1604 | wileyonlinelibrary.com/journal/mrm Magn Reson Med. 2019;82:1604–1616.© 2019 International Society for Magnetic Resonance in Medicine

Received: 19 April 2018 | Revised: 9 April 2019 | Accepted: 24 April 2019

DOI: 10.1002/mrm.27815

F U L L P A P E R

Banding‐free balanced SSFP cardiac cine using frequency modulation and phase cycle redundancy

Anjali Datta | Dwight G. Nishimura | Corey A. Baron

Electrical Engineering, Stanford University, Stanford, California

CorrespondenceAnjali Datta, David Packard Electrical Engineering, Stanford University, 350 Serra Mall, Rm. 308, Stanford, CA 94305‐4027.Email: adatta@utexas.eduTwitter: @adatta92

Purpose: To develop a method for banding‐free balanced SSFP cardiac cine imaging in a single breath‐hold.Methods: A frequency modulation scheme was designed for cardiac applications to eliminate the time normally required for steady‐state stabilization between multiple phase‐cycled acquisitions. Highly undersampled acquisitions were reconstructed using a model‐based reconstruction that exploits redundancy both over time and between phase cycles. Performance of the methods was evaluated using both retrospective and prospec­tive undersampling in scans with and without frequency modulation from four subjects.Results: The proposed methods enabled balanced SSFP cardiac cine with three effective phase cycles in only 10 heartbeats. Images acquired with frequency modu­lation and with standard phase cycling were of similar quality. The combination of temporal and inter‐acquisition similarity constraints reduced errors by approximately 45% compared to enforcing similarity constraints over time alone.Conclusions: In off‐resonance conditions that preclude the acquisition of single‐ acquisition balanced SSFP, phase cycling can eliminate the dark bands in balanced SSFP cine cardiac imaging at the expense of some SNR efficiency. The proposed techniques permit these types of acquisitions in a single breath‐hold.

K E Y W O R D Sbalanced SSFP, banding, cine, compressed sensing, frequency modulation, phase cycle, redundancy, regularization, SSFP

1 | INTRODUCTION

Cardiac cine MRI is the gold standard for ventricular func­tion and wall motion assessment. Balanced steady‐state free precession (SSFP) is the preferred sequence for cine imag­ing due to its high SNR, strong blood myocardium contrast, and short acquisition times.1 The spectral profile of balanced SSFP (bSSFP) has periodic nulls spaced at the repetition rate (i.e., TR−1), leading to dark band artifacts. Furthermore, severe hyperintense artifacts can result from flow near dark bands, which is a concern in cardiac applications.2 This sensi­tivity to off‐resonance hampers its use at high field (e.g., 3T)

or in the vicinity of unmovable metallic objects like implants or wires.

For cine at 3T, Schar et al3 recommended careful, locali­zed second‐order shimming and center frequency selection to decrease the off‐resonance over the region‐of‐interest. However, these methods are time‐consuming and system‐dependent. In addition, a minimal TR was recommended to widen the band‐to‐band spacing, but the requirement of a TR short enough to avoid artifacts may limit the resolution and the use of time‐efficient acquisition schemes. Wideband SSFP4 increases the central passband width, thus easing the TR limitation for cine at 3T, at the cost of worsened temporal

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resolution and contrast‐to‐noise ratio (CNR) and increased RF heating. In addition, the amount of off‐resonance toler­ated, and thus the TR, is still limited.

In multiple‐acquisition SSFP, bSSFP images are acquired with different RF phase increments (i.e., “phase cycles”), which shifts the frequency profile and thus the bands.5,6 When the multiple acquisitions are combined, the signal nulls are removed. Therefore, in situations where the contrast and high SNR efficiency of bSSFP is desired but off‐resonance precludes the acquisition of a single bSSFP image, multi­ple‐acquisiton bSSFP trades some of the SNR efficiency in the passband for robustness to off‐resonance. This, however, comes at the cost of substantially increased scan time for ac­quisition of the additional phase‐cycled images and stabiliza­tion of the steady‐state signal between each phase cycle.

