Mathematical Sciences - Doctoral Theses
http://hdl.handle.net/10468/859
2018-11-20T21:11:00ZStatistical analysis of positron emission tomography data
http://hdl.handle.net/10468/6282
Statistical analysis of positron emission tomography data
Mou, Tian
Positron emission tomography (PET) is a noninvasive medical imaging tool that produces sequences of images describing the distribution of radiotracers in the object. PET images can be processed to evaluate functional, biochemical, and physiological parameters of interest in human body. However, images generated by PET are generally noisy, thereby complicating their geometric interpretation and affecting the precision. The use of physical models to simulate the performance of PET scanners is well established. Such techniques are particularly useful at the design stage as they allow alternative specifications to be examined. When a scanner is installed and begins to be used operationally, its actual performance may deviate somewhat from the predictions made at the design stage. Thus it is recommended that routine quality assurance (QA) measurements could be used to provide an operational understanding of scanning properties. While QA data are primarily used to evaluate sensitivity and bias patterns, there is a possibility to also make use of such data sets for a more refined understanding of the 3-D scanning properties. Therefore, a comprehensive understanding of the noise characteristics in PET images could lead to improvements in clinical decision making. The main goals of this thesis are to develop model-based approaches for describing and evaluating the statistical properties of noise and a practical approach for simulation of an operational PET scanner. We began with the empirical analysis of statistical characteristics—bias, variance and correlation patterns in a series of operational scanning data. A multiplicative Gamma model had been developed for representing the structure of reconstructed PET data. The novel iteratively re-weighted least squares (IRLS) techniques were proposed for the model fitting. These included the use of a Gamma-based probability transform for normalising residuals, which could be used for model diagnostics. Building on the Gamma based modelling and probability transformation, we developed a 3-D spatial autoregressive (SAR) model to represent the 3-D spatial auto-covariance structure within the normalised data. Auto-regressive coefficients were also estimated based on the minimisation of difference between 3-D auto-correlations calculated from the normalised data and model. Both traditional filtered back-projection (FBP) and expectation-maximisation (EM) reconstructions were considered. Numerical simulation studies were carried out to evaluate the performance of the above models. The proposed models led to a very trivial process for simulation of the scanner—one that can be implemented in R. This provided a very practical mechanism to be routinely used in clinical practice—assessing error characteristics associated with quantified PET measures. Moreover, this fast and simplified approach has a potential usage in enhancing the quality of inferences produced from operational clinical PET scanners.
2018-01-01T00:00:00ZHydrodynamic optimisation of an array of wave-power devices
http://hdl.handle.net/10468/6259
Hydrodynamic optimisation of an array of wave-power devices
McGuinness, Justin P. L.
Third generation wave-power devices are usually envisaged as being either a single large device or an array of smaller devices. The benefit of an array, compared to a single device, is that the individual components are relatively inexpensive to repair and replace; however issues arise due to interaction between the array members, which can lead to constructive or destructive interference of the wave-field, thus increasing or decreasing the power that can be absorbed. This thesis is concerned with the optimal formation and design of these arrays of wave-power devices from a hydrodynamic perspective. Previous literature has indicated that a deterministic optimisation of the array layout, which directly maximises the array performance, results in high sensitivity of the optimal performance to incoming wave parameters. This work considers a more robust optimisation, where the mean performance of the array is maximised. Determining the optimal array configuration is associated with numerical optimisation. Previous studies have shown that a balance must be struck between accurately modelling the devices of the array (including their interactions) and the requirement of establishing a reliable optimisation process. Thus, linear wave theory and the point absorber approximation are utilised within this work. Several array geometries are investigated, including linear and circular arrays, along with a general 2D optimisation without any imposed symmetry. Both constrained and unconstrained WEC motions are considered. Regular waves are assumed for the majority of this work, with a preliminary extension to irregular waves also investigated for elementary linear arrays. In general, it is shown that optimal unconstrained arrays tend to contain closely spaced groups of WECs, while constrained arrays are more spread out. A trade-off between peak performance and performance stability is identified for general WEC arrays, while linear arrays also exhibit a trade-off between stability to wavenumber variations and incident wave angle variations. Overall, it is shown that linear arrays perform poorly for some orientations, regardless of the array layout. Better constructive interaction can be achieved in beam seas for unconstrained motions, while head seas allow for the best interaction when WEC motions constraints are applied. As expected, better interaction can be achieved for more general array layouts, without a prescribed geometry.
2018-01-01T00:00:00ZRandom walks on finite quantum groups
http://hdl.handle.net/10468/4033
Random walks on finite quantum groups
McCarthy, J.P.
Of central interest in the study of random walks on finite groups are ergodic random walks. Ergodic random walks converge to random in the sense that as the number of transitions grows to infinity, the state-distribution converges to the uniform distribution on G. The study of random walks on finite groups is generalised to the study of random walks on quantum groups. Quantum groups are neither groups nor sets and rather what are studied are finite dimensional algebras that have the same properties as the algebra of functions on an actual group — except for commutativity. The concept of a random walk converging to random — and a metric for measuring the distance to random after k transitions — is generalised from the classical case to the case of random walks on quantum groups. A central tool in the study of ergodic random walks on finite groups is the Upper Bound Lemma of Diaconis and Shahshahani. The Upper Bound Lemma uses the representation theory of the group to generate upper bounds for the distance to random and thus can be used to determine convergence rates for ergodic walks. The representation theory of quantum groups is very well understood and is remarkably similar to the representation theory of classical groups. This allows for a generalisation of the Upper Bound Lemma to an Upper Bound Lemma for quantum groups. The Quantum Diaconis–Shahshahani Upper Bound Lemma is used to study the convergence of ergodic random walks on classical groups Zn, Z n 2 , the dual group Scn as well as the ‘truly’ quantum groups of Kac and Paljutkin and Sekine. Note that for all of these generalisations, restricting to commutative subalgebras gives the same definitions and results as the classical theory.
2017-01-01T00:00:00ZStatistical considerations in the kinetic analysis of PET-FDG brain tumour studies
http://hdl.handle.net/10468/2779
Statistical considerations in the kinetic analysis of PET-FDG brain tumour studies
Hawe, David
Dynamic positron emission tomography (PET) imaging can be used to track the distribution of injected radio-labelled molecules over time in vivo. This is a powerful technique, which provides researchers and clinicians the opportunity to study the status of healthy and pathological tissue by examining how it processes substances of interest. Widely used tracers include 18F-uorodeoxyglucose, an analog of glucose, which is used as the radiotracer in over ninety percent of PET scans. This radiotracer provides a way of quantifying the distribution of glucose utilisation in vivo. The interpretation of PET time-course data is complicated because the measured signal is a combination of vascular delivery and tissue retention effects. If the arterial time-course is known, the tissue time-course can typically be expressed in terms of a linear convolution between the arterial time-course and the tissue residue function. As the residue represents the amount of tracer remaining in the tissue, this can be thought of as a survival function; these functions been examined in great detail by the statistics community. Kinetic analysis of PET data is concerned with estimation of the residue and associated functionals such as ow, ux and volume of distribution. This thesis presents a Markov chain formulation of blood tissue exchange and explores how this relates to established compartmental forms. A nonparametric approach to the estimation of the residue is examined and the improvement in this model relative to compartmental model is evaluated using simulations and cross-validation techniques. The reference distribution of the test statistics, generated in comparing the models, is also studied. We explore these models further with simulated studies and an FDG-PET dataset from subjects with gliomas, which has previously been analysed with compartmental modelling. We also consider the performance of a recently proposed mixture modelling technique in this study.
2016-01-01T00:00:00Z