Non-Negative Spherical Deconvolution (NNSD) for estimation of fiber Orientation Distribution Function in single-/multi-shell diffusion MRI

Optimized rectification of fiber orientation density function with background threshold

PURPOSE

To describe an optimized fiber orientation density function (fODF) rectification procedure that removes negative values and absorbs all features below a specified threshold into a constant background.

THEORY AND METHODS

The fODF for a white matter imaging voxel describes the angular density of axons. Because of signal noise and Gibbs ringing, fODFs estimated with diffusion MRI may take on unphysical negative values in some directions and contain spurious peaks. In order to suppress such artifacts, an fODF rectification procedure is proposed that both eliminates all negative values and incorporates all features below a specified threshold, η, into a constant background while at the same time minimizing the mean square deviation from the original, unrectified fODF. Calculating this fODF is straightforward, and the directions and shapes of peaks not absorbed into the background are preserved. The rectification method is illustrated for an analytic fODF model and for experimental diffusion MRI data obtained in healthy human brain, with the original fODFs being obtained from fiber ball imaging.

RESULTS

Examples of optimal rectified fODFs are given for three choices of the background threshold referred to as minimal rectification (η = 0), average-level rectification (η ≈ 0.08), and fractional-anisotropy-axonal-based rectification (η ≈ 0.1). As η is increased, artifacts and other small features are more strongly suppressed, but the major fODF peaks are largely unaffected for the range of η values illustrated by these three alternatives.

CONCLUSION

Artifactual features of fODFs estimated with diffusion MRI can be effectively suppressed by applying the proposed optimized rectification procedure. Since it minimizes fODF distortion in the mean square sense, it may be useful in the study of how fODF fine structure is affected by aging and disease.

CHIASM, the human brain albinism and achiasma MRI dataset

We describe a collection of T1-, diffusion- and functional T2*-weighted magnetic resonance imaging data from human individuals with albinism and achiasma. This repository can be used as a test-bed to develop and validate tractography methods like diffusion-signal modeling and fiber tracking as well as to investigate the properties of the human visual system in individuals with congenital abnormalities. The MRI data is provided together with tools and files allowing for its preprocessing and analysis, along with the data derivatives such as manually curated masks and regions of interest for performing tractography.

Orientational changes of white matter fibers in Alzheimer's disease and amnestic mild cognitive impairment

White matter abnormalities represent early neuropathological events in neurodegenerative diseases such as Alzheimer's disease (AD), investigating these white matter alterations would likely provide valuable insights into pathological changes over the course of AD. Using a novel mathematical framework called "Director Field Analysis" (DFA), we investigated the geometric microstructural properties (i.e., splay, bend, twist, and total distortion) in the orientation of white matter fibers in AD, amnestic mild cognitive impairment (aMCI), and cognitively normal (CN) individuals from the Alzheimer's Disease Neuroimaging Initiative 2 database. Results revealed that AD patients had extensive orientational changes in the bilateral anterior thalamic radiation, corticospinal tract, inferior and superior longitudinal fasciculus, inferior fronto-occipital fasciculus, and uncinate fasciculus in comparison with CN. We postulate that these orientational changes of white matter fibers may be partially caused by the expansion of lateral ventricle, white matter atrophy, and gray matter atrophy in AD. In contrast, aMCI individuals showed subtle orientational changes in the left inferior longitudinal fasciculus and right uncinate fasciculus, which showed a significant association with the cognitive performance, suggesting that these regions may be preferential vulnerable to breakdown by neurodegenerative brain disorders, thereby resulting in the patients' cognitive impairment. To our knowledge, this article is the first to examine geometric microstructural changes in the orientation of white matter fibers in AD and aMCI. Our findings demonstrate that the orientational information of white matter fibers could provide novel insight into the underlying biological and pathological changes in AD and aMCI.

Diffusion interactions between crossing fibers of the brain

PURPOSE

Recent observations of several preferred orientations of diffusion in deep white matter may indicate either (a) that axons in different directions are independently bundled in thick sheets and function noninteractively, or more interestingly, (b) that the axons are closely interwoven and would exhibit branching and sharp turns. This study aims to investigate whether the dependence of dMRI Q-ball signal on the interpulse time Δ can decode the smaller-than-voxel-size brain structure, in particular, to distinguish scenarios (a) and (b).

