Reduced cross-scanner variability using vendor-agnostic sequences for single-shell diffusion MRI.

PURPOSE: To reduce the inter-scanner variability of diffusion MRI (dMRI) measures between scanners from different vendors by developing a vendor-neutral dMRI pulse sequence using the open-source vendor-agnostic Pulseq platform. METHODS: We implemented a standard EPI based dMRI sequence in Pulseq. We tested it on two clinical scanners from different vendors (Siemens Prisma and GE Premier), systematically evaluating and comparing the within- and inter-scanner variability across the vendors, using both the vendor-provided and Pulseq dMRI sequences. Assessments covered both a diffusion phantom and three human subjects, using standard error (SE) and Lin's concordance correlation to measure the repeatability and reproducibility of standard DTI metrics including fractional anisotropy (FA) and mean diffusivity (MD). RESULTS: Identical dMRI sequences were executed on both scanners using Pulseq. On the phantom, the Pulseq sequence showed more than a 2.5× reduction in SE (variability) across Siemens and GE scanners. Furthermore, Pulseq sequences exhibited markedly reduced SE in-vivo, maintaining scan-rescan repeatability while delivering lower variability in FA and MD (more than 50% reduction in cortical/subcortical regions) compared to vendor-provided sequences. CONCLUSION: The Pulseq diffusion sequence reduces the cross-scanner variability for both phantom and in-vivo data, which will benefit multi-center neuroimaging studies and improve the reproducibility of neuroimaging studies.

Accelerating joint relaxation-diffusion MRI by integrating time division multiplexing and simultaneous multi-slice (TDM-SMS) strategies.

PURPOSE: To accelerate the acquisition of relaxation-diffusion imaging by integrating time-division multiplexing (TDM) with simultaneous multi-slice (SMS) for EPI and evaluate imaging quality and diffusion measures. METHODS: The time-division multiplexing (TDM) technique and SMS method were integrated to achieve a high slice-acceleration (e.g., 6×) factor for acquiring relaxation-diffusion MRI. Two variants of the sequence, referred to as TDM3e-SMS and TDM2s-SMS, were developed to simultaneously acquire slice groups with three distinct TEs and two slice groups with the same TE, respectively. Both sequences were evaluated on a 3T scanner with in vivo human brains and compared with standard single-band (SB) -EPI and SMS-EPI using diffusion measures and tractography results. RESULTS: Experimental results showed that the TDM3e-SMS sequence with total slice acceleration of 6 (multiplexing factor (MP) = 3 × multi-band factor (MB) = 2) provided similar image intensity and microstructure measures compared to standard SMS-EPI with MB = 2, and yielded less bias in intensity compared to standard SMS-EPI with MB = 4. The three sequences showed a similar positive correlation between TE and mean kurtosis (MK) and a negative correlation between TE and mean diffusivity (MD) in white matter. Multi-fiber tractography also shows consistency of results in TE-dependent measures between different sequences. The TDM2s-SMS sequence (MP = 2, MB = 2) also provided imaging measures similar to standard SMS-EPI sequences (MB = 2) for single-TE diffusion imaging. CONCLUSIONS: The TDM-SMS sequence can provide additional 2× to 3× acceleration to SMS without degrading imaging quality. With the significant reduction in scan time, TDM-SMS makes joint relaxation-diffusion MRI a feasible technique in neuroimaging research to investigate new markers of brain disorders.

Deep learning based segmentation of brain tissue from diffusion MRI.

