Repeat it without me: Crowdsourcing the T1 mapping common ground via the ISMRM reproducibility challenge.

PURPOSE: T1 mapping is a widely used quantitative MRI technique, but its tissue-specific values remain inconsistent across protocols, sites, and vendors. The ISMRM Reproducible Research and Quantitative MR study groups jointly launched a challenge to assess the reproducibility of a well-established inversion-recovery T1 mapping technique, using acquisition details from a seminal T1 mapping paper on a standardized phantom and in human brains. METHODS: The challenge used the acquisition protocol from Barral et al. (2010). Researchers collected T1 mapping data on the ISMRM/NIST phantom and/or in human brains. Data submission, pipeline development, and analysis were conducted using open-source platforms. Intersubmission and intrasubmission comparisons were performed. RESULTS: Eighteen submissions (39 phantom and 56 human datasets) on scanners by three MRI vendors were collected at 3 T (except one, at 0.35 T). The mean coefficient of variation was 6.1% for intersubmission phantom measurements, and 2.9% for intrasubmission measurements. For humans, the intersubmission/intrasubmission coefficient of variation was 5.9/3.2% in the genu and 16/6.9% in the cortex. An interactive dashboard for data visualization was also developed: https://rrsg2020.dashboards.neurolibre.org. CONCLUSION: The T1 intersubmission variability was twice as high as the intrasubmission variability in both phantoms and human brains, indicating that the acquisition details in the original paper were insufficient to reproduce a quantitative MRI protocol. This study reports the inherent uncertainty in T1 measures across independent research groups, bringing us one step closer to a practical clinical baseline of T1 variations in vivo.

Tractography passes the test: Results from the diffusion-simulated connectivity (disco) challenge.

Estimating structural connectivity from diffusion-weighted magnetic resonance imaging is a challenging task, partly due to the presence of false-positive connections and the misestimation of connection weights. Building on previous efforts, the MICCAI-CDMRI Diffusion-Simulated Connectivity (DiSCo) challenge was carried out to evaluate state-of-the-art connectivity methods using novel large-scale numerical phantoms. The diffusion signal for the phantoms was obtained from Monte Carlo simulations. The results of the challenge suggest that methods selected by the 14 teams participating in the challenge can provide high correlations between estimated and ground-truth connectivity weights, in complex numerical environments. Additionally, the methods used by the participating teams were able to accurately identify the binary connectivity of the numerical dataset. However, specific false positive and false negative connections were consistently estimated across all methods. Although the challenge dataset doesn't capture the complexity of a real brain, it provided unique data with known macrostructure and microstructure ground-truth properties to facilitate the development of connectivity estimation methods.

Differentiation of white matter histopathology using b-tensor encoding and machine learning.

Diffusion-weighted magnetic resonance imaging (DW-MRI) is a non-invasive technique that is sensitive to microstructural geometry in neural tissue and is useful for the detection of neuropathology in research and clinical settings. Tensor-valued diffusion encoding schemes (b-tensor) have been developed to enrich the microstructural data that can be obtained through DW-MRI. These advanced methods have proven to be more specific to microstructural properties than conventional DW-MRI acquisitions. Additionally, machine learning methods are particularly useful for the study of multidimensional data sets. In this work, we have tested the reach of b-tensor encoding data analyses with machine learning in different histopathological scenarios. We achieved this in three steps: 1) We induced different levels of white matter damage in rodent optic nerves. 2) We obtained ex vivo DW-MRI data with b-tensor encoding schemes and calculated quantitative metrics using Q-space trajectory imaging. 3) We used a machine learning model to identify the main contributing features and built a voxel-wise probabilistic classification map of histological damage. Our results show that this model is sensitive to characteristics of microstructural damage. In conclusion, b-tensor encoded DW-MRI data analyzed with machine learning methods, have the potential to be further developed for the detection of histopathology and neurodegeneration.

Multi-tensor diffusion abnormalities of gray matter in an animal model of cortical dysplasia.

