Accelerated diffusion spectrum imaging in the human

Title: Accelerated diffusion spectrum imaging in the human brain using compressed sensing Authors: 5

Marion I. Menzel (1), Ek T. Tan (2), Kedar Khare (2), Jonathan I. Sperl (1), Kevin F. King (3), Xiaodong Tao (2), Christopher J. Hardy (2), Luca Marinelli (2)

(1) GE Global Research, Munich, Germany. 10

(2) GE Global Research, Niskayuna, NY, USA. (3) GE Healthcare, Waukesha, WI, USA.

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Corresponding author: Marion I. Menzel GE Global Research Freisinger Landstr. 50 D-85748 Garching b. München

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Germany [email protected] Tel. +49 (0)89 55283-730 Fax. +49 (0)89 55283-180

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word count: 3132 30

Running head: Accelerated DSI by compressed sensing

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ABSTRACT We developed a novel method to accelerate diffusion spectrum imaging (DSI) using compressed sensing (CS). The method can be applied to either reduce acquisition time of DSI acquisition without losing critical information or to improve the resolution in diffusion space without 35

increasing scan time. Unlike parallel imaging, compressed sensing can be applied to reconstruct a sub-Nyquist sampled dataset in domains other than the spatial one. Simulations of fiber crossings in 2D and 3D were performed to systematically evaluate the effect of CS reconstruction with different types of undersampling patterns (random, Gaussian, Poisson disk) and different acceleration factors on radial and axial diffusion information. Experiments in brains

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of healthy volunteers were performed, where diffusion space was under-sampled with different sampling patterns and reconstructed using CS. Essential information on diffusion properties, such as orientation distribution function (ODF), diffusion coefficient, and kurtosis is preserved up to an acceleration factor of R = 4.

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KEYWORDS compressed sensing, q-space, diffusion spectrum imaging, kurtosis, undersampling, orientation distribution function

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INTRODUCTION 50

Over the last decade the application of diffusion weighted MR imaging to the central nervous system has gained significant attention. Recently, Inglese et al. (1) reviewed the importance of diffusion in clinical evaluation of multiple sclerosis. Similarly, earlier studies indicated that diffusion tensor imaging (DTI) could be used to detect evidence of traumatic brain injury (2). DTI samples only a very small subset of the full diffusion information encoded in q-space and

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describes diffusion as single compartment Gaussian (3). This assumption however fails short for instance in fiber crossings or in biological tissue (4), which may exhibit restricted, non-Gaussian diffusion. The concept of full q-space imaging to study molecular diffusion and tissue microstructure, was introduced by Callaghan et al. (5) and first applied to brain tissue by King et al. (6); its modulus Fourier transform variant is known as Diffusion Spectrum magnetic

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resonance Imaging (DSI) (7). DSI does not assume any underlying model, i.e. it is model free and samples the full q-space. Despite the large information content of DSI, its high dimensionality (3 dimensions in the spatial domain (k-space) and 3 dimensions in the q-space) leading to very long acquisition times, severely limited its clinical application in-vivo. And indeed the application of DSI has been reported only a few times in biological systems (6,8),

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although the non-localized analysis of q-space is commonly employed in porous media (9). It can however be envisioned that utilizing the full potential of diffusion information of full q-space to derive and evaluate surrogate markers for multiple sclerosis MS and traumatic brain injury would add significant clinical benefit and indeed more extended sampling of diffusion space (10) was reported useful in multiple sclerosis.