Segmentation of brain tissue types from diffusion MRI (dMRI) is an important task, required for quantification of brain microstructure and for improving tractography. Current dMRI segmentation is mostly based on anatomical MRI (e.g., T1- and T2-weighted) segmentation that is registered to the dMRI space. However, such inter-modality registration is challenging due to more image distortions and lower image resolution in dMRI as compared with anatomical MRI. In this study, we present a deep learning method for diffusion MRI segmentation, which we refer to as DDSeg. Our proposed method learns tissue segmentation from high-quality imaging data from the Human Connectome Project (HCP), where registration of anatomical MRI to dMRI is more precise. The method is then able to predict a tissue segmentation directly from new dMRI data, including data collected with different acquisition protocols, without requiring anatomical data and inter-modality registration. We train a convolutional neural network (CNN) to learn a tissue segmentation model using a novel augmented target loss function designed to improve accuracy in regions of tissue boundary. To further improve accuracy, our method adds diffusion kurtosis imaging (DKI) parameters that characterize non-Gaussian water molecule diffusion to the conventional diffusion tensor imaging parameters. The DKI parameters are calculated from the recently proposed mean-kurtosis-curve method that corrects implausible DKI parameter values and provides additional features that discriminate between tissue types. We demonstrate high tissue segmentation accuracy on HCP data, and also when applying the HCP-trained model on dMRI data from other acquisitions with lower resolution and fewer gradient directions.

Accelerated diffusion and relaxation-diffusion MRI using time-division multiplexing EPI.

PURPOSE: To develop a time-division multiplexing echo-planar imaging (TDM-EPI) sequence for approximately two- to threefold acceleration when acquiring joint relaxation-diffusion MRI data with multiple TEs. METHODS: The proposed TDM-EPI sequence interleaves excitation and data collection for up to 3 separate slices at different TEs and uses echo-shifting gradients to disentangle the overlapping echo signals during the readout period. By properly arranging the sequence event blocks for each slice and adjusting the echo-shifting gradients, diffusion-weighted images from separate slices can be acquired. Therefore, we present 2 variants of the sequence. A single-TE TDM-EPI is presented to demonstrate the concept. Next, a multi-TE TDM-EPI is presented to highlight the advantages of the TDM approach for relaxation-diffusion imaging. These sequences were evaluated on a 3 Tesla scanner with a water phantom and in vivo human brain data. RESULTS: The single-TE TDM-EPI sequence can simultaneously acquire 2 slices with a maximum b value of 3000 s/mm2 and 2.5 mm isotropic resolution using interleaved readout windows with TE ≈ 78 ms. With the same b value and resolution, the multi-TE TDM-EPI sequence can simultaneously acquire 2 or 3 separate slices using interleaved readout sections with shortest TE ≈ 70 ms and ΔTE ≈ 30 ms. Phantom and in vivo experiments have shown that the proposed TDM-EPI sequences can provide similar image quality and diffusion measures as conventional EPI readouts with multiple echoes but can reduce the overall relaxation-diffusion protocol scan time by approximately two- to threefold. CONCLUSION: TDM-EPI is a novel approach to acquire diffusion imaging data at multiple TEs. This enables a significant reduction in acquisition time for relaxation-diffusion MRI experiments but without compromising image quality and diffusion measurements, thus removing a significant barrier to the adoption of relaxation-diffusion MRI in clinical research studies of neurological and mental disorders.

Quantifying Genetic and Environmental Influence on Gray Matter Microstructure Using Diffusion MRI.

Early neuroimaging work in twin studies focused on studying genetic and environmental influence on gray matter macrostructure. However, it is also important to understand how gray matter microstructure is influenced by genes and environment to facilitate future investigations of their influence in mental disorders. Advanced diffusion MRI (dMRI) measures allow more accurate assessment of gray matter microstructure compared with conventional diffusion tensor measures. To understand genetic and environmental influence on gray matter, we used diffusion and structural MRI data from a large twin and sibling study (N = 840) and computed advanced dMRI measures including return to origin probability (RTOP), which is heavily weighted toward intracellular and intra-axonal restricted spaces, and mean squared displacement (MSD), more heavily weighted to diffusion in extracellular space and large cell bodies in gray matter. We show that while macrostructural features like brain volume are mainly genetically influenced, RTOP and MSD can together tap into both genetic and environmental influence on microstructure.

