Diffusion Orientation Distribution Function Research Articles

Voxel-Wise Fusion of 3T and 7T Diffusion MRI Data to Extract more Accurate Fiber Orientations.

While 7T diffusion magnetic resonance imaging (dMRI) has high spatial resolution, its diffusion imaging quality is usually affected by signal loss due to B1 inhomogeneity, T2 decay, susceptibility, and chemical shift. In contrast, 3T dMRI has relative higher diffusion angular resolution, but lower spatial resolution. Combination of 3T and 7T dMRI, thus, may provide more detailed and accurate information about the voxel-wise fiber orientations to better understand the structural brain connectivity. However, this topic has not yet been thoroughly explored until now. In this study, we explored the feasibility of fusing 3T and 7T dMRI data to extract voxel-wise quantitative parameters at higher spatial resolution. After 3T and 7T dMRI data was preprocessed, respectively, 3T dMRI volumes were coregistered into 7T dMRI space. Then, 7T dMRI data was harmonized to the coregistered 3T dMRI B0 (b = 0) images. Last, harmonized 7T dMRI data was fused with 3T dMRI data according to four fusion rules proposed in this study. We employed high-quality 3T and 7T dMRI datasets (N = 24) from the Human Connectome Project to test our algorithms. The diffusion tensors (DTs) and orientation distribution functions (ODFs) estimated from the 3T-7T fused dMRI volumes were statistically analyzed. More voxels containing multiple fiber populations were found from the fused dMRI data than from 7T dMRI data set. Moreover, extra fiber directions were extracted in temporal brain regions from the fused dMRI data at Otsu's thresholds of quantitative anisotropy, but could not be extracted from 7T dMRI dataset. This study provides novel algorithms to fuse intra-subject 3T and 7T dMRI data for extracting more detailed information of voxel-wise quantitative parameters, and a new perspective to build more accurate structural brain networks.

Voxel-Wise Brain Graphs From Diffusion MRI: Intrinsic Eigenspace Dimensionality and Application to Functional MRI

<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Goal:</i> Structural brain graphs are conventionally limited to defining nodes as gray matter regions from an atlas, with edges reflecting the density of axonal projections between pairs of nodes. Here we explicitly model the entire set of voxels within a brain mask as nodes of high-resolution, subject-specific graphs. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Methods:</i> We define the strength of local voxel-to-voxel connections using diffusion tensors and orientation distribution functions derived from diffusion MRI data. We study the graphs' Laplacian spectral properties on data from the Human Connectome Project. We then assess the extent of inter-subject variability of the Laplacian eigenmodes via a procrustes validation scheme. Finally, we demonstrate the extent to which functional MRI data are shaped by the underlying anatomical structure via graph signal processing. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Results:</i> The graph Laplacian eigenmodes manifest highly resolved spatial profiles, reflecting distributed patterns that correspond to major white matter pathways. We show that the intrinsic dimensionality of the eigenspace of such high-resolution graphs is only a mere fraction of the graph dimensions. By projecting task and resting-state data on low-frequency graph Laplacian eigenmodes, we show that brain activity can be well approximated by a small subset of low-frequency components. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Conclusions:</i> The proposed graphs open new avenues in studying the brain, be it, by exploring their organisational properties via graph or spectral graph theory, or by treating them as the scaffold on which brain function is observed at the individual level.

Manifold-aware synthesis of high-resolution diffusion from structural imaging.

