On approximation of orientation distributions by means of spherical ridgelets

Q-space imaging based on Gaussian radial basis function with Laplace regularization

PURPOSE

To introduce the diffusion signal characteristics presented by spherical harmonics (SH) basis into the q-space imaging method based on Gaussian radial basis function (GRBF) to robustly reconstruct ensemble average diffusion propagator (EAP) in diffusion MRI (dMRI).

METHODS

We introduced the Laplacian regularization of the signal into the dMRI imaging method based on GRBF, and derived the relevant indicators of microstructure imaging and the orientation distribution function (ODF) providing fiber bundle direction information based on EAP. In addition, this method is combined with a multi-compartment model to calculate the diameter of fiber bundle axons. The evaluation of the results included qualitative comparisons and quantitative assessments of the signal fitting.

RESULTS

The results show that the proposed method achieves the more significant accuracy improvement in reconstructing signal. Meanwhile, ODFs estimated by the proposed method show the sharper profiles and less spurious peaks, even under the sparse and noisy conditions. In the 36 sets of axon diameter estimation experiments, 34 and 30 sets of results showed that the proposed method reduced the mean and SD of axon diameter estimates, respectively. Moreover, compared with the current state-of-the-art method, the mean and SD of axon diameter estimated by the proposed method are mostly lower, with 32 and 29 of 36 groups.

CONCLUSION

The proposed method outperforms the GRBF regarding signal fitting and the estimation of the EAP and ODF with multi-shell sparse samples. Moreover, it shows the potential to recover important features of microstructures with less uncertainty by using proposed method together with multi-compartment models.

Neural orientation distribution fields for estimation and uncertainty quantification in diffusion MRI

Inferring brain connectivity and structure in-vivo requires accurate estimation of the orientation distribution function (ODF), which encodes key local tissue properties. However, estimating the ODF from diffusion MRI (dMRI) signals is a challenging inverse problem due to obstacles such as significant noise, high-dimensional parameter spaces, and sparse angular measurements. In this paper, we address these challenges by proposing a novel deep-learning based methodology for continuous estimation and uncertainty quantification of the spatially varying ODF field. We use a neural field (NF) to parameterize a random series representation of the latent ODFs, implicitly modeling the often ignored but valuable spatial correlation structures in the data, and thereby improving efficiency in sparse and noisy regimes. An analytic approximation to the posterior predictive distribution is derived which can be used to quantify the uncertainty in the ODF estimate at any spatial location, avoiding the need for expensive resampling-based approaches that are typically employed for this purpose. We present empirical evaluations on both synthetic and real in-vivo diffusion data, demonstrating the advantages of our method over existing approaches.

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.

Accelerating in vivo fast spin echo high angular resolution diffusion imaging with an isotropic resolution in mice through compressed sensing

PURPOSE

Echo planar imaging (EPI) is commonly used to acquire the many volumes needed for high angular resolution diffusion Imaging (HARDI), posing a higher risk for artifacts, such as distortion and deformation. An alternative to EPI is fast spin echo (FSE) imaging, which has fewer artifacts but is inherently slower. The aim is to accelerate FSE such that a HARDI data set can be acquired in a time comparable to EPI using compressed sensing.

METHODS

Compressed sensing was applied in either q-space or simultaneously in k-space and q-space, by undersampling the k-space in the phase-encoding direction or retrospectively eliminating diffusion directions for different degrees of undersampling. To test the replicability of the acquisition and reconstruction, brain data were acquired from six mice, and a numerical phantom experiment was performed. All HARDI data were analyzed individually using constrained spherical deconvolution, and the apparent fiber density and complexity metric were evaluated, together with whole-brain tractography.

RESULTS

The apparent fiber density and complexity metric showed relatively minor differences when only q-space undersampling was used, but deteriorate when k-space undersampling was applied. Likewise, the tract density weighted image showed good results when only q-space undersampling was applied using 15 directions or more, but information was lost when fewer volumes or k-space undersampling were used.

CONCLUSION

It was found that acquiring 15 to 20 diffusion directions with a full k-space and reconstructed using compressed sensing could suffice for a replicable measurement of quantitative measures in mice, where areas near the sinuses and ear cavities are untainted by signal loss.

