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.     

基于高斯平均的DTI脑模板构建方法

在获取被试的张量数据后通常对其进行多通道线性平均以得到张量模板.但线性平均不仅会忽略张量中的向量信息,还会使灰质和白质的交界处过于平滑,降低模板的分辨率.为了解决以上问题,本文引入了四元数及高斯加权平均来构建高斯扩散张量成像（Diffusion Tensor Imaging,DTI）脑模板.本文首先对55个健康被试的DTI数据进行预处理,使得数据伪影最小化;再通过扩散张量成像工具包（Diffusion Tensor Imaging ToolKit,DTI-TK）将预处理后的数据进行初步空间标准化;然后将张量通过特征分解得到特征向量和特征值;最后,将由特征向量转化的四元数标量和特征值分别进行高斯加权平均得到平均后的特征向量和特征值,并对其进行重建得到张量模板.实验结果表明相比于线性DTI模板,高斯DTI模板在DTED、COH、DVED、OVL、corr_FA评估指标上表现更优,而IA指标较差,说明本文提出的高斯DTI模板在整体信息保留方面有所优化,但方向信息有所丢失.     

Tensor diffusion level set method for infrared targets contours extraction

A tensor diffusion level set method is presented to extract infrared (IR) targets contour under a sky-mountain-water complex background. The proposed model combines tensor diffusion operator and the eigenvalues of tensor-image into a common energy minimization level set framework. By incorporating the information of image tensor diffusion operator into the external energy term, the level set function can move in a specific way. And eigenvalues of tensor-image are used for the regularization of zero level curves in order to diminish the influence of image ‘clutter’ and noise. An additional benefit of the proposed method is robust to initial conditions. Experimental results show very good performance of the tensor diffusion level set method for IR targets contours extraction.     

Separation of variables in the diffusion equation. I

For both stationary and nonstationary cases, all the coefficients of the linear diffusion equation with a constant diffusion tensor are enumerated for which total separation of the variables is possible by means of differential symmetry operators of a definite structure.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 46–53, November, 1976.     

Excitonic diffusion in thin molecular films

The analyses of exciton diffusion in thin molecular films have shown that the diagonal elements of the diffusion tensor, usually called diffusion coefficients, depend on the layer index labeling layers in the direction of disturbed symmetry. The particular analysis was done for a thin film having four layers. In this structure only two layers are occupied by optical excitations. It means that in the four layer film two films occur in which optical excitations can travel. The subfilm contains a boundary layer that noticeably differs from the subfilm with internal layers. If the subfilm contains the boundary layer, the diffusion coefficient of this layer differs from the diffusion coefficient of any internal layer. If the subfilm contains two internal layers, the diffusion coefficient of these layers are equal, expectably from the viewpoint of physics. The exciton diffusion is very low due to the high exciton energies. This work was supported by the Serbian Ministry of Science and Technology: Grant No 1895.     

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.     

磁共振扩散张量成像中扩散敏感梯度磁场方向分布方案的研究进展

磁共振扩散张量成像可以定量无创研究人体内水分子在三维空间中的各向异性扩散规律,进而获取重要的病理及生理信息.为了得到水分子各向异性扩散信息,需要按照一定的方案依次施加不同方向的扩散敏感梯度磁场,测量水分子在这些方向上的扩散系数用以估算扩散张量.扩散张量成像测量结果的准确程度受梯度磁场方向分布方案的影响,本文对扩散敏感梯度磁场方向分布方案进行综述,包括完全随机方案、启发式方案、规则多面体式方案和数值优化方案等,分析这些方案的优势与局限性,并提出需进一步研究的问题.     

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.     

A note on confined diffusion

The random motion of a Brownian particle confined in some finite domain is considered. Quite generally, the relevant statistical properties involve infinite series, whose coefficients are related to the eigenvalues of the diffusion operator. Because the latter depend on space dimensionality and on the particular shape of the domain, an analytical expression is in most circumstances not available. In this article, it is shown that the series may in some circumstances sum up exactly. Explicit calculations are performed for 2D diffusion restricted to a circular domain and 3D diffusion inside a sphere. In both cases, the short-time behaviour of the mean square displacement is obtained.     

Theory of intramolecular spin relaxation by translational diffusion in locally ordered fluids

In locally ordered fluids, such as macromolecular solutions, clays and lyotropic liquid crystals, nuclear spin relaxation can be induced by modulation, through translational diffusion of the fluid molecules, of the magnitude and orientation of the residual intramolecular spin-lattice coupling tensor, which is only partially averaged by local molecular motions near an interface. A theory of spin relaxation in locally ordered fluids bounded by planar interfaces is developed, with special emphasis on effects of translational diffusion. The theory is based on a continuous diffusion model (CDM) which, in contrast to the commonly adopted discrete exchange model (DEM), treats equilibrium and time-dependent distribution functions in a self-consistent way. A striking feature of translational diffusion in heterogeneous systems is the abundance of reencounters with previously visited interfacial regions. It is demonstrated that these diffusional reencounters, which are inherent in the CDM theory, may lead to a relaxation behaviour which is qualitatively different from that predicted by the DEM theory. Furthermore, it is seen that the widespread concept of intrinsic relaxation rate (associated with a spatial region) and the fast/slow exchange classification are not generally valid. The formal framework of the CDM theory allows molecular interactions of any complexity to be introduced. In this paper a mean-field model based on the nonlinear Poisson-Boltzmann equation is used to obtain analytic expressions for the spectral density functions that determine the relaxation behaviour in the presence and in the absence of spectral line splittings.     