Cross-scanner and cross-protocol multi-shell diffusion MRI data harmonization: Algorithms and results.

Cross-scanner and cross-protocol variability of diffusion magnetic resonance imaging (dMRI) data are known to be major obstacles in multi-site clinical studies since they limit the ability to aggregate dMRI data and derived measures. Computational algorithms that harmonize the data and minimize such variability are critical to reliably combine datasets acquired from different scanners and/or protocols, thus improving the statistical power and sensitivity of multi-site studies. Different computational approaches have been proposed to harmonize diffusion MRI data or remove scanner-specific differences. To date, these methods have mostly been developed for or evaluated on single b-value diffusion MRI data. In this work, we present the evaluation results of 19 algorithms that are developed to harmonize the cross-scanner and cross-protocol variability of multi-shell diffusion MRI using a benchmark database. The proposed algorithms rely on various signal representation approaches and computational tools, such as rotational invariant spherical harmonics, deep neural networks and hybrid biophysical and statistical approaches. The benchmark database consists of data acquired from the same subjects on two scanners with different maximum gradient strength (80 and 300 â€‹mT/m) and with two protocols. We evaluated the performance of these algorithms for mapping multi-shell diffusion MRI data across scanners and across protocols using several state-of-the-art imaging measures. The results show that data harmonization algorithms can reduce the cross-scanner and cross-protocol variabilities to a similar level as scan-rescan variability using the same scanner and protocol. In particular, the LinearRISH algorithm based on adaptive linear mapping of rotational invariant spherical harmonics features yields the lowest variability for our data in predicting the fractional anisotropy (FA), mean diffusivity (MD), mean kurtosis (MK) and the rotationally invariant spherical harmonic (RISH) features. But other algorithms, such as DIAMOND, SHResNet, DIQT, CMResNet show further improvement in harmonizing the return-to-origin probability (RTOP). The performance of different approaches provides useful guidelines on data harmonization in future multi-site studies.

High-fidelity, accelerated whole-brain submillimeter in vivo diffusion MRI using gSlider-spherical ridgelets (gSlider-SR).

PURPOSE: To develop an accelerated, robust, and accurate diffusion MRI acquisition and reconstruction technique for submillimeter whole human brain in vivo scan on a clinical scanner. METHODS: We extend the ultra-high resolution diffusion MRI acquisition technique, gSlider, by allowing undersampling in q-space and radiofrequency (RF)-encoding space, thereby dramatically reducing the total acquisition time of conventional gSlider. The novel method, termed gSlider-SR, compensates for the lack of acquired information by exploiting redundancy in the dMRI data using a basis of spherical ridgelets (SR), while simultaneously enhancing the signal-to-noise ratio. Using Monte Carlo simulation with realistic noise levels and several acquisitions of in vivo human brain dMRI data (acquired on a Siemens Prisma 3T scanner), we demonstrate the efficacy of our method using several quantitative metrics. RESULTS: For high-resolution dMRI data with realistic noise levels (synthetically added), we show that gSlider-SR can reconstruct high-quality dMRI data at different acceleration factors preserving both signal and angular information. With in vivo data, we demonstrate that gSlider-SR can accurately reconstruct 860 µm diffusion MRI data (64 diffusion directions at b=2000s/mm2 ), at comparable quality as that obtained with conventional gSlider with four averages, thereby providing an eight-fold reduction in scan time (from 1 hour 20 to 10 minutes). CONCLUSIONS: gSlider-SR enables whole-brain high angular resolution dMRI at a submillimeter spatial resolution with a dramatically reduced acquisition time, making it feasible to use the proposed scheme on existing clinical scanners.

Smooth interpolation of covariance matrices and brain network estimation: Part II.

