SENSE and Simultaneous Multi Slice Imaging

Abstract

Purpose

Simultaneous multi-slice (SMS) acquisitions play an import role in the challenge of increasing single-shot imaging speed. We show that 2D-SENSE can be used to reconstruct SMS acquisitions with periodic but otherwise arbitrary undersampling patterns.

Theory and Methods

By adopting a 3D k-space representation of the SMS sampling process, the accelerated in-plane and slice encoding directions form a 2D reconstruction problem that is equivalent to volumetric CAIPIRINHA. 2D-SENSE does otherwise not distinguish between standard volumetric and SMS imaging with arbitrary CAIPIRINHA sampling.

Results

Use of the SENSE algorithm is demonstrated for in-vivo brain data obtained with blipped-CAIPRINHA sampling in 2D SMS-EPI and RARE acquisitions and 3D EPI with various in-plane and through-plane acceleration factors and CAIPIRINHA shifts. The proposed SENSE reconstruction works for any combination of SMS-factor and CAIPIRINHA shift by the addition of “dummy slices”, allowing for non-integer undersampling in the slice direction. Images with commonly used Slice-GRAPPA reconstruction are shown for reference.

Conclusion

SENSE is conceptually simple and provides a one-step reconstruction along both undersampled dimensions. It also provides a “contrast independent” parallel imaging reconstruction for SMS.

Introduction

Parallel undersampled simultaneous multi-slice (SMS) imaging (1,2) has recently gained popularity, especially for 2D single-shot sequence like Echo-planar imaging (EPI), where in-plane parallel imaging only results in marginal reductions of volume TR. The initial demonstrations of SMS-EPI (2) use a SENSE/GRAPPA approach to separate the slices (3). Blipped-CAIPIRINHA (2,4) for SMS acquisitions is generally preferred since it reduces g-noise penalty by distributing the aliasing energy more evenly in image space. However it is incompatible with SENSE/GRAPPA since sharp signal discontinuities arise when concatenating the FOV-shifted reference slices to form the SENSE/GRAPPA calibration data. These signal discontinuities can lead to phase errors in the reconstructed image. As a solution, Setsompop has proposed a multi-kernel GRAPPA method (“slice-GRAPPA”) where a separate kernel is fitted for every slice (2). Crucially, with the exception of abstracts by refs (5–7), most SMS reconstructions employ a two-step approach to first disentangle the aliased slices, and then perform in-plane parallel reconstruction, or vice versa. Here we propose to use SENSE (8) along two undersampled directions (2D-SENSE (9)) for reconstruction of simultaneously excited slices, as a general one-step approach to reconstruct SMS data with arbitrarily undersampled Cartesian k-space in phase and/or slice directions. A SENSE approach is conceptually straightforward, easy to implement and well understood. Because contrast independent coil sensitivities from a reference scan are used, a SENSE approach can easily be employed for cases where the contrast of the target image considerably differs from the reference scan. Furthermore, by using SENSE (or equivalently a one-step, 3D-GRAPPA (5,6)), the same reconstruction algorithm can be used to reconstruct SMS and conventional 3D imaging with or without CAIPIRINHA k-space patterns. This will facilitate SNR comparisons between SMS and 3D imaging modalities (e.g. SMS-EPI (2) and 3D-EPI (10–12)) since no algorithm implementation dependent bias is introduced.

In this work, a SENSE reconstruction is applied to blipped-CAIPI SMS-EPI acquisitions as well as a SMS-RARE sequence using CAIPIRINHA sampling and PINS excitation (13,14). Furthermore, as an example of 3D encoding following volumetric slab excitation, we show results from a 3D-EPI acquisition with 2D undersampling using a CAIPIRINHA pattern.

