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Adiabatic Pulses

Alberto Tanns and Michael Garwood

Center for Magnetic Resonance Research and Department of Radiology, University of Minnesota, Minneapolis, MN 55455, USAand Departamento de Fsica e Informtica (DFI-IFSC), Universidade de Sao Paulo, Sao Carlos, SP 13560-970, Brazil

Adiabatic pulses are sometimes considered to be mysterious and exotic entities which are difficult to understand,

complex to generate and impractical to implement. This work is an attempt to bring familiarity and to fulfill the

preliminary needs of anyone interested in learning more about this subject. The response of magnetization to

stimuli produced by adiabatic pulses is analyzed using vector representations in a frequency modulated rotating

frame. The first section deals with basic principles of amplitude and frequency modulated pulses and a vector

representation in a second rotating frame is used to explain how the adiabatic condition can be satisfied. The

subsequent section explains the principles of offset independent adiabaticity. These principles are then used to

design optimal functions for the amplitude, frequency, and magnetic field gradient modulations for adiabatic

inversion pulses. The last section considers some practical aspects for those who want to develop methodologies

involving adiabatic pulses. 1997 John Wiley & Sons, Ltd.

NMR in Biomed. 10, 423434 (1997) No. of Figures: 8 No. of Tables: 2 No. of References: 59

Keywords: radiofrequency pulse; NMR; adiabatic pulse

Received 6 June, 1997; accepted 4 August 1997

INTRODUCTION

When NMR was first performed,1, 2 resonance was achievedby sweeping the amplitude of the polarizing magnetic field

B0 in the presence of a perpendicular field B1 whichoscillated at a constant radio frequency (RF). This con-tinuous wave (CW) approach has since been replaced by thepulsed NMR experiment3 which is performed in a static B0and uses a pulsed B1 to excite the full band of spectralfrequencies simultaneously. Typically, the carrier frequencyof the pulse remains constant and is applied at the center ofthe spectral region of interest. In this review, we examine analternative approach in which the carrier frequency varieswith time during the pulse. These frequency-swept pulses,known as adiabatic pulses, have benefitted from recentadvances which have expanded their capabilities andpopularity in applications ranging from in vivo imaging tohigh resolution spectroscopy of isolated molecules. In

analogy to the classical CW experiment, the differentspectral components are rotated in succession during theadiabatic frequency sweep. When the total sweep time isshort relative to T1 , the transient response of the spin systemcan be induced, which allows observation of NMRphenomena (e.g. FIDs or echoes) related to the pulsedmethod. By rapidly sweeping the frequency of the adiabaticpulses, NMR experiments can be performed in the samemanner as the pulsed experiment (i.e. the length of adiabatic

pulses can be short enough to permit their use in most pulsesequences). In this manner, the advantages of both classicalCW and pulsed NMR approaches can be exploited.

In a sweep of eitherB0 (classical experiment) or RF pulsefrequency (adiabatic experiment) from one side of reso-nance to the other, the net rotation of the magnetizationvector M is highly insensitive to changes in B1 amplitude.This desirable property has led to the common use ofadiabatic pulses in NMR experiments performed withsurface coils. Although the B1 produced by a surface coilvaries throughout space, the sensitivity gain provided bysuch coils is a major advantage for many in vivo NMRapplications. Within the sensitive volume of a typicalsurface coil, the amplitude ofB1 varies by >10-fold, whichmeans the flip angle also varies by >10-fold across thesample when conventional (constant frequency) pulses aretransmitted with this coil. In many experiments, flip angleerrors cause sensitivity losses, quantification errors, andartifacts (e.g. undesirable coherences). Adiabatic pulses

offer a means to rotate M by a constant flip angle, evenwhenB1 is extremely inhomogeneous.Across the spectral bandwidth of interest, spins with

different precession frequencies (isochromats) are sequen-tially rotated as the frequency sweep RF(t) approaches theresonance frequency 0 of each isochromat. With sometypes of adiabatic pulse, such as adiabatic full passage(AFP), the bandwidth is dictated solely by the range ofthe frequency sweep. For the spins precessing within thisfrequency band, the flip angle will be uniform, provided thatthe orientation of the effective magnetic field changesslower than the rotation ofM about this effective field. Thisrequirement, which is known as the adiabatic condition, canbe satisfied by using a sufficiently high B1 amplitude or bya slow frequency sweep. With the latter method, can bearbitrarily wide, even when using low peak RF power,provided that the pulse length Tp can be sufficiently long.The ability to achieve uniform flip angles over broad

