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 indexes are sensitive to selecting the EPI readout-encoding bandwidth at high-field MRI

Diffusion tensor magnetic resonance imaging (DT-MRI) is generally performed using an echo planar imaging (EPI) acquisition to map directional water diffusion. However, the oscillating magnetic field gradients of the EPI acquisition can result in considerable mechanical vibrations, which lead, in turn, to magnetic field fluctuations causing Nyquist ghosting in the EPI data. The objective of this study was to investigate effects of EPI readout gradient modulation frequency, which is directly associated with the EPI readout bandwidth (BW), on the accuracy of DT-MRI measurements in a high magnetic field system. A spherical water phantom was used to study the relationship between the EPI BW and the Nyquist ghost for a spin-echo EPI acquisition with a matrix size of 128x128, complemented by diffusion sensitization gradients of up to b=800 s/mm(2) along six directions for DT-MRI. Nine volunteers (four males and five females) were studied using EPI at different BW acquisitions. Analysis of variance was used to investigate the EPI BW effects. The phantom studies demonstrated a systematic relationship between BWs and the intensities of Nyquist ghosts. In the human brain studies, EPI BW variations substantially corrupted diffusion anisotropy indexes (i.e., fractional anisotropy and relative anisotropy) (F=10.5, P=.0001) but were unrelated to diffusion-encoding directions (F=0.14, P=.98). It was possible to minimize BW dependence (F=1.48, P=.25) by tuning the modulation frequency of the EPI readout gradient. In conclusion, diffusion anisotropic indexes are sensitive to the readout BW of EPI due to associated Nyquist ghosting. However, the effect can be minimized by tuning the modulation frequency of the EPI readout gradient, that is, the EPI BW, to a range outside the harmonics of mechanical gradient vibrations.     

Microstructural and physiological features of tissues elucidated by quantitative-diffusion-tensor MRI. 1996

Quantitative-diffusion-tensor MRI consists of deriving and displaying parameters that resemble histological or physiological stains, i.e., that characterize intrinsic features of tissue microstructure and microdynamics. Specifically, these parameters are objective, and insensitive to the choice of laboratory coordinate system. Here, these two properties are used to derive intravoxel measures of diffusion isotropy and the degree of diffusion anisotropy, as well as intervoxel measures of structural similarity, and fiber-tract organization from the effective diffusion tensor, D, which is estimated in each voxel. First, D is decomposed into its isotropic and anisotropic parts, 〈D〉 I and D – 〈D〉 I, respectively (where 〈D〉 = Trace(D)/3 is the mean diffusivity, and I is the identity tensor). Then, the tensor (dot) product operator is used to generate a family of new rotationally and translationally invariant quantities. Finally, maps of these quantitative parameters are produced from high-resolution diffusion tensor images (in which D is estimated in each voxel from a series of 2D-FT spin-echo diffusion-weighted images) in living cat brain. Due to the high inherent sensitivity of these parameters to changes in tissue architecture (i.e., macromolecular, cellular, tissue, and organ structure) and in its physiologic state, their potential applications include monitoring structural changes in development, aging, and disease.     

A quantitative comparison of two methods to correct eddy current-induced distortions in DT-MRI

Eddy current-induced geometric distortions of single-shot, diffusion-weighted, echo-planar (DW-EP) images are a major confounding factor to the accurate determination of water diffusion parameters in diffusion tensor MRI (DT-MRI). Previously, it has been suggested that these geometric distortions can be removed from brain DW-EP images using affine transformations determined from phantom calibration experiments using iterative cross-correlation (ICC). Since this approach was first described, a number of image-based registration methods have become available that can also correct eddy current-induced distortions in DW-EP images. However, as yet no study has investigated whether separate eddy current calibration or image-based registration provides the most accurate way of removing these artefacts from DT-MRI data. Here we compare how ICC phantom calibration and affine FLIRT (http://www.fmrib.ox.ac.uk), a popular image-based multi-modal registration method that can correct both eddy current-induced distortions and bulk subject motion, perform when registering DW-EP images acquired with different slice thicknesses (2.8 and 5 mm) and b-values (1000 and 3000 s/mm(2)). With the use of consistency testing, it was found that ICC was a more robust algorithm for correcting eddy current-induced distortions than affine FLIRT, especially at high b-value and small slice thickness. In addition, principal component analysis demonstrated that the combination of ICC phantom calibration (to remove eddy current-induced distortions) with rigid body FLIRT (to remove bulk subject motion) provided a more accurate registration of DT-MRI data than that achieved by affine FLIRT.     

