Minimum TR radiofrequency‐pulse design for rapid gradient echo sequences

A framework to design radiofrequency (RF) pulses specifically to minimize the TR of gradient echo sequences is presented, subject to hardware and physiological constraints.


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ABO SEADA Et Al. requiring high flip angles (FAs) or using high-energy pulses, for instance, when multiband (MB) pulses are used for simultaneous multislice (SMS) imaging. [5][6][7][8][9][10] For example, SAR is usually the limiting factor for cardiac bSSFP at 3T and above, and can prevent sequences from using FAs that would maximize contrast between myocardium and blood. 11,12 Much recent RF pulse design development has focused on minimum-duration techniques subject to hardware constraints, which are shown to benefit greatly from timevariable selection gradients designed using time-optimal variable rate selective excitation (VERSE), 13 nonuniform Shinnar-Le Roux design, 14 or optimal control. 15 This has been particularly effective in improving MB RF pulses. [16][17][18] However, the problem of minimizing RF pulse duration and minimizing TR are not equivalent; indeed, reducing pulse duration can actually increase the minimum achievable TR because it increases the pulse energy. Beqiri et al 19 proposed a strategy for minimizing the TR of rapid sequences in the context of RF shimming with parallel transmission, using a nested optimization that directly minimizes the TR. In this work, we propose a similar strategy for the design of RF excitation pulses with time-variable selection gradient waveforms for single-channel RF transmission. Time-variable selection gradients also effect PNS calculations, which we investigate. This is applied to both single and MB pulse designs, and their application to SMS cardiac bSSFP imaging is demonstrated in vivo. This work has partially been presented as conference proceedings. 20,21 2 | THEORY A TR of a rapid gradient echo sequence consists of an excitation and an encoding period. The image encoding gradients can be altered by changing the image resolution, read-out bandwidth, the maximum gradient amplitude, and slew rate. The duration of the encoding period T enc depends directly on scan-specific parameters as mentioned above, and so is considered to be outside of the scope of the optimization proposed in this work. The overall TR is then determined by the sum of T enc (which is fixed) and the RF pulse duration T pulse such that For an RF pulse waveform B 1 (t) we may define the following measure that is proportional to RF energy: expressed in μT 2 ms. Assuming a steady-state sequence with constant FAs, RF energy can be related to TR and SAR using the relationship where SAR is expressed in W/kg −1 , and k is a conversion factor in W/kg −1 , μT −2 relating to the efficiency of the RF chain. 19 This factor is typically determined from simulations. In addition, RF power amplifiers typically have duty-cycle constraints on the fraction of "on-time," 0 , which can be related to sequence TR as Combining the limitations on TR from Equations (1), (3) and (4), the minimum achievable TR is then determined by: where SAR lim is the regulatory set SAR-limit. A crucial observation from this equation is that RF pulse designs, which solely aim to minimize T pulse , may not lead to the minimum sequence TR because the SAR restrictions may force a longer TR, whereas on increasing T pulse can induce RF duty-cycle restrictions.
For on-resonance RF pulses with fixed shape, it can be shown 19 that where is the FA, is the gyromagnetic ratio, and 1 and 2 are the normalized integrals of the pulse waveform and the square of the pulse waveform, respectively. These dimensionless values are properties of the pulse shape and are defined as In the common case of fixed RF shapes with constant valuedselection gradients, these values do not vary with pulse duration; therefore, selecting the minimum TR is a straightforward problem that can be solved by finding the optimal T pulse that minimizes TR min via Equation (5). Recent "minimum-duration" RF-pulse design methods use time-variable gradient waveforms to maximize performance within gradient and RF constraints. 16,17,22 In this case, E RF is no longer a simple function of T pulse because the pulse shape parameters 1 and 2 themselves change as the peak amplitude changes. This is illustrated in Figure 1A-C.

| Peripheral nerve stimulation
The use of time-variable selection gradients raises the question of whether PNS should be considered as a physiological constraint in addition to SAR. This was a topic in a recent article on minimizing TR for rapid phase-contrast imaging. 4 PNS is caused by fast switching gradients and depends on the rate and strength of time-varying fields, as well as directionality with respect to anatomy. 23 The "SAFE" model is often used in practice to predict stimulation. 24,25 The model considers the first-derivative of a gradient waveform G (t) as a peripheral nerve stimulus and lowpass filters this using a combination of different weightings: The threshold for PNS is assumed to have been exceeded when the SAFE model output exceeds some set limit. The coefficients used are prescribed by the vendor implementation of this method and relate to the hardware.