Contract grant sponsor: NIHContract grant number: RR08079Contract grant number: CA64338Contract grant sponsor: FAPESP (Brazil)

Abbreviations used: AFP, adiabatic full passage; AHP, adiabatic half-

passage; AM, amplitude-modulated; BIR, B1-insensitive rotation; CW,continuous wave; FM, frequency-modulated; FOCI, frequency offsetcorrected inversion; GOIA, gradient-modulated offset independent adiaba-ticity; HS, hyperbolic secant; NOM, numerically optimized modulations;OIA, offset independent adiabaticity; PSD, power spectral density.

1997 John Wiley & Sons, Ltd. CCC 09523480/97/08042312 $17.50

NMR IN BIOMEDICINE, VOL. 10, 423434 (1997)


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bandwidths with low B1 amplitude is a unique feature ofthese adiabatic pulses. With conventional constant-fre-quency pulses, is always inversely proportional to Tp ,whereas and Tp are independent parameters in certaintypes of adiabatic pulses. The ability to invert magnetizationuniformly across wide bandwidths with arbitrarily low B1amplitude has led to a major advance in broadband

decoupling with minimal sample heating in high resolutionNMR applications.48 With these broadband pulses, in vivoNMR can also benefit from reduced peak RF powerrequirements and the ability to minimize voxel displace-ment for different chemical shifts.

For almost two decades, major efforts in NMR researchhave focused on the design of complex RF pulses tocompensate for changes inB1 amplitude and/or to increasebandwidths. A close relative of adiabatic pulses is thecomposite pulse, which consists of a train of rectangularpulses of different phases.9,10 Although composite pulsescan be derived to compensate for >10-fold variation ofB1(e.g. see procedure in Ref. 10), adiabatic pulses generallyoffer the greatest combined immunity to B1 inhomogeneity

and resonance offsets for a given amount of RF power.In this review, adiabatic pulses will be analyzed theoreti-cally and explored with vector diagrams. Our purpose is toprovide an understanding of how magnetization vectors canbe rotated by a constant angle, even when B1 is variable.Pulses to be described include adiabatic half-passage (AHP)and full-passage (AFP), both of which are commonlyexploited to generate uniform excitation (90) and inversion(180) in surface coil applications. The discussion willinclude a relatively new class of composite adiabatic pulses(BIR-1 and BIR-4) which can uniformly rotate magnetiza-tion vectors by any desired angle. Also included is anextensive discussion on the design of modulation functionsto be used in pulses to allow rotations that are invariant withfrequency offsets. A general goal is to provide an under-standing of the most common types of adiabatic pulses, withmention of how these pulses can be advantageous in someexperimental applications.

BASIC PRINCIPLES

Visualizing adiabatic pulses

A classical description of these adiabatic pulses can be

understood by considering the components of the magneticfields and M in a reference frame that rotates at theinstantaneous frequency RF(t). By convention, this frameis called the frequency-modulated (FM) frame with axislabelsx, y, z. In the FM frame, the direction of RF fieldvector B1(t ) remains fixed during an adiabatic passage.When the frequency of the pulse deviates from the Larmorfrequency 0 , a magnetic field with amplitude equal to/ is encountered along the z-axis, where=0RF. Figures 1(a) and 1(b) show examples of

B1(t ) and (t ) modulation functions for adiabatic passages(AHP and AFP).