The physical and clinical constraints surrounding diffusion-weighted imaging (DWI) often limit the spatial resolution of the produced images to voxels up to eight times larger than those of T1w images. The detailed information contained in accessible high-resolution T1w images could help in the synthesis of diffusion images with a greater level of detail. However, the non-Euclidean nature of diffusion imaging hinders current deep generative models from synthesizing physically plausible images. In this work, we propose the first Riemannian network architecture for the direct generation of diffusion tensors (DT) and diffusion orientation distribution functions (dODFs) from high-resolution T1w images. Our integration of the log-Euclidean Metric into a learning objective guarantees, unlike standard Euclidean networks, the mathematically-valid synthesis of diffusion. Furthermore, our approach improves the fractional anisotropy mean squared error (FA MSE) between the synthesized diffusion and the ground-truth by more than 23% and the cosine similarity between principal directions by almost 5% when compared to our baselines. We validate our generated diffusion by comparing the resulting tractograms to our expected real data. We observe similar fiber bundles with streamlines having <3% difference in length, <1% difference in volume, and a visually close shape. While our method is able to generate diffusion images from structural inputs in a high-resolution space within 15 s, we acknowledge and discuss the limits of diffusion inference solely relying on T1w images. Our results nonetheless suggest a relationship between the high-level geometry of the brain and its overall white matter architecture that remains to be explored.

A framework to construct a longitudinal DW-MRI infant atlas based on mixed effects modeling of dODF coefficients.

Building of atlases plays a crucial role in the analysis of brain images. In scenarios where early growth, aging or disease trajectories are of key importance, longitudinal atlases become necessary as references, most often created from cross-sectional data. New opportunities will be offered by creating longitudinal brain atlases from longitudinal subject-specific image data, where explicit modeling of subject's variability in slope and intercept leads to a more robust estimation of average trajectories but also to estimates of confidence bounds. This work focuses on a framework to build a continuous 4D atlas from longitudinal high angular resolution diffusion images (HARDI) where, unlike atlases of derived scalar diffusion indices such as FA, statistics on dODFs is preserved. Multi-scalar images obtained from DW images are used for geometric alignment, and linear mixed-effects modeling from longitudinal diffusion orientation distribution functions (dODF) leads to estimation of continuous dODF changes. The proposed method is applied to a longitudinal dataset of HARDI images from healthy developing infants in the age range of 3 to 36 months. Verification of mixed-effects modeling is obtained by voxel-wise goodness of fit calculations. To demonstrate the potential of our method, we display changes of longitudinal atlas using dODF and derived generalized fractional anisotropy (GFA) of dODF. We also investigate white matter maturation patterns in genu, body, and splenium of the corpus callosum. The framework can be used to build an average dODF atlas from HARDI data and to derive subject-specific and population-based longitudinal change trajectories.

Hierarchical geodesic modeling on the diffusion orientation distribution function for longitudinal DW-MRI analysis.

The analysis of anatomy that undergoes rapid changes, such as neuroimaging of the early developing brain, greatly benefits from spatio-temporal statistical analysis methods to represent population variations but also subject-wise characteristics over time. Methods for spatio-temporal modeling and for analysis of longitudinal shape and image data have been presented before, but, to our knowledge, not for diffusion weighted MR images (DW-MRI) fitted with higher-order diffusion models. To bridge the gap between rapidly evolving DW-MRI methods in longitudinal studies and the existing frameworks, which are often limited to the analysis of derived measures like fractional anisotropy (FA), we propose a new framework to estimate a population trajectory of longitudinal diffusion orientation distribution functions (dODFs) along with subject-specific changes by using hierarchical geodesic modeling. The dODF is an angular profile of the diffusion probability density function derived from high angular resolution diffusion imaging (HARDI) and we consider the dODF with the square-root representation to lie on the unit sphere in a Hilbert space, which is a well-known Riemannian manifold, to respect the nonlinear characteristics of dODFs. The proposed method is validated on synthetic longitudinal dODF data and tested on a longitudinal set of 60 HARDI images from 25 healthy infants to characterize dODF changes associated with early brain development.

Generalized diffusion spectrum magnetic resonance imaging (GDSI) for model-free reconstruction of the ensemble average propagator

