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.Thesis (Ph.D.) -- University of Adelaide, School of Mathematical Sciences, 201
