Optimal control design of band-selective excitation pulses that accommodate relaxation and RF inhomogeneity
Thomas E. Skinner, Naum I. Gershenzon, Manoj Nimbalkar, Steffen J. Glaser
aa r X i v : . [ phy s i c s . i n s - d e t ] N ov Optimal control design of band-selective excitation pulsesthat accommodate relaxation and RF inhomogeneity
Thomas E. Skinner a, ∗ , Naum I. Gershenzon a , Manoj Nimbalkar b , Steffen J. Glaser b a Physics Department, Wright State University, Dayton, OH 45435, USA b Department of Chemistry, Technische Universit¨at M¨unchen, Lichtenbergstr. 4, 85747 Garching, Germany
Abstract
Existing optimal control protocols for mitigating the effects of relaxation and/or RF inhomogeneity on broad-band pulse performance are extended to the more difficult problem of designing robust, refocused, frequencyselective excitation pulses. For the demanding case of T and T equal to the pulse length, anticipated signallosses can be significantly reduced while achieving nearly ideal frequency selectivity. Improvements in perfor-mance are the result of allowing residual unrefocused magnetization after applying relaxation-compensatedselective excitation by optimized pulses (RC-SEBOP). We demonstrate simple pulse sequence elements foreliminating this unwanted residual signal. Keywords: selective excitation; RC-SEBOB; Relaxation; T relaxation; T relaxation; optimal controltheory PACS:
1. Introduction
Frequency-selective pulses have widespread util-ity in magnetic resonance and have motivated sig-nificant efforts towards their design [1, 2, 3, 4, 5,6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47]. Inmany useful cases, the resulting methodologies canachieve the best approximation to the ideal rect-angular profile for spin response as a function offrequency offset.However, in all of these approaches to pulse de-sign, performance criteria that can be included inthe design protocol are restricted either by analyt-ical procedures of highly specific scope or by nu-merical methods that are limited by the efficiencyof the optimizations employed. As a result, pulseresponse is typically optimized only for the nom-inal ideal RF pulse values. In addition, althoughthe length of pulses required for narrowband ap-plications can significantly reduce their effective-ness when relaxation times are comparable to the ∗ Corresponding author.
Email addresses: [email protected] (Thomas E. Skinner), [email protected] (Steffen J. Glaser) pulse length [48, 49], the solution to the problem—selective pulses which are less sensitive to relaxationeffects—can also be demanding for standard designmethods [50, 51, 33, 52, 53, 54].To make these design challenges tractable, thespace of possible pulse shapes is often reduced byforcing the solution to be a member of a particu-lar family of functional forms (for example, finiteFourier series). Thus, potentially, there are impor-tant solutions that are missed.Over the past decade, we have shown opti-mal control theory to be an efficient and power-ful method that can be applied to a wide rangeof challenging NMR pulse design problems withoutrestricting the space of possible solutions [55, 56,57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69,70, 71, 72, 73]. It is capable of designing broad-band pulses [66] and selective pulses [74, 75] thatare simultaneously tolerant to RF inhomogeneityand relaxation effects, which we develop further inthe present work.