METHODS

High-resolution Q-ball images of a healthy brain taken with b = 8000  s/mm² for 3 different values of Δ were analyzed. The exchange of water molecules between crossing fibers was characterized by the fourth Fourier coefficient f 4 ( Δ ) of the signal profile in the plane of crossing. To interpret the empirical results, a model consisting of differently oriented parallel sheets of cylinders was developed. Diffusion of water molecules inside and outside cylinders was simulated by the Monte Carlo method.

RESULTS

Simulations predict that f 4 ( Δ ) , agreeing with the empirical results, must increase with Δ for large b-values, but may peak at a typical Δ that depends on the thickness of the cylinder sheets for intermediate b-values. Thus, the thickness of axon layers in voxels with 2 predominant orientations can be detected from empirical f 4 ( Δ ) taken at smaller b-values.

CONCLUSION

Based on the simulation results, recommendations are made on how to design a dMRI experiment with optimal b-value and range of Δ in order to measure the thickness of axon sheets in the white matter, hence to distinguish (a) and (b).

On the cortical connectivity in the macaque brain: A comparison of diffusion tractography and histological tracing data

Diffusion-weighted magnetic resonance imaging (DW-MRI) tractography is a non-invasive tool to probe neural connections and the structure of the white matter. It has been applied successfully in studies of neurological disorders and normal connectivity. Recent work has revealed that tractography produces a high incidence of false-positive connections, often from "bottleneck" white matter configurations. The rich literature in histological connectivity analysis studies in the macaque monkey enables quantitative evaluation of the performance of tractography algorithms. In this study, we use the intricate connections of frontal, cingulate, and parietal areas, well established by the anatomical literature, to derive a symmetrical histological connectivity matrix composed of 59 cortical areas. We evaluate the performance of fifteen diffusion tractography algorithms, including global, deterministic, and probabilistic state-of-the-art methods for the connectivity predictions of 1711 distinct pairs of areas, among which 680 are reported connected by the literature. The diffusion connectivity analysis was performed on a different ex-vivo macaque brain, acquired using multi-shell DW-MRI protocol, at high spatial and angular resolutions. Across all tested algorithms, the true-positive and true-negative connections were dominant over false-positive and false-negative connections, respectively. Moreover, three-quarters of streamlines had endpoints location in agreement with histological data, on average. Furthermore, probabilistic streamline tractography algorithms show the best performances in predicting which areas are connected. Altogether, we propose a method for quantitative evaluation of tractography algorithms, which aims at improving the sensitivity and the specificity of diffusion-based connectivity analysis. Overall, those results confirm the usefulness of tractography in predicting connectivity, although errors are produced. Many of the errors result from bottleneck white matter configurations near the cortical grey matter and should be the target of future implementation of methods.

Enforcing necessary non-negativity constraints for common diffusion MRI models using sum of squares programming

In this work we investigate the use of sum of squares constraints for various diffusion-weighted MRI models, with a goal of enforcing strict, global non-negativity of the diffusion propagator. We formulate such constraints for the mean apparent propagator model and for spherical deconvolution, guaranteeing strict non-negativity of the corresponding diffusion propagators. For the cumulant expansion similar constraints cannot exist, and we instead derive a set of auxiliary constraints that are necessary but not sufficient to guarantee non-negativity. These constraints can all be verified and enforced at reasonable computational costs using semidefinite programming. By verifying our constraints on standard reconstructions of the different models, we show that currently used weak constraints are largely ineffective at ensuring non-negativity. We further show that if strict non-negativity is not enforced then estimated model parameters may suffer from significant errors, leading to serious inaccuracies in important derived quantities such as the main fiber orientations, mean kurtosis, etc. Finally, our experiments confirm that the observed constraint violations are mostly due to measurement noise, which is difficult to mitigate and suggests that properly constrained optimization should currently be considered the norm in many cases.

