Mapping Tensor Research Articles

Analysis ofb-value calculations in diffusion weighted and diffusion tensor imaging

Diffusion weighted imaging has opened new diagnostic possibilities by using microscopic diffusion of water molecules as a means of image contrast. The directional dependence of diffusion has led to the development of diffusion tensor imaging, which allows us to characterize microscopic tissue geometry. The link between the measured NMR signal and the self-diffusion tensor is established by the so-called b matrices that depend on the gradient's direction, strength, and timing. However, in the calculation of b-matrix elements, the influence of imaging gradients on each element of the b matrix is often neglected. This may cause errors, which in turn leads to an incorrect extraction of diffusion coefficients. In cases where the imaging gradients are high (high spatial resolution), these errors may be substantial. Using a generic pulsed gradient spin-echo (PGSE) imaging sequence, the effects of neglecting the imaging gradients on the b-matrix calculation are demonstrated. By measuring an isotropic phantom with this sequence it can be analytically as well as experimentally shown that large deviations in single b-matrix elements are generated. These deviations are obtained by applying the diffusion weighting in the readout direction of the imaging dimension in combination with relatively large imaging gradients. The systematic errors can be avoided by a full b-matrix calculation considering all the gradients of the sequence or by generating cross-term free signals using the geometric average of two diffusion weighted images with opposite polarity. The importance of calculating the exact b matrices by the proposed methods is based on the fact that more precise diffusion parameters are obtained for extracting correct property maps, such as fractional anisotropy, volume ratio, or conductivity tensor maps. © 2005 Wiley Periodicals, Inc. Concepts Magn Reson Part A 25A: 53–66, 2005

In vivo diffusion tensor mapping of the brain of squirrel monkey, rat, and mouse using single-shot STEAM MRI

The purpose was to assess the potential of half Fourier diffusion-weighted single-shot STEAM MRI for diffusion tensor mapping of animal brain in vivo. A STEAM sequence with image acquisition times of about 500 ms was implemented at 2.35 T using six gradient orientations and b values of 200, 700, and 1200 s mm(-2). The use of half Fourier phase-encoding increased the signal-to-noise ratio by 45% relative to full Fourier acquisitions. Moreover, STEAM-derived maps of the relative anisotropy and main diffusion direction were completely free of susceptibility-induced signal losses and geometric distortions. Within measuring times of 3 h, the achieved resolution varied from 600x700x1000 microm3 for squirrel monkeys to 140x280x720 microm3 for mice. While in monkeys the accessible white matter fiber connections were comparable to those reported for humans, detectable fiber structures in mice focused on the corpus callosum, anterior commissure, and hippocampal fimbria. In conclusion diffusion-weighted single-shot STEAM MRI allows for in vivo diffusion tensor mapping of the brain of squirrel monkeys, rats, and mice without motion artifacts and susceptibility distortions.

SENSE-DTI at 3 T.

While holding vast potential, diffusion tensor imaging (DTI) with single-excitation protocols still faces serious challenges. Limited spatial resolution, susceptibility to magnetic field inhomogeneity, and low signal-to-noise ratio (SNR) may be considered the most prominent limitations. It is demonstrated that all of these shortcomings can be effectively mitigated by the transition to parallel imaging technology and high magnetic field strength. Using the sensitivity encoding (SENSE) technique at 3 T, brain DTI was performed in nine healthy volunteers. Despite enhanced field inhomogeneity, parallel acquisition permitted both controlling geometric distortions and enhancing spatial resolution up to 0.8 mm in-plane. Heightened SNR requirements were met in part by high base sensitivity at 3 T. A further significant increase in SNR efficiency was accomplished by SENSE acquisition, exploiting enhanced encoding speed for echo time reduction. Based on the resulting image data, high-resolution tensor mapping is demonstrated.

Introduction to Symmetry Analysis

Introduction to Symmetry Analysis

Nonlinear FE analysis of residual stresses induced by dislocations in heterostructures

In this paper the field theory of dislocations is used in the finite element analysis of residual stresses in epitaxial layers. By digital processing of the HRTEM image of a GaAs/ZnTe/CdTe system the tensor maps of dislocation distribution are extracted. Such obtained maps are used as the input data to the finite element code. The mathematical foundations of this code are based on the compatibility equations for lattice distortions. The surface tension induced by misfit dislocations is considered here in terms of a 3D boundary-value problem for stress equilibrium in the interfacial zone. The numerical results show how strongly the surface tension depends on the nonlinear elastic behaviour of the crystal structure.