Effects of random subject rotation on optimised diffusion gradient sampling schemes in diffusion tensor MRI

The choice of the number (N) and orientation of diffusion sampling gradients required to measure accurately the water diffusion tensor remains contentious. Monte Carlo studies have suggested that between 20 and 30 uniformly distributed sampling orientations are required to provide robust estimates of water diffusions parameters. These simulations have not, however, taken into account what effect random subject motion, specifically rotation, might have on optimised gradient schemes, a problem which is especially relevant to clinical diffusion tensor MRI (DT-MRI). Here this question is investigated using Monte Carlo simulations of icosahedral sampling schemes and in vivo data. These polyhedra-based schemes, which have the advantage that large N can be created from optimised subsets of smaller N, appear to be ideal for the study of restless subjects since if scanning needs to be prematurely terminated it should be possible to identify a subset of images that have been acquired with a near optimised sampling scheme. The simulations and in vivo data show that as N increases, the rotational variance of fractional anisotropy (FA) estimates becomes progressively less dependent on the magnitude of subject rotation (), while higher FA values are progressively underestimated as increases. These data indicate that for large subject rotations the B-matrix should be recalculated to provide accurate diffusion anisotropy information.     

Influence of the Structure of Field Invariants on the Kinetics of Forming Two-Dimensional Dissipative Atomic Lattices

The properties of field invariants (ellipticity, intensity, total phase, and rotation angles of the polarization ellipse) for 3- and 4-beam 2D configurations are examined. In the sub-Doppler cooling approximation, the structure of the radiation force and diffusion tensor resolved into spatial gradients of field invariants is presented. The atomic dynamics is modeled numerically based on the Langevin equation. The dependence of the spatial structure of field invariants on the spatial structure of the atomic lattice is demonstrated on examples of specific models.     

Study of a positive corona discharge in argon at different pressures

The corona discharge in argon at atmospheric pressure has been studied by means of a 2D model. The reduced characteristic derived from the experimental data has been described by linear regressions for the different pressures and the two studied inter-electrode distances thus confirming the validity of Townsend's approximation also in case of point to plane configuration and argon as process gas. The model validated this hypothesis which has been attributed to the minor influence of space charge in the ionization zone. Its effect is, on the other hand, more significant in the drift zone where the electric field is greatly enhanced, leading, for higher currents, to the formation of a spark gap. Electron and ion distributions allow the influence of structural (electrode configurations and distance) and operative (pressure and discharge current) parameters to be evaluated including the current loss due to diffusion through different confining boundaries.     

Analytical computation of the eigenvalues and eigenvectors in DT-MRI

In this paper a noniterative algorithm to be used for the analytical determination of the sorted eigenvalues and corresponding orthonormalized eigenvectors obtained by diffusion tensor magnetic resonance imaging (DT-MRI) is described. The algorithm uses the three invariants of the raw water spin self-diffusion tensor represented by a 3 x 3 positive definite matrix and certain math functions that do not require iteration. The implementation requires a positive definite mask to preserve the physical meaning of the eigenvalues. This algorithm can increase the speed of eigenvalue/eigenvector calculations by a factor of 5-40 over standard iterative Jacobi or singular-value decomposition techniques. This approach may accelerate the computation of eigenvalues, eigenvalue-dependent metrics, and eigenvectors especially when having high-resolution measurements with large numbers of slices and large fields of view.     

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.     

Segmenting brain white matter, gray matter and cerebro-spinal fluid using diffusion tensor-MRI derived indices.

