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Magn Reson Med. Author manuscript; available in PMC 2013 May 1.

Published in final edited form as:

PMCID: PMC3323676

NIHMSID: NIHMS306253

Kawin Setsompop,^{1,}^{2} Borjan A. Gagoski,^{3} Jonathan R. Polimeni,^{1,}^{2} Thomas Witzel,^{1,}^{4} Van J. Wedeen,^{1,}^{2,}^{4} and Lawrence L. Wald^{1,}^{2,}^{4}

Corresponding author: Kawin Setsompop, Athinoula A. Martinos Center for Biomedical Imaging, Department of Radiology, Massachusetts General Hospital, Charlestown, MA 02129, USA, Email: ude.dravrah.hgm.rmn@niwak

The publisher's final edited version of this article is available free at Magn Reson Med

See other articles in PMC that cite the published article.

Simultaneous multi-slice EPI acquisition using parallel imaging can decrease the acquisition time for diffusion imaging and allow full-brain, high resolution fMRI acquisitions at a reduced TR. However, the unaliasing of simultaneously acquired, closely spaced slices can be difficult, leading to a high g-factor penalty. We introduce a method to create inter-slice image shifts in the phase encoding direction to increase the distance between aliasing pixels. The shift between the slices is induced using sign- and amplitude-modulated slice-select gradient blips simultaneous with the EPI phase encoding blips. This achieves the desired shifts but avoids an undesired “tilted voxel” blurring artifact associated with previous methods.

We validate the method in 3× slice-accelerated spin-echo and gradient-echo EPI at 3T and 7T using 32-channel RF coil brain arrays. The Monte-Carlo simulated average g-factor penalty of the 3-fold slice accelerated acquisition with inter-slice shifts is <1% at 3T (compared to 32% without slice-shift). Combining 3× slice acceleration with 2× in-plane acceleration, the g-factor penalty becomes 19% at 3T and 10% at 7T (compared to 41% and 23% without slice-shift). We demonstrate the potential of the method for accelerating diffusion imaging by comparing the fiber orientation uncertainty, where the three-fold faster acquisition showed no noticeable degradation.

Diffusion weighted imaging and fMRI are widely used for studying the structure and function of the brain. These studies typically rely on rapid, single-shot 2D EPI acquisition methods. However, for high resolution imaging where a large numbers of slices are needed to cover the brain, a long TR is required. This renders the method inefficient compared to 3D encoding methods (1). Conventional accelerated 2D parallel imaging approaches (2-4) can greatly increase the speed of the EPI encoding by eliminating phase encoding steps. While beneficial for other reasons, this does not translate to a significant reduction in TR or acquisition time, since diffusion and fMRI sequences contain large, fixed time blocks that cannot be shortened, such as the time for diffusion encoding or the time to a suitable TE for T_{2}* contrast.

In comparison, accelerating the data acquisition using the simultaneous acquisition of multiple slices can be very effective since it directly reduces the amount of time needed to acquire a fixed number of slices. For example, if three imaging slices are acquired per shot instead of one, the total acquisition time decreases directly by a factor of three. When TR T_{1}, e.g., if full-brain coverage is required at high resolution, then the acquisition can maintain a nearly fully-relaxed equilibrium magnetization even for 3× acceleration. Furthermore, unlike standard parallel imaging techniques, simultaneous multi-slice acquisition methods do not shorten the readout period or omit k-space samples. Therefore, they are not subject to a
$\sqrt{R}$ penalty on SNR (where *R* is the acceleration factor) found in conventional parallel imaging acceleration.

Various methods have been proposed for single-shot simultaneous multi-planar imaging including Echo Volumar Imaging (EVI) and its variants (1,5-8). Multi-slice methods using slice selection to excite multiple slices simultaneously include the “Wideband” imaging (9-11), Simultaneous Echo Refocusing (SER) (12) sometimes referred to as Simultaneous Image Refocused (SIR) (13) and parallel image reconstruction based multi-slice imaging (14-17). However, each of these methods has limitations. EVI and its variants offer the advantages of true Fourier encoded 3D single-shot imaging, but are limited by susceptibility distortions in the slowly encoded k-space direction as well as the difficulty performing the entire 3D trajectory within the T_{2}* decay time. Thus EVI requires significant parallel imaging acceleration to begin to overcome these issues (8). The Wideband technique applies a slice gradient (*G _{z}*) during the readout gradient (

A second class of simultaneous multi-slice methods uses parallel imaging concepts to unalias the pixels from slices excited and encoded simultaneously (14). Unfortunately the aliased slices are generally close to each other due to a comparatively smaller FOV in the slice direction. For example, a 60 slice 2 mm isotropic acquisition with 3× multi-slice acceleration would require unaliasing pixels 4 cm apart in the slice direction. This pixel separation is equivalent to that of a conventional parallel imaging reconstruction for a brain image with a 20 cm FOV_{inplane} and *R*=5 acceleration. The short distances between aliased pixels places a high demand on the spatial variations in the coil sensitivities and results in unaliasing artifacts and a high g-factor penalty. Parallel imaging based methods have recently been combined with the SER method to extend the number of simultaneous slices (18,19) and this approach has been recently explored for fMRI and diffusion applications (19).

Modifications to the multi-slice parallel imaging technique have been proposed to mitigate the high g-factor issue. The “Controlled Aliasing In Parallel Imaging Results In Higher Acceleration” (CAIPIRINHA) technique (15) introduces an in-plane image shift between the simultaneously acquired slices to increase the distance between aliasing voxels and thereby make them easier to separate. For the conventional FLASH acquisitions for which the method was demonstrated, this is achieved by using a different RF pulse for every other k-space line. The multiband pulse modulates the phase of the magnetization excited in the individual slices for each k-space line. For example, alternating the phase of every other k-space line’s excitation by π will result in a spatial shift of FOV/2 in the phase encoded direction for that slice. Unfortunately, this technique is not applicable to EPI where all the phase encoded lines are read out after a single RF excitation. An alternative approach, suitable to EPI, was proposed based on the Wideband method (10,16). In the first method (10), the Wideband technique is not used as the principal method of separating the aliased slices, but is used to assist the parallel imaging method by introducing relative in-plane shifts in the readout direction between the simultaneously acquired slices (by applying a constant slice-select gradient during the readout). In the second method (16), a shift in the phase encoded direction is also applied to introduce further distance between aliasing pixels. The shift in the phase encoded direction is achieved by applying *G _{z}* blips simultaneous with the EPI phase encoding blips. In either case, the information from the parallel acquisition is used to de-alias the slices and the CAIPIRINHA-style relative image shifts are used to reduce the g-factor. Because the shift is not used to fully separate the slices, it can be smaller than the whole FOV shift that is normally used in the Wideband method—which reduces, but does not eliminate, the voxel tilting artifact that results from having the slice gradient and encoding gradient on simultaneously. For example, applying

In this study, we extend the method of Nunes and Larkman (16) by introducing a technique we term “blipped-CAIPI”, which can be used in EPI to achieve spatial shifts in the phase encoded direction between simultaneously excited slices but without the voxel tilt problem. This allows for unrestricted application of image shifts in the phase encoded direction to gain the maximal benefit in g-factor reduction. We show that the method introduces negligible loss of signal or blurring compared to the wide-band or blipped wide-band methods. We estimate the resulting g-factor using measured noise covariance information and a Monte Carlo pseudo-multiple replicate time-series analysis and demonstrate 3× simultaneous multi-slice acquisitions in GRE- and SE-EPI sequences. We also combine the method with *R*=2 in-plane GRAPPA and/or a 2× SER excitation. The combination with the SER method achieved 6 slices per single-shot (echo train) in a GE-EPI acquisition.

