High Angular Resolution Diffusion Imaging Signals Research Articles

TL-HARDI: Transform learning based accelerated reconstruction of HARDI data

Diffusion magnetic resonance imaging (dMRI) is being extensively used to study the neural architecture of the brain. High angular resolution diffusion imaging (HARDI), a variant of diffusion MRI, measures the diffusion of water molecules along the angular gradient directions in the q-space. It provides better estimates of fiber orientations compared to the traditionally used diffusion tensor imaging (DTI). However, HARDI requires acquisition of relatively large number of samples leading to longer scanning times. Several approaches based on compressive sensing (CS) have been proposed to accelerate HARDI acquisition, leveraging on the sparse representation of the HARDI signal in a pre-specified sparsifying basis. In this paper, we propose to carry out reconstruction of compressively sensed HARDI data using an adaptively learned transform. The transform is learned (i) from the compressive measurements on-the-fly, thereby, eliminating the overhead of choosing fixed sparsifying transforms, and (ii) on overlapping patches of the data, thereby, capturing local image structure effectively. Experiments are conducted on multiple real HARDI data for varying sampling ratios and sampling schemes. The performance of the proposed “TL-HARDI” method is compared with the state-of-the-art methods on various known image quality metrics as well as on dMRI feature maps derived from the reconstructed images. The proposed method is observed to yield better reconstruction than the existing state-of-the-art methods in both quantitative and qualitative comparisons.

Quantitative evaluation of fiber tractography with a Delaunay triangulation-based interpolation approach.

The recent challenge in high angular resolution diffusion imaging (HARDI) is to find a tractography process that provides information about the neural architecture within the white matter of the brain in a clinically feasible measurement time. The great success of the HARDI technique comes from its capability to overcome the problem of crossing fiber detection. However, it requires a large number of diffusion-weighted (DW) images which is problematic for clinical time and hardware. The main contribution of this paper is to develop a full tractography framework that gives an accurate estimate of the crossing fiber problem with the aim of reducing data acquisition time. We explore the interpolation in the gradient direction domain as a method to estimate the HARDI signal from a reduced set of DW images. The experimentation was performed in a first time on simulated data for a quantitative evaluation using the Tractometer system. We used, also, in vivo human brain data to demonstrate the potential of our pipeline. Results on both simulated and real data illustrate the effectiveness of our approach to perform the brain connectivity. Overall, we have shown that the proposed approach achieves competitive results to other tractography methods according to Tractometer connectivity metrics. Graphical Abstract.

A localized Richardson-Lucy algorithm for fiber orientation estimation in high angular resolution diffusion imaging.

Diffusion tensor imaging is widely used for studying neural fiber trajectories in white matter and for quantifying changes in tissue using diffusion properties at each voxel in the brain. To better model the nature of crossing fibers within complex architectures, rather than using a simplified tensor model that assumes only a single fiber direction at each image voxel, a model mixing multiple diffusion tensors is used to profile diffusion signals from high angular resolution diffusion imaging (HARDI) data. Based on the HARDI signal and a multiple tensors model, spherical deconvolution methods have been developed to overcome the limitations of the diffusion tensor model when resolving crossing fibers. The Richardson-Lucy algorithm is a popular spherical deconvolution method used in previous work. However, it is based on a Gaussian distribution, while HARDI data are always very noisy, and the distribution of HARDI data follows a Rician distribution. This current work aims to present a novel solution to address these issues. By simultaneously considering both the Rician bias and neighbor correlation in HARDI data, the authors propose a localized Richardson-Lucy (LRL) algorithm to estimate fiber orientations for HARDI data. The proposed method can simultaneously reduce noise and correct the Rician bias. Mean angular error (MAE) between the estimated Fiber orientation distribution (FOD) field and the reference FOD field was computed to examine whether the proposed LRL algorithm offered any advantage over the conventional RL algorithm at various levels of noise. Normalized mean squared error (NMSE) was also computed to measure the similarity between the true FOD field and the estimated FOD filed. For MAE comparisons, the proposed LRL approach obtained the best results in most of the cases at different levels of SNR and b-values. For NMSE comparisons, the proposed LRL approach obtained the best results in most of the cases at b-value = 3000 s/mm(2), which is the recommended schema for HARDI data acquisition. In addition, the FOD fields estimated by the proposed LRL approach in regions of fiber crossing regions using real data sets also showed similar fiber structures which agreed with common acknowledge in these regions. The novel spherical deconvolution method for improved accuracy in investigating crossing fibers can simultaneously reduce noise and correct Rician bias. With the noise smoothed and bias corrected, this algorithm is especially suitable for estimation of fiber orientations in HARDI data. Experimental results using both synthetic and real imaging data demonstrated the success and effectiveness of the proposed LRL algorithm.

