Q-space Data Research Articles

Attention-based q-space Deep Learning Generalized for Accelerated Diffusion Magnetic Resonance Imaging.

Diffusion magnetic resonance imaging (dMRI) is a non-invasive method for capturing the microanatomical information of tissues by measuring the diffusion weighted signals along multiple directions, which is widely used in the quantification of microstructures. Obtaining microscopic parameters requires dense sampling in the q space, leading to significant time consumption. The most popular approach to accelerating dMRI acquisition is to undersample the q-space data, along with applying deep learning methods to reconstruct quantitative diffusion parameters. However, the reliance on a predetermined q-space sampling strategy often constrains traditional deep learning-based reconstructions. The present study proposed a novel deep learning model, named attention-based q-space deep learning (aqDL), to implement the reconstruction with variable q-space sampling strategies. The aqDL maps dMRI data from different scanning strategies onto a common feature space by using a series of Transformer encoders. The latent features are employed to reconstruct dMRI parameters via a multilayer perceptron. The performance of the aqDL model was assessed utilizing the Human Connectome Project datasets at varying undersampling numbers. To validate its generalizability, the model was further tested on two additional independent datasets. Our results showed that aqDL consistently achieves the highest reconstruction accuracy at various undersampling numbers, regardless of whether variable or predetermined q-space scanning strategies are employed. These findings suggest that aqDL has the potential to be used on general clinical dMRI datasets.

A microstructure estimation Transformer inspired by sparse representation for diffusion MRI

Diffusion magnetic resonance imaging (dMRI) is an important tool in characterizing tissue microstructure based on biophysical models, which are typically multi-compartmental models with mathematically complex and highly non-linear forms. Resolving microstructures from these models with conventional optimization techniques is prone to estimation errors and requires dense sampling in the q-space with a long scan time. Deep learning based approaches have been proposed to overcome these limitations. Motivated by the superior performance of the Transformer in feature extraction than the convolutional structure, in this work, we present a learning-based framework based on Transformer, namely, a Microstructure Estimation Transformer with Sparse Coding (METSC) for dMRI-based microstructural parameter estimation. To take advantage of the Transformer while addressing its limitation in large training data requirement, we explicitly introduce an inductive bias-model bias into the Transformer using a sparse coding technique to facilitate the training process. Thus, the METSC is composed with three stages, an embedding stage, a sparse representation stage, and a mapping stage. The embedding stage is a Transformer-based structure that encodes the signal in a high-level space to ensure the core voxel of a patch is represented effectively. In the sparse representation stage, a dictionary is constructed by solving a sparse reconstruction problem that unfolds the Iterative Hard Thresholding (IHT) process. The mapping stage is essentially a decoder that computes the microstructural parameters from the output of the second stage, based on the weighted sum of normalized dictionary coefficients where the weights are also learned. We tested our framework on two dMRI models with downsampled q-space data, including the intravoxel incoherent motion (IVIM) model and the neurite orientation dispersion and density imaging (NODDI) model. The proposed method achieved up to 11.25 folds of acceleration while retaining high fitting accuracy for NODDI fitting, reducing the mean squared error (MSE) up to 70% compared with the previous q-space learning approach. METSC outperformed the other state-of-the-art learning-based methods, including the model-free and model-based methods. The network also showed robustness against noise and generalizability across different datasets. The superior performance of METSC indicates its potential to improve dMRI acquisition and model fitting in clinical applications.

Jointly estimating parametric maps of multiple diffusion models from undersampled q-space data: A comparison of three deep learning approaches.

While advanced diffusion techniques have been found valuable in many studies, their clinical availability has been hampered partly due to their long scan times. Moreover, each diffusion technique can only extract a few relevant microstructural features. Using multiple diffusion methods may help to better understand the brain microstructure, which requires multiple expensive model fittings. In this work, we compare deep learning (DL) approaches to jointly estimate parametric maps of multiple diffusion representations/models from highly undersampled q-space data. We implement three DL approaches to jointly estimate parametric maps of diffusion tensor imaging (DTI), diffusion kurtosis imaging (DKI), neurite orientation dispersion and density imaging (NODDI), and multi-compartment spherical mean technique (SMT). A per-voxel q-space deep learning (1D-qDL), a per-slice convolutional neural network (2D-CNN), and a 3D-patch-based microstructure estimation with sparse coding using a separable dictionary (MESC-SD) network are considered. The accuracy of estimated diffusion maps depends on the q-space undersampling, the selected network architecture, and the region and the parameter of interest. The smallest errors are observed for the MESC-SD network architecture (less than 10 normalized RMSE in most brain regions). Our experiments show that DL methods are very efficient tools to simultaneously estimate several diffusion maps from undersampled q-space data. These methods can significantly reduce both the scan ( 6-fold) and processing times ( 25-fold) for estimating advanced parametric diffusion maps while achieving a reasonable accuracy.