Diffusion spectrum MRI (DSI) provides model-free estimation of the diffusion ensemble average propagator (EAP) and orientation distribution function (ODF) but requires the diffusion data to be acquired on a Cartesian q-space grid. Multi-shell diffusion acquisitions are more flexible and more commonly acquired but have, thus far, only been compatible with model-based analysis methods. Here, we propose a generalized DSI (GDSI) framework to recover the EAP from multi-shell diffusion MRI data. The proposed GDSI approach corrects for q-space sampling density non-uniformity using a fast geometrical approach. The EAP is directly calculated in a preferable coordinate system by multiplying the sampling density corrected q-space signals by a discrete Fourier transform matrix, without any need for gridding. The EAP is demonstrated as a way to map diffusion patterns in brain regions such as the thalamus, cortex and brainstem where the tissue microstructure is not as well characterized as in white matter. Scalar metrics such as the zero displacement probability and displacement distances at different fractions of the zero displacement probability were computed from the recovered EAP to characterize the diffusion pattern within each voxel. The probability averaged across directions at a specific displacement distance provides a diffusion property based image contrast that clearly differentiates tissue types. The displacement distance at the first zero crossing of the EAP averaged across directions orthogonal to the primary fiber orientation in the corpus callosum is found to be larger in the body (5.65 ± 0.09 μm) than in the genu (5.55 ± 0.15 μm) and splenium (5.4 ± 0.15 μm) of the corpus callosum, which corresponds well to prior histological studies. The EAP also provides model-free representations of angular structure such as the diffusion ODF, which allows estimation and comparison of fiber orientations from both the model-free and model-based methods on the same multi-shell data. For the model-free methods, detection of crossing fibers is found to be strongly dependent on the maximum b-value and less sensitive compared to the model-based methods. In conclusion, our study provides a generalized DSI approach that allows flexible reconstruction of the diffusion EAP and ODF from multi-shell diffusion data and data acquired with other sampling patterns.

Low Rank plus Sparse decomposition of ODFs for improved detection of group-level differences and variable correlations in white matter

A novel approach is presented for group statistical analysis of diffusion weighted MRI datasets through voxelwise Orientation Distribution Functions (ODF).Recent advances in MRI acquisition make it possible to use high quality diffusion weighted protocols (multi-shell, large number of gradient directions) for routine in vivo study of white matter architecture. The dimensionality of these data sets is however often reduced to simplify statistical analysis. While these approaches may detect large group differences, they do not fully capitalize on all acquired image volumes. Incorporation of all available diffusion information in the analysis however risks biasing the outcome by outliers.Here we propose a statistical analysis method operating on the ODF, either the diffusion ODF or fiber ODF. To avoid outlier bias and reliably detect voxelwise group differences and correlations with demographic or behavioral variables, we apply the Low-Rank plus Sparse (L+S) matrix decomposition on the voxelwise ODFs which separates the sparse individual variability in the sparse matrix S whilst recovering the essential ODF features in the low-rank matrix L.We demonstrate the performance of this ODF L+S approach by replicating the established negative association between global white matter integrity and physical obesity in the Human Connectome dataset. The volume of positive findings p<0.01,227cm3, agrees with and expands on the volume found by TBSS (17 cm3), Connectivity based fixel enhancement (15 cm3) and Connectometry (212 cm3). In the same dataset we further localize the correlations of brain structure with neurocognitive measures such as fluid intelligence and episodic memory.The presented ODF L+S approach will aid in the full utilization of all acquired diffusion weightings leading to the detection of smaller group differences in clinically relevant settings as well as in neuroscience applications.

Empirical consideration of the effects of acquisition parameters and analysis model on clinically feasible q-ball imaging

