Mapping the orientation of intravoxel crossing fibers based on the phase information of diffusion circular spectrum

Linear transforms for Fourier data on the sphere: application to high angular resolution diffusion MRI of the brain

This paper presents a novel family of linear transforms that can be applied to data collected from the surface of a 2-sphere in three-dimensional Fourier space. This family of transforms generalizes the previously-proposed Funk-Radon Transform (FRT), which was originally developed for estimating the orientations of white matter fibers in the central nervous system from diffusion magnetic resonance imaging data. The new family of transforms is characterized theoretically, and efficient numerical implementations of the transforms are presented for the case when the measured data is represented in a basis of spherical harmonics. After these general discussions, attention is focused on a particular new transform from this family that we name the Funk-Radon and Cosine Transform (FRACT). Based on theoretical arguments, it is expected that FRACT-based analysis should yield significantly better orientation information (e.g., improved accuracy and higher angular resolution) than FRT-based analysis, while maintaining the strong characterizability and computational efficiency of the FRT. Simulations are used to confirm these theoretical characteristics, and the practical significance of the proposed approach is illustrated with real diffusion weighted MRI brain data. These experiments demonstrate that, in addition to having strong theoretical characteristics, the proposed approach can outperform existing state-of-the-art orientation estimation methods with respect to measures such as angular resolution and robustness to noise and modeling errors.

Computational Representation of White Matter Fiber Orientations

We present a new methodology based on directional data clustering to represent white matter fiber orientations in magnetic resonance analyses for high angular resolution diffusion imaging. A probabilistic methodology is proposed for estimating intravoxel principal fiber directions, based on clustering directional data arising from orientation distribution function (ODF) profiles. ODF reconstructions are used to estimate intravoxel fiber directions using mixtures of von Mises-Fisher distributions. The method focuses on clustering data on the unit sphere, where complexity arises from representing ODF profiles as directional data. The proposed method is validated on synthetic simulations, as well as on a real data experiment. Based on experiments, we show that by clustering profile data using mixtures of von Mises-Fisher distributions it is possible to estimate multiple fiber configurations in a more robust manner than currently used approaches, without recourse to regularization or sharpening procedures. The method holds promise to support robust tractographic methodologies and to build realistic models of white matter tracts in the human brain.

Anisotropy in high-resolution diffusion-weighted MRI and anomalous diffusion

It is shown below that complex diffusion anisotropy observed in diffusion-weighted MRI can be fully accounted for by allowing for non-locality of the spatial operator in the diffusion equation. The anisotropy is represented by a distribution over directions on a sphere. It allows recognition of fiber tracts crossing at arbitrary angles. A simple generalization of the Stejskal-Tanner equation for the determination of the ODF is presented. Furthermore, an explicit solution of the Bloch-Torrey equation for an anisotropic time-fractional diffusion equation is obtained in terms of a generalized Mittag-Leffler type function.

Diffeomorphic image registration of diffusion MRI using spherical harmonics

Nonrigid registration of diffusion magnetic resonance imaging (MRI) is crucial for group analyses and building white matter and fiber tract atlases. Most current diffusion MRI registration techniques are limited to the alignment of diffusion tensor imaging (DTI) data. We propose a novel diffeomorphic registration method for high angular resolution diffusion images by mapping their orientation distribution functions (ODFs). ODFs can be reconstructed using q-ball imaging (QBI) techniques and represented by spherical harmonics (SHs) to resolve intra-voxel fiber crossings. The registration is based on optimizing a diffeomorphic demons cost function. Unlike scalar images, deforming ODF maps requires ODF reorientation to maintain its consistency with the local fiber orientations. Our method simultaneously reorients the ODFs by computing a Wigner rotation matrix at each voxel, and applies it to the SH coefficients during registration. Rotation of the coefficients avoids the estimation of principal directions, which has no analytical solution and is time consuming. The proposed method was validated on both simulated and real data sets with various metrics, which include the distance between the estimated and simulated transformation fields, the standard deviation of the general fractional anisotropy and the directional consistency of the deformed and reference images. The registration performance using SHs with different maximum orders were compared using these metrics. Results show that the diffeomorphic registration improved the affine alignment, and registration using SHs with higher order SHs further improved the registration accuracy by reducing the shape difference and improving the directional consistency of the registered and reference ODF maps.

