Metabonomics is one of the main technologies used in biomedical sciences to improve understanding of how various biological processes of living organisms work. It is considered a more advanced technology than e.g. genomics and proteomics, as it can provide important evidence of molecular biomarkers for the diagnosis of diseases and the evaluation of beneficial adverse drug effects, by studying the metabolic profiles of living organisms. This is achievable by studying samples of various types such as tissues and biofluids. The findings of a metabonomics study for a specific disease, disorder or drug effect, could be applied to other diseases, disorders or drugs, making metabonomics an important tool for biomedical research. This thesis aims to review and study various multivariate statistical techniques which can be used in the exploratory analysis of metabonomics data. To motivate this research, a metabonomics data set containing the metabolic profiles of a group of patients with epilepsy was used. More specifically, the metabolic fingerprints (proton NMR spectra) of 125 patients with epilepsy, of blood serum type, have been obtained from the Western Infirmary, Glasgow, for the purposes of this project. These data were originally collected as baseline data in a study to investigate if the treatment with Anti-Epileptic Drugs (AEDs), of patients with pharmacoresistant epilepsy affects the seizure levels of the patients. The response to the drug treatment in terms of the reduction in seizure levels of these patients enabled two main categories of response to be identified, i.e. responders and the non-responders to AEDs. We explore the use of statistical methods used in metabonomics to analyse these data. Novel aspects of the thesis are the use of Self Organising Maps (SOM) and of Fuzzy Clustering Methods to pattern recognition in metabonomics data. Part I of the thesis defines metabonomics and the other main "omics" technologies, and gives a detailed description of the metabonomics data to be analysed, as well as a description of the two main analytical chemical techniques, Mass Spectrometry (MS) and Nuclear Magnetic Resonance Spectroscopy (NMR), that can be used to generate metabonomics data. Pre-processing and pre-treatment methods that are commonly used in NMR-generated metabonomics data to enhance the quality and accuracy of the data, are also discussed. In Part II, several unsupervised statistical techniques are reviewed and applied to the epilepsy data to investigate the capability of these techniques to discriminate the patients according to their type of response. The techniques reviewed include Principal Components Analysis (PCA), Multi-dimensional scaling (both Classical scaling and Sammon's non-linear mapping) and Clustering techniques. The latter include Hierarchical clustering (with emphasis on Agglomerative Nesting algorithms), Partitioning methods (Fuzzy and Hard clustering algorithms) and Competitive Learning algorithms (Self Organizing maps). The advantages and disadvantages of the different methods are examined, for this kind of data. Results of the exploratory multivariate analyses showed that no natural clusters of patients existed with regards to th eir response to AEDs, therefore none of these techniques was capable of discriminating these patients according to their clinical characteristics. To examine the capability of an unsupervised technique such as PCA, to identify groups in such data as the data based on metabolic fingerprints of patients with epilepsy, a simulation algorithm was developed to run a series of experiments, covered in Part III of the thesis. The aim of the simulation study is to investigate the extent of the difference in the clusters of the data, and under what conditions this difference is detectable by unsupervised techniques. Furthermore, the study examines whether the existence or lack of variation in the mean-shifted variables affects the discriminating ability of the unsupervised techniques (in this case PCA) or not. In each simulation experiment, a reference and a test data set were generated based on the original epilepsy data, and the discriminating capability of PCA was assessed. A test set was generated by mean-shifting a pre-selected number of variables in a reference set. Three methods of selecting the variables to meanshift (maximum and minimum standard deviations and maximum means), five subsets of variables of sizes 1, 3, 20, 120 and 244 (total number of variables in the data sets) and three sample sizes (100, 500 and 1000) were used. Average values in 100 runs of an experiment for two statistics, i.e. the misclassification rate and the average separation (Webb, 2002) were recorded. Results showed that the number of mean-shifted variables (in general) and the methods used to select the variables (in some cases) are important factors for the discriminating ability of PCA, whereas the sample size of the two data sets does not play any role in the experiments (although experiments in large sample sizes showed greater stability in the results for the two statistics in 100 runs of any experiment). The results have implications for the use of PCA with metabonomics data generally.