Construction of brain structural connectivity network using a novel integrated algorithm based on ensemble average propagator

An important task for neuroscience is to accurately construct structural connectivity network of human brain. Tractography constructed based on high angular resolution diffusion imaging (HARDI) provides valuable information of human brain structural connections. Existing algorithms, mainly categorized as deterministic or probabilistic, come with inherent limitations (e.g., fiber direction uncertainty induced by noise, or anatomically unreasonable connections and heavy computational cost). In this study, a novel integrated algorithm was proposed to construct brain structural connectivity network by incorporating the deterministic path planning and probabilistic connection strength estimation, based on ensemble average propagator (EAP). We first estimated EAPs from multi-shell samples using the spherical polar Fourier imaging (SPFI), and then extracted diffusion orientations coinciding with neural fiber tracts. Only under angular constraints, the deterministic path planning algorithm was subsequently used to find all reasonable pathways between pairwise white matter (WM) voxels in different regions of interest (ROIs). Consequently, a train of consecutive WM voxels along each of the identified pathways was determined, and the connection strength of these pathways was computed by integrating their EAP alignment over a solid angle. The connection strength of a pair of WM voxels was assigned as the connection strength with the largest connection possibility. Finally, the connection strength between two ROIs was calculated as the sum of all the connection probabilities of each pair of WM voxels in the ROIs. A comparison against voxel-graph based probabilistic tractography method was performed on Fibercup phantom dataset, and the results demonstrated that the proposed method can produce better structural connection and is more computationally economical. Lastly, three datasets from Human Connectome Project (HCP) S1200 group were tested and their structural connectivity networks were constructed for topological analysis. The results showed great consistency in network metrics with previous WM network studies in healthy adults.

Histologically derived fiber response functions for diffusion MRI vary across white matter fibers-An ex vivo validation study in the squirrel monkey brain

Understanding the relationship between the diffusion-weighted MRI signal and the arrangement of white matter fibers is fundamental for accurate voxel-wise reconstruction of the fiber orientation distribution (FOD) and subsequent fiber tractography. Spherical deconvolution reconstruction techniques model the diffusion signal as the convolution of the FOD with a response function that represents the signal profile of a single fiber orientation. Thus, given the signal and a fiber response function, the FOD can be estimated in every imaging voxel by deconvolution. However, the selection of the appropriate response function remains relatively under-studied, and requires further validation. In this work, using 3D histologically defined FODs and the corresponding diffusion signal from three ex vivo squirrel monkey brains, we derive the ground truth response functions. We find that the histologically derived response functions differ from those conventionally used. Next, we find that response functions statistically vary across brain regions, which suggests that the practice of using the same kernel throughout the brain is not optimal. We show that different kernels lead to different FOD reconstructions, which in turn can lead to different tractography results depending on algorithmic parameters, with large variations in the accuracy of resulting reconstructions. Together, these results suggest there is room for improvement in estimating and understanding the relationship between the diffusion signal and the underlying FOD.

Fiber orientation distribution function from non-negative sparse recovery with quantitative analysis of local fiber orientations and tractography using DW-MRI datasets

Diffusion weighted MRI (DW-MRI) is the unique non-invasive imaging modality capable of estimating in vivo the structure of the white matter. In this paper, we propose, evaluate and validate a new DW-MRI method to model and recover high quality tractogram even with multiple fiber populations in a voxel and from a limited number of acquisitions. Our method relies on the estimation of the Fiber Orientation Distribution (FOD) function, parameterized as a non-negative sum of rank-1 tensors and the use of a non-negative sparse recovery scheme to efficiently recover the tensors, and their number. Each fiber population of a voxel is characterized by the orientation and the weight of a rank-1 tensor. Using both deterministic and probabilistic tractography algorithms, we show that our method is able to accurately reconstruct narrow crossing fibers and obtain a high quality connectivity reconstruction even from a limited number of acquisitions. To this end, a validation scheme based on the connectivity recovered from tractography is developed to quantitatively evaluate and analyze the performance of our method. The tractometer tool is used to quantify the tractography obtained from a simulated DW-MRI dataset including a high angular resolution dataset of 60 gradient directions and a dataset of 30 gradient directions, each of them corrupted with Rician noise of SNR 10 and 20. The performance of our FOD model and its impact on the tractography results are also demonstrated and illustrated on in vivo DW-MRI datasets with high and low angular resolutions.

