Radiofrequency pulse designs for three-dimensional MRI providing uniform tipping in inhomogeneous B1...

Radiofrequency Pulse Designs for Three-DimensionalMRI Providing Uniform Tipping in Inhomogeneous B1

Fields

Hui Liu1,2 and Gerald B. Matson1,3*

Although high-field MRI offers increased signal-to-noise, thenonuniform tipping produced by conventional radiofrequency(RF) pulses leads to spatially dependent contrast and subopti-mal signal-to-noise, thus complicating the interpretation ofthe MR images. For structural imaging, three-dimensionalsequences that do not make use of frequency-selective RFpulses have become popular. Therefore, the aim of thisresearch was to develop non-slice-selective (NSS) RF pulseswith immunity to both amplitude of (excitation) RF field (B1)inhomogeneity and resonance offset. To accomplish this, anoptimization routine based on optimal control theory was usedto design new NSS pulses with desired ranges of immunity toB1 inhomogeneity and resonance offset. The design allows thephase of transverse magnetization produced by the pulses tovary. Although the emphasis is on shallow tip designs, newdesigns for 30�, 60�, 90�, and 180� NSS RF pulses are also pro-vided. These larger tip angle pulses are compared withrecently published NSS pulses. Evidence is presented that thepulses presented in this article have equivalent performancebut are shorter than the recently published pulses. Althoughthe NSS pulses generate higher specific absorption rates andlarger magnetization transfer effects than the rectangularpulses they replace, they nevertheless show promise for three-dimensional MRI experiments at high field. Magn Reson Med66:1254–1266, 2011. VC 2011 Wiley Periodicals, Inc.

Key words: RF pulse; high field; MRI; optimization

Although high field MRI on the human head offers anumber of advantages including increased signal-to-noiseand higher contrast for many applications, there are alsosignificant difficulties. Chief among these are the non-uniform excitation (i.e., spatial variation of the flipangle), which occurs when conventional radiofrequency(RF) pulses are used. The nonuniform tipping occursdue to the inhomogeneous amplitude of (excitation) RFfields (B1) intrinsic in tissue at high magnetic fields(1,2). The nonuniform tipping leads to undesirable, spa-tially dependent image contrast and suboptimal signal-to-noise, thus complicating the interpretation of MR

images and resulting in errors of quantification. Theneed to keep specific absorption rates (SAR) low furthercomplicates RF pulse designs for high field.

Conventional approaches to overcome the nonuniformtipping associated with inhomogeneous B1 fields includecomposite RF pulse designs (3) and composite adiabaticpulse designs such as the BIR-4 pulse (4). Most compos-ite pulse designs are aimed at large tip angles (90� or180�). Furthermore, in general, composite pulse designshave been designed with ex vivo samples in mind andhave large total rotation values (the total rotation an on-resonance spin would experience), thus generating largeSAR. Although the BIR-4 pulse can be used for shallowtips (5), it is designed for large ranges of B1 inhomogene-ity and resonance offsets and also has a large total rota-tion value and generates considerable SAR. The hyper-bolic secant and related pulses (4) can be used forinversion and are less SAR intensive, but still have totalrotation values in the order of 800� or more. Morerecently, the teams of Skinner and Glaser (Skinner et al.(6–8)) have collaborated to produce a large variety ofpulses for high resolution NMR spectroscopy, using opti-mal control optimization methods. However, their pulsesare aimed at much larger resonance offsets than neededfor MRI and also generate high SAR. In general, the rangeof resonance offsets covered by their pulses is larger thanthe B1 fields (in Hz) used to generate their pulses.

Other methods for combating the B1 inhomogeneity athigh field worthy of note include RF shimming (9,10)and multidimensional RF pulses. The RF shimmingrequires multiple RF sources applied to a coil with mul-tiple RF channels and still leaves some B1 inhomogeneitywhen applied over the human head at high field (9). Toachieve reasonable duration, multidimensional RF pulsesalso require multiple RF sources applied to a coil withmultiple RF channels (11,12). The methodology used forthese multidimensional pulses has also been referred toas dynamic shimming (13). In addition, the B1 inhomoge-neity may be mitigated by sophisticated RF coil design(14), although the feasibility of this approach is yet to bedemonstrated experimentally. In a new and interestingapproach, Collins et al. (15) have demonstrated a 90�

composite pulse in which the different pulse segmentsare applied to a multiple channel transmit coil with dif-ferent amplitudes and phases on the different channels.

In this work, we concentrate on shallow tip, non-slice-selective (NSS) RF pulses that are suitable for three-dimensional (3D) MRI sequences such as 3D magnetiza-tion prepared rapid acquisition gradient echo (MPRAGE). However, we also show that our tools for RFpulse design can be used for larger tip angles, including30�, 60�, 90�, and 180� NSS pulses. These pulses are

1Department of Veterans Affairs Medical Center, Center for Imaging ofNeurodegenerative Diseases (114M), University of California, San Francisco,California, USA.2Northern California Institute for Research and Education, San Francisco,California, USA.3Department of Pharmaceutical Chemistry, University of California, SanFrancisco, California, USA.

Grant sponsor: NIH; Grant numbers: 5R01EB000766, 1P41RR023953.

*Correspondence to: Gerald B. Matson, Ph.D., Veterans Affairs MedicalCenter, Center for Imaging of Neurodegenerative Diseases (114M), 4150Clement Street, San Francisco, CA 94121. E-mail: [email protected]

Received 20 August 2010; revised 8 February 2011; accepted 14 February2011.

DOI 10.1002/mrm.22913Published online 26 April 2011 in Wiley Online Library (wileyonlinelibrary.com).