The extra time required for multiple acquisitions can be at least partially mitigated by undersampling data, which is facil­itated by appropriate reconstruction strategies. Parallel imag­ing7,8 is already used in standard cardiac cine acquisitions, so methods to extract even more acceleration are required to enable multiple‐acquisition bSSFP with similar parameters as single‐ acquisition bSSFP. k‐t BLAST and k‐t SENSE exploit cor­relations in time to improve acceleration rates.9 However, this requires the acquisition of additional training data. Alternatively, the VISTA approach enforces wavelet sparsity along the tempo­ral dimension, which requires no extra acquisition time.10,11

In this work, we employ a combined acquisition and re­construction strategy to acquire multiple‐acquisition SSFP in clinically relevant scan times. Frequency modulation obviates the need for stabilization periods between phase cycles, thus resulting in a shorter scan time than a comparable standard phase‐cycled sequence.12 Accordingly, we have developed a frequency modulation scheme for use in cardiac cine, shorten­ing the required scan time for multiple‐acquisition SSFP cine images while achieving almost identical images as standard phase cycling. Our scheme also causes the acquisition of the phase‐cycled images to be interleaved, ensuring that the mul­tiple acquisitions are co‐registered. To enable acquisition of three phase cycles within a breath‐hold (ten heartbeats), we have highly undersampled the data and employed a regular­ized reconstruction that exploits redundancy both over time and between phase cycles. This method is similar to VISTA,11 with the addition of a regularization that exploits redundan­cies between the multiple acquisitions. Thus, we illustrate the feasibility of banding‐free breath‐held bSSFP cine.

2 | METHODS

2.1 | Frequency modulationIn a frequency‐modulated acquisition, to maintain bSSFP contrast and avoid undersampling artifacts, it is necessary for

each image to contain only a small range in the RF phase increments. For example, if the spectral profile is shifted through a full 360◦ during a projection reconstruction acqui­sition of one image, the contrast of the net image (assuming a standard gridding reconstruction) is a complex sum over all phase cycles,13 which is similar to the contrast of gradi­ent‐spoiled gradient echo.6 In addition, since only a fraction of the lines are acquired for any given small range of RF phase increments, an “effective phase cycle,” the image from such an acquisition may suffer from streaking artifacts.13,14 In this work, the acquisition of multiple images of each car­diac phase, each containing only a limited variation in the shift of the spectral profile, is proposed to maintain bSSFP contrast and make the phase cycling method trajectory‐ independent. While four phase cycles have been recom­mended for multiple‐acquisition SSFP to reduce the appear­ance of residual darkened regions,6 we developed a sequence that acquires three effective phase cycles to improve the efficacy of breath‐held scans.

To acquire the data for a particular cardiac phase at a specific RF phase increment over several heart beats, the phase cycling must be synchronized to the cardiac cycle. In our modified frequency modulation scheme (Figure 1A), the phase cycling modulates slowly for a time less than the shortest expected R wave to R wave (RR) interval, after which the phase increment is held constant until the next cardiac trigger. Before the scan, a number of TRs, nFM, with total duration less than the shortest expected RR interval is chosen based on the average heart rate. For three effective phase cycles, the phase increment increases by 120◦∕nFM each TR for nFM repetitions (i.e., until the next phase cycle is reached) and then remains constant until the next trigger. This scheme handles heart rate variability—it ensures that any particular cardiac phase is acquired with the same av­erage phase cycling during every third heartbeat, and the average phase cycling during the other heartbeats is offset by exactly 120◦ and 240◦. Therefore, acquisition of the three effective phase cycles is interleaved throughout the scan. For nFM =180 (given the scan parameters here, this is ap­propriate for a heart rate not exceeding 100 beats per min­ute), the modulation rate would be less than a 0.7◦ increase in the RF phase increment per TR. This is well under the 3◦ guideline stated by Foxall for the upper limit of frequency modulation that the steady‐state tolerates,12 so the distor­tion to the balanced SSFP spectral profile is expected to be minimal. For nine views per segment, each effective phase cycle image would contain less than a 6◦ range in the phase increment. The simulated point spread function for TR = 3.4 ms, T1 = 850 ms, T2 = 50 ms and the phase cyling in Figure 2A is shown in Figure 2B. As expected, the small range in phase increment has negligible effect on the point spread function.

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F I G U R E 1 A, For a scheme designed for three effective phase cycles, the RF phase increment is swept through 360◦ every three heartbeats. The phase increment desired for the next trigger is reached after n

FM repetitions. B, The undersampling scheme is shown, where the sample

locations are shifted for each cardiac phase and effective phase cycle to enable computation of time‐ and phase‐cycle‐averaged calibration data. The data are fully sampled along the frequency encode direction. Only a subset of the samples along the phase‐encode and cardiac phase axes is shown. C, The image reconstruction uses time and phase cycle averaging to calibrate receiver sensitivity and phase cycle banding profiles

F I G U R E 2 A, The simulated phase cycling used to determine the point spread function. The acquisition contains a range of 5.33◦ in the phase cycling (0.67◦ change each TR and nine views‐per‐segment). The other simulation parameters were T

1=850 ms, T

2=50 ms, and TR = 3.4 ms.