Optic Radiation Fiber Tractography in Glioma Patients Based on High Angular Resolution Diffusion Imaging with Compressed Sensing Compared with Diffusion Tensor Imaging - Initial Experience

ObjectiveUp to now, fiber tractography in the clinical routine is mostly based on diffusion tensor imaging (DTI). However, there are known drawbacks in the resolution of crossing or kissing fibers and in the vicinity of a tumor or edema. These restrictions can be overcome by tractography based on High Angular Resolution Diffusion Imaging (HARDI) which in turn requires larger numbers of gradients resulting in longer acquisition times. Using compressed sensing (CS) techniques, HARDI signals can be obtained by using less non-collinear diffusion gradients, thus enabling the use of HARDI-based fiber tractography in the clinical routine.MethodsEight patients with gliomas in the temporal lobe, in proximity to the optic radiation (OR), underwent 3T MRI including a diffusion-weighted dataset with 30 gradient directions. Fiber tractography of the OR using a deterministic streamline algorithm based on DTI was compared to tractography based on reconstructed diffusion signals using HARDI+CS.ResultsHARDI+CS based tractography displayed the OR more conclusively compared to the DTI-based results in all eight cases. In particular, the potential of HARDI+CS-based tractography was observed for cases of high grade gliomas with significant peritumoral edema, larger tumor size or closer proximity of tumor and reconstructed fiber tract.ConclusionsOvercoming the problem of long acquisition times, HARDI+CS seems to be a promising basis for fiber tractography of the OR in regions of disturbed diffusion, areas of high interest in glioma surgery.

Modeling diffusion-weighted MRI as a spatially variant gaussian mixture: application to image denoising.

This work describes a spatially variant mixture model constrained by a Markov random field to model high angular resolution diffusion imaging (HARDI) data. Mixture models suit HARDI well because the attenuation by diffusion is inherently a mixture. The goal is to create a general model that can be used in different applications. This study focuses on image denoising and segmentation (primarily the former). HARDI signal attenuation data are used to train a Gaussian mixture model in which the mean vectors and covariance matrices are assumed to be independent of spatial locations, whereas the mixture weights are allowed to vary at different lattice positions. Spatial smoothness of the data is ensured by imposing a Markov random field prior on the mixture weights. The model is trained in an unsupervised fashion using the expectation maximization algorithm. The number of mixture components is determined using the minimum message length criterion from information theory. Once the model has been trained, it can be fitted to a noisy diffusion MRI volume by maximizing the posterior probability of the underlying noiseless data in a Bayesian framework, recovering a denoised version of the image. Moreover, the fitted probability maps of the mixture components can be used as features for posterior image segmentation. The model-based denoising algorithm proposed here was compared on real data with three other approaches that are commonly used in the literature: Gaussian filtering, anisotropic diffusion, and Rician-adapted nonlocal means. The comparison shows that, at low signal-to-noise ratio, when these methods falter, our algorithm considerably outperforms them. When tractography is performed on the model-fitted data rather than on the noisy measurements, the quality of the output improves substantially. Finally, ventricle and caudate nucleus segmentation experiments also show the potential usefulness of the mixture probability maps for classification tasks. The presented spatially variant mixture model for diffusion MRI provides excellent denoising results at low signal-to-noise ratios. This makes it possible to restore data acquired with a fast (i.e., noisy) pulse sequence to acceptable noise levels. This is the case in diffusion MRI, where a large number of diffusion-weighted volumes have to be acquired under clinical time constraints.