Q-ball imaging (QBI) is a popular high angular resolution diffusion imaging (HARDI) technique used to study brain architecture in vivo. Simulation and phantom-based studies suggest that QBI results are affected by the b-value, the number of diffusion weighting directions, and the signal-to-noise ratio (SNR). However, optimal acquisition schemes for QBI in clinical settings are largely undetermined given empirical (observed) imaging considerations. In this study, we acquire a HARDI dataset at five b-values with 11 repetitions on a single subject to investigate the effects of acquisition scheme and subsequent analysis models on the accuracy and precision of measures of tissue composition and fiber orientation derived from clinically feasible QBI at 3T. Clinical feasibility entails short scan protocols - less than 5minutes in the current study - resulting in lower SNR, lower b-values, and fewer diffusion directions than are typical in most QBI protocols with research applications, where time constraints are less prevalent. In agreement with previous studies, we find that the b-value and number of diffusion directions impact the magnitude and variation of QBI indices in both white matter and gray matter regions; however, QBI indices are most heavily dependent on the maximum order of the spherical harmonic (SH) series used to represent the diffusion orientation distribution function (ODF). Specifically, to ensure numerical stability and reduce the occurrence of false peaks and inflated anisotropy, we recommend oversampling by at least 8–12 more diffusion directions than the number of estimated coefficients for a given SH order. In addition, in an equal scan time comparison of QBI accuracy, we find that increasing the directional resolution of the HARDI dataset is preferable to repeating observations; however, our results indicate that as few as 32 directions at a low b-value (1000s/mm2) captures most of the angular information in the q-ball ODF. Our findings provide guidance for determining an optimal acquisition scheme for QBI in the low SNR and low scan time regime, and suggest that care must be taken when choosing the basis functions used to represent the QBI ODF.

Optimal DSI reconstruction parameter recommendations: Better ODFs and better connectivity

Diffusion Spectrum Imaging (DSI) has been used for tractography in several publicly available software and a number of recent high impact publications. However, there are several important theoretical, numerical and practical considerations that are often ignored. We revisit the theoretical and state-of-the-art processing steps necessary to go from the DSI signal to the diffusion orientation distribution function (dODF) used by tractography. We show that the parameters in the reconstruction have huge impact on the reconstruction quality and that, while there is no consensus about what they should be, the parameters we most often see in the literature are not optimal. We provide applicable recommendations that improve the accuracy of extracted local orientations and improve accuracy of global connectivity as measured by the Tractometer, a tractography online evaluation system. These recommendations come for “free” as they are applicable to all existing DSI data and do not require a significant increase in computation time. Hence, this paper highlights the do's and dont's of DSI reconstruction.

Compressive Sensing Based Q-Space Resampling for Handling Fast Bulk Motion in Hardi Acquisitions.

Diffusion-weighted (DW) MRI has become a widely adopted imaging modality to reveal the underlying brain connectivity. Long acquisition times and/or non-cooperative patients increase the chances of motion-related artifacts. Whereas slow bulk motion results in inter-gradient misalignment which can be handled via retrospective motion correction algorithms, fast bulk motion usually affects data during the application of a single diffusion gradient causing signal dropout artifacts. Common practices opt to discard gradients bearing signal attenuation due to the difficulty of their retrospective correction, with the disadvantage to lose full gradients for further processing. Nonetheless, such attenuation might only affect limited number of slices within a gradient volume. Q-space resampling has recently been proposed to recover corrupted slices while saving gradients for subsequent reconstruction. However, few corrupted gradients are implicitly assumed which might not hold in case of scanning unsedated infants or patients in pain. In this paper, we propose to adopt recent advances in compressive sensing based reconstruction of the diffusion orientation distribution functions (ODF) with under sampled measurements to resample corrupted slices. We make use of Simple Harmonic Oscillator based Reconstruction and Estimation (SHORE) basis functions which can analytically model ODF from arbitrary sampled signals. We demonstrate the impact of the proposed resampling strategy compared to state-of-art resampling and gradient exclusion on simulated intra-gradient motion as well as samples from real DWI data.

Resolving power for the diffusion orientation distribution function.

The diffusion orientation distribution function (dODF) is primarily used for white matter fiber tractography. Here the resolving power of the dODF is investigated for a simple diffusion model of two intersecting axonal fiber bundles. The resolving power for the dODF is evaluated using the Sparrow criterion. This is determined for the exact dODF and also for q-space imaging (QSI), q-ball, and kurtosis approximations. Based on theoretical and numerical calculations, the resolving power is found to depend on the eigenvalues of the diffusion model and on the degree of radial weighting for the dODF. The resolving powers of the QSI and q-ball dODFs improve with increased b-value. The kurtosis dODF has a resolving power similar to that of the exact dODF. The dODFs, whether exact or approximate, have finite resolving powers that limit their sensitivity to fiber crossings. The resolving powers for the different dODFs considered here provide convenient benchmarks for assessing and comparing their performance. Magn Reson Med 76:679-688, 2016. © 2015 Wiley Periodicals, Inc.