Joint RElaxation-Diffusion Imaging Moments to Probe Neurite Microstructure

Joint relaxation-diffusion measurements can provide new insight about the tissue microstructural properties. Most recent methods have focused on inverting the Laplace transform to recover the joint distribution of relaxation-diffusion. However, as is well-known, this problem is notoriously ill-posed and numerically unstable. In this work, we address this issue by directly computing the joint moments of transverse relaxation rate and diffusivity, which can be robustly estimated. To zoom into different parts of the joint distribution, we further enhance our method by applying multiplicative filters to the joint probability density function of relaxation and diffusion and compute the corresponding moments. We propose an approach to use these moments to compute several novel scalar indices to characterize specific properties of the underlying tissue microstructure. Furthermore, for the first time, we propose an algorithm to estimate diffusion signals that are independent of echo time based on the moments of the marginal probability density function of diffusion. We demonstrate its utility in extracting tissue information not contaminated with multiple intra-voxel relaxation rates. We compare the performance of four types of filters that zoom into tissue components with different relaxation and diffusion properties and demonstrate it on an in-vivo human dataset. Experimental results show that these filters are able to characterize heterogeneous tissue microstructure. Moreover, the filtered diffusion signals are also able to distinguish fiber bundles with similar orientations but different relaxation rates. The proposed method thus allows to characterize the neural microstructure information in a robust and unique manner not possible using existing techniques.

Advances in computational and statistical diffusion MRI

Computational methods are crucial for the analysis of diffusion magnetic resonance imaging (MRI) of the brain. Computational diffusion MRI can provide rich information at many size scales, including local microstructure measures such as diffusion anisotropies or apparent axon diameters, whole-brain connectivity information that describes the brain's wiring diagram and population-based studies in health and disease. Many of the diffusion MRI analyses performed today were not possible five, ten or twenty years ago, due to the requirements for large amounts of computer memory or processor time. In addition, mathematical frameworks had to be developed or adapted from other fields to create new ways to analyze diffusion MRI data. The purpose of this review is to highlight recent computational and statistical advances in diffusion MRI and to put these advances into context by comparison with the more traditional computational methods that are in popular clinical and scientific use. We aim to provide a high-level overview of interest to diffusion MRI researchers, with a more in-depth treatment to illustrate selected computational advances.

Joint spatial-angular sparse coding for dMRI with separable dictionaries

Diffusion MRI (dMRI) provides the ability to reconstruct neuronal fibers in the brain, in vivo, by measuring water diffusion along angular gradient directions in q-space. High angular resolution diffusion imaging (HARDI) can produce better estimates of fiber orientation than the popularly used diffusion tensor imaging, but the high number of samples needed to estimate diffusivity requires longer patient scan times. To accelerate dMRI, compressed sensing (CS) has been utilized by exploiting a sparse dictionary representation of the data, discovered through sparse coding. The sparser the representation, the fewer samples are needed to reconstruct a high resolution signal with limited information loss, and so an important area of research has focused on finding the sparsest possible representation of dMRI. Current reconstruction methods however, rely on an angular representation per voxel with added spatial regularization, and so, for non-zero signals, one is required to have at least one non-zero coefficient per voxel. This means that the global level of sparsity must be greater than the number of voxels. In contrast, we propose a joint spatial-angular representation of dMRI that will allow us to achieve levels of global sparsity that are below the number of voxels. A major challenge, however, is the computational complexity of solving a global sparse coding problem over large-scale dMRI. In this work, we present novel adaptations of popular sparse coding algorithms that become better suited for solving large-scale problems by exploiting spatial-angular separability. Our experiments show that our method achieves significantly sparser representations of HARDI than is possible by the state of the art.

Estimating fiber orientation distribution from diffusion MRI with spherical needlets

We present a novel method for estimation of the fiber orientation distribution (FOD) function based on diffusion-weighted magnetic resonance imaging (D-MRI) data. We formulate the problem of FOD estimation as a regression problem through spherical deconvolution and a sparse representation of the FOD by a spherical needlets basis that forms a multi-resolution tight frame for spherical functions. This sparse representation allows us to estimate the FOD by ℓ1-penalized regression under a non-negativity constraint on the estimated FOD. The resulting convex optimization problem is solved by an alternating direction method of multipliers (ADMM) algorithm. The proposed method leads to a reconstruction of the FOD that is accurate, has low variability and preserves sharp features. Through extensive experiments, we demonstrate the effectiveness and favorable performance of the proposed method compared to three existing methods. Specifically, we demonstrate that the proposed method is able to successfully resolve fiber crossings at small angles and automatically identify isotropic diffusion. We also apply the proposed method to real 3T D-MRI data sets of healthy individuals. The results show realistic depictions of crossing fibers that are more accurate, less noisy, and lead to superior tractography results compared to competing methods.