2. Selective pulse design
Optimal control (including similar, related op-timization procedures) was originally introduced
Preprint submitted to Elsevier November 8, 2018 nto magnetic resonance for the design of band-selective pulses, primarily for imaging [76, 77,78, 79, 80, 81, 82]. It was quickly supplantedby the efficient Shinnar-LeRoux (SLR) algorithm[17, 18, 19, 20, 21], which establishes a correspon-dence between frequency-selective pulse design anddigital filter design. There are fast, non-iterativealgorithms for the ideal filter and, hence, the idealpulse. Unfortunately, the algorithm does not ac-commodate additional criteria, such as tolerance toRF inhomogeneity (included in some of the earli-est optimal control-related approaches [76, 80]) orrelaxation effects. In addition, the most applica-ble and widely used form of the algorithm derivespulses which produce a specific linear phase dis-persion in the spectral response. Pulses producingno phase dispersion, suitable for spectroscopy, aremore problematic for the SLR algorithm.We first provide an overview of well-known issuesrelevant to selective pulse design, since there is con-siderably less freedom in the choice of parameterscompared to broadband pulses. For example, in de-signing broadband pulses, we have shown [60] thereis at best only a marginal relation between the max-imum amplitude, RF max , of a pulse and the achiev-able excitation or inversion bandwidth, as long asthe pulse length, T p , is allowed to increase suffi-ciently. Alternatively, increasing RF max for a given T p can improve performance for a given bandwidthor increase the bandwidth. There can also be in-numerable broadband pulses that provide approx-imately equal performance for a given RF max , T p ,and bandwidth.Selective pulses are far more constrained, witha well-known relation between the selective band-width and T p , and much tighter limits on the choiceof RF max for a given product of bandwidth and T p [21]. We provide only a simple overview of the op-timal control methodology, emphasizing the modi-fications necessary for the present work. The basicalgorithm for optimizing pulse performance over arange of resonance offsets and RF inhomogeneity isdescribed fully in [57]. Details related to incorpo-rating relaxation [66] and specific dispersion in thephase of the final magnetization[61, 67, 83], whichwe refer to as the phase slope, are provided in theassociated references. Values of the phase slope, R , at each offset [67]characterize the net phase dispersion that accumu-lates during a pulse of length T p . The phase slope is defined relative to the net precession of trans-verse magnetization that would be produced solelyby chemical-shift evolution during the same timeinterval, T p . A pulse that produces focused mag-netization of fixed phase for all spins in the offsetrange of interest would have constant R = 0 (i.e.,a self-refocused pulse). Many selective pulses aresymmetric, R = 1 / M ( t ) is driven by the RF controls to afinal magnetization F that is defined for each res-onance offset in the desired range. To excite trans-verse magnetization of linear phase slope R , we con-sider target states for each offset ω in the excitationbandwidth of the form [67] F = [ cos( ϕ ) , sin( ϕ ) , ϕ = R ωT p gives a linear phase slope, butany function can be considered to define a usefultarget phase, such as quadratic or higher order.Selective excitation most simply requires chang-ing the target to F = [ 0 , , ∼ For design conditions employing ideal RF in theabsence of relaxation, selective pulse performanceis completely determined by the desired passbandwidth B , pulse length T p , transition width W join-ing the passband and stopband, and residual signalfluctuation or ripple δ and δ about the ideal targetamplitude for the passband and stopband, respec-tively.The passband frequency ν p and stopband fre-quency ν s are defined where the magnitude of themagnetization response becomes less than the as-sociated fluctuations 1 − δ and | δ | , as illustratedin Fig.1 (adopted from Ref. [21]). The frequencywhere the amplitude drops to one-half is approxi-mately the average of these two frequencies. Thefull width of the filter is defined as twice this value,giving a bandwidth B = ν s + ν p and a fractionaltransition width W = ( ν s − ν p ) /B .More specifically (and again emphasizing the de-sign conditions stated at the beginning of the sec-tion), selective pulse performance is constrained byrelations for optimal FIR filters of the form W T p B = f ( δ , δ ) , (2)in terms of an empirically derived function f ( δ , δ )[86]. For a given value of f = W T p B , smaller(larger) δ gives larger (smaller) δ . Alternatively,for fixed δ or fixed δ , values of f increase as δ or δ , respectively, decrease. Flexibility in selec-tive pulse design is thus purchased at the cost oftrade-offs among bandwidth, pulse length, transi-tion width, and ripple amplitudes. Choosing anyfour of the set determines the fifth.This relation appears to have been little used inthe spectroscopic community. Although the pre-cise form of the function f ( δ , δ ) holds only for R = 1 / R . One im-portant implication is that pulse performance for agiven absolute transition width BW = ν s − ν p canbe made independent of the passband width, B .Fixed T p results in the same performance in termsof residual signal (ripple) for different B as long asthe transition width BW is constant. This was ob-served empirically and noted in [47]. We thus useEq. [ 2 ] to inform our optimal control design. The approximation to the ideal rectangular fre-quency response profile, as illustrated in Fig.1, canbe readily obtained using a variety of methods—among them, the Parks-McClellan (PM) algorithmfor linear-phase FIR digital filters [87]. After calcu-lating the PM polynomial, the standard SLR algo-rithm effectively inverts the frequency response toproduce a linear phase ( R = 1 /
2) RF pulse. Theequivalent optimal control approach would be touse the polynomial response function as the targetresponse and efficiently modify the RF controls toachieve the allowed performance.However, as noted already, the target responsederived using the PM algorithm applies only to R = 1 /
2, using the ideal RF amplitudes, in the ab-sence of relaxation. More significantly, optimal con-trol does not need to know the polynomial response.One can simply define the ideal passband/stopbandfrequency response, and the algorithm will find theresponse allowed by the particular choice of band-width, pulse length, and transition width. Differentfunctional forms can be defined for the response inthe transition region to provide additional flexibil-ity. The response at each frequency can also begiven weights to fine-tune the final excitation pro-file.We now proceed with the design of more robustselective excitation pulses. In what follows, we letthe frequency response to the pulse (ie, the tar-get function) be undefined in the transition region.The pulse has the flexibility to do anything it wantsthere. An important addition to the algorithm isan adjustable weight function which changes theweight for each particular offset depending on thedeviation of intermediate results from the desiredperformance. During a given iteration, if the de-viation of magnetization from the target for a par-ticular offset is larger than the allowed ripple am-plitude, then the weight for this offset increases;otherwise it decreases. This is an extension andgeneralization of the method introduced in [61].