Spatial HARDI: improved visualization of complex white matter architecture with Bayesian spatial regularization

Imaging of water diffusion using magnetic resonance imaging has become an important noninvasive method for probing the white matter connectivity of the human brain for scientific and clinical studies. Current methods, such as diffusion tensor imaging (DTI), high angular resolution diffusion imaging (HARDI) such as q-ball imaging, and diffusion spectrum imaging (DSI), are limited by low spatial resolution, long scan times, and low signal-to-noise ratio (SNR). These methods fundamentally perform reconstruction on a voxel-by-voxel level, effectively discarding the natural coherence of the data at different points in space. This paper attempts to overcome these tradeoffs by using spatial information to constrain the reconstruction from raw diffusion MRI data, and thereby improve angular resolution and noise tolerance. Spatial constraints are specified in terms of a prior probability distribution, which is then incorporated in a Bayesian reconstruction formulation. By taking the log of the resulting posterior distribution, optimal Bayesian reconstruction is reduced to a cost minimization problem. The minimization is solved using a new iterative algorithm based on successive least squares quadratic descent. Simulation studies and in vivo results are presented which indicate significant gains in terms of higher angular resolution of diffusion orientation distribution functions, better separation of crossing fibers, and improved reconstruction SNR over the same HARDI method, spherical harmonic q-ball imaging, without spatial regularization. Preliminary data also indicate that the proposed method might be better at maintaining accurate ODFs for smaller numbers of diffusion-weighted acquisition directions (hence faster scans) compared to conventional methods. Possible impacts of this work include improved evaluation of white matter microstructural integrity in regions of crossing fibers and higher spatial and angular resolution for more accurate tractography.

Symmetric positive-definite Cartesian tensor orientation distribution functions (CT-ODF)

A novel method for estimating a field of orientation distribution functions (ODF) from a given set of DW-MR images is presented. We model the ODF by Cartesian tensor basis using a parametrization that explicitly enforces the positive definite property to the computed ODF. The computed Cartesian tensors, dubbed Cartesian Tensor-ODF (CT-ODF), are symmetric positive definite tensors whose coefficients can be efficiently estimated by solving a linear system with non-negative constraints. Furthermore, we show how to use our method for converting higher-order diffusion tensors to CT-ODFs, which is an essential task since the maxima of higher-order tensors do not correspond to the underlying fiber orientations. We quantitatively evaluate our method using simulated DW-MR images as well as a real brain dataset from a post-mortem porcine brain. The results conclusively demonstrate the superiority of the proposed technique over several existing multi-fiber reconstruction methods.

Diffusion spectrum magnetic resonance imaging (DSI) tractography of crossing fibers

MRI tractography is the mapping of neural fiber pathways based on diffusion MRI of tissue diffusion anisotropy. Tractography based on diffusion tensor imaging (DTI) cannot directly image multiple fiber orientations within a single voxel. To address this limitation, diffusion spectrum MRI (DSI) and related methods were developed to image complex distributions of intravoxel fiber orientation. Here we demonstrate that tractography based on DSI has the capacity to image crossing fibers in neural tissue. DSI was performed in formalin-fixed brains of adult macaque and in the brains of healthy human subjects. Fiber tract solutions were constructed by a streamline procedure, following directions of maximum diffusion at every point, and analyzed in an interactive visualization environment (TrackVis). We report that DSI tractography accurately shows the known anatomic fiber crossings in optic chiasm, centrum semiovale, and brainstem; fiber intersections in gray matter, including cerebellar folia and the caudate nucleus; and radial fiber architecture in cerebral cortex. In contrast, none of these examples of fiber crossing and complex structure was identified by DTI analysis of the same data sets. These findings indicate that DSI tractography is able to image crossing fibers in neural tissue, an essential step toward non-invasive imaging of connectional neuroanatomy.

The effect of the rotational angle on MR diffusion indices in nerves: is the rms displacement of the slow-diffusing component a good measure of fiber orientation?

In recent years, much effort has been made to increase our ability to infer nerve fiber direction through the use of diffusion MR. The present study examines the effect of the rotational angle (alpha), i.e. the angle between the diffusion sensitizing gradients and the main axis of the fibers in the nerves, on different NMR indices. The indices examined were the apparent diffusion coefficient (ADC), extracted from low b-values (b(max) approximately 1200 s/mm(2)), and the root mean square (rms) displacement of the fast and the slow-diffusing components extracted from high b-value q-space diffusion MR data. In addition, the effect of both the diffusion time and myelination was evaluated. We found that the most sensitive index to the rotational angle is the rms displacement of the slow-diffusing component extracted from the high b-value q-space diffusion MR experiment. For this component the rms displacement was nearly constant for alpha values ranging from -10 degrees to +80 degrees (where alpha=0 degrees is the z direction), but it changed dramatically when diffusion was measured nearly perpendicular to the nerve fiber direction, i.e., for alpha=90+/-10 degrees. The ADC and the rms displacement of the fast-diffusing component exhibited only gradual changes, with a maximal change at alpha=45+/-15 degrees. The sensitivity of the rms displacement of the slow-diffusing component to the rotational angle was found to be higher at longer diffusion times and in mature fully myelinated nerves. The relevance of these observations for determining the fiber direction is briefly discussed.

