Tensor based analysis of diffusion weighted magnetic resonance images

Diffusion Weighted Magnetic Resonance Imaging (DW-MRI) is a non-invasive and in vivo medical imaging technique that allows neural tissue architecture to be probed at a microscopic scale. This is possible due to the diffusion of hydrogen atoms within water molecules in the imaging body; thus capturing the microstructure of the underlying tissues. DW-MRI adds to conventional MRI the capability of measuring this diffusion of water molecules by applying strong magnetic field along several gradient directions in order to measure the apparent diffusion coefficients along those directions. In this thesis, we look at modeling diffusion of water molecules with Cartesian Tensors: a model known as Diffusion Tensor Magnetic Resonance Imaging (DT-MRI). We begin with 2nd order tensor model which results in an image where at each voxel the preferred direction of diffusion is locally modeled by a 3 x 3 symmetric positive definite matrix whose coefficients are estimated from the DW-MR data. After briefly reviewing anisotropy and distance measures of 2nd order tensors, we extend these ideas to develop a novel anisotropy measure. Tensor distance measures are then used to extend scalar image segmentation algorithms in order to segment tensor images. Next, we present a clinical application of DT-MRI to investigate various features of white matter fiber tracts in the cortico-striatal region of the brain for the diagnosis of Parkinson's disease. Finally, we investigate the limitations of the 2nd order tensor model and extend it to higher order tensors in order to correctly depict crossing fiber tracts. In particular, we develop a new technique to model fiber orientation distribution functions using higher order tensors and develop a novel anisotropy measure derived directly from fiber orientation distribution functions.