This work focuses on the modeling of time-varying covariance matrices using the state covariance of linear systems. Following concepts from optimal mass transport, we investigate and compare three types of covariance paths which are solutions to different optimal control problems. One of the covariance paths solves the Schrödinger bridge problem (SBP). The other two types of covariance paths are based on generalizations of the Fisher-Rao metric in information geometry, which are the major contributions of this work. The general framework is an extension of the approach in [1] which focuses on linear systems without stochastic input. The performances of the three covariance paths are compared using synthetic data and a real-data example on the estimation of dynamic brain networks using functional magnetic resonance imaging.

MK-curve - Characterizing the relation between mean kurtosis and alterations in the diffusion MRI signal.

Diffusion kurtosis imaging (DKI) is a diffusion MRI (dMRI) technique to quantify brain microstructural properties. While DKI measures are sensitive to tissue alterations, they are also affected by signal alterations caused by imaging artifacts such as noise, motion and Gibbs ringing. Consequently, DKI often yields output parameter values (e.g. mean kurtosis; MK) that are implausible. These include implausible values that are outside of the range dictated by physics/biology, and visually apparent implausible values that form unexpected discontinuities, being too high or too low comparing with their neighborhood. These implausible values will introduce bias into any following data analyses (e.g. between-population statistical computation). Existing studies have attempted to correct implausible DKI parameter values in multiple ways; however, these approaches are not always effective. In this study, we propose a novel method for detecting and correcting voxels with implausible values to enable improved DKI parameter estimation. In particular, we focus on MK parameter estimation. We first characterize the relation between MK and alterations in the dMRI signal including diffusion weighted images (DWIs) and the baseline (b0) images. This is done by calculating MK for a range of synthetic DWI or b0 for each voxel, and generating curves (MK-curve) representing how alterations to the input dMRI signals affect the resulting output MK. We find that voxels with implausible MK values are more likely caused by artifacts in the b0 images than artifacts in DWIs with higher b-values. Accordingly, two characteristic b0 values, which define a range of synthetic b0 values that generate implausible MK values, are identified on the MK-curve. Based on this characterization, we propose an automatic approach for detection of voxels with implausible MK values by comparing a voxel's original b0 signal to the identified two characteristic b0 values, along with a correction strategy to replace the original b0 in each detected implausible voxel with a synthetic b0 value computed from the MK-curve. We evaluate the method on a DKI phantom dataset and dMRI datasets from the Human Connectome Project (HCP), and we compare the proposed correction method with other previously proposed correction methods. Results show that our proposed method is able to identify and correct most voxels with implausible DKI parameter values as well as voxels with implausible diffusion tensor parameter values.

Retrospective harmonization of multi-site diffusion MRI data acquired with different acquisition parameters.

A joint and integrated analysis of multi-site diffusion MRI (dMRI) datasets can dramatically increase the statistical power of neuroimaging studies and enable comparative studies pertaining to several brain disorders. However, dMRI data sets acquired on multiple scanners cannot be naively pooled for joint analysis due to scanner specific nonlinear effects as well as differences in acquisition parameters. Consequently, for joint analysis, the dMRI data has to be harmonized, which involves removing scanner-specific differences from the raw dMRI signal. In this work, we propose a dMRI harmonization method that is capable of removing scanner-specific effects, while accounting for minor differences in acquisition parameters such as b-value, spatial resolution and number of gradient directions. We validate our algorithm on dMRI data acquired from two sites: Philadelphia Neurodevelopmental Cohort (PNC) with 800 healthy adolescents (ages 8-22 years) and Brigham and Women's Hospital (BWH) with 70 healthy subjects (ages 14-54 years). In particular, we show that gender and age-related maturation differences in different age groups are preserved after harmonization, as measured using effect sizes (small, medium and large), irrespective of the test sample size. Since we use matched control subjects from different scanners to estimate scanner-specific effects, our goal in this work is also to determine the minimum number of well-matched subjects needed from each site to achieve best harmonization results. Our results indicate that at-least 16 to 18 well-matched healthy controls from each site are needed to reliably capture scanner related differences. The proposed method can thus be used for retrospective harmonization of raw dMRI data across sites despite differences in acquisition parameters, while preserving inter-subject anatomical variability.