Focal cortical dysplasias are a type of malformations of cortical development that are a common cause of drug-resistant focal epilepsy. Surgical treatment is a viable option for some of these patients, with their outcome being highly related to complete surgical resection of lesions visible in magnetic resonance imaging (MRI). However, subtle lesions often go undetected on conventional imaging. Several methods to analyze MRI have been proposed, with the common goal of rendering subtle cortical lesions visible. However, most image-processing methods are targeted to detect the macroscopic characteristics of cortical dysplasias, which do not always correspond to the microstructural disarrangement of these cortical malformations. Quantitative analysis of diffusion-weighted MRI (dMRI) enables the inference of tissue characteristics, and novel methods provide valuable microstructural features of complex tissue, including gray matter. We investigated the ability of advanced dMRI descriptors to detect diffusion abnormalities in an animal model of cortical dysplasia. For this purpose, we induced cortical dysplasia in 18 animals that were scanned at 30 postnatal days (along with 19 control animals). We obtained multi-shell dMRI, to which we fitted single and multi-tensor representations. Quantitative dMRI parameters derived from these methods were queried using a curvilinear coordinate system to sample the cortical mantle, providing inter-subject anatomical correspondence. We found region- and layer-specific diffusion abnormalities in experimental animals. Moreover, we were able to distinguish diffusion abnormalities related to altered intra-cortical tangential fibers from those associated with radial cortical fibers. Histological examinations revealed myelo-architectural abnormalities that explain the alterations observed through dMRI. The methods for dMRI acquisition and analysis used here are available in clinical settings and our work shows their clinical relevance to detect subtle cortical dysplasias through analysis of their microstructural properties.

Diffusion MRI approaches for investigating microstructural complexity in a rat model of traumatic brain injury.

Our study explores the potential of conventional and advanced diffusion MRI techniques including diffusion tensor imaging (DTI), and single-shell 3-tissue constrained spherical deconvolution (SS3T-CSD) to investigate complex microstructural changes following severe traumatic brain injury in rats at a chronic phase. Rat brains after sham-operation or lateral fluid percussion (LFP) injury were scanned ex vivo in a 9.4 T scanner. Our region-of-interest-based approach of tensor-, and SS3T-CSD derived fixel-, 3-tissue signal fraction maps were sensitive to changes in both white matter (WM) and grey matter (GM) areas. Tensor-based measures, such as fractional anisotropy (FA) and radial diffusivity (RD), detected more changes in WM and GM areas as compared to fixel-based measures including apparent fiber density (AFD), peak FOD amplitude and primary fiber bundle density, while 3-tissue signal fraction maps revealed distinct changes in WM, GM, and phosphate-buffered saline (PBS) fractions highlighting the complex tissue microstructural alterations post-trauma. Track-weighted imaging demonstrated changes in track morphology including reduced curvature and average pathlength distal from the primary lesion in severe TBI rats. In histological analysis, changes in the diffusion MRI measures could be associated to decreased myelin density, loss of myelinated axons, and increased cellularity, revealing progressive microstructural alterations in these brain areas five months after injury. Overall, this study highlights the use of combined conventional and advanced diffusion MRI measures to obtain more precise insights into the complex tissue microstructural alterations in chronic phase of severe brain injury.

A computational method for the covariance matrix associated with extracellular diffusivity on disordered models of cylindrical brain axons.

This study developed a method to approximate the covariance matrix associated with the simulation of water molecular diffusion inside the brain tissue. The computation implements the Discontinuous Galerkin method of the diffusion equation. A physically consistent numerical flux is applied to model the interaction between the axon walls and extracellular regions. This numerical flux yields an efficient GPU-CUDA implementation. We consider the two-dimensional case of high axon pack density, valid, for instance, in the brain's corpus callosum region.

Robust Monte-Carlo Simulations in Diffusion-MRI: Effect of the Substrate Complexity and Parameter Choice on the Reproducibility of Results.