When tractography meets tracer injections: a systematic study of trends and variation sources of diffusion-based connectivity

Tractography is a powerful technique capable of non-invasively reconstructing the structural connections in the brain using diffusion MRI images, but the validation of tractograms is challenging due to lack of ground truth. Owing to recent developments in mapping the mouse brain connectome, high-resolution tracer injection-based axonal projection maps have been created and quickly adopted for the validation of tractography. Previous studies using tracer injections mainly focused on investigating the match in projections and optimal tractography protocols. Being a complicated technique, however, tractography relies on multiple stages of operations and parameters. These factors introduce large variabilities in tractograms, hindering the optimization of protocols and making the interpretation of results difficult. Based on this observation, in contrast to previous studies, in this work we focused on quantifying and ranking the amount of performance variation introduced by these factors. For this purpose, we performed over a million tractography experiments and studied the variability across different subjects, injections, anatomical constraints and tractography parameters. By using N-way ANOVA analysis, we show that all tractography parameters are significant and importantly performance variations with respect to the differences in subjects are comparable to the variations due to tractography parameters, which strongly underlines the importance of fully documenting the tractography protocols in scientific experiments. We also quantitatively show that inclusion of anatomical constraints is the most significant factor for improving tractography performance. Although this critical factor helps reduce false positives, our analysis indicates that anatomy-informed tractography still fails to capture a large portion of axonal projections.

Single- and Multiple-Shell Uniform Sampling Schemes for Diffusion MRI Using Spherical Codes

In diffusion MRI (dMRI), a good sampling scheme is important for efficient acquisition and robust reconstruction. Diffusion weighted signal is normally acquired on single or multiple shells in q-space. Signal samples are typically distributed uniformly on different shells to make them invariant to the orientation of structures within tissue, or the laboratory coordinate frame. The Electrostatic Energy Minimization (EEM) method, originally proposed for single shell sampling scheme in dMRI, was recently generalized to multi-shell schemes, called Generalized EEM (GEEM). GEEM has been successfully used in the Human Connectome Project (HCP). However, EEM does not directly address the goal of optimal sampling, i.e., achieving large angular separation between sampling points. In this paper, we propose a more natural formulation, called Spherical Code (SC), to directly maximize the minimal angle between different samples in single or multiple shells. We consider not only continuous problems to design single or multiple shell sampling schemes, but also discrete problems to uniformly extract sub-sampled schemes from an existing single or multiple shell scheme, and to order samples in an existing scheme. We propose five algorithms to solve the above problems, including an incremental SC (ISC), a sophisticated greedy algorithm called Iterative Maximum Overlap Construction (IMOC), an 1-Opt greedy method, a Mixed Integer Linear Programming (MILP) method, and a Constrained Non-Linear Optimization (CNLO) method. To our knowledge, this is the first work to use the SC formulation for single or multiple shell sampling schemes in dMRI. Experimental results indicate that SC methods obtain larger angular separation and better rotational invariance than the state-of-the-art EEM and GEEM. The related codes and a tutorial have been released in DMRITool.

Connectome imaging for mapping human brain pathways

With the fast advance of connectome imaging techniques, we have the opportunity of mapping the human brain pathways in vivo at unprecedented resolution. In this article we review the current developments of diffusion magnetic resonance imaging (MRI) for the reconstruction of anatomical pathways in connectome studies. We first introduce the background of diffusion MRI with an emphasis on the technical advances and challenges in state-of-the-art multi-shell acquisition schemes used in the Human Connectome Project. Characterization of the microstructural environment in the human brain is discussed from the tensor model to the general fiber orientation distribution (FOD) models that can resolve crossing fibers in each voxel of the image. Using FOD-based tractography, we describe novel methods for fiber bundle reconstruction and graph-based connectivity analysis. Building upon these novel developments, there have already been successful applications of connectome imaging techniques in reconstructing challenging brain pathways. Examples including retinofugal and brainstem pathways will be reviewed. Finally, we discuss future directions in connectome imaging and its interaction with other aspects of brain imaging research.