B, Simulated point spread function plotted on a log scale. The small range in RF phase increments within the acquisition has negligible effects of the point spread function

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2.2 | Undersampling and image reconstructionA sequential, segmented 2DFT acquisition with uniform un­dersampling is used. Instead of repeating each k‐space seg­ment until the next trigger (as is typically done), here, the samples are shifted along the phase‐encode direction every segment so that different k‐space positions are sampled dur­ing each cardiac phase (Figure 1B). Every Rth cardiac phase has the same sampling pattern, where R is the undersampling rate. The sampling positions are also shifted between differ­ent effective phase cycles. This scheme allows averaging over the time and phase cycle dimensions to generate fully sampled data for parallel imaging calibration. Note that this sampling pattern is not specific to the frequency‐modulated sequence we propose but can be used for any multiple‐ acquisiton bSSFP cine sequence.

A numerical phantom that looks like a profile from a car­diac cine over the cardiac phases (i.e., there is one spatial di­mension and one temporal dimension) is used to illustrate the effects of undersampling with different sampling patterns. The shifting sampling pattern in k‐t space causes undersam­pling artifacts to drastically reduce wavelet sparsity (Figure 3). Based on this, we hypothesize that ℓ1‐wavelet regularization will help suppress undersampling artifacts, even though the sampling pattern is not pseudorandom.

To reconstruct the images, all cardiac phases and phase cycles are simultaneously determined using the following optimization:

where x ∈ ℂNxNyNtNp is a vectorized representation of the

desired 4D image (two spatial dimensions with size Nx and Ny, one time dimension with size Nt, and one dimension for the Np effective phase cycles), CR applies receiver sensitivity profiles, F is a two‐dimensional FFT in the spatial dimen­sions, S is k‐space sampling, y is the acquired k‐space data, P is a transform that exploits redundancy between phase cycling banding profiles (defined below), W is a three‐ dimensional undecimated wavelet transform in space and time, and �p and �w are parameters that control regulariza­tion strength. P is defined under the assumption that bSSFP banding can be approximated by multiplication of the “true” images by banding profiles. That is, xi = CPim, where xi ∈ℂ

NxNyNt is the cardiac cine for the ith phase cycle (x is the concatenation of all xi), m is the underlying cine with no banding and CPi ∈ ℂ

NxNyNt ×NxNyNt is a diagonal matrix that applies the ith phase cycle’s banding profile. If the profiles are normalized such that

∑j CH

PjCPj = I, it follows that

Therefore, ideally, CPi

∑j CH

Pjxj−xi = 0. Accordingly, we de­

fine P such that

The solution of Equation 1 requires estimation of both CR and CPi. Fully sampled calibration data for the determination

(1)arg minx

��SFCRx−y��2

2+�p ‖Px‖1+�w ‖Wx‖1

(2)CPi

Np∑j=1

CHPj

xj =CPi

Np∑j=1

CHPj

CPjm=CPim= xi

(3)Px=

⎡⎢⎢⎢⎣

CP1

∑j CH

Pjxj

⋮

CPNp

∑j CH

Pjxj

⎤⎥⎥⎥⎦−x

F I G U R E 3 1D single‐level undecimated wavelet transforms in the cardiac phase (horizontal) direction for a numerical phantom (A) sampled with the three different k‐t patterns shown in (B). A 1D Fourier transform is done along the phase encode (vertical) dimension prior to the wavelet transform, resulting in the images in (C). After full sampling (left), the original image is recovered, while for the two undersampled cases, the zero‐filled reconstructions suffer from undersampling artifacts. (D) The wavelet transforms are composed of a low‐pass filtered (labeled “low freq.”) image beside a high‐pass filtered image (“high freq.”). For regular undersampling (center), wavelet sparsity is similar to the fully sampled case (left), with the high‐pass filtered component (right half of the wavelet transform) containing little signal. With the shifted sampling pattern in k‐t space used in this work (right), the wavelet transform becomes less sparse after undersampling, suggesting that ℓ1‐wavelet regularization will help suppress undersampling artifacts

(A)

(B)

(C)

(D)

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of CR is generated by averaging all cardiac phases.9 We also average all phase cycles because the underlying receiver sensitivities should be the same. After determination of this calibration data, CR is determined using ESPIRiT.15 For es­timation of CPi, the R nearest cardiac phases are combined to generate fully sampled calibration data y ∈ ℂ