Based on its ability to provide quantitative information about tissue microstructure, diffusion tensor magnetic resonance imaging (DT-MRI) might be a valuable approach to improve the reliability of segmentation of the various brain tissues.In this study, a fully automated and easy-to-implement technique based on 2D histogram analysis of DT-MRI derived images was used to segment white and gray matter of the brain from 10 healthy subjects (aged = 27-56 years). The results obtained with this novel segmentation strategy were compared to those achieved by two experienced observers using an operator-dependent segmentation on the dual-echo scans.Visual inspection of the segmented tissues from a third senior observer disclosed that the automated technique worked properly on all images from all subjects and was more accurate than the human raters in defining thalamus white and gray matter portions as well as in tissue classification at the external brain edge. In addition, this segmentation technique resulted in an average gray/white matter ratio similar to that reported by post-mortem assessment. The application of the operator-dependent segmentation strategy was extremely time-consuming and the two observers achieved poorly reproducible results.Segmenting brain white and gray matter using information from DT-MRI proved to be an accurate approach with the potential for improving the understanding of the pathophysiology of many neurologic conditions.     

CPMG relaxation by diffusion with constant magnetic field gradient in a restricted geometry: numerical simulation and application

Carr-Purcell-Meiboom-Gill (CPMG) measurements are the primary nuclear magnetic resonance (NMR) technique used for evaluating formation properties and reservoir fluid properties in the well logging industry and laboratory sample analysis. The estimation of bulk volume irreducible (BVI), permeability, and fluid type relies on the accurate interpretation of the spin-spin relaxation time (T(2)) distribution. The interpretation is complicated when spin's self-diffusion in an inhomogeneous field and restricted geometry becomes dominant. The combined effects of field gradient, diffusion, and a restricted geometry are not easily evaluated analytically. We used a numerical method to evaluate the dependence of the free and restricted diffusion on the system parameters in the absence of surface relaxation, which usually can be neglected for the non-wetting fluids (e.g., oil or gas). The parameter space that defines the relaxation process is reduced to two dimensionless groups: D* and tau*. Three relaxation regimes: free diffusion, localization, and motionally averaging regimes are identified in the (log(10)D*, log(10)tau*) domain. The hypothesis that the normalized magnetization, M*, relaxes as a single exponential with a constant dimensionless relaxation time T*(2) is justified for most regions of the parameter space. The numerical simulation results are compared with the analytical solutions from the contour plots of T*(2). The locations of the boundaries between different relaxation regimes, derived from equalizing length scales, are challenged by observed discrepancies between numerical and analytical solutions. After adjustment of boundaries by equalizing T*(2), numerical simulation result and analytical solution match each other for every relaxation regime. The parameters, fluid diffusivity and pore length, can be estimated from analytical solutions in the free diffusion and motionally averaging regimes, respectively. Estimation of the parameters near the boundaries of the regimes may require numerical simulation.     

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.     

物质纯重力场部分的能量-动量张量研究

认为物质的质量(能量)存在形式可分为两部分，一部分是以纯物质形式存在的，另一部分是以纯重力场形式存在的.物质质量(能量)这两种形式各自对应着相应的能量 动量张量，物质总的能量-动量张量可表示为Tμν=T(Ⅰ)μν+T(Ⅱ)μν,这里，T(Ⅰ)μν，T（Ⅱ）μν分别代表物质纯物质部分和纯重力场部分的能量-动量张量.通过类比电磁理论，定义：ωμ≡-c2gμ0/g00,并引入一个反对称张量Dμν=ωμ/xν-ων/xμ，则物质纯重力场部分的能量-动量张量为T(Ⅱ)μν=(DμρDρν-gμνDαβDαβ/4 关键词： 能量-动量张量 纯重力场 重力场方程 标量重力势 矢量重力势     

Effects of permeable boundaries on the diffusion-attenuated MR signal: insights from a one-dimensional model

The local magnetization distribution M(x,t) and the net MR signal S arising from a one-dimensional periodic structure with permeable barriers in a Tanner-Stejskal pulsed-field gradient experiment are considered. In the framework of the narrow pulse approximation, the general expressions for M(x,t) and S as functions of diffusion time and the bipolar field gradient strength are obtained and analyzed. In contrast to a system with impermeable boundaries, the signal S as a function of the b-value is modeled well as a bi-exponential decay not only in the short-time regime but also in the long-time regime. At short diffusion times, the local magnetization M(x,t) is strongly spatially inhomogeneous and the two exponential components describing S have a clear physical interpretation as two "population fractions" of the slow- and fast-diffusing quasi-compartments (pools). In the long-diffusion time regime, the two exponential components do not have clear physical meaning but rather serve to approximate a more complex functional signal form. The average diffusion propagator, obtained by means of standard q-space analysis procedures in the long-diffusion time regime is explored; its structure creates the deceiving appearance of a system with multiple compartments of different sizes, while in reality, it reflects the permeable nature of boundaries in a system with multiple compartments all of the same size.     