Fiber ball imaging

By modeling axons as thin cylinders, it is shown that the inverse Funk transform of the diffusion MRI (dMRI) signal intensity obtained on a spherical shell in q-space gives an estimate for a fiber orientation density function (fODF), where the accuracy improves with increasing b-value provided the signal-to-noise ratio is sufficient. The method is similar to q-ball imaging, except that the Funk transform of q-ball imaging is replaced by its inverse. We call this new approach fiber ball imaging. The fiber ball method is demonstrated for healthy human brain, and fODF estimates are compared to diffusion orientation distribution function (dODF) approximations obtained with q-ball imaging. The fODFs are seen to have sharper features than the dODFs, reflecting an enhancement of the higher degree angular frequencies. The inverse Funk transform of the dMRI signal intensity data provides a simple and direct method of estimating a fODF. In addition, fiber ball imaging leads to an estimate for the ratio of the fraction of MRI visible water confined to the intra-axonal space divided by the square root of the intra-axonal diffusivity. This technique may be useful for white matter fiber tractography, as well as other types of microstructural modeling of brain tissue.

Quantifying the number of crossing fibers in the aging brain using dki tractography

Tractography is a technique that visualizes white matter (WM) fibers based on diffusion MRI data. A shortcoming of DTI is the inability to directly resolve fiber crossings. Here we reconstructed WM tracts using diffusional kurtosis imaging (DKI), which is an advanced diffusion MRI technique that quantifies the non-Gaussianity of diffusion and, allows for the detection of crossing fibers. Because aging is associated with decreased microstructural complexity in the brain, we hypothesize that this would be reflected by a decrease in the number of detected crossing fibers. The study sample is composed of 48 cognitively intact older adults (age = 68.0 ± 5.6y, range = 60-80y; MoCA score = 26.9 ± 1.8; 62.5% Female; 95.8% White). DKI was acquired for all individuals on a 3T Siemens Tim Trio. All diffusion metrics were calculated with DKE (Diffusional Kurtosis Estimator). Using in-house software, the diffusion orientation distribution function (dODF) was constructed for every voxel. The dODF quantifies the extent of diffusion as a function of direction. The local maxima of the dODF provide estimates for the fiber directions, and the number of local maxima determines the number of crossing fibers detected. The JHU WM atlas was used to identify the regions of interest. There was a significant negative correlation (r=-0.46, p<0.05) between the number of detected crossing fibers and age (Fig. 1). To verify our measurements, we compared the number of crossing fibers in areas with different tract morphologies: the corona radiata (CR) and the genu of the corpus callosum. As expected, more crossing fibers were detected in the CR, a region with fiber orientation heterogeneity, than in the more homogeneous genu (Table 1). Figure. 2 shows the fiber complexity at the level of the superior CR calculated with both DKI and DTI. Since many fibers cross the superior longitudinal fasciculus here a large part remains undetected by DTI. Advanced diffusion MRI techniques like DKI enable detection of fibers in heterogeneous regions with crossing fibers. The number of fiber crossings detected with diffusion MRI decreases with age. This may be relevant for the interpretation of tractography in older subjects.

Estimating diffusion propagator and its moments using directional radial basis functions.

The ensemble average diffusion propagator (EAP) obtained from diffusion MRI (dMRI) data captures important structural properties of the underlying tissue. As such, it is imperative to derive an accurate estimate of the EAP from the acquired diffusion data. In this work, we propose a novel method for estimating the EAP by representing the diffusion signal as a linear combination of directional radial basis functions scattered in q-space. In particular, we focus on a special case of anisotropic Gaussian basis functions and derive analytical expressions for the diffusion orientation distribution function (ODF), the return-to-origin probability (RTOP), and mean-squared-displacement (MSD). A significant advantage of the proposed method is that the second and the fourth order moment tensors of the EAP can be computed explicitly. This allows for computing several novel scalar indices (from the moment tensors) such as mean-fourth-order-displacement (MFD) and generalized kurtosis (GK)-which is a generalization of the mean kurtosis measure used in diffusion kurtosis imaging. Additionally, we also propose novel scalar indices computed from the signal in q-space, called the q-space mean-squared-displacement (QMSD) and the q-space mean-fourth-order-displacement (QMFD), which are sensitive to short diffusion time scales. We validate our method extensively on data obtained from a physical phantom with known crossing angle as well as on in-vivo human brain data. Our experiments demonstrate the robustness of our method for different combinations of b-values and number of gradient directions.