High-fidelity approximation of grid- and shell-based sampling schemes from undersampled DSI using compressed sensing: Post mortem validation

While many useful microstructural indices, as well as orientation distribution functions, can be obtained from multi-shell dMRI data, there is growing interest in exploring the richer set of microstructural features that can be extracted from the full ensemble average propagator (EAP). The EAP can be readily computed from diffusion spectrum imaging (DSI) data, at the cost of a very lengthy acquisition. Compressed sensing (CS) has been used to make DSI more practical by reducing its acquisition time. CS applied to DSI (CS-DSI) attempts to reconstruct the EAP from significantly undersampled q-space data. We present a post mortem validation study where we evaluate the ability of CS-DSI to approximate not only fully sampled DSI but also multi-shell acquisitions with high fidelity. Human brain samples are imaged with high-resolution DSI at 9.4T and with polarization-sensitive optical coherence tomography (PSOCT). The latter provides direct measurements of axonal orientations at microscopic resolutions, allowing us to evaluate the mesoscopic orientation estimates obtained from diffusion MRI, in terms of their angular error and the presence of spurious peaks. We test two fast, dictionary-based, L2-regularized algorithms for CS-DSI reconstruction. We find that, for a CS acceleration factor of R=3, i.e., an acquisition with 171 gradient directions, one of these methods is able to achieve both low angular error and low number of spurious peaks. With a scan length similar to that of high angular resolution multi-shell acquisition schemes, this CS-DSI approach is able to approximate both fully sampled DSI and multi-shell data with high accuracy. Thus it is suitable for orientation reconstruction and microstructural modeling techniques that require either grid- or shell-based acquisitions. We find that the signal-to-noise ratio (SNR) of the training data used to construct the dictionary can have an impact on the accuracy of CS-DSI, but that there is substantial robustness to loss of SNR in the test data. Finally, we show that, as the CS acceleration factor increases beyond R=3, the accuracy of these reconstruction methods degrade, either in terms of the angular error, or in terms of the number of spurious peaks. Our results provide useful benchmarks for the future development of even more efficient q-space acceleration techniques.

Characterizing pore-scale structure-flow correlations in sedimentary rocks using magnetic resonance imaging.

Quantitative, three-dimensional (3D) spatially resolved magnetic resonance flow imaging (flow MRI) methods are presented to characterize structure-flow correlations in a 4-mm-diameter plug of Ketton limestone rock using undersampled k- and q-space data acquisition methods combined with compressed sensing (CS) data reconstruction techniques. The acquired MRI data are coregistered with an X-ray microcomputed tomography (μCT) image of the same rock sample, allowing direct correlation of the structural features of the rock with local fluid transport characteristics. First, 3D velocity maps acquired at 35 μm isotropic spatial resolution showed that the flow was highly heterogeneous, with ∼10% of the pores carrying more than 50% of the flow. Structure-flow correlations were found between the local flow velocities through pores and the size and topology (coordination number) associated with these pores. These data show consistent trends with analogous data acquired for flow through a packing of 4-mm-diameter spheres, which may be due to the microstructure of Ketton rock being a consolidation of approximately spherical grains. Using two-dimensional and 3D visualization of coregistered μCT images and velocity maps, complex pore-scale flow patterns were identified. Second, 3D spatially resolved propagators were acquired at 94 μm isotropic spatial resolution. Flow dispersion within the rock was examined by analyzing each of the 331 776 local propagators as a function of observation time. Again, the heterogeneity of flow within the rock was shown. Quantification of the mean and standard deviation of each of the local propagators showed enhanced mixing occurring within the pore space at longer observation times. These spatially resolved measurements also enable investigation of the length scale of a representative elementary volume. It is shown that for a 4-mm-diameter plug this length scale is not reached.