Phase contrast MRI of myocardial 3D strain by encoding contiguous slices in a single shot.

Quantitative measurements of inherently three-dimensional (3D) cardiac strain and strain rate require 3D data; MRI provides uniquely high sensitivity to material strain by combining phase contrast with single-shot acquisition methods, such as echo-planar imaging (EPI). Previous MRI methods applied to 3D strain used multiple two-dimensional (2D) acquisitions and suffered loss of sensitivity due to magnification within the strain calculation of physiologic noise related to cardiac beat-to-beat variability. In the present work, each single-shot acquisition generates 3D image data by acquiring two contiguous 2D Fourier transform (FT) images in a single echo train of an EPI readout. Although strain encoding divides across multiple EPI shots, each strain component is computed only within single-shot data, avoiding noise magnification. Strain tensor maps are displayed using iconic 3D graphics or a simple color code of tensor shape. In a deforming gel phantom, gradient-recalled echo (GRE) MRI movies of 3D strain rates match expected strain fields. In normal human subjects, 3D strain rate tensor movies of heart and brain comprising seven slices in each of seven cardiac phases were completed in 56 heartbeats. Stimulated echo (STE) MRI of net systolic 3D strain was also demonstrated. Two-slices-in-one-shot spatial encoding permits a complete quantitative survey of ventricular 3D strain in under a minute, with routine patient supervision and turnkey image processing.

Detecting dynamic (4D) profiles of degenerative rates in Alzheimer's disease patients, using high-resolution tensor mapping and a brain atlas encoding atrophic rates in a population

Early detection and monitoring of Alzheimer’s Disease (AD) requires tools with unprecedented sensitivity for mapping its dynamic progression. We developed a dynamic brain atlas to store spatially complex profiles of degenerative rates and gray matter loss throughout the brain. Our goals were to (1) create detailed maps of degenerative change and (2) use high-dimensional flows [l] to compare individual patients’ dynamic maps with probabilistic loss rate data in the atlas. We further aimed to define robust measures of change in AD and controls and assess how change rates in different systems are correlated.

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.

Gradient preemphasis calibration in diffusion‐weighted echo‐planar imaging

This article describes a method which enables fast and objective pulse-sequence-specific preemphasis calibration, using standard pulse sequences and system hardware. The method is based on a k-space measurement technique, and has been applied to single-shot, diffusion-weighted, spin-echo, echo-planar imaging (DW-SE-EPI), which is particularly sensitive to eddy-current-induced image distortions. The efficiency of the technique was demonstrated not only by the reduction of eddy-current fields to a negligible level using full preemphasis compensation, but also by the fact that adjustment of the slow time-base alone sufficed for the practical elimination of image distortions in the DW-SE-EPI images and the subsequent diffusion tensor maps (in a phantom and a human brain). By seeking to eliminate directly the effect of eddy-current-induced phase shifts during the EPI data collection, the method is free of the complications and restrictions associated with other eddy-current correction techniques for DW-SE-EPI (such as acquisition of additional calibration scans, intense postprocessing, extensive pulse-sequence modifications), making their use redundant. Magn Reson Med 44:616–624, 2000. © 2000 Wiley-Liss, Inc.

Growth tensor mapping: A rostro-caudal wave of peak growth rates detected in the developing human brain in the first 15 years of life

Growth tensor mapping: A rostro-caudal wave of peak growth rates detected in the developing human brain in the first 15 years of life

Growth patterns in the developing brain detected by using continuum mechanical tensor maps.

The dynamic nature of growth and degenerative disease processes requires the design of sensitive strategies to detect, track and quantify structural change in the brain in its full spatial and temporal complexity. Although volumes of brain substructures are known to change during development, detailed maps of these dynamic growth processes have been unavailable. Here we report the creation of spatially complex, four-dimensional quantitative maps of growth patterns in the developing human brain, detected using a tensor mapping strategy with greater spatial detail and sensitivity than previously obtainable. By repeatedly scanning children (aged 3-15 years) across time spans of up to four years, a rostro-caudal wave of growth was detected at the corpus callosum, a fibre system that relays information between brain hemispheres. Peak growth rates, in fibres innervating association and language cortices, were attenuated after puberty, and contrasted sharply with a severe, spatially localized loss of subcortical grey matter. Conversely, at ages 3-6 years, the fastest growth rates occurred in frontal networks that regulate the planning of new actions. Local rates, profiles, and principal directions of growth were visualized in each individual child.