A joint space-angle regularization approach for single 4D diffusion image super-resolution

PURPOSE

Low signal-to-noise-ratio and limited scan time of diffusion magnetic resonance imaging (dMRI) in current clinical settings impede obtaining images with high spatial and angular resolution (HSAR) for a reliable fiber reconstruction with fine anatomical details. To overcome this problem, we propose a joint space-angle regularization approach to reconstruct HSAR diffusion signals from a single 4D low resolution (LR) dMRI, which is down-sampled in both 3D-space and q-space.

METHODS

Different from the existing works which combine multiple 4D LR diffusion images acquired using specific acquisition protocols, the proposed method reconstructs HSAR dMRI from only a single 4D dMRI by exploring and integrating two key priors, that is, the nonlocal self-similarity in the spatial domain as a prior to increase spatial resolution and ridgelet approximations in the diffusion domain as another prior to increase the angular resolution of dMRI. To more effectively capture nonlocal self-similarity in the spatial domain, a novel 3D block-based nonlocal means filter is imposed as the 3D image space regularization term which is accurate in measuring the similarity and fast for 3D reconstruction. To reduce computational complexity, we use the L₂ -norm instead of sparsity constraint on the representation coefficients.

RESULTS

Experimental results demonstrate that the proposed method can obtain the HSAR dMRI efficiently with approximately 2% per-voxel root-mean-square error between the actual and reconstructed HSAR dMRI.

CONCLUSION

The proposed approach can effectively increase the spatial and angular resolution of the dMRI which is independent of the acquisition protocol, thus overcomes the inherent resolution limitation of imaging systems.

Model-based denoising in diffusion-weighted imaging using generalized spherical deconvolution

PURPOSE

Diffusion MRI often suffers from low signal-to-noise ratio, especially for high b-values. This work proposes a model-based denoising technique to address this limitation.

METHODS

A generalization of the multi-shell spherical deconvolution model using a Richardson-Lucy algorithm is applied to noisy data. The reconstructed coefficients are then used in the forward model to compute denoised diffusion-weighted images (DWIs). The proposed method operates in the diffusion space and thus is complementary to image-based denoising methods.

RESULTS

We demonstrate improved image quality on the DWIs themselves, maps of neurite orientation dispersion and density imaging, and diffusional kurtosis imaging (DKI), as well as reduced spurious peaks in deterministic tractography. For DKI in particular, we observe up to 50% error reduction and demonstrate high image quality using just 30 DWIs. This corresponds to greater than fourfold reduction in scan time if compared to the widely used 140-DWI acquisitions. We also confirm consistent performance in pathological data sets, namely in white matter lesions of a multiple sclerosis patient.

CONCLUSION

The proposed denoising technique termed generalized spherical deconvolution has the potential of significantly improving image quality in diffusion MRI. Magn Reson Med 78:2428-2438, 2017. © 2017 International Society for Magnetic Resonance in Medicine.

Tight Graph Framelets for Sparse Diffusion MRI q-Space Representation

In diffusion MRI, the outcome of estimation problems can often be improved by taking into account the correlation of diffusion-weighted images scanned with neighboring wavevectors in q-space. For this purpose, we propose in this paper to employ tight wavelet frames constructed on non-flat domains for multi-scale sparse representation of diffusion signals. This representation is well suited for signals sampled regularly or irregularly, such as on a grid or on multiple shells, in q-space. Using spectral graph theory, the frames are constructed based on quasi-affine systems (i.e., generalized dilations and shifts of a finite collection of wavelet functions) defined on graphs, which can be seen as a discrete representation of manifolds. The associated wavelet analysis and synthesis transforms can be computed efficiently and accurately without the need for explicit eigen-decomposition of the graph Laplacian, allowing scalability to very large problems. We demonstrate the effectiveness of this representation, generated using what we call tight graph framelets, in two specific applications: denoising and super-resolution in q-space using ℓ₀ regularization. The associated optimization problem involves only thresholding and solving a trivial inverse problem in an iterative manner. The effectiveness of graph framelets is confirmed via evaluation using synthetic data with noncentral chi noise and real data with repeated scans.