A brief explanation of the Wideband approach and its associated tilted voxel artifact is provided in the context of achieving an inter-slice image shift in the EPI phase encoded (PE) direction, i.e., blipped-Wideband (16). The blipped-CAIPI scheme will then be introduced as a modification to the blipped-Wideband approach to remove the tilted voxel artifact. The basic method of imparting a FOV/2 shift in the PE direction will be described using an example of a two simultaneously excited slices, one of which is at isocenter, as shown in Fig. 1A. However, a similar analysis applies to cases at off-isocenter locations and to cases with differing numbers of simultaneous slices and inter-slice image shifts.

A description of the blipped-Wideband and blipped-CAIPI methods for creating FOV/2 inter-slice image shift between two simultaneously excited sliced (the bottom slice at isocenter) and the source of the blurring artifact (tilted voxel) of the blipped-Wideband **...**

The blipped-Wideband gradient scheme is shown in Fig. 1B (left). A train of constant gradient blips in the slice-select direction (*G _{z}*) is applied simultaneously with the conventional

The voxel tilting artifact associated with this technique arises from the finite thickness, Δ*z*, of the slices. Each *G _{z}* blip introduces a ±

A diagram of the undesirable additional phase at the edges of each slice is shown in Fig. 1B. Fig. 1C shows the phase variation *θ*(*z′, N _{y}*) within the slices as a function of

The blipped-CAIPI technique aims to achieve the same inter-slice image shift as the blipped-Wideband technique but without the undesirable blurring/tilting artifact. This is achieved by modifying the *G _{z}* blips so that they impart the desired phase modulation along

The scheme presented in Fig. 1 illustrates the blipped-CAIPI technique for a special case where one of the excited slices is at the isocenter position. Fig. 2 also shows the case where neither slice is at isocenter (case 2; *orange slices* in Fig. 2A) as well as the case described above where one of the slices is at isocenter (case 1; red slices). Again, the phase vectors in Fig. 2B represent the signal phase of even and odd k-space lines for each slice. Here, only the slice center is considered (as the through-slice phase variation is unchanged from before). For the non-isocenter case, an application of a single positive *G _{z}* blip adds

Generalization of the blipped-CAIPI 2-slice FOV/2 shift method to the case where neither slice is at isocenter. **(A)** Case 1, where one slice at isocenter (*red*), and Case 2, the general case with neither slice at isocenter (*orange*). **(B)** The phase at the **...**

Fig. 3 shows the blip schemes and phase diagrams depicting the relative phase between the excited slices for spins at the slice center for various types of inter-slice shifts (FOV/3, FOV/4 in addition to the FOV/2 shift). The FOV/2 diagram shows the blip pattern and resulting phase of the even and odd lines, which produces a FOV/2 shift for every other slice. This pattern is clearly useful for two simultaneously excited slices, but can also be used for 3 simultaneously excited slices. In this case the shifts would be 0, FOV/2 and 0 for the 3 slices. Thus, a *z* variation in coil sensitivity patterns is needed to un-alias the first and third slices. For a coil without a *z* variation in coil sensitivity patterns (elements distributed in *x* and *y* only), it is desirable to give each slice a unique shift relative to the bottom slice. For example, if 3 slices are simultaneously excited, then the shifts might be 0, FOV/3 and 2FOV/3. The aliased pixels are then always separated in *y*. This scheme is shown in the FOV/3 diagram shown in Fig. 3. Similarly, the FOV/4 shifts are useful when 4 slices are excited and no *z* variation is present in the coil array, or if 5 slices are simultaneously excited and the array has some *z* variation.

Generalization of the blipped-CAIPI method to FOV/3 and FOV/4 shifts between successive slices. Each panel shows: *top, G*_{z} gradient scheme; and *bottom*, corresponding phase diagrams between excited slices and at slice edges of blipped-CAIPI acquisition **...**

Fig. 3 also shows the extra phase accrued at the edges of the slices. Due to periodic rewinding of the gradient moment in blipped-CAIPI, no significant phase accrual is allowed to build up over the EPI readout. Furthermore with the balancing blip in place, the phase states are always centered on the zero phase position. For the FOV/2 shift this was shown to result in complete cancelation of the image ghost when the signal is uniformly distributed through the slice. For other shift factors, the balancing blip reduces the ghost artifact but does not eliminate it. Nonetheless, the ghost level is negligible for typical acquisitions.

Since the phase per mm imparted by the blip scheme depends on the distance between simultaneously excited slices (e.g. two simultaneously excited slices must receive a π relative phase shift for a FOV/2 shift), the phase incurred across the slice is dependent on the ratio of the slice thickness to the distance between simultaneously excited slices. To assess this effect, we analyze a nearly worst-case example. We assess the ghost level for the somewhat extreme case where a FOV/3 shift is applied to an acquisition with a slice thickness of 2 mm and a 6mm spacing between simultaneously excited slices. A small distance between simultaneously excited slices is potentially useful for cardiac imaging where FOVz is small. This resulted in a delta point-spread function with ghosting artifacts at ~ ± FOV/6 locations. When the balancing blip was employed in the acquisition, the resulting ghost level was modeled to be 0.6% and the delta response signal level was found to be 98.5% of the expected (ideal) value when no through-slice signal variation was present. In contrast, the ghost level without the balancing blip was 2.2% and the delta response was 96.4%. We also modeled the case with through-slice signal variation assuming that the signal has a 45° linear phase variation over the slice thickness. Using the balancing blip, the ghost and signal levels were 2.3% and 96% in this case. Without the balancing blip they were 4.2% and 89%.

Blipped-CAIPI was implemented in both SE-EPI and GRE-EPI sequences to assess the technique for both diffusion weighted imaging and fMRI. All acquisitions were obtained from healthy subjects after obtaining informed consent using an institutionally approved protocol. Imaging was performed using a Siemens scanner (Siemens Healthcare Erlangen, Germany) equipped with a 32-channel head array coil. The 3T acquisitions used a Siemens whole-body TIM Trio scanner and the commercial 32-channel head array coil and the 7T acquisitions used a 7T Siemens whole-body system equipped with AC84 head gradients (80 mT/m maximum gradient strength and 400 T/m/s maximum slew rate). A custom-built 32-channel RF head array was used for reception, and a custom-built detunable band-pass birdcage coil for transmit (20).