Estimating Tissue Microstructure with Undersampled Diffusion Data via Graph Convolutional Neural Networks.

Advanced diffusion models for tissue microstructure are widely employed to study brain disorders. However, these models usually require diffusion MRI (DMRI) data with densely sampled q-space, which is prohibitive in clinical settings. This problem can be resolved by using deep learning techniques, which learn the mapping between sparsely sampled q-space data and the high-quality diffusion microstructural indices estimated from densely sampled data. However, most existing methods simply view the input DMRI data as a vector without considering data structure in the q-space. In this paper, we propose to overcome this limitation by representing DMRI data using graphs and utilizing graph convolutional neural networks to estimate tissue microstructure. Our method makes full use of the q-space angular neighboring information to improve estimation accuracy. Experimental results based on data from the Baby Connectome Project demonstrate that our method outperforms state-of-the-art methods both qualitatively and quantitatively.

Simultaneous NODDI and GFA parameter map generation from subsampled q-space imaging using deep learning.

To develop a robust multidimensional deep-learning based method to simultaneously generate accurate neurite orientation dispersion and density imaging (NODDI) and generalized fractional anisotropy (GFA) parameter maps from undersampled q-space datasets for use in stroke imaging. Traditional diffusion spectrum imaging (DSI) capable of producing accurate NODDI and GFA parameter maps requires hundreds of q-space samples which renders the scan time clinically untenable. A convolutional neural network (CNN) was trained to generated NODDI and GFA parameter maps simultaneously from 10× undersampled q-space data. A total of 48 DSI scans from 15 stroke patients and 14 normal subjects were acquired for training, validating, and testing this method. The proposed network was compared to previously proposed voxel-wise machine learning based approaches for q-space imaging. Network-generated images were used to predict stroke functional outcome measures. The proposed network achieves significant performance advantages compared to previously proposed machine learning approaches, showing significant improvements across image quality metrics. Generating these parameter maps using CNNs also comes with the computational benefits of only needing to generate and train a single network instead of multiple networks for each parameter type. Post-stroke outcome prediction metrics do not appreciably change when using images generated from this proposed technique. Over three test participants, the predicted stroke functional outcome scores were within 1-6% of the clinical evaluations. Estimates of NODDI and GFA parameters estimated simultaneously with a deep learning network from highly undersampled q-space data were improved compared to other state-of-the-art methods providing a 10-fold reduction scan time compared to conventional methods.

Pulsed second order fields for parallel acquisition of q-space

Acquisition of displacement space with NMR, the conjugate of q-space, has many applications including diffusion, dispersion, velocimetry, tortuosity, and surface to volume ratio measurements in porous media and emulsions. Normally, acquisition of q-space data requires a series of scans over which current pulsed through a gradient coil producing a constant magnetic field gradient is varied to step through q-space. By replacing the constant magnetic field gradient with a second order field whose gradient strength varies in space, a range of gradient values are applied over the sample in a single experiment, and q-space encoded into real space. Using slice selection and a read gradient in conjunction with the second order field, a modified PGSE sequence has been developed which allows the parallel acquisition of q-space in a single-shot experiment for a homogeneous medium. A proof of concept is presented for the parallel acquisition of q-space under the diffusive process, allowing a single-shot, single-excitation, single observation time diffusion measurement to be made. In addition to this, the mapping between real space and q-space leads to a mapping between displacement space and time. This means that for this experiment, without the use of any Fourier transforms, the average propagator may be obtained simply by normalising the acquired echo and plotting it against displacement space.

Parallel acquisition of q-space using second order magnetic fields for single-shot diffusion measurements

A proof of concept is presented for the parallel acquisition of q-space under diffusion using a second order magnetic field. The second order field produces a gradient strength which varies in space, allowing a range of gradients to be applied in a single pulse, and q-space encoded into real space. With the use of a read gradient, the spatial information is regained from the NMR signal, and real space mapped onto q-space for a thin slice excitation volume. As the diffusion encoded image for a thin slice can be mapped onto q-space, and the average propagator is the inverse Fourier transform of the q-space data, it follows that the acquisition of the echo is a direct measurement of the average propagator. In the absence of a thin slice selection, the real space to q-space mapping is lost, but the ability to measure the diffusion coefficient retained with an increase in signal to noise.