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

PURPOSE

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.

METHODS

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.

RESULTS

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.

CONCLUSIONS

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.

Assessment of the Characteristics of Orientation Distribution Functions in HARDI Using Morphological Metrics

Orientation distribution functions (ODFs) are widely used to resolve fiber crossing problems in high angular resolution diffusion imaging (HARDI). The characteristics of the ODFs are often assessed using a visual criterion, although the use of objective criteria is also reported, which are directly borrowed from classic signal and image processing theory because they are intuitive and simple to compute. However, they are not always pertinent for the characterization of ODFs. We propose a more general paradigm for assessing the characteristics of ODFs. The idea consists in regarding an ODF as a three-dimensional (3D) point cloud, projecting the 3D point cloud onto an angle-distance map, constructing an angle-distance matrix, and calculating metrics such as length ratio, separability, and uncertainty. The results from both simulated and real data show that the proposed metrics allow for the assessment of the characteristics of ODFs in a quantitative and relatively complete manner.

A joint compressed-sensing and super-resolution approach for very high-resolution diffusion imaging

Diffusion MRI (dMRI) can provide invaluable information about the structure of different tissue types in the brain. Standard dMRI acquisitions facilitate a proper analysis (e.g. tracing) of medium-to-large white matter bundles. However, smaller fiber bundles connecting very small cortical or sub-cortical regions cannot be traced accurately in images with large voxel sizes. Yet, the ability to trace such fiber bundles is critical for several applications such as deep brain stimulation and neurosurgery. In this work, we propose a novel acquisition and reconstruction scheme for obtaining high spatial resolution dMRI images using multiple low resolution (LR) images, which is effective in reducing acquisition time while improving the signal-to-noise ratio (SNR). The proposed method called compressed-sensing super resolution reconstruction (CS-SRR), uses multiple overlapping thick-slice dMRI volumes that are under-sampled in q-space to reconstruct diffusion signal with complex orientations. The proposed method combines the twin concepts of compressed sensing and super-resolution to model the diffusion signal (at a given b-value) in a basis of spherical ridgelets with total-variation (TV) regularization to account for signal correlation in neighboring voxels. A computationally efficient algorithm based on the alternating direction method of multipliers (ADMM) is introduced for solving the CS-SRR problem. The performance of the proposed method is quantitatively evaluated on several in-vivo human data sets including a true SRR scenario. Our experimental results demonstrate that the proposed method can be used for reconstructing sub-millimeter super resolution dMRI data with very good data fidelity in clinically feasible acquisition time.

Strengths and weaknesses of state of the art fiber tractography pipelines--A comprehensive in-vivo and phantom evaluation study using Tractometer

Many different tractography approaches and corresponding isolated evaluation attempts have been presented over the last years, but a comparative and quantitative evaluation of tractography algorithms still remains a challenge, particularly in-vivo. The recently presented evaluation framework Tractometer is the first attempt to approach this challenge in a quantitative, comparative, persistent and open-access way. Tractometer is currently based on the evaluation of several global connectivity and tract-overlap metrics on hardware phantom data. The work presented in this paper focuses on extending Tractometer with a metric that enables the assessment of the local consistency of tractograms with the underlying image data that is not only applicable to phantom dataset but allows the quantitative and purely data-driven evaluation of in-vivo tractography. We furthermore present an extensive reference-based evaluation study of 25,000 tractograms obtained on phantom and in-vivo datasets using the presented local metric as well as all the methods already established in Tractometer. The experiments showed that the presented local metric successfully reflects the behavior of in-vivo tractography under different conditions and that it is consistent with the results of previous studies. Additionally our experiments enabled a multitude of conclusions with implications for fiber tractography in general, including recommendations regarding optimal choice of a local modeling technique, tractography algorithm, and parameterization, confirming and complementing the results of earlier studies.

Sparse Reconstruction Challenge for diffusion MRI: Validation on a physical phantom to determine which acquisition scheme and analysis method to use?