Multi-Tissue Decomposition of Diffusion MRI Signals via Sparse-Group Estimation

Sparse estimation techniques are widely utilized in diffusion magnetic resonance imaging (DMRI). In this paper, we present an algorithm for solving the ℓ₀ sparse-group estimation problem and apply it to the tissue signal separation problem in DMRI. Our algorithm solves the ℓ₀ problem directly, unlike existing approaches that often seek to solve its relaxed approximations. We include the mathematical proofs showing that the algorithm will converge to a solution satisfying the firstorder optimality condition within a finite number of iterations. We apply this algorithm to DMRI data to tease apart signal contributions from white matter, gray matter, and cerebrospinal fluid with the aim of improving the estimation of the fiber orientation distribution function (FODF). Unlike spherical deconvolution approaches that assume an invariant fiber response function (RF), our approach utilizes an RF group to span the signal subspace of each tissue type, allowing greater flexibility in accounting for possible variations of the RF throughout space and within each voxel. Our ℓ₀ algorithm allows for the natural groupings of the RFs to be considered during signal decomposition. Experimental results confirm that our method yields estimates of FODFs and volume fractions of tissue compartments with improved robustness and accuracy. Our ℓ₀ algorithm is general and can be applied to sparse estimation problems beyond the scope of this paper.

Greedy Algorithms for Nonnegativity-Constrained Simultaneous Sparse Recovery

This work proposes a family of greedy algorithms to jointly reconstruct a set of vectors that are (i) nonnegative and (ii) simultaneously sparse with a shared support set. The proposed algorithms generalize previous approaches that were designed to impose these constraints individually. Similar to previous greedy algorithms for sparse recovery, the proposed algorithms iteratively identify promising support indices. In contrast to previous approaches, the support index selection procedure has been adapted to prioritize indices that are consistent with both the nonnegativity and shared support constraints. Empirical results demonstrate for the first time that the combined use of simultaneous sparsity and nonnegativity constraints can substantially improve recovery performance relative to existing greedy algorithms that impose less signal structure.

Estimation of fiber orientations using neighborhood information

Data from diffusion magnetic resonance imaging (dMRI) can be used to reconstruct fiber tracts, for example, in muscle and white matter. Estimation of fiber orientations (FOs) is a crucial step in the reconstruction process and these estimates can be corrupted by noise. In this paper, a new method called Fiber Orientation Reconstruction using Neighborhood Information (FORNI) is described and shown to reduce the effects of noise and improve FO estimation performance by incorporating spatial consistency. FORNI uses a fixed tensor basis to model the diffusion weighted signals, which has the advantage of providing an explicit relationship between the basis vectors and the FOs. FO spatial coherence is encouraged using weighted ℓ1-norm regularization terms, which contain the interaction of directional information between neighbor voxels. Data fidelity is encouraged using a squared error between the observed and reconstructed diffusion weighted signals. After appropriate weighting of these competing objectives, the resulting objective function is minimized using a block coordinate descent algorithm, and a straightforward parallelization strategy is used to speed up processing. Experiments were performed on a digital crossing phantom, ex vivo tongue dMRI data, and in vivo brain dMRI data for both qualitative and quantitative evaluation. The results demonstrate that FORNI improves the quality of FO estimation over other state of the art algorithms.