Evaluation of fiber tracking from subsampled q-space data in diffusion spectrum imaging

Diffusion spectrum imaging (DSI) is capable of resolving crossing and touching fiber bundles in a given voxel. Acquisition of DSI data involves sampling large number of points in the q-space which significantly increases scan times. The scan times can be reduced by exploiting the symmetry of the q-space. In this study the fiber pathways for five (fornix, cingulum, superior longitudinal fasciculus, corticospinal tract, and crossing fibers in the centrum semiovale region) fiber bundles derived using three subsampled data sets of different sizes derived from the 257 samples in the q-space are compared. The coefficient of variation of the ratio of the number of fiber pathways for each subsample data set to the original data points, averaged over all the 10 subjects, was used for quantitatively investigating the effect of subsampling on the tractography. The effect of threshold angles on tractography is also investigated. The effect of subsampling on the orientation distribution function (ODF) was quantitatively evaluated using both scalar and vector measures derived from the ODF. A streamline tractography method that improves the curvature problem and reduces the local truncation error to further improve the mapping of fiber pathways is adapted. Analysis of the fiber pathways in ten normal subjects, based on qualitative and quantitative methods, shows that the 129 and 198 q-space points provide very similar result with angle of threshold between 41° and 45°. Based on the scan time advantage, 129 subsampled points appear to be adequate for tractography.

Probing tissue microstructure with restriction spectrum imaging: Histological and theoretical validation.

Water diffusion magnetic resonance imaging (dMRI) is a powerful tool for studying biological tissue microarchitectures in vivo. Recently, there has been increased effort to develop quantitative dMRI methods to probe both length scale and orientation information in diffusion media. Diffusion spectrum imaging (DSI) is one such approach that aims to resolve such information based on the three-dimensional diffusion propagator at each voxel. However, in practice, only the orientation component of the propagator function is preserved when deriving the orientation distribution function. Here, we demonstrate how a straightforward extension of the linear spherical deconvolution (SD) model can be used to probe tissue orientation structures over a range (or "spectrum") of length scales with minimal assumptions on the underlying microarchitecture. Using high b-value Cartesian q-space data on a rat brain tissue sample, we demonstrate how this "restriction spectrum imaging" (RSI) model allows for separating the volume fraction and orientation distribution of hindered and restricted diffusion, which we argue stems primarily from diffusion in the extraneurite and intraneurite water compartment, respectively. Moreover, we demonstrate how empirical RSI estimates of the neurite orientation distribution and volume fraction capture important additional structure not afforded by traditional DSI or fixed-scale SD-like reconstructions, particularly in gray matter. We conclude that incorporating length scale information in geometric models of diffusion offers promise for advancing state-of-the-art dMRI methods beyond white matter into gray matter structures while allowing more detailed quantitative characterization of water compartmentalization and histoarchitecture of healthy and diseased tissue.

Compartment-specific q-space analysis of diffusion-weighted data from isolated rhesus optic and sciatic nerves

We investigated compartment-specific water diffusion properties in two widely structurally different isolated bovine nerves. Sciatic and optic nerves were immersed in saline containing Gd-DTPA(2+). Consequently, T(1) became non-monoexponential and fit well to a biexponential function. q-Space diffusion data were collected for each component. In the sciatic nerve, the slow-decaying component (T(1s)) was considerably more restricted and directional than the fast-decaying component (T(1f)). In the optic nerve, fractional anisotropy of both components was comparable and similar to that of the total H(2)O signal. The root mean square of the displacement distribution functions of T(1s) correlated well with the widely different axonal diameters of both nerves. Possibly, the source of T(1s) is the intra-axonal compartment and that of T(1f) is associated with the inter-axonal space. The compartment specificity of the method shown makes it useful for the investigation of the contribution of each nerve compartment to diffusion tensor imaging measurements and other diffusion-based methods.