Diffusion magnetic resonance imaging (dMRI) is the modality of choice for investigating in-vivo white matter connectivity and neural tissue architecture of the brain. The diffusion-weighted signal in dMRI reflects the diffusivity of water molecules in brain tissue and can be utilized to produce image-based biomarkers for clinical research. Due to the constraints on scanning time, a limited number of measurements can be acquired within a clinically feasible scan time. In order to reconstruct the dMRI signal from a discrete set of measurements, a large number of algorithms have been proposed in recent years in conjunction with varying sampling schemes, i.e., with varying b-values and gradient directions. Thus, it is imperative to compare the performance of these reconstruction methods on a single data set to provide appropriate guidelines to neuroscientists on making an informed decision while designing their acquisition protocols. For this purpose, the SPArse Reconstruction Challenge (SPARC) was held along with the workshop on Computational Diffusion MRI (at MICCAI 2014) to validate the performance of multiple reconstruction methods using data acquired from a physical phantom. A total of 16 reconstruction algorithms (9 teams) participated in this community challenge. The goal was to reconstruct single b-value and/or multiple b-value data from a sparse set of measurements. In particular, the aim was to determine an appropriate acquisition protocol (in terms of the number of measurements, b-values) and the analysis method to use for a neuroimaging study. The challenge did not delve on the accuracy of these methods in estimating model specific measures such as fractional anisotropy (FA) or mean diffusivity, but on the accuracy of these methods to fit the data. This paper presents several quantitative results pertaining to each reconstruction algorithm. The conclusions in this paper provide a valuable guideline for choosing a suitable algorithm and the corresponding data-sampling scheme for clinical neuroscience applications.

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.

A Compressed-Sensing Approach for Super-Resolution Reconstruction of Diffusion MRI

We present an innovative framework for reconstructing high-spatial-resolution diffusion magnetic resonance imaging (dMRI) from multiple low-resolution (LR) images. Our approach combines the twin concepts of compressed sensing (CS) and classical super-resolution to reduce acquisition time while increasing spatial resolution. We use subpixel-shifted LR images with down-sampled and non-overlapping diffusion directions to reduce acquisition time. The diffusion signal in the high resolution (HR) image is represented in a sparsifying basis of spherical ridgelets to model complex fiber orientations with reduced number of measurements. The HR image is obtained as the solution of a convex optimization problem which can be solved using the proposed algorithm based on the alternating direction method of multipliers (ADMM). We qualitatively and quantitatively evaluate the performance of our method on two sets of in-vivo human brain data and show its effectiveness in accurately recovering very high resolution diffusion images.

Fiber estimation and tractography in diffusion MRI: development of simulated brain images and comparison of multi-fiber analysis methods at clinical b-values

Advances in diffusion-weighted magnetic resonance imaging (DW-MRI) have led to many alternative diffusion sampling strategies and analysis methodologies. A common objective among methods is estimation of white matter fiber orientations within each voxel, as doing so permits in-vivo fiber-tracking and the ability to study brain connectivity and networks. Knowledge of how DW-MRI sampling schemes affect fiber estimation accuracy, tractography and the ability to recover complex white-matter pathways, differences between results due to choice of analysis method, and which method(s) perform optimally for specific data sets, all remain important problems, especially as tractography-based studies become common. In this work, we begin to address these concerns by developing sets of simulated diffusion-weighted brain images which we then use to quantitatively evaluate the performance of six DW-MRI analysis methods in terms of estimated fiber orientation accuracy, false-positive (spurious) and false-negative (missing) fiber rates, and fiber-tracking. The analysis methods studied are: 1) a two-compartment "ball and stick" model (BSM) (Behrens et al., 2003); 2) a non-negativity constrained spherical deconvolution (CSD) approach (Tournier et al., 2007); 3) analytical q-ball imaging (QBI) (Descoteaux et al., 2007); 4) q-ball imaging with Funk-Radon and Cosine Transform (FRACT) (Haldar and Leahy, 2013); 5) q-ball imaging within constant solid angle (CSA) (Aganj et al., 2010); and 6) a generalized Fourier transform approach known as generalized q-sampling imaging (GQI) (Yeh et al., 2010). We investigate these methods using 20, 30, 40, 60, 90 and 120 evenly distributed q-space samples of a single shell, and focus on a signal-to-noise ratio (SNR = 18) and diffusion-weighting (b = 1000 s/mm(2)) common to clinical studies. We found that the BSM and CSD methods consistently yielded the least fiber orientation error and simultaneously greatest detection rate of fibers. Fiber detection rate was found to be the most distinguishing characteristic between the methods, and a significant factor for complete recovery of tractography through complex white-matter pathways. For example, while all methods recovered similar tractography of prominent white matter pathways of limited fiber crossing, CSD (which had the highest fiber detection rate, especially for voxels containing three fibers) recovered the greatest number of fibers and largest fraction of correct tractography for complex three-fiber crossing regions. The synthetic data sets, ground-truth, and tools for quantitative evaluation are publically available on the NITRC website as the project "Simulated DW-MRI Brain Data Sets for Quantitative Evaluation of Estimated Fiber Orientations" at http://www.nitrc.org/projects/sim_dwi_brain.