How Does B-Value Affect HARDI Reconstruction Using Clinical Diffusion MRI Data?

BackgroundA number of imaging factors can affect the orientation distribution function (ODF) reconstruction in high angular resolution diffusion imaging (HARDI). The aim of this study was to investigate the effect of the b-value on the HARDI reconstruction and to seek for the appropriate b-value for ODF reconstruction from clinical HARDI data.MethodsDiffusion MRI data with various b-values were collected on a GE 3T MRI scanner. To reconstruct the diffusion ODF and fiber ODF, decomposition-based spherical polar Fourier imaging and deconvolution-based constrained spherical deconvolution approaches were applied separately. The full width at half maximum (FWHM) of the ODF and the angular difference of the peaks extracted from ODF were measured to investigate the effect of b-value on the ODF reconstruction. Visual inspection of the ODF was used to evaluate the reconstructions.ResultsThe FWHM of the ODFs in the corpus callosum, which was chosen as the region of interest (ROI), decreased with increasing b-values. The differences in the FWHM for the diffusion ODF and the fiber ODF between the b-values of 2000 s/mm2 and 2500 s/mm2 were not significant. The angular differences of the ODF between 2000 s/mm2 and 2500 s/mm2 were lowest in both single-directional and two-directional situations. The ODFs became sharper and crossing-fiber situations were detected with an increase in b-value. B = 2000 s/mm2 and above revealed most of the two-way or three-way crossing-fiber structures.ConclusionsConsidering both the signal-to-noise ratio and the acquisition time, b = 2000 s/mm2 is the basic requirement for ODF reconstruction using current HARDI methods on clinical data. This study can provide a useful reference for researchers and clinicians attempting to set appropriate scan protocols for specific HARDI experiments.

7T Multi-shell Hybrid Diffusion Imaging (HYDI) for Mapping Brain Connectivity in Mice.

Diffusion weighted imaging (DWI) is widely used to study microstructural characteristics of the brain. High angular resolution diffusion imaging (HARDI) samples diffusivity at a large number of spherical angles, to better resolve neural fibers that mix or cross. Here, we implemented a framework for advanced mathematical analysis of mouse 5-shell HARDI (b=1000, 3000, 4000, 8000, 12000 s/mm2), also known as hybrid diffusion imaging (HYDI). Using q-ball imaging (QBI) at ultra-high field strength (7 Tesla), we computed diffusion and fiber orientation distribution functions (dODF, fODF) to better detect crossing fibers. We also computed a quantitative anisotropy (QA) index, and deterministic tractography, from the peak orientation of the fODFs. We found that the signal to noise ratio (SNR) of the QA was significantly higher in single and multi-shell reconstructed data at the lower b-values (b=1000, 3000, 4000 s/mm2) than at higher b-values (b=8000, 12000 s/mm2); the b=1000 s/mm2 shell increased the SNR of the QA in all multi-shell reconstructions, but when used alone or in <5-shell reconstruction, it led to higher angular error for the major fibers, compared to 5-shell HYDI. Multi-shell data reconstructed major fibers with less error than single-shell data, and was most successful at reducing the angular error when the lowest shell was excluded (b=1000 s/mm2). Overall, high-resolution connectivity mapping with 7T HYDI offers great potential for understanding unresolved changes in mouse models of brain disease.

A 4D hyperspherical interpretation of q-space.