Mapping the intracellular fraction of water by varying the gradient pulse length in q-space diffusion MRI

Finite gradient pulse lengths are traditionally considered a nuisance in q-space diffusion NMR and MRI, since the simple Fourier relation between the acquired signal and the displacement probability is invalidated. Increasing the value of the pulse length leads to an apparently smaller value of the estimated compartment size. We propose that q-space data at different gradient pulse lengths, but with the same effective diffusion time, can be used to identify and quantify components with free or restricted diffusion from multiexponential echo decay curves obtained on cellular systems. The method is demonstrated with experiments on excised human brain white matter and a series of model systems with well-defined free, restricted, and combined free and restricted diffusion behavior. Time-resolved diffusion MRI experiments are used to map the spatial distribution of the intracellular fraction in a yeast cell suspension during sedimentation, and observe the disappearance of this fraction after a heat treatment.

Assessment of axonal fiber tract architecture in excised rat spinal cord by localized NMR q-space imaging: simulations and experimental studies.

NMR q-space imaging is a method designed to obtain information from porous materials where diffusion-diffraction phenomena were observed from which pore size was derived. Recently, the technique has been applied to the study of biological structures as well. Although diffusive diffraction has so far not been observed in multicellular systems, displacement profiles have been used with some success as a means to estimate structure size. However, there have been no quantitative correlations of the retrieved structure sizes with histology. Clearly, the complexity of tissue architecture poses significant challenges to the interpretation of q-space data. In this work, simulations were first performed on a two-compartment model to demonstrate the effects of interference of the diffraction patterns arising from intra and extra-axonal compartments and finite boundary permeability on q-space data. Second, q-space echo attenuation was simulated on the basis of histologic images of various rat spinal cord fiber tracts and the information obtained from the displacement profiles were compared with structural parameters computed from the histologic images. The results show that calculated mean displacements and kurtosis parallel mean axon size and axonal density. Finally, spatially localized q-space measurements were carried out at the locations where simulations had previously been performed, resulting in displacement data that support those obtained by simulation. The data suggest the NMR q-space approach has potential for nondestructive analysis of the axonal architecture in the mammalian spinal cord.

Simulations of NMR-detected diffusion in suspensions of red cells: the effects of variation in membrane permeability and observation time.

Monte Carlo random-walk simulations of diffusion in virtual lattices of cells have been used to study and characterize diffusion-coherence phenomena that arise when pulsed field-gradient spin-echo (PGSE) nuclear magnetic resonance (NMR) experiments are conducted on human red blood cell (RBC; erythrocytes) suspensions. These coherence effects are manifest as diffraction-like patterns when the normalized PGSE signal intensities are plotted as a function of the spatial wave vector q in so-called q-space plots. q-Space analysis is sensitive to small changes in cell morphology, cell size, membrane transport rates, hematocrit, and packing arrangement. In the present study we used simulations to predict the effect of varying the time over which diffusion is measured (the "observation time" or "diffusion time") and the permeability of the membrane on the form of q-space plots. Thus we predict that inhibiting water exchange across the human RBC membrane, such that the value of the permeability coefficient is reduced by approximately an order of magnitude below the normal physiological value, will effectively render the membrane impermeable on the timescale of the PGSE NMR experiment; further inhibition will therefore result in negligible reduction in the measured root-mean-square displacement (r.m.s.d.) of diffusing water as a function of the observation time. The work also underscores the importance of using an appropriate experimental observation time if q-space data are to be used to estimate compartment dimensions and interbarrier spacing, and illustrates an expeditious method for determining this value.

Simulations of Molecular Diffusion in Lattices of Cells: Insights for NMR of Red Blood Cells

The pulsed field-gradient spin-echo (PGSE) nuclear magnetic resonance (NMR) experiment, conducted on a suspension of red blood cells (RBC) in a strong magnetic field yields a q-space plot consisting of a series of maxima and minima. This is mathematically analogous to a classical optical diffraction pattern. The method provides a noninvasive and novel means of characterizing cell suspensions that is sensitive to changes in cell shape and packing density. The positions of the features in a q-space plot characterize the rate of exchange across the membrane, cell dimensions, and packing density. A diffusion tensor, containing information regarding the diffusion anisotropy of the system, can also be derived from the PGSE NMR data. In this study, we carried out Monte Carlo simulations of diffusion in suspensions of “virtual” cells that had either biconcave disc (as in RBC) or oblate spheroid geometry. The simulations were performed in a PGSE NMR context thus enabling predictions of q-space and diffusion tensor data. The simulated data were compared with those from real PGSE NMR diffusion experiments on RBC suspensions that had a range of hematocrit values. Methods that facilitate the processing of q-space data were also developed.

