In array imaging (1, 2), partial phase encoding steps can be replaced by the spatially localized sensitivity encoding of each component coil in the phased array (3). Various reconstruction algorithms, like SENSE (sensitivity encoding) (4) or GRAPPA (generalized autocalibrating partially parallel acquisitions) (5), can be then used to recover the missing data in this partially parallel imaging (PPI) technology. With a phased array consisting of four or eight surface receive coils, two- to four- or even higher fold acceleration has previously been obtained in 2D imaging using parallel imaging alone without perceptible aliasing artifacts and significant signal-to-noise-ratio (SNR) penalty(6). Limited to the phase encoding (PE) direction, the maximal achievable acceleration factor (AF) is generally limited by the ill-posed inverse problem involved in the data reconstruction(4), originating from the geometric properties of the coil setup; the number of component coils in the array also sets a theoretical AF upper bound to 1D PPI.
Several benefits can be achieved by combining PPI with 3D Fourier MRI. First, applying PPI along the two PE directions separately, a net acceleration can be obtained as the product of the two independent AFs along the two PE directions(7). In the case of high AF PPI, the inverse problem conditioning of PPI reconstruction can be improved by splitting the high net AF into two smaller AFs in each PE(7). The SNR degradation can then be improved, compared to using the same net AF along one PE in 1D PPI. Second, the SNR loss issue of PPI in 3D MRI may be less important due to the intrinsic high SNR of 3D MRI. Third, PPI can directly relieve the prominent long scan time issue of 3D MRI, compared to the corresponding 2D version.
2D PPI for 3D MRI was first reported in (7). Called 2D SENSE therein, this 2D PPI works in the image domain using the SENSE (4) unfolding method for each slice along each acceleration direction separately. An improvement was subsequently presented by Breuer et al. using an optimal sampling strategy, CAIPIRINHA (controlled aliasing in parallel imaging results in higher acceleration) (8–10), to control the aliasing along the PPI direction, which does not have enough sensitivity variation for an effective PPI reconstruction. Inherited from SENSE, the 2D SENSE reconstruction quality depends on the explicitly estimated sensitivity map, which may be hard to obtain in some regions with very low SNR in whole brain imaging, and its estimation error will be inevitably transmitted into the final reconstruction. Without explicit sensitivity estimation, Blaimer et al. (11) gave a k-space-based 2D PPI using the GRAPPA operator (12). In this approach, the missing k-space data were recovered by applying GRAPPA reconstruction along the two PE directions sequentially, and part of the precedently recovered data were used in the second PPI reconstruction step, which may cause extra noise due to error propagation from the imperfect reconstruction in the first PPI reconstruction step. Moreover, using the original GRAPPA reconstruction approach, it only considered the correlation information carried by the neighbors from one k-space dimension, while ignoring the abundant information from other surrounding neighbors in the 3D k-space (13, 14).
In this paper, a previously proposed improved GRAPPA data reconstruction method, multicolumn multiline interpolation (MCMLI) (13, 14) was generalized to the surrounding neighbors-based autocalibrating partially parallel imaging (SNAPPI) and used in 2D PPI reconstruction for 3D MRI. Several 2D PPI reconstruction strategies were presented, using separable 2D PPI reconstruction through SNAPPI reconstruction along each PE direction independently as in (11) and nonseparable 2D PPI reconstruction also through SNAPPI, in the 3D k-space or the hybrid k and image space (hybrid space). In the case where one PE direction does not have enough sensitivity variation for regular PPI, CAIPIRINHA type sampling strategy (8–10) was also used to control the aliasing along that direction. Both simulations and in vivo experiments were conducted to evaluate the presented 2D PPI approaches.
MCMLI (13, 14) was previously proposed to improve the GRAPPA reconstruction by including all surrounding acquired neighbors to reconstruct a missing k-space data point. Originally applied to 1D PPI in Fourier MR imaging, this concept can be generalized to the surrounding neighbors-based autocalibrating partially parallel imaging as shown in Fig. 1. Denoting the two PE directions and readout as kz, ky, and kx, to recover a missing k-space data point of a component coil, SNAPPI forms an interpolation net (14) first, consisting of a missing data point and its surrounding neighboring points from all component coils along kx, ky, and kz directions as in MCMLI, which can be treated as a special case of SNAPPI in 1D PPI. Each circle in Fig. 1 represents data points from all component coils with the same k-space coordinate, though the root node of the interpolation net only corresponds to a missing data point from one specific component coil. The interpolation net weights (or SNAPPI reconstruction parameters) are then estimated using the additionally acquired reference data as marked by crosshair filled circles in Fig. 1.