Theory

Simultaneous multi-slice imaging and 3D k-space

The motivation for 2D-SENSE reconstruction becomes obvious when adopting a 3D k-space representation of the SMS sampling process as described in (5) and (15). The read direction is assumed to be fully sampled, so the two remaining phase encode directions (in-plane and slice direction) form a 2D reconstruction problem that is mathematically equivalent to that of volumetric CAIPIRINHA (4). In this framework, the reconstruction algorithm does not distinguish between standard volumetric imaging with 3D Fourier encoding and SMS imaging with optimized k-space sampling pattern.

Given an undersampled k-space, the resulting image aliasing pattern is obtained by the Fourier Transform of the sampling pattern, with zeros and ones encoding skipped and acquired samples, respectively. The aliasing pattern is the point spread function (PSF) of a particular undersampling pattern in k-space, see Fig 1a and 1b. The total number of non-zero entries of the aliasing pattern defines the total acceleration factor AF_tot. For practical purposes, the aliasing pattern for a single voxel at location r_i= (r_xi, r_yi, r_zi) directly provides the matrix indices of the aliased voxels, ${\stackrel{\sim }{r}}_{\text{ij}}\text{ with j}=1\dots A{F}_{\mathit{tot}}$, that have to be separated. This allows for a most general and convenient implementation of the 2D-SENSE algorithm. For a more detailed description of SENSE with arbitrary undersampling patterns along two dimensions we refer to the literature, especially (9) and (4).

Figure 1.

Sampling pattern (AF_pe=2, AF_z=4, shift 2, matrix 96×96, 36 slices) and 2D-SENSE aliasing pattern for and SMS-EPI acquisition with in-plane undersampling. The grid size in slice direction is given by the slice acceleration factor of the multiband pulse (one SMS group). A total acceleration factor of 8 results in 8 non-zero PSF entries as displayed in b. Slice-GRAPPA and 2D-SENSE reconstructions are shown in c and d, respectively.

Fig 1a shows an example of four simultaneously excited slices (AF_z=4), factor-2 in-plane undersampling (AF_pe=2) and CAIPI-shift 2Δkz. The SENSE aliasing pattern in Fig 1b has AF_tot=2×4=8 non zero entries where the y-axis scales with the image resolution. Typically, the number of slices in an SMS group is on the same order or even smaller than the total acceleration factor. The definition of a quadratic sampling cell with length equal to the total acceleration factor as defined in (4) is therefore ambiguous and can not be readily applied to yield an analytical formulation of the coil sensitivity indices. Because of the periodicity of k-space it is possible to define dummy slices with zero signal intensity and fill up the sampling cell along the slice direction. However, for the sake of clarity we are using sampling cells (representing k-space undersampling) with the dimension along the slice direction equal to the number of slices in an SMS group.

Non-integer slice undersampling

Cartesian SENSE (8) generally is taken to imply a regular under-sampling pattern leading to a discrete image domain aliasing pattern (or PSF), and is mostly intuitively implemented with integer under-sample factors. For SMS Imaging and its typically low number of k-space points along the slice direction, equivalent to the number of simultaneously excited slices, non-integer slice undersampling factors occur frequently. Note that a distinction needs to be made between “slice acceleration” which is given by the number of simultaneously excited slices and hence determines the temporal speed-up of the acquisition, and “slice undersampling” factor which denotes the ratio between the sampled reduced FoV and the nominal full FoV (Nslices*slice gap) along the slice direction. For instance, AF_z=8, CAIPI-3 corresponds to 8/3-fold slice undersampling because only three out of eight planes in k-space are sampled (see Figure 2). The k-space increment in slice direction (CAIPI blip moment) is then given as $\mathrm{\Delta }{k}_{zR}=2\frac{2}{3}\mathrm{\Delta }{k}_z$, where Δk_z is the k-space increment that fulfills the Nyquist condition. This results in acquired samples that do not fall onto Cartesian grid points for a k-space with N=8 as shown in Figure 2a. Slightly changing the applied k-space moment, so that the acquired points lie on the grid, destroys the periodic sampling pattern under circular shifts of k-space. By adding a “dummy slice” with zero intensity (Figure 2b), we effectively increase the FOV in the slice direction and therefore decrease the k-space increment $\mathrm{\Delta }{k}_z^{\mathit{ext}}$ for full sampling. The k-space maximum, however, remains constant because it only depends on the slice separation d and is given as $k_{\mathit{zmax}}=k_{\mathit{zmax}}^{\mathit{ext}}=\frac{1}{2d}$. Because the number dummy slices was chosen to result in an integer slice acceleration factor, the acquired k-space points (identical to Figure 2a) now fall onto the Cartesian grid for N_ext=9 slices and the problem becomes trivial to solve with SENSE. This approach of increasing the FOV by adding virtual slices in image space is called extended FOV reconstruction.