Magnetic Resonance in Medicine 66:1254–1266 (2011)

VC 2011 Wiley Periodicals, Inc. 1254

designed over a range of B1 inhomogeneity and reso-nance offsets suitable for human head imaging at highfield strengths (from 4.0 to 7.0 T). Advantages of theapproach include the fact that prior B1 field measure-ments on individuals are not required, and the pulsescan be used with conventional head coils. Finally, wenote that our NSS pulses refer to NSS pulses for MRIpurposes (noncoalesced magnetization) that perform wellover resonance offsets ranging up to a few 100 Hz.

Very recently, efficient NSS pulses of different tipangles suitable for human head MRI at high field havebeen presented by Boulant et al. (16,17) and by Mooreet al. (18). Similar to our NSS pulses, these pulses areoptimized for immunity to B1 inhomogeneity and to res-onance offset, and also produce noncoalesced magnetiza-tion. We provide some comparisons of our pulses tothese recent developments later in this work. Frequency-selective RF pulses have traditionally been comparedthrough their performance as judged by their frequencyselection profile and amount of SAR generated. How-ever, the NSS pulses can in general be applied at differ-ent power levels and different lengths to affect the sameperformance. For the sake of simplicity, to compare ourNSS pulses with recently published NSS pulses, we gen-erate the shortest pulse with the same maximum B1 field(B1max) that generates equivalent performance and com-pare pulse lengths. In addition, for the Moore et al. (18)pulses, we are able to compare the relative SAR.

MATERIALS AND METHODS

Shallow Tip NSS RF Pulse Design

We previously developed a short composite pulsesequence for generation of shallow tips using the optimi-zation package Global Optimization 6.0 by Loehle Enter-prises in Mathematica (Wolfram Research) (19). The opti-mization was over a cascade of four on-resonancesubpulses, where the tips and phases of the subpulseswere allowed to vary. The composite pulse was opti-mized for a shallow tip pulse with immunity to B1 inho-mogeneity and to resonance offset. The optimizationreturned four 90� (nominal tip) subpulses with variousphases. The operation of this cascade is most easily under-stood from trace diagrams (traces of magnetization trajec-tories over the unit sphere) of the null tip version of this

pulse. Figure 1a shows the action of the first subpulse ofthe cascade on magnetizations experiencing different B1

field strengths (color) and magnetizations at different reso-nance offsets (gray). The red arrow depicts the B1 field.Following the first pulse, magnetizations experiencingdifferent B1 field strengths are spread out in the yz-plane,whereas magnetizations at different resonance offsets arespread out in the xy-plane. Although the second pulsecoalesces the magnetizations at different B1 strengths, themagnetizations at different resonance offsets continue tospread out. The action of the third and fourth pulses coa-lesce magnetizations at both different B1 strengths and dif-ferent resonance offsets back along the z-axis. Thus, thenull pulse could be used for inducing magnetization trans-fer effects. The null pulse is converted to a shallow tippulse by moving the B1 field of the latter pulses slightlyaway from the x-axis. However, the pulse is rather ineffi-cient, requiring a total rotation of 360� to achieve a net tipof 10� or 20�. This pulse is termed the parent pulse.

To obtain a more efficient pulse, we developed anoptimization routine based on optimal control theory asdescribed by Skinner et al. (6–8,20) and incorporated itinto MatPulse (21,22). The parent pulse was reformu-lated into 400 subpulses to form the initiating pulse(initial pulse submitted to the optimization routine) forthe optimal control routine in MatPulse. Inputs for theoptimization included the desired tip angle, maximumB1 strength, and additional parameters for the optimiza-tion including the desired range of immunity to B1 inho-mogeneity and to resonance offset. Optimization wasdone with a B1max of 7.0 mT (�300 Hz) over a B1 fieldvariation of 620% and over a resonance offset range of6100 Hz, to make it suitable for use on the human headat 4.0 T. The overall length of the pulse was varied byvarying the dwell time (duration of the subpulses) togenerate a pulse that produced a 10� tip within a fewpercent over this range. Additional details of the optimalcontrol routine in MatPulse are available in Ref. 22. Thepulse shapes and contour plots of pulse performancewere generated with MatPulse.

Interrelationships of Shallow Tip Pulse Parameters

To get some sense of how pulse length, pulse B1max, pulseimmunity to B1 inhomogeneity, and pulse immunity toresonance offset were interrelated in our shallow tip

FIG. 1. Depiction of the action of the parent pulse (null tip). a: Following the first nominal tip of 90�, the spins experiencing different B1

strengths (in color) are spread out in the yz plane, whereas spins at different resonance offsets (gray) are spread out in the xy plane.b: Following the second nominal tip of 90�, the spins at different B1 strengths are coalesced, but the spins at different resonance offsets

continue to spread out. c: Following the third nominal tip of 90�, all spins are spread out. d: Following the fourth nominal tip of 90�, allspins are coalesced (approximately). The red arrow depicts the direction of the B1 field.

RF Pulse Providing Uniform Tipping in B1 Fields 1255

design, a number of related optimizations were done toobtain the shortest pulse length that produced the desiredperformance: (i) The initial 10� pulse was optimized tonew tips of 10�, 15�, 20�, and 30�, with the B1max at15 mT, to obtain the shortest length pulse whose tip waswithin 63% of the desired tip, over a B1 range of 620%,and over a resonance offset range of 6100 Hz. (ii) The 20�

pulse obtained in the above step was optimized at differ-ent values of B1max (7, 12, 15, and 25 mT), again for thesame level of performance (tip within 63% of the desiredtip, over a B1 range of 620%, and over a resonance offsetrange of 6100 Hz). (iii) The 20� pulse with B1max at 15 mTwas optimized for different ranges of immunity to B1

inhomogeneity (610%, 620%, 625%, and 630%) againwith the same level of performance (tip within 63% ofthe desired tip and a resonance offset range of 6100 Hz.(iv) The 20� pulse with B1max at 15 mT was optimized overdifferent ranges of resonance offsets (6100, 6200, and6250 Hz) again with the same level of performance (tipwithin 63% of the desired tip, over a B1 range of 620%).

