A new framework for MR diffusion tensor distribution

The ability to characterize heterogeneous and anisotropic water diffusion processes within macroscopic MRI voxels non-invasively and in vivo is a desideratum in biology, neuroscience, and medicine. While an MRI voxel may contain approximately a microliter of tissue, our goal is to examine intravoxel diffusion processes on the order of picoliters. Here we propose a new theoretical framework and efficient experimental design to describe and measure such intravoxel structural heterogeneity and anisotropy. We assume that a constrained normal tensor-variate distribution (CNTVD) describes the variability of positive definite diffusion tensors within a voxel which extends its applicability to a wide range of b-values while preserving the richness of diffusion tensor distribution (DTD) paradigm unlike existing models. We introduce a new Monte Carlo (MC) scheme to synthesize realistic 6D DTD numerical phantoms and invert the MR signal. We show that the signal inversion is well-posed and estimate the CNTVD parameters parsimoniously by exploiting the different symmetries of the mean and covariance tensors of CNTVD. The robustness of the estimation pipeline is assessed by adding noise to calculated MR signals and compared with the ground truth. A family of invariant parameters and glyphs which characterize microscopic shape, size and orientation heterogeneity within a voxel are also presented.

Eigenvalues of Random Matrices with Isotropic Gaussian Noise and the Design of Diffusion Tensor Imaging Experiments

Tensor-valued and matrix-valued measurements of different physical properties are increasingly available in material sciences and medical imaging applications. The eigenvalues and eigenvectors of such multivariate data provide novel and unique information, but at the cost of requiring a more complex statistical analysis. In this work we derive the distributions of eigenvalues and eigenvectors in the special but important case of m×m symmetric random matrices, D, observed with isotropic matrix-variate Gaussian noise. The properties of these distributions depend strongly on the symmetries of the mean tensor/matrix, D̄. When D̄ has repeated eigenvalues, the eigenvalues of D are not asymptotically Gaussian, and repulsion is observed between the eigenvalues corresponding to the same D̄ eigenspaces. We apply these results to diffusion tensor imaging (DTI), with m = 3, addressing an important problem of detecting the symmetries of the diffusion tensor, and seeking an experimental design that could potentially yield an isotropic Gaussian distribution. In the 3-dimensional case, when the mean tensor is spherically symmetric and the noise is Gaussian and isotropic, the asymptotic distribution of the first three eigenvalue central moment statistics is simple and can be used to test for isotropy. In order to apply such tests, we use quadrature rules of order t ≥ 4 with constant weights on the unit sphere to design a DTI-experiment with the property that isotropy of the underlying true tensor implies isotropy of the Fisher information. We also explain the potential implications of the methods using simulated DTI data with a Rician noise model.

A note on the validity of statistical bootstrapping for estimating the uncertainty of tensor parameters in diffusion tensor images

Diffusion tensors are estimated from magnetic resonance images (MRIs) that are diffusion-weighted, and those images inherently contain noise. Therefore, noise in the diffusion-weighted images produces uncertainty in estimation of the tensors and their derived parameters, which include eigenvalues, eigenvectors, and the trajectories of fiber pathways that are reconstructed from those eigenvalues and eigenvectors. Although repetition and wild bootstrap methods have been widely used to quantify the uncertainty of diffusion tensors and their derived parameters, we currently lack theoretical derivations that would validate the use of these two bootstrap methods for the estimation of statistical parameters of tensors in the presence of noise. The aim of this paper is to examine theoretically and numerically the repetition and wild bootstrap methods for approximating uncertainty in estimation of diffusion tensor parameters under two different schemes for acquiring diffusion weighted images. Whether these bootstrap methods can be used to quantify uncertainty in some diffusion tensor parameters, such as fractional anisotropy (FA), depends critically on the morphology of the diffusion tensor that is being estimated. The wild and repetition bootstrap methods in particular cannot quantify uncertainty in the principal direction (PD) of isotropic (or oblate) tensor. We also examine the use of bootstrap methods in estimating tensors in a voxel containing multiple tensors, demonstrating their limitations when quantifying the uncertainty of tensor parameters in those locations. Simulation studies are also used to understand more thoroughly our theoretical results. Our findings raise serious concerns about the use of bootstrap methods to quantify the uncertainty of fiber pathways when those pathways pass through voxels that contain either isotropic tensors, oblate tensors, or multiple tensors.