This thesis endeavours to develop and implement new and improved methods for the characterisation of defects embedded in steel welds through the analysis of data collected by ultrasonic phased array inspections. A factor common to the existing imaging techniques used for flaw characterisation is the subjective thresholding required to estimate the size of the flaw. The work contained in this thesis uses the mathematics of inverse problems and scattering theory to extract information about such defects and puts forward an objective approach which employs a mathematical model. A relationship between the pulse-echo response curve of a scattering matrix and the size and orientation of a flaw is derived analytically via the Born approximation and results in a completely objective approach to crack sizing. Further expansion of these relationships allows for expressions to be formulated concerning the minimum resolvable crack length and the effects of array pitch and flaw depth on the accuracy of the algorithm. The methodology is then extended and tested on experimental data collected from welded austenitic steel plates containing a lack of fusion crack. In the latter part of this thesis, work focusses on the exploration of the fractional Fourier transform and coded excitations. The fractional Fourier transform allows for retention of both time and frequency domain information simultaneously and permits the in homogeneous wave equation (with a forcing function prescribed as a linear chirp modulated by a Gaussian envelope) to be solved in time-frequency space. This in turn facilitates a comparison between a gated continuous wave excitation and a Gaussian modulated linear chirp. It is observed that the Gaussian modulated linear chirp results in a marked increase in the scattering amplitude.

In this thesis the evolution of sessile droplets in different modes of evaporation and their lifetimes are investigated. The thesis focuses on situations in which the diffusion of vapour into the surrounding atmosphere is the rate-limiting mechanism of evaporation. First, we describe the evolution of droplets evaporating in the two extreme modes, namely the constant contact radius mode, in which the contact line of the droplet is always pinned, and the constant contact angle mode, in which the contact line of the droplet is always de-pinned. In particular, we demonstrate how these two modes converge on strongly hydrophobic substrates. Next we study the evolution of droplets evaporating in the stick-slide mode, in which the contact line is initially pinned and the contact angle decreases to the receding contact angle, but thereafter the contact line is de-pinned and the contact radius decreases to zero. The lifetimes of droplets evaporating in the stick-slide mode are investigated in two situations, namely when the initial and receding contact angles are independent and when there is a simple relationship between them based on the assumption of a constant maximum pinning force. In particular, it is shown that the lifetimes of droplets evaporating in this mode may be longer than those of initially identical droplets evaporating in the two extreme modes. Finally, we develop a model for the evolution of droplets evaporating in a stick-jump mode, in which the contact line pins, de-pins and re-pins multiple times. It is shown that the lifetimes of droplets evaporating in this mode may be longer or shorter than those of initially identical droplets evaporating in the two extreme modes. Good agreement is found between the predicted lifetimes of droplets in both the stick-slide and the stick-jump modes and the lifetimes of droplets determined from relevant experiments in the literature.

Synchronisation is an important process for different kinds of systems, such as biological, chemical, physical and social. Among the related synchronisation problems, consensus has received high attention because of the distributed properties shown by its models and the possibility they offer for controlling complex systems. When dealing with consensus processes in social networks, we known from empirical evidence that the formation of opinions is not free from being influenced by people around every actor, and more, it is well known that some of the actors may play a leading role and guide a social system to a final state different from the pure average consensus. A main paradigm while modelling interactions among actors in social networks is that every actor receives and transmits information from and to her nearest neighbours, thus implicitly assuming that the decisions of a given actor only are influenced by their directly connected peers, and not tking into account indirect influences coming from not directly connnected peers in the same social network, for example, the influence coming from the friend's friend of a friend. Our work studies consensus processes in the presence of influence coming from not only those directly connected actors, but from other ones in the same network. We call this influence peer pressure (PP). We propose a consensus model that takes into account direct and indirect PP modelled as a function of the social distance among actors. We apply this consensus model to different real social networks assuming three different decay laws for the strength of PP, and in the presence of leaders and without them. We choose those nodes acting as leaders according to different centrality criteria, as well as randomly, and compare thier performance for driving the system. Since it is natural that different leaders may diverge in their positions, we introduce a divergence parameter among the initial states of the leaders with respect to the avreage consensus of the system, to take the feature into account in our model. We then analyse the effects of PP on two different real cases of diffusion of innovation processes. We show that as the strength of indirect PP increases, the centrality criteria used to select the leaders has a decaying effect on the effectiveness of such leaders to better drive a consensus process, allowing random leaders to be as good as those with better centrality. Our work also shows that, despite divergence among leaders induces higher times for reaching consensus, this effect is reduced for stronger levels of PP present in the system. For the case of diffusion innovations our model reproduces the behaviour of the empirical data, and we demonstrate that certainlevels of PP are necessary to match the results coming from two different studies, supporting our hypothesis that indirect PP is an important factor to be taken into account when modelling opinion formations in social networks. Leaders emerging by global centrality criteria in networks with tightly connected groups can be counterproductive. This can be tackled by selecting node-leaders in a local basis. This effect is also reduced when indirect PP is allowed to be higher. This finding points to the fact that distance among nodes is an important characteristic for consenus processes. For the purpose of studying this structural feature, we propose a distance-sum heterogeneity index based on a fictional consensus process. We conjecture that an special type of graph, that we call complete split graph, is related with the maximization of the index, and based on this conjecture we study the relative distance-sum heterogeneity of random graphs and different real-world networks, which allows us to characterise them. We propose a spectral representation of the distance-sum heterogeneity index for networks that we call S-plots. We also study the relation between the time for consensus and the distance-sum heterogeneities in complex networks from different nature.