NxNyNtNpNr, where Nr is the number of receivers (i.e., a moving average with length equal to the undersampling rate generates cal­ibration data, as proposed by Kellman et al16). Notably, an average over all cardiac phase is not possible because the phase cycling varies slowly over the cardiac cycle. The three phase cycles are concatenated in time before performing the moving average with cyclic boundary conditions, because the phase cycling varies smoothly from the last time position of one effective phase cycle into the first time position of the next effective phase cycle (Figure 1A). This concatenation in time is also used before computing the 3D wavelet transform W in optimization iterations. Following the computation of y, the receivers are combined to form images for calibration, x ∈ ℂ

NxNyNtNp, using

Finally, the nth diagonal entry of CPi is determined using

which gives the proper normalization required for Equation 2. Notably, ESPIRiT is not used to compute CPi because high spatial frequencies are expected near the bands, which violates assumptions implicitly made in ESPIRiT because a small con­volution kernel in k‐space is used.

Wavelet regularization is typically implemented for com­pressed sensing applications that have pseudorandom sam­pling patterns. Here, wavelet regularization enforces slow time variation, similar to in the VISTA method.10 Equation 1 is solved using balanced FISTA11 (see Appendix A). After solving Equation 1, the final time‐series of images is ob­tained by performing a root‐sum‐of‐squares over the three phase cycles.6

2.3 | Experimental methodsExperiments were performed on a 1.5T GE Signa Excite scanner (GE Healthcare, Waukesha, WI) using an 8‐channel cardiac receive coil. Phantoms (a small bottle with ap­proximate T1/T2 = 1000 ms/200 ms and a standard GE phantom) as well as four normal subjects who had given informed signed consent were scanned. The subjects were

instructed to hold their breath near end‐expiration for every scan, and a pulse plethysmograph was used for cardiac triggering. In all subjects, cine loops were acquired in two axial scan planes (during separate scans) using a single‐acquisition bSSFP sequence, a multiple‐acquisition stand­ard phase‐cycled sequence (i.e., static phase cycling), and the previously described frequency‐modulated sequence (i.e., dynamic phase cycling). The phase cycling scheme used in the static phase‐cycled multiple‐acquisition sequence is shown in Supporting Information Table S1. The k‐t undersampling pattern and all other scan param­eters were identical between the two sequences. The scan parameters were: slice thickness = 8 mm, flip angle = 60◦, field of view = 24‐28 cm isotropic (dependent on sub­ject size), matrix size = 192 × 162, receiver bandwidth = ± 125 kHz, TE = 1.4 ms, TR = 3.3–3.4 ms, temporal resolution = 29.4–30.6 ms. The R = 2 single‐acquisi­tion bSSFP sequence was 10 heartbeats long. For the phase‐cycled sequences, scans using undersampling rates of R  =  2 (scan lengths of 30 heartbeats for static phase cycling and 28 heartbeats for dynamic) and R = 6 (12 heartbeats for static and 10 heartbeats for dynamic) were performed. All of the sequences began with a heart­beat of discarded acquisitions, which is included in heart­beat counts above, to allow the signal to reach steady state. The static phase cycling cases required two addi­tional heartbeats compared to dynamic because of the sta­bilization periods needed between each of the three phase cycles. Although catalyzation schemes can shorten the duration needed for stabilization,17,18 their application to cardiac cine poses some challenges (simply playing the catalyzation pulses and then acquiring data for the rest of the heartbeat would cause the first few cardiac phases to be missing data, which would have to be acquired during a subsequent heartbeat.) Therefore, for simplicity, we as­sumed a full heartbeat for stabilization and thus didn’t use any specialized catalyzation scheme.

We used partial dephasing in the slice direction to miti­gate the severe artifacts that result from through‐plane flow near dark bands without notably affecting the passband sig­nal.19 A 30◦ range in the phase accrual during a TR was cre­ated over the voxel by shortening the slice‐select rephaser after the readout, slightly unbalancing the gradients on that axis. Based on the Bloch simulations in the prior work, with 30◦ of partial dephasing, the signal from on‐resonant static spins is more than 99% of that in standard balanced SSFP.

Linear shim values were determined automatically using the scanner software. The gradient in the phase‐encode di­rection was then deliberately offset from the optimal shim to create increased off‐resonance to better assess the perfor­mance in a challenging setting. Using this “detuned shim­ming” method, a range of approximately 290 Hz was added

(4)x=CHR

F−1y

(5)CPi,n =xi,n�∑i x∗

i,nxi,n

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over the field of view, which caused dark bands to be present in all phase cycles.