The SE-EPI diffusion weighted imaging was performed with a twice-refocused sequence (21,22) using either conventional single-slice imaging or with a 3× slice-accelerated simultaneous multi-slice acquisition. No in-plane acceleration was used. Imaging parameters were: resolution=2 mm isotropic; FOV = 208×208×120 mm; Partial Fourier = 6/8; Bandwidth = 1658 Hz, *b* = 1000 s/mm^{2}, 64 directions, one *b* = 0 image, 60× 2 mm slices and TE=96 ms. For the conventional acquisition, TR = 9s and the total acquisition time (*T*_{acq}) was 9.75 min. For the simultaneous multi-slice acquisition, 3 slices separated by 4 cm were simultaneously excited with an FOV/2 shift applied to the middle slice (i.e., the scheme depicted in Fig. 3A). The 3× slice accelerated sequence resulted in TR=3 s and *T*_{acq}= 3.25 min.

A GRE-EPI sequence with 1 mm isotropic spatial resolution was acquired at 7T to illustrate the potential benefit of blipped-CAIPI technique for high resolution fMRI. The 3× slice-accelerated simultaneous multi-slice acquisition allowed the whole-brain acquisition (120 slices) with a TR of 2.88 s. For this acquisition, 2× in-plane acceleration was employed to reduce EPI image distortion. Imaging parameters were: resolution = 1 mm isotropic; FOV = 200×200×120 mm; Partial Fourier = 6/8; TR/TE = 2.88 s / 24 ms; effective echo-spacing = 0.32 ms, Bandwidth = 1786 Hz, 120 1 mm slices, flip angle = 80° (approximately the Ernst angle). The simultaneous multi-slice acquisition used 3 simultaneously excited slices separated by 4 cm. Due to the 2× in-plane acceleration, an intra-slice image shift of FOV/4 was employed to prevent voxels with same in-plane (*x,y*) locations of adjacent image slices from overlapping. This was achieved by applying the FOV/2 Blipped-CAIPI *G _{z}* train to the reduced FOV acquisition.

In addition to these acquisitions, the blipped-CAIPI method was also demonstrated in combination with Simultaneous Echo Refocusing (SER) technique to achieve 6 slices per shot in a 3T GRE-EPI acquisition. This acquisition used *R*=3 blipped-CAIPI together with SER to achieve an extra 2× slice acceleration factor. An *R*=2 in-plane acceleration was employed to counteract the lengthened echo-spacing produced by the SER method. The SER method increased the echo spacing by ~50%, yielding a net ~25% improvement in susceptibility induced distortion given the *R*=2 in-plane acceleration. The parameters for the acquisition were: resolution = 2 mm isotropic; FOV = 208×208×120 mm; Partial Fourier = 6/8; Bandwidth = 1093 Hz, TR/TE = 0.77 s / 33 ms; effective echo-spacing = 0.525 ms, 60 2 mm slices, flip angle = 50° (approximately the Ernst angle). A FOV/4 intra-slice image shift was used due to the in-plane acceleration.

To perform simultaneous multi-slice excitation, conventional slice-selective RF pulses were frequency modulated and summed. The VERSE method (11) was used to reduce the peak RF voltage and thus the SAR of the excitation pulses. Bloch simulation was employed to assess the slice selection performance of the multi-slice RF pulses in comparison to the standard RF pulses normally used by the system. To compensate for the degradation in the slice selection profile at off-resonance frequencies due to VERSE, the Time BandWidth (TBW) product was increased compared to the conventional pulses. The VERSEed multi-slice RF pulse was deemed acceptable when: (*i*) the root mean square error of the excited slice profile was approximately equal to or less than that of the standard RF pulse over an off resonance frequency range of +/- 50Hz, (*ii*) the sequence would run without exceeding the SAR threshold, and (*iii*) the pulse duration was below 5.5 ms.

For the SE-EPI acquisitions, the pulses were designed using the SLR algorithm (10). The VERSE method was applied more heavily to the 180° pulses. For the 90° excitation, the TBW product, VERSE factor and pulse-length were 6, 3× and 4.8 ms respectively. For the 180° refocusing pulses, these parameters were 6, 6× and 5.38 ms. The normalized root mean square error (nRMSE) from the ideal profile (normalized by the area under the ideal profile) for the 90°-180°-180° sequence was 17.6% at 0Hz and 28.5% at +/-50 Hz off resonance for our designed RF pulses. This was an improvement over the standard RF pulse train (27.4% and 32.1% for spins off resonant by 0Hz and 50Hz). For the GRE-EPI acquisitions, modulated Hanning-windowed sinc pulses were used with a TBW product, VERSE factor and pulse-length of 6, 3× and 4.8ms for both the 3T blipped+SER acquisition and the 7T acquisition. For the GRE-EPI the nRMSE of the slice profile was 20.7% for on resonant spins and 27% for +/-50 Hz off resonance spins. The nRMSE of the standard pulses were 23% and 26%. For the blipped+SER acquisition the pulse duration reported here represents the duration of each sub-pulse in the two SER excitations module.

The SENSE algorithm (3) can be directly adapted for simultaneous multi-slice imaging acquisition (14,15). On the other hand, direct application of the GRAPPA method is less straightforward and previous applications of GRAPPA to slice-aliased data have used a SENSE/GRAPPA combination method (23). Although the method works well for standard multi-slice acquisitions, direct application of this method causes significant aliasing artifact in multi-slice acquisitions with inter-slice image shifts. To overcome this issue, we developed an alternative k-space based parallel imaging approach for multi-slice imaging, which we term *slice-GRAPPA*. In brief, the slice-GRAPPA algorithm fits a GRAPPA-like kernel to each slice of a pre-scan calibration data set acquired one slice at a time and then applies these kernels to estimate the k-space data of each individual imaging slice from the collapsed slices. Thus, for the three-fold slice-accelerated acquisition, three separate sets of GRAPPA kernels were fitted and applied, one for each imaging slice. The issues in applying SENSE/GRAPPA to CAIPIRINHA shifted slices and the slice-GRAPPA algorithm are described in detail in Appendix A.

For the acquisitions with both in-plane and slice acceleration, the reconstructions were performed in sequential steps: first the slice-GRAPPA was applied to separate the aliased slices, then conventional GRAPPA was used to generate the missing k-space lines for the in-plane under-sampled slices. For acquisitions at 3T, the reconstructed images from the individual coil channels were combined using the root Sum-of-Square (rSoS) method. For acquisition at 7T where the coil channels are more prone to coupling, a coil noise-covariance-weighted rSoS method was applied to improve SNR.