Figure 2.

Extended FOV reconstruction for non-integer slice undersampling. Adding a dummy slice in image space reduces the k-space increment so that the acquired samples fall onto the Cartesian grid. The dummy slices are removed in a post-processing step.

Similar in implementation, the extended FOV reconstruction must not be confused with conventional zero-padding of k-space in order to virtually increase the spatial resolution. In case of zero-padding, the k-space increment is maintained and k-space is extended beyond its original maximum thereby increasing the apparent spatial resolution but keeping the FOV constant. In contrast, the extended FOV reconstruction changes the FOV and thereby the target k-space increment while maintaining the k-space maximum and therefore the spatial resolution. The actual acquisition is not affected, and only the target k-space into which the acquired samples are sorted, is adapted in order to achieve periodic k-space undersampling on a Cartesian grid.

An extended FOV reconstruction is only necessary in the context of a Cartesian SENSE type reconstruction that operates on periodic aliasing in the image domain. A non-Cartesian, iterative reconstruction approach (e.g. CG-SENSE (16)) is able to directly reconstruct the acquired k-space pattern regardless of whether it is on a Cartesian grid or not. The latter is also true for GRAPPA type methods that operate directly in k-space, such as the Slice-GRAPPA.

Methods

Image acquisition

Experiments were performed on the Siemens 3T Tim Trio at the MR Research Center of the University of Hawaii and the Siemens Magnetom 7T scanner at Scannexus (www.scannexus.nl). Both scanners are equipped with 32-channel head receive arrays (3T : Siemens Healthcare, Erlangen, Germany; 7T: Nova Medical, Wilmington, MA). Data were acquired in N=4 healthy subjects after obtaining informed consent under the institutionally approved protocols. Image reconstructions were performed offline using Matlab (The Mathworks, Natick, MA).

Image acquisition and reconstruction

We show two examples of SMS reconstruction with SENSE. A single-shot blipped-CAIPI SMS-EPI acquisitions at 7T with matrix 96×96, 2.5 mm isotropic voxels, 36 slices in 9 SMS slice groups, AF_z=4, AF_pe=2, CAIPI-factor 2, slice separation d=20 mm; and an SMS TSE/RARE acquisition at 3T with PINS (power independent number of slices) multi-slice excitation (14) with matrix 256×256, 56 slices in 7 SMS-slice groups, AF_z=8, AF_pe=1, CAIPI-factor 3, slice thickness 3mm, SMS slice distance d=21 mm, FoV = 250×250mm, in-plane resolution 0.98mm, BW=130Hz/pix that was acquired in 8 shots with ETL=32 (see ref (14) for further details).

To demonstrate the applicability of the reconstruction approach to volumetric as well as SMS acquisitions, two CAIPIRINHA 3D-EPI and blipped-CAIPI SMS-EPI datasets with corresponding geometry and undersampling parameters were acquired at 3T from the same subject in immediate succession. Identical acquisition parameters were used and the k-space undersampling pattern was matched so the total number of slices in case of the SMS-EPI acquisition was identical to the number of phase encoding steps along the slice direction in case of 3D-EPI. The acquisitions hence employ the exact same blipped EPI readout trajectory and differed only in slice phase encoding / slice selection and excitation TR. The imaging parameters were: matrix size 96×96, FoV = 250×250mm², 2.3×2.3×3 mm³ voxels, 72 slices in 18 SMS slice groups (or 72 partitions acquired with 18 excitations in case of 3D-EPI), slice acceleration AF_z=4, in-plane acceleration AF_pe=2, CAIPI-factor 2, SMS slice separation d=41.5 mm. For details on the CAIPIRINHA 3D implementation, see ref (10).