Two additional healthy volunteers were scanned at 3T using a GE Signa HDx scanner. Each slice was lin­early shimmed and was not detuned. Cine loops of two axial slices were acquired using the proposed frequency‐ modulated sequence (with R = 6), as well as three R = 2 reference sequences: single‐acquisition bSSFP without par­tial dephasing, single‐acquisition bSSFP with 30◦ of partial dephasing, and spoiled gradient echo (SPGR). A 40◦ flip angle was used for all of the bSSFP acquisitions, and a 30◦ flip angle was used for the SPGR acquisitions. The other scan parameters were: slice thickness = 8 mm, field of view = 26 cm isotropic, matrix size = 192 × 162, receiver bandwidth = ±111 kHz, TE = 1.7 ms, TR = 4.0 ms, tempo­ral resolution = 35.6 ms.

Reconstructions were performed using MATLAB on an Intel Xeon E5‐2650 v4 2.20 GHz workstation with 100 GB RAM and an NVidia GTX 1080 with 8 GB RAM. While most operations were computed on the GPU, the multipli­cations with (ΨHΨ)−1 (see Appendix A) were performed on the CPU due to a current lack of built‐in MATLAB GPU support for this type of operation. The R  =  2 scans were reconstructed using SENSE. In addition, the R = 2 scans were retrospectively undersampled to a net undersampling rate of 6, and Equation 1 was solved using regularization parameters �w and �p that were determined using a course‐to‐fine grid‐based search that minimized the mean squared error relative to the known ground truth reconstructed from R = 2 using SENSE (optimal parameters were determined separately for each subject and for static versus dynamic). The same �w and �p were used for reconstruction of the prospectively undersampled R = 6 scans. For all cases, 100 iterations were used for the FISTA‐based computation of Equation 1.

3 | RESULTS

Images reconstructed from phantom scans for static and dynamic phase cycling are shown in Figure 4. While dynamic phase cy­cling results in an asymmetrical spectral profile,12 which affects the appearance of banding, the lack of artifacts suggests that the phase cycling modulation is sufficiently small to avoid transient effects. This is expected given the very slow frequency modula­tion of less than 0.7◦ per TR for the dynamic case.

The ground truth dynamic phase‐cycled images for one subject reconstructed using SENSE at R = 2 are shown in Figure 5, along with the images and the difference from ground truth for reconstructions performed with retrospective undersampling to R = 6. (The analogous static phase‐cycled images are in Supporting Information Figure S1.) Results for both using only wavelet regularization (i.e., 𝜇w > 0, �p = 0) and using combined wavelet and phase cycle redundancy regularization (i.e., 𝜇w, p > 0) are shown, where substantially lower error is observed when phase cycle redundancy regu­larization is used. For both cases, optimal regularization pa­rameters that minimize mean squared error from ground truth were used. Similar results are also observed in the other sub­jects (Supporting Information Figure S1). The net normali­zed mean squared error was calculated for each slice using ‖x−xT‖2

2∕‖xT‖2

2, where xT is the ground truth image. The

mean error over all cardiac phases and subjects is reduced by almost 50% when phase cycle redundancy is used (Figure 6). A time series of an anterior‐posterior line in an atrial location with rapid motion is included in the Supporting Information (Supporting Information Figure S3).

When Equation 1 is used to reconstruct data acquired with an undersampling rate of R = 6, high quality images are obtained for all four subjects even though the data was acquired with only eight receiver channels (Figure 7). See Supporting Information for a representative example of

F I G U R E 4 Three effective phase cycles for SENSE reconstructions of data undersampled with R = 2 in phantoms for both static and dynamic phase cycling. While dynamic phase cycling has an effect on the spectral profile, the lack of artifacts suggests that the acquisition is in the slow frequency modulation regime for static spins

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cines acquired with R = 2 and R = 6 undersampling in the same subject (Supporting Information Videos S1 and S2). Example images of a standard bSSFP acquisition compared to the proposed method with dynamic phase cycling show mitigation of catastrophic off‐resonance artifacts, both dark bands and hyperintensities from near‐band flow (Figure 8). Some remaining blood pool heterogeneity is observed in the proposed method, which likely stems from the over‐ representation of hyper‐intensities from flow into dark bands in the sum‐of‐squares combination of the separate phase cycles. Calibration data that were generated from the undersampled data using temporal averaging, along with the computed CR and CP, are shown in Figure 9 for one of the subjects for dynamic phase cycling. High spa­tial frequencies are observed in CP, which precludes using ESPIRiT for calibration. See Supporting Information for a comparison of R = 2 ground truth images before combina­tion of the phase cycles to the banding profiles estimated after retrospectively undersampling to R  =  6 (Supporting Information Video S3).