3. Results and Discussion
We present several examples illustrating the ad-vantages of relaxation-compensated selective exci-tation by optimized pulses (RC-SEBOP). As dis-cussed, possible performance improvements includeincreased tolerance to RF inhomogeneity and com-pensation for relaxation effects.3 .1. Compensation for relaxation and RF inhomo-geneity
The algorithm for generating relaxation compen-sated broadband pulses (RC-BEBOP) has alreadybeen developed [66]. It only needs inclusion ofthe modified target functions described in Section2.1 for application to selective pulses (RC-SEBOP).The primary result of that work is that substan-tial signal gains are possible relative to expectedlosses from short T , T by finding trajectories forthe transformed magnetization that minimize theselosses, even if this requires a longer pulse length.Relaxation losses are minimized by keeping spinsnear the z-axis and orienting them so that all off-sets can be transformed to the transverse plane ina very short time at the end of the pulse. This so-lution not only mitigates the effects of transverserelaxation, but short T becomes an advantage dueto repolarization of z -magnetization during the rel-atively long time the spins are aligned close to thelongitudinal axis. The net affect is that almost allthe RF power is applied at the end of our relax-ation compensated pulses [66]. This is fortunatelyalso consistent with a particular family of solutionsfor broadband R = 0 (refocused) pulses [67].There are also solutions for refocused selectiveexcitation pulses [47] that employ significant RFpower throughout the pulse and therefore do notlend themselves to relaxation compensation. Onthe other hand, these pulses do an excellent job ofminimizing the residual off-axis magnetization inboth the passband and stopband. Consistent withthis result, we find empirically that minimizing re-laxation losses for selective pulses competes withminimizing residual off-axis magnetization. Thetrajectories that reduce relaxation effects are notcompatible with those that achieve good refocus-ing. We therefore consider a strategy that maxi-mizes x -magnetization in the passband, minimizesit in the stopband, i.e., the standard procedure,but removes any explicit restrictions on residual y -magnetization. This allows the optimal control al-gorithm to emphasize relaxation compensation andfind solutions with considerably enhanced perfor-mance compared to those that give highest priorityto minimizing the y -component. As we will show,this undesirable residual magnetization can be elim-inated after the pulse without significantly affectingperformance. In addition to causing signal losses, relaxationcan also degrade the uniformity of the excitationprofile [48]. SLURP pulses [50] were developedspecifically to obtain more uniform response overthe selected bandwidth while accepting attenuationdue to short T , T . SLURP-1 pulses address theparticular case T = T , and were derived for vari-ous values of the ratio T / T p .For the demanding case T p = T = T = 1 ms,theoretical performance of SLURP-1 is comparedto RC-SEBOP in Fig. 2 for RF miscalibrationsof − x -magnetizationover the optimized bandwidth of 4 kHz, in spite ofshort T , T equal to the pulse length. It is rela-tively insensitive to RF inhomogeneity of ± M x magnetization in the stopband, and a narrowertransition width.Analogous investigations have optimizedrelaxation-compensated pulses by applying simu-lated annealing to RF waveforms represented usingfinite Fourier series ( ∼ <
