Yunsong Liu^{1}, Kawin Setsompop^{2}, and Justin P. Haldar^{1}

^{1}Signal and Image Processing Institute, University of Southern California, Los Angeles, CA, United States, ^{2}Department of Radiology, Stanford University, Stanford, CA, United States

gSlider is an efficient technique for diffusion MRI that uses multiple RF encodings to encode high-resolution spatial information along the slice dimension. In this work, we investigate whether smooth-phase constraints can be used to reduce the required number of RF encodings. Although smooth-phase constraints are classically used to reduce k-space sampling (partial Fourier acquisition), we believe that their use to reduce RF encoding requirements is novel. Theoretical and simulation results demonstrate that, if optimized RF encodings are used, phase constraints can successfully be used to reduce the number of required RF encodings in image regions where the phase is smooth.

In this work, we consider a novel and complementary approach to reducing the number of RF encodings. Specifically, we consider the use of smooth phase constraints. While phase constraints are classical for accelerated k-space acquisitions (i.e., partial Fourier acquisition

$$b_{r}(\mathbf{x}_{n})= e^{i \phi_r(\mathbf{x}_n)} \sum_{s=1}^Sg_{rs}m_s(\mathbf{x}_n)+z_{rn},$$

where $$$b_{r}(\mathbf{x}_{n})$$$ is the thick-slab data measured for the $$$r$$$th RF encoding ($$$r=1,\ldots,N_{RF}$$$) at voxel location $$$\mathbf{x}_{n}$$$ ($$$n=1,\ldots, N_v$$$), $$$g_{rs}$$$ represents the RF encoding applied to the $$$s$$$th thin slice ($$$s=1,\ldots,N_s$$$) by the $$$r$$$th RF encoding, $$$m_s(\mathbf{x}_n)$$$ represents the magnitude of the $$$s$$$th thin slice at voxel location $$$\mathbf{x}_n$$$, $$$\phi_r(\mathbf{x}_n)$$$ represents the measured image phase for the $$$r$$$th RF encoding (which is random due to the physics of diffusion acquisition), and $$$z_{rn}$$$ represents measurement noise.

In conventional gSlider, the number of RF encodings $$$N_{RF}$$$ is taken to be greater than or equal to the number of thin slices $$$N_s$$$. In this work, we evaluate whether assuming that the phase is smooth can enable reduced RF encoding.

We approach this question in two different ways. First, we evaluate the Cramer-Rao bound (CRB), which is an estimation-theoretic tool that can be used to estimate uncertainty in parameter estimation and can also be used to optimize acquisition strategies

Figure 3 shows representative simulation results (based on complex-valued T2-weighted brain data) using 4 optimized RF encodings to reconstruct 5 slices. It can be observed that the brain parenchyma (inside the green outline) is reconstructed well when using phase constraints, while the minimum-norm reconstruction unsurprisingly has noticeable errors in this underdetermined setting. The phase constrained reconstruction has more substantial errors outside the brain parenchyma, which we believe occurs because the phase becomes non-smooth as we move from the brain into the skull and scalp.

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[2] Liao C, Stockmann J, Tian Q, Bilgic B, Arango NS, Manhard MK, Huang SY, Grissom WA, Wald LL, Setsompop K. High‐fidelity, high‐isotropic‐resolution diffusion imaging through gSlider acquisition with and corrections and integrated shim array. Magn Reson Med 83:56-67, 2020.

[3] Haldar JP, Setsompop K. “Fast high-resolution diffusion MRI using gSlider-SMS, interlaced subsampling, and SNR-enhancing joint reconstruction.” Proc. ISMRM 2017, p. 175.

[4] Liu Y, Liao C, Setsompop K, Haldar JP. “An evaluation of q-space regularization strategies for gSlider with interlaced subsampling.” Proc. ISMRM 2020, p. 4368.

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[9] Liu Y, Haldar JP. NAPALM: An Algorithm for MRI Reconstruction with Separate Magnitude and Phase Regularization. Proc. ISMRM 2019, p. 4764.

[10] Haldar JP, Liu Y, Liao C, Fan Q, Setsompop K. Fast submillimeter diffusion MRI using gSlider-SMS and SNR-enhancing joint reconstruction. Magn Reson Med 84:762-776, 2020.

Figure 1. RF excitation profiles corresponding to (top)
the original gSlider RF encoding scheme to resolve 5 thin slices from 5 RF
encodings, and (bottom) the CRB-optimized RF encoding scheme to resolve 5 thin
slices from 4 RF encodings with phase constraints. Note that all sub-slices are
excited with a full magnitude of 1 in all cases.

Figure 2. CRBs for resolving 5 thin
slices as a function of the number of voxels sharing the same phase value,
plotted for several different RF encoding strategies. Note that without phase constraints, the CRB
blows up to infinity whenever the number of RF encodings is smaller than the
number of thin slices (5 in this case).

Figure 3. Simulated reconstruction of 5 thin slices from
the set of 4 optimized RF encodings, for both phase-constrained and minimum
norm reconstruction. For ease of visualization, we only show two out of the five
reconstructed slices for each case.