A tensor approach to double wave vector diffusion-weighting experiments on restricted diffusion

Previously, it has been shown theoretically that in case of restricted diffusion, e.g. within isolated pores or cells, a measure of the pore size, the mean radius of gyration, can be estimated from double wave vector diffusion-weighting experiments. However, these results are based on the assumption of an isotropic orientation distribution of the pores or cells which hampers the applicability to samples with anisotropic or unknown orientation distributions, such as biological tissue. Here, the theoretical considerations are re-investigated and generalized in order to describe the signal dependency for arbitrary orientation distributions. The second-order Taylor expansion of the signal delivers a symmetric rank-2 tensor with six independent elements if the two wave vectors are concatenated to a single six-element vector. With this tensor approach the signal behavior for arbitrary wave vectors and orientation distributions can be described as is demonstrated by numerical simulations. The rotationally invariant trace of the tensor represents a pore size measure and can be determined from three orthogonal directions with parallel and antiparallel orientation of the two wave vectors. Thus, the presented tensor approach may help to improve the applicability of double wave vector diffusion-weighting experiments to determine pore or cell sizes, in particular in biological tissue.     

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.     

Covariant stringlike dynamics of scroll wave filaments in anisotropic cardiac tissue

It has been hypothesized that stationary scroll wave filaments in cardiac tissue describe a geodesic in a curved space whose metric is the inverse diffusion tensor. Several numerical studies support this hypothesis, but no analytical proof has been provided yet for general anisotropy. In this Letter, we derive dynamic equations for the filament in the case of general anisotropy. These equations are covariant under general spatial coordinate transformations and describe the motion of a stringlike object in a curved space whose metric tensor is the inverse diffusion tensor. Therefore the behavior of scroll wave filaments in excitable media with anisotropy is similar to the one of cosmic strings in a curved universe. Our dynamic equations are valid for thin filaments and for general anisotropy. We show that stationary filaments obey the geodesic equation.     

A Monte Carlo simulation of image misalignment effects in diffusion tensor imaging

Diffusion tensor imaging (DTI) and tractography are noninvasive MRI methods, providing an insight on microscopic structural information of anisotropic tissues in vivo. The success of this technique stems on a watchful choice of imaging parameters and post-acquisition reconstruction. In the present work, we have focused on the problem of residual linear image misalignment in the DTI data and its effects on the parameters of the diffusion tensor and fiber tracking in human brain. We demonstrate substantial sensitivity of the reconstructed diffusion tensor and fiber tractography on increasing amplitude of artificially induced random image misalignment in the DTI. We show that already a submillimeter image misalignment in the DTI is an important source of error, which may potentially mask pathological presentations of the diseases and may partially explain variations in the results obtained from the DTI. Finally, we evaluated four implementations of image registrations and demonstrate their variable performance. This further supports the fact that a robust image registration must be performed to ensure reliable and reproducible diffusion tensor mapping and reconstruction of white matter (WM) fibers.     

Numerical simulations of short-mixing-time double-wave-vector diffusion-weighting experiments with multiple concatenations on whole-body MR systems

Double- or two-wave-vector diffusion-weighting experiments with short mixing times in which two diffusion-weighting periods are applied in direct succession, are a promising tool to estimate cell sizes in the living tissue. However, the underlying effect, a signal difference between parallel and antiparallel wave vector orientations, is considerably reduced for the long gradient pulses required on whole-body MR systems. Recently, it has been shown that multiple concatenations of the two wave vectors in a single acquisition can double the modulation amplitude if short gradient pulses are used. In this study, numerical simulations of such experiments were performed with parameters achievable with whole-body MR systems. It is shown that the theoretical model yields a good approximation of the signal behavior if an additional term describing free diffusion is included. More importantly, it is demonstrated that the shorter gradient pulses sufficient to achieve the desired diffusion weighting for multiple concatenations, increase the signal modulation considerably, e.g. by a factor of about five for five concatenations. Even at identical echo times, achieved by a shortened diffusion time, a moderate number of concatenations significantly improves the signal modulation. Thus, experiments on whole-body MR systems may benefit from multiple concatenations.     