Monte-Carlo Diffusion Simulations (MCDS) have been used extensively as a ground truth tool for the validation of microstructure models for Diffusion-Weighted MRI. However, methodological pitfalls in the design of the biomimicking geometrical configurations and the simulation parameters can lead to approximation biases. Such pitfalls affect the reliability of the estimated signal, as well as its validity and reproducibility as ground truth data. In this work, we first present a set of experiments in order to study three critical pitfalls encountered in the design of MCDS in the literature, namely, the number of simulated particles and time steps, simplifications in the intra-axonal substrate representation, and the impact of the substrate's size on the signal stemming from the extra-axonal space. The results obtained show important changes in the simulated signals and the recovered microstructure features when changes in those parameters are introduced. Thereupon, driven by our findings from the first studies, we outline a general framework able to generate complex substrates. We show the framework's capability to overcome the aforementioned simplifications by generating a complex crossing substrate, which preserves the volume in the crossing area and achieves a high packing density. The results presented in this work, along with the simulator developed, pave the way toward more realistic and reproducible Monte-Carlo simulations for Diffusion-Weighted MRI.

Histological and diffusion-weighted magnetic resonance imaging data from normal and degenerated optic nerve and chiasm of the rat.

Diffusion-weighted magnetic resonance imaging (dMRI) is widely used to infer microstructural characteristics of tissue, particularly in cerebral white matter. Histological validation of the metrics derived from dMRI methods are needed to fully characterize their ability to capture biologically-relevant histological features non-invasively. The data described here were used to correlate metrics derived from dMRI and quantitative histology in an animal model of axonal degeneration ("Histological validation of per-bundle water diffusion metrics within a region of fiber crossing following axonal degeneration" [1]). Unilateral retinal ischemia/reperfusion was induced in 10 rats, by the elevation of pressure of the anterior chamber of the eye for 90 min. Five rats were used as controls. After five weeks, injured animals were intracardially perfused to analyze the optic nerves and chiasm with dMRI and histology. This resulted in 15 brain scans, each with 80 diffusion-sensitizing gradient directions with b = 2000 and 2500 s/mm2 and 20 non-diffusion-weighted images (b = 0 s/mm2), with isometric voxel resolution of 125 µm3. Histological sections were obtained after dMRI. Optical microscopy photomicrographs of the optic nerves (stained with toluidine blue) are available, as well as their corresponding automatic segmentations of axons and myelin.

Histological validation of per-bundle water diffusion metrics within a region of fiber crossing following axonal degeneration.

Micro-architectural characteristics of white matter can be inferred through analysis of diffusion-weighted magnetic resonance imaging (dMRI). The diffusion-dependent signal can be analyzed through several methods, with the tensor model being the most frequently used due to its straightforward interpretation and low requirements for acquisition parameters. While valuable information can be gained from the tensor-derived metrics in regions of homogeneous tissue organization, this model does not provide reliable microstructural information at crossing fiber regions, which are pervasive throughout human white matter. Several multiple fiber models have been proposed that seem to overcome the limitations of the tensor, with few providing per-bundle dMRI-derived metrics. However, biological interpretations of such metrics are limited by the lack of histological confirmation. To this end, we developed a straightforward biological validation framework. Unilateral retinal ischemia was induced in ten rats, which resulted in axonal (Wallerian) degeneration of the corresponding optic nerve, while the contralateral was left intact; the intact and injured axonal populations meet at the optic chiasm as they cross the midline, generating a fiber crossing region in which each population has different diffusion properties. Five rats served as controls. High-resolution ex vivo dMRI was acquired five weeks after experimental procedures. We correlated and compared histology to per-bundle descriptors derived from three methodologies for dMRI analysis (constrained spherical deconvolution and two multi-tensor representations). We found a tight correlation between axonal density (as evaluated through automatic segmentation of histological sections) with per-bundle apparent fiber density and fractional anisotropy (derived from dMRI). The multi-fiber methods explored were able to correctly identify the damaged fiber populations in a region of fiber crossings (chiasm). Our results provide validation of metrics that bring substantial and clinically useful information about white-matter tissue at crossing fiber regions. Our proposed framework is useful to validate other current and future dMRI methods.

Estimation of individual axon bundle properties by a Multi-Resolution Discrete-Search method.