Experimental Tests of the Initial 10� Pulse

All experiments were performed on a 4.0-T Bruker MedSpecinstrument equipped with a 7-kW RF amplifier, providing amaximum B1 field of� 25 mT in a transmit/receive head coilloaded with a normal human head. The head coils usedwere birdcage excite and eight receive channel coil and aTransverse electromagnetic (TEM) coil (23). To verify thetheoretical simulations of the NSS 10� pulse obtained withMatPulse, we performed measurements using the birdcageexcite and eight receive channel coil on a small silicon oilphantom (to avoid B1 inhomogeneities arising due to the sam-ple). The sequence was a short spin echo (SE) sequence,where the SE pulse (180�) was the same in all experiments,and the maximum of the echo was measured. We measuredthe maximum signal intensity as the voltage of the initialpulse was varied using first a rectangular 10� pulse as the ini-tial pulse and second the NSS 10� pulse as the initial pulse.We also varied the frequency of the NSS 10� pulse to verifythe immunity to resonance offset. Three readings of signalamplitude were taken and averaged for each measurement.

To measure the range of B1 inhomogeneity over thehuman brain, we used a Siemens service sequence(RF_MAP) to measure the tip angle variation on a humanvolunteer with the TEM coil. The TEM coil was used forthe human experiments because of the more uniform sen-sitivity profile of this coil as compared to the eight receivechannel coil, and also because the receive sensitivity pro-file was expected to mimic the transmit sensitivity (1) forthis coil. The RF_MAP sequence was composed of an ini-tial slice-selective pulse of � 90�, followed by a 2x alpharectangular pulse and then an alpha rectangular pulse, togenerate a SE followed by a stimulated echo. The tipangle alpha was calculated from:

cosðalphaÞ ¼ stimulated echo=spin echo;

where stimulated echo and spin echo represent the max-imum value of the stimulated echo and SE signals. Theexperiment was run with a nominal alpha of 60�, with apulse repetition time of 2 s.

We then compared the use of a rectangular 10� pulsewith the NSS 10� pulse in a 3D FLASH experiment on ahuman brain. The 3D FLASH experiment used a rela-tively long delay between RF pulses (100 ms) to notexceed SAR constraints and also to minimize partial satu-ration and magnetization transfer effects. Subjects used inthe RF_MAP and FLASH experiments provided writteninformed consent before participating in the studies,which were approved by the VA Committee on HumanResearch. The parameters for the FLASH experimentwere: pulse repetition time/echo time ¼ 5000 ms/1.45 ms,tip angle ¼ 10�, interpulse delay ¼ 100 ms, 44 phase enc-odes (producing 44 slices), and 300 mm field of view.

Comparison with Other Pulse Designs

Moore et al. (18) very recently showed 30�, 60�, 90�, and180� NSS pulses suitable for MRI at 7 T. Although theyentitled their article in part ‘‘Composite RF pulses.’’ theirpulses were composed of 64 subpulses of varying ampli-tudes and phases, and thus could be considered asshaped pulses. The simulated performances of 4.096 mslong pulses with B1max values of 15 mT were character-ized by standard deviations of tip angles over a grid witha relative B1 range from 0.35 to 1.30 in increments of0.05, and a resonance offset range of 6250 Hz in incre-ments of 25 Hz. Moore et al. also compared the powerrequired by these pulses with a rectangular pulse of am-plitude 15 mT that generated the same nominal tip astheir pulses (18) (Table 1). For our comparison pulses, wegenerated the shortest pulses that provided essentially thesame standard deviations over the same grid as used byMoore et al. (18). This involved altering the dwell timeand to some extent the optimization parameters to achievethe desired comparisons. Except for the 180� comparisonpulse, we used a lower tip angle pulse as the initiatingpulse for the higher tip angle comparison pulse. For the180� pulse, we used a hyperbolic secant shape as the ini-tiating pulse. However, as the optimization was only overa relatively small B1 range (630%), the resulting pulseshape was modified from the initiating pulse shape andthe pulse operated well out of the adiabatic range. Wealso compared the SAR of our pulses with the 4.096 mspulses of Moore et al. Finally, we provide plots of the realand imaginary components of these pulses, along withcontour plots of simulated performance of our pulses overthe grid used for the standard deviation calculations.

Boulant et al. (17) have also presented NSS pulses suit-able for MRI at 7 T, which they label strongly modulatedpulses (SMP) with results for 12�, 30�, 60�, and 90�

pulses. A comparison with the 12� SMP pulse was notattempted because of the rather sparse information pro-vided on this pulse. Our comparison pulses for the 30�,60�, and 90� SMP pulses were necessarily somewhat ap-proximate, as simulated performances for these SMPpulses were not provided. Although standard deviationsof the sine of the tip (i.e., the relative signal level) wereshown, these were calculated over the entire head, butthe distributions of B1 and B0 field inhomogeneities,along with the relative amount of tissue experiencingthese inhomogeneities, were not provided in a quantita-tive format. However, the distributions were shown in

1256 Liu and Matson

graphical form for a sagittal slice (17) (Fig. 1) along witha graphical display of the signal level over the same slicefor the 30�, 60�, and 90� SMP pulses (17) (Figs. 2–4). Inaddition, the pulse lengths and B1max values were pro-vided (17) (Table 1). The distribution of resonance offsetsover the sagittal slice appeared to go from approximately�100 to 150 Hz, whereas the range over B1 went fromapproximately 7 to 20 mT. As with the Moore et al. com-parisons, we attempted to find the shortest pulses withthe same B1max that we judged matched or slightlyexceeded the performance of the SMP 30�, 60�, and 90�

pulses. We then compared our pulse lengths with theirs.However, SAR comparisons could not be done.

