continuous and discrete models, which highlights the importance of considering the underlying biological constraints in using the continuous approximated models. In particular, we show that the derivation of the approximate continuum model from the ABM introduces terms representing cell-size effects. These terms can lead to the emergence of stripes in cases where they would not be predicted in the similar continuum model of Painter et al. (2015), which does not include these terms. The combination of spatial quantification and mathematical modelling (using both continuous and discrete methods) developed in this work helps us to gain a better understanding of tissue development. Our approach provides a novel means to investigate the underpinning mechanisms of tissue development by combining model simulations with analysis of biological and synthetic data using the pair-correlation functions.