To utilize all bSSFP phase cycles simultaneously in GRAPPA calibration.

The GRAPPA calibration is performed via the analytical solution of ¹

$$g_i=\left(S^*S+\lambda I\right)^{-1}S^*s_i$$

where $$$i=$$$ receiver channel index, $$$s_i=$$$ all k-space samples in the calibration region reshaped into a vector, $$$\lambda=$$$ factor for Tikhonov regularization, and each row of $$$S$$$ contains all the samples around $$$s_i$$$ that are chosen by the GRAPPA kernel (in all receiver channels). Instead of solving this equation separately for each phase cycle in a bSSFP acquisition, we propose that the $$$s_i$$$ and $$$S$$$ from each phase cycle should be concatenated vertically, so that

$$S=\begin{bmatrix}S^1 \\S^2 \\ \vdots \end{bmatrix}, s_i=\begin{bmatrix}s_i^1 \\s_i^2 \\ \vdots \end{bmatrix}$$

where the superscripts correspond to each phase cycle. This will be referred to as the “jointPC” method (PC for phase-cycle).

To test this method, fully sampled phase-cycled bSSFP scans in a phantom and brain of a volunteer were performed. The phantom data was acquired on a 1.5 T GE Signa Excite with 8 receiver channels, phase cycles = {0°, 90°, 180°, 270°}, TR = 4.6 ms, FOV = 26 x 26 cm², 5 mm slices (x36), flip angle 30°, BW = ±125 kHz. The brain data was acquired using a 3D multi-shot fly-back EPI sequence with an echo train length of 4 on a 3 T GE Discovery MR750 with 32 receiver channels, phase cycles = {0°, 180°}, TR = 14.3 ms, FOV = 24 x 24 x 11 cm³, flip angle 35°, BW = ±62.5 kHz. A Fourier transform along z was performed and 2D undersampling with individual slices was tested.

GRAPPA calibration was performed on 10 central phase encode lines of the fully sampled data (from all phase cycles) with a 7x7 kernel. Three calibration methods were tested: (1) separate kernels for each phase cycle (sepPC), (2) separately calculated GRAPPA weights that are averaged to form a single set of weights (avePC), similar to what has been proposed for multiple calibration scans in phase-contrast MRI ², and (3) the jointPC method proposed here. The data was retrospectively undersampled along the phase-encode direction by a factor of 4 in the phantom and 6 in the brain (no fully sampled calibration region was kept in the undersampled data). Reconstructions were performed using SPIRiT ³, and g-factor maps were computed using the pseudo-multiple replica method with 30 replicas ⁴. The separate receivers and phase cycles were combined using sum-of-squares.

For both the phantom and brain, the highest agreement with the fully sampled data was with jointPC (Figure 1,2). Similar results were obtained in other slices. While the averaged weights (avePC) performed better than the separately calculated weights (sepPC), the images still had residual aliasing. The errors likely stem from mis-calibration in the bSSFP dark bands, where there is little signal to provide accurate estimation of the underlying receiver sensitivities. When the weights are averaged for avePC, the sensitivities in the low signal dark band regions are over-represented compared to the brighter signal in other phase cycles. By using the jointPC method, the low signal dark band regions are properly de-emphasized in the least-square solution for the weights. The g-factor maps appeared similar for avePC and sepPC, while g-factors were larger for jointPC (Figure 3).

This method requires only a simple modification to the GRAPPA calibration algorithm. Even with the large undersampling factors and relatively small calibration region, reconstruction of the undersampled data was relatively accurate. Notably, the jointPC method results in a larger g-factor because it more completely unfolds aliasing (i.e., it is a necessary trade-off to remove residual aliasing). In other trials with lower accelerations where residual aliasing was low for all cases, the g-factors were similar (not shown). The jointPC method may be sensitive to motion that occurs between the phase-cycled acquisitions, in particular with surface coils where the sensitivity profiles would also change slightly. Nevertheless, this approach could be used in other scenarios where multiple sets of calibration data are acquired, such as in self-calibrated phase-contrast MRI.