As an intrinsic process of modulating spatially localized sensitivity waveform to synthesize R − 1 (R ≥ 1 is the AF along certain PE direction) adjacent spatial harmonics corresponding to those skipped phase encoding steps (3), SNAPPI has only R − 1 independent sets of reconstruction parameters to be estimated (14), each corresponding to a relative offset in the PE direction. For an easy description, this feature can be described by a label system as shown in Fig. 2a, which is a planar view of 2D SNAPPI with AF of 2 along each PE direction using regular 2D PPI Cartesian sampling paradigm. A number from 0 to R − 1 is used to denote the data status along each PE direction in 2D PPI Cartesian Fourier imaging, with 0 means acquired, and m (1 ≤ m ≤ R − 1) means skipped PE steps with an offset of mΔk (Δk is the k-space sampling interval) to their preceding acquired lines marked with 0. For the example shown in Fig. 2a, there are only two independent labels for each PE direction. The dash lines marked by 0 in Fig. 2a represent acquired PE steps with the acquired data points marked by solid circles; dotted lines marked by 1 mean skipped PE steps. Using the label system, it is easy to note that there are only three independent states (represented by (labelPE1, labelPE2)) of the missing data in the 2D PPI example in Fig. 2a: (0,1),(1,1), and (1,0), as marked by the three stars. The missing data of each component coil with the same status label can be recovered using the same interpolation net. For simplicity, the same neighbors can be used to form the interpolation net for each of these missing points, as highlighted by the large gray square in Fig. 2a, which represents the acquired neighbor searching region and can be of arbitrary shape.
Denoting the signal value as S and the sampling intervals along kx, ky, and kz coordinates as Δkx, Δky, and Δkz, respectively, the 2D SNAPPI data reconstruction process can be described by
where L is the number of component coils in the array and j represents the jth coil. Ry (Ry ≥ 1) and Rz (Rz ≥ 1) are the AF along ky and kz, respectively; ry (1 ≤ ry < Ry) and rz (1 ≤ rz < Rz) are the relative offset along ky and kz, respectively. Ω indicates the index set of the acquired surrounding neighbors of location (kx, ky + ryΔky, kz + rzΔkz). W refers to the interpolation net weights from the lth component coil for the target missing data points with label (ry, rz) in the jth coil.
Using additionally acquired reference data with full sampling density in the central kz − ky plane, the interpolation weights are estimated by solving the inverse problem of Eq. . The floating net based fitting concept (13, 14) is used to increase the number of training sets by shifting the net (represented by the star and the small solid circles within the gray square in Fig. 2a) to all possible positions in the reference data matrix space.
Although this work focuses on the Cartesian sampling based Fourier imaging, SNAPPI applies to other k-space sampling strategies if the sampling patterns of the calibration scan and the PPI scan have the same k-space topology, i.e., the distance between two neighboring points along a certain direction is the same in both calibration data and PPI data. An example of non-Cartesian sampling is the equivalent distance radial sampling as implemented in radial GRAPPA (15). Other non-Cartesian sampling like spiral trajectories could be also fitted into this paradigm using some interpolation-based approaches such as regridding(16).
2D Separable SNAPPI Reconstruction
The separately applied PE along each direction in 3D Fourier MR imaging renders an alternative 2D SNAPPI reconstruction by applying 1D SNAPPI reconstruction along each PE direction separately as stated in (11, 12). The reconstruction coefficients of each PPI reconstruction step are estimated independently using the reference data. Part of the reconstructed k-space data will be used for other missing data reconstruction in the second SNAPPI reconstruction step, as can be seen from the sampling paradigm in Fig. 2a.
2D SNAPPI Reconstruction in the Hybrid Space
Limited to the PE directions, 2D PPI reconstruction can be applied either in the 3D k-space before any inverse Fourier transform or in the hybrid space after inverse Fourier transform along the readout. Compared to the 3D k-space reconstruction approach, the hybrid space approach needs more reference data to estimate SNAPPI parameters for each kz − ky plane along the readout direction.