As with any parallel imaging method, simultaneous multi-slice technique brings with it non-uniformity in noise enhancement which complicates the interpretation of the statistical analysis of fMRI activation and diffusion imaging. The SNR performance of the simultaneous multi-slice technique was calculated using the pseudo-multiple replica method (24) and compared to the SNR of the unaccelerated acquisition also assessed with the pseudo-multiple replica method. We start by applying the pseudo-multiple replica method to fully slice-sampled data (also with no in-plane acceleration). The signal level was taken as the average image signal in a 20 image time series taken from this fully sampled data (one slice per EPI readout), which was taken to be a “noise-free” estimate of the signal. Noise was added to each of these images by drawing from the noise distribution defined by the measured noise covariance matrix. The SNR is then assessed by computing the mean value of each pixel in the 1000 image pseudo time series of the coil-combined images and dividing by its calculated standard deviation over the pseudo time series.

For the slice-accelerated case (but without in-plane acceleration), the pseudo time-series of k-space data was formed by aliasing the slice data with or without spatial shifts prior to adding the noise. In both cases, the appropriate slice-GRAPPA kernel was fit and applied to the pseudo time series. For the slice-accelerated case with in-plane acceleration, the signal model was derived from the time average of 20 images acquired with conventional (*R*=2) in-plane acceleration and GRAPPA reconstruction (and no-slice acceleration). This was done to insure that the object shape in all models contained identical B_{0} distortions.

The ratio between SNR maps of the slice-accelerated and conventional acquisitions allowed us to create a map of the “*SNR retained*” by the image reconstruction. In general, the SNR retained is
$1/g\sqrt{{R}_{\mathrm{inplane}}}$ and for case of no in-plane acceleration, the *SNR retained* is just 1/*g*. Thus, for all acquisitions, we plot the 1/*g* maps as determined by the *SNR retained* calculation. The g-factor calculations were obtained for simultaneous multi-slice acquisition both with and without inter-slice image shift in order to assess the effect of CAIPIRINHA shifts. Values for 1/*g*_{ave} were obtained from a brain-like ROI.

Non-accelerated and 3× slice-accelerated diffusion weighted imaging acquisitions were compared using qualitative and quantitative metrics from FSL (25). First, tensor-derived, directionally-encoded colormaps for both datasets were generated and compared. Furthermore, bedpost (26) was used to estimate samples from the posterior probability density function (PDF) of the principal and crossing fiber orientations of both acquisitions. These estimated samples of the PDFs were used to calculate and map the 95% uncertainty angle for the first and second fiber orientations.

Fig. 4A shows the results of the 3× slice-accelerated blipped CAIPIRINHA SE-EPI acquisition, including views of an unaliased 3D stack of slices and the corresponding unaliased slice group. Fig. 4B shows the blipped-CAIPI aliased slice and the corresponding g-factor maps. Fig. 4C shows the same information for the non-blipped acquisition. The 1/*g* average ± standard deviation in the brain region was 0.997 ± 0.08 for the blipped-CAIPI and 0.68 ± 0.14 for non-blipped simultaneous multi-slice method indicating an average SNR improvement of 47%. Furthermore, the blipped-CAIPI acquisition reduced the peak g-factor penalty (taken from the peak g-factor after smoothing the g-maps with a 5×5 voxel square kernel) by more than a factor of 2 (1/*g*_{peak} of 0.84 vs. 0.41) and thereby provided an SNR improvement of >100% in the high noise enhancement region. For this acquisition the blipped-wideband technique (16) would have produced a voxel tilting artifact of ~3.5 voxels.

Results from 3× slice-accelerated SE-EPI acquisition with FOV/2 inter-slice shift. **(A)** Unfolded images of the unaliased 3D volume; *left*: coronal and sagittal views, *right*: axial views of an unalised slice group. **(B)**
*Left*: aliased image of blipped-CAIPI **...**

Fig. 5 shows the diffusion orientation color-coded Fractional Anisotropy (FA) maps for conventional non-accelerated slices and a blipped-CAIPI 3× slice-accelerated acquisition. Fig. 5 also shows maps of the 95% uncertainty angle for the estimation of the principal (fiber 1) and crossing (fiber 2) fiber orientations for the two acquisitions. As expected, the uncertainty of the principal fiber orientation is low in white matter. Furthermore, the uncertainty of the estimates of the secondary crossing fiber (where determined to exist) are low in putative crossing fiber regions such as the anterior region of the corona radiata (indicated by white arrows). These metrics show nearly identical uncertainties for the non-accelerated and accelerated acquisitions despite the 3× shorter acquisition time of the simultaneous multi-slice method.

Diffusion imaging results from a conventional unaccelerated acquisition (*top row*) and 3× slice-accelerated acquisition with FOV/2 inter-slice shift (*bottom row*). *Left*: directionally encoded diffusion color-maps; *right*: maps of the 95% uncertainty **...**

Fig. 6 shows a whole-brain 7T GRE-EPI acquisition at 1 mm isotropic resolution which utilized 3× slice and 2× in-plane acceleration. The top panel displays the coronal, sagittal and transverse views of a single reconstructed volume acquired at a TR of 2.88 s using the blipped-CAIPI method. The bottom panel shows the corresponding 1/g-factor map for a representative slice group. The 1/*g*_{ave} in the brain region was 0.90 ± 0.06 for the blipped-CAIPI and 0.77 ± 0.08 for non-blipped simultaneous multi-slice method indicating an average SNR improvement of 17%. Peak 1/g was reduced from 0.57 vs. 0.8 by the blip scheme suggesting an SNR improvement 40% in these high noise enhancement regions.

Example 1 mm isotropic whole-brain GRE-EPI blipped-CAIPI acquisition at 7T with 3× slice and 2× inplane acceleration (TR = 2.88s). **(A)** Coronal, sagittal and axial views of the unaliased 3D volume. (**B** and **C**) Corresponding 1/*g* map of a representative **...**

Fig. 7 shows the results of the 2 mm isotropic resolution GRE-EPI acquired with 6 slices per shot (i.e., 3× slices unaliased with the blipped-CAIPI parallel imaging method and 2× slices separated with the SER method). Thus, two groups of slices are shown marked in red and yellow showing the two SER groups. The coronal and sagittal views of a blipped-CAIPI unaliased slice stack are shown with the slice locations of the 6 slices sharing the EPI readout. Reduced contrast between gray and white matter is observed as a result of the short TR (0.77 s). The 1/*g* maps of the blipped and non-blipped acquisitions are shown for all 6 slices. The blipped-CAIPI acquisition results in substantially higher average SNR, with 1/*g*_{ave} = 0.81 ± 0.12 compared to 0.59 ± 0.12 for the non-blip case showing a 37% slice-average SNR improvement by utilizing the blipped-CAIPIRHINA method. Peak 1/g was reduced from 0.33 to 0.57 by the blip scheme suggesting an SNR improvement of 73% in these high noise enhancement regions.