RESULTS


The real and imaginary components of the 10� NSS pulse,optimized for a B1max of 7 mT and a length of 1.47 ms, areshown in Fig. 2a, whereas the theoretical performance is

shown in the contour plot of Fig. 2b. The vertical dashedred lines at 5.6 and 8.4 mT represent the 620% B1

strengths. The contour plot shows the pulse generates thenominal tip (10�) within 64% over a B1 range of 620%and over a resonance offset range of 6100 Hz. This rangewas suitable for human head MRI at 4.0 T.

In general, the MatPulse optimal control routinetended to optimize the NSS pulses at the maximumallowed B1 field (B1max) for the majority of the pulse du-ration if the optimization were allowed to continue longenough. Thus, the magnitude of the pulse is a constant 7mT. Figure 3 shows the frequency sweep of the pulse.Figure 4 shows the trajectory of spins during the pulseover the unit sphere, as viewed from the z-axis. Spinsexperiencing different B1 strengths are displayed incolor, whereas spins at the nominal B1 strength, but atdifferent resonance offsets, are displayed in gray.

FIG. 2. a: The real and imaginary components of the 10� NSS pulse with a length of 1.47 ms. b: Contour plot showing the performanceof the NSS 10� pulse. The red, vertical dashed lines represent the 620% demarcation of the B1 field.

FIG. 3. The frequency sweep of the NSS 10� pulse. The frequencyspikes represent phase jumps.

FIG. 4. Trace diagram of the NSS 10� pulse showing the paths ofvarious magnetizations over the unit sphere. Spins experiencingdifferent B1 fields are shown in color, whereas spins at different

resonance offsets are depicted in gray. The red arrow depicts thedirection of the B1 field at the end of the pulse.



The interrelationships of tip parameters for shallow tips areshown in Fig. 5, which shows pulse length vs. tip angle,B1max, immunity to B1, and immunity to resonance offset.

Experimental Tests of the Initial 10� Pulse

The results of experiments conducted with a TEM headcoil on a small silicon oil phantom to compare rectangu-lar pulse results with our NSS 10� pulse are shown inTables 1 and 2. Table 1 displays the results of varyingthe voltage of a rectangular 10� pulse by 610% and620%. Table 2 shows experimental results for varyingboth the voltage and the resonance offset (frequency off-set from the Larmor frequency) for the NSS 10� pulse.The experimental performance of the 10� NSS pulse is tobe compared with the theoretical performance shown bythe contour plot of Fig. 2b.

RF_MAP Results

The results of running the RF_MAP sequence on a trans-verse slice of a human brain are shown in Fig. 6, which

displays the intensity of the slice in terms of the tipangle. The sequence was taken with our birdcage excite–eight channel receive head coil. The figure was gener-ated by routines developed in Matlab (The MathWorks).The figure shows that the B1 inhomogeneity over a cen-tral slice in the head varied by just under 620%.

In Vivo 3D FLASH Results

The results of one of the central slices from the 3DFLASH experiment, along with the intensity profilealong the dotted line in the image, are shown in Fig. 7.Figure 7a,b shows the results from the use of the 10�

FIG. 5. a: The increase in pulse length required as the tip angle is increased to maintain constant pulse performance. b: The increase in

pulse length required to maintain constant pulse performance as the B1max of the pulse is reduced. c: The increase in pulse lengthrequired to maintain constant pulse performance as the immunity to B1 inhomogeneity is increased. The plot shows that little reductionin pulse length is possible as the immunity to B1 is reduced below 20%. d: The increase in pulse length required to maintain constant

pulse performance as the immunity to off resonance is increased. The plot shows that the immunity cannot be increased much beyond6250 Hz for a B1max of 7 mT (� 300 Hz). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Table 1Signal Change with Rectangular Pulse

Transmitter voltage (changefrom nominal value; %)

Signal change withrectangular pulse (%)

�20 �20.1�10 �11.9

60 0.0þ10 8.0þ20 18.7

1258 Liu and Matson

rectangular pulse, whereas Fig. 7c,d shows the resultsfrom the same slice using the NSS 10� pulse.


NSS Pulses to Compare with Moore et al.

A comparison of our NSS 30�, 60�, 90�, and 180� pulsesthat obtained equivalent standard deviations as the 4.096ms Moore et al. (18) pulses is provided in Table 3. Allpulses had a B1max of 15 mT. The table includes the stand-ard deviations of the tip angle (normalized to the targettip) over the grid specified by Moore et al. The table alsoincludes the mean tip (normalized to the target tip) alongwith the pulse length, and relative SAR compared with arectangular pulse with the same B1max and producing thesame target tip. For the 180� tip, net z-magnetization ofabsolute magnitude <1 was assumed to have been tipped<180�, as this is what we presumed was done in theresults of Moore et al. The results for the pulses of Mooreet al. were taken from their Table 1 (18).

The real and imaginary components of our comparisonpulses with Moore et al., along with contour plots of thepulse performance over the grid, are shown in Fig. 8a–h.However, the contour plots are given in terms of relativemagnetization, not tip angle, as we consider the signallevel to be the more relevant parameter.