In the FM frame (Fig. 2(a)), the effective field Beff(t ) isthe vector sum of the longitudinal field /and B1(t ). In

an adiabatic passage, RF is time dependent, and thereforeBeff(t) changes its orientation at the instantaneous angularvelocity, d/dt, where

(t)=arctanB1(t )(t) (1)At the beginning of the pulse, RF0 , and is at itsmaximum value (max). Initially / is very largerelative to B1; thus, the initial orientation of Beff isapproximately collinear with z. As RF(t ) begins toincrease, (t ) decreases and Beff rotates toward thetransverse plane. When

RF

(t)=0

, Beff

equals B1

. At thispoint, an AHP (90 excitation) has been completed. Toperform an AFP, the frequency sweep continues pastresonance toward max , leading to a final Befforientationalong z. During the adiabatic passage, M follows Beff(t ),

Figure 1. Examples of four adiabatic pulses. Adiabatic passages: (a) AHP and (b) AFP; andB1-insensitive rotations, (c) BIR-1 and (d) BIR-4.

A. TANNS AND M. GARWOOD424

1997 John Wiley & Sons, Ltd. NMR IN BIOMEDICINE, VOL. 10, 423434 (1997)


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provided that the adiabatic condition, |Beff(t)| |d/dt|, issatisfied for all t.

Vector description of adiabatic passage: FM and Beffframes

The vector analysis in the FM frame is inadequate to explainwhy M follows Beff(t) during an adiabatic sweep. For thispurpose, it is necessary to define a second frame ofreference which rotates with Beff(t) called the Beff framewith axis labels x, y, z. Figure 2(a) depicts therelationships beween the FM and Beff frames during anadiabatic passage. Initially, the two frames are super-imposable. During the pulse, the Beff frames changes itsorientation relative to the FM frame at the rate d/dt, andBeff remains collinear with thez-axis of theBeff frame. Bythe end of the pulse, the two frames are related by an anglewhich is the net Beff sweep angle (and flip angle). Figure2(b) shows the relationships between M and the magneticfield components in the Beff frame. The rotation of the Beffframe about the y-axis results in a magnetic field withinstantaneous magnitude (d/dt)/along they-axis. Thus,in the Beff frame, the resultant magnetic field E(t) is thevector sum of Beff(t) and

1(d/dt)y. For simplicity, lettheB1(t) and (t) functions be such thatBeff and d/dtareconstants. In this case, M simply precesses about E in the

cone of angle . As shown in Fig. 2(b), M never straysbeyond an angle 2 ofBeff, and therefore, M also remainswithin an angle 2 of Beff in the FM frame. When theadiabatic condition is well satisfied, is small andE(t)Beff(t). Furthermore, for components of M that areinitially perpendicular to Beff(0), the angle between Beff(t)and M remains within 90 (i.e. M remains approximatelyperpendicular to Beff(t)).

The classical adiabatic passage is insensitive to changesin the magnitude ofB1 only when the net rotation angle isa multiple of 90, since the final orientation of Beff isinsensitive to changes in B1 amplitude only when the finalvalue of |(t)| is zero (e.g. AHP) or is large relative to thefinal value ofB1(t) (e.g. AFP). In addition, the adiabaticpassages (AHP and AFP) are useful only for rotating asingle component of M from one point to another andcannot accomplish the plane rotations that many experi-ments require. For example, to generate optimal spin echo

(SE) signal, the refocusng pulse used in a SE pulse sequencemust rotate the components ofM in thexy-plane about anaxis which is invariant to changes in 0 andB1 amplitude.Although an AFP pulse can sometimes produce an observ-able echo, such applications generally yield poor resultssince the phase of the resultant magnetization varies as afunction of0 andB1 amplitude. Alternatively, as describedbelow, the use of Beffflips allows the creation of compositeadiabatic pulses which can accomplish plane rotation.1117 Inaddition, adjustable phase shifts can be introduced in someof these composite adiabatic pulses to allow the formationof any desired flip angle, while retaining B1 insensitiv-ity.1416, 18 This latter type of pulse is often called a universalrotator since any specified flip angle can be induced for all

components ofM lying perpendicular to a constant rotationaxis (i.e. a plane rotation of any desired angle).