A model-based deconvolution approach to solve fiber crossing in diffusion-weighted MR imaging

A deconvolution approach is presented to solve fiber crossing in diffusion magnetic resonance imaging. In order to provide a direct physical interpretation of the signal generation process, we started from the classical multicompartment model and rewrote this in terms of a convolution process, identifying a significant scalar parameter alpha to characterize the physical system response. Deconvolution is performed by a modified version of the Richardson-Lucy algorithm. Simulations show the ability of this method to correctly separate fiber crossing, even in the presence of noisy data, with lower signal-to-noise ratio, and imprecision in the impulse response function imposed during deconvolution. The in vivo data confirms the efficacy of this method to resolve fiber crossing in real complex brain structures. These results suggest the usefulness of our approach in fiber tracking or connectivity studies.

Regularized, fast, and robust analytical Q-ball imaging

We propose a regularized, fast, and robust analytical solution for the Q-ball imaging (QBI) reconstruction of the orientation distribution function (ODF) together with its detailed validation and a discussion on its benefits over the state-of-the-art. Our analytical solution is achieved by modeling the raw high angular resolution diffusion imaging signal with a spherical harmonic basis that incorporates a regularization term based on the Laplace-Beltrami operator defined on the unit sphere. This leads to an elegant mathematical simplification of the Funk-Radon transform which approximates the ODF. We prove a new corollary of the Funk-Hecke theorem to obtain this simplification. Then, we show that the Laplace-Beltrami regularization is theoretically and practically better than Tikhonov regularization. At the cost of slightly reducing angular resolution, the Laplace-Beltrami regularization reduces ODF estimation errors and improves fiber detection while reducing angular error in the ODF maxima detected. Finally, a careful quantitative validation is performed against ground truth from synthetic data and against real data from a biological phantom and a human brain dataset. We show that our technique is also able to recover known fiber crossings in the human brain and provides the practical advantage of being up to 15 times faster than original numerical QBI method.

Development and initial evaluation of 7-T q-ball imaging of the human brain

Diffusion tensor imaging (DTI) noninvasively depicts white matter connectivity in regions where the Gaussian model of diffusion is valid but yields inaccurate results in those where diffusion has a more complex distribution, such as fiber crossings. q-ball imaging (QBI) overcomes this limitation of DTI by more fully characterizing the angular dependence of intravoxel diffusion with larger numbers of diffusion-encoding directional measurements at higher diffusion-weighting factors (b values). However, the former technique results in longer acquisition times and the latter technique results in a lower signal-to-noise ratio (SNR). In this project, we developed specialized 7-T acquisition methods utilizing novel radiofrequency pulses, eight-channel parallel imaging EPI and high-order shimming with a phase-sensitive multichannel B0 field map reconstruction. These methods were applied in initial healthy adult volunteer studies, which demonstrated the feasibility of performing 7-T QBI. Preliminary comparisons of 3 T with 7 T within supratentorial crossing white matter tracts documented a 79.5% SNR increase for b=3000 s/mm2 (P=.0001) and a 38.6% SNR increase for b=6000 s/mm2 (P=.015). With spherical harmonic reconstruction of the q-ball orientation distribution function at b=3000 s/mm2, 7-T QBI allowed for accurate visualization of crossing fiber tracts with fewer diffusion-encoding acquisitions as compared with 3-T QBI. The improvement of 7-T QBI at b factors as high as 6000 s/mm2 resulted in better angular resolution as compared with 3-T QBI for depicting fibers crossing at shallow angles. Although the increased susceptibility effects at 7 T caused problematic distortions near brain-air interfaces at the skull base and posterior fossa, these initial 7-T QBI studies demonstrated excellent quality in much of the supratentorial brain, with significant improvements as compared with 3-T acquisitions in the same individuals.

A Deconvolution Approach Based on Multi-Tensor Model to Solve Fiber Crossing in Diffusion-MRI

A deconvolution approach, based on a multi-tensor model, is presented to solve fiber crossing in diffusion MRI. In order to provide a direct physical interpretation of the signal generation process, we re-wrote the classical multi-tensor model, identifying a significant scalar parameter alpha to characterize the deconvolution process. Simulations show that, in presence of noise, the method is able to correctly separate fiber crossing. Application on in-vivo data highlights the ability of our approach to distinguish more than two fibers within the same voxel, suggesting its application in fiber tracking or connectivity studies even of complex brain structures.