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.     

Adaptive registration of diffusion tensor images on lie groups

With diffusion tensor imaging (DTI), more exquisite information on tissue microstructure is provided for medical image processing. In this paper, we present a locally adaptive topology preserving method for DTI registration on Lie groups. The method aims to obtain more plausible diffeomorphisms for spatial transformations via accurate approximation for the local tangent space on the Lie group manifold. In order to capture an exact geometric structure of the Lie group, the local linear approximation is efficiently optimized by using the adaptive selection of the local neighborhood sizes on the given set of data points. Furthermore, numerical comparative experiments are conducted on both synthetic data and real DTI data to demonstrate that the proposed method yields a higher degree of topology preservation on a dense deformation tensor field while improving the registration accuracy.     

Importance of exactb-tensor calculation for quantitative diffusion tensor imaging and tracking of neuronal fiber bundles

Quantitative diffusion tensor imaging (DTI) is a novel method of magnetic resonance (MR) imaging providing information on the brain’s microstructure in vivo. DTI can be effectively measured with modern clinical MR scanners. However, imaging sequence details required for accurateb matrix calculation and for following DTI quantification are normally unknown to the user. In this work, we investigated the accuracy ofb value approximation if theb matrix is calculated without taking into account the effect of imaging gradients. It was found that an error of more than 4% in DTI estimation arises for a quite typical brain imaging protocol. The errors in mean diffusivity and fractional anisotropy index depend on diffusion tensor shape and eigenvectors orientation and exceed noise level in DTI quantification. These errors however have a strong impact on fiber tracking — up to 30% difference was found between the fiber tracks corresponding to exact and approximate calculated DTI data. Since these errors are dependent on imaging parameters and sequence implementation, accurateb matrix calculations are important for adequate comparison between data acquired on different MR scanners and also for data measured with the different imaging protocols.     

Sum-Frequency Generation from a Thin Cylindrical Layer

In the Rayleigh–Gans–Debye approximation, we have solved the problem of the sum-frequency generation by two plane elliptically polarized electromagnetic waves from the surface of a dielectric particle of a cylindrical shape that is coated by a thin layer possessing nonlinear optical properties. The formulas that describe the sum-frequency field have been presented in the tensor and vector forms for the second-order nonlinear dielectric susceptibility tensor, which was chosen in the general form, containing chiral components. Expressions describing the sum-frequency field from the cylindrical particle ends have been obtained for the case of a nonlinear layer possessing chiral properties. Three-dimensional directivity patterns of the sum-frequency radiation have been analyzed for different combinations of parameters (angles of incidence, degrees of ellipticity, orientations of polarization ellipses, cylindrical particle dimensions). The mathematical properties of the spatial distribution functions of the sum-frequency field, which characterize the symmetry of directivity patterns, have been revealed.     

含物质纯重力场部分贡献的静态和稳态重力场研究

给出了包含重力场贡献在内具有宇宙因子项最普遍形式的重力场方程为Rμν-gμνR/2+λgμν=8πG(T(Ⅰ)μν+T(Ⅱ)μν)/c4，这里λ为Einstein宇宙常数，Ｔ（Ⅰ）μν，Ｔ（Ⅱ）μν分别代表物质纯物质部分和纯重力场部分的能量-动量张量.物质纯重力场部分的能量-动量张量表述为T(Ⅱ)μν=(DμρＤρν-gμνＤαβＤαβ/4)/4πG，式中Dμν的定义为Ｄμν＝ωμ／xν-ων／xμ，ωμ≡-c2gμ0/g00.并用重力场贡献在内最普遍形式的重力场方程分别研究了几个大家所熟悉的静态和稳态重力场，像带有Einstein宇宙因子λ项球对称纯物质球外部静态度规、静态荷电球外部度规、匀速转动星体外部度规及理想纯物质星体内部静态平衡等,并进行了讨论. 关键词： 能量动量张量 重力场方程 静态重力场 稳态重力场