Adiabatic plane rotation pulses: BIR-1 and BIR-4

Universal rotations can be accomplished with the class ofcomposite adiabatic pulses known asB1-insensitive rotation(BIR). BIR pulses (Figs 1(c) and 1(d)) can uniformly rotateall components of magnetization lying in a plane perpendi-cular to the rotation axis which remains constant despitechanges inB1 amplitude. The first generation plane rotationpulse known as BIR-1 has the disadvantage of producing

nonuniform response for Larmor frequencies 0 not equal tothe central frequency c in the sweep range of RF(t).However, a composite adiabatic pulse known as BIR-4,14,15

which consists of double BIR-1, provides a constantrotation axis for moderate resonance offsets(| |= |0c |>0). Although BIR-1 does not possess thislatter advantage, this pulse is simpler than BIR-4. Thus, webegin with a vector description of BIR-1.

The motions ofBeff and M during a 90 BIR-1 of lengthTp are shown in Figs 3(ad). The initial phase of B1 isarbitrarily chosen to coincide with x. At the beginning ofBIR-1, RF is applied on resonance ((0)=0), Thus,unlike the classical passage described in the previoussection, initially Beff lies in the transverse plane and isperpendicular to M (i.e. (0)=90). A vector analysissimilar to that performed in the previous section can be usedto prove that the angle between Beff and M will remainbetween 90 and 90+ during the pulse. To simplify

Figure 2. Vector diagrams showing the effective field and its components in two rotatingframes of reference. (a) Relationship between the FM frame, x, y, z (thin axes) and the Beffframe, x, y, z (thick axes). (b) Magnetic field components and evolution of themagnetization vector (M ) in the Beff frame.

ADIABATIC PULSES 425



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the problem, here we assume that the adiabatic condition iswell satisfied (|Beff(t) | | (d/dt)|), so that can be set to

zero. During the first half of BIR-1, Beff sweeps fromx toz, while M rotates about Beff (Fig. 3(a) and (b)). At t=Tp/2,the orientation ofM can be obtained from the solution of theBloch equation,

dM

dt =[MBeff]=

M

d

dt

(2)

where d/dt is the rate of change ofM around Beffsince the

Figure 3. (a)(d) Evolution of Beff and M in the FM frame during a 90 BIR-1 (and thefirst half of a 180 BIR-4). Thick, curved arrows represent precession of M around Beff.The evolution of Beff is implied by the thin, curved arrows and the orientation of the Beffframe at each time point (t=0, Tp /2 and Tp) is indicated by the double primed axes inparentheses. Beff evolves towards z (a), while M remains perpendicular to it anddisperses due to B1 inhomogeneity (b). At t=Tp/2, Beff is instantaneously inverted andthe transverse component of Beff (i.e., B1) is phase shifted by 270 (c). During the secondhalf of BIR-1, Beff evolves towards y and drives M back to its initial coherence, whichnow takes place along x of the FM frame (d). This represents the final stateproduced by a 90 BIR-1 and the halfway condition for a 180 BIR-4.

Figure 3. (e)(h) Second half of a 180 BIR-4. Beff continues to evolve from the previousstate (d) towards z (e). The process of dispersion (f) and coherence recovery (g)repeats with the second Beff flip accompanied by a 270 phase shift. The final state (h)is achieved with Beff along its initial orientation, but with M inverted in the FM frame.




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equation has the form of the angular momentum precessionequation.19 Provided that the adiabatic condition wassatisfied throughout the first segment of the pulse, M is inthexy plane at t=Tp/2 and the accumulated rotation angle() about Beff at this time is simply:

(Tp/2)=

Tp/2

0

Beff(t)dt (3)

At t=Tp /2, an instantaneous Beff flip is created by jumpingthe frequency from max to max (see BIR-1 in Fig. 1).To produce the desired flip angle , the phase of B1 issimultaneously shifted by =180+. In the presentexample (90 BIR-1), =270. During the second half ofthe pulse (Figs 3(c) and 3(d)), Beffsweeps from z to y,while M rotates in the opposite sense about x, so that thenet rotation for the pulse is zero. Thus, the centralBefffliptogether with the subsequent time reversal of the modula-tion functions, compensate (refocus) the rotations whichtook place during the first and second halves of the pulse(i.e. tot=0). This phenomenon is equivalent to a rotary