In this thesis a Finite Element solver package called FEDomain is developed for C++ finite element software developers. It is focused on solving the Finite Element problem on shared memory as well as distributed memory architectures. The FEDomain package segregates the finite element software into two phases. The first phase includes defining the finite element problem. The user selects the mathematical problem, domain shape, domain dimensions, triangulation of the domain and formulations to compute elements' data. The second phase comprises assembly of system of equations and computing its solution. The FEDomain package concentrates on the second phase. It facilitates the user by providing the efficient implementation of the second stage using parallel algorithms. This design allows the C++ finite element application developers, with no knowledge and experience of parallel computing, to implement parallel finite element application for shared and distributed memory architectures. More specifically, FEDomain package is focused on introducing a new type of user interface. The interface requires the user to provide the mathematical problem and domain related data in terms of C++ element objects. The FEDomain assembles the system of equations, computes its solution, and provides it back to the user through element objects. The FEDomain package computes the residual vector and solution for the system of equations on shared memory and distributed memory architectures.

The mountain hare population currently appears to be under threat in Scotland. The natural population cycles exhibited by this species are thought to be, at least in part, due to its infestation by a parasitic worm. We seek to gain an understanding of these population dynamics through a mathematical model of this system and so determine whether low population levels observed in the field are a natural trough associated with this cycling, or whether they point to a more serious decline in overall population densities. A generic result, that can be used to predict the presence of periodic travelling waves (PTWs) in a spatially heterogeneous system, is reported. This result is applicable to any two population host-parasite system with a supercritical Hopf bifurcation in the reaction kinetics. Application of this result to two examples of well studied host-parasite systems, namely the mountain hare and the red grouse systems, predicts and illustrates, for the first time, the existence of PTWs as solutions for these reaction advection diffusion schemes. One method for designing bone scaffolds involves the acoustic irradiation of a reacting polymer foam resulting in a final sample with graded porosity. The work in this thesis represents the first attempt to derive a mathematical model, for this empirical method, in order to inform the experimental design and tailor the porosity profile of samples. We isolate and study the direct effect of the acoustic pressure amplitude as well as its indirect effect on the reaction rate. We demonstrate that the direct effect of the acoustic pressure amplitude is negligible due to a high degree of attenuation by the sample. The indirect effect, on reaction rate, is significant and the standing wave is shown to produce a heterogeneous bubble size distribution. Several suggestions for further work are made.

One of the biggest challenges in building intricate realistic real world models is to incorporate data and their subsequent analysis. When analysing such a system, researchers typically only use one of two modelling methodologies. The first modelling methodology is “keep it simple stupid” (KISS), which aims to capture the simplified, sometimes extremely abstract behaviours of the real world. By not including all of the intricate features of a system, leads to the problem of having to justify an abstract model to represent the real world and for the results to be verified through theoretical reasoning. However, this method is often easier to construct and yields a clear overview of the system’s behaviour. The second modelling approach is the “keep it descriptive stupid” (KIDS) approach that aims to include more vital features or behaviours of a system. The justification of using these highly descriptive models is easier, as it captures more intricate behaviours, but are often significantly more difficult to build and to analyse. This thesis shows that by using the KISS methodology to analyse the system as a whole, vital information about the build of the KIDS model, i.e. which behaviours need to be simulated, can be obtained. This simplifies the process for building the KIDS model and ensures that the general behaviour of the system is included. The KIDS model is then used to analyse how the intricate behaviours influence the system. I demonstrate this approach on two case studies, where the first investigates how impacts such as a leader’s reputation and family’s party preference influencing an individual voter alters the re-election rate of a leader or party. The second case study analyses how policies impact the UK phosphorus and nitrogen flows.