Directional sensitivity of anomalous diffusion in human brain assessed by tensorial fractional motion model

Anisotropic diffusion in the nervous system is most commonly modeled by apparent diffusion tensor, which is based on regular diffusion theory. However, the departure of diffusion-induced signal attenuation from a mono-exponential form implies that there is anomalous diffusion. Recently, a novel diffusion NMR theory based on the fractional motion (FM) model, which is an anomalous diffusion model, has been proposed. While the FM model has been applied to both healthy subjects and tumor patients, its anisotropy in the nervous system remains elusive. In this study, this issue was addressed by measuring the FM-related parameters in 12 non-collinear directions. A metric to quantify the directional deviation was derived. Furthermore, the FM-related parameters were modeled as tensors and analyzed in analogy with the conventional diffusion tensor imaging (DTI). Experimental results, which were obtained for 15 healthy subjects at 3T, exhibited pronounced anisotropy of the FM-related parameters, although the effects were smaller than the apparent diffusion coefficient (ADC). The tensorial nature for α, which is the Noah exponent in the FM model, showed behavior similar to the ADC, especially the principal eigenvector for α aligned with the dominant white matter fiber directions. The Hurst exponent H in the FM model, however, showed no correlation with the major fiber directions. The anisotropy of the FM model may provide complementary information to DTI and may have potential for tractography and detecting brain abnormalities.     

Unified boundary integral equation for the scattering of elastic and acoustic waves: solution by the method of moments

A unified boundary integral equation (BIE) is developed for the scattering of elastic and acoustic waves. Traditionally, the elastic and acoustic wave problems are solved separately with different BIEs. The elastic wave case is represented in a vector BIE with the traction and displacement vectors as unknowns whereas the acoustic wave case is governed by a scalar BIE with velocity potential or pressure as unknowns. Although these two waves can be unified in the form of a partial differential equation, the unified form in its BIE counterpart has not been reported. In this work, we derive the unified BIE for these two waves and then show that the acoustic wave case can be derived from this BIE by introducing a shielding loss for small shear modulus approximation; hence only one code needs to be maintained for both elastic and acoustic wave scattering. We also derive the asymptotic Green's tensor for zero shear modulus and solve the corresponding vector equation. We employ the method of moments, which has been widely used in electromagnetics, as a numerical tool to solve the BIEs involved. Our numerical experiments show that it can also be used robustly in elastodynamics and acoustics.     

Nonergodicity mimics inhomogeneity in single particle tracking

Most statistical theories of anomalous diffusion rely on ensemble-averaged quantities such as the mean squared displacement. Single molecule tracking measurements require, however, temporal averaging. We contrast the two approaches in the case of continuous-time random walks with a power-law distribution of waiting times psi(t) proportional to t{-1-alpha}, with 0相似文献   

Unified boundary integral equation for the scattering of elastic and acoustic waves: solution by the method of moments

A unified boundary integral equation (BIE) is developed for the scattering of elastic and acoustic waves. Traditionally, the elastic and acoustic wave problems are solved separately with different BIEs. The elastic wave case is represented in a vector BIE with the traction and displacement vectors as unknowns whereas the acoustic wave case is governed by a scalar BIE with velocity potential or pressure as unknowns. Although these two waves can be unified in the form of a partial differential equation, the unified form in its BIE counterpart has not been reported. In this work, we derive the unified BIE for these two waves and then show that the acoustic wave case can be derived from this BIE by introducing a shielding loss for small shear modulus approximation; hence only one code needs to be maintained for both elastic and acoustic wave scattering. We also derive the asymptotic Green's tensor for zero shear modulus and solve the corresponding vector equation. We employ the method of moments, which has been widely used in electromagnetics, as a numerical tool to solve the BIEs involved. Our numerical experiments show that it can also be used robustly in elastodynamics and acoustics.     

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.     