NSS Pulses to Compare with SMP Pulses

A comparison of our NSS 30�, 60�, and 90� pulses thatwe judged to be comparable to or slightly better than the

SMP pulses (17) generating the same tip angles is pro-vided in Table 4. The NSS pulses were optimized withthe same B1max as the SMP counterparts. The tableincludes the value of the B1max, the estimated B1 rangeas estimated from the Fig. 1 of Boulant et al. (17), thenominal signal, and the estimated range of signal inten-sities as judged from the Figs. 2–4 of Boulant et al. (17).The other results for the Boulant et al. SMP pulses weretaken from their Table 1 (17).

The real and imaginary components of our SMP com-parison pulses, along with contour plots of the pulseperformance over the grid, are shown in Figs. 9a–f. Inthis case, the contour plots, which are given in terms ofrelative magnetization, correspond to the Figs. 2–4 ofBoulant et al. (17).

DISCUSSION

Overview

Optimal control theory, with some of the innovationsdeveloped by Skinner et al., was used for the optimiza-tions as this method enabled efficient optimization witha relatively large number of subpulses. In particular, ouroptimization included the vector cross product methodsuggested by Skinner et al. (6–8) for correcting the RFpulse and also their method for limiting the B1 maxi-mum (B1max). However, the optimization only returnedresults from local minima and did not guarantee globaloptimization. Our previous experience suggested that, atleast in some cases, the initiating pulse was important toobtain the best result (22). On the other hand, the experi-ence of Skinner et al. (6–8) was that, for the most part,random initiating pulses in general produced similarresults.

There were several important differences in the opti-mizations used for the pulses in this article, and theoptimizations performed by Skinner et al. First, Skinneret al. optimized pulses for large bandwidths, so thebandwidth exceeded the strength of the B1 field (in Hz).Second, as these pulses were developed for high resolu-tion NMR, the optimization included criteria for coales-cence of the magnetizations at the end of the pulse.

FIG. 6. This figure shows the tip angle over a transverse slice in ahuman brain at 4.0 T. The nominal tip was 60�.

Table 2Signal Change with NSS Pulse

Transmittervoltage change

from nominalvalue (%)

Offset

frequency(Hz)

Normalized

signalchange (%)

Normalizedchange

over �50, 0,and 50 Hz (%)

�20 �100 Hz �3.8�20 �50 Hz �5.2�20 0 Hz �4.5

�20 50 Hz �4.5�20 100 Hz �1.7

�10 �100 Hz 0.9�10 �50 Hz �1.2�10 0 Hz 0.8 �0.10

�10 50 Hz �0.1�10 100 Hz 3.2

0 �100 Hz 3.40 �50 Hz 0.90 0 Hz 1.4 1.30

0 50 Hz 1.50 100 Hz 4.0

þ10 �100 Hz 3.0

þ10 �50 Hz �0.5þ10 0 Hz 0.3 0.20

þ10 50 Hz 0.7þ10 100 Hz 5.4þ20 �100 Hz �0.8

þ20 �50 Hz �4.1þ20 0 Hz �3.4

þ20 50 Hz �2.4þ20 100 Hz 2.3


Finally, the high resolution NMR pulses were not opti-mized for shallow tip angles. In contrast, the pulses inthis article were optimized for relatively small ranges ofresonance offsets, so the bandwidth was always less thanthe maximum B1 strength (B1max). In addition, our opti-mizations did not require the magnetizations to be coa-lesced at the end of the pulse. As the optimal controlmethod compares the forward path (the path of the ini-tial magnetization toward the desired result) with the

reverse path (path of the desired result with reversed fre-quency isochromats toward the initial magnetization bythe time-reversed pulse) to correct the RF pulse, this wasaccomplished by applying the phase of the last step ofthe forward result to the desired result before computingthe reverse path. Finally, although our initial emphasiswas on shallow tip pulses, we also investigated the abil-ity of our method to produce larger tip angle pulses.However, we had previously shown that our version of

FIG. 7. a: Image of a transverse slice through a human brain from a FLASH experiment executed with a 10� rectangular pulse. b: Profile along

the dotted line in the image. c: Image of a transverse slice through a human brain from a FLASH experiment executed with the 10� NSS pulse.d: Profile along the dotted line in the image. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Table 3

Comparison with Pulses of Moore et al.

Tip Pulse Performance Length (ms) P/P blocka

30 degrees Moore et al. 0.98 6 0.07 4.10 21.1This article 1.00 6 0.07 2.71 19.8



180 degrees Moore et al. 0.92 6 0.10 4.10 4.3

This article. 0.91 6 0.09 3.20 4.1aPower deposition compared with a rectangular pulse of the same B1max and producing the same tip.

1260 Liu and Matson

FIG. 8. a: Real and imaginary components of the NSS pulse to be compared with the 30� pulse of Moore et al. b: Contour plot showing

the performance of the NSS 30� comparison pulse. c: Real and imaginary components of the NSS pulse to be compared with the 60�

pulse of Moore et al. d: Contour plot showing the performance of the NSS 60� comparison pulse. e: Real and imaginary components ofthe NSS pulse to be compared with the 90� pulse of Moore et al. f: Contour plot showing the performance of the NSS 90� comparison

pulse. g: Real and imaginary components of the NSS pulse to be compared with the 180� pulse of Moore et al. h: Contour plot showingthe performance of the NSS 180� comparison pulse.


optimal control theory could generate an efficient broad-band inversion pulse (22).