70 Common techniques of image acceleration based in k-space such as parallel imaging are not applicable to accelerate DSI acquisition. In DSI usually a single-shot echoplanar readout is employed for the imaging component of the pulse sequence. Parallel imaging may be used to reduce the geometric distortion of the echoplanar readout, however not to reduce acquisition time 75

or to accelerate in the diffusion domain. To overcome the limitation of impractical scan time of full DSI for routine clinical use, we propose to accelerate DSI acquisition by using the technique of compressed sensing (11,12) on the three q-space dimensions, without changing the k-space acquisition. With the combination of DSI and CS (13,14), a conceptually different approach to assess diffusion information in a clinically relevant time is proposed. This approach does not

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assume any underlying model for diffusion, i.e. it is model free and reconstructs the full q-space from under-sampled acquisitions. To evaluate the accuracy of compressed sensing and to study its applicability to in-vivo DSI applications, the effect of CS on derived metrics of diffusion, such as ODF, diffusion or kurtosis (15) is studied. Using different under-sampling schemes, with acceleration factor R ranging from 2 to 8, the effect of CS for DSI on simulated fiber crossings

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and on the brains of healthy volunteers is investigated. The results of CS reconstructed undersampled data sets are validated by comparison (correlation) to the corresponding fully sampled dataset (ground truth) with respect to ODF (angular information) and kurtosis (radial information).

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THEORY Following the Stejskal-Tanner analysis (16) and using the kurtosis expansion (15), the MR signal S(b) for a diffusion experiment is expressed as a function of b-values along a given gradient g with direction n: 3 3 3 3 3 3 1 ln[S (b)] = ln[S0 ] - båå ni n j Dij + b 2 D 2 åååå ni n j nk nlWijkl 6 i =1 j =1 i =1 j =1 k =1 l =1

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(Eqn. 1)

where S0 is the signal in the absence of diffusion encoding gradients g. This equation has a diffusion tensor term (linear in b), which spans a symmetric 3x3 matrix of the diffusion tensor Dij (6 independent elements, positive definite) and a kurtosis tensor term (quadratic in b), which spans a symmetric 3x3x3x3 matrix Wijkl (15 independent elements). Let q=gdg/2p be the wave vector in q-space, the amplitude of the MR signal S(q) is related to the spin displacement r by

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the Fourier transform F of the diffusion propagator P(r) (17):

S (q) = S 0 ò P(r )e iq r d 3r = S 0 F [P(r )] T

(Eqn. 2)

In Eq. 2, D (diffusion mixing time)-dependence is omitted for simplicity. The diffusion propagator in q-space can be analyzed in angular and radial terms. A common representation of the angular information of the diffusion propagator, which reflects the underlying tissue 105

anisotropy, is the orientation distribution function (ODF) (7). ODF is computed as the integral over a set of weighted radial projections of the (magnitude of the) diffusion propagator P(r):

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ODF (u) = ò P( ru) r 2 dr

(Eqn. 3)

as a function of direction unit vector u (the weighting factor being the radial component of the Jacobian in spherical coordinates). The ODF inherently retains only the angular component of 110

the diffusion propagator in each voxel, while averaging the weighted radial dependence. An approximation of this angular function can be measured using methods like HARDI (18), HYDI (19) or q-ball imaging (20), which represent subsets of q-space imaging with acquisition on one spherical single shell (HARDI) or multiple concentric spherical shells (HYDI) instead of a full Cartesian grid. The radial component of the diffusion propagator encodes information about the

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physical nature of the diffusion process, which can be analyzed according to a variety of models (such as Gaussian (3), bi-exponential (4), higher order tensors (21), kurtosis (15)). In the long D limit, the information encoded in the radial part of the diffusion propagator is the underlying structure pattern, which was described as diffusion-diffraction (22) for regularly spaced layers. Morphologically, biological tissue is assumed to be an ensemble average over differently sized

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and shaped cells embedded in extra-cellular space, ranging between free water and diffusion diffraction (23). In a first approximation, the signal in q-space can be regarded as Gaussian distributed, which admits an approximately sparse representation in an appropriate domain (i.e. wavelet or total variation domain). Compressed Sensing reconstruction is formulated as a problem of minimizing

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a cost function consisting of a mean squared error term for data consistency and one or more sparsity constraints in an appropriate transform domain (e.g. wavelets, Total Variation, etc.). In an iterative framework, a fast converging Nesterov-type updating scheme (24) was used for minimizing the data consistency term along with wavelet threshholding and a Total Variation reducing step. As described elsewhere (25) the implementation employed uses standard adaptive

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image denoising approaches for imposing sparsity constraints on the solution at each update, and does not require manual tuning of weights associated with the sparsity terms. The iterative process is stopped when the resultant image solution does not change by more than 0.1% in L2norm in consecutive iterations. Typically 15-20 iterations are required for reconstructing the qspace for each voxel.