A stable, accurate and robust-to-noise method for the estimation of the intra-voxel bundle-wise diffusion properties for diffusion-weighted magnetic resonance imaging is presented. The proposed method overcomes some of the limitations of most of the multi-fiber algorithms in the literature and extends them to estimate the diffusion profiles, improving the estimation of the intra-voxel geometry at challenging microstructure configurations, that is to say: relatively small crossing angles, different voxel-wise anisotropic diffusion profiles and low SNR. The proposed methodology is based on four key novel ideas: (i) A Multi-Resolution Discrete-Search determines the orientation of the fiber bundles accurately and naturally constrains the sparsity on the recovered solutions; (ii) the determination of the number of fiber bundles using the F-test combined with a Rician bias correction; (iii) a Simultaneous Denoising and Fitting procedure that exploits the spatial redundancy of the axon bundles to achieve robustness with respect to noise; and (iv) a general framework for the estimation of the axial and radial diffusivity parameters independently for each voxel. A new useful evaluation metric is also proposed, which combines the information of the success rate in the estimated number of bundles and the angular error, avoiding in this way, some of the limitations these metrics have individually. A novel methodology for the evaluation of the methods on in-vivo data is also proposed. This work presents an extensive evaluation: the proposed methodology has been tested on state-of-the-art biophysical synthetic data for a variety of conditions, on the challenging spatially coherent phantom used on the HARDI reconstruction Challenge 2012, and on the recently released in-vivo MASSIVE data-set. Our results present significant improvements on the estimation of the number and orientation of the fiber bundles over the Spherical Deconvolution algorithm for multi-shell data, which is one of the most widely used multi-fiber algorithm. The results also show that, by the voxel-wise estimation of the diffusion profiles, the axial and radial diffusivity parameters are robustly estimated, being this essential for a better understanding of the individual bundle diffusion properties at challenging structural configurations.

Diffusion MRI microstructure models with in vivo human brain Connectome data: results from a multi-group comparison.

A large number of mathematical models have been proposed to describe the measured signal in diffusion-weighted (DW) magnetic resonance imaging (MRI). However, model comparison to date focuses only on specific subclasses, e.g. compartment models or signal models, and little or no information is available in the literature on how performance varies among the different types of models. To address this deficiency, we organized the 'White Matter Modeling Challenge' during the International Symposium on Biomedical Imaging (ISBI) 2015 conference. This competition aimed to compare a range of different kinds of models in their ability to explain a large range of measurable in vivo DW human brain data. Specifically, we assessed the ability of models to predict the DW signal accurately for new diffusion gradients and b values. We did not evaluate the accuracy of estimated model parameters, as a ground truth is hard to obtain. We used the Connectome scanner at the Massachusetts General Hospital, using gradient strengths of up to 300 mT/m and a broad set of diffusion times. We focused on assessing the DW signal prediction in two regions: the genu in the corpus callosum, where the fibres are relatively straight and parallel, and the fornix, where the configuration of fibres is more complex. The challenge participants had access to three-quarters of the dataset and their models were ranked on their ability to predict the remaining unseen quarter of the data. The challenge provided a unique opportunity for a quantitative comparison of diverse methods from multiple groups worldwide. The comparison of the challenge entries reveals interesting trends that could potentially influence the next generation of diffusion-based quantitative MRI techniques. The first is that signal models do not necessarily outperform tissue models; in fact, of those tested, tissue models rank highest on average. The second is that assuming a non-Gaussian (rather than purely Gaussian) noise model provides little improvement in prediction of unseen data, although it is possible that this may still have a beneficial effect on estimated parameter values. The third is that preprocessing the training data, here by omitting signal outliers, and using signal-predicting strategies, such as bootstrapping or cross-validation, could benefit the model fitting. The analysis in this study provides a benchmark for other models and the data remain available to build up a more complete comparison in the future.

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.

A hierarchical algorithm for molecular similarity (H-FORMS).

A new hierarchical method to determine molecular similarity is introduced. The goal of this method is to detect if a pair of molecules has the same structure by estimating a rigid transformation that aligns the molecules and a correspondence function that matches their atoms. The algorithm firstly detect similarity based on the global spatial structure. If this analysis is not sufficient, the algorithm computes novel local structural rotation-invariant descriptors for the atom neighborhood and uses this information to match atoms. Two strategies (deterministic and stochastic) on the matching based alignment computation are tested. As a result, the atom-matching based on local similarity indexes decreases the number of testing trials and significantly reduces the dimensionality of the Hungarian assignation problem. The experiments on well-known datasets show that our proposal outperforms state-of-the-art methods in terms of the required computational time and accuracy.