In a proof‐of‐concept in two volunteers at 3T (Figure 10), shimming was insufficient to eliminate banding in a stan­dard bSSFP acquisition. In both volunteers, the proposed

F I G U R E 5 Data acquired with an undersampling rate of R = 2 using dynamic phase cycling. Three cardiac phases are shown for each case. The left column of images (i.e., “Ground Truth”) was reconstructed using SENSE. The other columns show the images and difference from ground truth (scaled by a factor of three) for reconstructions after retrospective undersampling to R = 6. When both wavelet and phase cycle redundancy regularization are used (Wavelet + PC), errors are lower compared to when only wavelet regularization is used

F I G U R E 6 The mean difference from ground truth over all cardiac phases and subjects for images reconstructed after retrospective undersampling to R = 6. When both wavelet and phase cycle redundancy regularization are used (Wavelet + PC), errors are 41% lower for static and 48% lower for dynamic phase cycling compared to when only wavelet regularization is used (P < 0.001 for each of the static and dynamic cases using a paired students t‐test)

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acquisition and reconstruction produced phase‐cycled im­ages with no additional scan time, mitigating signal loss in off‐resonant regions.

4 | DISCUSSION

In this work, we have introduced a combined acquisition and reconstruction strategy that allows the acquisition of bSSFP cardiac cine with three phase cycles in only 10 heartbeats without sacrificing temporal resolution. Notably, this is the same length as a conventional single‐acquisition bSSFP with R = 2 and otherwise matched parameters. Therefore, in conditions where off‐resonance results in banding in single‐ acquisition bSSFP images, the proposed method may provide a feasible alternative. (Since multiple‐acquisition bSSFP sacrifices some of the SNR efficiency of single‐acquisition bSSFP in the passband, this method is not intended in situa­tions where a single bSSFP cine is possible). For the acqui­sition, a modified frequency modulation scheme eliminated

the need for steady‐state stabilization between phase cycles. Over all subjects, similar results were observed between static and dynamic phase cycling (Supporting Information Figure S2 and Figure 6), suggesting that the frequency mod­ulation did not introduce artifacts. In addition to shortening the breath‐hold time, the proposed frequency modulation scheme interleaves the acquisition of the phase cycles, en­suring co‐registration. This may be an advantage with less compliant subjects but was not evaluated here. A potential drawback with dynamic phase cycling is that the residual ripple in the combined images varies over the cardiac cycle. Therefore, while we illustrate the feasibility of this dynamic phase cycling method, other methods, such as static phase cycling with catalyzation, and their trade‐offs may merit investigation.

For the reconstruction, a regularized reconstruction enforcing sparse wavelet coefficients in space and time as well as redundancy between phase cycles enabled a high undersampling rate. Note that this reconstuction is compat­ible with both dynamic and static phase cycling and should

F I G U R E 7 Cardiac cine reconstructed using Equation 1 for data acquired with three effective phase cycles and a rate of undersampling R = 6. Select cardiac phases spaced evenly throughout the cardiac cycle are shown. The scan required 10 heartbeats. High‐quality images were observed in all the subjects

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also be compatible with other acquisition techniques, such as catalyzation. With a higher number of receiver elements than the eight used here, even higher rates of undersampling may be possible. The proposed method was primarily evaluated at 1.5T, with a detuned shim applied to increase inhomogeneity. The images from two volunteers at 3T (Figure 10) suggest that the amount of inhomogeneity from detuned shimming at 1.5T is comparable to the inhomogeneity at 3T accordingly, this approach should have little difficulty scaling to higher fields or longer TRs, but further study is warranted.

Using a 3D wavelet transform gave substantially lower mean squared error compared to using a 2D transform over only the phase encode dimension and time or a 1D trans­form only over time (data not shown). This behavior was somewhat unexpected given the lack of randomness in the

sampling pattern, but may have occurred because 3D wavelet regularization helped suppress undersampling artifacts in the final solution. Nevertheless, improved performance would possibly be observed with pseudorandom undersampling; however, incorporating this into a design that still permits de­termination of CP is nontrivial.

Here, multiple phase cycles were combined using root‐sum‐of‐squares, which generally has poor performance in hyper‐intense regions near the bands.2,6 However, the pro­posed methods are compatible with any combination method because the solution of Equation 1 retains the separate phase cycles. Accordingly, a combination that gives lower weight to hyper‐intense regions near the bands could improve the gen­eral image quality (e.g., the homogeneity of the blood pool), but is beyond the scope of this work.

