SENSE-LORAKS: Phase-Constrained Parallel MRI without Phase Calibration
Tae Hyung Kim1, Kawin Setsompop2, and Justin P. Haldar1

1Electrical Engineering, University of Southern California, Los Angeles, CA, United States, 2Radiology, Harvard Medical School, Boston, MA, United States

### Synopsis

We introduce a novel framework called SENSE-LORAKS for partial Fourier phase-constrained parallel MRI reconstruction. SENSE-LORAKS combines classical SENSE data modeling with advanced regularization based on the novel low-rank modeling of local k-space neighorhoods (LORAKS) framework. Unlike conventional phase-constrained SENSE techniques, SENSE-LORAKS enables use of phase constraints without requiring a prior estimate of the image phase or a fully sampled region of k-space that could be used for phase autocalibration. Compared to previous SENSE-based and LORAKS-based reconstruction approaches, SENSE-LORAKS is compatible with a much wider range of sampling trajectories, which can be leveraged to achieve much higher acceleration rates.

### Purpose

SENSE [1,2] and phase-constrained partial Fourier [3,4] reconstruction are two powerful approaches to accelerated MRI, that are even more successful when combined together [5-7]. However, previous phase-constrained SENSE approaches require an estimate of the image phase. As a result, specialized k-space trajectories are often used that densley sample the center of k-space to enable phase calibration. This can lead to non-uniform k-space sampling, which may introduce undesirable artifacts when used with sequences like EPI and balanced SSFP [8-9]. In addition, measuring a densely-sampled calibration region can also place restrictions on trajectory design and limit potential acceleration factors.

In this work, we introduce a novel image reconstruction approach, called SENSE-LORAKS, that enables phase-constrained parallel MRI reconstruction without prior phase estimation. SENSE-LORAKS uses the implicit regularization-based phase, support, and parallel imaging constraints of LORAKS [10,11] to regularize conventional SENSE reconstruction. LORAKS regularization is based on the observation that it is possible to embed k-space data into higher-dimensional low-rank matrices for images that possess limited image support, slowly-varying phase, and/or correlations between different parallel imaging receiver channels. This low-rank matrix embedding is powerful, because low-rank matrices have relatively few degrees of freedom and can be recovered from limited data using regularization techniques. Additionally, LORAKS is compatible with a wider range of sampling trajectories than conventional sparsity-based reconstruction [10,11].

Compared to previous fully-calibrationless LORAKS-based methods [10,11], SENSE-LORAKS has access to additional information about the coil sensitivity profiles. As a result, the SENSE-LORAKS inverse problem is better posed, and SENSE-LORAKS is compatible with a wider range of k-space trajectories that are less appropriate for fully calibrationless reconstruction (e.g., the uniform undersampling trajectories that are most appropriate for EPI and balanced SSFP).

### Method

Because LORAKS [10] and P-LORAKS [11] apply their constraints as a form of regularization, it can easily be combined with approaches like SENSE. The objective function of SENSE-LORAKS is $\hat{\mathbf{\rho}}=\arg\min_\mathbf{\rho}\|\mathbf{E}\mathbf{\rho}-\mathbf{d}\|_{\ell_2}^2+\lambda_S \mathbf{J}(\mathbf{S}_P(\mathbf{\rho}))+\lambda_T\|\mathbf{\rho}\|_{\ell_2}^2$ which is composed of the SENSE data consistency term, P-LORAKS regularization that encourages a specially-constructed matrix to have low-rank [11], and additional Tikhonov regularization [12]. In this equation, $\mathbf{d}$ is the acquired multi-channel k-space data, $\mathbf{\rho}$ is the unknown image, $\mathbf{E}$ is the SENSE encoding matrix, and $\mathbf{S}_P(\mathbf{\rho})$ is one of the P-LORAKS matrices constructed from local k-space neighborhoods. The optimization problem is solved using a majorize-minimize algorithm similar to [10,11].

### Results

A fully sampled brain dataset was acquired using a T2-weighted turbo spin echo (TSE) sequence with a 12 channel headcoil on a 3T scanner. The data was retrospectively undersampled using uniform 1D k-space trajectories that are not well-suited to conventional phase-constrained reconstruction. Conventional uniform sampling was simulated with sample spacing 5$\times$ larger than Nyquist, leading to a total of 5.1$\times$ acceleration (noninteger because the image matrix is not divisible by the undersampling factor). We also applied uniform partial Fourier sampling to achieve the same total 5.1$\times$ acceleration but with higher sampling density. Results are shown in Fig. 1. Conventional Tikhonov-SENSE reconstruction has large error due to the poor g-factor at this level of acceleration, and SENSE with total variation (TV) [13] has limited capabilities to address the coherent aliasing artifacts. P-LORAKS [11] is unable to reconstruct this data successfully, because it lacks prior sensitivity map information and cannot successfully estimate the intracoil correlations without more nonuniformity in the sampling pattern. In contrast, the proposed SENSE-LORAKS reconstruction yields much more accurate reconstruction results using both full and partial Fourier k-space trajectories. Importantly, the partial Fourier SENSE-LORAKS reconstruction leads to the best results, which may be explained by the fact that it achieves higher sampling density than the other trajectories with the same acceleration factors.

Fig. 3 shows a similar comparison with 64-channel uniformly undersampled EPI data at 3T. Notably, SENSE-LORAKS reconstruction with a partial Fourier trajectory can achieve a similar artifact level at 7.7$\times$ acceleration as compared to conventional Tikhonov-SENSE [12] reconstruction at 5$\times$ acceleration without partial Fourier acquisition. Moreover, the use of partial Fourier trajectories with EPI enables shorter minimum echo time [14].

### Conclusions

We present a novel MRI reconstruction framework called SENSE-LORAKS. The proposed method does not require prior phase estimation, and utilizes the complementary benefits of SENSE and LORAKS to achieve high acceleration factors with flexibly-chosen k-space trajectories. We have shown results using the calibrationless uniform undersampling trajectories appropriate for EPI and balanced SSFP acquisitions, though SENSE-LORAKS is very flexible and can also be used advantageously with alternative trajectories. In addition, since LORAKS is imposed as regularization, it is easily combined with other data acquisition models (e.g., simultaneous multi-slice) and regularization constraints (e.g., sparsity).

### Acknowledgements

This work was supported in part by NSF CAREER award CCF-1350563 and NIH grants R01-NS089212, R24-MH106096, and R01-EB019437.

### References

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### Figures

Figure 1: Reconstruction results with 5.1× acceleration. Numbers below the results are normalized root-mean-square error (NRMSE). Error images are also shown in color. (a) Gold Standard images (magnitude and phase). With uniform sampling, (b) SENSE, (c) SENSE+TV, (d) SENSE-LORAKS, and (e) P-LORAKS. With partial Fourier uniform undersampling, (f) P-LORAKS, and (g) SENSE-LORAKS.

Figure 2: Reconstruction results with 64-channel EPI data. (a) Gold Standard images. (b) 5.0× SENSE with uniform sampling, (c) 5.0× SENSE-LORAKS with uniform sampling. (d) 7.7× SENSE-LORAKS with partial Fourier uniform undersampling.

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