The 90� pulses of Skinner et al. were invariably charac-terized by a large number of rapid phase variations. Incontrast, the pulses in this article show much fewer phasevariations, although they often displayed some regions ofrapid phase variation. It appears that the larger bandwidthof the pulses of Skinner et al. required numerous phasevariations, whereas our different optimization parametersdid not require such perfusion of phase transitions. Infact, using one of the 90� pulses of Skinner et al. as theinitiating pulse for the 90� comparison pulse of Mooreet al. did not produce superior results, but instead pro-duced a pulse with far fewer phase variations, with somesimilarities to the 90� comparison pulse shown in Fig. 8e.

Although Skinner et al. (6–8) and Moore et al. (18) usedrandom initiating pulses, we were unable to obtain a shal-low tip pulse with performance similar to the 10� pulseusing random initiating pulses. Except for the 180� inver-sion pulse, the larger tip angle pulses were designed usingthe previously designed, lower tip angle pulse as the ini-tiating pulse. For the inversion pulse, a version of a hyper-bolic secant was used as the initiating pulse, as this hadpreviously produced good results (22). Random initiatingpulses were not explored for these larger tip angle pulses.

With the exception of the actual optimization routine,our overall RF design approach was quite similar to that ofMoore et al. (18). However, we believe that an important dif-ference between our pulses and those of Moore et al. wasthe number and length of the subpulses. We typically used400 subpulses for our shallow tip pulses, which yieldedsubpulse durations of less than 10 ms. This facilitated rapidfrequency sweeps and phase jumps, which may be one ofthe reasons for the differences in pulse performances. Incontrast, Moore et al. used 64 subpulses, so that their 4.096ms long pulses used subpulse durations of 64 ms. Thislength of subpulse may have inhibited rapid frequencysweeps. On the other hand, the difference in results maysimply be due to the larger number of degrees of freedomprovided by the larger number of subpulses. In contrast, theBoulant et al. SMP pulse design allowed each subpulse (orsegment) to have its own initial phase, frequency, andlength (17). It appears that the pulses of Boulant et al. weredesigned with several segments. Although the number ofsegments for a previous 60� pulse designed for operation at3.0 T was three (16), and the number of segments used forthe pulses designed for 7.0 T was unspecified (17). How-ever, it does appear that, with only a few segments, the Bou-lant pulses were also limited in terms of frequency sweepsand phase jumps. However, to be fair, the Boulant et al.design had the requirement that the pulses be designed and

optimized at run time, so the number of parameters had tobe rather limited to achieve short optimization times.

Perhaps the most surprising result to us was the highdegree of symmetry in the majority of the optimized pulses,and in the case of the comparison pulses to the SMP pulses,the general preservation of the 10� pulse shape to higher tipangle pulses (� 90�). However, the pulse shapes did morphinto more complicated and nonsymmetrical shapes, partic-ularly at the larger tip angles, and as the pulse was opti-mized to perform over larger ranges of B1 inhomogeneityand resonance offset. However, this trend did not alwayshold; for example, the comparison pulse for the Mooreet al. 90� pulse was decidedly unsymmetrical, whereas thecomparison pulse for the SMP 90� pulse retained a highdegree of symmetry. In addition, the lower tip angles SMPcomparison pulses deviated from symmetry. Even in thenonsymmetrical pulse shapes, the deviation from symme-try was not great, and one is tempted to describe them asdistorted symmetrical shapes.


Figure 2a shows the optimized 10� NSS pulse (obtainedusing the parent pulse (Fig. 1) as the initiating pulse)with a B1max of 7 mT (�300 Hz), whereas the contourplot of Fig. 2b shows that the tip over the range of opti-mization is within approximately 64% of 10�, where thecontours are displayed in terms of relative signalstrength. Thus, the variation in tip is reduced over thatproduced by a rectangular pulse over this same range byapproximately a factor of 5.

Figure 3 shows the pulse in terms of a frequency sweep.However, the initial sweep is to a frequency of over 1 kHzin less than half a millisecond. As the on-resonance B1

strength is of the order of 300 Hz (7 mT), the sweep is welloutside of the adiabatic limit, and the angle of the effectivefield with the magnetization being tipped does not remainconstant. The sweep has some similarities with that of theBIR-4 pulse (5) except that the sign of the frequency doesnot reverse at the position of the phase jumps.

Figure 4, which shows a trace diagram of spins experi-encing different B1 field strengths (in color) and differentresonance offsets (in gray), indicates the 400 subpulseoptimized pulse remains remarkably similar to theparent pulse, in that the magnetizations experiencing dif-ferent B1 strengths are coalesced halfway through thepulse, and again at the end of the pulse, while magnet-izations experiencing different resonance offsets are notcoalesced until the end of the pulse. Finally, we notethat the coalescence is not complete, as the optimizationdid not specify the spins to be coalesced.

Table 4Comparison with pulses of Boulant et al.

Pulse B1max B1 range (mT) Nominal signal Signal range (nominal %) Length (ms)

SMP 30� 7.0 3.40–10.60 0.500 80–140 3.30

This article 7.0 3.40–10.60 0.500 80–140 2.56SMP 60� 7.4 3.55–11.25 0.866 86–115 4.78

This article 7.4 3.55–11.25 0.866 86–115 2.96SMP 90� 6.6 3.17–10.03 1.00 90–100 6.70This article 6.6 3.17–10.03 1.00 90–100 3.20

1262 Liu and Matson


Tip Angle vs. Pulse Length

Figure 5a shows the lengthening required to retain the sameperformance as the tip angle was increased. The increaseneeded with increasing tip angle is relatively small. Note thatthis relationship was found for the shallow tip angle regime,but is not expected to hold as the tip angle continues to be

increased. As discussed below, the pulse length actuallydecreased as the tip angle approached 90�.

