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.     

Study of pediatric brain development using magnetic resonance imaging of anisotropic diffusion

The properties of water diffusion in human brain tissue can be characterized by diffusion tensors computed from diffusion weighted magnetic resonance images. Since these properties are strongly determined by the structural and geometrical characteristics of the tissue, the maturation process of white matter and gray matter tissue can be expected to be reflected in these images and derived tensor quantities. The purpose of this work was therefore to study the development of pediatric brain in terms of changes occurring in the observed diffusion behavior. Echo planar diffusion tensor imaging was performed on 22 (10 females and 12 males) full term newborn and infant patients, diagnosed in retrospect as neurologically healthy. The subjects were subdivided in three age categories. A number of quantities based on the diffusion images were calculated for each tissue type and age category, and the ability of these quantities to provide sensitive and consistent information about the tissue differences and evolution was evaluated. The results clearly illustrate that the rotationally invariant quantities (e.g., the highest diffusivity, anisotropy ratio and volume ratio) are superior to the rotationally variant ones (e.g., ADCs measured along the three axes of the magnet) often used in the clinic. On the basis of the anisotropy ratio and the volume ratio indices, a correlation between the white matter maturation and the evolution of the diffusion anisotropy could be established. The same quantities did not exhibit any age dependence for the gray matter tissues.     

On the use of water phantom images to calibrate and correct eddy current induced artefacts in MR diffusion tensor imaging

The accurate determination of absolute measures of diffusion anisotropy in vivo using single-shot, echo-planar imaging techniques requires the acquisition of a set of high signal-to-noise ratio, diffusion-weighted images that are free from eddy current induced image distortions. Such geometric distortions can be characterized and corrected in brain imaging data using magnification (M), translation (T), and shear (S) distortion parameters derived from separate water phantom calibration experiments. Here we examine the practicalities of using separate phantom calibration data to correct high b-value diffusion tensor imaging data by investigating the stability of these distortion parameters, and hence the eddy currents, with time. It is found that M, T, and S vary only slowly with time (i.e., on the order of weeks), so that calibration scans need not be performed after every patient examination. This not only minimises the scan time required to collect the calibration data, but also the computational time needed to characterize these eddy current induced distortions. Examples of how measurements of diffusion anisotropy are improved using this post-processing scheme are also presented.     

Diffusion tensor image up-sampling: a registration-based approach

Diffusion weighted images (DWI), from which the corresponding diffusion tensor images (DTI) are estimated, are commonly acquired with anisotropic discretizations. Traditional methods to up-sample diffusion weighted images generally rely on scene-based interpolation and do not exploit structural information from the images. In this study, a DTI up-sampling framework is presented that incorporates the underlying anatomical shape information by means of non-rigid inter-slice registration. A strategy is proposed to reorient the interpolated tensor in order to maintain its proper orientation. Tests on phantom as well as on real data sets show that the proposed method is able to produce better results compared to scene based interpolation methods in terms of the accuracy of DWI/DTI interpolation, especially when diffusion tensor orientation is taken into account.     

A continuous tensor field approximation of discrete DT-MRI data for extracting microstructural and architectural features of tissue

The effective diffusion tensor of water, D, measured by diffusion tensor MRI (DT-MRI), is inherently a discrete, noisy, voxel-averaged sample of an underlying macroscopic effective diffusion tensor field, D(x). Within fibrous tissues this field is presumed to be continuous and smooth at a gross anatomical length scale. Here a new, general mathematical framework is proposed that uses measured DT-MRI data to produce a continuous approximation to D(x). One essential finding is that the continuous tensor field representation can be constructed by repeatedly performing one-dimensional B-spline transforms of the DT-MRI data. The fidelity and noise-immunity of this approximation are tested using a set of synthetically generated tensor fields to which background noise is added via Monte Carlo methods. Generally, these tensor field templates are reproduced faithfully except at boundaries where diffusion properties change discontinuously or where the tensor field is not microscopically homogeneous. Away from such regions, the tensor field approximation does not introduce bias in useful DT-MRI parameters, such as Trace(D(x)). It also facilitates the calculation of several new parameters, particularly differential quantities obtained from the tensor of spatial gradients of D(x). As an example, we show that they can identify tissue boundaries across which diffusion properties change rapidly using in vivo human brain data. One important application of this methodology is to improve the reliability and robustness of DT-MRI fiber tractography.     