Learning-based structurally-guided construction of resting-state functional correlation tensors

Functional magnetic resonance imaging (fMRI) measures changes in blood-oxygenation-level-dependent (BOLD) signals to detect brain activities. It has been recently reported that the spatial correlation patterns of resting-state BOLD signals in the white matter (WM) also give WM information often measured by diffusion tensor imaging (DTI). These correlation patterns can be captured using functional correlation tensor (FCT), which is analogous to the diffusion tensor (DT) obtained from DTI. In this paper, we propose a noise-robust FCT method aiming at further improving its quality, and making it eligible for further neuroscience study. The novel FCT estimation method consists of three major steps: First, we estimate the initial FCT using a patch-based approach for BOLD signal correlation to improve the noise robustness. Second, by utilizing the relationship between functional and diffusion data, we employ a regression forest model to learn the mapping between the initial FCTs and the corresponding DTs using the training data. The learned forest can then be applied to predict the DTI-like tensors given the initial FCTs from the testing fMRI data. Third, we re-estimate the enhanced FCT by utilizing the DTI-like tensors as a feedback guidance to further improve FCT computation. We have demonstrated the utility of our enhanced FCTs in Alzheimer's disease (AD) diagnosis by identifying mild cognitive impairment (MCI) patients from normal subjects.     

Anisotropic velocity correlation spectrum at small scales in a homogeneous turbulent shear flow

A simple theoretical analysis and direct numerical simulations on 512(3) grid points suggest that the velocity correlation spectrum tensor in the inertial subrange of homogeneous turbulent shear flow at high Reynolds number is given by a simple form that is an anisotropic function of the wave vector. The tensor is determined by the rate of the strain tensor of the mean flow, the rate of energy dissipation per unit mass, the wave vector, and two nondimensional constants.     

Simultaneous magnetic resonance imaging of diffusion anisotropy and diffusion gradient

The theory of diffusion gradient-weighted MRI (DGWI) is presented in this paper. The Bloch-Torrey equation was modified to include the effect of intravoxel spatial-location variation of water diffusion (diffusion gradient) on MRI signal, in addition to the effect of intravoxel spatial-direction variation of water diffusion (diffusion anisotropy). An analytical solution for a diffusion-encoding spin-echo pulse sequence was derived. Unlike water diffusion which attenuates the image signal intensity, this newly derived solution relates the spatial gradient of the water diffusion with the phase of the image signal. This novel MRI technique directly measures both the water diffusion and its spatial gradient, and thus offers a noninvasive imaging tool to simultaneously investigate the intravoxel inhomogeneity and anisotropy of tissue structures. In addition, as demonstrated with our preliminary data, this new method may be utilized to delineate the interfaces of tissues with different diffusion. This method is an extension of the successful diffusion tensor MRI (DTI), but requires no additional data acquisition. In addition to the measured diffusion tensor, this new method provides measurements of the spatial derivatives of the three principal diffusivities of the tensor, thereby providing additional information for improving white matter fiber tractography.     

A geometric theory for scroll wave filaments in anisotropic excitable media

Scroll waves are an important example of self-organisation in excitable media. In cardiac tissue, scroll waves of electrical activity underlie lethal ventricular arrhythmias and fibrillation. They rotate around a topological line defect which has been termed the filament. Numerical investigation has shown that anisotropy can substantially affect the dynamics of scroll waves. It has recently been hypothesised that stationary scroll wave filaments in cardiac tissue describe geodesics in a space whose metric is the inverse diffusion tensor. Several computational studies have validated this hypothesis, but until now no quantitative theory has been provided to study the effects of anisotropy on scroll wave filaments. Here, we review in detail the recently developed covariant formalism for scroll wave dynamics in general anisotropy and derive the equations of motion of filaments. These equations are fully covariant under general spatial coordinate transformations and describe the motion of filaments in a curved space whose metric tensor is the inverse diffusion tensor. Our dynamic equations are valid for thin filaments and for general anisotropy and we show that stationary filaments obey the geodesic equation. We extend previous work by allowing spatial variations in the determinant of the diffusion tensor and the reaction parameters, leading to drift of the filament.     

Diffusion-direction-dependent imaging: a novel MRI approach for peripheral nerve imaging

A novel magnetic resonance imaging approach, called diffusion-direction-dependent imaging (DDI), is introduced. Due to inherent anisotropic diffusion properties, peripheral nerves can be visualized on diffusion tensor imaging (DTI). The largest signal attenuation on DTI correlates with the direction of a nerve fiber, and the least signal attenuation correlates with the direction perpendicular to the nerve fiber. Since low signal-to-noise ratio is a concern in peripheral nerve DTI, we explored a new approach focusing on the perpendicular diffusion direction. A 36-gradient diffusion direction scheme was used. A mean expected curve specific for peripheral nerves was calculated based on the sciatic nerve and its division into the common peroneal nerve and the tibial nerve in three healthy volunteers. By a simple postprocessing method, a comparison of the mean expected curve and the measured curve was made voxel by voxel, and the sciatic nerve and its division were reconstructed, excluding other tissues. More studies are needed to investigate whether other postprocessing methods or other diffusion direction schemes are more suited for peripheral nerve imaging with DDI. Further studies may also be of interest to investigate whether DDI can be a complementary method to conventional T(1)-weighted and T(2)-weighted sequences in the imaging of peripheral nerve pathology or even in the visualization of other tissues, possibly with different diffusion direction schemes.     