Spatially Regularized Compressed Sensing for High Angular Resolution Diffusion Imaging

Despite the relative recency of its inception, the theory of compressive sampling (aka compressed sensing) (CS) has already revolutionized multiple areas of applied sciences, a particularly important instance of which is medical imaging. Specifically, the theory has provided a different perspective on the important problem of optimal sampling in magnetic resonance imaging (MRI), with an ever-increasing body of works reporting stable and accurate reconstruction of MRI scans from the number of spectral measurements which would have been deemed unacceptably small as recently as five years ago. In this paper, the theory of CS is employed to palliate the problem of long acquisition times, which is known to be a major impediment to the clinical application of high angular resolution diffusion imaging (HARDI). Specifically, we demonstrate that a substantial reduction in data acquisition times is possible through minimization of the number of diffusion encoding gradients required for reliable reconstruction of HARDI scans. The success of such a minimization is primarily due to the availability of spherical ridgelet transformation, which excels in sparsifying HARDI signals. What makes the resulting reconstruction procedure even more accurate is a combination of the sparsity constraints in the diffusion domain with additional constraints imposed on the estimated diffusion field in the spatial domain. Accordingly, the present paper describes an original way to combine the diffusion- and spatial-domain constraints to achieve a maximal reduction in the number of diffusion measurements, while sacrificing little in terms of reconstruction accuracy. Finally, details are provided on an efficient numerical scheme which can be used to solve the aforementioned reconstruction problem by means of standard and readily available estimation tools. The paper is concluded with experimental results which support the practical value of the proposed reconstruction methodology.

On approximation of orientation distributions by means of spherical ridgelets.

Visualization and analysis of the micro-architecture of brain parenchyma by means of magnetic resonance imaging is nowadays believed to be one of the most powerful tools used for the assessment of various cerebral conditions as well as for understanding the intracerebral connectivity. Unfortunately, the conventional diffusion tensor imaging (DTI) used for estimating the local orientations of neural fibers is incapable of performing reliably in the situations when a voxel of interest accommodates multiple fiber tracts. In this case, a much more accurate analysis is possible using the high angular resolution diffusion imaging (HARDI) that represents local diffusion by its apparent coefficients measured as a discrete function of spatial orientations. In this note, a novel approach to enhancing and modeling the HARDI signals using multiresolution bases of spherical ridgelets is presented. In addition to its desirable properties of being adaptive, sparsifying, and efficiently computable, the proposed modeling leads to analytical computation of the orientation distribution functions associated with the measured diffusion, thereby providing a fast and robust analytical solution for q-ball imaging.

How does angular resolution affect diffusion imaging measures?

A key question in diffusion imaging is how many diffusion-weighted images suffice to provide adequate signal-to-noise ratio (SNR) for studies of fiber integrity. Motion, physiological effects, and scan duration all affect the achievable SNR in real brain images, making theoretical studies and simulations only partially useful. We therefore scanned 50 healthy adults with 105-gradient high-angular resolution diffusion imaging (HARDI) at 4T. From gradient image subsets of varying size (6≤N≤94) that optimized a spherical angular distribution energy, we created SNR plots (versus gradient numbers) for seven common diffusion anisotropy indices: fractional and relative anisotropy (FA, RA), mean diffusivity (MD), volume ratio (VR), geodesic anisotropy (GA), its hyperbolic tangent (tGA), and generalized fractional anisotropy (GFA). SNR, defined in a region of interest in the corpus callosum, was near-maximal with 58, 66, and 62 gradients for MD, FA, and RA, respectively, and with about 55 gradients for GA and tGA. For VR and GFA, SNR increased rapidly with more gradients. SNR was optimized when the ratio of diffusion-sensitized to non-sensitized images was 9.13 for GA and tGA, 10.57 for FA, 9.17 for RA, and 26 for MD and VR. In orientation density functions modeling the HARDI signal as a continuous mixture of tensors, the diffusion profile reconstruction accuracy rose rapidly with additional gradients. These plots may help in making trade-off decisions when designing diffusion imaging protocols.