How accurately can the diffusion profiles indicate multiple fiber orientations? A study on general fiber crossings in diffusion MRI

The q-space imaging techniques and high angular resolution diffusion (HARD) imaging have shown promise to identify intravoxel multiple fibers. The measured orientation distribution function (ODF) and apparent diffusion coefficient (ADC) profiles can be used to identify the orientations of the actual intravoxel fibers. The present study aims to examine the accuracy of these profile-based orientation methods by comparing the angular deviations between the estimated local maxima of the profiles and the real fiber orientation for a fiber crossing simulated with various intersection angles under different b values in diffusion-weighted MRI experiments. Both noisy and noise-free environments were investigated. The diffusion spectrum imaging (DSI), q-ball imaging (QBI), and HARD techniques were used to generate ODF and ADC profiles. To provide a better comparison between ODF and ADC techniques, the phase-corrected angular deviations were also presented for the ADC method based on a circular spectrum mapping method. The results indicate that systematic angular deviations exist between the actual fiber orientations and the corresponding local maxima of either the ADC or ODF profiles. All methods are apt to underestimation of acute intersection and overestimation of obtuse intersection angle. For a typical slow-exchange fiber crossing, the ODF methods have a non-deviation zone around the 90 degrees intersection. Before the phase-correction, the deviation of ADC profiles approaches a peak at the 90 degrees intersection, while after the correction the ADC deviations are significantly reduced. When the b factor is larger than 1000 s/mm2, the ODF methods have smaller angular deviations than the ADC methods for the intersections close to 90 degrees . QBI method demonstrates a slight yet consistent advantage over the DSI method under the same conditions. In the noisy environment, the mean value of the deviation angles shows a high consistency with the corresponding deviation in the nose-free condition.

Q-ball reconstruction of multimodal fiber orientations using the spherical harmonic basis

Diffusion tensor imaging (DTI) accurately delineates white matter pathways when the Gaussian model of diffusion is valid. However, DTI yields erroneous results when diffusion takes on a more complex distribution, as is the case in the brain when fiber tracts cross. High angular resolution diffusion imaging (HARDI) overcomes this limitation of DTI by more fully characterizing the angular dependence of intravoxel diffusion. Among the various HARDI methods that have been proposed, QBI offers advantages such as linearity, model independence, and relatively easy implementation. In this work, reconstruction of the q-ball orientation distribution function (ODF) is reformulated in terms of spherical harmonic basis functions, yielding an analytic solution with useful properties of a frequency domain representation. The harmonic basis is parsimonious for typical b-values, which enables the ODF to be synthesized from a relatively small number of noisy measurements and thus brings the technique closer to clinical feasibility from the standpoint of total imaging time. The proposed method is assessed using Monte Carlo computer simulations and compared with conventional q-ball reconstruction using spherical RBFs. In vivo results from 3T whole-brain HARDI of adult volunteers are also provided to verify the underlying mathematical theory.

Hybrid visualization for white matter tracts using triangle strips and point sprites

Diffusion tensor imaging is of high value in neurosurgery, providing information about the location of white matter tracts in the human brain. For their reconstruction, streamline techniques commonly referred to as fiber tracking model the underlying fiber structures and have therefore gained interest. To meet the requirements of surgical planning and to overcome the visual limitations of line representations, a new real-time visualization approach of high visual quality is introduced. For this purpose, textured triangle strips and point sprites are combined in a hybrid strategy employing GPU programming. The triangle strips follow the fiber streamlines and are textured to obtain a tube-like appearance. A vertex program is used to orient the triangle strips towards the camera. In order to avoid triangle flipping in case of fiber segments where the viewing and segment direction are parallel, a correct visual representation is achieved in these areas by chains of point sprites. As a result, a high quality visualization similar to tubes is provided allowing for interactive multimodal inspection. Overall, the presented approach is faster than existing techniques of similar visualization quality and at the same time allows for real-time rendering of dense bundles encompassing a high number of fibers, which is of high importance for diagnosis and surgical planning.