Fiber Orientation and Compartment Parameter Estimation From Multi-Shell Diffusion Imaging

Diffusion MRI offers the unique opportunity of assessing the structural connections of human brains in vivo. With the advance of diffusion MRI technology, multi-shell imaging methods are becoming increasingly practical for large scale studies and clinical application. In this work, we propose a novel method for the analysis of multi-shell diffusion imaging data by incorporating compartment models into a spherical deconvolution framework for fiber orientation distribution (FOD) reconstruction. For numerical implementation, we develop an adaptively constrained energy minimization approach to efficiently compute the solution. On simulated and real data from Human Connectome Project (HCP), we show that our method not only reconstructs sharp and clean FODs for the modeling of fiber crossings, but also generates reliable estimation of compartment parameters with great potential for clinical research of neurological diseases. In comparisons with publicly available DSI-Studio and BEDPOSTX of FSL, we demonstrate that our method reconstructs sharper FODs with more precise estimation of fiber directions. By applying probabilistic tractography to the FODs computed by our method, we show that more complete reconstruction of the corpus callosum bundle can be achieved. On a clinical, two-shell diffusion imaging data, we also demonstrate the feasibility of our method in analyzing white matter lesions.

Sparse and Adaptive Diffusion Dictionary (SADD) for recovering intra-voxel white matter structure

On the analysis of the Diffusion-Weighted Magnetic Resonance Images, multi-compartment models overcome the limitations of the well-known Diffusion Tensor model for fitting in vivo brain axonal orientations at voxels with fiber crossings, branching, kissing or bifurcations. Some successful multi-compartment methods are based on diffusion dictionaries. The diffusion dictionary-based methods assume that the observed Magnetic Resonance signal at each voxel is a linear combination of the fixed dictionary elements (dictionary atoms). The atoms are fixed along different orientations and diffusivity profiles. In this work, we present a sparse and adaptive diffusion dictionary method based on the Diffusion Basis Functions Model to estimate in vivo brain axonal fiber populations. Our proposal overcomes the following limitations of the diffusion dictionary-based methods: the limited angular resolution and the fixed shapes for the atom set. We propose to iteratively re-estimate the orientations and the diffusivity profile of the atoms independently at each voxel by using a simplified and easier-to-solve mathematical approach. As a result, we improve the fitting of the Diffusion-Weighted Magnetic Resonance signal. The advantages with respect to the former Diffusion Basis Functions method are demonstrated on the synthetic data-set used on the 2012 HARDI Reconstruction Challenge and in vivo human data. We demonstrate that improvements obtained in the intra-voxel fiber structure estimations benefit brain research allowing to obtain better tractography estimations. Hence, these improvements result in an accurate computation of the brain connectivity patterns.

Non-Negative Spherical Deconvolution (NNSD) for estimation of fiber Orientation Distribution Function in single-/multi-shell diffusion MRI

Spherical Deconvolution (SD) is commonly used for estimating fiber Orientation Distribution Functions (fODFs) from diffusion-weighted signals. Existing SD methods can be classified into two categories: 1) Continuous Representation based SD (CR-SD), where typically Spherical Harmonic (SH) representation is used for convenient analytical solutions, and 2) Discrete Representation based SD (DR-SD), where the signal profile is represented by a discrete set of basis functions uniformly oriented on the unit sphere. A feasible fODF should be non-negative and should integrate to unity throughout the unit sphere S(2). However, to our knowledge, most existing SH-based SD methods enforce non-negativity only on discretized points and not the whole continuum of S(2). Maximum Entropy SD (MESD) and Cartesian Tensor Fiber Orientation Distributions (CT-FOD) are the only SD methods that ensure non-negativity throughout the unit sphere. They are however computational intensive and are susceptible to errors caused by numerical spherical integration. Existing SD methods are also known to overestimate the number of fiber directions, especially in regions with low anisotropy. DR-SD introduces additional error in peak detection owing to the angular discretization of the unit sphere. This paper proposes a SD framework, called Non-Negative SD (NNSD), to overcome all the limitations above. NNSD is significantly less susceptible to the false-positive peaks, uses SH representation for efficient analytical spherical deconvolution, and allows accurate peak detection throughout the whole unit sphere. We further show that NNSD and most existing SD methods can be extended to work on multi-shell data by introducing a three-dimensional fiber response function. We evaluated NNSD in comparison with Constrained SD (CSD), a quadratic programming variant of CSD, MESD, and an L1-norm regularized non-negative least-squares DR-SD. Experiments on synthetic and real single-/multi-shell data indicate that NNSD improves estimation performance in terms of mean difference of angles, peak detection consistency, and anisotropy contrast between isotropic and anisotropic regions.