In vivo diffusion tensor imaging of rat spinal cord at 7 T

In vivo diffusion tensor imaging of normal rat spinal cord was performed using a multi-segmented, blipped EPI sequence at 7 T field strength. At high diffusion weighting, the signal exhibited a non-monoexponential decay that was fitted to a biexponential function, associated with the fast and slow components of diffusion in the cord tissue, using a nonlinear regression analysis along with a constrained optimization procedure. From the measured tensors, the eigenvalues and the maps of invariant scalar measures (fractional anisotropy, relative anisotropy, volume ratio, and trace) were calculated and analyzed statistically. The results were combined to quantitatively characterize the anisotropic properties of the fast and slow diffusions in white- and gray matter of live spinal cords.     

A linear relationship exists among brain diffusion eigenvalues measured by diffusion tensor magnetic resonance imaging

Diffusion in biological tissues can be measured by magnetic resonance diffusion tensor imaging The complex nature of anisotropic diffusion in the brain has been described by a diffusion tensor which contains information about the magnitude of diffusion in different directions. Each tensor contains a set of three eigenvalues which are related to the major, intermediate, and minor axes of a diffusion ellipsoid. This investigation demonstrates that the various sets of diffusion eigenvalues from different regions of the brain lie along a line in ordered eigenvalue space. Sets of ordered diffusion eigenvalues were considered points in ordered eigenvalue space. The line which best fit the data by minimizing the total squared deviations was determined. A new coordinate system was constructed through translation and rotation which spanned ordered eigenvalue space. Eigenvalues from both monkey brain and human brain were studied. It was found that the sets of eigenvalues from both species have significant linear trends. Moreover, the same line may describe the brain eigenvalues from both species. It is likely that this linear relationship of the eigenvalues observed in an ordered eigenvalue plot is related to a combination of (1) conservation of total isotropic diffusion and (2) the degree of orientational dispersion of the microfibers within each voxel.     

The ellipsoidal area ratio: an alternative anisotropy index for diffusion tensor imaging

In the processing and analysis of diffusion tensor imaging (DTI) data, certain predefined morphological features of diffusion tensors are often represented as simplified scalar indices, termed diffusion anisotropy indices (DAIs). When comparing tensor morphologies across differing voxels of an image, or across corresponding voxels in different images, DAIs are mathematically and statistically more tractable than are the full tensors, which are probabilistic ellipsoids consisting of three orthogonal vectors that each has a direction and an associated scalar magnitude. We have developed a new DAI, the "ellipsoidal area ratio" (EAR), to represent the degree of anisotropy in the morphological features of a diffusion tensor. The EAR is a normalized geometrical measure of surface curvature in the 3D diffusion ellipsoid. Monte Carlo simulations and applications to the study of in vivo human data demonstrate that, at low noise levels, EAR provides a similar contrast-to-noise ratio (CNR) but a higher signal-to-noise ratio (SNR) than does fractional anisotropy (FA), which is currently the most popular anisotropy index in active use. Moreover, at the high noise levels encountered most commonly in real-world DTI datasets, EAR compared with FA is consistently much more robust to perturbations from noise and it provides a higher CNR, features useful for the analysis of DTI data that are inherently noise sensitive.     

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 theoretical study of the effect of experimental noise on the measurement of anisotropy in diffusion imaging

Diffusion tensor imaging (DTI) is a modality known to be highly sensitive to the detrimental effects of experimental noise. Here, using Monte Carlo simulations, we compare and contrast how noise complicates the measurement of diffusion anisotropy in diffusion tensor and conventional diffusion-weighted imaging (DWI). As the signal-to-noise ratio (SNR) decreases below a value of approximately 20, the eigenvalues (λ_i) of the diffusion tensor D are found to diverge rapidly from their true values, with the result that the measured anisotropy can be significantly in error and isotropic structures falsely assigned a high level of anisotropy. The effect of noise on the rotationally variant indices, calculated from a conventional diffusion-weighted imaging experiment, is found to be much less insidious, because the apparent diffusion coefficients (ADCs) diverge only slowly as the signal-to-noise decreases. Thus, although rotationally variant indices almost always underestimate the true diffusion anisotropy, they show only a small susceptibility to experimental noise and hence, are preferred to their rotationally invariant counterparts when the signal-to-noise ratio is small.     

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.     