Efficient anisotropic filtering of diffusion tensor images

To improve the accuracy of structural and architectural characterization of living tissue with diffusion tensor imaging, an efficient smoothing algorithm is presented for reducing noise in diffusion tensor images. The algorithm is based on anisotropic diffusion filtering, which allows both image detail preservation and noise reduction. However, traditional numerical schemes for anisotropic filtering have the drawback of inefficiency and inaccuracy due to their poor stability and first order time accuracy. To address this, an unconditionally stable and second order time accuracy semi-implicit Craig-Sneyd scheme is adapted in our anisotropic filtering. By using large step size, unconditional stability allows this scheme to take much fewer iterations and thus less computation time than the explicit scheme to achieve a certain degree of smoothing. Second-order time accuracy makes the algorithm reduce noise more effectively than a first order scheme with the same total iteration time. Both the efficiency and effectiveness are quantitatively evaluated based on synthetic and in vivo human brain diffusion tensor images, and these tests demonstrate that our algorithm is an efficient and effective tool for denoising diffusion tensor images.     

Correction of eddy current-induced artefacts in diffusion tensor imaging using iterative cross-correlation.

Geometric distortions of echo-planar images produced by the strong eddy currents present in the diffusion tensor imaging experiment are a major confound to the accurate quantification of diffusion coefficients, and measures of diffusion anisotropy based upon them. Here we investigate how the method of iterative cross-correlation (ICC) of baseline and diffusion-weighted images (DWIs) originally proposed by Haselgrove and Moore (Magn. Reson. Med. 36:960-964; 1996) can be extended to correct high b-value DWIs, without the need for extrapolation of distortion parameters determined from low b-value images. Monte Carlo simulations of synthetic brain images show that the maximum value of the trace of the b-matrix, Tr(b), at which distorted DWIs can be accurately corrected by direct comparison with the undistorted baseline image is approximately 300 s mm(-2). Removal of the cerebrospinal fluid signal greatly extends this value of Tr(b) (up to approximately 2000 s mm(-2)), thereby allowing direct comparison of baseline and distorted images. The use of ICC distortion parameters determined from separate calibrations of water phantom images is also investigated, and found to be effective in correcting geometric distortions observed in the DWIs collected as part of a human volunteer diffusion tensor imaging study. This work suggests that distorted DWIs acquired at high values of b may be corrected using the ICC algorithm without collecting additional low b-value images, thus allowing simplified methods of measuring the apparent diffusion tensor D, based on collecting a small number of DWIs, to be implemented in quantitative patient examinations.     

Parametric and non-parametric statistical analysis of DT-MRI data

In this work parametric and non-parametric statistical methods are proposed to analyze Diffusion Tensor Magnetic Resonance Imaging (DT-MRI) data. A Multivariate Normal Distribution is proposed as a parametric statistical model of diffusion tensor data when magnitude MR images contain no artifacts other than Johnson noise. We test this model using Monte Carlo (MC) simulations of DT-MRI experiments. The non-parametric approach proposed here is an implementation of bootstrap methodology that we call the DT-MRI bootstrap. It is used to estimate an empirical probability distribution of experimental DT-MRI data, and to perform hypothesis tests on them. The DT-MRI bootstrap is also used to obtain various statistics of DT-MRI parameters within a single voxel, and within a region of interest (ROI); we also use the bootstrap to study the intrinsic variability of these parameters in the ROI, independent of background noise. We evaluate the DT-MRI bootstrap using MC simulations and apply it to DT-MRI data acquired on human brain in vivo, and on a phantom with uniform diffusion properties.     