Gradient preemphasis calibration in diffusion-weighted echo-planar imaging.

This article describes a method which enables fast and objective pulse-sequence-specific preemphasis calibration, using standard pulse sequences and system hardware. The method is based on a k-space measurement technique, and has been applied to single-shot, diffusion-weighted, spin-echo, echo-planar imaging (DW-SE-EPI), which is particularly sensitive to eddy-current-induced image distortions. The efficiency of the technique was demonstrated not only by the reduction of eddy-current fields to a negligible level using full preemphasis compensation, but also by the fact that adjustment of the slow time-base alone sufficed for the practical elimination of image distortions in the DW-SE-EPI images and the subsequent diffusion tensor maps (in a phantom and a human brain). By seeking to eliminate directly the effect of eddy-current-induced phase shifts during the EPI data collection, the method is free of the complications and restrictions associated with other eddy-current correction techniques for DW-SE-EPI (such as acquisition of additional calibration scans, intense postprocessing, extensive pulse-sequence modifications), making their use redundant.

Characterization of and correction for eddy current artifacts in echo planar diffusion imaging.

Magnetic resonance diffusion imaging is potentially an important tool for the noninvasive characterization of normal and pathological tissue. The technique, however, is prone to a number of artifacts that can severely affect its ability to provide clinically useful information. In this study, the problem of eddy current-induced geometric distortions that occur in diffusion images acquired with echo planar sequences was addressed. These geometric distortions produce artifacts in computed maps of diffusion parameters and are caused by misalignments in the individual diffusion-weighted images that comprise the diffusion data set. A new approach is presented to characterize and calibrate the eddy current effects, enabling the eddy current distortions to be corrected in sets of interleaved (or snapshot) echo planar diffusion images. Correction is achieved by acquiring one-dimensional field maps in the read and phase encode direction for each slice and each diffusion step. The method is then demonstrated through the correction of distortions in diffusion images of the human brain. It is shown that by using the eddy current correction scheme outlined, the eddy current-induced artifacts in the diffusion-weighted images are almost completely eliminated. In addition, there is a significant improvement in the quality of the resulting diffusion tensor maps.

Magnetic resonance imaging shows orientation and asymmetry of white matter fiber tracts

Apparent diffusion tensor maps of the human brain were acquired with a magnetic resonance imaging sequence (Gudbjartsson, H., Maier, S.E., Mulkern, R.V., Mórocz, I.A., Patz, S., Jolesz, F.A., Magn. Reson. Med. 36 (1996) 509–519). It was shown that the geometric nature of the apparent diffusion tensors can quantitatively characterize the tissue structure. Display of the orientation and directional uniformity of the water diffusion in the brain demonstrated most of the known major anatomical constituents of human white matter. A comparison of corresponding anatomic regions in the white matter of both hemispheres in 24 healthy volunteers revealed that fiber tracts within the anterior limb of the internal capsule have a significantly higher ( P<0.01) measure of alignment in the right hemisphere. This method offers a unique tool for the in vivo demonstration of neural connectivity in healthy and diseased brain.

FUSION AND TENSORING OF CONFORMAL FIELD THEORY AND COMPOSITE FERMION PICTURE OF FRACTIONAL QUANTUM HALL EFFECT

We propose a new way for describing the transition between two quantum Hall effect states with different filling factors within the framework of rational conformal field theory. Using a particular class of nonunitary theories, we explicitly recover Jain’s picture of attaching flux quanta by the fusion rules of primary fields. Filling higher Landau levels of composite fermions can be described by taking tensor products of conformal theories. The usual projection to the lowest Landau level corresponds then to a simple coset of these tensor products with several U(1)-theories divided out. These two operations — the fusion map and the tensor map — can explain the Jain series and all other observed fractions as exceptional cases. Within our scheme of transitions we naturally find a field with the experimentally observed universal critical exponent 7/3.

Substitution of matrices over rings

For a given commutative ring R with an identity element, we define and study the substitution of a matrix with entries in R into a matrix polynomial or rational function over R . A Bezout-type remainder theorem and a “partial-substitution rule” are derived and used to obtain a number of results. The tensor map is introduced and used to investigate the solvability of linear matrix equations.

Operator substitution

Substitution of an operator into an operator-valued map is defined and studied. A Bezout-type remainder theorem is used to derive a number of results. The tensor map is used to formulate solvability conditions for linear matrix equations. Some applications to system theory are given, in particular an application to the regulator problem.