F I G U R E 8 Cardiac cine reconstructed from standard bSSFP (R = 2; SENSE) and the proposed dynamic phase‐cycled data (R = 6; Equation 1). A sum‐of‐squares (SOS) combination of effective phase cycles was used for the proposed method, and the individual phase cycles (PC1‐3) are shown to the right. Yellow boxes indicate regions with banding and near‐band flow artifacts (hyperintensity and transient‐related artifacts), which are mitigated by the proposed method

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In some cases, there appeared to be additional error in re­gions of high flow (e.g., the descending aorta for static phase cycling in Supporting Information Figure S2), which could have stemmed from the temporal averaging in the calibra­tion of CP being unable to keep up with rapid signal changes. These errors were generally localized to regions with high flow near dark bands; that is, they did not introduce alias­ing errors throughout the image and were only present in one of the three phase cycles at any particular temporal frame. Accordingly, a more sophisticated phase cycle combination method would likely also decrease the net contribution of these more error‐prone regions to the final image.

Although it is possible to include CP in the data consis­tency term of Equation 1 instead of in the regularization

transform P (which would result in a solution that already has the phase cycles combined), this results in solutions with higher error than the proposed methods (data not shown). This likely occurs because it is difficult to obtain a calibra­tion of CP of sufficient accuracy to strictly enforce the band­ing profiles due to rapid signal fluctuations near the bands from flow. Notably, the ℓ1‐regularization of the phase cycle term only enforces a sparse Px and permits some high spatial frequency deviations from the true profiles. A similar recon­struction using an ℓ2‐regularization on Px instead resulted in higher mean square error due to these flow effects (data not shown). It is also noteworthy that the partial dephasing19 used here to reduce flow artifacts likely increased the quality of CP calibration.

A potential limitation of the reconstruction approach is that two regularization parameters need to be deter­mined. In this study, the optimal parameters were found using mean squared error because we had datasets with lower acceleration that could be retrospectively under­sampled. While this may not be possible in some appli­cations, methods that do not require knowledge of the ground truth could be applied instead.20 In addition, there was little variation in optimal regularization parameters between subjects (�p = 0.033 ± 0.006, �w = 0.07 ± 0.01). When parameters were offset from their optimal values, mean squared error (MSE) increased slowly (Supporting Information Figure S4), with two standard deviations of offset resulting in MSE increases of less than 1.5% and 4% for �p and �w, respectively. The 3T reconstruc­tion used the mean (taken over all subjects and slices) of the parameters determined from the 1.5T retrospective reconstructions. The success of the same parameters on a different scanner at a different field strength provides further evidence of robustness in parameter choice. Given the large margin of error for nonoptimal parameters and the consistency between subjects, it would be feasible to empirically determine suitable parameters by qualita­tively assessing the images for a small grid of parameters for a single subject and then use the same parameters for all other subjects.

The reconstructions required approximately 2.7 minutes when both wavelet and phase cycle redundancy regulariza­tion was used, and 50 s when only wavelet regularization was used. For both cases, ESPIRiT calibration was performed on the CPU and required approximately 30 s, which could be greatly reduced via implementation on a GPU. For phase cycle redundancy regularization, the LU decomposition on the CPU required approximately 20 s, and the evaluations of (ΨHΨ)−1 (see Appendix A) were performed on the CPU due to a current lack of built‐in MATLAB support. Therefore, it is expected that, with full implementation on a GPU, clini­cally relevant reconstruction times of less than 1 minute are feasible.

F I G U R E 9 Calibration data and profiles for one of the subjects for both (A) receiver sensitivity profiles, C

R, and (B) phase cycle

banding profiles, CP, when dynamic phase cycling was used. The

calibration data were generated using temporal averaging of the undersampled data

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5 | CONCLUSION

Balanced SSFP is an essential clinical tool for the evalua­tion of cardiac function. However, banding artifacts hamper its use in the presence of off‐resonance. In this work, we il­lustrated a combined acquisition and reconstruction approach that allows acquisition of banding‐free cardiac cine in a short breath‐hold. A frequency modulation scheme for cardiac cine eliminated the time normally required for steady‐state stabilization between multiple acquisitions while producing images comparable to standard phase cycling. Exploiting re­dundancy both over time and between multiple phase‐cycled acquisitions enabled reconstruction of highly undersampled acquisitions. Together, these strategies allowed acquisi­tion of bSSFP cardiac cine with three phase cycles in only 10 heartbeats. Thus, banding‐free balanced SSFP in a single breath‐hold (of the same duration as conventional single‐ acquisition bSSFP with R = 2) is possible using the proposed techniques.