In this work, we introduce a blipped-CAIPIRIHINA EPI acquisition to shift the image slices of successive simultaneously acquired slices in order to improve the ability of parallel imaging to unalias the simultaneously excited slices. The method was demonstrated for gradient-and spin-echo EPI acquisitions and assessed using Monte Carlo estimated g-factor maps as well as visual comparisons and comparisons of the uncertainty of the principal diffusion directions in a diffusion acquisition. The method achieves a three-fold reduction of TR and thus a three-fold reduction in the time needed to acquire a whole-brain data set. For the acquisitions demonstrated (and many applications requiring a large number of slices), the reduction of TR still maintains a TR>T_{1}. If TR was reduced significantly below T_{1} (such as the 6 slice acquisition utilizing 3× simultaneous multi-slice and 2× SER slice acceleration), reducing the flip angle from 90° to the Ernst angle will improve the SNR of gradient echo acquisitions. For SE acquisitions, SNR per unit time is maximized for TR = 1.25 T_{1} and reductions in TR below this are not advantageous from an SNR perspective. For SE-EPI, the conventional HARDI acquisition acquired in 10 minutes was obtained in just over 3 minutes with no appreciable artifact or loss in SNR. Thus, the blipped-CAIPI method may facilitate acquisition of HARDI/Q-ball or DSI-based diffusion imaging in a clinically relevant time frame.

For a GRE-EPI sequence suitable for fMRI acquisition, the method was used to demonstrate whole-brain 1-mm isotropic acquisition at a TR of less than 3 s. Finally, blipped-CAIPI was combined with the SER technique to achieve a rapid 6 slices per shot acquisition that provides 2-mm isotropic whole-brain coverage in 0.77 s, which improves the ability to perform event-related studies (27) and aids functional imaging experiments desiring rapid temporal sampling of the hemodynamic response (e.g., in order to better observe its initial transient onset) and/or experiments requiring a fully Nyquist-sampled respiratory cycle.

Recently, Feinberg et al. (19) applied a combination of the SER method with parallel imaging based methods to separate the simultaneously acquired slices for a 6-fold higher sampling rate in resting-state functional brain networks study, and demonstrated a notable 60% increase in peak functional sensitivity. However, the multiplexed EPI method of Feinberg et al. (19) does not attempt to mitigate the high g-factor penalties incurred with high acceleration factors. The proposed blipped-CAIPI method explicitly addresses this noise enhancement by shifting the superimposed slices with a phase ramp, thus reducing the g-factor penalty. This benefit should be especially pronounced for applications such as diffusion imaging where thermal noise dominates.

To obtain an absolute SNR comparison between accelerated blipped-CAIPI and non-accelerated acquisitions, differences in TR and hence T_{1} recovery period need to be accounted for. The 3× accelerated SE-EPI acquisition used in this work has a TR of 3 s, while the non-accelerated acquisition has a TR of 9 s. White matter (T_{1} of ~850 ms at 3T (28)) is the main focus of diffusion imaging. Therefore, the signal loss due to saturation effects in the accelerated acquisition is ~3%. With low g-factor penalty, the SNR of the slice-accelerated acquisition is very similar to the non-accelerated case (despite the 3× shorter acquisition time). This was confirmed by the nearly identical directionally diffusion metrics of the accelerated and non-accelerated acquisitions in Fig. 5. For acquisitions with even shorter TR, the saturation effect lowers the SNR in a given acquisition, but the SNR *per unit time* is still improved until TR<1.25T_{1}. For GRE-EPI acquired at the Ernst angle, the SNR per unit time (Signal strength divided by
$\sqrt{\mathit{TR}}$) increases as TR is reduced (29,30), until the g-factor penalty overcomes the increasingly marginal SNR per unit time gains as TR is brought below T_{1}. We also note that as TR decreases (TR<T_{1}), an interleaved slice excitation pattern become less effective at reducing saturation effects in the overlap region of the imperfect slices. This will result in undesirable signal loss that can be partly mitigated by employing RF pulses with sharper slice excitation profiles. This is less of a problem for applications such as high resolution fMRI and Q-ball diffusion where the number of slices needed is high (60 to 120). In this case even with R=3 simultaneous multi-slice the TR will still be relatively high (>2s) and the interleaves will be separated by more than T_{1}.

The blipped-CAIPI technique can be thought of as a way to modify the aliasing pattern of simultaneous multi-slice EPI acquisition to better suit the geometry of typical receive array coils. For the 32-channel coil array used in this work, the coil elements are approximately uniformly distributed around the head (with some reduction in ability to unalias pixels aliased in the *z* direction). In such a configuration, the parallel imaging performance is similar in all spatial directions. For a 2× slice accelerated case, it is clear that the best shift is FOV/2. Also, if one only considers unaliasing between adjacent slices in the 3× slice accelerated case, the same holds. But, considering all three slices, there may be a tradeoff where a smaller shift between slices, but one extending across all slices, allows a better unaliasing of slice 1 and 3. For example a scheme where slice 1,2 and 3 in an simultaneously acquired slice group have center shifts of (0, FOV/3, 2FOV/3) instead of the (0, FOV/2, 0) strategy used might perform better since slices 1 and 3 will have a relative shift. In picking the optimal inter-slice shift for our acquisitions, we perform g-factor performance analysis for different choices of inter-slice shift using the pseudo-multiple replica method (24). For our 3T, 32 channel acquisition with 3× slice acceleration, we tested the g-factor performance of both FOV/2 and FOV/3 inter-slice shift and found the performance of the FOV/2 shift to be marginally better (1/g_{ave} of 0.997 vs 0.96). It seems that for the FOV/3 shift case, the benefit of the increased distance between the most distant slices is out weighted by the degradation to the center slice which is now shifted less compared to its neighbors. We note that the FOV/3 inter-slice image shift will cause a 2FOV/3 image shift differences between the top and bottom slice. However, because of the wrap around effect, the actual distance between the aliased pixel pairs from these two slices will be 2FOV/3 for only 1/3 of all the voxels and FOV/3 for the rest. Similar results were also obtained for the 7T acquisition with 3× slice and 2× inplane. Here, the FOV/4 inter-slice shift marginally outperformed the FOV/6 shift (1/g_{ave} of 0.9 vs 0.87).

For a coil array with elements distributed along the *x-y* plane with no *z*-variation, the aliased voxels need unique *x-y* locations to be separable. In such case, the largest feasible choice of inter-slice image shift for axial slices is FOV/3 for a 3× simultaneous multi-slice acquisition. This will be the optimal choice. For sagittal and coronal acquisitions using an array with no *z*-axis distribution, the in-plane image shift is not helpful when the PE direction is along *z*. For other coil geometries, the g-factor performance of the blipped-CAIPI acquisition with a given inter-slice image shift depends on the coil geometry and the distance between aliasing pixels in a non-trivial manner. Nonetheless, the pseudo-multiple replica method can be use to determine the ideal choice of inter-slice image shift in a case by case basis.