Analysis of Fiber Reconstruction Accuracy in High Angular Resolution Diffusion Images (HARDI)

Introduction: High angular resolution diffusion imaging (HARDI) is a powerful extension of MRI that maps the directional diffusion of water in the brain. With more diffusion gradients and directions, fiber directions may be tracked with greater angular precision, fiber crossings can be resolved, and anisotropy measures can be derived from the full fiber orientation density function. To better reconstruct HARDI, we recently introduced the tensor distribution function (TDF), which models multidirectional diffusion as a probabilistic mixture of all symmetric positive definite tensors [1]. The TDF overcomes limitations of several HARDI reconstruction methods (e.g., q-ball imaging, DOT, PAS) which restrict all component fibers in a voxel to have the same anisotropy profile. The TDF models the HARDI signal more flexibly, as a unit-mass probability density on the 6D manifold of symmetric positive definite tensors, yielding a TDF, or continuous mixture of tensors, at each point in the brain. From the TDF, one can derive analytic formulae for the orientation distribution function (ODF), tensor orientation density (TOD), and their corresponding anisotropy measures. Because this model can accurately resolve sharp signal peaks in angular space where fibers cross, we studied how many gradients are required in practice to compute accurate orientation density functions, as more gradients require longer scanning times. In simulated two-fiber systems with varying Rician noise, we assessed how many diffusionsensitized gradients were sufficient for (1) accurately resolving the diffusion profile, and (2) measuring the exponential isotropy (EI), a TDF-derived measure of fiber integrity that exploits the full multidirectional HARDI signal.

Validating the Tensor Distribution Function for Fiber Reconstruction in HARDI (High-Angular Resolution Diffusion Imaging)

Introduction: Diffusion tensor imaging (DTI) reveals white matter microstructure and fiber pathways in the living brain by examining the 3D diffusion profile of water molecules in brain tissue. Even so, DTI-derived measures will be incorrect where fibers cross or mix, as the single tensor model cannot resolve these more complicated white matter configurations. High-angular resolution diffusion imaging (HARDI) addresses this problem by applying more than 6 independent diffusion-sensitized gradients. Many HARDI reconstruction methods (e.g., q-ball imaging, DOT, PAS) impose restrictive assumptions on fibers, e.g., all fiber tracts must have the same anisotropy profile. Here we model the HARDI signal more flexibly, as in [1], as a unit-mass probability density on the 6D manifold of symmetric positive definite tensors, yielding a Tensor Distribution Function (TDF), or continuous mixture of tensors, at each point in the brain. The TDF can model fiber crossing and non-Gaussian diffusion. From the TDF, one can derive analytic formulae for the water displacement probability function, orientation distribution function (ODF), tensor orientation distribution function (TOD), and their corresponding anisotropy measures. Here we further develop the TDF framework, verifying its accuracy in revealing fiber crossings in human brain HARDI data.

Fully automated classification of HARDI in vivo data using a support vector machine

The purpose of this study is the classification of high angular resolution diffusion imaging (HARDI) in vivo data using a model-free approach. This is achieved by using a Support Vector Machine (SVM) algorithm taken from the field of supervised statistical learning. Six classes of image components are determined: grey matter, parallel neuronal fibre bundles in white matter, crossing neuronal fibre bundles in white matter, partial volume between white and grey matter, background noise and cerebrospinal fluid. The SVM requires properties derived from the data as input, the so called feature vector, which should be rotation invariant. For our application we derive such a description from the spherical harmonic decomposition of the HARDI signal. With this information the SVM is trained in order to find the function for separating the classes. The SVM is systematically tested with simulated data and then applied to six in vivo data sets. This new approach is data-driven and enables fully automatic HARDI data segmentation without employing a T1 MPRAGE scan and subjective expert intervention. This was demonstrated on five test in vivo data sets giving robust results. The segmentation results could be used as a priori knowledge for increasing the performance of fibre tracking as well as for other clinical and diagnostic applications of diffusion weighted imaging (DWI).

