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Type: Theses
Title: On the derivation and application of closure approximations of cellular automata models
Author: Davies, Kale James
Issue Date: 2016
School/Discipline: School of Mathematical Sciences
Abstract: We consider discrete agent-based models known as a cellular automata (CA). CA models are lattice-based systems, in which each position on the lattice, referred to as sites, may take on a number of states. The overall dynamics of the CA model are dictated by events which allow the states of lattice sites to change over time based on the states of neighbouring sites. As such, CA models are particularly good at modelling physical systems where the spatial distributions are of interest. In particular, we are interested in exploring CA models that are derived from applications of cellular biology such as tissue and tumour growth or wound healing. As such, we consider a CA model with two possible states: occupied, referring to sites where a cell is present, and vacant, referring to sites where there is no cell present. We then describe two events for our CA model, based on dynamics that are generally present in such cell-based problems: motility and proliferation. A motility event will swap the states of two adjacent sites, corresponding to the movement of a cell from one position to another. A proliferation event will cause a previously vacant site to become occupied due to an interaction with an adjacent occupied site, corresponding to cell division. Initially, each site is designated either an occupied or vacant state. Given a lattice with some initial condition, we wish to know the probability of each site being occupied at some later time. The CA model is a continuous-time Markov chain and as such, a solution to the model exists and can theoretically be obtained. Unfortunately, in order to obtain this solution, we must compute a corresponding system of differential equations based on the size of the state space of the Markov chain. The state space of the Markov chain is based on the number of different configurations of the states of the sites in the CA model, hence for a domain with X sites, we must numerically solve a system of 2X [X superscript] differential equations. As the time taken to compute the solution to this system will increase with the number of equations in the system, it is not computationally viable to solve these problems, even when the total number of sites X is relatively small. In order to solve the system, approximations are made, reducing the number of equations, resulting in a system of differential equations that are computationally manageable. The simplest existing approximation is known as mean-field (MF) theory. By applying MF theory, an assumption of independence is made between the state of each site. This reduces the system to only X differential equations at the expense of no longer modelling the correlation between the states of each site, however we find that this yields inaccurate approximations in most cases. Some more sophisticated approximations have been considered in the literature in which some of the correlation betweens sites is modelled. These are also considered in this thesis. We formalise an approach that allows us to derive a class of approximate systems based on a statistical technique called moment closure. Moment closure aims to describe approximations to statistical distributions using finitely many moments of a system (where in reality there are infinitely many moments). In the same manner, applying a closure approximation to a CA model reduces the system so that we need only consider a much reduced number of differential equations. While the concept of closure approximations has been applied to CA models, pre-existing approaches have had unaddressed issues that prevent them from performing optimally. Our formal approach outlines these issues as they appear by making a distinction between the differential equations governing the physical system and the differential equations governing the approximate system. Our approach is applied to MF theory, as well as the other pre-existing approximations. After demonstrating the functionality of the formal closure approach, we begin to develop our own closure approximations. This allows us to develop and analyse new approximate systems with the goal of accurately approximating the motility and proliferation CA model. By exploring a variety of approximations, we come to understand the desirable and undesirable properties of the closures. By enforcing the desirable properties and rejecting closures with undesirable properties, we are able to restrict our search and develop an approach for deriving closure approximations. We demonstrate this approach, showing some interesting cases before drawing general conclusions about closure approximations when applied to CA models with different rules. Furthermore, we make conclusions about the performance of the closure approximations derived throughout the thesis when applied to the motility and proliferation CA model.
Advisor: Bean, Nigel Geoffrey
Binder, Benjamin
Joshua, Ross
Dissertation Note: Thesis (Ph.D.) -- University of Adelaide, School of Mathematical Sciences, 2016.
Keywords: cellular automata
moment closure
mathematical modelling
cellular biology
Provenance: This electronic version is made publicly available by the University of Adelaide in accordance with its open access policy for student theses. Copyright in this thesis remains with the author. This thesis may incorporate third party material which has been used by the author pursuant to Fair Dealing exceptions. If you are the owner of any included third party copyright material you wish to be removed from this electronic version, please complete the take down form located at:
DOI: 10.4225/55/582cf8d107285
Appears in Collections:Research Theses

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