A flocking based method for brain tractography.

We propose a new method to estimate axonal fiber pathways from Multiple Intra-Voxel Diffusion Orientations. Our method uses the multiple local orientation information for leading stochastic walks of particles. These stochastic particles are modeled with mass and thus they are subject to gravitational and inertial forces. As result, we obtain smooth, filtered and compact trajectory bundles. This gravitational interaction can be seen as a flocking behavior among particles that promotes better and robust axon fiber estimations because they use collective information to move. However, the stochastic walks may generate paths with low support (outliers), generally associated to incorrect brain connections. In order to eliminate the outlier pathways, we propose a filtering procedure based on principal component analysis and spectral clustering. The performance of the proposal is evaluated on Multiple Intra-Voxel Diffusion Orientations from two realistic numeric diffusion phantoms and a physical diffusion phantom. Additionally, we qualitatively demonstrate the performance on in vivo human brain data.

Quantitative comparison of reconstruction methods for intra-voxel fiber recovery from diffusion MRI.

Validation is arguably the bottleneck in the diffusion magnetic resonance imaging (MRI) community. This paper evaluates and compares 20 algorithms for recovering the local intra-voxel fiber structure from diffusion MRI data and is based on the results of the "HARDI reconstruction challenge" organized in the context of the "ISBI 2012" conference. Evaluated methods encompass a mixture of classical techniques well known in the literature such as diffusion tensor, Q-Ball and diffusion spectrum imaging, algorithms inspired by the recent theory of compressed sensing and also brand new approaches proposed for the first time at this contest. To quantitatively compare the methods under controlled conditions, two datasets with known ground-truth were synthetically generated and two main criteria were used to evaluate the quality of the reconstructions in every voxel: correct assessment of the number of fiber populations and angular accuracy in their orientation. This comparative study investigates the behavior of every algorithm with varying experimental conditions and highlights strengths and weaknesses of each approach. This information can be useful not only for enhancing current algorithms and develop the next generation of reconstruction methods, but also to assist physicians in the choice of the most adequate technique for their studies.

Heavy periodane.

The potential energy surface of the hypothetical NaMgAlSiPSCl system (heavy periodane) is exhaustively analyzed via the gradient embedded genetic algorithm (GEGA) in combination with density functional theory (DFT) computations. The electronegativity differences among the elements in both the second and third rows of the periodic table indicate that low-energy heavy periodane structures are obtained when highly electronegative and electropositive elements are bound together, but the global minimum of the heavy periodane system is completely different to its second-row analog (LiBeBCNOF).

Resolving axon fiber crossings at clinical b-values: an evaluation study.

PURPOSE: Diffusion tensor magnetic resonance imaging is widely used to study the structure of the fiber pathways of brain white matter. However, the diffusion tensor cannot capture complex intravoxel fiber architecture such as fiber crossings of bifurcations. Consequently, a number of methods have been proposed to recover intravoxel fiber bundle orientations from high angular resolution diffusion imaging scans, optimized to resolve fiber crossings. It is important to improve the brain tractography by applying these multifiber methods to diffusion tensor protocols with a clinical b- value (low), which are optimized on computing tensor scalar statistics. In order to characterize the variance among different methods, consequently to be able to select the most appropriate one for a particular application, it is desirable to compare them under identical experimental conditions. METHODS: In this work, the authors study how QBall, spherical deconvolution, persistent angular structure, stick and ball, diffusion basis functions, and analytical QBall methods perform under clinically-realistic scanning conditions, where the b-value is typically lower (around 1000 s∕mm(2)), and the number of diffusion encoding orientations is fewer (30-60) than in dedicated high angular resolution diffusion imaging scans. To characterize the performance of the methods, they consider the accuracy of the estimated number of fibers, the relative contribution of each fiber population to the total magnetic resonance signal, and the recovered orientation error for each fiber bundle. To this aim, they use four different sources of data: synthetic data from Gaussian mixture model, cylinder restricted model, and in vivo data from two different acquisition schemes. RESULTS: Results of their experiments indicate that: (a) it is feasible to apply only a subset of these methods to clinical data sets and (b) it allows one to characterize the performance of each method. In particular, two methods are not feasible to the kind of magnetic resonance diffusion data they test. By the characterization of their systematic behavior, among other conclusions, they report the method which better performs for the estimation of the number of diffusion peaks per voxel, also the method which better estimates the diffusion orientation. CONCLUSIONS: The framework they propose for comparison allows one to effectively characterize and compare the performance of the most frequently used multifiber algorithms under realistic medical settings and realistic signal-to-noise ratio environments. The framework is based on several crossings with a non-orientational bias and different signal models. The results they present are relevant for medical doctors and researchers, interested in the use of the multifiber solution for tractography.