A framework for quality control and parameter optimization in diffusion tensor imaging: theoretical analysis and validation

In this communication, a theoretical framework for quality control and parameter optimization in diffusion tensor imaging (DTI) is presented and validated. The approach is based on the analytical error propagation of the mean diffusivity (D(av)) obtained directly from the diffusion-weighted data acquired using rotationally invariant and uniformly distributed icosahedral encoding schemes. The error propagation of a recently described and validated cylindrical tensor model is further extrapolated to the spherical tensor case (diffusion anisotropy approximately 0) to relate analytically the precision error in fractional tensor anisotropy (FA) with the mean diffusion-to-noise ratio (DNR). The approach provided simple analytical and empirical quality control measures for optimization of diffusion parameter space in an isotropic medium that can be tested using widely available water phantoms.     

Translation invariant tensor product states in a finite lattice system

We show that the matrix (or more generally tensor) product states in a finite translation invariant system can be accurately constructed from a same set of local matrices (or tensors) that are determined from an infinite lattice system in one or higher dimensions. This provides an efficient approach for studying translation invariant tensor product states in finite lattice systems. Two methods are introduced to determine the size-independent local tensors.     

Gauge invariants and Killing tensors in higher-spin gauge theories

In free completely symmetric tensor gauge field theories on Minkowski space–time, all gauge invariant functions and Killing tensor fields are computed, both on-shell and off-shell. These problems are addressed in the metric-like formalisms.     

An invariant theory of the constitutive parameters of a plasma and a ferrite

Under the influence of a constant magnetic field, the electric property of a plasma and the magnetic property of a ferrite are anisotropic. In this paper, the general coordinatefree invariant forms of the dielectric tensor of a plasma and the permeability tensor of a ferrite are obtained. The tensors are expressed explicitly as a sum of three tensors: a unit tensor, a symmetric tensor and an antisymmetric tensor, each of which is weighted by different constants. The symmetric and antisymmetric tensors are related to the unit vector of the constant magnetic field. The invariant forms in terms of the sum of the projectors of the tensors are also derived. When a Cartesian coordinate system is introduced, the invariant forms are easily reduced to the commonly used matrix representations. The invariant forms clearly show the effects of the constant magnetic field on the anisotropies of the media. Moreover, they effectuate and simplify the deduction of the general solutions of problems involving wave propagation and excitation in plasma and ferrite and thus facilitate interpretations of the final results.     

Constrained metric variations and emergent equilibrium surfaces

Any surface is completely characterized by a metric and a symmetric tensor satisfying the Gauss–Codazzi–Mainardi equations (GCM), which identifies the latter as its curvature. We demonstrate that physical questions relating to a surface described by any Hamiltonian involving only surface degrees of freedom can be phrased completely in terms of these tensors without explicit reference to the ambient space: the surface is an emergent entity. Lagrange multipliers are introduced to impose GCM as constraints on these variables and equations describing stationary surface states derived. The behavior of these multipliers is explored for minimal surfaces, showing how their singularities correlate with surface instabilities.     

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.     

Geometric analysis of the b-dependent effects of Rician signal noise on diffusion tensor imaging estimates and determining an optimal b value

The optimal diffusion weighting (DW) factor, b, for use in diffusion tensor imaging (DTI) studies remains uncertain. In this study, the geometric relations of DW quantities are examined, in particular, the effects of Rician noise in the measured magnetic resonance signal. This geometric analysis is used to make theoretical predictions for selecting a b value to reduce the influence of noise. It is shown that the optimal b value for DTI studies in healthy human parenchyma is approximately b=1200 s mm⁻², with a simple relation given as well for a given expected apparent diffusion coefficient. Monte-Carlo simulations on sets of realistic DTI measures are then performed, verifying the optimal DW for minimizing estimate errors. The effects of noise on various DTI parameters such as anisotropy indices (fractional anisotropy and scaled relative anisotropy), mean diffusivity, radial diffusivity, eigenvalues and the direction of the first eigenvector are investigated as well.     

Condition number as a measure of noise performance of diffusion tensor data acquisition schemes with MRI

Diffusion tensor mapping with MRI can noninvasively track neural connectivity and has great potential for neural scientific research and clinical applications. For each diffusion tensor imaging (DTI) data acquisition scheme, the diffusion tensor is related to the measured apparent diffusion coefficients (ADC) by a transformation matrix. With theoretical analysis we demonstrate that the noise performance of a DTI scheme is dependent on the condition number of the transformation matrix. To test the theoretical framework, we compared the noise performances of different DTI schemes using Monte-Carlo computer simulations and experimental DTI measurements. Both the simulation and the experimental results confirmed that the noise performances of different DTI schemes are significantly correlated with the condition number of the associated transformation matrices. We therefore applied numerical algorithms to optimize a DTI scheme by minimizing the condition number, hence improving the robustness to experimental noise. In the determination of anisotropic diffusion tensors with different orientations, MRI data acquisitions using a single optimum b value based on the mean diffusivity can produce ADC maps with regional differences in noise level. This will give rise to rotational variances of eigenvalues and anisotropy when diffusion tensor mapping is performed using a DTI scheme with a limited number of diffusion-weighting gradient directions. To reduce this type of artifact, a DTI scheme with not only a small condition number but also a large number of evenly distributed diffusion-weighting gradients in 3D is preferable.     

