For parallel magnetic resonance imaging (pMRI) with 2D or 3D generalized auto-calibrating partially parallel acquisition (GRAPPA) algorithms, not-acquired k-space data are estimated based on the auto-calibration signal (ACS) lines [1,2]. In case of the 3D (i.e., volumetric) GRAPPA, the identical number of acquired ACS lines for phase-encoding and partition-encoding directions are generally used even though the acquired images have different encoding steps along the phase- and partition-encoding directions [3-5]. Since it is important to find appropriate selection criteria for ACS lines to optimize the image quality, images were reconstructed using various selections of ACS lines, reflecting the geometrical shape of the image matrix in this study.

3D MR data were acquired from a phantom using the 3D-gradient-echo sequence from a 3T-scanner (Verio, Simens) with a 12-channel head coil with the following parameters: TR = 20 ms, TE = 14 ms, flip angle = 25º, FOV = 210 × 210 mm², slice thickness = 0.8 mm, and matrix size = 512 × 256 × 208. To demonstrate the performance of 3D-GRAPPA for different ACS lines, the fully acquired MR data were undersampled with reduction factors of two for both phase-encoding (ky) and partition-encoding (kz) directions, thereby generating a k-space dataset having a matrix size of 512 × 128 × 104 (kx × ky × kz).

The undersampled 3D data were then reconstructed using two different volumetric GRAPPA algorithms: extension (EX)-3D-GRAPPA and single kernel (SK)-3D-GRAPPA [4,5]. For EX-3D-GRAPPA and SK-3D-GRAPPA algorithms, not-acquired lines were estimated using the kernels as shown in Fig.1(a) and (b), respectively. As illustrated, EX-3D-GRAPPA uses three different 3D-kernels and SK-3D-GRAPPA uses only one 3D-composite kernel.

A schematic diagram for acquisition of ACS lines is illustrated in Fig. 2. If the ACS lines are acquired to form a square in the ky-kz plane, the number of ACS lines are identical for phase-encoding and partition-encoding directions (Fig. 2(a)). When the dimensions of phase-encoding and partition-encoding steps are considered, the ACS lines can be acquired by reflecting the aspect ratio of phase-encoding and partition-encoding steps (Fig. 2(b)).

To demonstrate the effects of ACS geometry (i.e., the aspect ratio of acquired ACS lines in the phase-encoding and the partition-encoding directions), the number of ACS lines were differently selected, while keeping the total number of ACS lines constant. The images were then reconstructed from the undersampled dataset using the following selections of ACS lines: (number of ACS lines in ky) × (number of ACS in kz) = 48 × 40, 40 × 48, 60 × 32, 32 × 60, 80 × 24, 24 × 80. As a reference, 44 × 44 ACS lines, which contain a slightly larger number of ACS lines than the rest, were also selected. The reconstruction experiments were performed using the Matlab (R2015a).

A reference image was reconstructed from the fully sampled k-space data as presented in Fig. 3(a). Using different ACS lines, images were also reconstructed from the undersampled data. Figs. 3 (b-h) show the difference images calculated from the reference image (Fig. 3(a)) and the reconstructed images using different ACS lines. The upper and lower images were reconstructed using EX-3D-GRAPPA and SK-3D-GRAPPA algorithms, respectively. While the difference images were generally similar, the difference image calculated from the image reconstructed with the 48 × 40 ACS lines shows slightly better quality. In Fig.4, the root-mean-squared-error (RMSE) [6] and the standard deviation (STD) values of the difference calculated from the entire image slices were plotted with respect to the ACS geometry for each 3D-GRAPPA algorithm. The RMSE increased when the ACS geometry did not follow the image matrix as long as the same total number of ACS lines was used. Compared to 44 × 44 case, which actually used a slightly larger number of ACS lines, following the aspect ratio (48 × 40 ACS lines) resulted in better performance.Generally, an identical number of ACS lines is used for the phase- and partition-encoding directions although the matrix size of the k-space is different for the phase- and partition-encoding directions. When the number of ACS lines in each encoding direction followed the ratio of phase- and partition-encoding steps in this study, however, the image quality was improved. Thus, it can be concluded that the ACS lines should be selected to follow the aspect ratio of the image matrix for efficient 3D GRAPPA reconstruction.