Multi-shell diffusion signal recovery from sparse measurements

For accurate estimation of the ensemble average diffusion propagator (EAP), traditional multi-shell diffusion imaging (MSDI) approaches require acquisition of diffusion signals for a range of b-values. However, this makes the acquisition time too long for several types of patients, making it difficult to use in a clinical setting. In this work, we propose a new method for the reconstruction of diffusion signals in the entire q-space from highly undersampled sets of MSDI data, thus reducing the scan time significantly. In particular, to sparsely represent the diffusion signal over multiple q-shells, we propose a novel extension to the framework of spherical ridgelets by accurately modeling the monotonically decreasing radial component of the diffusion signal. Further, we enforce the reconstructed signal to have smooth spatial regularity in the brain, by minimizing the total variation (TV) norm. We combine these requirements into a novel cost function and derive an optimal solution using the Alternating Directions Method of Multipliers (ADMM) algorithm. We use a physical phantom data set with known fiber crossing angle of 45° to determine the optimal number of measurements (gradient directions and b-values) needed for accurate signal recovery. We compare our technique with a state-of-the-art sparse reconstruction method (i.e., the SHORE method of Cheng et al. (2010)) in terms of angular error in estimating the crossing angle, incorrect number of peaks detected, normalized mean squared error in signal recovery as well as error in estimating the return-to-origin probability (RTOP). Finally, we also demonstrate the behavior of the proposed technique on human in vivo data sets. Based on these experiments, we conclude that using the proposed algorithm, at least 60 measurements (spread over three b-value shells) are needed for proper recovery of MSDI data in the entire q-space.

Estimation of the CSA-ODF using Bayesian compressed sensing of multi-shell HARDI

PURPOSE

Diffusion MRI provides important information about the brain white matter structures and has opened new avenues for neuroscience and translational research. However, acquisition time needed for advanced applications can still be a challenge in clinical settings. There is consequently a need to accelerate diffusion MRI acquisitions.

METHODS

A multi-task Bayesian compressive sensing (MT-BCS) framework is proposed to directly estimate the constant solid angle orientation distribution function (CSA-ODF) from under-sampled (i.e., accelerated image acquisition) multi-shell high angular resolution diffusion imaging (HARDI) datasets, and accurately recover HARDI data at higher resolution in q-space. The proposed MT-BCS approach exploits the spatial redundancy of the data by modeling the statistical relationships within groups (clusters) of diffusion signal. This framework also provides uncertainty estimates of the computed CSA-ODF and diffusion signal, directly computed from the compressive measurements. Experiments validating the proposed framework are performed using realistic multi-shell synthetic images and in vivo multi-shell high angular resolution HARDI datasets.

RESULTS

Results indicate a practical reduction in the number of required diffusion volumes (q-space samples) by at least a factor of four to estimate the CSA-ODF from multi-shell data.

CONCLUSION

This work presents, for the first time, a multi-task Bayesian compressive sensing approach to simultaneously estimate the full posterior of the CSA-ODF and diffusion-weighted volumes from multi-shell HARDI acquisitions. It demonstrates improvement of the quality of acquired datasets by means of CS de-noising, and accurate estimation of the CSA-ODF, as well as enables a reduction in the acquisition time by a factor of two to four, especially when "staggered" q-space sampling schemes are used. The proposed MT-BCS framework can naturally be combined with parallel MR imaging to further accelerate HARDI acquisitions.