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METHODS Simulations in 2D/3D Simulations of 2D and 3D diffusion propagator data for crossed fibers were performed using 140

Matlab (Mathworks, Natick, MA, USA). The displacement probability density S for a two-fiber crossing was simulated with a sum of two separate Gaussian propagators, assuming no diffusional exchange between the crossing fibers, a fractional anisotropy (FA) of 0.85 and typical gradient performance of a clinical MRI system (gmax = 40 mT/m). The eigenvalues and orientation of diffusion for both fibers were chosen such that a 70° crossing was realized; rx, ry,

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rz denote the Cartesian components. The displacement probability density S for a single fiber according to the sum of Gaussian model was simulated as

ry2 rx2 rz2 1 S= Dxx D yy D yzz exp{-( + + )} 4pD 4 Dxx D 4 D yy D 4 Dzz D

(Eqn. 4)

on a 33x33 (2D) and 11x11x11 (3D) q-space grid. Subsequent summation of the two fiber signals S1 and S2 with equal weights was performed according to 150

(Eqn. 5) The simulation parameters were: mixing time Δ = 66 ms, diffusion gradient duration d = 60 ms, Dxx = 2.00 x 10-9 m2s-1, Dyy = Dzz = 0.21 x 10-9 m2s-1, and qmax = gdgmax/2p; samples in q-space 33x33 / 11x11x11, which corresponds to bmax = 18.962 s mm-2.

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q-space under-sampling schemes Using the simulated fiber crossing, a set of different under-sampling schemes was tested in combination of CS (25) and compared to fully sampled data as ground truth. In the 2D simulations, three different schemes for under-sampling of q-space with acceleration factors up to R = 8, were employed. These were uniform random distribution, Poisson disk (a uniform

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random distribution with specified minimal distance constraint between two samples (26)) and Gaussian. Contrary to the first two schemes, the random Gaussian distribution samples the center of q-space more densely. To ensure the fairness of the comparison, the central 5x5 square (3x3x3 cube in 3D) was always fully sampled. Samples were only allowed within a fixed radius in qspace up to the maximum b-value, which implies that the corners of the q-space were never

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sampled. For a comparison of exemplary under-sampling schemes see, Fig. 1. For each of the

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undersampling patterns, the effect of the under-sampling and subsequent CS reconstruction on the ODF was compared to the ground truth for a 70° fiber crossing (Fig. 2). Reproducibility/Repeatability 170

Compressed Sensing requires incoherent aliasing, mostly achieved by random undersampling. For very small grids however, ensuring incoherence is not a trivial task (27). To evaluate a potential dependence of CS reconstruction performance and quality on the particular instance of the stochastic/random sampling pattern, 1000 different instances of Gaussian sampling patterns with identical parameters were evaluated with respect to their performance in q-space

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reconstruction and retention of maxima of the ODF. To this end, the pairwise linear correlation of q-space (and ODF respectively) for ground truth and CS reconstructed data was compared for every voxel. Sampling in q-space

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The number of samples in q-space is determined by two limiting factors: the Nyquist sampling rate and the image acquisition time. We selected bmax = 10.000 s mm-2 for experiments in volunteers to ensure proper signal decay at the edges of q-space as to avoid Gibbs ringing in the displacement domain, which would affect ODF calculation. To avoid aliasing artifacts at the edges of displacement space, fast diffusing species must be sampled densely enough. Such

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aliasing artifacts are most likely to appear at brain locations dominated by cerebrospinal fluid, which are of limited interest in this study. Therefore the number of q-space samples and sampling density was optimized for gray and white matter. Real DSI experiments / Measurement Protocol