80% for thecase T p = T = T over a less selective bandwidththan obtained here. Some of our performancegains may be due to a more efficient optimizationprocedure that does not restrict the solution space.However, we expect the largest gains are due tothe flexibility of allowing residual y -magnetization, M y , which can be quite large, as shown in thebottom panel of Fig. 2. This is easily eliminated, asshown later, and allows a much more ideal, rectan-gular excitation profile than previously consideredpossible for short T , T [48], with minimal loss ofsignal. Alternatively, if we choose to minimize theresidual M y during the optimization, this requirestrajectories that sacrifice signal enhancement andselectivity of the passband, consistent with theother cited studies. A simple strategy for reducing relaxation effectsis to reduce the pulse length, but according toEq.[ 2 ], achieving acceptable performance for a nar-row excitation bandwidth then becomes more prob-lematic. A given time-bandwidth product requires4rade-offs in the sharpness of the excitation profile(transition width) and the signal variation (ripple)in both the passband and the stopband. Still, onecan optimize performance for a desired low valueof the time-bandwidth product, as accomplished inthe SNOB family of pulses [33]. Including relax-ation compensation beyond what is accomplishedby a short pulse length alone and including toler-ance to RF inhomogeneity provide additional op-portunities for improved performance.For e-SNOB, T p = 1 ms, selective bandwidth ± . T = T = T p and toleranceto RF inhomogeneity of ± M x mag-netization in the stopband. To minimize relaxationlosses, RC-SEBOP delivers most of its RF powerat the end of the pulse. As in the previous exam-ple, these enhanced performance features are pur-chased at the cost of a larger residual M y comparedto e-SNOB. In the next section, we present meth-ods for selecting only the desired M x magnetizationwhile maintaining the performance advantages ofRC-SEBOP.
4. Experimental
All the selective pulses considered so far, bothtraditional pulses and optimal control pulses, pro-duce significant residual M y magnetization in thepassband at non-ideal RF calibration. Methods forremoving this unwanted signal therefore have moregeneral applicability.To destroy undesired M y after selective excita-tion in a single-acquistion experiment, a hard 90 ◦− y flip-back pulse can be applied to store M x alongthe z -axis. A gradient pulse is then employed todephase the M y component, followed by a hard90 ◦ y pulse for acquisition of the signal due to M x .This “crusher” sequence, implemented in Fig. 4aas a more general phase-cycled sequence (to bediscussed below), was first tested for the case ofunlimited-bandwidth hard pulses by applying thehard pulses on resonance. The transmitter fre-quency was shifted only for the RC-SEBOP pulseto measure its off-resonance performance. This se-quence is insensitive to the actual T of the sample,since there is minimal relaxation during the shorthard pulses and no T relaxation of magnetization stored along the z -axis during the gradient pulse. Inaddition, for typical samples with T ≫ T , thereare minimal T effects as a result of the sequence.However, for very short T , repolarization of stop-band magnetization during the gradient pulse canlead to slightly more M x magnetization in the stop-band than expected from the theoretical selectiv-ity profile of RC-BEBOP, previously illustrated inFig. 3.To eliminate extra stopband signal in cases where T is too short for ideal performance of the originalcrusher sequence, it can be phase-cycled as shownin the figure. The two acquisitions add construc-tively in the passband. Repolarization at stopbandfrequencies leads to + z -magnetization after the firsthard pulse in each cycle which is canceled by addi-tion of the two acquisitions after the second hardpulse. As a general strategy, this sequence worksvery well for the case of ideal hard pulses with nobandwidth limitations. It therefore also works wellover a bandwidth of ∼ z -component, and the y -magnetizationwill no longer remain untouched. However, the gra-dient still dephases the transverse magnetization re-maining after the first hard pulse. To a first approx-imation, there is only z -magnetization prior to thesecond hard pulse, and the phase cycle producestransverse components of opposite sign that cancelon addition.Experimental off-resonance performance of RC-SEBOP and the phase-cycled crusher sequence isdemonstrated in Fig. 5 for the residual HDO sig-nal in a sample of 99.96% D O, doped with CuSO to relaxation times of T = 1 .
345 ms and T =1 .