Fibertract segmentation in position orientation space from high angular resolution diffusion MRI

In diffusion MRI, standard approaches for fibertract identification are based on algorithms that generate lines of coherent diffusion, currently known as tractography. A tract is then identified as a set of such lines selected on some criteria. In the present study, we investigate whether fibertract identification can be formulated as a segmentation task that recognizes a fibertract as a region where diffusion is intense and coherent. Indeed, we show that it is possible to segment efficiently well-known fibertracts with classical image processing methods provided that the problem is formulated in a five-dimensional space of position and orientation. As an example, we choose to adapt to this newly defined high-dimensional non-Euclidean space, called position orientation space, an algorithm based on the hidden Markov random field framework. Structures such as the cerebellar peduncles, corticospinal tract, association bundles can be identified and represented in three dimensions by a back projection technique similar to maximum intensity projection. Potential advantages and drawbacks as compared to classical tractography are discussed; for example, it appears that our formulation handles naturally crossing tracts and is not biased by human intervention.

Potential of diffusion tensor MRI in the assessment of periventricular leukomalacia

AIM

To investigate magnetic resonance (MR) diffusion tensor imaging (DTI) and fibre tractography in the assessment of altered major white matter (WM) fibre tracts in periventricular leukomalacia (PVL).

MATERIALS AND METHODS

Twelve children (male:female = 7:5, age range 3-10 years; mean age = 6.5 years) who had suffered PVL were included in this study. Meanwhile, Twelve age-matched normal controls (male:female = 6:6, age range 4-12 years; mean age = 7.3 years) with normal MRI findings and no neurological abnormalities were recruited for comparison. DTI was performed with 15 different diffusion gradient directions and DTI colour maps were created from fractional anisotropy (FA) values and the three vector elements. To identify alteration of WM fibre tracts in patient of PVL quantitatively, FA values on diffusion tensor colour maps were compared between the patients and controls. Quantitative analysis was performed using the regions of interest (ROI) method settled on the central part of all identifiable WM fibres, including the corticospinal tract (CST) in the brainstem, middle cerebellar peduncle (MCP), medial lemniscus (ML), anterior/posterior limb of internal capsule (ICAL/ICPL), arcuate fasciculus (AF), posterior thalamic radiation (PTR), genu of corpus callosum (GCC), splenium of corpus callosum (SCC), corona radiata (CR), cingulum (CG), and superior longitudinal fasciculus (SLF). The averaged FA value of each WM fibre was measured and summarized as the mean +/- standard deviation (SD). All data were analysed by paired Student's t-test. A p-value of less than 0.05 was considered to indicate statistical significance.

RESULTS

Visual investigation of WM fibre tracts showed that the ICAL, brainstem CST, ML, MCP, and external capsule (EC) was similar in controls and subjects. However, the ICPL, AF, PTR, CR, CG, SLF and corpus callosum, were all attenuated in size. All 12 cases of PVL showed a significant mean FA reduction in the ICPL, AF, PTR, CR, CG, SLF, SCC, and GCC in comparison with the ipsilateral regions of healthy controls (p<0.05). However, there were no statistically significant differences of the ICAL, ML, MCP, and brainstem CST when analysed using a two-tailed Student's t-test for paired data (p>0.01).

CONCLUSION

DTI can provide more information for understanding the pathophysiology of motor disability and associated sensory handicap in PVL.

A rotation-invariant spherical harmonic decomposition method for mapping intravoxel multiple fiber structures

A new rotation-invariant spherical harmonic decomposition (SHD) method is proposed in this paper for analyzing high angular resolution diffusion (HARD) imaging. Regular SHD methods have been used to characterize the features of the apparent diffusion coefficient (ADC) profile measured by the HARD technique. However, these regular SHD methods are rotation-variant, i.e., the magnitude and/or the phase of the harmonic components changes with the rotation of the ADC profile. We propose a new rotation-invariant SHD (RI-SHD) method based on the rotation-invariant property of a diffusion tensor model. The basic idea of the proposed method is to reorient the measured ADC profile into a local coordinate system determined by the three eigenvectors of the diffusion tensor in each imaging voxel, and then apply a SHD to the ADC profile. Both simulations and in vivo experiments were carried out to validate the method. Comparisons were made between the component maps from a regular SHD method, diffusion circular spectrum mapping (DCSM) method and the proposed RI-SHD method. The results indicate that the regular SHD maps vary significantly with the rotation of the diffusion-encoding scheme, whereas the maps of the DCSM and the proposed method remain unchanged. In particular, the (0,0)-th, (2,2)-th and (4,4)-th component maps from the RI-SHD method exhibited good consistency with the 0th, 2nd and 4th order maps of the DCSM method, respectively. Compared with the regular SHD methods used in HARD imaging, the proposed RI-SHD method is superior in characterizing the diffusion patterns of multiple fiber structures between different brain regions or across subjects.