Permeability Coefficients from NMR q-Space Data: Models with Unevenly Spaced Semi-permeable Parallel Membranes

The NMR “q-space” experiment conducted on water provides information on the sizes of repeated structures on the micrometer-length scale in heterogeneous samples, including cell suspensions or tissues. Under some circumstances these plots display coherence peaks, and it has been implied theoretically that the position of the peaks will vary with the rate of molecular exchange across the membranes. This has been demonstrated (qualitatively) with human erythrocytes in suspension. Thus, in the quest for a quantitative approach to the interpretation of such data, we address here the “inverse problem,” namely the estimate of the permeability coefficient of membranes from q-space experiments. The present work describes theoretical predictions of q-space plots from molecules diffusing in a simple system of parallel semi-permeable membranes arranged with separations that alternate between two different values; this was designed to (loosely) mimic the intra- and extracellular compartments in a suspension of cells or a tissue. The development of the theory was facilitated by symbolic computation, and the analysis of synthetic data was shown to be achievable by the use of a three-layer back-propagation artificial neural network.

The Structure of Amorphous CrS3 Containing [CrIII2((S−I)2)3]x Chains: An X-Ray Diffraction Modeling Study

The amorphous chromium sulfide, CrS3, has no crystalline counterpart. A model is constructed to account for the interference function, Qi(Q), and the real space total correlation function, T(r), obtained from X-ray diffraction measurements. The structural unit (S−I)2/2CrIII((S−I)2))2CrIII(S−I)2/2 gives a good fit to the Q-space data over the range 2–20 Å−1 and explains features in T(r) to 3.6 Å. The unprecedented coordination geometry around chromium explains the unusual magnetic moment of the compound, corresponding to one unpaired electron per CrIII. The individual structural units are linked by the terminal disulfide groups to give [–Cr(S2)2Cr(S2)–]n chains. The flexibility of the single disulfide linkages within the chains and the difficulty of rearranging these chains to form an ordered structure after the rapid formation of CrS3 from solution explains the amorphous nature of CrS3.

Microstructural characterization of starch systems by NMR relaxation and q-space microscopy

Nuclear magnetic resonance (NMR) relaxation and q-space diffusion measurements have been used to probe the microscopic water distribution in a variety of starch-based systems, including packed beds of native starch granules with varying water contents, starch gels, and freeze-dried starch gels. The q-space data for the granular beds is compared with a variety of theoretical models and conforms best to unbounded diffusion in a lower dimensional space. In contrast to some earlier reports, the data for the gelatinized samples are not anomalous and conform to simple unrestricted diffusion in a three-dimensional space. The paper concludes with a novel method for probing pore size distribution in freeze-dried starch gels by infusion of acetone.

Measurement of pulsatile flow using MRI and a Bayesian technique of probability analysis.

This work shows that complete spatial information of periodic pulsatile fluid flows can be rapidly obtained by Bayesian probability analysis of flow encoded magnetic resonance imaging data. These data were acquired as a set of two-dimensional images (complete two-dimensional sampling of k-space or reciprocal position space) but with a sparse (six point) and nonuniform sampling of q-space or reciprocal displacement space. This approach enables more precise calculation of fluid velocity to be achieved than by conventional two q-sample phase encoding of velocities, without the significant time disadvantage associated with the complete flow measurement required for Fourier velocity imaging. For experimental comparison with the Bayesian analysis applied to nonuniformly sampled q-space data, a Fourier velocity imaging technique was used with one-dimensional spatial encoding within a selected slice and a uniform sampling of q-space using 64 values of the pulsed gradients to encode fluid flow. Because the pulsatile flows were axially symmetric within the resolution of the experiment, the radial variation of fluid velocity, in the direction of the pulsed gradients, was reconstructed from one-dimensional spatial projections of the velocity by exploiting the central slice theorem. Data were analysed for internal consistency using linearised flow theories. The results show that nonuniform q-space sampling followed by Bayesian probability analysis is at least as accurate as the combined uniform q-space sampling with Fourier velocity imaging and projection reconstruction method. Both techniques give smaller errors than a two-point sampling of q-space (the conventional flow encoding experiment).