Exploiting Spatial Information to Enhance DTI Segmentations via Spatial Fuzzy c-Means with Covariance Matrix Data and Non-Euclidean Metrics

A diffusion tensor models the covariance of the Brownian motion of water at a voxel and is required to be symmetric and positive semi-definite. Therefore, image processing approaches, designed for linear entities, are not effective for diffusion tensor data manipulation, and the existence of artefacts in diffusion tensor imaging acquisition makes diffusion tensor data segmentation even more challenging. In this study, we develop a spatial fuzzy c-means clustering method for diffusion tensor data that effectively segments diffusion tensor images by accounting for the noise, partial voluming, magnetic field inhomogeneity, and other imaging artefacts. To retain the symmetry and positive semi-definiteness of diffusion tensors, the log and root Euclidean metrics are used to estimate the mean diffusion tensor for each cluster. The method exploits spatial contextual information and provides uncertainty information in segmentation decisions by calculating the membership values for assigning a diffusion tensor at one voxel to different clusters. A regularisation model that allows the user to integrate their prior knowledge into the segmentation scheme or to highlight and segment local structures is also proposed. Experiments on simulated images and real brain datasets from healthy and Spinocerebellar ataxia 2 subjects showed that the new method was more effective than conventional segmentation methods.

A noninvasive fluorescence imaging-based platform measures 3D anisotropic extracellular diffusion

Diffusion is a major molecular transport mechanism in biological systems. Quantifying direction-dependent (i.e., anisotropic) diffusion is vitally important to depicting how the three-dimensional (3D) tissue structure and composition affect the biochemical environment, and thus define tissue functions. However, a tool for noninvasively measuring the 3D anisotropic extracellular diffusion of biorelevant molecules is not yet available. Here, we present light-sheet imaging-based Fourier transform fluorescence recovery after photobleaching (LiFT-FRAP), which noninvasively determines 3D diffusion tensors of various biomolecules with diffusivities up to 51 µm2 s−1, reaching the physiological diffusivity range in most biological systems. Using cornea as an example, LiFT-FRAP reveals fundamental limitations of current invasive two-dimensional diffusion measurements, which have drawn controversial conclusions on extracellular diffusion in healthy and clinically treated tissues. Moreover, LiFT-FRAP demonstrates that tissue structural or compositional changes caused by diseases or scaffold fabrication yield direction-dependent diffusion changes. These results demonstrate LiFT-FRAP as a powerful platform technology for studying disease mechanisms, advancing clinical outcomes, and improving tissue engineering.

Diffusion Tensor Imaging Group Analysis Using Tract Profiling and Directional Statistics

Group analysis in diffusion tensor imaging is challenging. Comparisons of tensor morphology across groups have typically been performed on scalar measures of diffusivity, such as fractional anisotropy (FA), disregarding the complex three-dimensional morphologies of diffusion tensors. Scalar measures consider only the magnitude of the diffusion but not directions. In the present study, we have introduced a new approach based on directional statistics to use directional information of diffusion tensors in statistical group analysis based on Bingham distribution. We have investigated different directional statistical models to find the best fit. During the experiments, we confirmed that carrying out directional statistical analysis along the tract is much more effective than voxel- or skeleton-guided directional statistics. Hence, we propose a new method called tract profiling and directional statistics (TPDS) applicable to fiber bundles. As a case study, the method has been applied to identify connectivity differences of patients with major depressive disorder. The results obtained with the directional statistic-based analysis are consistent with those of NBS, but additionally, we found significant changes in the right hemisphere striatum, ACC, and prefrontal, parietal, temporal, and occipital connections as well as left hemispheric differences in the limbic areas such as the thalamus, amygdala, and hippocampus. The results are also evaluated with respect to fiber lengths. Comparison with the output of the network-based statistical toolbox indicated that the benefit of the proposed method becomes much more distinctive as the tract length increases. The likelihood of finding clusters of voxels that differ in long tracts is higher in TPDS, while that relationship is not clearly established in NBS.

A generalised complete flux scheme for anisotropic advection-diffusion equations

In this paper, we consider separating the discretisation of the diffusive and advective fluxes in the complete flux scheme. This allows the combination of several discretisation methods for the homogeneous flux with the complete flux (CF) method. In particular, we explore the combination of the hybrid mimetic mixed (HMM) method and the CF method, in order to utilise the advantages of each of these methods. The usage of HMM allows us to handle anisotropic diffusion tensors on generic polygonal (polytopal) grids, whereas the CF method provides a framework for the construction of a uniformly second-order method, even when the problem is advection dominated.