024 ms at 298 ◦ K.Signals representing offsets between −
15 kHz to15 kHz were obtained in steps of 200 Hz by offset-ting the transmitter in this fashion for all appliedpulses. The sequence was first implemented using10 µ s hard pulses, which resulted in stopband signalof ∼
5% relative to the centerband. It was thenfine-tuned by increasing the hard-pulse length to10.68 µ s in the first cycle. The resulting selectivity5rofile is in very good agreement with the simula-tions for eSNOB and RC-BEBOP performance inFig.3. We obtain a signal of 0.83 on resonance usingthe nominal ideal RF values for RC-SEBOP com-pared to the theoretical value 0.89. All values arerelative to an ideal signal of 1 for the case of norelaxation. The small disagreement between exper-iment and simulation can be attributed to RF in-homogeneity/miscalibration in the flip-back pulse,which will leave some small fraction of the desiredpassband signal in the transverse plane to be de-stroyed by the gradient.Since relaxation compensation functions by keep-ing spins close to the z -axis as long as possible dur-ing the pulse, the performance of RC-SEBOP is nothighly specific to particular T , T values. Althoughoptimized for T = T = 1 ms, the T p = 1 msRC-SEBOP pulse of Fig. 3 performs well for muchshorter relaxation of T = 708 µ s and T = 527 µ s,as shown in Fig. 6. A signal of 0.73 is obtained onresonance using the nominal ideal RF values for thepulse and the phase-cycled crusher sequence com-pared to a theoretical value of 0.81 for on-resonanceexcitation by RC-SEBOP alone. The pulse couldalso be optimized for faster relaxation to improveperformance further.If a single-acquisition sequence is preferred, thesequence of Fig. 4b can be used to more completelyeliminate stopband signal for the case of short T .The first gradient is followed by a hard 180 ◦ y pulse. M z is flipped to − z where any additional magne-tization repolarized during the first gradient pulsecan relax to zero during a subsequent delay of thesame length as the first. This extra delay can alsobe used for additional dephasing of unwanted trans-verse magnetization by a gradient pulse of oppositesign to the first gradient. The sequence ends witha hard 90 ◦ y pulse followed by signal acquisition.However, performance depends more sensitively onany RF inhomogeneity or miscalibration of the hardpulses, since there are now two opportunities to de-stroy leftover transverse magnetization due to im-perfect rotation and storage along the z -axis. Off-resonance effects of the hard pulses, which now in-clude a 180 ◦ pulse, further degrade performance.For the experiment described in Fig. 6, a signalof 0.68 is obtained using the single-acquisition se-quence of Fig. 4b. The scheme is included as an op-tion for narrow passband applications. In addition,one might want to explore possibilities using broad-band, shaped 90 ◦ and 180 ◦ pulses. These could alsobe incorporated into a simpler sequence that elimi- nates the gradient in Fig.4a and cycles according to[ ± y , y , Acq( x, − x )]. These topics are beyond thescope of the present article.
5. Conclusion
Pulses which provide robust and enhanced per-formance despite RF inhomogeneity/miscalibrationand relaxation effects are highly desirable. The op-timal control approach to designing refocused se-lective excitation pulses with these compensatorymechanisms has been presented. The examples con-sidered the standard selectivity profile comprised ofa passband, transition region, stopband, and vari-ations (ripple) in signal uniformity, as illustratedin Fig. 1. Constraints and trade-offs in the perfor-mance among these fundamental parameters wereemphasized. In particular, we found relaxationcompensation and null excitation of magnetizationin the stopband to be competing goals. Consid-erable improvements in selectivity and relaxation-compensation for short T , T were obtained by al-lowing residual unrefocused magnetization in boththe passband and stopband. This residual mag-netization can be readily eliminated without sig-nificantly diminishing performance using additionalpulse elements. Acknowledgments
T.E.S. acknowledges support from the NationalScience Foundation under Grant CHE-0943441.S.J.G. acknowledges support from the DFG (GL203/6-1), SFB 631 and the Fonds der Chemis-chen Industrie. Experiments were performed at theBavarian NMR center at TU M¨unchen.
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Med. δ ν p ν s δ Figure 1: (Adapted from Ref.[21]) A finite length selective pulse can only approximate the ideal rectangular frequency response.Residual signal or ripple amplitude in the selected frequency spectrum (passband) is denoted by δ , with δ representing theripple over the frequency range where the signal should be nulled (stopband). The positive frequencies ν p and ν s define thepassband and stopband, respectively. The plotted response is symmetric about the zero frequency.