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DSI experiments on healthy volunteers were performed using a 3T GE MR750 clinical MR scanner (GE Healthcare, Milwaukee, WI, USA) which is capable of a max gradient strength of 50 mT/m; equipped with a GE 8-Channel brain phased-array coil. Images were acquired using a single-shot echo planar imaging sequence (TE =141 ms, TR = 3 s, 128x128, FOV = 25 cm, slice = 4 mm, b-max = 10,000 s/mm2). For each voxel, q-space was sampled on Cartesian grid points

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within a 3D sphere with diameter of 11 or 17 samples. For the small q-space coverage (i.e. 11x11x11 grid), we collected data sets both full sampled and under-sampled with a random

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Gaussian distribution with an acceleration factor of R=4. The image acquisition time was 26 minutes and 6.42 minutes, respectively. For the large q-space coverage (i.e. 17x17x17 grid), we only acquired a Gaussian under-sampled data set with an acceleration factor of R=4. The image 200

acquisition time was 26 minutes.

Data processing and visualization Fully sampled datasets were reconstructed using the modulus Fourier transform (Eqn. 2), while under-sampled datasets were reconstructed using compressed sensing with a sparsity constraint 205

in the wavelet domain (using Matlab’s commercially available 2D wavelet package) in 2D and with a minimum Total Variation constraint in the 3D case, including the volunteer data (the total variation approach is computationally more efficient in 3D and no significant variations to wavelets were observed). ODFs were calculated according to Eqn. 3 with zero-filling prior to FFT and 3D Gaussian-weighted interpolation for the spherical, radially weighted projections.

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After min/max normalization, ODFs were visualized as colored glyphs, with conventional color scheme where red, green, and blue indicate diffusion maximum in the directions of left-right, anterior-posterior, and superior-inferior, respectively. Diffusion tensors Dij, and kurtosis tensors Wijkl, were computed by fitting the fully sampled data and the CS reconstructed data to Eqn. 1. Different approaches for sequential fitting of diffusion

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and kurtosis using linear, weighted linear, and non-linear approaches were considered. For computational efficiency, a linear approach with constraints was used (implemented in Matlab). Furthermore, the validity of kurtosis as a Taylor expansion in b-values reaches its limits with bvalues reported as ranging between 3,000 s mm-2 (28) and even 8,000 s mm-2 (29). Empirically it was found that limiting the kurtosis fit to bmax = 7,000 s mm-2 yielded best results. Diffusion

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tensor fitting was always limited to the central and fully sampled 3x3x3 cube (5x5 square respectively). Based on diffusion tensor and kurtosis tensor fitting, diffusion eigenvalues, diffusion invariants as well as D-, Kelvin-eigenvalues and principal invariants for kurtosis were calculated according to (30). For clarity of interpretation, only orthogonal (K⊥) and parallel (K║) kurtosis, which share the coordinate system with the diffusion eigenvalues, are presented here.

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RESULTS Simulation In Fig. 2 the results of the simulated fiber experiments are presented. Different under-sampling 230

schemes (Fig. 1) were compared qualitatively and the CS reconstruction performance on the reconstructed ODFs was evaluated. We noticed that the main features of ODFs were retained for acceleration factors smaller than 4. At higher accelerations, the shape of the ODFs gradually degraded, leading to ambiguities in assigning major orientations of diffusion and number of fiber bundles per voxel. The comparison of different under-sampling schemes (uniform random,

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Poisson disk, and Gaussian) revealed that for q-space data sets, which have nearly Gaussian shape, the Gaussian under-sampling scheme performed best. In Fig. 3 simulations in 3D (11x11x11) are presented which lead to similar observations. Even with the small 3D sampling grids, Gaussian undersampling and subsequent reconstruction using CS resulted in acceptable estimation of ODF maxima for acceleration factors up to R=4.

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Alternatively, it was demonstrated that a larger sampling grid (17x17x17) with acceleration factor R=4 could be used to improve separation of ODF minima and maxima and therefore higher angular resolution, which would allow us to distinguish fiber bundles intersecting each other at shallower angles.