Cross-scanner and cross-protocol diffusion MRI data harmonisation: A benchmark database and evaluation of algorithms.

Diffusion MRI is being used increasingly in studies of the brain and other parts of the body for its ability to provide quantitative measures that are sensitive to changes in tissue microstructure. However, inter-scanner and inter-protocol differences are known to induce significant measurement variability, which in turn jeopardises the ability to obtain 'truly quantitative measures' and challenges the reliable combination of different datasets. Combining datasets from different scanners and/or acquired at different time points could dramatically increase the statistical power of clinical studies, and facilitate multi-centre research. Even though careful harmonisation of acquisition parameters can reduce variability, inter-protocol differences become almost inevitable with improvements in hardware and sequence design over time, even within a site. In this work, we present a benchmark diffusion MRI database of the same subjects acquired on three distinct scanners with different maximum gradient strength (40, 80, and 300 mT/m), and with 'standard' and 'state-of-the-art' protocols, where the latter have higher spatial and angular resolution. The dataset serves as a useful testbed for method development in cross-scanner/cross-protocol diffusion MRI harmonisation and quality enhancement. Using the database, we compare the performance of five different methods for estimating mappings between the scanners and protocols. The results show that cross-scanner harmonisation of single-shell diffusion data sets can reduce the variability between scanners, and highlight the promises and shortcomings of today's data harmonisation techniques.

Smooth Interpolation of Covariance Matrices and Brain Network Estimation.

We propose an approach to use the state covariance of autonomous linear systems to track time-varying covariance matrices of nonstationary time series. Following concepts from the Riemannian geometry, we investigate three types of covariance paths obtained by using different quadratic regularizations of system matrices. The first quadratic form induces the geodesics based on the Hellinger-Bures metric related to optimal mass transport (OMT) theory and quantum mechanics. The second type of quadratic form leads to the geodesics based on the Fisher-Rao metric from information geometry. In the process, we introduce a weighted-OMT interpretation of the Fisher-Rao metric for multivariate Gaussian distributions. A main contribution of this work is the introduction of the third type of covariance paths, which are steered by system matrices with rotating eigenspaces. The three types of covariance paths are compared using two examples with synthetic data and real data from resting-state functional magnetic resonance imaging, respectively.

Suprathreshold fiber cluster statistics: Leveraging white matter geometry to enhance tractography statistical analysis.

This work presents a suprathreshold fiber cluster (STFC) method that leverages the whole brain fiber geometry to enhance statistical group difference analyses. The proposed method consists of 1) a well-established study-specific data-driven tractography parcellation to obtain white matter tract parcels and 2) a newly proposed nonparametric, permutation-test-based STFC method to identify significant differences between study populations. The basic idea of our method is that a white matter parcel's neighborhood (nearby parcels with similar white matter anatomy) can support the parcel's statistical significance when correcting for multiple comparisons. We propose an adaptive parcel neighborhood strategy to allow suprathreshold fiber cluster formation that is robust to anatomically varying inter-parcel distances. The method is demonstrated by application to a multi-shell diffusion MRI dataset from 59 individuals, including 30 attention deficit hyperactivity disorder patients and 29 healthy controls. Evaluations are conducted using both synthetic and in-vivo data. The results indicate that the STFC method gives greater sensitivity in finding group differences in white matter tract parcels compared to several traditional multiple comparison correction methods.

Cumulant expansions for measuring water exchange using diffusion MRI.