| Proposed minimum TR design approach
The following section introduces a general optimization framework to the problem of minimum-TR RF pulse design, followed by a description of the more limited implementation used in this work.
The TR can be minimized by the appropriate choice of RF pulse waveform B 1 (t) and corresponding gradient G (t), which together are written as control variable x. Assuming a fixed encoding time T enc , the minimization of TR is achieved by minimizing the RF pulse-duration subject to hardware | 185 where B max 1 , G max , S max correspond to hardware limitations on RF amplitude, gradient amplitude, and gradient slew-rate, respectively. PNS model represents a function, such as SAFE, which returns a PNS evaluation based on the selection gradient G (t) and the remaining sequence gradients G sequence (t), and PNS max is a set threshold value. The constraint in Equation (10.4) could be considered a hardware limitation; however, because it depends on the sequence parameter T enc , we consider it a sequence-level constraint.
Rather than approach this as a general optimization problem in which both RF pulse and gradient waveforms are designed, we propose a simple method to optimize performance for a given initial RF pulse shape. For an input RF pulse (typically designed for a constant gradient), the time-optimal VERSE 13 method is used to design a minimum-duration gradient waveform (and reshape the RF pulse waveform) subject to the hardware constraints in Equations (10.1)-(10.3). To satisfy the remaining constraints, a family of solutions is generated by altering the maximum B 1 specified for the VERSE algorithm. We refer to this limit as B control 1 to distinguish this from the hardware-related maximum B max 1 . From this family, the solution that minimizes T pulse while satisfying constraints (10.4)-(10.6) is selected. The PNS constraint in Equation (10.6) can be assumed inactive for most practical imaging scenarios, which we will discuss later. A list of steps for our implementation is summarized in Table 1.
Although not general in scope, the advantages of this method are that we can start with an existing RF pulse waveform that has desired characteristics, and that VERSE (which is a deterministic algorithm) avoids convergence issues caused by local minima. Figure 1 illustrates a single-band (SB) design case (FA = 40°, time bandwidth product [TBP] = 4) using this method for minimizing TR in a constant gradient and VERSE (time-variable gradient) case. Figure 1A shows different constant gradient pulses; Figure 1B shows different VERSE pulses computed for a range of B control 1 . Figure 1C demonstrates that the pulse shape properties 1 and 2 , relating to E RF , change significantly for the VERSE'd pulses as B control 1 is changed, and Figure 1D illustrates the application of sequence-level constraints in Equations (10.4) and (10.5). The minimum TR for each case is marked by a circle; in this case, using VERSE results in a reduction in minimum TR of approximately 20% compared with a constant gradient. To achieve this result, the required value of B control 1 was 7.08 μT, which would not have been obvious a priori. Simply minimizing pulse duration subject to hardware constraints would yield a solution with a TR over 7 ms, after sequence-level constraints are applied. This illustrates the importance of including sequence level constraints as part of the RF pulse design. The PNS constraint in Equation (10.6) is not shown; it was not a limiting factor for the optimal solution-this will be addressed later.

| RF pulse design
This work considered both SB-and MB-pulse design and has used the time-optimal VERSE 13 method and the recently proposed VERSE-MB method 17 , respectively, for these design problems. Within the context of this article, these algorithms are examples of general design methods to evaluate a pulse set x: B 1 (t) , G (t) given pulse properties such as flip-angle, time-bandwidth product, and number of excited slices. The resulting TR-optimized designs as a consequence of VERSE will be referred to as "TR-optimal," and TR-optimized constant gradient pulses will be referred to as "constant gradient." For all examples, the limits G max = 31 mT/m −1 and S max = 200 mT/m −1 /ms −1 were used, reflecting the performance (10) min x T pulse (x) subject to limits of the scanner used for experiments. Both design methods start with a SB constant gradient RF pulse waveform, and then compress this in time using VERSE for the SB case, and then further apply temporal modulation after VERSE for the MB case. 17 Amplitude-only modulation was used to avoid known hardware issues with RF fidelity. 26,27 The TR-optimal pulses were always compared against matched constant gradient pulse waveforms (ie, same starting pulse shape), which were also optimized to minimize the TR by adjusting their peak B 1 amplitude accordingly. In all designs, we constrained the gradient amplitude to start and end at zero. Quoted pulse durations include the selection gradient ramp-up and rampdown times for TR-optimal designs but exclude them for constant gradient pulses. This is because in the latter case the ramps can be overlapped with other gradients in the pulse sequence, whereas this is not true for TR-optimal designs as the RF is generally on during this time.