Exactly Solvable Model of Quantum Diffusion

We study the transport property of diffusion in a finite translationally invariant quantum subsystem described by a tight-binding Hamiltonian with a single energy band. The subsystem interacts with its environment by a coupling expressed in terms of correlation functions which are delta-correlated in space and time. For weak coupling, the time evolution of the subsystem density matrix is ruled by a quantum master equation of Lindblad type. Thanks to the invariance under spatial translations, we can apply the Bloch theorem to the subsystem density matrix and exactly diagonalize the time evolution superoperator to obtain the complete spectrum of its eigenvalues, which fully describe the relaxation to equilibrium. Above a critical coupling which is inversely proportional to the size of the subsystem, the spectrum at given wave number contains an isolated eigenvalue describing diffusion. The other eigenvalues rule the decay of the populations and quantum coherences with decay rates which are proportional to the intensity of the environmental noise. An analytical expression is obtained for the dispersion relation of diffusion. The diffusion coefficient is proportional to the square of the width of the energy band and inversely proportional to the intensity of the environmental noise because diffusion results from the perturbation of quantum tunneling by the environmental fluctuations in this model. Diffusion disappears below the critical coupling     

Robust tensor estimation in diffusion tensor imaging

The signal response measured in diffusion tensor imaging is subject to detrimental influences caused by noise. Noise fields arise due to various contributions such as thermal and physiological noise and sources related to the hardware imperfection. As a result, diffusion tensors estimated by different linear and non-linear least squares methods in absence of a proper noise correction tend to be substantially corrupted. In this work, we propose an advanced tensor estimation approach based on the least median squares method of the robust statistics. Both constrained and non-constrained versions of the method are considered. The performance of the developed algorithm is compared to that of the conventional least squares method and of the alternative robust methods proposed in the literature. Two examples of simulated diffusion attenuations and experimental in vivo diffusion data sets were used as a basis for comparison. The robust algorithms were shown to be advantageous compared to the least squares method in the cases where elimination of the outliers is desirable. Additionally, the constraints were applied in order to prevent generation of the non-positive definite tensors and reduce related artefacts in the maps of fractional anisotropy. The developed method can potentially be exploited also by other MR techniques where a robust regression or outlier localisation is required.     

Assessment of subgrid-scale stress statistics in non-premixed turbulent wall-jet flames

We investigate the heat-release effects on the characteristics of the subgrid-scale (SGS) stress tensor and SGS dissipation of kinetic energy and enstrophy. Direct numerical simulation data of a non-premixed reacting turbulent wall-jet flow with and without substantial heat release is employed for the analysis. This study comprises, among others, an analysis of the eigenvalues of the resolved strain rate and SGS stress tensors, to identify the heat-release effects on their topology. An assessment of the alignment between the eigenvectors corresponding to the largest eigenvalues of these two tensors is also given to provide further information for modelling of the SGS stress tensor. To find out the heat-release effects on the dynamics of the turbulent kinetic energy and enstrophy dissipation, probability density functions (PDFs) and mean values are analysed. The mean SGS shear stress and turbulent kinetic energy both slightly increase in the buffer layer and substantially decrease further away from the wall, due to the heat-release effects. Contrary to the kinetic energy, heat release decreases the mean SGS dissipation of enstrophy in the near-wall region. Moreover, differences in the shapes of the PDFs between the isothermal and exothermic cases indicate changes in the intermittency level of both SGS dissipations. Heat release also increases the SGS stress anisotropy in the near-wall region. Although, the structure of the mean resolved strain-rate tensor only marginally differs between the isothermal and exothermic cases in the near-wall region, substantial differences are observed in the jet area, where compressibility effects are important and heat-release effects are found to promote compression states. The differences in the relative alignment between the SGS stress and resolved strain-rate tensors in the isothermal and exothermic cases are discussed in connection with the differences in the SGS dissipation of kinetic energy.