In the blipped-CAIPI method, in-plane acceleration places a constraint on the image shift. This constraint is required to prevent the in-plane shift from over-shifting the already aliased in-plane pixels so that they superimpose in the collapsed slices. For this reason we used a FOV/4 shift for *R*_{inplane} = 2. This limits the maximal distances between the aliasing voxels and reduces the g-factor mitigation benefit of blipped-CAIPI. In this respect slice acceleration and in-plane acceleration begin to compete with one another. Nonetheless, good SNR performance can still be achieved—as evident in the g-factor maps of the 3T and 7T GRE-EPI acquisitions with 3× slice-GRAPPA and 2× in-plane acceleration (see Figs. 6 and and7).7). We note that for R_{inplane}= 2, 3 and 4, a FOV/4, FOV/6 and FOV/8 shift are needed respectively. However, the blip scheme needed for these shifts is the same as that used for a FOV/2 shift for the R_{inplane}= 1 case. For example, Fig. 6 shows the R_{inplane}= 2 image with a FOV/4 shift implemented with the same blip scheme as the R_{inplane}= 1 FOV/2 shift case.

The average g-factor (1/g_{ave} =77%; g_{ave}=1.3) of our non-blipped 7T R_{slice}×R_{inplane} = 3×2 axial plane acquisition agrees well with that of Moeller et al (17). They report an average g-factor of ~ 1.5 in a non-blipped coronal plane acquisition with R_{slice}×R_{inplane} = 4×4 but with a coil configuration with more elements aiding the acceleration (circumferentially distributed) and a FOV in the phase encode and slice direction chosen so the degree of aliasing was similar to what our axial acquisition would experience for a 3×3 acceleration.

Simultaneous multi-slice acquisitions require the use of simultaneous multi-slice RF modulated excitation pulses, which can lead to high SAR levels. In this work, the VERSE algorithm was used to alleviate this issue, at a cost of degraded slice selection profile at off-resonance frequencies (which was in turn mitigated by increasing the TBW product). To limit slice degradation, minimal amount of VERSEing should be use to allow the sequence to stay within the SAR limits. It is important to note that the SAR limits as well as the voltage requirement (for a given flip angle) are both patient specific. Therefore, a sequence with a given RF pulse would be under the SAR limits for one subject and might not be for another. For example, the R_{multislice} =3 accelerated GRE-EPI acquisition at 7T used in this work required a VERSE factor of 3 to ensure sufficient SAR margin such that this acquisition could routinely run in extended runs for all subjects. In contrast, a VERSE factor of 2 appears to be borderline. i.e. it triggers the SAR monitor for about half the subjects tried.

Parallel imaging based simultaneous multi-slice acquisition provides an efficient way to dramatically accelerate EPI acquisitions. The proposed blipped-CAIPI technique explicitly reduces the noise amplification incurred by such methods, allowing dramatic scan time reduction with very minor noise penalty. For diffusion imaging, this enables data acquisition with an increased number of diffusion directions and/or higher spatial resolution. In clinical settings, the scan time reduction could facilitate a wider adoption of HARDI/Q-ball or DSI-based diffusion imaging—techniques which provide more robust and detailed information about the local diffusion environment but require acquisitions that are typically too long for many clinical applications. For fMRI acquisitions, the reduction in scan time translates to an increase temporal sampling and/or an increase in spatial coverage. This is particularly important for high-resolution resting-state fMRI studies, where whole-brain acquisitions are required (19).

This work was supported in part by the Human Connectome Project (1U01MH093765) from the 16 National Institutes of Health Institutes and Centers that support the NIH Blueprint for Neuroscience Research and by NIH NIBIB Grants No. K99EB012107, R01EB006847, R01EB007942, and NCRR Grant P41RR14075.

First, we briefly describe the SENSE/GRAPPA approach previously used for unaliasing simultaneous multi-slice acquisitions (23) to explain the artifact which arises when it is applied to acquisitions with inter-slice image shifts. We then show that the slice-GRAPPA method works well for multi-slice acquisitions in general—with or without inter-slice image shifts. For illustration, the formalism presented describes a two-slice acquisition, but the method is easily generalized to more simultaneous slices.

The SENSE/GRAPPA technique relies on a relationship illustrated in Fig. 8A. The left of the figure shows a concatenation of two imaging slices as they would appear if acquired by conventional single-slice excitation. Following Fig. 8A from left to right, the k-space data of the concatenated slices is under-sampled by 2× and reconstructed with the inverse Discrete Fourier Transform (iDFT) to produce the aliased multi-slice image. The SENSE/GRAPPA approach exploits this relationship in the reverse direction. Viewing Fig. 8A from right to left, the k-space data of the acquired aliased multi-slice image is viewed as a 2× under-sampled data set. A GRAPPA kernel trained on concatenated reference images (acquired one slice at a time) is applied to the under-sampled k-space data to generate the full k-space which can be converted to an unaliased, but concatenated, image of the slices by applying the DFT operator.

The equation for the fitting or application of the SENSE/GRAPPA kernel is given by

$${\stackrel{\u02c6}{S}}_{j}\left({k}_{x},{\stackrel{\u02c6}{k}}_{y}-m\Delta {\stackrel{\u02c6}{k}}_{y}\right)={\sum}_{\ell =1}^{L}{\sum}_{{b}_{x}=-{B}_{x}}^{{B}_{x}}{\sum}_{{b}_{y}=-{B}_{y}}^{{B}_{y}}{n}_{j,m,\ell}^{{b}_{x},{b}_{y}}\phantom{\rule{0.2em}{0ex}}{\stackrel{\u02c6}{S}}_{\ell}\left({k}_{x}-{b}_{x}\Delta {k}_{x},{\stackrel{\u02c6}{k}}_{y}-{b}_{y}{N}_{\mathrm{slice}}\Delta {\stackrel{\u02c6}{k}}_{y}\right)$$

(1)

where *ŷ* and * _{y}* are the image and k-space coordinates of the PE direction of the concatenated image,

$$\begin{array}{l}\underset{x}{\mathit{\int}}\underset{\stackrel{\u02c6}{y}}{\mathit{\int}}{\stackrel{\u02c6}{C}}_{j}\left(x,\stackrel{\u02c6}{y}\right)\stackrel{\u02c6}{\rho}\left(x,\stackrel{\u02c6}{y}\right){e}^{-i2\pi ({k}_{x}x+({\stackrel{\u02c6}{k}}_{y}-m\Delta {\stackrel{\u02c6}{k}}_{y})\stackrel{\u02c6}{y})}\mathit{dxd}\stackrel{\u02c6}{y}\\ \phantom{\rule{0.9em}{0ex}}=\sum _{\ell =1}^{L}\sum _{{b}_{x}=-{B}_{x}}^{{B}_{x}}\sum _{{b}_{x}=-{B}_{y}}^{{B}_{y}}{n}_{j,m,\ell}^{{b}_{x},{b}_{y}}\underset{x}{\mathit{\int}}\underset{\stackrel{\u02c6}{y}}{\mathit{\int}}{\stackrel{\u02c6}{C}}_{\ell}\left(x,\stackrel{\u02c6}{y}\right)\stackrel{\u02c6}{\rho}\left(x,\stackrel{\u02c6}{y}\right){e}^{-i2\pi (({k}_{x}-{b}_{x}\Delta {k}_{x})x+({\stackrel{\u02c6}{k}}_{y}-{b}_{y}{N}_{\mathrm{slice}}\Delta {\stackrel{\u02c6}{k}}_{y})\stackrel{\u02c6}{y})}\mathit{dxd}\stackrel{\u02c6}{y}\end{array}$$