echo20

in theBeff frame. The phase shift determines theflip angle and the final orientation of theBeff frame relativeto the FM frame. After this 90 BIR-1, the Beff frame isrelated to the FM frame by a 90 rotation abouty and a90 rotation aboutx. In the FM frame, the net rotation ofM is equivalent to a 90 rotation aboutx, followed by a 90rotation (phase shift) about z. It is also possible to showthat BIR-1 can induce any flip angle by setting=180+, in which case the net rotation of M isequivalent to a rotation aboutx, followed by a phase shiftof . Although Fig. 3 shows only the motions for amagnetization vector initially oriented along z, the planerotation properties of the pulse can be revealed byperforming similar vector analyses using other initialorientations ofM.

In the presence of a resonance offset (=0RF(0)0), the performance of BIR1 degrades fortwo reasons. First, a rotary echo in theBeffframe may not beachieved, since tot depends onB1(t) and 0 according to:

tot=Tp/20

(1(t))2 +(0RF(t))2 dt

+TpTp/2

(1(t))2 +(0RF(t))2 dt (4)

where 1(t)=B1(t). When 0, the first and second

integrals in eq. (4) are unequal, and therefore, the netrotation of M about Beff is no longer zero. As describedfurther below, this first problem is eliminated with BIR-4,which is essentially double BIR-1. Secondly, in the presenceof a resonance offset, the initial orientation of Beff is nolonger perpendicular toz, but is given by the angle:

(0)=arctan 1(0)(0RF(0)) (5)As |0RF(0)| increases, Beff(0) acquires an increasinglongitudinal component in the FM frame, and as a result, anincreasing component of the initial longitudinal magnetiza-tion is spin locked to Beff. In BIR-4, the component of Mthat becomes spin-locked to Beff is returned toz by the endof the pulse, since the initial (t=0) and final (t=Tp)orientations of Beff are the same. As a consequence, the

desired flip angle is not acheived for an increasing fractionof M as frequency offset increases. Like conventionalpulses, the only way to alleviate this problem is to increase

B1 amplitude.The complete sequence of diagrams in Fig. 3 depicts the

vector motions of BIR-4. BIR-4 is a composite adiabaticpulse consisting of four segments and two Beffflips, creating

a double rotary echo in the Beff frame. The third segmentbegins (Fig. 3(e)) where BIR-1 ends (Fig. 3(d)). BIR-4 usestwo phase shifts (1 and 2) to produce a flip angleequal to , where

1 =180+

2(6)

2 =180+

2(7)

With BIR-4, the rotation always takes place about an axiswhich coincides with the initial direction of B1 (x in the

present example). Furthermore, tot is always zero followingBIR-4, since 1 =3 and 2 =4 (subscripts denote thevalues accumulated in each of the four segments of BIR-4). Of course, satisfactory performance requires that themodulation functions fulfill the adiabatic condition and that0 is contained in the frequency sweep RF(t).

Some applications: adiabatic solvent suppression andspectral editing

One of the unique features of BIR-4 (and BIR-1 onresonance) is the fact that the rotation angle is determinedsimply in the phase shifts. These phase shifts can beachieved not only by shifting B1 phase, but also by allowingthe magnetization vectors to precess in the transverse planeduring finite delays inserted in the pulse at the points where1 and 2 normally occur. In the absence of RFirradiation, evolution during each of these delays takesplace according to the rotating-frame Hamiltonian describ-ing the spin system. The net transformation achieved byBIR-4 with one or more delays then depends on thecombined effects of the shifts of B1 and the spinevolution that occurred during the delay(s). This generalprinciple forms the basis for a whole series of methods thatdiscriminate based on the spin evolution during the delay(s).These include solvent-suppressive adiabatic pulses,2124

adiabatic spectral editing based on spinspin coupling(BISEP),25,26 adiabatic multiple quantum coherence filter-ing,26,27 and adiabatic polarization transfer.26,2830 Thesepulses and pulse sequences accomplish their tasks whileproviding a high degree of compensation for B1 inhomoge-neity and are, therefore, particularly advantageous for invivo surface coil studies.