For all SENSE based reconstructions, coil sensitivity maps were derived from fully sampled short-TE-TR low resolution GRE scans covering the acquisition volume, subsequently dividing each coil by the sum-of-squares combination of all coil images and additional spatial smoothing with a 3×3×3 Gaussian kernel.

For comparison, the 2D SMS data were also reconstructed using the slice-GRAPPA method (2) with a 3×4 kernel, followed by a regular GRAPPA (17) reconstruction with 3×4 kernel in case of additional in-plane undersampling.

Results

SMS-EPI

Fig 1 shows the reconstruction of a 96×96 blipped-EPI reconstruction with twofold in-plane undersampling (AF_pe=2). The slice-GRAPPA reconstruction in the left column (c) and the SENSE reconstruction in the right column (d) result in images with minor differences. The two-fold in-plane undersampling results in an aliasing pattern with more than one non-zero entry per slice. The combined effect of in-plane and slice undersampling, however, does not result in a FOV/2 shift between neighboring slices. Only a shift by one quarter of the FOV is present for neighboring slices. This observation does not affect the actual reconstruction, neither for the slice-GRAPPA method, nor for the SENSE approach. However it provides an instructive illustration that the reconstruction problem is best described with two dimensions, assuming that the read direction is fully sampled.

SMS-RARE

Fig 3 shows the extended FoV SENSE reconstruction of the RARE (13) data. The sampling pattern and the aliased PSF are shown in Fig 3a and b. The dummy slices, added to facilitate integer SENSE reconstruction, and the resulting virtual entry in the aliasing pattern are indicated in gray. In Figure 3c one reconstructed SMS group (sagittal orientation) is shown, including the dummy slice which is reconstructed as a regular slice with zero intensity. The dummy slices are then removed in a post-processing step. Fig 3d displays the reconstructed volume as three orthogonal slices.

Figure 3.

Reconstruction of a PINS-RARE dataset with dummy slices. The extended FoV undersampling and aliasing patterns for (AF_pe=1, AF_z=8, CAIPI-3) are shown in a and b, respectively. One SMS group reconstructed with 2D-SENSE including the additional dummy slice is shown in c. Three orthogonal slices out of the full volume are shown in d.

Common reconstruction pipeline for 3D- and SMS-EPI

Fig 4 demonstrates the ability of the SENSE approach to reconstruct 3D datasets either acquired with conventional slab or SMS excitation. In Fig 4a the k-space sampling patterns for both imaging modalities are shown (orange for SMS-EPI and blue for 3D EPI). Because the acceleration parameters are identical in both cases (AF_z=4, AF_pe=2, CAIPI-shifts 2) the 3D k-space for the scan with volume excitation and partition encoding can be obtained by “stacking” replicas of the SMS k-space (depicted with gray background) 18 times. The effective volume TR is consequently the same for both scans (TR_vol=600ms, 18 × the slice excitation TR which was TR=50ms). Exactly the same reconstruction pipeline and coil sensitivity estimation was used for both acquisitions with the obvious exception of the k-space dimension along the slice direction (4 in case of SMS-EPI and 72 in case of 3D-EPI). The corresponding reconstructions are shown in Fig 4b for the SMS-EPI scan and in Fig 4c for the 3D-EPI scan. The expected contrast difference between the 2D and 3D EPI scans naturally results from the difference in tissue excitation rate (600ms vs. 50ms). The identical volume TR however makes them directly comparable choices for fMRI use. The SENSE reconstruction approach performs equally well for 3D and SMS data with no visible artifacts due to the slice or in-plane acceleration. Also, since both methods share the same coil sensitivity data (derived from a short TR prescan) and undersampling pattern, the pixel locations that fold onto each other are also identical and therefore each sub-problem of the SENSE algorithm has identical coil sensitivity weighting coefficients. Since the g-factor penalty only depends on the coil sensitivity weighting coefficients and potential noise-correlations between the coils ψ by $g_p=\sqrt{{(S^H{\mathrm{\Psi }}^{-1}S)}_{\mathit{pp}}^{-1}{(S^H{\mathrm{\Psi }}^{-1}S)}_{\mathit{pp}}}$ (see ref. (8)) it is also identical for both modalities. Any observable difference in SNR or temporal SNR between 2D and 3D-EPI can thus be attributed to sequence specific contributions like 2D vs. 3D excitation, in-flow effects, effective TR difference etc.