The rate of water exchange across cell membranes is a parameter of biological interest and can be measured by diffusion magnetic resonance imaging (dMRI). In this work, we investigate a stochastic model for the diffusion-and-exchange of water molecules. This model provides a general solution for the temporal evolution of dMRI signal using any type of gradient waveform, thereby generalizing the signal expressions for the Kärger model. Moreover, we also derive a general nth order cumulant expansion of the dMRI signal accounting for water exchange, which has not been explored in earlier studies. Based on this analytical expression, we compute the cumulant expansion for dMRI signals for the special case of single diffusion encoding (SDE) and double diffusion encoding (DDE) sequences. Our results provide a theoretical guideline on optimizing experimental parameters for SDE and DDE sequences, respectively. Moreover, we show that DDE signals are more sensitive to water exchange at short-time scale but provide less attenuation at long-time scale than SDE signals. Our theoretical analysis is also validated using Monte Carlo simulations on synthetic structures.

Precise Inference and Characterization of Structural Organization (PICASO) of tissue from molecular diffusion.

Inferring the microstructure of complex media from the diffusive motion of molecules is a challenging problem in diffusion physics. In this paper, we introduce a novel representation of diffusion MRI (dMRI) signal from tissue with spatially-varying diffusivity using a diffusion disturbance function. This disturbance function contains information about the (intra-voxel) spatial fluctuations in diffusivity due to restrictions, hindrances and tissue heterogeneity of the underlying tissue substrate. We derive the short- and long-range disturbance coefficients from this disturbance function to characterize the tissue structure and organization. Moreover, we provide an exact relation between the disturbance coefficients and the time-varying moments of the diffusion propagator, as well as their relation to specific tissue microstructural information such as the intra-axonal volume fraction and the apparent axon radius. The proposed approach is quite general and can model dMRI signal for any type of gradient sequence (rectangular, oscillating, etc.) without using the Gaussian phase approximation. The relevance of the proposed PICASO model is explored using Monte-Carlo simulations and in-vivo dMRI data. The results show that the estimated disturbance coefficients can distinguish different types of microstructural organization of axons.

New insights about time-varying diffusivity and its estimation from diffusion MRI.

PURPOSE: Characterizing the relation between the applied gradient sequences and the measured diffusion MRI signal is important for estimating the time-dependent diffusivity, which provides important information about the microscopic tissue structure. THEORY AND METHODS: In this article, we extend the classical theory of Stepisnik for measuring time-dependent diffusivity under the Gaussian phase approximation. In particular, we derive three novel expressions which represent the diffusion MRI signal in terms of the mean-squared displacement, the instantaneous diffusivity, and the velocity autocorrelation function. We present the explicit signal expressions for the case of single diffusion encoding and oscillating gradient spin-echo sequences. Additionally, we also propose three different models to represent time-varying diffusivity and test them using Monte-Carlo simulations and in vivo human brain data. RESULTS: The time-varying diffusivities are able to distinguish the synthetic structures in the Monte-Carlo simulations. There is also strong statistical evidence about time-varying diffusivity from the in vivo human data set. CONCLUSION: The proposed theory provides new insights into our understanding of the time-varying diffusivity using different gradient sequences. The proposed models for representing time-varying diffusivity can be utilized to study time-varying diffusivity using in vivo human brain diffusion MRI data. Magn Reson Med 78:763-774, 2017. © 2016 International Society for Magnetic Resonance in Medicine.

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.

On Matrix-Valued Monge-Kantorovich Optimal Mass Transport.

We present a particular formulation of optimal transport for matrix-valued density functions. Our aim is to devise a geometry which is suitable for comparing power spectral densities of multivariable time series. More specifically, the value of a power spectral density at a given frequency, which in the matricial case encodes power as well as directionality, is thought of as a proxy for a "matrix-valued mass density." Optimal transport aims at establishing a natural metric in the space of such matrix-valued densities which takes into account differences between power across frequencies as well as misalignment of the corresponding principle axes. Thus, our transportation cost includes a cost of transference of power between frequencies together with a cost of rotating the principle directions of matrix densities. The two endpoint matrix-valued densities can be thought of as marginals of a joint matrix-valued density on a tensor product space. This joint density, very much as in the classical Monge-Kantorovich setting, can be thought to specify the transportation plan. Contrary to the classical setting, the optimal transport plan for matrices is no longer supported on a thin zero-measure set.

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.