Two example applications were designed for in this work: (1) bSSFP with low TBP pulses (TBP = 2.13) and (2) high TBP (TBP = 4, 6, 8, 10) for imaging where spatial selectivity may be more important, such as slab-selective excitation for three-dimensional (3D) encoding. For both cases, we investigated the number of simultaneously excited slices N Sl from 1-6 (1 being equivalent to single-slice imaging), with slice-thickness 5 mm. For N sl > 1, interslice spacing (centerto-center) was computed to cover an imaging field of view of 100 mm in the through-slice direction. Both applications were designed for FAs from 25°-90°. All initial pulses on which the study was based were taken from the vendor pulse library.
To examine off-resonance effects on the slice-excitation profile because of the use of time-variable selection gradients, off-resonance analysis was conducted using Bloch equation simulations. Results are shown in Supporting Information Figure S1.

| Minimum TR calculation
Equation (10) could be approached as a nonlinear optimization problem; however, because there is only a single design variable used in this work (B control 1 ), we have taken the simpler approach of tabulating the relationship between E RF and T pulse for each design scenario by precomputing RF pulse designs for a range of B control 1 from 2 μT to 13 μT ( Table 1). The TR-optimal design can be found by searching for the intersection between the curves indicated by Equation (5) plotted on Figure 1D. The curves were precomputed using 20 points, after which the optimal B control 1 was determined to higher precision by interpolation. Once determined, an RF pulse specifically corresponding to this optimal B control 1 was designed. The optimal solution will depend on the T enc prescribed. An advantage of this ad hoc tabulation approach is that the same precalculated family of pulses can be used with different readout geometries that alter T enc , thus saving computational cost to redesign new pulses at run-time.
From earlier experience, we found PNS not to be an issue for typical imaging sequences; therefore, we did not explicitly enforce constraint [Equation (10.6)] for the design problems presented and did not find that PNS limits were violated by doing this. Instead, we have investigated the PNS behavior for a single imaging scenario between a constant gradient and a TR-optimal example.
All simulations used the International Electrotechnical Commission (IEC) 10-second local SAR limit (SAR lim = 20 W/kg −1 ) 28 ; the factor k was computed to relate pulse energy to maximum local SAR for the transmit coil used. The encoding time T enc was set to 1.79 ms in the low-TBP-bSSFP case, which was the optimized encoding time for the acquisition described in the methods below, and in the high TBP cases for slab-selective imaging the encoding time was 3.39 ms, which was also chosen for an optimized 3D bSSFP sequence. An RF duty-cycle limit of 50% "on time" (ie, 0 = 0.5) was used, corresponding to the scanner used for experimental work.

| Slice-shifting and nonideal gradient behavior
The time-variable selection gradients required by VERSE can lead to imperfection issues caused by limited temporal bandwidth of gradient systems. 15 Such imperfections can be incorporated into a gradient impulse-response function (GIRF) model, which characterizes the gradient system response as a linear time-invariant system. 29 In this work, a measured GIRF 30 was approximated using a Lorentzian function with a time constant of 42 μs (Supporting Information Figure S2) and used to evaluate adapted RF pulses that correct for the gradient errors. Details of this process are given in Appendix A, and phantom images showing its imaging effects are shown in Supporting Information Figures S3 and S4. The predicted gradient distortion was also used to spatially shift slices away from isocenter, using the shift-theorem described in Conolly et al. 31 The corrected RF waveform had a negligible effect on TR optimality, as the increase in RF energy was insignificant.

| Peripheral nerve stimulation
The SAFE model was used with parameters a 1 = 0.78, a 2 = 0.22, 1 = 0.32 ms, and 2 = 4.1 ms. PNS is most likely induced by the gradient along the anteroposterior (AP) direction when humans are scanned in the supine position. To account for this, each gradient direction was evaluated with the SAFE model in Equation (9) and further scaled by 1,

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ABO SEADA Et Al. 0.83 and 0.61 for anterior→posterior (AP), left→right (LR), and foot→head (FH) direction, respectively. It is assumed, that when the weighted output of the SAFE model exceeds a preset stimulation level PNS max , PNS will occur.