2D SNAPPI with an Optimal Sampling Pattern
CAIPIRINHA (8–10) was originally proposed to control the 2D SENSE reconstruction aliasing in a PE direction that does not have enough sensitivity variation for applying PPI. This optimal sampling strategy can be also used in 2D SNAPPI. As shown in Fig. 2b, the sampling points along PE 1 are shifted with an additional offset every second phase encoding step in PE 2. Using the same label system as for the regular 2D PPI sampling paradigm (Fig. 2a), each missing data point can be assigned to one of the several independent data states. In the example shown in Fig. 2b, there are only three independent missing data states (represented by (labelPE1, labelPE2)): (0,1),(1,1), and (1,0), and the missing data of each component coil with the same status label can be recovered using the same SNAPPI interpolation net. The large gray cycle in Fig. 2b represents a neighbor searching region for the three missing data, as marked by the stars. Note that different neighbors can be used to reconstruct one of the three data points, and the neighbor searching region can also be in any other shape rather than a cycle (or cylinder), as shown in Fig. 2b.
3D Cartesian Fourier MRI acquisitions with and without 2D PPI were performed on a Siemens Trio 3-T whole-body MR scanner with a product eight-channel array coil (Siemens, Erlangen, Germany). The two directions that have enough sensitivity variation for applying PPI with this array coil are the anterior–posterior (y) and left–right (x) directions of a subject lying supine in the scanner. Five subjects were scanned with written informed consent following an institutional review board approved protocol for the associated experiments.
Four scans with a 3D MPRAGE sequence were performed. (1) sagittal full-FOV scan with PE along y and x, TR/TE/TI = 1620/3.88/950 ms, matrix readout × PE1 × PE2 = 128 × 128 × 60, FOV = 250 × 250 × 180, flip angle = 15°, number of subjects = 5; (2) axial full-FOV scan with PE along z (the inferior–superior direction, also the B0 field direction) and y, TR/TE/TI = 1620/3.87/950 ms, matrix = 128 × 128 × 96, FOV = 250 × 250 × 160, flip angle = 15°, number of subjects = 1; (3) in vivo 2D PPI sagittal scan using the regular 3D cartesian sampling paradigm (Fig. 2) with PE along y with an AF of 3 and x with an AF of 2, number of subjects = 1; (4) reference Full-FOV scan for the in vivo 2D PPI data reconstruction with TR/TE/TI = 1620/3.43/950 ms, matrix = 128 × 128 × 88, FOV = 250 × 250 × 198, flip angle = 15°.
Simulations were performed by subsampling the sagittal and axial full-FOV 3D MRI volumes with the sampling paradigm as shown in Figs. 2a and b using an AF of 2 along each PE direction. 1D PPI was also simulated on the sagittal 3D MRI volume by subsampling ky with a factor of 4. Various SNAPPI reconstruction were simulated in the 3D k-space and the hybrid space with separable or nonseparable SNAPPI kernels (interpolation nets), which were called separable SNAPPI and nonseparable SNAPPI in the following.
(A)1DPPI: PPI along ky direction with an AF of 4, data reconstruction with a ky × kx = 4 × 6 2D kernel (4 × 6 neighbors from each component coil) in the hybrid space. For each image slice, 30 central fully sampled k-space lines were used for estimating SNAPPI reconstruction coefficients.
(B)2DPPI-sep-k: PPI along kz and ky using the 2D PPI sampling paradigm as shown in Fig. 2, separable SNAPPI reconstruction with a kz × ky × kx = 4 × 4 × 3 3D kernel along ky and kz separately in the 3D k-space. A total of 24 × 24 × 100 fully sampled central k-space reference data were used for estimating the reconstruction coefficients.
(C)2DPPI-sep-h: the same 2D PPI data as in (B) but reconstruction in the hybrid space with a kz × ky = 4 × 4 2D kernel along ky and kz separately. For each kz − ky plane, 50 × 60 fully sampled central k-space reference data were used for estimating the reconstruction coefficients.
(D)2DPPI-nonSep-k: the same 2D PPI data as in (B) but using nonseparable SNAPPI reconstruction in the 3D k-space with a kz × ky × kx = 4 × 4 × 3 3D kernel for each independently labeled missing point using the label system. A total of 12 × 12 × 100 fully sampled central k-space reference data were used for estimating the reconstruction coefficients.
(E)2DPPI-nonSep-h: different from (D) only by reconstruction in the hybrid space using a kz × ky = 4 × 4 2D kernel for each independently labeled missing point using the label system. For each kz − ky plane, 50 × 60 fully sampled central k-space reference data were used for estimating the reconstruction coefficients.
(F)CAIPIRINHA-k: 2D PPI using the CAIPIRINHA sampling paradigm as shown in Fig. 2b, data reconstruction in 3D k-space with a 3D kernel defined by a cylinder as shown in Fig. 2b for each independently labeled missing point using the label system. A total of 12 × 12 × 100 fully sampled central k-space reference data were used for estimating the reconstruction coefficients.