1/B1max vs. Pulse Length

The increase required in pulse length of the NSS 20�

pulse with decreasing B1max is shown in Fig. 5b. Itappeared that the primary reason for the increased length

FIG. 9. a: Real and imaginary components of the NSS pulse to be compared with the 30� pulse of Boulant et al. b: Contour plot show-ing the performance of the NSS 30� comparison pulse. c: Real and imaginary components of the NSS pulse to be compared with the

60� pulse of Boulant et al. d: Contour plot showing the performance of the NSS 60� comparison pulse. e: Real and imaginary compo-nents of the NSS pulse to be compared with the 90� pulse of Boulant et al. f: Contour plot showing the performance of the NSS 90�

comparison pulse.


was the reduction in immunity to resonance offset as theB1max value was reduced.

B1 Immunity vs. Pulse Length

Figure 5c shows the increase required in pulse length toextend the range of B1. The first point (for B1 immunityof 610%), which does not follow the trend of the otherpoints, indicates that there is little reduction to begained in pulse length for reducing the B1 immunitybelow 620%.

Immunity to Resonance Offset vs. Pulse Length

Figure 5d shows the increase required in pulse length toextend the range of immunity to resonance offset for theNSS 20� pulse. The flattening of the curve near an offsetof 250 Hz reflects the fact that this style pulse did notperform well for offsets above 6250 Hz.

Experimental Results

Verification of 10� Tip Angle

Table 1 shows the performance of a rectangular pulseapplied to a silicon oil phantom. The table demonstratesthe expected linear relationship between the voltage andsignal intensity for the shallow tip (10� nominal tip) rec-tangular pulse, with the signal varying by approximately610% as the voltage is changed by 610% and varyingby approximately 620% as the voltage is changed by620%. The table results are averages over three measure-ments. However, the results indicate that the measure-ments are only reliable to 1 or 2%.

Table 2 shows the performance of the NSS 10� pulseas the voltage is changed by 610% and 620% and asthe frequency offset of the pulse is varied by 650 and6100 Hz. As the contour plot showing the expected per-formance of the pulse (Fig. 2b) indicated that the tipangle at 610% of nominal voltage and resonance offsetsof �50, 0, and þ50 Hz were all very close to 10�, the sig-nals from these six points were averaged and used as thesignal value representing a tip of 10�. Although the con-tour plot indicates that the signal level is expected tovary by no more than 64%, the table shows variations ofup to 5.4%. However, this tolerance is within the reli-ability of the measurements. As the contour plot alsoshows very little variation is expected over resonanceoffset values of 650 Hz, values for �50, 0, and þ50 Hzwere also averaged and are displayed in the last column.The averages show that the tip expected over this rangeof resonance offsets and at 7 mT is 1.3% above 10�, andat 610% values is within 60.2% of 10�, which is invery good agreement with the contour plot. The measure-ments at 6100 Hz are all at more rapidly varying tipareas, where more variance is to be expected. Overall,Table 2 shows that our experimental results were in rea-sonable agreement with the simulated results of Fig. 2b.

In Vivo 3D FLASH Results

Figure 7 shows a slice from the in vivo 3D FLASHexperiment obtained with the TEM coil and executedwith a 10� rectangular pulse (Fig. 7a) and with the NSS

10� pulse (Fig. 7c). Because of the shallow tip angle andrelatively slow repetition rate, the gray matter is seen inthese figures with greater intensity than the white matterdue to its greater water content. As expected, the experi-ment with the rectangular pulse showed higher signal in-tensity in the center of the head. This is in part due tothe increased sensitivity of the TEM coil and tip angle atthe center of the head. More importantly, the profile inFig. 7b shows that the intensity distinction between thegray and white matter is somewhat washed out. Webelieve that some of this is because of the increased tipat the center of the head. By contrast, the NSS 10� pulseproduced a more uniform signal intensity over the slice,with the pronounced differences in intensity betweengray and white matter retained even at the center of thebrain (Fig. 7c,d).

In summary, Fig. 7 demonstrates the degradation incontrast that can occur due to nonuniform excitationover the brain. This can result in difficulties in segmen-tation, and uncertainties in determining gray matter–white matter boundaries. This difficulty has also beenpointed out in the articles of Moore et al. and Boulantet al. (17,18).


NSS Pulses to Compare with Moore et al.

As the Moore et al. (18) article provided the grid overwhich the standard deviations for the 4.095 ms pulseswere calculated (Table 1), we could calculate the standarddeviation over the identical grid for our pulses to obtainan accurate comparison. By trial and error, we adjustedour pulse lengths to obtain the shortest pulse that com-pared favorably with their 4.096 ms length pulse. We alsofollowed their lead and compared the SAR generated byour pulses to a rectangular pulse with the same B1max (15mT) that produced the same tip. Although Table 3 showsour pulses were substantially shorter than the pulses ofMoore et al., and thus more efficient in terms of pulselength, the improvement was not dramatic, and wasalways considerably less than a factor of two. Theimprovement in SAR was even less significant.

As did our optimization procedure, the Moore et al.procedure produced pulses that operated at or near theB1max value for most of the pulse duration. However,their 4.096 ms pulses did not display the symmetric (ordistorted symmetric) features of our pulses, althoughthey did indicate rather smooth variations in phase, sug-gesting that, as remarked by Moore et al., that they couldbe considered as frequency swept pulses. Both proce-dures did produce significant phase variations of mag-netizations with different B1 strengths and resonance off-sets. However, our phase variations were fairly smoothand were well within 180� over the range over which theoptimization was performed. Finally, we did not see anythreshold effect, such as evidenced by adiabatic pulseswhere a threshold in B1 strength is needed to achieveadiabaticity for a given frequency sweep. However, thisis consistent with our analysis that our comparisonpulses were not adiabatic in character.

