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Making Reconstruction WORK (Weighted Optimized Reconstruction of K-Space): Improving CNR/SNR via non-FFT Weighted Reconstruction
Vincent Jerome Schmithorst1 and Ashok Panigrahy1

1Radiology, UPMC Pittsburgh Children's Hospital, Pittsburgh, PA, United States

### Synopsis

FFT-based reconstruction is suboptimal in the presence of signal decay during acquisition and between-excitation (shot-to-shot) variance in relaxation parameters. We present a novel reconstruction algorithm, Weighted Optimized Reconstruction of K-space (WORK), which weights each k-space point differently, optimizing for all sources of variance. Simulation results demonstrate the potential for 2X temporal SNR improvement in gradient-echo EPI acquisitions compared to standard FFT reconstruction while preserving spatial information. Substantial SNR improvement is also demonstrated for a pCASL 2D GE-EPI-SMS acquisition.

### Introduction

While MRI has been greatly improved via improved hardware and optimization of pulse sequences, limited attention has been paid to optimization of reconstruction algorithms. However, FFT-based methods are not optimal in the presence of signal decay during acquisition and “shot-to-shot” (between-excitation) variance in relaxation parameters such as R1, R2, or R2*, or when the relaxation parameter itself is the variable of interest. With improvements in hardware leading to reductions in thermal noise, temporal or image SNR is often dominated by variance in relaxation parameters. We present the theory behind WORK (Weighted Optimized Reconstruction of K-Space) and the example of 2-D SMS-GE-EPI applied to a PCASL acquisition. WORK produces substantial CNR gains with minimal additional computational demand.

### Theory

We formulate the reconstruction problem as follows. Assume true MR intensities $V_{i}$ for each excited voxel i. The acquired k-space data $k_{j}$ is given via the forward transform $k_{j} = V_{i} A_{ij}$. The forward transform incorporates signal relaxation, phase encoding gradients, and all other gradients such as SMS (Blipped-Caipi) gradients [1]. The key to WORK is that the voxel intensities are real-valued at TE = 0, which means the inverse problem may be solved in the real domain as opposed to the complex domain. For a given voxel k, we construct a linear estimator $\hat{V}_{k} = k_{j} B_{j} = V_{i} A_{ij} B_{j} = V_{i} C_{i}$ where $C_{i}$ is the contribution from each voxel. For an unbiased estimator we constrain $C_{i} = \delta_{ik}$. We then carefully model all sources of noise, including shot-to-shot variance in relaxation parameters from all voxels. We use an FFT-based algorithm for correcting for voxelwise B0 inhomogeneity, as the ΔB0 is proportional to the phase of the FFT-reconstructed voxel.

### Materials and Methods

Simulation data was generated using routines in IDL. Parameters were taken to be comparable with a 64-X-64 2-D GE-EPI acquisition: dwell time = 0.5 ms, R2* = 28 Hz, ΔR2* = 0.7 Hz, ΔB0 = 0.1 Hz, thermal SNR = 500.

PCASL data was acquired on a normal adult volunteer using 2D GE-EPI with 4 X SMS acceleration on a Siemens 3T Skyra system. A FFT was performed and the slices unaliased. PCASL contrast (label-control) was computed. An inverse FFT was performed and the WORK algorithm performed using the k-space data. The WORK algorithm included line-by-line ghost correction and ΔB0 correction. pCASL contrast was again computed from the WORK reconstructed images.

### Results and Discussion

The simulation results (Figures 1, 2) show that approximately 2X CNR improvement is available using the optimized WORK reconstruction. For the PCASL acquisition, there is excellent agreement between the pCASL contrasts (Figure 3), while the WORK technique displays substantial CNR gain (Figure 4). This implementation of WORK used the simulation values for R2* and ΔR2* for each voxel. Further improvement is evidently available via estimation of R2 and ΔR2* on a voxelwise basis.

Thus, WORK can provide substantial gain for 2D GE-EPI acquisitions with minimal computational cost. While background suppression is typically used to suppress R2* variation during a PCASL acquisition, our results show that substantial improvement is available even without background suppression, and our clinical experience in children shows that background suppression is less robust to patient motion.

This implementation of WORK is simplified such that only image data (magnitude/phase) need be acquired of the scanner instead of the raw data. This is the case for SMS acquisitions and also undersampled in-plane acquisitions, as the unaliasing occurs in the spatial and not the time dimension, so it may be done in a separate step.

The WORK algorithm may be extended to anatomical (e.g. non-dynamic) multi-shot acquisitions where there is shot-to-shot variance in relaxation parameters, such as T2-weighted FSE or T1-weighted MPRAGE sequences, as well as single-voxel spectroscopy and spectroscopic imaging. It may also be extended to sequences such as BOLD-EPI where the relaxation parameter is in the fact the variable of interest; in this case, the optimized reconstruction will more heavily weight the later-acquired datapoints. Further research will investigate the application of WORK to magnetic resonance fingerprinting (MRF) applications [2], as shot-to-shot parameter relaxation variance may also greatly affect the results.

### Conclusion

Standard FFT-based reconstruction techniques are suboptimal in the presence of shot-to-shot variation in relaxation parameters. We introduce an optimized reconstruction technique, WORK, which can provide substantial improvement for a variety of anatomical and dynamic sequences and demonstrates 2X CNR improvement for 2D GE-EPI acquisitions. WORK will likely become more relevant at higher field strength and with improvements in scanner hardware [3] and image (thermal) SNR.

### Acknowledgements

No acknowledgement found.

### References

[1] Setsompop K et al., Magn Reson Med 2012; 67(5): 1210-24.

[2] Ma D et al., Nature 2013; 495: 187-192.

[3] Fan Q et al., Brain Connect 2014; 4(9): 718–726.

### Figures

Figure 1. As proof-of-concept, a 64-point FID is simulated without any phase encoding (1000 repetitions). Left: Correctly modeling the covariance due to shot-to-shot R2* variance results in an estimator highly weighted towards the earlier time points which produces an approximately 2X increase in temporal SNR. Simulation repeated with misestimations of R2* and ΔR2*; Middle: results are quite robust to underestimation of R2*; Right: results are robust to overestimation of DR2*.

Figure 2. A 64-point k-space acquisition is simulated with 40 voxels with magnitudes between 0.8 and 1.2. Top Left: The WORK algorithm (dot-dashed line) results in an almost 2X increase in temporal SNR compared to a standard FFT (solid line: deconvolved to account for spatial blurring due to signal decay; dashed line: not deconvolved FFT); Top Right: the ground truth voxel intensities are correctly recovered (expected since the WORK estimator is unbiased); Bottom Left: The WORK estimator (20th voxel; real component) is again weighted to earlier time points; Bottom Right: Improvement available from WORK varies approximately linearly with image SNR.

Figure 3. Comparison of pCASL contrast from standard FFT-based reconstruction (top) and WORK (bottom) from a normal adult volunteer using a 2D GE-SMS-EPI acquisition. Results show excellent agreement between the two techniques.

Figure 4. The relative improvement in T-score for the pCASL contrast produced via the WORK reconstruction compared to the standard reconstruction for the acquisition shown in Figure 3. Results show substantial improvement throughout the image even with use in each voxel of the R2* and ΔR2* values used for the simulations; further improvement is available via voxelwise estimation.

Proc. Intl. Soc. Mag. Reson. Med. 27 (2019)
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