In Diffusion Weighted Magnetic Resonance Image (DW-MRI) processing, a 2 nd order tensor has been commonly used to approximate the diffusivity function at each lattice point of the DW-MRI data. From this tensor approximation, one can compute useful scalar quantities (e.g. anisotropy, mean diffusivity) which have been clinically used for monitoring encephalopathy, sclerosis, ischemia and other brain disorders. It is now well known that this 2 nd -order tensor approximation fails to capture complex local tissue structures, e.g. crossing fibers, and as a result, the scalar quantities derived from these tensors are grossly inaccurate at such locations. In this paper we employ a 4 th order symmetric positive-definite (SPD) tensor approximation to represent the diffusivity function and present a novel technique to estimate these tensors from the DW-MRI data guaranteeing the SPD property. Several articles have been reported in literature on higher order tensor approximations of the diffusivity function but none of them guarantee the positivity of the estimates, which is a fundamental constraint since negative values of the diffusivity are not meaningful. In this paper we represent the 4 th -order tensors as ternary quartics and then apply Hilbert's theorem on ternary quartics along with the Iwasawa parametrization to guarantee an SPD 4 th -order tensor approximation from the DW-MRI data. The performance of this model is depicted on synthetic data as well as real DW-MRIs from a set of excised control and injured rat spinal cords, showing accurate estimation of scalar quantities such as generalized anisotropy and trace as well as fiber orientations.
Background Development of valid, non-invasive biomarkers for parkinsonian syndromes is crucially needed. We aimed to assess whether non-invasive diffusion-weighted MRI can distinguish between parkinsonian syndromes using an automated imaging approach. MethodsWe did an international study at 17 MRI centres in Austria, Germany, and the USA. We used diffusion-weighted MRI from 1002 patients and the Movement Disorders Society Unified Parkinson's Disease Rating Scale part III (MDS-UPDRS III) to develop and validate disease-specific machine learning comparisons using 60 template regions and tracts of interest in Montreal Neurological Institute space between Parkinson's disease and atypical parkinsonism (multiple system atrophy and progressive supranuclear palsy) and between multiple system atrophy and progressive supranuclear palsy. For each comparison, models were developed on a training and validation cohort and evaluated in an independent test cohort by quantifying the area under the curve (AUC) of receiving operating characteristic curves. The primary outcomes were free water and free-water-corrected fractional anisotropy across 60 different template regions. Findings In the test cohort for disease-specific comparisons, the diffusion-weighted MRI plus MDS-UPDRS III model (Parkinson's disease vs atypical parkinsonism had an AUC 0•962; multiple system atrophy vs progressive supranuclear palsy AUC 0•897) and diffusion-weighted MRI only model had high AUCs (Parkinson's disease vs atypical parkinsonism AUC 0•955; multiple system atrophy vs progressive supranuclear palsy AUC 0•926), whereas the MDS-UPDRS III only models had significantly lower AUCs (Parkinson's disease vs atypical parkinsonism 0•775; multiple system atrophy vs progressive supranuclear palsy 0•582). These results indicate that a non-invasive imaging approach is capable of differentiating forms of parkinsonism comparable to current gold standard methods.Interpretations This study provides an objective, validated, and generalisable imaging approach to distinguish different forms of parkinsonian syndromes using multisite diffusion-weighted MRI cohorts. The diffusion-weighted MRI method does not involve radioactive tracers, is completely automated, and can be collected in less than 12 min across 3T scanners worldwide. The use of this test could positively affect the clinical care of patients with Parkinson's disease and parkinsonism and reduce the number of misdiagnosed cases in clinical trials.
Cartesian tensors of various orders have been employed for either modeling the diffusivity or the orientation distribution function in Diffusion-Weighted MRI datasets. In both cases, the estimated tensors have to be positive-definite since they model positive-valued functions. In this paper we present a novel unified framework for estimating positive-definite tensors of any order, in contrast to the existing methods in literature, which are either order-specific or fail to handle the positive-definite property. The proposed framework employs a homogeneous polynomial parametrization that covers the full space of any order positive-definite tensors and explicitly imposes the positive-definite constraint on the estimated tensors. We show that this parametrization leads to a linear system that is solved using the non-negative least squares technique. The framework is demonstrated using synthetic and real data from an excised rat hippocampus.
In Diffusion Weighted Magnetic Resonance Image (DW-MRI) processing a 2nd order tensor has been commonly used to approximate the diffusivity function at each lattice point of the DW-MRI data. It is now well known that this 2nd-order approximation fails to approximate complex local tissue structures, such as fibers crossings. In this paper we employ a 4th order symmetric positive semi-definite (PSD) tensor approximation to represent the diffusivity function and present a novel technique to estimate these tensors from the DW-MRI data guaranteeing the PSD property. There have been several published articles in literature on higher order tensor approximations of the diffusivity function but none of them guarantee the positive semi-definite constraint, which is a fundamental constraint since negative values of the diffusivity coefficients are not meaningful. In our methods, we parameterize the 4th order tensors as a sum of squares of quadratic forms by using the so called Gram matrix method from linear algebra and its relation to the Hilbert's theorem on ternary quartics. This parametric representation is then used in a nonlinear-least squares formulation to estimate the PSD tensors of order 4 from the data. We define a metric for the higher-order tensors and employ it for regularization across the lattice. Finally, performance of this model is depicted on synthetic data as well as real DW-MRI from an isolated rat hippocampus.
In this paper, we present novel algorithms for statistically robust interpolation and approximation of diffusion tensors-which are symmetric positive definite (SPD) matrices-and use them in developing a significant extension to an existing probabilistic algorithm for scalar field segmentation, in order to segment diffusion tensor magnetic resonance imaging (DT-MRI) datasets. Using the Riemannian metric on the space of SPD matrices, we present a novel and robust higher order (cubic) continuous tensor product of B-splines algorithm to approximate the SPD diffusion tensor fields. The resulting approximations are appropriately dubbed tensor splines.Next, we segment the diffusion tensor field by jointly estimating the label (assigned to each voxel) field, which is modeled by a Gauss Markov measure field (GMMF) and the parameters of each smooth tensor spline model representing the labeled regions. Results of interpolation, approximation, and segmentation are presented for synthetic data and real diffusion tensor fields from an isolated rat hippocampus, along with validation. We also present comparisons of our algorithms with existing methods and show significantly improved results in the presence of noise as well as outliers.
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