Quantitative evaluation of 10 tractography algorithms on a realistic diffusion MR phantom.

As it provides the only method for mapping white matter fibers in vivo, diffusion MRI tractography is gaining importance in clinical and neuroscience research. However, despite the increasing availability of different diffusion models and tractography algorithms, it remains unclear how to select the optimal fiber reconstruction method, given certain imaging parameters. Consequently, it is of utmost importance to have a quantitative comparison of these models and algorithms and a deeper understanding of the corresponding strengths and weaknesses. In this work, we use a common dataset with known ground truth and a reproducible methodology to quantitatively evaluate the performance of various diffusion models and tractography algorithms. To examine a wide range of methods, the dataset, but not the ground truth, was released to the public for evaluation in a contest, the "Fiber Cup". 10 fiber reconstruction methods were evaluated. The results provide evidence that: 1. For high SNR datasets, diffusion models such as (fiber) orientation distribution functions correctly model the underlying fiber distribution and can be used in conjunction with streamline tractography, and 2. For medium or low SNR datasets, a prior on the spatial smoothness of either the diffusion model or the fibers is recommended for correct modelling of the fiber distribution and proper tractography results. The phantom dataset, the ground truth fibers, the evaluation methodology and the results obtained so far will remain publicly available on: http://www.lnao.fr/spip.php?rubrique79 to serve as a comparison basis for existing or new tractography methods. New results can be submitted to fibercup09@gmail.com and updates will be published on the webpage.

A comparison of methods for recovering intra-voxel white matter fiber architecture from clinical diffusion imaging scans.

Diffusion tensor magnetic resonance imaging is widely used to study the structure of the fiber pathways of brain white matter. However, the diffusion tensor cannot capture complex intra-voxel fiber architecture such as fiber crossings. Consequently, a number of methods have been proposed to recover intra-voxel fiber bundle orientations from high angular-resolution diffusion imaging scans, which are optimized to resolve fiber crossings. In this work we study how multi-tensor, spherical deconvolution, analytical QBall and diffusion basis function methods perform under clinical scanning conditions. Our experiments indicate that it is feasible to apply some of these methods in clinical data sets.

Diffusion basis functions decomposition for estimating white matter intravoxel fiber geometry.

In this paper, we present a new formulation for recovering the fiber tract geometry within a voxel from diffusion weighted magnetic resonance imaging (MRI) data, in the presence of single or multiple neuronal fibers. To this end, we define a discrete set of diffusion basis functions. The intravoxel information is recovered at voxels containing fiber crossings or bifurcations via the use of a linear combination of the above mentioned basis functions. Then, the parametric representation of the intravoxel fiber geometry is a discrete mixture of Gaussians. Our synthetic experiments depict several advantages by using this discrete schema: the approach uses a small number of diffusion weighted images (23) and relatively small b values (1250 s/mm2), i.e., the intravoxel information can be inferred at a fraction of the acquisition time required for datasets involving a large number of diffusion gradient orientations. Moreover our method is robust in the presence of more than two fibers within a voxel, improving the state-of-the-art of such parametric models. We present two algorithmic solutions to our formulation: by solving a linear program or by minimizing a quadratic cost function (both with non-negativity constraints). Such minimizations are efficiently achieved with standard iterative deterministic algorithms. Finally, we present results of applying the algorithms to synthetic as well as real data.