(2)

where *Ĉ _{j}(x, ŷ*) is the coil sensitivity profile of the

Utilizing the orthogonality property of the Fourier transform to remove the integrals and cancelling out common terms results in the following governing equation:

$$\begin{array}{lll}{\stackrel{\u02c6}{C}}_{j}\left(x,\stackrel{\u02c6}{y}\right){e}^{im\Delta {\stackrel{\u02c6}{k}}_{y}\stackrel{\u02c6}{y}}& =& \sum _{\ell =1}^{L}{\stackrel{\u02c6}{C}}_{\ell}\left(x,\stackrel{\u02c6}{y}\right)[\sum _{{b}_{x}=-{B}_{x}}^{{B}_{x}}\sum _{{b}_{x}=-{B}_{y}}^{{B}_{y}}{n}_{j,m,\ell}^{{b}_{x},{b}_{y}}{e}^{i2\pi ({b}_{x}\Delta {k}_{x}x+{b}_{y}{N}_{\mathrm{slice}}\Delta {k}_{y})}]\\ & =& \sum _{\ell =1}^{L}{\stackrel{\u02c6}{C}}_{\ell}\left(x,\stackrel{\u02c6}{y}\right){\stackrel{\u02c6}{K}}_{j,m,\ell}\left(x,\stackrel{\u02c6}{y}\right)\end{array}$$

(3)

where * _{j,m,}*(

This governing equation (Eqn. 3) is not used for fitting or applying the GRAPPA kernel, but represents the relationship between the coil sensitivity profiles and the GRAPPA kernel. The ability of the SENSE/GRAPPA technique to unalias a particular collapsed image is determined by how well the governing equation can be satisfied for that dataset. Thus, the details of how the phase factor *e*^{imΔyŷ} on the left-hand side of the governing equation varies across the concatenated and separated images are critical in determining how well the GRAPPA reconstruction will unalias the slices. Fig. 8B illustrates the phase roll of this θ_{desired} = *m*Δ* _{y}ŷ* for the two slice (

The discontinuity in θ_{desired}(*y*), however, is extremely difficult to produce using the combination of slowly varying coil profiles and the low frequency modulation in the image-domain GRAPPA operator. As a result, the SENSE/GRAPPA technique is ill-equipped to satisfy the GRAPPA governing equation for the FOV/2 simultaneous multi-slice case. This is illustrated in Fig. 9 where the SENSE/GRAPPA algorithm is used to unalias standard, non-shifted multi-slice (top) and a FOV/2 image-shifted multi-slice acquisition (bottom). Both were obtained using a 32-channel coil array at 3T. For standard, non-shifted multi-slice acquisition, the slice separation is very clean. In contrast, the FOV/2 image-shifted acquisition shows significant artifact in the areas of the phase discontinuity.

The slice-GRAPPA algorithm is outlined in Fig. 10A. Here, GRAPPA-like kernels are fit using data acquired from separately excited conventional single-slice data. The kernel sets are applied directly to the k-space data of the collapsed images. Two sets of GRAPPA kernels are used to generate each of the two imaging slices. The fitting equation for the algorithm is:

$${S}_{j,z}({k}_{x},{k}_{y})=\sum _{\ell =1}^{L}\sum _{{b}_{x}=-{B}_{x}}^{{B}_{x}}\sum _{{b}_{y}=-{B}_{y}}^{{B}_{y}}{n}_{j,z,\ell}^{{b}_{x},{b}_{y}}\phantom{\rule{0.2em}{0ex}}{S}_{\ell ,\mathrm{collapse}}({k}_{x}-{b}_{x}\Delta {k}_{x},{k}_{y}-{b}_{y}\Delta {k}_{y}).$$

(4)

Thus, each of the two kernel sets
${n}_{j,z,\ell}^{{b}_{x},{b}_{y}}$ (one for each slice) is a set of 32×32 kernels (32 kernels to generate data for each of the 32 coil elements), *S _{j,z}*(

Similar to the analysis performed for the SENSE/GRAPPA algorithm, the governing equation for slice-GRAPPA can be obtained by expanding and simplifying the fitting equation (Eqn. 4) which results in:

$$\begin{array}{lll}{C}_{j,z}\left(x,y\right){\rho}_{z}\left(x,y\right)& =& \sum _{\ell =1}^{L}{C}_{l,\mathrm{collapse}}\left(x,y\right){\rho}_{\mathrm{collapse}}\left(x,y\right){K}_{\ell ,j,z}\left(x,y\right)\\ & =& \sum _{\ell =1}^{L}\left\{\sum _{\mathit{sl}=1}^{{N}_{\mathrm{slice}}}{C}_{\ell ,\mathit{sl}}\left(x,y\right){\rho}_{\mathit{sl}}\left(x,y\right)\right\}{K}_{\ell ,j,z}\left(x,y\right)\\ & =& \sum _{\mathit{sl}=1}^{{N}_{\mathrm{slice}}}\left\{\sum _{\ell =1}^{L}{C}_{\ell ,\mathit{sl}}\left(x,y\right){K}_{\ell ,j,z}\left(x,y\right)\right\}{\rho}_{\mathit{sl}}\left(x,y\right)\end{array}$$

(5)

where *C _{j,z}*(

$${C}_{j,z}\left(x,y\right){\rho}_{z}\left(x,y\right)=\left\{{{\sum}_{\ell =1}^{L}C}_{\ell ,1}\left(x,y\right){K}_{\ell ,j,z}\left(x,y\right)\right\}{\rho}_{1}\left(x,y\right)+\left\{{{\sum}_{\ell =1}^{L}C}_{\ell ,2}\left(x,y\right){K}_{\ell ,j,z}\left(x,y\right)\right\}{\rho}_{2}\left(x,y\right).$$

(6)