REFERENCES

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F I G U R E 1 0 Cardiac cine at 3T reconstructed from SPGR data (R = 2; SENSE), standard bSSFP data without and with 30◦ of partial dephasing (R = 2; SENSE), and the proposed dynamic phase‐cycled data (R = 6; Equation 1). The effective phase cycles produced by the proposed method were combined using sum‐of‐squares (SOS). Despite shimming, the bSSFP images have banding and therefore near‐band flow artifacts (yellow arrow). The bSSFP images with partial dephasing have reduced flow artifacts and similar CNR as the bSSFP images, but still suffer from banding, which is mitigated by the proposed method (yellow boxes). The SPGR data have lower SNR than all of the other images. Therefore, the proposed method may provide a feasible alternative to SPGR when linear shimming is insufficient to avoid banding in single‐acquisition bSSFP images

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SUPPORTING INFORMATION

Additional supporting information may be found online in the Supporting Information section at the end of the article.

VIDEO S1 Representative cine loop acquired with an accel­eration factor of 2 using the proposed acquisition with dy­namic phase cycling (sosCineR2.avi)VIDEO S2 Representative cine loop acquired with R  =  6 prospective undersampling in the same subject as Supporting Information Video S1. sosCineR6.avi was acquired using the proposed acquisition with dynamic phase cycling and recon­structed using the proposed reconstruction with both wavelet and phase cycle consistency regularizationVIDEO S3 Representative cine loops for all three effective phase cycles of an R = 2 SENSE reconstruction with dynamic phase cycling (top row), and the corresponding CP calibrated after retrospectively undersampling to R  =  6 (cinesWith­BandingProfiles.gif)FIGURE S1 Data acquired with an undersampling rate of R = 2 using static phase cycling. Three cardiac phases are shown for each case. The left column of images (i.e., “Ground Truth”) was reconstructed using SENSE. The other columns show the images and difference from ground truth (scaled by a factor of three) for reconstructions after retrospective un­dersampling to R = 6. When both wavelet and phase cycle re­dundancy regularization are used (Wavelet + PC), errors are lower compared to when only wavelet regularization is usedFIGURE S2 Error from ground truth for data acquired using both static and dynamic phase cycling. Two slice locations

are shown for each of four subjects. Three cardiac phases of each cine are shown. The data were acquired with an under­sampling rate of R = 2 and reconstructed using SENSE to obtain the ground truth images, which was subtracted from reconstructions after the data were retrospectively undersam­pled to R = 6. Generally, when both wavelet and phase cycle consistency regularization are used (Wavelet + PC), errors are lower compared to when only wavelet regularization is usedFIGURE S3 Time evolution of an anterior‐posterior line through an atrial location with rapid movement for both the ground truth (R  =  2 SENSE) and proposed Wavelet + PC method after retrospectively undersampling to R = 6. Good correspondence of time‐varying features is observed between the two casesFIGURE S4 Percent increase in mean square error (MSE) as compressed sensing parameters are adjusted from the optimal value for �

p (left) and �

w (right). The results shown are the

mean over all subjects, and little variation was observed be­tween the subjects. The horizontal axis is shown using a log­arithmic scale, which spans a factor of 0.5 to factor of 2 change in each parameter. The standard deviation of optimal parameters across subjects and slices corresponds to approx­imately 0.05 on the horizontal axis for each parameterTABLE S1 Phase cycling scheme used in the standard, static phase‐cycled multiple‐acquisition sequence

How to cite this article: Datta A, Nishimura DG, Baron CA. Banding‐free balanced SSFP cardiac cine using frequency modulation and phase­cycle redundancy. Magn Reson Med. 2019;82:1604–1616. https ://doi.org/10.1002/mrm.27815

APPENDIX ABalanced FISTA was used to solve Equation 1, where the following iterative procedure is used11:

where xn is a specific linear combination of xn and xn−1 re­quired for FISTA,21 T is the shrinkage operator, γ is the gradi­ent step size, μ is a diagonal matrix of regularization factors, Ψ is a sparsifying transform, and E=SFCR is the MRI sam­pling operation. Here, the net sparsifying transform is the concatenation of W and P

and

(A1)xn+1 = (ΨHΨ)−1ΨHT𝛾𝜇[Ψ(xn−𝛾(ΨHΨ)−1EH(Exn−y))]

(A2)Ψ=

[W

P

]

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In Ting et al. Ψ = W; thus, (ΨHΨ)−1 = I and Equation A1 simplifies considerably.11 However, this simplification

is not possible here. Nevertheless, ΨHΨ = I + PHP is sparse and can be explicitly (and rapidly) computed before beginning FISTA iterations. To accelerate multiplications with (ΨHΨ)−1 during each iteration, an LU factorization of ΨHΨ was also computed in advance of the iterative procedure.

(A3)�=

[�wI 0

0 �pI

]