(G)CAIPIRINHA-h: different from (F) by reconstruction in the hybrid space using a 2D kernel defined by a cycle as shown in Fig. 2b for each independently labeled missing point using the label system. For each kz − ky plane, 50 × 60 fully sampled central k-space reference data were used for estimating the reconstruction coefficients.
Simulations on 2DPPI-nonSep-k/h and CAIPIRINHA-k/h were also performed on axial 3D MRI data (with PE along y and z) to evaluate the sampling scheme for aliasing control when one PE direction does not have enough sensitivity variation to allow PPI.
In Vivo 2D PPI Data Reconstruction
The in vivo 2D PPI data were reconstructed using 2DPPI-sep-k, 2DPPI-nonSep-k, and 2DPPI-nonSep-h as stated above, except that 2DPPI-nonSep-k used 12 × 16 × 100 reference data.
To quantify the reconstruction performance of each method, the relative root of the mean squared distance (RRMS) between the simulated PPI images and the reference images were calculated for different reconstruction approaches by
where M is the number of voxels, Ireference is the reference image, and Irecon is the reconstructed image.
Figure 3 shows the RRMS performance of different SNAPPI reconstructions in the 2D PPI simulations using five subjects' sagittal full-FOV scan with PE along y and x. All 2D SNAPPI reconstruction methods yielded significantly (P < 0.001) reduced RRMS compared to 1D SNAPPI. Nonseparable 2D SNAPPI reconstruction (2DPPI-nonSep-k and 2DPPI-nonSep-h) outperformed separable SNAPPI reconstruction (2DPPI-Sep-k, 2DPPI-Sep-h) (P < 0.001), and 2D SNAPPI with CAIPIRINHA (CAIPIRINHA-k and CAIPIRINHA-h) outperformed 2D SNAPPI with regular 2D PPI sampling paradigm (2DPPI-nonSep-k and 2DPPI-nonSep-h). No significant improvement was found by performing either separable or nonseparable SNAPPI with or without CAIPIRINHA reconstruction in the hybrid space (2DPPI-Sep-h, 2DPPI-nonSep-h, and CAIPIRINHA-h) compared to the 3D k-space (2DPPI-Sep-k, 2DPPI-nonSep-k, and CAIPIRINHA-k).
Figure 4, shows an axial slice of the reference image and the simulated 1D and 2D PPI images reconstructed by various SNAPPI reconstruction strategies using a representative subject's 3D fully sampled sagittal MRI data. Figure 4a is the reference, and Figs. 4b–h are images reconstructed by 1DPPI, 2DPPI-Sep-k, 2DPPI-Sep-h, 2DPPI-nonSep-k, 2DPPI-nonSep-h, CAIPIRINHA-k, and CAIPIRINHA-h, respectively. Below each reconstructed image is the difference image (from the reference) displayed using the same grayscale window. All reconstructed images present no perceptible aliasing artifact, while yield different noise patterns as shown in the difference images. Consistent with Fig. 3, 1DPPI (Fig. 4b), 2DPPI-Sep-k (Fig. 4c), and 2DPPI-Sep-h (Fig. 4d) yielded higher residual reconstruction errors than other methods. Except for the different artifacts indicated by arrows, there are no other obvious differences between the difference maps of 2DPPI-nonSep-k (Fig. 4e), 2DPPI-nonSep-h (Fig. 4f), CAIPIRINHA-k (Fig. 4g), and CAIPIRINHA-h (Fig. 4h).
Figure 5 shows the simulation results of 2D PPI using the axial full-FOV 3D MRI data with PE along z and y (dataset 2). Figures 5a–c are the three views of the reference image. Each column represents a different reconstruction approach as noted by the name underneath. The difference images are also shown right below the corresponding reconstructed images. Different from the simulation results using the sagittal 3D MRI data, significant reconstruction quality improvement was found in the reconstructed PPI images using CAIPIRINHA sampling paradigm (Figs. 5f, g, j, k, n, o) compared to those using regular 2D PPI sampling (Figs. 5d, e, h, i, l, m), especially along the z direction, which does not have much coil sensitivity variation for an effective PPI reconstruction for the product receiver array coil used in this paper. Only a slight difference was found between the images reconstructed in the hybrid space (Figs. 5e, g, i, k, m, o) and those reconstructed in the 3D k-space (Figs. 5d, f, h, j, l, n).