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Brain data The experiments on healthy volunteers confirmed our findings with the simulated data. Acquisition of fully sampled (11x11x11 grid, 515 points in q-space) and under-sampled DSI datasets (11x11x11 grid, 128 points with R=4) allowed for a direct comparison of the fully sampled and the under-sampled reconstructions. The comparison of the ODF for both fully

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sampled (treated as ground truth) and under-sampled CS reconstructions revealed that essential features of the ODFs are retained for an acceleration factor up to R = 4. As we increase the acceleration factor, the peaks of the ODFs, corresponding to the principal directions of the local diffusion, gradually diminish and side peaks caused by noise and reconstruction errors become more apparent. From Fig. 4, we can also see that larger q-space coverage (17x17x17) allows

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better separation of fiber bundles intersecting at shallow angles. For particularly noisy voxels, ODFs after CS reconstruction appear smoother than the corresponding ground truth.

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To avoid preference of CS reconstruction to a particular orientation of fibers in brain, also the uniformity of the correlation coefficients for the whole brain was regarded. Fig. 5 depicts the voxel-by-voxel correlation of fully sampled q-space (treated as the ground truth) versus a CS 260

reconstruction of the identical data, artificially under-sampled prior to CS reconstruction. 1000 different patterns were evaluated. Based on the correlation coefficient in the corpus callosum regions of interest, two instances were chosen as examples of particularly good sampling (Fig.5 (a)) and one less desirable sampling pattern (Fig. 5 (c)). A pattern was considered preferable, if it resulted in a high and spatially homogeneous correlation. Direct comparison of Fig. 5 (a) and (c)

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reveal that pattern (a) results in higher correlation coefficients most pronounced for the corpus callosum region, but also overall for the white matter, as compared to pattern (c). The standard deviation map, which describes the deviation between fully sample ground truth and CS reconstructed data for all 1000 patterns, is depicted in Fig. 5 (d). It reveals that larger standard deviations, which imply low correlation between ground truth and CS reconstructed data, are

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preferably found in regions of the brain that exhibit highly oriented structures, such as in the corpus callosum region. These regions systematically depict lower correlation coefficients than isotropic gray matter, which implies that for a good CS reconstruction of such oriented structures the choice of undersampling pattern instance is more critical. Next, the effect of CS reconstruction (using the identified “good” and “less desirable” patterns)

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on kurtosis fitting was evaluated. Based on that, scalar invariants were calculated and compared for different instances of the undersampling pattern. Fig. 6 depicts the comparison of orthogonal (K⊥) and parallel kurtosis (K║) for both fully sampled and CS reconstructed cases: most notably, the orthogonal kurtosis, which by definition is orthogonal to the direction corresponding to the largest eigenvalue, is lower for the CS reconstructed cases as compared to the fully sampled; this

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effect is not noticeable for parallel kurtosis.

DISCUSSION AND CONCLUSION We proposed a novel method for accelerating Diffusion Spectrum MR Imaging using compressed sensing for reconstruction of sub-Nyquist sampled q-space. Simulations and 285

experiments on human volunteers demonstrated that random Gaussian undersampling in q-space with CS reconstruction can be applied to either reduce acquisition time of DSI acquisition without losing critical information or to improve the resolution in diffusion space without

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increasing scan time. We could demonstrate that both axial (ODF) and radial (kurtosis) metrics, used to evaluate local diffusion properties of tissue can be retained with acceptable accuracy for 290

an acceleration up to factors of R=4. For both metrics a slight denoising effect in the CS reconstructed data was observed. However the apparent underestimation of the orthogonal kurtosis in the CS reconstructed case could also be a consequence of overestimation of kurtosis of the noisy ground truth data. It is therefore difficult to separate unambiguously the effect of CS reconstruction on kurtosis properties from

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potential fitting errors. Especially for low SNR data fitting errors could be the dominant component in evaluation of CS performance. Furthermore, kurtosis fitting is performed on magnitude data, and the Rician character of the noise may artificially result in a larger kurtosis component for the ground truth data. As a consequence for the present work it is difficult to identify the true source of deviation between ground truth and CS reconstructed data.

