G-factor describes the signal-to-noise ratio (SNR) loss in parallel imaging. In simultaneous multi-slice (SMS) imaging, a commonly used method to compute the g-factor is the pseudo multiple replica $$$^{1,2}$$$ method, whose accuracy depends on the number of simulated replicas. In this work, we derive analytical g-factor maps for SMS acquisitions with arbitrary Cartesian undersampling patterns basing on a hybrid-space SENSE reconstruction.

The hybrid-space SENSE reconstruction adopts a 3D representation $$$^{3,4}$$$ of SMS acquisitions. The 3D k-space data (Fig. 1a) is transformed into the hybrid $$$x$$$-$$$k_y$$$-$$$k_z$$$ space (Fig. 1b) by a 1D inverse FFT. For each point $$$x$$$ = $$$x_0$$$, the unknown magnetization on the $$$y$$$-$$$z$$$ plane is solved by $$${\bf m}_{x_0}$$$ = $$$pinv( {\bf E}_{x_0}) ~ {\bf s}_{x_0}$$$, where $$${\bf E}_{x_0}$$$ is the encoding matrix consisting of sensitivity and $$$k_y$$$-$$$k_z$$$ phase encodings, and $$${\bf s}_{x_0}$$$ is a vector containing the measured signals on the $$$k_y$$$-$$$k_z$$$ plane.

The analytical g-factor is calculated as $$$g_\rho = {\text{SNR}_\rho^\text{full}} / {\text{SNR}_\rho^\text{SMS} / \sqrt{R}} = {\sqrt{{\bf X}_{(\rho, \rho)}^\text{SMS}}}/ {\sqrt{{\bf X}_{(\rho, \rho)}^\text{single-slice}}}$$$ for the $$$\rho$$$-th pixel $$$^5$$$. The data reduction factor $$$R=1$$$ when single-slice data are used as the reference fully sampled data. The image noise covariance matrix, $$${\bf X}$$$, is calculated by tracking the noise propagation through the reconstruction pipeline. In general, for a linear transform $$${\bf y}={\bf A x}$$$, the noise covariance matrix of the output vector $$${\bf y}$$$ is $$${\bf X_{y}}={\bf A} {\bf X _x} {\bf A^H}$$$, where $$${\bf X_x}$$$ is the noise covariance matrix of the input vector $$${\bf x}$$$. In particular, our image noise covariance matrix calculation includes the following linear reconstruction steps: (1) Data whitening; (2) Nyquist ghosting correction (for EPI acquisition); (3) Ramp sampling correction; (4) 1D inverse FFT along $$$k_x$$$; (5) Geometric-decomposition coil compression $$$^6$$$; and (6) Solving sensitivity and $$$k_y$$$-$$$k_z$$$ encodings.

Brain images are acquired on a GE 3T MR750 scanner (GE Healthcare, Waukesha, WI) with a SMS gradient echo EPI sequence using a 32-channel head coil (Nova Medical, Wilminton, MA). Experiments are approved by our university's institutional review board. The coil noise covariance matrix is estimated from 4096 noise data points. Images are reconstructed offline in Matlab (the MathWorks, Natick, MA). Geometric-decomposition coil compression is applied to compress the 32 channels into 10 channels $$$^6$$$. Sensitivity maps are computed using an eigenvalue approach $$$^7$$$. A CAIPI $$$^{2,8}$$$ acquisition with 3$$$\times$$$ slice and 2$$$\times$$$ in-plane acceleration, no partial Fourier and FOV$$$_y$$$/4 shift between adjacent slices is conducted. The calibration data, which are fully sampled SMS data, are used to compute the g-factor maps. Fully sampled single-slice data are also acquired to calculate the analytical g-factor maps for different slice acceleration factors and different undersampling patterns. Both CAIPI and multi-slice acquisition with incoherent aliasing (MICA) $$$^9$$$ undersampling patterns are simulated. MICA randomizes the $$$k_y$$$-$$$k_z$$$ undersampling by traversing $$$k_z$$$ with a bit-reversal ordering.The analytical retained SNR (one over g-factor) maps agree with maps from the pseudo multiple replica (PMR) simulation (Fig. 2). In contrast to the PMR maps, whose accuracy improves with increased number of replicas $$$^1$$$, the analytical maps are noiseless. The retained SNR value is therefore always below 100% in the analytical maps but may exceed 100% in the pseudo multiple replica maps. With our straightforward implementations, the map computation time for one group of simultaneous slices in Fig. 2 is approximately 40 s for PMR with 50 replicas, 426 s for PMR with 500 replicas and 22 s for the analytical maps. The proposed analytical g-factor calculation is compatible with SMS acquisition with arbitrary acceleration factor and arbitrary undersampling pattern (Fig. 3).The analytical method is faster than PMR to generate high quality maps under the hybrid-space SENSE reconstruction framework. The analytical g-factor maps can assist a fair comparison between both coherent and incoherent Cartesian SMS undersampling patterns.