Variance of estimated DTI-derived parameters via first-order perturbation methods

In typical applications of diffusion tensor imaging (DTI), DT-derived quantities are used to make a diagnostic, therapeutic, or scientific determination. In such cases it is essential to characterize the variability of these tensor-derived quantities. Parametric and empirical methods have been proposed to estimate the variance of the estimated DT, and quantities derived from it. However, the former method cannot be generalized since a parametric distribution cannot be found for all DT-derived quantities. Although powerful empirical methods, such as the bootstrap, are available, they require oversampling of the diffusion-weighted imaging (DWI) data. Statistical perturbation methods represent a hybrid between parametric and empirical approaches, and can overcome the primary limitations of both methods. In this study we used a first-order perturbation method to obtain analytic expressions for the variance of DT-derived quantities, such as the trace, fractional anisotropy (FA), eigenvalues, and eigenvectors, for a given experimental design. We performed Monte Carlo (MC) simulations of DTI experiments to test and validate these formulae, and to determine their range of applicability for different experimental design parameters, including the signal-to-noise ratio (SNR), diffusion gradient sampling scheme, and number of DWI acquisitions. This information should be useful for designing DTI studies and assessing the quality of inferences drawn from them.

Quantitative analysis along the pyramidal tract by length-normalized parameterization based on diffusion tensor tractography: application to patients with relapsing neuromyelitis optica

In this study, we introduced a length-normalized parameterization method to establish anatomical correspondence of white matter fiber tracts across subjects and applied this method to investigate the presence of abnormal diffusion along the pyramidal tract (PYT) of relapsing neuromyelitis optica (RNMO) patients without visible brain lesions. In this approach, the part of the PYT between the lowest slice of the cerebral peduncle and the uppermost slice of the lateral ventricle was reconstructed to establish the anatomical correspondence across subjects using diffusion tensor tractography. Then it was parameterized by normalizing its length and dividing equally the normalized length into a certain number of segments, so that the comparability of each segment across subjects along the PYT was established. Tract-specific diffusion indices, including directionally averaged diffusivity (D(av)), fractional anisotropy (FA), primary diffusivity (lambda(1)) and transverse diffusivity (lambda(23)), were obtained from each segment. Thus, the distribution maps of these indices along the PYT were obtained. The distribution maps of D(av), FA, and lambda(23) of RNMO patients were significantly different from those of healthy controls, especially in the lower part of the PYT. The differences may be caused by secondary degeneration to lesions in the spinal cord. In conclusion, a length-normalized parameterization method is proposed to establish anatomical correspondence for the PYT. Compared with existed methods, a major merit of our method is to provide comparability across subjects along the PYT on the basis of diffusion tensor tractography and to make it possible for the quantitative analysis along the fiber tract. This method can also be used to quantitatively analyze other white matter fiber tracts between two definite anatomic landmarks in many neurological or psychiatric diseases.

A statistical framework for the classification of tensor morphologies in diffusion tensor images

Tractography algorithms for diffusion tensor (DT) images consecutively connect directions of maximal diffusion across neighboring DTs in order to reconstruct the 3-dimensional trajectories of white matter tracts in vivo in the human brain. The performance of these algorithms, however, is strongly influenced by the amount of noise in the images and by the presence of degenerate tensors-- i.e., tensors in which the direction of maximal diffusion is poorly defined. We propose a simple procedure for the classification of tensor morphologies that uses test statistics based on invariant measures of DTs, such as fractional anisotropy, while accounting for the effects of noise on tensor estimates. Examining DT images from seven human subjects, we demonstrate that this procedure validly classifies DTs at each voxel into standard types (nondegenerate DTs, as well as degenerate oblate, prolate or isotropic DTs), and we provide preliminary estimates for the prevalence and spatial distribution of degenerate tensors in these brains. We also show that the P values for test statistics are more sensitive tools for classifying tensor morphologies than are invariant measures of anisotropy alone.

A method for obtaining tract-specific diffusion tensor MRI measurements in the presence of disease: application to patients with clinically isolated syndromes suggestive of multiple sclerosis

The aim of this study was to investigate whether neurological symptoms related to a specific axonal fiber tract in brain white matter were associated with a higher degree of tissue damage in that region, in patients at presentation with clinically isolated syndromes (CIS) suggestive of multiple sclerosis. To this end, a magnetic resonance imaging (MRI) method to segment and evaluate the fiber bundle of interest was implemented, taking care to circumvent the problems caused by pathology. Diffusion tensor (DT) MRI tractography was used to construct, from healthy volunteer data, a probability map for the pyramidal tract (PYT), and this map was applied to patients to calculate DT-derived metrics inside the PYT. In CIS patients with clinical symptoms related to motor function, the DT-derived mean diffusivity and the lesion volume in the PYT were found to be increased, while the fractional anisotropy was no different, when compared to those patients without motor symptoms. These results may be explained by several microstructural changes in the damaged tissue, such as changes in the permeability of axonal cell membranes, decreases of axonal density and edema. The approach taken to analyze a specific fiber tract was possible because the axons in the tract have a high orientational coherence, allowing tissue structure changes to be isolated from the tissue architecture. Its extension to other white matter fiber bundles is therefore limited to bundles with high orientational coherence.