10 0 1000.20.40.60.81 M x −10 0 1000.20.40.60.81 −10 0 1000.20.40.60.81−10 0 10−1−0.500.51 M y −10 0 10−1−0.500.51 Frequency (kHz) −10 0 10−1−0.500.51 B /B =0.9 B /B =1 B /B =1.1 Figure 2: Excited magnetization M x (top panels) and M y (bottom panels) is plotted as a function of resonance offset forSLURP-1 (green) and RC-SEBOP (blue) for T p = T = T = 1 ms and RF inhomogeneity/miscalibration of − >
90% of the desired M x while achieving anearly rectangular profile that is relatively insensitive to ±
10% RF inhomogeneity over the optimized excitation bandwidthof 4 kHz. The signal gain on resonance for the nominal ideal RF is a factor of 1.6. Minimal relaxation losses are achievedby allowing large residual M y , particularly in the stopband. This unwanted signal can be subsequently eliminated withoutsignificantly affecting performance, as described in the text and Figs.4–6. R F A m p li t ud e ( k H z ) Time (ms) −15 −10 −5 0 5 10 1500.20.40.60.81 M x Frequency (kHz) −15 −10 −5 0 5 10 1500.20.40.60.81 M x Frequency (kHz) −15 −10 −5 0 5 10 1500.20.40.60.81
Frequency (kHz)
A BC D
Figure 3: Selective pulses eSNOB (green) and RC-SEBOP (blue) of length T p = 1 ms are plotted in panel A. RC-SEBOPutilizes less RF power, which is applied primarily at the end of the pulse to reduce relaxation losses, as described in thetext. Panels B, C, and D show the frequency response of the pulses for RF miscalibaration/inhomogeneity of − T = T = 1 ms. RC-SEBOP significantly reduces relaxation losses and provides a sharper and morerectangular selectivity profile over the designed excitation bandwidth of 3 kHz. The signal gain in this example is a factor of 2. H G G G Z SP H G G Z SP ± ± yy (a)(b)x,x Figure 4: (a) Phase-cycled sequence consisting of excitation by selective pulse (SP), hard 90 ◦ pulse, gradient pulse, hard 90 ◦ pulse, and acquisition. The sequence is designed to eliminate residual y -magnetization produced either by standard selectivepulses in the presence of RF inhomogeneity or produced by design in the case of RC-SEBOP. If T is sufficiently long so thereis no repolarization during the 90 ◦ –G—90 ◦ sequence, the first cycle is not needed, and the second cycle can be employed as asingle-acquisition sequence. (b) a single-acquisition sequence designed as an alternative to (a) for the case of short T . Furtherdetails for both sequences are provided in the Experimental secion. .91.01.1-15 0 15 -15 0 15 Dn ( ) kHz
Dn ( ) kHz M x BB eSNOB RC-SEBOPT = 1 ms, T = 1.345 ms, T = 1.024 ms p 1 2 Figure 5: Experimental selectivity profiles of RC-BEBOP (right) and eSNOB (left), pulse length T p = 1 ms, using the phase-cycled crusher sequence of Fig. 4a applied to a strongly relaxing sample with T = 1.345 ms and T = 1.024 ms. Furtherexperimental details are in the text. Results are shown for three values of the RF calibration relative to the ideal value B for each pulse, showing insensitivity of RC-BEBOP to RF miscalibration of ± M y preserves 83% of the initial magnetization on-resonance for the case B /B = 1 compared to the theoretical value of 89% for M x alone shown in Fig. 3. .91.01.1-15 0 15 -15 0 15 Dn ( ) kHz
Dn ( ) kHz M x BB eSNOB RC-SEBOPT = 1ms, T = 0.708 ms, T = 0.527 ms p 1 2 Figure 6: Same as Fig. 5, but applied to a faster-relaxing sample with T = 708 µ s, T = 527 µ s. Although RC-SEBOP wasoptimized for T = T = T p = 1 ms, it provides excellent resistance to relaxation for much shorter values, preserving 75% ofthe initial magnetization on-resonance for the case B /B = 1.= 1.