Joint K-space Trajectory and Parallel Imaging Optimization for Auto-calibrated Image Reconstruction

Stephen Cauley^{1,2}, Kawin Setsompop^{1,2}, Berkin Bilgic^{1}, Himanshu Bhat^{3}, Borjan Gagoski^{2,4}, Thomas Witzel^{1,2}, and Lawrence L. Wald^{1,2,5}

Knowledge of gradient trajectories is critical for the accurate reconstruction of images for many fast MRI acquisitions. Several non-Cartesian acquisition schemes, e.g. Spiral[1], Bunched Phase Encode[2], Wave-CAIPI[3] imaging, often require separate pre-scans to estimate the trajectory for a given protocol. There have been attempts to fully model the gradient behavior using system theory [4,5], but they need to be re-tuned and are approximations based upon the system model. In this work, we apply the concept of joint optimization in order to find the best model parameters that describe the non-ideal trajectory along with the corresponding image to be reconstructed through parallel imaging. The optimization considers the quality of both the trajectory and the image simultaneously in order to ensure robust reconstruction.

SENSE [6] parallel imaging minimizes $$$\sum_{i=1}^N \|| F C_{i} x - k_{i} \||_{2}$$$, where F is a Fourier operator that describes the trajectory, $$$C_{i}$$$ is a parallel receive channel, and $$$k_{i}$$$ is the observed k-space data, and x is the image. In the case of trajectory errors F can be replaced with F(t), where the mapping to k-space is now a function of gradient parameters t. Here, the objective is to minimize $$$\sum_{i=1}^N \|| F(t) C_{i} x - k_{i} \||_{2}$$$ across both x and t [7,8]. There are a variety of non-linear least squares techniques that can be utilized to solve this joint problem, e.g. levenberg-marquardt, trust-regions, etc. However, these algorithms can be computationally prohibitive when the number of parameters grows and may suffer from convergence issues. We exploit reduced models to make this joint optimization computationally practical and ensure accurate reconstruction.

We demonstrate our model reduction scheme for the Wave-CAIPI method that utilizes sinusoidal gradient trajectories to efficiently encode k-space. Fig. 1 shows the gradient diagrams, the k-space coverage and image aliasing pattern for Wave-CAIPI. Modeling of this aliasing pattern, i.e. the point spread function (PSF) resulting from the sinusoidal gradients, is critical for accurate reconstruction. As shown in [3], a full pre-scan can be used to accurately estimate the PSF, in the presence of gradient and system imperfections, for a fixed FOV and set of protocol parameters. However, this single measurement does not generalize to different orientations and protocols that would arise in practice.

Fig. 2(top) illustrates the data consistency model for accelerated Wave-CAIPI data. Fig. 2(middle) shows a PSF from a full pre-scan (corresponding to a fixed FOV/protocol). Note that only a sparse set of Fourier coefficients accurately capture the PSF and represent a good basis for PSF estimation. Fig. 2(bottom) illustrates image-space test locations that could be selectively reconstructed based upon iterative changes to the PSF estimate. The computational cost of this operation is much smaller than a complete reconstruction, facilitating efficient optimization of the trajectory parameters. This concept of selective reconstruction can also be extended to sampling patterns that fully couple image-space (e.g. Spiral), through the use of domain decomposition iterative methods [9].

Fig. 3(top-left) shows the smooth change in data consistency RMSE across 0.35% of the imaging voxels, as the Fourier coefficients used to describe the PSF are manipulated. We employ an efficient multi-pass greedy search in order to reduce the RMSE. Fig. 3(top-right) shows reconstruction of a uniform brain phantom imaged on a 3T Siemens Skyra. Here, joint optimization across less than 1% of the voxels achieved lower RMSE than the full pre-scan approach. An additional benefit of the sparse modeling/optimization of the PSF is the denoising effect on the final parallel imaging encoding matrix (see Fig. 3-bottom). This allows for the use of a sparse approximate pre-conditioner to guide the iterative reconstruction efficiently toward the final image.

Fig. 4 shows online reconstruction results of a 3D GRE Wave-CAIPI acquisition at 9x acceleration on a 3 T Siemens Skyra, with imaging parameters: 1mm iso, 256x192x120mm3 FOV, R=3x3, TR=43ms, TE=9, 22, and 35ms, and total acquisition time of 113s. The magnitude, phase, echo combined magnitude and a SWI image averaged across 8mm are provided. In this clinically relevant acquisition, the joint optimization converged in only 30s, and the efficiency of our sparse pre-conditioner allowed for each of the 3 echoes to be reconstructed in 26s (>3x faster than standard iterative CG).

Figure 1: (top) Wave-CAIPI gradient diagram and associated k-space coverage are illustrated. Image-space aliasing across 6x over-sampled readout is shown (bottom).

Figure 2: (top) Data consistency forward model for Wave-CAIPI that relates underlying image, PSF phase scaling, and collapsed data. (middle) Sparse frequency coefficient modeling of PSF for reduced model optimization. (bottom) Selective parallel imaging reconstruction for comparing data consistency associated with iteratively updated PSF estimate.

Figure 3: (top-left) Smooth variation in RMSE across small subset of voxels (< 1%) when varying PSF estimate, clear local minima are observed. (top-right) Comparison of final parallel imaging reconstruction quality using full pre-scan and reduced model joint optimization. (bottom) Denoised parallel imaging encoding matrix that enables sparse approximation for fast pre-conditioned iterative reconstruction.

Figure 4: Online reconstruction of 9x accelerated 3D-GRE
Wave-CAIPI data with imaging parameters: 1mm iso, 256x192x120mm^{3} FOV, R=3x3, TR=43ms, TE=9, 22, and 35ms. (left) Magnitude and phase images from each of the 3 echoes. (right) Weighted echo combination magnitude and a SWI image averaged across 8mm.

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

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