Diffusion Acceleration with Gaussian process Estimated Reconstruction (DAGER)

Purpose Image acceleration provides multiple benefits to diffusion MRI, with in‐plane acceleration reducing distortion and slice‐wise acceleration increasing the number of directions that can be acquired in a given scan time. However, as acceleration factors increase, the reconstruction problem becomes ill‐conditioned, particularly when using both in‐plane acceleration and simultaneous multislice imaging. In this work, we develop a novel reconstruction method for in vivo MRI acquisition that provides acceleration beyond what conventional techniques can achieve. Theory and Methods We propose to constrain the reconstruction in the spatial (k) domain by incorporating information from the angular (q) domain. This approach exploits smoothness of the signal in q‐space using Gaussian processes, as has previously been exploited in post‐reconstruction analysis. We demonstrate in‐plane undersampling exceeding the theoretical parallel imaging limits, and simultaneous multislice combined with in‐plane undersampling at a total factor of 12. This reconstruction is cast within a Bayesian framework that incorporates estimation of smoothness hyper‐parameters, with no need for manual tuning. Results Simulations and in vivo results demonstrate superior performance of the proposed method compared with conventional parallel imaging methods. These improvements are achieved without loss of spatial or angular resolution and require only a minor modification to standard pulse sequences. Conclusion The proposed method provides improvements over existing methods for diffusion acceleration, particularly for high simultaneous multislice acceleration with in‐plane undersampling.

problematic. At lower field, acquisition of navigators is more feasible, incurring a slight loss of efficiency due to longer TR.

Supporting Information Discussion S3: GP hyper-parameter estimation
The hyper-parameters in the GP model are estimated from the data based on a Bayesian formulation. One subtle but important detail is that image aliasing, which the reconstruction aims to remove based on smoothness, has a detrimental effect on hyper-parameter estimation.
An iterative updating scheme is adopted in DAGER, where the hyper-parameter estimates and the image reconstruction are consecutively updated such that errors in both procedures are gradually suppressed over multiple iterations.
It is worth noting that this reconstruction scheme differs from classical model fitting procedure where the model is identical in each iteration. Instead, in DAGER reconstruction the model is refined in each iteration, which might lead to a non-monotonically decreasing cost-function. In our experiments, we find the reconstruction typically converges after 10-15 iterations and the model becomes relatively stable afterwards. This is demonstrated in simulation (Fig. 2), where the iteration converges after 13 iterations. However, under certain circumstances, the proposed algorithm might fail to converge, such as strong subject motion, severe eddy current distortion and a low number of diffusion directions. In future work, we aim to integrate the motion and distortion correction, which could aid in convergence under these conditions. Supporting Information Discussion S4: Effects of the number of diffusion directions on DAGER reconstruction As DAGER relies on the joint information between q-space neighbours, the number of diffusion directions is expected to affect the reconstruction performance. In this work, we found that DAGER reconstruction with 64 diffusion directions is very close to SENSE reconstruction, indicating low covariance between the available q-space samples (the average angular difference between each point and its nearest neighbour is about 20 degree). This result suggests that a relatively large number of directions are needed for DAGER reconstruction. However, given that DAGER is intended to support higher SMS acceleration factors, the burden of time required to reach a target number of directions goes down, leading to a synergy between the needs of the GP and the benefits it delivers. Here, we were able to achieve high quality reconstructions by acquiring 128 directions in ~7.5min. Additionally, once distortion correction is incorporated in DAGER, we may be able to exploit q-space symmetry (the equivalence of q-space samples on opposite sides of the sphere) to increase the number of 'effective' q-space neighbours. This will be investigated in future work.

Supporting Information Figures
Supporting Information Figure S1. Left: An L-curve shows the data fidelity term for different regularization parameters used in the reconstruction of in vivo dMRI data. From left to right, the images correspond to regularization parameters (λ) that are too small (noisy images), L-curve optimal and too large (aliased images). The regularization term at the "corner" (λ ( ) provides an acceptable compromise between these two error metrics, which is used for SENSE and SMS-SENSE reconstruction. To achieve minimal aliasing artefacts, a lower regularization parameter (λ & ) is used for phase error estimation and DAGER initialization. Right: SMS-SENSE reconstruction with different regularization parameters as shown in the L-curve.
Supporting Information Figure S2. Effects of k-q sampling and phase error correction on DAGER reconstruction of simulated data. Two data sets are simulated with R=6: one containing no phase errors and acquired using a fixed k-q sampling (k-space sampling pattern is identical for all q-space points); the other containing phase errors and acquired using the variable k-q sampling (different k-space sampling patterns are used within local neighborhood in q-space). Top row: the data set with fixed k-q sampling but without phase errors is reconstructed using SENSE and DAGER. Bottom row: the data set with k-q sampling and simulated phase errors is reconstructed using DAGER with and without phase suggesting that the reconstruction fidelity is not overly sensitive to the estimate of this hyperparameter, particularly if the true noise variance is low (i.e. SNR=30 and 60). For low SNR data, deviation from the true variance by more than a factor of ~3 can lead to a sharp increase of reconstruction errors (i.e. SNR=6 and 10). Most importantly, the DAGER enables noise variance estimates (solid circles) that provide similar or better NRMSE as compared to the true noise variance, enabling a robust reconstruction.
Supporting Information Figure S4. Reconstruction of in-plane under-sampled data acquired from subject 1. The R=6 data is reconstructed with SENSE and DAGER (with and without navigator acquisition).
Supporting Information Figure S5. Supporting Information Figure S7. Reconstruction with 64 diffusion directions using SMS-SENSE and three SMS-DAGER configurations, where smoothness hyper-parameter a is estimated from the data (0.7) and manually set to 1 and π, respectively. Supporting Information Table S1. Analysis of fiber orientation estimations based on the SB-1ave data and the SMS-DAGER data. The SB-2ave data is used as a reference here. The absolute angular difference tells how much SB-1ave or SMS-DAGER data deviate from the reference in the estimation of fiber orientations. The deviations for these two data sets are compared and tested.