To investigate whether the minimum TR would be affected because of the insertion of a time-variable gradient, we implemented the SAFE model off-line and repeated this for encoding oriented along all three physical gradients axes independently. 32,33 This way, each encoding gradient (frequency, phase, slice) was simulated on each physical gradient (AP, LR, FH).

| Experiments
Cardiac imaging was performed using a breath-hold retrospective vectorcardiogram-gated cine sequence on a 3T Philips Achieva (Philips Healthcare) and a 32-channel receiver coil. Data were collected on a healthy volunteer (27-year-old man) following written informed consent under local ethical guidelines. MB imaging was performed using blipped-controlled aliasing in parallel imaging (CAIPI) as previously described for cine bSSFP. 6 N Sl = 2 RF pulses were designed for a FA of 40°, TBP = 2.13, slice thicknesses = 7 mm, and an interslice spacing (center-to-center) of 49 mm. The in-plane resolution was 1.6 × 2 mm, with 120 phase-encoding steps and 20 cardiac phases. No forms of in-plane acceleration (sensitivity encoding [SENSE] or half-Fourier) were used. B 0 -shimming was performed using a custom slice-by-slice tool. Aliased MB images were unfolded and reconstructed with a SENSE-based algorithm in ReconFrame (GyroTools GmbH). Software issues arose with user-defined RF and gradient shapes close to the maximum SAR limit of 20 W/kg −1 because the custom shapes were strictly forbidden from being stretched and shaped as usual trapezoids, which interfered with the scanner's original TR minimization. These issues were not present when using the scanner's native RF pulses. To make a fair comparison for in vivo experiments, all pulses were optimized for a limit of 18.6 W/kg −1 , which worked reliably. An additional software limitation, when incorporating the arbitrary-shaped gradient waveforms required for VERSE, meant that for the implemented constant gradient pulses, the ramp periods of the selection gradient could no longer be overlapped with the adjacent gradient objects in the pulse sequence. For in vivo comparisons using constant gradient pulses, the minimum TR was calculated accounting for this. Figure 2 shows the minimized TR for the case of low TBP (2.13) MB-pulse design, with slice thickness 5 mm for both constant and TR-optimal designs in Figure 2A,B, respectively. Relative performance between constant gradient and TR-optimal designs are shown in Figure 2C. Although both are optimized for TR, using TR-optimal designs reductions in TR of approximately 10% can be achieved for moderate FAs and number of excited slices (N Sl ), and, for example, 14.2% for N Sl = 6 and a 90° FA.

| RESULTS
An example is the reduction in TR of 12.3% from 4.93 ms to 4.32 ms, which can be achieved for a N Sl = 3 and 60° excitation, the example pulses for which are shown in Figure 3A,B. However, it is notable that for the case of a 25° FA and SB excitation, there is a longer TR for the TRoptimal case. This case is further examined in Figure 3C,D, which plots this example alongside the 60°, N Sl = 3 case. The reason for the lengthening of the TR for 25° is that this pulse is essentially limited by gradient amplitude and slew rate. Our implementation cannot improve on this because F I G U R E 2 Results for minimum TR optimization when using low time bandwidth product (TBP; 2.13) pulse and T enc = 1.79 ms, which is relevant for cardiac balanced steady-state free-precession. Minimum achievable TR for (A) constant gradient radiofrequency (RF) pulses, and (B) TR-optimal pulses. (C) Percentage TR reduction from optimization. For low TBP, the use of TR-optimal pulses can reduce minimum TR by 10% or more for higher flip angles and multiband factors. The negative values are further investigated in Figure 3C,D the TR-optimal design cannot overlap the gradient ramps whereas the constant gradient could, and so the TR actually increases. The examples in Figure 3A,B are operating at the SAR limit and RF duty-cycle limit [ie, Equations (10.5) and (10.4), respectively], whereas the examples in Figure 3C,D are operating at the SAR limit and sequence-timing limit [Equations (10.5) and (1)]. Figure 4 shows minimum TR results for TBP 4-10. It suggests that much larger improvements can be obtained if higher TBP pulses are used, particularly at higher FAs. For SB pulses, TR reduction of 10%-40% is possible. Similar performance was found for slice thicknesses up to 60 mm (not shown), making these results applicable to slab-selective imaging. For a MB (N Sl > 1) sequence, a TR reduction of 40% is commonly possible and can reach 72% in the case of N Sl = 6 and a 90° FA. Figure 5 shows an example for a SB TBP = 6 example at 60° excitation. The TR can be reduced from 5.95 to 5.05 (15%) as shown in Figure 5A,B. An example with N Sl = 3 reduces TR from 10.24 ms to 6.55 ms (36%) as shown in Figure 5C,D. Note that all pulses shown here are optimized for minimum TR, but a shorter TR is achieved when using time-variable gradients. A minimum-duration approach to design time-variable pulses would, for instance, use the peak B 1 from the constant gradient to design a VERSE pulse. If that were the case, the TR would increase to 7.93 ms and 13.1 ms as shown in 5B,D, respectively. The example given in Figure 5C operates at the SAR and RF duty-cycle limit [i.e., Equations (10.5) and (10.4), respectively], the remaining examples operate at the SAR and sequence-timing limit [Equations (1) and (10.5)]. Figure 6 shows cardiac bSSFP images in a healthy volunteer: T enc = 1.79 ms with TBP = 2.13 pulses in the diastole phase. On the left is a standard SB acquisition using a single 14.4-second breath-hold. The column to the right shows a N Sl = 2 accelerated acquisition of this using a constant gradient pulse, acquired in an 8.4-second breath-hold. In the next column, a minimum-duration RF pulse was designed to have the same peak-RF amplitude as the constant gradient pulse, leading to a short RF pulse that leads to high-RF F I G U R E 3 Two example cases of TR optimized pulses. Each subfigure contains gradient waveforms in orange and radiofrequency (RF) pulse in blue. All pulses shown are optimized for minimum TR, but an overall lower sequence TR is achieved using variable selection gradients. (A) This graph shows an N Sl = 3 pulse for a constant gradient case (flip angle = 60°, slice-thickness = 5 mm), which would result in a TR of 4.93 ms adhering to all limitations in Equation (10). For the same design, a TR-optimized approach reduces the TR to 4.32 ms with the pulses shown in (B). In these cases, the designs were limited by specific absorption rate (SAR) and RF duty cycle [Equations (10.5) and (10.4)]. (C,D) Examples (flip angle = 25°, slice thickness = 5 mm) are shown where the TR-optimal approach fails to produce a shorter TR. This is ultimately because the time-variable gradient waveforms cannot be overlapped with neighboring gradients; for the TR-optimal waveforms, the full gradient duration is counted, whereas in the constant gradient case the "ramps" can be overlapped and are not counted. In these two cases, designs were limited by SAR and sequence timing [Equations (1)  energy. However, this leads to a 0.9-ms increase in TR, increasing banding artifacts and the breath-hold period. Using our proposed framework, TR-optimal RF pulse and gradient waveforms allow imaging with a TR of 2.9 ms, leading to a breath-hold of 7.8 seconds. An animated version of this CINE dataset is available in Supporting Information Video S1. Figure 7 shows an annotated analysis from the same dataset as Figure 6, near the end-systole phase. Banding artifacts F I G U R E 4 Results for minimum TR optimization when using high time bandwidth product (TBP) 4-10 and T enc = 3.39 ms, as would be suitable for slab-selective excitation for three-dimensional encoding or multiband (MB) imaging with high selectivity. TR-optimal solutions are most effective at high TBP, MB factors, and flip angle. Moderate improvements can reach 40% reduction in TR, whereas an example with N Sl = 6, TBP = 10, and flip angle of 90° showed a reduction of 72%. A higher TBP leads to a higher TR in constant gradient designs, whereas for TRoptimal designs an increase in TBP comes at an insignificant extra cost in TR. N Sl = 1, N Sl = 3, and TBP = 6 pulses are shown in Figure 5 for a 15% and 36% TR reduction, respectively become clear at the edges of the myocardium in the constant gradient MB acquisition in the second column (red arrows) caused by the increase in TR. In the third column, an acquisition with minimum-duration VERSE leads to a non-TRoptimal acquisition with stronger artifacts on the outside of the myocardium, as well as in the blood pool within it, as is indicated by the red arrows. As shown in the fourth column, TR-optimal acquisition performs better, with some banding artifacts still visible at the top of the myocardium.