Diffusion anisotropy in excised normal rat spinal cord measured by NMR microscopy

A conventional spin-echo NMR imaging pulse sequence was used to obtain high-resolution images of excised normal rat spinal cord at 7 and 14 T. It was observed that the large pulsed-field gradients necessary for high-resolution imaging caused a diffusion weighting that dominated the image contrast and that could be used to infer microscopic structural organization beyond that defined by the resolution of the image matrix (i.e., fiber orientation could be assigned based on diffusion anisotropy). Anisotropic diffusion coefficients were therefore measured using apparent diffusion tensor (ADT) imaging to assess more accurately fiber orientations in the spinal cord; structural anisotropy information is portrayed in the six unique images of the complete ADT. To reduce the dimensionality of the data, a trace image was generated using a separate color scale for each of the three diagonal element images of the ADT. This new image retains much of the invariance of the trace to the relative orientations of laboratory and sample axes (inherent to a greyscale trace image) but provides, by the use of color, contrast reflecting diffusion anisotropy. The colored trace image yields a pseudo-three-dimensional view of the rat spinal cord, from which it is possible to deduce fiber orientations.     

A diffusion gradient optimization framework for spinal cord diffusion tensor imaging

The uncertainty in the estimation of diffusion model parameters in diffusion tensor imaging (DTI) can be reduced by optimally selecting the diffusion gradient directions utilizing some prior structural information. This is beneficial for spinal cord DTI, where the magnetic resonance images have low signal-to-noise ratio and thus high uncertainty in diffusion model parameter estimation. Presented is a gradient optimization scheme based on D-optimality, which reduces the overall estimation uncertainty by minimizing the Rician Cramer-Rao lower bound of the variance of the model parameter estimates. The tensor-based diffusion model for DTI is simplified to a four-parameter axisymmetric DTI model where diffusion transverse to the principal eigenvector of the tensor is assumed isotropic. Through simulations and experimental validation, we demonstrate that an optimized gradient scheme based on D-optimality is able to reduce the overall uncertainty in the estimation of diffusion model parameters for the cervical spinal cord and brain stem white matter tracts.     

Efficient cardiac diffusion tensor MRI by three-dimensional reconstruction of solenoidal tensor fields

Tensor tomography is being investigated as a technique for reconstruction of in vivo diffusion tensor fields that can potentially be used to reduce the number of magnetic resonance imaging (MRI) measurements. Specifically, assessments are being made of the reconstruction of cardiac diffusion tensor fields from 3D Radon planar projections using a filtered backprojection algorithm in order to specify the helical fiber structure of myocardial tissue. Helmholtz type decomposition is proposed for 3D second order tensor fields. Using this decomposition a Fourier projection theorem is formulated in terms of the solenoidal and irrotational components of the tensor field. From the Fourier projection theorem, two sets of Radon directional measurements, one that reconstructs the solenoidal component and one that reconstructs the irrotational component of the tensor field, are prescribed. Based on these observations filtered backprojection reconstruction formulae are given for the reconstruction of a 3D second order tensor field and its solenoidal and irrotational components from Radon projection measurements. Computer simulations demonstrate the validity of the mathematical formulations and demonstrate that a realistic model of the helical fiber structure of the myocardial tissue specifies a diffusion tensor field for which the first principal vector (the vector associated with the maximum eigenvalue) of the solenoidal component accurately approximates the first principal vector of the diffusion tensor. A priori knowledge of this allows the orientation of the myocardial fiber structure to be specified utilizing one half of the number of MRI measurements of a normal diffusion tensor field study.     

Variational multiple-tensor fitting of fiber-ambiguous diffusion-weighted magnetic resonance imaging voxels

Partial volume effects are often experienced in diffusion-weighted MRI of biologic tissue. This is when the signal attenuation reflects a mixture of diffusion processes, originating from different tissue compartments, residing in the same voxel. Decomposing the mixture requires elaborated models that account for multiple compartments, yet the fitting problem for those models is usually ill posed. We suggest a novel approach for stabilizing the fitting problem of the multiple-tensors model by a variational framework that adds biologically oriented assumption of neighborhood alignments. The framework is designed to address fiber ambiguity caused by a number of neuronal fiber compartments residing in the same voxel. The method requires diffusion data acquired by common, clinically feasible MRI sequences, and is able to derive familiar tensor quantities for each compartment. Neighborhood alignment is performed by adding piece-wise smooth regularization constraints to an energy function. Minimization with the gradient descent method produces a set of diffusion-reaction partial differential equations that describe a tensor-preserving flow towards a best approximation of the data while maintaining the constraints. We analyze fiber compartment separation capabilities on a synthetic model of crossing fibers and on brain areas known to have crossing fibers. We compare the results with diffusion tensor imaging analysis and discuss applications for the framework.     