MODULATION FUNCTIONS

The recent popularity of adiabatic pulses in MRI andspectroscopy is a predominant factor leading to the rapidprogress in the development of these types of pulses. As anexample, an analytical solution of the Bloch equationswritten in the Ricatti form yielded an amplitude andfrequency modulated pulse known as the hyperbolic secant




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pulse31,32 (hereafter called HS pulse), which now enjoyswide popularity in high resolution and in vivo NMR. TheHS pulse is an adiabatic full passage composed of ahyperbolic secant function for amplitude modulation and ahyperbolic tangent function for frequency modulation. Anextraordinary property of the HS pulse is its insensitivity tovariations ofB1 intensity approaching several orders of

magnitude. The desirable plateau of the inversion profileand the sharpness of its transition zones were thought to beunique and curious features of these particular modulationfunctions. The flat response of the inversion profile was thefirst known manifestation of offset independent adiabaticity(OIA), although the reason for this was not well understooduntil recently. In this section we present the theoretical basisfor the generation of any pulse shape which can exhibitoffset independent adiabaticity, even when the resonanceoffset is time dependent. A similar development restricted toconstant resonance offsets can be found in Refs 33, 34.

Many different functions have been proposed to drive thefrequency ((t)) and amplitude (1(t)) modulations ofadiabatic pulses.6,15,16, 19, 31, 32, 3554 Many of these modulation

functions were derived from theoretical analyses consider-ing only the isochromat at the center of the excitation band(=0); consequently, the efficiency of these pulses gen-erally degrades as the resonance offset increases exceptwhen using the HS pulse.

Modulation functions optimized for specified ranges ofB1 amplitude and/or resonance offset have been derivedfrom considerations of the criterion for adiabaticity.55 Ananalytical method, known as NOM,40, 41 requires a numericalintegration which yields the optimal time dependence of thedriving functions for (t) and 1(t). Here we use a similarapproach to broaden the bandwidth of AFP pulses whileminimizing RF power.

Offset-independent adiabaticity (OIA)

For the sake of clarity, we first consider offsets, , producedfor example by a constantgradient or chemical shift. Ourapproach keeps the average RF power constant over thedesired bandwidth, . This condition is essential toperform rotations uniformly over a large using eitheramplitude modulation at a constant pulse frequency orcombined amplitude and frequency modulation. Amplitude-modulated (AM) pulses (e.g. sinc pulses) operate bydistributing equal amounts of power for every frequency inthe bandwidth at the same time. In the method used here, the

energy is distributed uniformly over the bandwidth, butsequentially in time.38 This requirement can be fulfilled byadiabatic pulses defined by an AM function:

1(t)=[0RF(t)z=B01F1(t)x (8)

and a frequency-modulated (FM) function;

(t)=[AF2(t)]z (9)

=[0RF(t)]z'

where B01 and A are the B1(t) and frequency sweepamplitudes, respectively, and is the gyromagnetic ratio inHz/Gauss. Equations (8) and (9) describe the componentsof the effective field experienced by an isochromat withLarmor frequency o, in a frame rotating around B0z withinstantaneous frequency RF(t) (i.e. the FM frame). As inRef. 40, we use these equations to express the adiabaticcondition as a ratio K, which here is a function of and t.

For the time interval equal to the pulse duration Tp and forthe offset interval | |A,

K(, t)= Beff(t) =A

2

B01[(B

01F1(t)/A)

2

+[F2(t)/A]2

]3/2

| (F2(t)/A)F1(t)F1(t)

F2(t)|

1 (10)

whereBeff(t) is the effective field as seen by an isochromatat offset frequency and is the rate of change of theBeff(t) orientation expressed in hertz. Our requirement is thatthe condition stated for K(, t) (eq. (10)) must be equallysatisfied for all values of inside the specified bandwidth.In other words, Kis defined to be constant in . It followsthat K(, t) can be calculated for all specific times t=twhen the isochromat at is on resonance,

F2(t)=/A (11)

giving;

K(t)=(B01F1(t))

2

AF2(t)

1 (12)

Hence, the identity;

K(t)AF2(t)=(B

01F1(t))

2 (13)

specifies the relationship between the two driving functionsF1(t) and F2(t). It states that, for all isochromats with| |A, 1(t)

2 must be much larger than the rate of changeof the frequency sweep (t) by the same factor K(t).