All TR-optimal pulse designs have been computed using maximum gradient amplitude and slew-rate constraints, raising the question of whether PNS ought to be considered as an additional constraint to achieving the minimum TR. Figure 8 shows example gradient waveforms for constant and TRoptimal gradient designs, their time derivatives, and predicted outputs from the SAFE model for a bSSFP example. The calculation uses repeated TR to account for nonsteadystate effects from Equation (9). This particular example roughly matched our in vivo demonstration, with TR = 2.81 ms, slice thickness = 7 mm, and in-plane resolution = 2 mm. The figure does not include phase-encoding gradients, as these vary through the sequence and in any case are not the major contributors to PNS. The shaded patches in Figure 8C account for the fact that the PNS predictions are orientation dependent because different gradient axes have different PNS sensitivity, AP and FH being the most and least sensitive directions, respectively. The biggest contributors towards PNS are the slopes on either side of the readout gradient. The SAFE-model output shows that replacing a constant gradient with a TR-optimal gradient increases the contribution of the slice-select gradient towards PNS prediction. However, it remains relatively low in comparison with the contribution from the readout gradient. In practice, for both low and high TBP experiments, we did not encounter any reasonable scenarios in which PNS would have limited the solutions obtained.
The off-resonance simulations in Supporting Information Figure S1 show that slice-shifting effects (calculated by maximizing cross correlation) are similar for both constant and TR-optimal pulses. Slice distortion (calculated by normalized root mean square error) on the other hand is worse for TR-optimal pulses than for the constant gradient pulses, and this effect is worse with increasing TBP caused by highly time-variable gradients.

F I G U R E 5
Two examples of higher time bandwidth product (TBP) pulses. All pulses shown are optimized for minimum pulse-repetition time (TR), but a lower-sequence TR is achieved when a variable selection gradient is used. A TR reduction of 15% can be achieved by moving from a constant gradient N Sl = 1 TB 6 case in (A) to a TR-optimal case in (B). In these cases, designs were limited by specific absorption rate (SAR) and sequence constraints [Equations (1)

| DISCUSSION
Rapid gradient echo sequences such as bSSFP and SPGR are widely used in clinical MR and research applications. In the case of two-dimensional (2D) sequences, typically low-timebandwidth-product RF pulses are used and pushed beyond their usual operating points when higher slice selectivity or MB imaging is desired. Much work has gone into optimizing RF pulses for minimum duration, 31,34 in particular for MB excitation using time-variable-selection gradients. 17,18,22 However, for rapid gradient echo sequences, SAR can quickly become a limiting factor; in this case, a shorter TR F I G U R E 6 Cardiac multiband2 (MB2) balanced steady-state free-precession experiments using conventional MB2 (N Sl = 2) and two different TR-optimal pulses (time bandwidth = 2.13, flip angle = 40°, slice thickness = 7 mm, 1.6 × 2 mm in-plane resolution. No in-plane acceleration). (A) Shows data acquired without MB using a 14.4-second breath-hold. (B) Using a MB2 pulse designed for minimum TR, this can be reduced to a 8.4-second breath-hold, which comes at a TR increase of 19.2%. The constant gradient MB2 pulse can be optimized in the same way as the N Sl = 1 pulse. (C) A minimum-duration radiofrequency (RF) pulse (non-TR-optimal) designed to have the same peak B 1 -amplitude as the constant gradient MB2 pulse, results in an RF pulse shorter than the MB2 pulse; however, this leads to a 0.8-ms increase in TR caused by high-RF energy. This also leads to a 10.6-second breath-hold. (D) A TR-optimal pulse designed using our minimum TR framework results in a reduced 7.8-second breathhold with only a 11.5% increase in TR compared with the data acquired with single-slice excitation. S, second F I G U R E 7 An annotated version of Figure 6 using multiband2 (MB2; N Sl = 2) showing just 1 slice, picked at the end-systole phase. (A) Shows the single-band image as a reference. Banding artifacts become most clear at the edges of the myocardium in the constant gradient MB acquisition in (B) (red arrows) caused by the increase in TR. In (C), a MB2 minimum-duration radiofrequency pulse (non-TR-optimal) acquisition leads to more detrimental artifacts at two ends of the myocardium (red arrows) as the band approaches the blood pool. (D) With TR-optimal, acquisition performs better, with minimal banding artifacts left in the myocardium can often be achieved by lengthening the RF pulse. This is already well-understood within the context of constant gradient RF pulses that can be manipulated by stretching in time to reduce the peak amplitude, but whose shapes remain fixed. The situation is more complicated for minimum-duration RF pulses because their shape properties change as a function of their duration (Figure 1). Hence, this work proposes a method for extending the logic for minimizing TR with constant gradient 19 pulses to VERSE pulses (or any pulses using timevariable-selection gradients), by employing an optimization framework that finds the optimal set of pulses to minimize TR. Figure 2 depicts the reductions in TR of approximately 10% that can be made for low TBP pulses (TBP = 2.13) typically used for rapid 2D imaging. This was demonstrated in vivo for MB cardiac bSSFP imaging (Figures 6 and 7) in which the TR was reduced from 3.1 ms to 2.9 ms, which helps by reducing the breath-hold duration and reducing the impact of bSSFP black-banding artifacts. These figures also show the results of applying minimum-duration pulse design directly-though the pulse is made shorter, in which case the TR increases to 4.0 ms.