Crossing fibers, diffractions and nonhomogeneous magnetic field: correction of artifacts by bipolar gradient pulses

In recent years, diffusion tensor imaging (DTI) and its variants have been used to describe fiber orientations and q-space diffusion MR was proposed as a means to obtain structural information on a micron scale. Therefore, there is an increasing need for complex phantoms with predictable microcharacteristics to challenge different indices extracted from the different diffusion MR techniques used. The present study examines the effect of diffusion pulse sequence on the signal decay and diffraction patterns observed in q-space diffusion MR performed on micron-scale phantoms of different geometries and homogeneities. We evaluated the effect of the pulse gradient stimulated-echo, the longitudinal eddy current delay (LED) and the bipolar LED (BPLED) pulse sequences. Interestingly, in the less homogeneous samples, the expected diffraction patterns were observed only when diffusion was measured with the BPLED sequence. We demonstrated the correction ability of bipolar diffusion gradients and showed that more accurate physical parameters are obtained when such a diffusion gradient scheme is used. These results suggest that bipolar gradient pulses may result in more accurate data if incorporated into conventional diffusion-weighted imaging and DTI.     

Extension of the double-wave-vector diffusion-weighting experiment to multiple concatenations

Experiments involving two diffusion-weightings in a single acquisition, so-called double- or two-wave-vector experiments, have recently been applied to measure the microscopic anisotropy in macroscopically isotropic samples or to estimate pore or compartment sizes. These informations are derived from the signal modulation observed when varying the wave vectors’ orientations. However, the modulation amplitude can be small and, for short mixing times between the two diffusion-weightings, decays with increased gradient pulse lengths which hampers its detectability on whole-body MR systems. Here, an approach is investigated that involves multiple concatenations of the two diffusion-weightings in a single experiment. The theoretical framework for double-wave-vector experiments of fully restricted diffusion is adapted and the corresponding tensor approach recently presented for short mixing times extended and compared to numerical simulations. It is shown that for short mixing times (i) the extended tensor approach well describes the signal behavior observed for multiple concatenations and (ii) the relative amplitude of the signal modulation increases with the number of concatenations. Thus, the presented extension of the double-wave-vector experiment may help to improve the detectability of the signal modulations observed for short mixing times, in particular on whole-body MR systems with their limited gradient amplitudes.     

基于外调制器的全双工无线信号光传输系统

提出了一种基于双平行马赫 增德尔调制器的全双工矢量信号传输系统。传统的光学倍频技术只适用于NRZ (not return to zero) 等强度调制格式,而不适用于无线通信中常用的MPSK、MQAM等矢量信号调制技术,用传统的倍频技术生成光毫米波后,用滤波器将其2个边带分开,在其中一个边带上调制矢量信号,这样幅度和相位信息被正确保留,且使用了8倍频模块,降低了传输过程中对光电器件的带宽要求。理论分析和仿真结果表明:通过此方法产生的携带在60 GHz载波上的625 Msymbol/s的4QPSK信号,经过20 km单模光纤传输后,误差向量幅度损耗可以忽略不计。     

对于“包含在连续谱量子体系中的决定论性”一文的讨论

对于具有连续能谱的单粒子量子体系,“包含在连续谱量子体系中的决定论性”一文用所谓“双波函数”来描述处于能量本征态的粒子系综中各粒子的量子行为,并且在所谓的“等价定理”中称:双波函数描述在经典极限下将化为经典力学描述.然而,此描述所给出的系综力学量观测值统计分布的预言与通常量子力学不相容;并且,该文对其“等价定理”的证明是不正确的,这个“定理”实际上不成立 关键词： 连续能谱量子体系 双波函数 经典极限