To illustrate the basis of OIA, Fig. 4 shows the fieldcomponents at three different times during an AFP pulse forthe case where 0. Far from resonance (Figs 4(a) and4(c)) the dominant contribution to Beff is given by1(t)=1[AF2(t)]z. When resonance is achievedat t (Fig. 4(b)), the effective field is solely determined byits transverse component, B1(t); therefore, t is the mostcritical time in regards to the adiabatic condition. Consider-ing that t is a function, eq. (13) must be used to calculatemodulation functions which yield a constant K(t) for allisochromats, since it represents the on-resonance adiabaticfactor for each one of them. Any pair of modulationfunctions that satisfy eq. (13) will achieve uniform adiaba-ticity asAF2(t) sweeps through consecutive values of.

Table 1 lists examples of OIA inversion pulses thatconform to eq. (13). To compare the performance of the

different OIA pulses, some useful parameters were defined.B01 (99%) is the minimumB01 needed to perform 99% inversion

at =0. Brms1 is the root mean square (rms) value ofB1(t).The last column lists the quality factor Q=90%/B

rms1 ,

where 90% is the effective bandwidth for > 90% inversionofMz (i.e. Mz/M0


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smoothness of the transition at the extremities of the AMfunction can easily be tuned by adjusting the parameter n.

Although the pulses in Table 1 were designed forbroadband applications (e.g. 13C decoupling), both and

B01 (99%) scale inversely with Tp. For example, when using theGaussian OIA pulse with Tp =10 ms, approximately thesame performance can be achieved for =10 kHz usingB01 (99%)=1.2 kHz. Figure 5 shows this Gaussian OIA pulse,along with a simulation of its inversion profile producedwith the latter parameters.

Gradient-modulated offset-independent adiabaticity(GOIA)

Here we continue the analysis considering the case of atime-dependent resonance offset resulting from a modulatedmagnetic field such as that produced by a gradient. For thiscase, the instantaneous frequency offset (t) of a givenisochromat at positionx is now defined as:

(t)=xGF3(t) (14)

Figure 4. Effective field components in the FM frame at three different times during anAFP pulse for the case in which =0c>0. (a) and (c) Far from resonance, the dominantcontribution to Beff is given by

1(t)=1[AF2(t)]. (b) When resonance is achieved att, [/]z and [AF2(t)/]z cancel each other; thus, the effective field is solely determinedby its transverse component, B1(t). The adiabatic condition at this time (| B1(t) | |d/dt|) limits the ability to invert the isochromat .

Table 1. Modulation functions and performance comparisona of OIA inversion pulsesB01 (99%) B

rms1

Pulseb F1() F2() (kHz) (kHz) Q

Lorentz1

1+2

1+2+

1

tan1() 11.49 3.25 15.08

HS sech()tanh()

tanh()7.56 3.28 14.81

Gaussc exp 222 erf()erf() 6.13 3.29 14.67

Hanning1+cos()

2+

4

3sin() 1+14 cos() 5.51 3.32 14.50

HSnc (n=8) sech(n) sech2(n) d 3.71 3.25 14.49

Sin40d (n=40) 1 sinn2 sinn2 1+cos22 d 3.61 3.29 14.20Chirp C (constant) 3.38 3.38 11.24

a All performance factors (B01 (99%), Brms1 and Q) were determined from simulations using Tp=2 ms

and A=25 kHz (=50 kHz).b The parameter was chosen to set the minimum value of F1() equal to 0.01.c These FM driving functions must be obtained by numerical integration of F1()2, =2t/Tp1,||1 for 0 tTp.d The second term of the FM function is more significant for lower n values and can be obtainedin closed form, although the numerical integration of F1()

2 is simpler to perform.