A case where the TR-optimal approach failed is shown in Figure 3C,D. This is because the constant gradient is already at its highest amplitude, and the ramps cannot be overlapped with other gradients for the TR-optimal approach as the RF is transmitted concurrently. This can occur generally when the desired pulses have low FAs (ie, B 1 limit is not reached) and thin slices (requiring high-excitation bandwidth). Our implementation did not allow the gradient ramps to be excluded F I G U R E 8 Time-resolved input and output waveforms of the SAFE model for peripheral nerve stimulation (PNS) prediction during a multi-TR sequence simulation, here only shown for one TR period. (A) Input gradient waveforms Gxyz (for gradient direction x, y, z) for a balanced steady-state free-precession sequence, for a TR-optimal and constant gradient case, selected to match the TR of the TR-optimal case for clarity. The phase-encoding gradients are omitted for clarity. (B) Time-derivative of the gradient waveforms (dGxyz) over time (dt), needed by the SAFE model in Equation (9). (C) SAFE model output; shaded areas represent variation expected based on different slice orientation (see text for details). The peak stimulation output relates to the readout gradient in the anteroposterior gradient direction for a supine positioned subject. The TR-optimal, time-variable selection gradient adds more to the model output than the constant gradient (green arrows). However, the readout gradient (blue arrows) has the larger effect towards PNS violation. Const, constant; GR, gradient | 193 ABO SEADA Et Al.
from the VERSE-based design; however, the algorithm does allow this flexibility. A future implementation could be improved by allowing ramps to be excluded when necessary, such that the start and end of the gradient pulse are simply concatenated with neighboring ramp gradients, giving them the same flexibility as the constant gradient case.
The benefit of time-variable gradient waveforms for pulse design becomes more significant when higher TBP and MB factors are employed as shown in Figures 4 and 5. The proposed TR-optimal approach could allow use of higher TBP pulses for less penalty in TR. For example, using constant gradient RF pulses, a 60° N Sl = 3 excitation with TBP = 2 gives minimum TR = 4.93 ms, increasing to 8.46 ms for TBP 4 and 10.24 ms for TBP 6, whereas the TR-optimized designs can achieve minimum TRs of 4.32 ms, 6.25 ms, and 6.43 ms, respectively. For high TBP pulses, the TR-optimal designs also have notably different operating points. For the constant gradient design the optimal scenario is to push to a higher peak B 1 , whereas the TR-optimal designs have a much lower peak B 1 than the system limit of 13 µT. Simply reducing the pulse duration by increasing the peak B 1 would not help here because the SAR limit would be exceeded. The benefits of high TBP slab selection have already been shown by Hargreaves et al 34 for cardiac imaging or high-field MR angiography. 35 With MB imaging, use of a higher TBP could aid multislab MB angiography, 36 improve simultaneous multislice magnetic resonance fingerprinting, 37,38 or potentially reduce leakage between slices. 39 Going further, because turbo spin-echo sequences can also be SAR-limited, an interesting further investigation could adapt this framework for multi-echo sequences. The common ground between rapid gradient echo and, for example, turbo spin echo sequences is that low-SAR pulses have long duration, which increases echo times. Such a framework would be useful for MB turbo spin-echo, 40 gradient-and spinecho, 41 and volumetric imaging techniques. 42 A recent study proposed a convex optimization framework to shorten TR for four-dimensional phase-contrast MRI acquisition. 4 VERSE was also used for shortening RF pulses; however, no SAR violation was reported. Their framework is conceptually similar to our work but differs by optimizing encoding gradients and derating these where necessary to meet PNS constraints, whereas we optimized selection pulses to meet encoding and SAR constraints. When PNS is not considered in optimization, the most time-efficient pulse will make the most use of the hardware until limited by peak-RF amplitude. We considered PNS using the SAFE model ( Figure 8) and concluded that the readout gradient is always the largest contributor. This is inherent when the readout resolution is higher than the through-slice resolution, which is most often the case. Stimulation overheads can be seen on either side of the readout ramps. Thus, when stimulation becomes a limiting factor, an effective approach would be to alter the readout-encoding gradients, such as changing the physical readout direction or reducing the readout bandwidth.
