Analytical G-factor for Cartesian Simultaneous Multi-Slice Imaging

Kangrong Zhu^{1}, Hua Wu^{2}, Robert F. Dougherty^{2}, Matthew J. Middione^{3}, John M. Pauly^{1}, and Adam B. Kerr^{1}

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.

Hybrid-space SENSE reconstruction for Cartesian simultaneous multi-slice imaging.

Hybrid-space SENSE reconstructed images and retained SNR maps of a CAIPI acquisition with 3$$$\times$$$ slice and 2$$$\times$$$ in-plane acceleration. The analytical maps are compared with maps calculated using pseudo multiple replica (PMR) simulation with either 50 or 500 replicas.

Analytical retained SNR maps of SMS acquisitions with different slice acceleration factors and $$$k_y$$$-$$$k_z$$$ undersampling patterns. No in-plane acceleration or partial Fourier is applied. The corresponding single-slice image is displayed in the lower left corner for reference.

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

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