W. Scott Hoge^{1,2} and Jonathan R. Polimeni^{2,3}

^{1}Radiology, Brigham and Women's Hospital, Boston, MA, United States, ^{2}Dept of Radiology, Harvard Medical School, Boston, MA, United States, ^{3}Athinoula A. Martinos Center for Biomedical Imaging, Massachusetts General Hospital, Charlestown, MA, United States

The structural relationship between the linear system of equations used to calibrate inplane-GRAPPA, slice-GRAPPA, and LeakBlock for SMS is analyzed. This analysis reveals that LeakBlock is structurally identical to inplane-GRAPPA in the specific case of a (synthetic) zero-slice-gap. Signal leakage in the original slice-GRAPPA formulation is thus revealed to be due to signal cancellation in the original formulation. This relationship between inplane-GRAPPA and LeakBlock further justifies its preferred usage over the original slice-GRAPPA formulation for the separation of slice-accelerated data.

Mathematically, the associated normal equations are as follows. From Cauley et al.

By contrast, the slice-GRAPPA LeakBlock implementation seeks to identify GRAPPA coefficients that both generate the target slice of interest, $$$A_t g_t = b_t$$$, while constraining the other slices to be identically zero, $$$A_s g_t = 0$$$ for $$$s \neq t$$$. In this case, the associated normal equations are $$ \left( \sum_{s=1}^S A_s^H A_s \right) g_t = A_t^H b_t . $$

To relate to inplane-GRAPPA, consider the MB=2 zero-slice-gap example above, with the corresponding normal equations

$$ (A_1 + A_2 )^H (A_1 + A_2 ) g_t = (A_1 + A_2 )^H b_t $$ for original slice-GRAPPA versus $$(A_1^H A_1 + A_2^H A_2 ) g_t = A_t^H b_t $$ for LeakBlock.

Structurally, these two linear system matrices are quite different. The phase change in alternate k-space lines of the 1/2 FOV-shifted slice result in signal cancellation in the ($$$A_1+A_2$$$) matrix that appears in the original variant. In LeakBlock, by contrast, each of the source data matrices are first multiplied by their conjugate-transpose (i.e., $$$A_1^H A_1$$$) prior to the summation. The ensures that the self-product of the 1/2 FOV-shifted slice, $$$A_2^H A_2$$$, identically matches that of the original slice because $$$\{+1,-1,+1,-1,\cdots\}^2 = \{+1,+1,+1,+1,\cdots\}$$$. This reveals that the LeakBlock formulation reduces to $$$2(A^H_t A_t) g_t = A_t b_t,$$$ which is identically equivalent structurally to the normal system of equations used for inplane-GRAPPA data recovery. This result can be generalized to state that in the zero-slice-gap case, a synthetic blipped-CAIPI acquisition will reduce to an accelerated acquisition when the acceleration rate,

This work was supported in part by the NIH NIBIB (grants P41-EB030006, R01-EB019437, R03-EB023489, and R03-EB030831), by the BRAIN Initiative (NIH NIMH grant R01-MH111419 and NIBIB grant U01-EB025162), and by the MGH/HST Athinoula A. Martinos Center for Biomedical Imaging.

- Griswold MA, Jakob PM, Heidemann RM, Nittka M, Jellus V, Wang J, Kiefer B, Haase A. Generalized autocalibrating partially parallel acquisitions (GRAPPA). Magn Reson in Med 2002;47(6):1202–1210.
- Setsompop K, Cohen-Adad J, Gagoski BA, Raij T, Yendiki A, Keil B, Wedeen VJ, Wald LL. Improving diffusion MRI using simultaneous multi-slice echo planar imaging. NeuroImage 2012;63(1):569 – 580.
- Cauley SF, Polimeni JR, Bhat H, Wald LL, Setsompop K. Interslice leakage artifact reduction technique for simultaneous multislice acquisitions. Magnetic Resonance in Medicine 2014;72(1):93–102.
- Setsompop K, Gagoski BA, Polimeni JR, Witzel T, Wedeen VJ, Wald LL. Blipped-controlled aliasing in parallel imaging for simultaneous multislice echo planar imaging with reduced g-factor penalty. Magn Reson Med 2012;67(5):1210–1224.
- Hoge WS, Polimeni JR. System conditioning during GRAPPA kernel training improves temporal SNR in accelerated EPI-based functional, diffusion, and perfusion MRI applications. in Proc Intl Soc Mag ResonMed. Montreal, Quebec, CA, 2019; 2402.

Figure 1: Visualization of the zero-slice-gap MB=2 simulation. The images show two simulated SMS experiments. On the left, a standard acquisition with two different slices simultaneously acquired is shown. On the right, the same slice is acquired twice. Blipped-CAIPI shifts one of the slices by 1/2-FOV and induces a phase change in the k-space data representation. A simulated slice group is then formed by adding the two slices together. The bottom row of the 0-slice gap group (right) reveals an R=2 subsampling pattern in k-space and the familiar R=2 aliasing pattern in the image domain.

Figure 2: Visualization of how the spectrum of the slice-GRAPPA system matrix evolves between the maximum separation, as measured in units of slices, and zero slice-gap. Note that the ratio between the smallest and largest singular values (e.g. the condition number) changes very little. Rather, the subspace of support---between the largest value and the bend of the elbow---grows more narrow as the slice gap decreases.

Figure 3: Spectrum plot showing the spectral relationship between slice-GRAPPA, LeakBlock, and inplane GRAPPA. Note that the inplane and LeakBlock spectra are identical.

Figure 4: Matrix illustration of the cross-correlation between the Eigen-vectors of (left) slice-GRAPPA versus inplane-GRAPPA and (right) LeakBlock versus inplane-GRAPPA. The subspace degradation in slice-GRAPPA appears as a correlation dispersion along the matrix diagonal in the left image. By contrast, the right image shows a 1-to-1 correlation between the LeakBlock system and the inplane-GRAPPA system. Colors correspond to the color wheel at the far right, which captures both magnitude and phase.

DOI: https://doi.org/10.58530/2022/4054