Figure 6 shows the in vivo 2D PPI imaging results. Both 2DPPI-nonSep-k (Fig. 6c) and 2DPPI-nonSep-h (Fig. 6d) yielded comparable image quality to the reference image (Fig. 6a).
Exploring k-space PPI reconstruction, this paper presented a surrounding neighbors-based PPI method as a generalization of a previously reported method, MCMLI (13, 14), which has demonstrated superiority to the original GRAPPA reconstruction by including more surrounding neighbors to recover the missing data(13, 14). Various 2D PPI reconstruction strategies were given by using separable or nonseparable SNAPPI in either 3D k-space or the hybrid space. The separable SNAPPI is similar to the 2D GRAPPA operator (11), while the latter used the GRAPPA reconstruction approach. Both simulations and in vivo imaging experiments demonstrated that nonseparable SNAPPI outperformed separable SNAPPI, which is similar to the 2D-GRAPPA operator for 3D MRI (11). Since separable SNAPPI uses precedently reconstructed missing data in the second data reconstruction step along another PPI direction, additional reconstruction errors may be introduced due to the imperfect reconstructions in the first PPI reconstruction step. For nonseparable SNAPPI, data reconstruction for each missing data point is based on acquired data, so there is no such issue of reentry of reconstruction errors. Also, with the same amount of calibration data the nonseparable SNAPPI can obtain more reference data sets than separable SNAPPI using the floating net based fitting approach (13, 14), which was used in this paper. By running nonseparable SNAPPI in the 3D k-space (2DPPI-nonSep-k), we assumed that the reconstruction coefficients for each readout were the same. When this ideal assumption is violated somehow in real PPI imaging, nonseparable SNAPPI in 3D k-space introduces a more complex noise pattern than nonseparable SNAPPI in the hybrid space, which applies SNAPPI fitting and reconstruction for each kz − ky plane so that the residual noise is confined to each image plane. However, no significant reconstruction quality difference was found in the simulations of nonseparable SNAPPI in the hybrid space and in the 3D k-space, even when an optimal sampling paradigm was used. This may be due to the amount of reference data used in the simulations. With the same amount of autocalibration data, SNAPPI in 3D k-space has inherently much more reference data than SNAPPI in the hybrid space, resulting in an improved reconstruction coefficient estimation, which may compensate the performance difference between SNAPPI in 3D k-space and SNAPPI in the hybrid space. Increasing the autocalibration data should manifest the performance difference but will also decrease the net AF. In this paper, we have already used a different amount of autocalibration data for SNAPPI in 3D k-space and in the hybrid space in this paper to partly compensate for the difference in real reference data amount between SNAPPI in 3D k-space ad SNAPPI in the hybrid space. Since the reconstruction coefficients must be estimated separately for each x plane in SNAPPI in the hybrid space, the total reconstruction coefficient estimation time is roughly max(x)/H times (max(x) is the image dimension along x, H is the number of neighboring points used along kx) that of SNAPPI in the 3D k-space. Considering the reconstruction quality, the reconstruction coefficient estimation time, and the amount of calibration data, nonseparable SNAPPI in 3D k-space (2DPPI-nonSep-k or CAIPIRINHA-k) is the best choice among all 2D SNAPPI variations presented in this paper. The in vivo imaging experiment demonstrated a successful 2D PPI with a net AF of 6 by applying PPI along y and x directions.
When one PPI direction does not have enough receiver coil sensitivity variation for an effective PPI reconstruction, the CAIPIRINHA sampling scheme (8–10) was used to control the aliasing along that direction by shifting an additional sampling step of every second PE step along the other PE direction. Simulations on PPI along y and x and especially on PPI along z and y showed better reconstruction quality by using the CAIPIRINHA sampling scheme rather than the regular Cartesian sampling paradigm.
In summary, this paper first gave a generalized framework of k-space autocalibrating PPI reconstruction, SNAPPI. Second, different k-space 2D PPI reconstruction strategies using separable or nonseparable SNAPPI in 3D k-space or the hybrid space were presented. Among those approaches, nonseparable SNAPPI in 3D k-space was demonstrated as a practical method in terms of reconstruction quality, the amount of reference data requested, and reconstruction coefficient estimation time. Third, in the case of applying PPI along one direction with insufficient receiver coil sensitivity variations, the CAIPIRINHA sampling paradigm is introduced to control the aliasing along that direction. Future work must be explored is SNAPPI for non-Cartesian sampling situations, SNAPPI kernel optimization, and noise features analysis.
The authors thank Dr. John A. Detre for manuscript revision and providing important comments.