Note that this governing equation, and hence the resulting slice-GRAPPA kernels, is dependent on the underlying images. This is in contrast to conventional GRAPPA and has several implications if this dependency is strong. For example, the same image contrast parameters must be used in the single-slice training data as in the slice-aliased data. This is particularly problematic for diffusion imaging where each diffusion direction could require a distinct kernel set. Fortunately, typical kernels fit to brain data satisfy a special case of Eqn. 6, which preserves the anatomy independence of the GRAPPA operators. We show that under typical imaging conditions, the governing equation simplifies to the following two conditions where the dependence on the underlying image is no longer present:

$${C}_{j,z}\left(x,y\right)={{\sum}_{\ell =1}^{L}C}_{\ell ,z}\left(x,y\right){K}_{\ell ,j,z}\left(x,y\right)$$

(7)

$$0={\sum}_{\ell =1}^{L}{C}_{\ell ,\mathit{sl}}\left(x,y\right){K}_{\ell ,j,z}\left(x,y\right),\phantom{\rule{0.2em}{0ex}}\mathrm{for}\phantom{\rule{0.2em}{0ex}}\mathrm{all}\phantom{\rule{0.2em}{0ex}}\mathit{sl}\ne z$$

(8)

For *N*_{slice} = 2 and *z* = 1 case, Eqns. 7 and 8 satisfy the Eqn. 6 by requiring the first term in Eqn. 6 to satisfy the equation and the second term to be zero. With these conditions, the GRAPPA kernels are no longer dependent on the underlying images and are similar to conventional GRAPPA where the kernels are based only on the relationships between the coil sensitivity patterns.

A simple analysis can be performed to test the validity of Eqns. 7 and 8. We will illustrate this on a standard two simultaneous multi-slice acquisition example. In this test the kernels used to generate the coil data for a given slice is applied to single-slice training data. First it is applied to the single-slice data of the slice it was intended to generate. If Eqn. 7 holds, it should re-generate this slice perfectly. Second it is applied to the single-slice data of the slice it was not intended to generate. If Eqn. 8 holds, this generated image should be 0. The test steps are as follows:

- Train the GRAPPA kernels using a training dataset (i.e., a fully sampled dataset). Here, two sets of kernels are created (
*K*_{,j,1}and*K*_{,j,2}), one for each imaging slice. **Test**Eqn. 7: Apply*K*_{,j,1}kernels to data from slice 1 and*K*_{,j,2}kernels to data from slice 2. This will result in the following set of images:where$$\begin{array}{ll}{I}_{j,(1,1)}(x,y)=& {\sum}_{\ell =1}^{L}{C}_{\ell ,1}\left(x,y\right){\rho}_{1}\left(x,y\right){K}_{\ell ,j,1}\left(x,y\right)\\ {I}_{j,(2,2)}(x,y)=& {\sum}_{\ell =1}^{L}{C}_{\ell ,2}\left(x,y\right){\rho}_{2}\left(x,y\right){K}_{\ell ,j,2}\left(x,y\right)\end{array}$$*I*,(_{j}*sl*_{kernel},*sl*_{image}) is the image from coil*j*^{th}generated by applying the GRAPPA kernel set*K*_{,j,slkernel}to the data from image slice.*sl*_{image}If Eqn. 7 is true then these images becomefor all$$\begin{array}{lll}{I}_{j,(1,1)}(x,y)& =& {C}_{j,1}\left(x,y\right){\rho}_{1}\left(x,y\right)\\ {I}_{j,(2,2)}(x,y)& =& {C}_{j,2}\left(x,y\right){\rho}_{2}\left(x,y\right)\end{array}$$*j*coils. The left of Fig. 10B illustrates the result of this test, where the sum of square images of slice 1 and 2 are shown before and after the application of the GRAPPA kernel. The before and after images appear to be near identical, confirming that Eqn. 7 is well satisfied.**Test**Eqn. 8: Apply*K*_{,j,2}kernels to data from slice 1 and*K*_{,j,1}kernels to data from slice 2. This will result in the following images:If Eqn. 8 is true then these images should all be zeros. The right of Fig. 10B illustrates this. Here the images after the application of the GRAPPA kernels are scale up by 20× to show very minor intensity deviation. Therefore, Eqn. 8 is also well satisfied.$$\begin{array}{lll}{I}_{j,(1,2)}(x,y)& =& {\sum}_{\ell =1}^{L}{C}_{\ell ,2}\left(x,y\right){\rho}_{2}\left(x,y\right){K}_{\ell ,j,1}\left(x,y\right)\\ {I}_{j,(2,1)}(x,y)& =& {\sum}_{\ell =1}^{L}{C}_{\ell ,1}\left(x,y\right){\rho}_{1}\left(x,y\right){K}_{\ell ,j,2}\left(x,y\right)\end{array}$$

The intuition behind why Eqns. 7 and 8 can be expected to hold is obtained by revisiting the original governing equation for *N*_{slice}=2 (in Eqn. 6). The governing equation for slice 1 is:
${C}_{j,1}\left(x,y\right){\rho}_{1}\left(x,y\right)=\left\{{{\sum}_{\ell =1}^{L}C}_{\ell ,1}\left(x,y\right){K}_{\ell ,j,1}\left(x,y\right)\right\}{\rho}_{1}\left(x,y\right)+\left\{{{\sum}_{\ell =1}^{L}C}_{\ell ,2}\left(x,y\right){K}_{\ell ,j,1}\left(x,y\right)\right\}{\rho}_{2}\left(x,y\right)$. For the second term to be non-zero, the factor
$\left\{{{\sum}_{\ell =1}^{L}C}_{\ell ,2}\left(x,y\right){K}_{\ell ,j,1}\left(x,y\right)\right\}$ needs to approximate *C*_{j,1}(*x, y*)*ρ*_{1}(*x, y*)/*ρ*_{2}(*x, y*). In general, the ratio of two slices, *ρ*_{1}(*x, y*)/*ρ*_{2}(*x, y*), is not smooth and contains high frequency components which cannot be well approximated by the GRAPPA kernels and coil sensitivity profiles. It is simpler for the fitting process to make
$\left\{{{\sum}_{\ell =1}^{L}C}_{\ell ,2}\left(x,y\right){K}_{\ell ,j,1}\left(x,y\right)\right\}$ fit a spatial constant (e.g., zero). Therefore, in the kernel fitting process, it is considerably easier to form GRAPPA kernels which make
$\left\{{{\sum}_{\ell =1}^{L}C}_{\ell ,2}\left(x,y\right){K}_{\ell ,j,1}\left(x,y\right)\right\}=0$ and thus satisfy the conditions of Eqn. 8. Once Eqn. 8 is satisfied, the governing equation reduces to
${C}_{j,1}\left(x,y\right)={\sum}_{\ell =1}^{L}{C}_{\ell ,1}\left(x,y\right){K}_{\ell ,j,1}\left(x,y\right)$, exactly that required by Eqn. 7.

This same analysis holds for the cases of simultaneous multi-slice acquisition with inter-slice image shift. In such cases, the only modification to the governing equation is that the coil sensitivities and underlying images of the different slices are now shifted relative to one another.

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