This thesis may be divided into two related parts. The first of which considers a population balance approach to modelling a population of cells, with particular emphasis on how the cells pass between the G1 and S phases of the cell cycle. In the second part of the thesis a model is described which combines a cell cycle model with a simple Pharmacokinetic/Pharmacodynamic (PKPD) drug model. This model is then discussed in detail. Knowledge of how a population of cancerous cells progress through the cell cycle is vital if the population is to be treated effectively, as treatment outcome is dependent on the phase distributions of the population. Estimates on the phase distribution may be obtained experimentally however the errors present in these estimates may effect treatment efficacy and planning. In this thesis mathematical models are used to explore the factors that effect the phase distributions of the population. In this thesis it is shown that two different transition rates at the G1-S checkpoint provide a good fit to a growth curve obtained experimentally. However, the different transition functions predict a different phase distribution for the population, but both lying within the bounds of experimental error. Since treatment outcome is effected by the phase distribution of the population this difference may be critical in treatment planning. Using an age-structured population balance approach the cell cycle is modelled with particular emphasis on the G1-S checkpoint. By considering the probability of cells transitioning at the G1-S checkpoint, different transition functions are obtained. A suitable finite difference scheme for the numerical simulation of the model is derived and shown to be stable. The model is then fitted using the different probability transition functions to experimental data and the effects of the different probability transition functions on the model's results are discussed. In contrast to the population balance approach a more simplistic compartmental model is also considered. This model results in a system of linear ordinary differential equations which, under specific circumstances may be solved analytically. It is shown that whilst not as accurate as the population balance model this model provides an adequate fit to experimental data with the results for the total cell population lying within the bounds of experimental error. The ODE compartment model is combined with a simple PKPD model to allow a detailed analysis of the equations for this combined model to be undertaken for different drug-cell interactions. These results are then discussed. As a tumour grows many of the cells receive oxygen and nutrients from blood vessels formed within the tumour, these provide a less than ideal supply, resulting in areas that are well perfused, hypoxic and necrotic. In hypoxic regions the lack of oxygen and nutrients limit the cells' growth by increasing their cycle time and also reducing the effects of radiation and chemotherapy. In the conclusion of this thesis the idea of separating a tumour into three regions, normoxic, hypoxic and necrotic is discussed. Each of these regions may then be modelled using three coupled compartments, each of which contain a cell cycle model, modelled using a set of ordinary differential equations. Additionally, the interaction of a simple (PKPD) drug model with these populations of cells may be considered.

The world faces an immense burden of hepatitis C virus (HCV) infection related morbidity and mortality. Transmission of HCV is ongoing, and the incidence of HCV infection has been increasing in recent years. Approximately 130-150 million people are estimated to be chronically infected with HCV and each year an estimated three to four million individuals are newly infected (WHO, 2013; Mohd Hanafiah et al., 2013). In developed countries, injecting drug users are considered as being at the highest risk of prevalence of HCV. Thus, this thesis describes the spread of HCV amongst injecting drug users. We use a mathematical model to study the effect of heterogeneity on the progress of the disease by dividing the population of addicts into p groups where they are sharing injecting needles in q shooting galleries and investigate the epidemic behavior of the virus. Moreover, we estimate the basic reproductive number R₀ and show analytically that HCV is controlled by this number R₀, if R₀ ≤ 1 then the disease dies out and if R₀ > 1 the disease takes off in both addicts and needles and there is a unique endemic equilibrium. We look at analytical results on the effect of heterogeneity on the spread of HCV and optimal control of the epidemic by needle exchange and needle cleaning. Simulations with realistic parameter values estimated from data and the literature confirm the theoretical results and we numerically investigate the effect of heterogeneity on the spread of HCV. Then we extend the basic model to more realistic assumptions where addicts move in and out of groups, and investigate the HCV dynamic behaviour. We obtain similar analytical results again validated by simulations with realistic parameter values estimated from data and the literature.

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