Figure 4.

Common reconstruction pipeline for SMS-EPI and volumetric 3D-EPI acquisitions. In a the sampling pattern for SMS-EPI is depicted with orange squares. Because both acquisition modalities use identical undersampling parameters (AF_z=4, AF_pe=2, CAIPI-shifts 2) the pattern for 3D-EPI (blue squares) can be generated by replicating the SMS-EPI pattern along the slice phase encoding dimension. In b and c the reconstructed volumes are shown for SMS-EPI and 3D-EPI, respectively.

Discussion

The present work demonstrates that a 2D-SENSE approach can be used for a wide variety of reconstructions with optimized k-space undersampling patterns for both volumetric and SMS excitation (i.e. blipped-CAIPI EPI, PINS-TSE, and 3D EPI with volumetric CAIPIRINHA). A 2D-SENSE approach is conceptually simple and easy to implement. In case of additional in-plane undersampling it provides a one-step reconstruction along both undersampled dimensions, which may be potentially numerically more stable than a two-step approach. It also provides a “contrast independent” parallel imaging reconstruction by using actual coil sensitivity maps instead of relying on a training dataset. The aim of this paper has not been to perform a direct comparison of SNR or other reconstruction properties of SENSE with other methods like slice-GRAPPA. Well tuned reconstructions of both types can be expected to deliver images of very comparable image quality. As in other GRAPPA type reconstructions, the g-noise distribution (18) in a two step slice-GRAPPA / GRAPPA reconstruction may be smoother than in a SENSE reconstruction, but with comparable values. GRAPPA reconstruction may be less sensitive to imperfections in the reference data, since the inverse problem is highly over-determined, whereas a good SENSE reconstruction stands and falls with the quality of the sensitivity map. A conceptual advantage of SENSE type approaches is the contrast-independence between the sensitivity maps and the undersampled data, which may lead to improved reconstructions for highly diffusion-weighted, perfusion weighted or angiographic acquisitions, whose contrast spatial frequency spectrum differs significantly from typical calibration data with flat contrast. Another method that deals with the contrast problem is given by the so called Split Slice-GRAPPA (19) approach which is based on optimized kernels. One practical advantage of SENSE reconstructions is that they are highly parallelizable on GPU hardware, with minimal working memory requirements, since each set of aliased voxels can be reconstructed completely independently of the rest. We also demonstrated that by using 2D-SENSE, a single reconstruction pipeline can be used to reconstruct volumetric (i.e. 3D-EPI) and SMS data (i.e. blipped-EPI). For the ongoing debate over the advantages/disadvantages of segmented volumetric EPI versus 2D single-shot EPI, this might prove very helpful to perform a fair comparison in terms of temporal SNR without introducing any bias from a sequence specific reconstruction method. Since g-noise will be the same in both, 2D-SMS and 3D reconstructions, any observed differences can unequivocally be attributed to differences in the imaging experiment itself, such as (functional) imaging contrast and sensitivity, or differing contributions of physiological fluctuations.