Statistical Analysis of Diffusion Tensor Imaging

This thesis considers the statistical analysis of diffusion tensor imaging (DTI). DTI is an advanced magnetic resonance imaging (MRI) method that provides a unique insight into biological microstructure \textit{in vivo} by directionally describing the water molecular diffusion. We firstly develop a Bayesian multi-tensor model with reparameterisation for capturing water diffusion at voxels with one or more distinct fibre orientations. Our model substantially alleviates the non-identifiability issue present in the standard multi-tensor model. A Markov chain Monte Carlo (MCMC) algorithm is then developed to study the uncertainty of the model parameters based on the posterior distribution. We apply the Bayesian method to Monte Carlo (MC) simulated datasets as well as a healthy human brain dataset. A region containing crossing fibre bundles is investigated using our multi-tensor model with automatic model selection. A diffusion tensor, a covariance matrix related to the molecular displacement at a particular voxel in the brain, is in the non-Euclidean space of 3x3 positive semidefinite symmetric matrices. We define the sample mean of tensor data to be the Frechet mean. We carry out the non-Euclidean statistical analysis of diffusion tensor data. The primary focus is on the use of Procrustes size-and-shape space. Comparisons are made with other non-Euclidean techniques, including the log-Euclidean, Riemannian, Cholesky, root Euclidean and power Euclidean methods. The weighted generalised Procrustes analysis has been developed to efficiently interpolate and smooth an arbitrary number of tensors with the flexibility of controlling individual contributions. A new anisotropy measure, Procrustes Anisotropy is defined and compared with other widely used anisotropy measures. All methods are illustrated through synthetic examples as well as white matter tractography of a healthy human brain. Finally, we use Gine’s statistic to design uniformly distributed diffusion gradient direction schemes with different numbers of directions. MC simulation studies are carried out to compare effects of Gine’s and widely used Jones' schemes on tensor estimation. We conclude by discussing potential areas for further research.