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We demonstrated that CS can be employed to shorten the acquisition times of DSI to an extent, which could be tolerated by volunteers and potentially patients. Shortening DSI acquisitions significantly by means of CS would open up the door to new contrasts that are truly based on underlying tissue properties. In future, we plan to apply the technology developed in this work to clinical applications such as traumatic brain injury and multiple sclerosis to prove the value of

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the technology in a clinical setting.

ACKNOWLEDGEMENT The authors thank Dimitrios Karampinos and Van J. Wedeen for helpful discussion.

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FIGURES

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Fig. 1: Exemplary 2D undersampling schemes (33x33 samples), central 5x5 cube fully sampled: Upper row (R = 2.3) a) Gaussian; b) Poisson disk (min. dist = 0.9); c) uniform random. Lower row (R = 4); d) Gaussian; e) Poisson disk (min. dist = 1.3); f) uniform random. All schemes sample equal number of points for the corresponding acceleration.

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Fig. 2: ODF of simulated fiber crossing (2D, 70° crossing, noiseless case, 33x33), under-sampled with different patterns and acceleration factors; reconstructed using CS. 1000 instances of each undersampling scheme were tested. Black: ground truth ODF (fully sampled), blue: mean ODF after reconstruction from 1000 different patterns, red: ODF for individual instances with correlation coefficients to fully sampled larger than specified threshold: a) uniform random R=2.3 (correlation coefficient > 0.9); b) Poisson disk R=2.3 (correlation coefficients > 0.93); c) Gaussian patterns R=2.3 (correlation coefficients > 0.99); d) Gaussian patterns R=4 (correlation coefficients > 0.90), e) Gaussian patterns R=6 (correlation coefficients > 0.85); f) Gaussian patterns R=8 (correlation coefficients > 0.80) Gaussian undersampling schemes perform best (compare a) - c)) in retention of ODF maxima and result in higher correlation coefficients than uniform random and Poisson disk patterns, compared to ground truth ODF. Gaussian random undersampling patterns retain ODF maxima for acceleration factors up to R=4 (compare d) – f)).

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Fig. 3: Rendering of ODF for simulated fiber crossing (3D, 70° crossing, 5% noise): a) fully sampled (11x11x11, 515 q-space samples, equiv. scan time: 26 min); b) under-sampled and CS reconstructed (11x11x11, R=4, Gaussian undersampling, 128 q-space samples, equivalent scan time: 6:42 min); c) fully sampled (17x17x17, 2060 q-space samples, equiv. scan time: >2hr); d) under-sampled and CS reconstructed (17x17x17, R=4, Gaussian undersampling, 515 q-space samples, equiv. scan time: 26 min). Compared to a) depicting the ground truth, the undersampled case b) still allows the identification of the two prevalent fiber orientations, even if slightly less pronounced in the accelerated case. With the denser sampling pattern c) ODF minima and maxima are more pronounced, which is retained in the accelerated case d).

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Fig. 4: Normal subject brain: a) DTI visualization; b) fully sampled dataset (11x11x11, 515 qspace samples, scan time: 26 min); c) under-sampled and CS reconstructed (11x11x11, R = 4, Gaussian, 128 q-space samples, scan time: 6:42 min); d) under-sampled and CS reconstructed (17x17x17, R = 4, Gaussian, 515 q-space samples, scan time: 26 min).

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Fig. 5: Evaluation of voxel wise correlation of q-space data (fully sampled ground truth vs. under-sampled (R=4) and CS reconstructed) with different instances of the undersampling pattern: a) good pattern; b) mean of correlation of all 1000 patterns; c) bad pattern; d) standard deviation of correlation of all patterns.

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Fig. 6: Comparison of fully sampled ground truth vs. under-sampled (R=4) and CS reconstructed dataset for orthogonal (K⊥) and parallel (K║) kurtosis. a) K⊥ ground truth; b) K⊥ with R=4, CS reconstructed; c) K║ ground truth; d) K║ with R=4, CS reconstructed. CS reconstruction leads to a slight underestimation of K⊥ while for K║ CS reconstruction has very little effect.

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