Tensor decomposition processes for interpolation of diffusion magnetic resonance imaging

Diffusion magnetic resonance imaging (dMRI) is an established medical technique used for describing water diffusion in an organic tissue. Typically, rank-2 or 2nd-order tensors quantify this diffusion. From this quantification, it is possible to calculate relevant scalar measures (i.e. fractional anisotropy) employed in the clinical diagnosis of neurological diseases. Nonetheless, 2nd-order tensors fail to represent complex tissue structures like crossing fibers. To overcome this limitation, several researchers proposed a diffusion representation with higher order tensors (HOT), specifically 4th and 6th orders. However, the current acquisition protocols of dMRI data allow images with a spatial resolution between 1 mm3 and 2 mm3, and this voxel size is much bigger than tissue structures. Therefore, several clinical procedures derived from dMRI may be inaccurate. Concerning this, interpolation has been used to enhance the resolution of dMRI in a tensorial space. Most interpolation methods are valid only for rank-2 tensors and a generalization for HOT data is missing. In this work, we propose a probabilistic framework for performing HOT data interpolation. In particular, we introduce two novel probabilistic models based on the Tucker and the canonical decompositions. We call our approaches: Tucker decomposition process (TDP) and canonical decomposition process (CDP). We test the TDP and CDP in rank-2, 4 and 6 HOT fields. For rank-2 tensors, we compare against direct interpolation, log-Euclidean approach, and Generalized Wishart processes. For rank-4 and 6 tensors, we compare against direct interpolation and raw dMRI interpolation. Results obtained show that TDP and CDP interpolate accurately the HOT fields in terms of Frobenius distance, anisotropy measurements, and fiber tracts. Besides, CDP and TDP can be generalized to any rank. Also, the proposed framework keeps the mandatory constraint of positive definite tensors, and preserves morphological properties such as fractional anisotropy (FA), generalized anisotropy (GA) and tractography

Nonparametric tests of structure for high angular resolution diffusion imaging in Q-space

High angular resolution diffusion imaging data is the observed characteristic function for the local diffusion of water molecules in tissue. This data is used to infer structural information in brain imaging. Nonparametric scalar measures are proposed to summarize such data, and to locally characterize spatial features of the diffusion probability density function (PDF), relying on the geometry of the characteristic function. Summary statistics are defined so that their distributions are, to first-order, both independent of nuisance parameters and also analytically tractable. The dominant direction of the diffusion at a spatial location (voxel) is determined, and a new set of axes are introduced in Fourier space. Variation quantified in these axes determines the local spatial properties of the diffusion density. Nonparametric hypothesis tests for determining whether the diffusion is unimodal, isotropic or multi-modal are proposed. More subtle characteristics of white-matter microstructure, such as the degree of anisotropy of the PDF and symmetry compared with a variety of asymmetric PDF alternatives, may be ascertained directly in the Fourier domain without parametric assumptions on the form of the diffusion PDF. We simulate a set of diffusion processes and characterize their local properties using the newly introduced summaries. We show how complex white-matter structures across multiple voxels exhibit clear ellipsoidal and asymmetric structure in simulation, and assess the performance of the statistics in clinically-acquired magnetic resonance imaging data.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS441 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Space-Varying Coefficient Models for Brain Imaging

The methodological development and the application in this paper originate from diffusion tensor imaging (DTI), a powerful nuclear magnetic resonance technique enabling diagnosis and monitoring of several diseases as well as reconstruction of neural pathways. We reformulate the current analysis framework of separate voxelwise regressions as a 3d space-varying coefficient model (VCM) for the entire set of DTI images recorded on a 3d grid of voxels. Hence by allowing to borrow strength from spatially adjacent voxels, to smooth noisy observations, and to estimate diffusion tensors at any location within the brain, the three-step cascade of standard data processing is overcome simultaneously. We conceptualize two VCM variants based on B-spline basis functions: a full tensor product approach and a sequential approximation, rendering the VCM numerically and computationally feasible even for the huge dimension of the joint model in a realistic setup. A simulation study shows that both approaches outperform the standard method of voxelwise regressions with subsequent regularization. Due to major efficacy, we apply the sequential method to a clinical DTI data set and demonstrate the inherent ability of increasing the rigid grid resolution by evaluating the incorporated basis functions at intermediate points. In conclusion, the suggested fitting methods clearly improve the current state-of-the-art, but ameloriation of local adaptivity remains desirable

High rank tensor and spherical harmonic models for diffusion MRI processing

Diffusion tensor imaging (DTI) is a non-invasive quantitative method of characterizing tissue micro-structure. Diffusion imaging attempts to characterize the manner by which the water molecules within a particular location move within a given amount of time. Measurement of the diffusion tensor (D) within a voxel allows a macroscopic voxel-averaged description of fiber structure, orientation and fully quantitative evaluation of the microstructural features of healthy and diseased tissue.;The rank two tensor model is incapable of resolving multiple fiber orientations within an individual voxel. This shortcoming of single tensor model stems from the fact that the tensor possesses only a single orientational maximum. Several authors reported this non-mono-exponential behavior for the diffusion-induced attenuation in brain tissue in water and N-Acetyl Aspartate (NAA) signals, that is why the Multi-Tensor, Higher Rank Tensor and Orientation Distribution Function (ODF) were introduced.;Using the higher rank tensor, we will propose a scheme for tensor field interpolation which is inspired by subdivision surfaces in computer graphics. The method applies to Cartesian tensors of all ranks and imposes smoothness on the interpolated field by constraining the divergence and curl of the tensor field. Results demonstrate that the subdivision scheme can better preserve anisotropicity and interpolate rotations than some other interpolation methods. As one of the most important applications of DTI, fiber tractography was implemented to study the shape geometry changes. Based on the divergence and curl measurement, we will introduce new scalar measures that are sensitive to behaviors such as fiber bending and fanning.;Based on the ODF analysis, a new anisotropy measure that has the ability to describe multi-fiber heterogeneity while remaining rotationally invariant, will be introduced, which is a problem with many other anisotropy measures defined using the ODF. The performance of this novel measure is demonstrated for data with varying Signal to Noise Ratio (SNR), and different material characteristics

Generalized Wishart processes for interpolation over diffusion tensor fields

Diffusion Magnetic Resonance Imaging (dMRI) is a non-invasive tool for watching the microstructure of fibrous nerve and muscle tissue. From dMRI, it is possible to estimate 2-rank diffusion tensors imaging (DTI) fields, that are widely used in clinical applications: tissue segmentation, fiber tractography, brain atlas construction, brain conductivity models, among others. Due to hardware limitations of MRI scanners, DTI has the difficult compromise between spatial resolution and signal noise ratio (SNR) during acquisition. For this reason, the data are often acquired with very low resolution. To enhance DTI data resolution, interpolation provides an interesting software solution. The aim of this work is to develop a methodology for DTI interpolation that enhance the spatial resolution of DTI fields. We assume that a DTI field follows a recently introduced stochastic process known as a generalized Wishart process (GWP), which we use as a prior over the diffusion tensor field. For posterior inference, we use Markov Chain Monte Carlo methods. We perform experiments in toy and real data. Results of GWP outperform other methods in the literature, when compared in different validation protocols