Evaluation of diffusional anisotropy and microscopic structure in skeletal muscles using magnetic resonance

The pulsed-gradient spin-echo (PGSE) nuclear magnetic resonance (NMR) method is used for detecting the diffusion of water molecules in biological tissues. Because tissues generally have diffusional anisotropy, their diffusion properties are denoted by a tensor. In this study, we evaluated the diffusional anisotropy and microscopic structure in atrophied skeletal muscles using the PGSE NMR method. The left sciatic nerve was severed in twelve 9-week-old rats. Neurotomy caused neurogenic muscular atrophy at the left gastrocnemius. At 2, 4 and 8 weeks after neurotomy, magnetic resonance signals were selectively acquired from a 2 x 2 x 2 mm(3) voxel, which was located on the left gastrocnemius. The diffusion tensor, the mean diffusivity (MD) and the fractional anisotropy (FA) were calculated from the signals. A theoretical model of the diffusion in muscles was derived from Tanner's equation. The muscle fiber diameter was estimated by fitting the model to the measured signals. The measurements were also performed for normal rats as controls. No significant difference was found in the MD and the estimated intracellular diffusion coefficient between the control group and the denervated group. The denervated group had significantly higher FA compared with the control group (P<.05). The estimated muscle fiber diameter of the denervated group was significantly smaller than the estimated value of the control group (P<.05). These differences were found at 8 weeks after neurotomy. The proposed method is effective for evaluating changes in the microscopic structure of skeletal muscles.     

A modified fuzzy clustering algorithm for operator independent brain tissue classification of dual echo MR images.

Methods for brain tissue classification or segmentation of structural magnetic resonance imaging (MRI) data should ideally be independent of human operators for reasons of reliability and tractability. An algorithm is described for fully automated segmentation of dual echo, fast spin-echo MRI data. The method is used to assign fuzzy-membership values for each of four tissue classes (gray matter, white matter, cerebrospinal fluid and dura) to each voxel based on partition of a two dimensional feature space. Fuzzy clustering is modified for this application in two ways. First, a two component normal mixture model is initially fitted to the thresholded feature space to identify exemplary gray and white matter voxels. These exemplary data protect subsequently estimated cluster means against the tendency of unmodified fuzzy clustering to equalize the number of voxels in each class. Second, fuzzy clustering is implemented in a moving window scheme that accommodates reduced image contrast at the axial extremes of the transmitting/receiving coil. MRI data acquired from 5 normal volunteers were used to identify stable values for three arbitrary parameters of the algorithm: feature space threshold, relative weight of exemplary gray and white matter voxels, and moving window size. The modified algorithm incorporating these parameter values was then used to classify data from simulated images of the brain, validating the use of fuzzy-membership values as estimates of partial volume. Gray:white matter ratios were estimated from 20 twenty normal volunteers (mean age 32.8 years). Processing time for each three-dimensional image was approximately 30 min on a 170 MHz workstation. Mean cerebral gray and white matter volumes estimated from these automatically segmented images were very similar to comparable results previously obtained by operator dependent methods, but without their inherent unreliability.     

MRI diffusion tensor reconstruction with PROPELLER data acquisition

MRI diffusion imaging is effective in measuring the diffusion tensor in brain, cardiac, liver, and spinal tissue. Diffusion tensor tomography MRI (DTT MRI) method is based on reconstructing the diffusion tensor field from measurements of projections of the tensor field. Projections are obtained by appropriate application of rotated diffusion gradients. In the present paper, the potential of a novel data acquisition scheme, PROPELLER (Periodically Rotated Overlapping ParallEL Lines with Enhanced Reconstruction), is examined in combination with DTT MRI for its capability and sufficiency for diffusion imaging. An iterative reconstruction algorithm is used to reconstruct the diffusion tensor field from rotated diffusion weighted blades by appropriate rotated diffusion gradients. DTT MRI with PROPELLER data acquisition shows significant potential to reduce the number of weighted measurements, avoid ambiguity in reconstructing diffusion tensor parameters, increase signal-to-noise ratio, and decrease the influence of signal distortion.     

