Many 3D MRI acquisitions necessitate free-breathing imaging for high resolution and large volumetric coverage and thus require efficient respiratory gating (RG) to suppress adverse motion effects. Conventional RG is performed by selection of an acceptance window (AW) for acceptance/rejection of data acquisition. Such a hard-threshholding scheme fully accepts data within the AW ignoring intra-window motion corruption and completely rejects out-of-AW data discarding their potential for improving reconstruction. Theoretically, respiratory soft gating (RSG) [1] that utilizes data with respiratory-position-based soft-thresholding can more efficiently suppress motion. However, RSG enforces soft-threshholding as a data consistency constraint and thus is limited to iterative processing with intensive computation. This work aimed to develop a non-iterative solution based on motion-error regularization for conventional auto-calibrating parallel imaging (ac-PI).ac-PI synthesizes missing k-space data from neighboring source k-space data (S_j, j=1,2,…). The reconstruction weights, x, for a synthesis pattern are calculated by solving Ax=b (1) [2], with A and b for the source and target calibration data matrixes, respectively. x is calculated by minimizing the L₂-norm error of ||Ax-b||² (2). The j-th column in A, A.,j, is comprised of concatenated calibration data with a k-space shifting corresponding to S_j in reconstruction. In free breathing imaging, S_j is collected with motion displacement and, accordingly, A_.,j needs to be updated with A_.,j+e_.,j to calculate the optimal x in the existence of motion, where e_j represents the change in calibration data with the corresponding motion of S_j. Alternatively, according to equation 1, motion at S_j would increase the L₂-norm error by δ_j=x_jŸ||e_.,j||² and the entire reconstruction error due to motion can then be approximated by Σ(δ_j). Therefore, the original ac-PI equation in (2) turns to min(||Ax-b||²+||Δx||²) (3), where Δ is a diagonal matrix with Δ_j,j=sqrt(δ_j). This is the well-known Tikhonov regularization with an analytical solution: x=(A^TA+Δ^TΔ)^-1A^Tb (4). Equation 4 in effect reduces x_j for “bad” data with large δ_j to suppress motion (min||Δx||²) and accordingly increases x_j for “good” data with small δ_j to improve data fitting (min||Ax-b||²).

The proposed Motion-Error Regularized Ac-PI (MERA) method is validated in free-breathing 3D cardiac CINE with k-t sampling and a pseudo-random vieworder. Acquisition of center 8% k-space was repeated by a factor of 4 for estimating Δ as follows (Fig.1). From the simultaneously recorded respiratory bellow signal, we generate a respiratory histogram and derive the most consistent position near end-expiration (P_EE) and end-inspiration (P_EI). In the repeatedly acquired center k-space, we generate two datasets at P_EE (K_EE) and P_EI (K_EI), respectively. The motion-induced error from P_EE to P_EI, ||K_EE-K_EI||², is calculated. Then, δ_P at a respiratory position, P, is estimated by δ_P= ||K_EE-K_EI||²Ÿ(P-P_EE)/(P_EI-P_EE), assuming δ_P increases linearly with off-P_EE displacement. Note that different coil channels sense motion differently and create different δ’s (e.g. lower for elements near dorsal and higher for elements near chest wall). Therefore, ||K_EE-K_EI||² is calculated individually for each coil channel to generate coil-specific δ_P‘s.

3D CINE was scanned on 4 healthy volunteers during free-breathing with 5× acceleration on a GE 3T (MR750). Data were processed using a k-t ac-PI method, kat ARC [3]. Static-tissue-removal (SSR) [4] was used to first identify and remove signals from static tissues (e.g. chest wall, dorsal) in the original data to improve kat ARC at high acceleration. The static tissue image was generated from a time-projection dataset with weighted averaging based on δ_P to obtain motion-suppressed reconstruction in static tissues. Next, in kat ARC, all k-space lines, including the acquired to correct motion corruption from acquisition, are synthesized based on equation 4 and Δ is constructed based on the P of each source line for each synthesis and the prior-calculated δ_P. Because each k-space neighborhood contains a mix of ‘good’ and ‘bad’ data, equation 4 synthesizes each line in the entire k-space with an optimal balance (min overall L₂-norm error) between data fitting and motion suppression.

As shown in Fig.2, MERA effectively suppresses motion ghosting and edge blurring that is apparent without gating. In comparison, conventional RG with an AW of 50% on the same dataset shows visible aliasing and motion artifacts. RG suffers from either increasing aliasing with smaller AW or increasing motion with larger AW. In the case of Fig.3, MERA suppresses motion in free-breathing acquisition and provides heart delineation similar to breathholding. This work incorporates a simplified model-error model into ac-PI and derived a new non-iterative respiratory soft gating technique. The proposed method can more effectively suppress motion than conventional RG. Our experiments show that MERA can also apply to full acquisition and non-k-t ac-PI and be used to suppress cardiac motion.