Optic radiation fiber tractography in glioma patients based on high angular resolution diffusion imaging with compressed sensing compared with diffusion tensor imaging - initial experience

OBJECTIVE

Up 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.

METHODS

Eight 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.

RESULTS

HARDI+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.

CONCLUSIONS

Overcoming 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.

Fiber tractography based on diffusion tensor imaging compared with high-angular-resolution diffusion imaging with compressed sensing: initial experience

BACKGROUND

The most frequently used method for fiber tractography based on diffusion tensor imaging (DTI) is associated with restrictions in the resolution of crossing or kissing fibers and in the vicinity of tumor or edema. Tractography based on high-angular-resolution diffusion imaging (HARDI) is capable of overcoming this restriction. With compressed sensing (CS) techniques, HARDI acquisitions with a smaller number of directional measurements can be used, thus enabling the use of HARDI-based fiber tractography in clinical practice.

OBJECTIVE

To investigate whether HARDI+CS-based fiber tractography improves the display of neuroanatomically complex pathways and in areas of disturbed diffusion properties.

METHODS

Six patients with gliomas in the vicinity of language-related areas underwent 3-T magnetic resonance imaging including a diffusion-weighted data set with 30 gradient directions. Additionally, functional magnetic resonance imaging for cortical language sites was obtained. Fiber tractography was performed with deterministic streamline algorithms based on DTI using 3 different software platforms. Additionally, tractography based on reconstructed diffusion signals using HARDI+CS was performed.

RESULTS

HARDI+CS-based tractography displayed more compact fiber bundles compared with the DTI-based results in all cases. In 3 cases, neuroanatomically plausible fiber bundles were displayed in the vicinity of tumor and peritumoral edema, which could not be traced on the basis of DTI. The curvature around the sylvian fissure was displayed properly in 6 cases and in only 2 cases with DTI-based tractography.

CONCLUSION

HARDI+CS seems to be a promising approach for fiber tractography in clinical practice for neuroanatomically complex fiber pathways and in areas of disturbed diffusion, overcoming the problem of long acquisition times.

Continuous diffusion signal, EAP and ODF estimation via Compressive Sensing in diffusion MRI

In this paper, we exploit the ability of Compressed Sensing (CS) to recover the whole 3D Diffusion MRI (dMRI) signal from a limited number of samples while efficiently recovering important diffusion features such as the Ensemble Average Propagator (EAP) and the Orientation Distribution Function (ODF). Some attempts to use CS in estimating diffusion signals have been done recently. However, this was mainly an experimental insight of CS capabilities in dMRI and the CS theory has not been fully exploited. In this work, we also propose to study the impact of the sparsity, the incoherence and the RIP property on the reconstruction of diffusion signals. We show that an efficient use of the CS theory enables to drastically reduce the number of measurements commonly used in dMRI acquisitions. Only 20-30 measurements, optimally spread on several b-value shells, are shown to be necessary, which is less than previous attempts to recover the diffusion signal using CS. This opens an attractive perspective to measure the diffusion signals in white matter within a reduced acquisition time and shows that CS holds great promise and opens new and exciting perspectives in diffusion MRI (dMRI).

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.

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.

Reconstruction of the orientation distribution function in single- and multiple-shell q-ball imaging within constant solid angle

q-Ball imaging is a high-angular-resolution diffusion imaging technique that has been proven very successful in resolving multiple intravoxel fiber orientations in MR images. The standard computation of the orientation distribution function (the probability of diffusion in a given direction) from q-ball data uses linear radial projection, neglecting the change in the volume element along each direction. This results in spherical distributions that are different from the true orientation distribution functions. For instance, they are neither normalized nor as sharp as expected and generally require postprocessing, such as artificial sharpening. In this paper, a new technique is proposed that, by considering the solid angle factor, uses the mathematically correct definition of the orientation distribution function and results in a dimensionless and normalized orientation distribution function expression. Our model is flexible enough so that orientation distribution functions can be estimated either from single q-shell datasets or by exploiting the greater information available from multiple q-shell acquisitions. We show that the latter can be achieved by using a more accurate multiexponential model for the diffusion signal. The improved performance of the proposed method is demonstrated on artificial examples and high-angular-resolution diffusion imaging data acquired on a 7-T magnet.