Error bounds in diffusion tensor estimation using multiple-coil acquisition systems

We extend the diffusion tensor (DT) signal model for multiple-coil acquisition systems. Considering the sum-of-squares reconstruction method, we compute the Cramér–Rao bound (CRB) assuming the widely accepted noncentral chi distribution. Within this framework, we assess the effect of noise in DT estimation and other measures derived from it, as a function of the number of acquisition coils, as well as other system parameters. We show the applications of CRB in many actual problems related to DT estimation: we compare different gradient field setup schemes proposed in the literature and show how the CRB can be used to choose a convenient one; we show that for fiber-type anisotropy tensors the ellipsoidal area ratio (EAR) can be estimated with less error than other scalar factors such as the fractional anisotropy (FA) or the relative anisotropy (RA), and that for this type of anisotropy tensors, increasing the number of coils is equivalent to increasing the signal-to-noise ratio, i.e., the information of the different coils can be regarded as independent. Also, we present results showing the CRB of several parameters for actual DT-MRI data. We conclude that the CRB is a valuable tool to optimal experiment design in DT-related studies.     

Fluctuation interaction of Ising subsystems

A phase diagram of two Ising subsystems σ and s has been constructed on a Bethe lattice with a coordination number 4 (a simplified analog of a square lattice). In contrast to the known Ashkin-Teller model, the interaction between these two subsystems is of a purely fluctuational nature; i.e., it does not manifest itself in the ground state and nullifies the sums of the products of average spins 〈σ〉〈s〉 (interactions of this type are realized in lattice-type adsorbed systems with dipolelike intermolecular interactions and strong azimuthal angular dependence of the adsorption potential of symmetry C₄). Apart from conventional states, i.e., a high-temperature disordered state (〈σ〉=〈s〉=0) and a low-temperature ordered state (〈σ〉 and 〈s〉 =? 0), this system can also exist in a correlated state (〈σs〉 =? 0 at 〈σ〉=〈s〉=0). In the theory of orientational phase transitions, this state corresponds to a fundamentally different, intermediate (on the temperature axis) phase in which a preferred direction of long molecule axes arises in the absence of spontaneous polarization. The results of Monte Carlo simulation on a square lattice agree with the conclusions obtained on a Bethe lattice. The characteristics of the orientational phase transition in a 2 × 1 monolayer of CO molecules adsorbed on the NaCl(100) surface are discussed.     

Dependence of the fractional anisotropy in cervical spine from the number of diffusion gradients,repeated acquisition and voxel size

The aim of this study is to investigate the consequences of using different gradient schemes, number of repeated measurements and voxel size on the fractional anisotropy (FA) value in a diffusion tensor imaging (DTI) sequence on the cervical tract of the spinal cord. Twenty healthy volunteers underwent a total of 86 DTI axial acquisitions performed by using different voxel size and number of diffusion gradient directions (NDGDs). Three different diffusion gradient schemes were applied, named 6, 15 and 32 according to the NDGD. Furthermore, some acquisitions were repeated to investigate the effects of image averaging on FA value.     

Threshold resummation for the production of a color sextet (antitriplet) scalar at the LHC

This paper is devoted to the study of warm inflation using vector fields in the background of a locally rotationally symmetric Bianchi type I model of the universe. We formulate the field equations, and slow-roll and perturbation parameters (scalar and tensor power spectra as well as their spectral indices) in the slow-roll approximation. We evaluate all these parameters in terms of the directional Hubble parameter during the intermediate and logamediate inflationary regimes by taking the dissipation factor as a function of the scalar field as well as a constant. In each case, we calculate the observational parameter of interest, i.e., the tensor–scalar ratio in terms of the inflaton. The graphical behavior of these parameters shows that the anisotropic model is also compatible with WMAP7 and the Planck observational data.