Directional functions for orientation distribution estimation

Computing the orientation distribution function (ODF) from high angular resolution diffusion imaging (HARDI) signals makes it possible to determine the orientation of fiber bundles of the brain. The HARDI signals are samples measured from a spherical shell and thus require processing on the sphere. Past work on ODF estimation involved using the spherical harmonics or spherical radial basis functions. In this work, we propose three novel directional functions able to represent the measured signals in a very compact manner, i.e., they require very few parameters to completely describe the measured signal. Analytical expressions are derived for computing the corresponding ODF. The directional functions can represent diffusion in a particular direction and mixture models can be used to represent multi-fiber orientations. We show how to estimate the parameters of this mixture model and elaborate on the differences between these functions. We also compare this general framework with estimation of ODF using spherical harmonics on some real and synthetic data. The proposed method could be particularly useful in applications such as tractography and segmentation. Details are also given on different ways in which interpolation can be performed using directional functions. In particular, we discuss a complete Euclidean as well as a "hybrid" framework, comprising of the Riemannian as well as Euclidean spaces, to perform interpolation and compute geodesic distances between two ODF's.

Deterministic and Probabilistic Tractography Based on Complex Fibre Orientation Distributions

We propose an integral concept for tractography to describe crossing and splitting fibre bundles based on the fibre orientation distribution function (ODF) estimated from high angular resolution diffusion imaging (HARDI). We show that in order to perform accurate probabilistic tractography, one needs to use a fibre ODF estimation and not the diffusion ODF. We use a new fibre ODF estimation obtained from a sharpening deconvolution transform (SDT) of the diffusion ODF reconstructed from q-ball imaging (QBI). This SDT provides new insight into the relationship between the HARDI signal, the diffusion ODF, and the fibre ODF. We demonstrate that the SDT agrees with classical spherical deconvolution and improves the angular resolution of QBI. Another important contribution of this paper is the development of new deterministic and new probabilistic tractography algorithms using the full multidirectional information obtained through use of the fibre ODF. An extensive comparison study is performed on human brain datasets comparing our new deterministic and probabilistic tracking algorithms in complex fibre crossing regions. Finally, as an application of our new probabilistic tracking, we quantify the reconstruction of transcallosal fibres intersecting with the corona radiata and the superior longitudinal fasciculus in a group of eight subjects. Most current diffusion tensor imaging (DTI)-based methods neglect these fibres, which might lead to incorrect interpretations of brain functions.

Q‐ball imaging

Magnetic resonance diffusion tensor imaging (DTI) provides a powerful tool for mapping neural histoarchitecture in vivo. However, DTI can only resolve a single fiber orientation within each imaging voxel due to the constraints of the tensor model. For example, DTI cannot resolve fibers crossing, bending, or twisting within an individual voxel. Intravoxel fiber crossing can be resolved using q-space diffusion imaging, but q-space imaging requires large pulsed field gradients and time-intensive sampling. It is also possible to resolve intravoxel fiber crossing using mixture model decomposition of the high angular resolution diffusion imaging (HARDI) signal, but mixture modeling requires a model of the underlying diffusion process.Recently, it has been shown that the HARDI signal can be reconstructed model-independently using a spherical tomographic inversion called the Funk-Radon transform, also known as the spherical Radon transform. The resulting imaging method, termed q-ball imaging, can resolve multiple intravoxel fiber orientations and does not require any assumptions on the diffusion process such as Gaussianity or multi-Gaussianity. The present paper reviews the theory of q-ball imaging and describes a simple linear matrix formulation for the q-ball reconstruction based on spherical radial basis function interpolation. Open aspects of the q-ball reconstruction algorithm are discussed.