Sparse multi-shell diffusion imaging

Diffusion magnetic resonance imaging (dMRI) is an important tool that allows non-invasive investigation of neural architecture of the brain. The data obtained from these in-vivo scans provides important information about the integrity and connectivity of neural fiber bundles in the brain. A multi-shell imaging (MSI) scan can be of great value in the study of several psychiatric and neurological disorders, yet its usability has been limited due to the long acquisition times required. A typical MSI scan involves acquiring a large number of gradient directions for the 2 (or more) spherical shells (several b-values), making the acquisition time significantly long for clinical application. In this work, we propose to use results from the theory of compressive sampling and determine the minimum number of gradient directions required to attain signal reconstruction similar to a traditional MSI scan. In particular, we propose a generalization of the single shell spherical ridgelets basis for sparse representation of multi shell signals. We demonstrate its efficacy on several synthetic and in-vivo data sets and perform quantitative comparisons with solid spherical harmonics based representation. Our preliminary results show that around 20-24 directions per shell are enough for robustly recovering the diffusion propagator.

A filtered approach to neural tractography using the Watson directional function

We propose a technique to simultaneously estimate the local fiber orientations and perform multi-fiber tractography. Existing techniques estimate the local fiber orientation at each voxel independently so there is no running knowledge of confidence in the measured signal or estimated fiber orientation. Further, to overcome noise, many algorithms use a filter as a post-processing step to obtain a smooth trajectory. We formulate fiber tracking as causal estimation: at each step of tracing the fiber, the current estimate of the signal is guided by the previous. To do this, we model the signal as a discrete mixture of Watson directional functions and perform tractography within a filtering framework. Starting from a seed point, each fiber is traced to its termination using an unscented Kalman filter to simultaneously fit the signal and propagate in the most consistent direction. Despite the presence of noise and uncertainty, this provides an accurate estimate of the local structure at each point along the fiber. We choose the Watson function since it provides a compact representation of the signal parameterized by the principal diffusion direction and a scaling parameter describing anisotropy, and also allows analytic reconstruction of the oriented diffusion function from those parameters. Using a mixture of two and three components (corresponding to two-fiber and three-fiber models) we demonstrate in synthetic experiments that this approach reduces signal reconstruction error and significantly improves the angular resolution at crossings and branchings. In vivo experiments examine the corpus callosum and internal capsule and confirm the ability to trace through regions known to contain such crossing and branching while providing inherent path regularization.

Fast and accurate reconstruction of HARDI data using compressed sensing

A spectrum of brain-related disorders are nowadays known to manifest themselves in degradation of the integrity and connectivity of neural tracts in the white matter of the brain. Such damage tends to affect the pattern of water diffusion in the white matter--the information which can be quantified by diffusion MRI (dMRI). Unfortunately, practical implementation of dMRI still poses a number of challenges which hamper its wide-spread integration into regular clinical practice. Chief among these is the problem of long scanning times. In particular, in the case of High Angular Resolution Diffusion Imaging (HARDI), the scanning times are known to increase linearly with the number of diffusion-encoding gradients. In this research, we use the theory of compressive sampling (aka compressed sensing) to substantially reduce the number of diffusion gradients without compromising the informational content of HARDI signals. The experimental part of our study compares the proposed method with a number of alternative approaches, and shows that the former results in more accurate estimation of HARDI data in terms of the mean squared error.

Neural tractography using an unscented Kalman filter

We describe a technique to simultaneously estimate a local neural fiber model and trace out its path. Existing techniques estimate the local fiber orientation at each voxel independently so there is no running knowledge of confidence in the estimated fiber model. We formulate fiber tracking as recursive estimation: at each step of tracing the fiber, the current estimate is guided by the previous. To do this we model the signal as a mixture of Gaussian tensors and perform tractography within a filter framework. Starting from a seed point, each fiber is traced to its termination using an unscented Kalman filter to simultaneously fit the local model and propagate in the most consistent direction. Despite the presence of noise and uncertainty, this provides a causal estimate of the local structure at each point along the fiber. Synthetic experiments demonstrate that this approach reduces signal reconstruction error and significantly improves the angular resolution at crossings and branchings. In vivo experiments confirm the ability to trace out fibers in areas known to contain such crossing and branching while providing inherent path regularization.