Modelling Complex-Valued Diffusion Tensor Imaging Data and Efficient Methods for Inference

The main focus of this project is the modelling and analysis of brain structure using Diffusion Tensor Imaging (DTI) measurements. DTI is an in vivo medical imaging technique, based on Magnetic Resonance Imaging (MRI) technology, that captures the diffusion of water molecules in tissue. The impediment of this diffusion process by nerves enables the characterisation of white matter structure and the measurement of quantitative descriptions of white matter integrity. DTI has identified white matter alterations for a large number of conditions including Alzheimer's disease, Parkinson's disease, schizophrenia, neurological complications of HIV infection, autism, multiple sclerosis etc. The potential of DTI to generate imaging biomarkers for disease progression opens the door to applications in the pharmaceutical industry for drug discovery and development. DTI stands as one of the most important new technologies that will help us to improve our understanding of the complex structure of the brain. DTI data takes the form of a complex-valued quantity but current practice in the analysis of such data involves naively converting the complex-valued data into a real-valued data set. This procedure is especially detrimental in the analysis of very noisy data sets, and may yield very poor analysis methods. To improve estimation both the amplitude and phase of the complex-valued data should be modelled and used. Once the complex-valued data is treated appropriately, estimation may be very much improved upon.By developing better estimation procedures the number of subjects required to find statistically significant changes between treatment groups, may be reduced. The development of more powerful inferential procedures with estimators that have better sensitivity and specificity may aid, for example, in the early identification of a reduction in atrophy using DTI data, and be utilised for routine characterisation of disease progression. Modern signal processing methods will also be used at two different levels, namely locally at a given location (voxel), and regionally to estimate structure across locations. Locally, the usage of multiscale methods will enable modelling the diffusion without artificial constraints in terms directional preference: in observed data brain fibres may cross, kiss and/or fork at a given voxel, and this must be incorporated in the model.Regionally methods will be developed to characterise structural features observed in the brain, for instance by designing fibre dictionaries, i.e. local decompositions that incorporate the characteristics of brain fibres and nerves. These decompositions can be used to estimate the presence of highly specific features, and enable good estimation even at very low signal to noise ratios. Furthermore as structural features are naturally represented in this framework, it is straightforward to test for potential structural degradation in longitudinal measurements, especially relevant for understanding the development of diseases exhibiting degradation such as Alzheimer's.The proposed methods are then the culmination of a research programme, starting from the basic problem of modelling the amplitude and phase of the DTI measurement in a given direction, building a coherent likelihood for the amplitude and phase, modelling the local structure whilst allowing for complicated fibre structure, and finally producing a model for the entire brain structure across voxels, where the latter can be used to answer questions of human physiognomy. The full tool-kit of methods represents a synthesis of state of the art developments in signal processing, statistics and MRI, and will help answer important physiological question of human disease progression.