Finally, we note that the Moore et al. (18) standarddeviations were calculated over the tip angles, so that, in

1264 Liu and Matson

terms of variation of signal, the higher tip angle pulseswere better performing pulses. This is also seen in thecontour plots of our comparison pulses, where the con-tours display relative signal levels. Thus, it is interestingthat the length required for our 90� comparison pulseactually decreased over that required for the 30� and 60�

comparison pulses. This is in part due to the fact thattips both larger and smaller than 90� give rise to lowersignal, and also to what Boulant et al. termed the sinecompression of the flip angle variability (17). However, itmay also indicate that the shallow tip angle pulses withimmunity to B1 inhomogeneity and resonance offset aremore difficult to generate. This may be true is also sug-gested by the relative SAR comparisons to rectangularpulses generating the same tip (Table 3).

NSS Pulses to Compare with SMP Pulses

The pulse comparisons for the 30�, 60�, and 90� SMPpulses were necessarily approximate, as simulated per-formances for the SMP pulses were not provided. As withthe Moore et al. comparisons, we attempted to find theshortest pulses which matched or slightly exceeded theperformance of the SMP 30�, 60�, and 90� pulses, and withthe same B1max. The results were presented in Table 4.

Similar to our comparison pulses with the pulses ofMoore et al., Table 4 shows that our comparison pulseswere somewhat shorter in length than the SMP pulses,indicating better tipping efficiency in terms of pulselength. However, unlike our comparison pulses to Mooreet al., we cannot conclude anything on the relative SARproduced by our pulses compared with the SMP pulses.In fact, if the SMP 3.0 T 30� pulse is indicative of SMPpulse shapes (16), the SMP pulses may play out formuch of their length at B1 values below their B1max, andthus may produce less SAR than our comparison pulses.

Our comparison pulses for the SMP pulses were opti-mized over a smaller range of resonance offsets than thecomparison pulses for the pulses of Moore et al., andshowed somewhat simpler patterns. In addition, unlikeour 90� Moore et al. comparison pulse, our 90� SMPcomparison pulse was fully symmetric, and retainedmuch of the general shape of the 10� pulse. Thus, it maybe that the complexity and distorted symmetry of ouroptimized pulses was governed in part by the range ofresonance offsets over which the pulse was optimized.

CONCLUSIONS

In this article, we demonstrated that our optimal controlroutine in MatPulse (22), when used with our previouslydeveloped four subpulse composite pulse developed forshallow tip angles (19), generated new pulse designs forshallow tip NSS pulses with immunity to both B1 inho-mogeneity and to resonance offset. Advantages of theapproach include the fact that prior B1 field measure-ments on individuals are not required, and the pulsescan be used with conventional head coils. The newdesign was composed of subpulses of varying amplitudeand phase, although the optimal control routine tendedto raise the amplitude of all the subpulses close to theB1max level. When used as the initiating pulse for larger

tips of 30�, 60�, and 90�, we found that our NSS pulseswere comparable in performance to previously publishedNSS pulses in the literature (17,18), but were more effi-cient (in terms of pulse length) than these pulses. How-ever, the improvements in efficiency were modest, andin all cases were less than a factor of two. In addition,the optimal control routine returned symmetrical or dis-torted symmetrical pulse shapes. As our general pulsedesign was similar to that of Moore et al., with theexceptions that we used more subpulses and a differentinitiating pulse, our designs had some resemblance totheirs. In contrast, the SMP design was quite different(17), and used subpulses with variable initial phase, am-plitude, and frequency. Although we provided evidencethat our SMP comparison pulse were more efficient interms of length, we could not rule out that the SMPpulses generated less SAR. It would be interesting to seeif using more segments in the SMP design would furtherimprove the efficiency of those pulses.

Although our comparison pulses with the SMP pulsesretained the general shape of the original 10� NSS pulse,the comparison pulses for the pulses of Moore et al.,which were optimized over larger ranges of B1 inhomo-geneity and resonance offsets than the SMP comparisonpulses, morphed into more complicated shapes, and inthe case of the 90� pulse, more distorted symmetry.

In agreement with the authors of the previously pub-lished NSS pulses (17,18), we believe these pulses havepotential application for MRI at high field. The compari-son of images taken with a conventional rectangularpulse and with the NSS 10� pulse at 4.0 T shows theimage degradation that can occur due to the nonuniformtipping generated at high field with conventional pulses.As B1 shimming is still expected to leave significant B1

inhomogeneity (9), a potential use of NSS pulses is inconjunction with B1 shimming, which would enabledesigns with less immunity to B1 inhomogeneity, andgenerating less SAR, to be used. However, our simula-tions with the 20� NSS pulse suggest that the reductionin SAR would only be modest (Fig. 5b). Thus, the highpower deposition (compared to rectangular pulses), dele-terious magnetization transfer effects (17,24), and lengthremain impediments to their incorporation into highfield MRI experiments. In addition, with versions of NSSpulses that exceed a few milliseconds, relaxation duringthe pulse may need to be taken into account (25). Takingadvantage of these pulses will also require the collectionof more k-space data following each pulse to avoidlengthening the duration of the MRI exam. Finally, wenote that the NSS pulse optimizations performed in thisarticle were not global, so in our view additional explo-ration of this style of pulse remains to be done.

A version of MatPulse with the optimal control rou-tines, the 10� NSS pulse, and the comparison NSS pulsesused in this article are available at www.cind.research.va.gov/.

ACKNOWLEDGMENTS

We thank Dr. Michael W. Weiner for his support of ourresearch at the Center for Imaging of NeurodegenerativeDiseases at the San Francisco VA Medical Center.


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