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     

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.     

On the definition of the natural frequency of oscillations in nonlinear large rotation problems

Computational multibody system algorithms allow for performing eigenvalue analysis at different time points during the simulation to study the system stability. The nonlinear equations of motion are linearized at these time points, and the resulting linear equations are used to determine the eigenvalues and eigenvectors of the system. In the case of linear systems, the system eigenvalues remain the same under a constant coordinate transformation; and zero eigenvalues are always associated with rigid body modes, while nonzero eigenvalues are associated with non-rigid body motion. These results, however, cannot in general be applied to nonlinear multibody systems as demonstrated in this paper. Different sets of large rotation parameters lead to different forms of the nonlinear and linearized equations of motion, making it necessary to have a correct interpretation of the obtained eigenvalue solution. As shown in this investigation, the frequencies associated with different sets of orientation parameters can differ significantly, and rigid body motion can be associated with non-zero oscillation frequencies, depending on the coordinates used. In order to demonstrate this fact, the multibody system motion equations associated with the system degrees of freedom are presented and linearized. The resulting linear equations are used to define an eigevalue problem using the state space representation in order to account for general damping that characterizes multibody system applications. In order to demonstrate the significant differences between the eigenvalue solutions associated with two different sets of orientation parameters, a simple rotating disk example is considered in this study. The equations of motion of this simple example are formulated using Euler angles, Euler parameters and Rodriguez parameters. The results presented in this study demonstrate that the frequencies obtained using computational multibody system algorithms should not in general be interpreted as the system natural frequencies, but as the frequencies of the oscillations of the coordinates used to describe the motion of the system.     

广义BVP电路系统的振荡行为及其非光滑分岔机理

探讨了具有分段线性特性的广义BVP电路系统随参数变化的复杂动力学演化过程. 其非光滑分界面将相空间划分成不同的区域, 分析了各区域中平衡点的稳定性, 得到其相应的简单分岔和Hopf分岔的临界条件. 给出了不同分界面处广义Jacobian矩阵特征值随辅助参数变化的分布情况, 讨论了分界面处系统可能存在的分岔行为, 指出当广义特征值穿越虚轴时可能引起Hopf分岔, 导致系统由周期振荡转变为概周期振荡, 而当出现零特征值时则导致系统的振荡在不同平衡点之间转换. 针对系统的两种典型振荡行为, 结合数值模拟验证了理论分析的结果.     

The reduction of the sorting bias in the eigenvalues of the diffusion tensor.

One of the most intrinsic quantities when measuring the diffusion properties of a system is the set of principal diffusivities, which represents diffusion along the fibre axes. System noise is a well-known cause of systematic sorting bias when closely spaced diffusivities are ordered according to their magnitude and leads to their inaccurate estimation. This paper describes a new method for the ordering of the principal diffusivities in which local fibre directional coherence was used as a basis for sorting. The method was applied and tested in computer simulations and experimental data acquired in an isotropic water phantom and healthy human brain. Our results demonstrate that this method leads to significant reduction in the sorting bias in comparison to other techniques and thus a more accurate estimation of the eigenvalues. The method is advantageous over other proposed alternatives to the conventional magnitude sorting method because it is not reliant on a large region-of-interest averaging scheme.     

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.     

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.     

A practical approach to in vivo high-resolution diffusion tensor imaging of rhesus monkeys on a 3-T human scanner

The nonhuman primate brain study provides important supplemental means for human brain exploration since the two species share close anatomical and functional similarities. MR diffusion tensor imaging (DTI) in human brain has revealed exquisite details of brain structures especially in the brain white matter. However, most previous monkey brain DTI results lack the spatial resolution in comparison to the conventional tracing and postmortem imaging methods, especially when it is acquired in commonly available human MRI scanners of field strength of 3 T or lower. To meet the increasing demands for nonhuman primate DTI studies, we proposed an in vivo high-resolution monkey DTI acquisition protocol that is practically feasible and combined it with an improved postprocessing procedure for a 3-T human scanner. The acquisition protocol, susceptibility distortion correction method with phase reversal acquisition, and postprocessing steps were proved to be effective in our study of rhesus monkeys. Results from diffusion tensor estimations and fiber tractography at 1 x 1 x 1 mm(3) resolution were found to be comparable to previous ex vivo DTI studies with much longer acquisition times. Effects of image resolution were evaluated and it was confirmed that the partial volume effect due to the larger voxel size in low-resolution data biased the diffusion tensor estimation and produced erroneous fiber tractography. Our results suggest that in vivo high-resolution monkey brain DTI can be achieved within practical time, which allows accurate diffusion tensor estimation and fiber tractography in monkey brains, so that the complex anatomical structures within many small but important anatomic structures can be delineated.