A new framework for MR diffusion tensor distribution

The ability to characterize heterogeneous and anisotropic water diffusion processes within macroscopic MRI voxels non-invasively and in vivo is a desideratum in biology, neuroscience, and medicine. While an MRI voxel may contain approximately a microliter of tissue, our goal is to examine intravoxel diffusion processes on the order of picoliters. Here we propose a new theoretical framework and efficient experimental design to describe and measure such intravoxel structural heterogeneity and anisotropy. We assume that a constrained normal tensor-variate distribution (CNTVD) describes the variability of positive definite diffusion tensors within a voxel which extends its applicability to a wide range of b-values while preserving the richness of diffusion tensor distribution (DTD) paradigm unlike existing models. We introduce a new Monte Carlo (MC) scheme to synthesize realistic 6D DTD numerical phantoms and invert the MR signal. We show that the signal inversion is well-posed and estimate the CNTVD parameters parsimoniously by exploiting the different symmetries of the mean and covariance tensors of CNTVD. The robustness of the estimation pipeline is assessed by adding noise to calculated MR signals and compared with the ground truth. A family of invariant parameters and glyphs which characterize microscopic shape, size and orientation heterogeneity within a voxel are also presented.

Probing tissue microstructure by diffusion skewness tensor imaging

Probing the cellular structure of in vivo biological tissue is a fundamental problem in biomedical imaging and medical science. This work introduces an approach for analyzing diffusion magnetic resonance imaging data acquired by the novel tensor-valued encoding technique for characterizing tissue microstructure. Our approach first uses a signal model to estimate the variance and skewness of the distribution of apparent diffusion tensors modeling the underlying tissue. Then several novel imaging indices, such as weighted microscopic anisotropy and microscopic skewness, are derived to characterize different ensembles of diffusion processes that are indistinguishable by existing techniques. The contributions of this work also include a theoretical proof that shows that, to estimate the skewness of a diffusion tensor distribution, the encoding protocol needs to include full-rank tensor diffusion encoding. This proof provides a guideline for the application of this technique. The properties of the proposed indices are illustrated using both synthetic data and in vivo data acquired from a human brain.

A New Framework for MR Diffusion Tensor Distribution

The ability to characterize heterogeneous and anisotropic water diffusion processes within macro-scopic MRI voxels non-invasively and in vivo is a desideratum in biology, neuroscience, and medicine. While an MRI voxel may contain approximately a microliter of tissue, our goal is to examine intravoxel diffusion processes on the order of picoliters. Here we propose a new theoretical framework and experimental design to describe and measure such intravoxel structural heterogeneity and anisotropy. We assume that a constrained normal tensor-variate distribution (CNTVD) describes the variability of positive definite diffusion tensors within a voxel which extends its applicability to a wide range of b-values unlike existing models. We use a Monte Carlo scheme to synthesize realistic numerical diffusion tensor distribution (DTD) phantoms and invert the MR signal. We show that the signal inversion is well-posed and estimate the CNTVD parameters parsimoniously by exploiting the different symmetries of the mean and covariance tensors of CNTVD. The robustness of the estimation pipeline is assessed by adding noise to calculated MR signals and compared with the ground truth. A family of invariant parameters which characterize microscopic shape, size and orientation heterogeneity within a voxel are also presented.

Diffusion Tensors of Arbitrary-Shaped Nanoparticles in Fluid by Molecular Dynamics Simulation

The anisotropic diffusive behavior of nanoparticles with complex shapes attracts great interest due to its potential applications in many fields ranging from bionics to aeronautic industry. Although molecular dynamics (MD) simulations are used widely to investigate nanoparticle diffusion properties, universal methods to describe the diffusion process comprehensively are still lacking. Here, we address this problem by introducing diffusion tensor as it can describe translational and rotational diffusion in three dimensions both individually and their coupling. We take carbon triple sphere suspended in argon fluid as our model system. The consistency of our results and velocity autocorrelation function(VAF) method validates our simulations. The coupling between translational and rotational diffusion is observed directly from analyzing diffusion tensor, and quantified by coupling diffusion coefficient. Our simulation reveals non-trivial effect of some factors in diffusion at nanoscale, which was not considered in previous theories. In addition to introducing an effective method to calculate the diffusion tensor in MD simulations, our work also provides insights for understanding the diffusion process of arbitrary-shaped particles in nanoengineering.