Other SMS cardiac studies have used RF phase-cycling schemes to perform controlled aliasing for anatomic 5,[8][9][10][43][44][45] and quantitative imaging. 46,47 Such methods generally require two unique RF pulses per shift in the slice direction. This study used blipped-CAIPI, as did another recent study 27 however, our presented framework with the VERSE-MB approach 17 is compatible with RF phase-cycling because the gradient is not further optimized after applying the CAIPI modulation. This is because once B control 1 is set, the pulse duration becomes fixed; therefore, a CAIPI phase offset does not change RF energy per Parseval's theorem. If the gradient is optimized after CAIPI, its optimal form will depend on the RF envelope; hence, a time-optimal solution will depend on the phase-cycling scheme.
A limitation of rapidly modulated time-variable gradients is that nonideal gradient performance may affect some types of scanner more than others. In this study, we have built on previous work that characterized these effects using an impulse-response function 29 and have used an inherently lower bandwidth design approach for constructing the MB pulses. 17 In addition, for this work we corrected the RF-pulse waveforms by resampling using the GIRF-predicted gradient waveform (see the Appendix A). We also used this to correct the frequency modulation that is required to shift the slices off-center. We used a Lorentzian function to approximate the GIRF measurement and found that a GIRF measurement is not required. Note that resampling the pulses does slightly change their energy-duration relationship, but this was not found to significantly change the solutions and so corrections were applied post hoc. Imaging results are shown in Supporting Information Figures S3 and S4.
The method presented for designing TR-optimal pulses can potentially be used with any minimum-duration RFpulse-design approach, and more efficient design algorithms than the ones used here, such as NU-SLR 14 or optimal control 15,22 could yield better results. We opted to precalculate a number of solutions, from which the optimum B control 1 can be identified quickly by interpolation, after which a single pulse can be designed in seconds. This reduces the need for a complex optimization at run-time, but does require a library of calculations to be performed for different parameters (FAs, TBP, etc.). An alternative would be to solve the minimization in Equation (10) directly. Going further, we did not experience TR improvement from lower limits on maximum gradient amplitude and slew rate; however, we did not rule out better solutions generally. We also did not experiment with gradient waveforms with nonzero start-and-end values. Therefore, in future work these could be introduced as control variables, which could avoid the issue seen in Figure 3C,D. More flexibility on gradient control is expected to become important as PNS becomes a limiting factor. Introducing more control variables and constraints would make our exhaustive tabulation approach less attractive. Instead, because the relation between pulse duration and RF energy ( Figure 1D) is smooth and changing gradient limits affects the pulse duration, a gradient-descent approach could be a plausible alternative.

| CONCLUSION
We propose a general framework and simple approach for designing TR-optimal RF pulses that minimize the overall sequence TR by considering hardware and sequence-level constraints. TR can be reduced by up to 14% and 72% when using low and high time-bandwidth profiles, respectively. The benefits are particularly strong for high FAs, MB acceleration factors, and time-bandwidth-product pulses. The flexibility of this framework was demonstrated in MB cardiac bSSFP and should benefit other applications such as quantitative imaging and MR angiography.