3D q-space can be viewed as the surface of a 4D hypersphere. In this paper, we seek to develop a 4D hyperspherical interpretation of q-space by projecting it onto a hypersphere and subsequently modeling the q-space signal via 4D hyperspherical harmonics (HSH). Using this orthonormal basis, we derive several well-established q-space indices and numerically estimate the diffusion orientation distribution function (dODF). We also derive the integral transform describing the relationship between the diffusion signal and propagator on a hypersphere. Most importantly, we will demonstrate that for hybrid diffusion imaging (HYDI) acquisitions low order linear expansion of the HSH basis is sufficient to characterize diffusion in neural tissue. In fact, the HSH basis achieves comparable signal and better dODF reconstructions than other well-established methods, such as Bessel Fourier orientation reconstruction (BFOR), using fewer fitting parameters. All in all, this work provides a new way of looking at q-space.

A Simplified Algorithm for Inverting Higher Order Diffusion Tensors

In Riemannian geometry, a distance function is determined by an inner product on the tangent space. In Riemann–Finsler geometry, this distance function can be determined by a norm. This gives more freedom on the form of the so-called indicatrix or the set of unit vectors. This has some interesting applications, e.g., in medical image analysis, especially in diffusion weighted imaging (DWI). An important application of DWI is in the inference of the local architecture of the tissue, typically consisting of thin elongated structures, such as axons or muscle fibers, by measuring the constrained diffusion of water within the tissue. From high angular resolution diffusion imaging (HARDI) data, one can estimate the diffusion orientation distribution function (dODF), which indicates the relative diffusivity in all directions and can be represented by a spherical polynomial. We express this dODF as an equivalent spherical monomial (higher order tensor) to directly generalize the (second order) diffusion tensor approach. To enable efficient computation of Riemann–Finslerian quantities on diffusion weighted (DW)-images, such as the metric/norm tensor, we present a simple and efficient algorithm to invert even order spherical monomials, which extends the familiar inversion of diffusion tensors, i.e., symmetric matrices.

Multi-directional anisotropy from diffusion orientation distribution functions.

To evaluate a model-independent, multi-directional anisotropy (MDA) metric that is analytically and experimentally equivalent to fractional anisotropy (FA) in single-direction diffusivity, but potentially superior to FA in its sensitivity to the underlying anisotropy of multi-directional diffusivity. An expression for MDA was defined from the orientation distribution function (ODF) and its analytical relation to FA was derived. Simulations of single and crossed double-fibers were performed using a compressed-sensing-accelerated diffusion-spectrum-imaging (CS-DSI) scheme. In vivo brain imaging using CS-DSI was performed on eight healthy subjects. MDA was compared with FA and with another ODF-based metric known as generalized FA (GFA). In simulated single-direction fibers, MDA was shown to be equivalent to FA (from FA = 0.2 to 0.8). In crossed fibers, MDA provided superior differentiation of the underlying anisotropy as compared to FA and GFA. In vivo analysis shows that the MDA was superior to both FA (P = 0.015) and GFA (P = 0.021) in terms of its relative accuracy in crossed fiber regions. MDA provides a potentially superior measure of fiber anisotropy relative to conventional FA or GFA, and may be used to improve the assessment of disease in regions with multi-directional brain fibers.

Leading non-Gaussian corrections for diffusion orientation distribution function.

An analytical representation of the leading non-Gaussian corrections for a class of diffusion orientation distribution functions (dODFs) is presented. This formula is constructed from the diffusion and diffusional kurtosis tensors, both of which may be estimated with diffusional kurtosis imaging (DKI). By incorporating model-independent non-Gaussian diffusion effects, it improves on the Gaussian approximation used in diffusion tensor imaging (DTI). This analytical representation therefore provides a natural foundation for DKI-based white matter fiber tractography, which has potential advantages over conventional DTI-based fiber tractography in generating more accurate predictions for the orientations of fiber bundles and in being able to directly resolve intra-voxel fiber crossings. The formula is illustrated with numerical simulations for a two-compartment model of fiber crossings and for human brain data. These results indicate that the inclusion of the leading non-Gaussian corrections can significantly affect fiber tractography in white matter regions, such as the centrum semiovale, where fiber crossings are common.