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where G is the amplitude of the gradient and the drivingfunction F3(t), like the other modulation functions, isnormalized to 1. The time dependent gradient affects theresponse of the magnetization in two different ways. Boththe Larmor frequency at a given positionx and the spectralwidth of a slab of magnetization with thickness x becomefunctions of time. To account for this new variability, theconcept of frequency sweep as used in the previous sectionmust now be redefined. The effective range of the frequency

sweep over x and the respective sweep rate are based onthe longitudinal (z) component of the effective field writtenas

(t)=[(t)AF2(t)]z=[xGF3(t)AF2(t)]z (15)

Previous attempts to enhance pulse performance usingmodulated gradients have been relatively successful, par-tially due to the robustness of adiabatic pulses. For example,Conolly et al.45 developed a time distortion method similarto the one proposed by Baum et al.,32 which in some casescan preserve the trajectory ofBeff(t). More recently, Ordidgeet al.52 devised a similar approach with FOCI pulses, wherethe Beff(t) trajectory was preserved by multiplying both its

longitudinal and transverse components by the sameweighting function W(t). Both of these methods allow thegeneration of slice selective AFP pulses that require less RFpower than the parent pulses for a specified bandwidth.However, these previous approaches were concerned withthe trajectory ofBeff(t) for the isochromat at =0, not its

power distribution among distinct isochromats contained inthe requested bandwidth. Consider, for example, an experi-ment where a FOCI pulse is applied with different values of

B01. AsB01 increases, the borders of the magnetization profile

become inverted sooner than the central region, which is aclear indication that the final result can be achieved withless RF energy. This phenomenon occurs because the FOCIamplitude modulation function is inadequate for the rate ofsweep determined by its frequency and gradient modulationfunctions. In the analysis that follows we provide ananalytical means to relate these functions based on theconcept of offset independent adiabaticity.

The adiabatic condition K(, t) is now defined as:

K(, t)= Beff(t) =A2

B01

[(B01F1(t)/A)2 +[F2(t)gF3(t)]

2]3/2

| (F2(t)gF3(t))F1(t)F1(t)[

F2(t)g

F3(t)]|

1 (16)

where g=xG/A. Once again we analyze the adiabaticcondition (eq. (16)) at its most critical time, when theeffective field is crossing the transverse plane (xy). For allspecific times t= t when the isochromat at (t) is onresonance, eq. [16] simplifies and K(, t) can be calculatedfrom:

K(t)=(B01F1(t))

2

A[F2(t)g

F3(t)]

1 (17)

The coordinate along the gradient direction is given by:

x(t)=AF2(t)

GF3(t)(18)

To achieve an effective sweep over spatially dispersedisochromats, eq. (18) demands that F3(t) is not proportionalto F2(t). The theoretical development for the case in whichthese functions are proportional to each other, as inGMAX43,56 and BISS-8,16 will be addressed in forthcomingwork.

Upon substituting g and x(t) into Eq. (17), the newidentity is

K(t)A F2(t)F2(t) F3(t)F3(t) =(B01F1(t))2 (19)The easiest way to generate OIA modulation functions fromeq. (19) is to calculate F1(t) given F2(t) and F3(t). This directapproach, however, is not satisfactory since the resultantshape ofB1(t) can have undesirable properties, such as high

Figure 5. (a) AM and (b) FM functions of the Gaussian OIA pulse with Tp=10 ms andA=5 kHz, and (c) its inversion profile obtained from the numerical solution of the Blochequations.




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peak power. Figure 6 shows the AM function of a pulseobtained using this OIA approach based on the F2(t) andF3(t) functions of FOCI. The new pulse requires less areaunder theB1(t) function, and thus lower RF energy, than theoriginal FOCI pulse, while the inversion profile remainsalmost unchanged. However, its AM shape is still far fromthe desirable plateau needed to reduce RF peak power.

Alternatively, a reverse GOIA approach supplies F1(t)and F3(t), and F2(t) is obtained by solving the differentialequation given by eq. (19) with initial condition F2(Tp/2)=0. An important point to be noticed here is that the shapeof the solution is no longer independent of the magnitude ofK, as it was in the previous section. Now Kmust be adjustedin order to allow the frequency sweep given by F2(t) to fallin the interval [A, A].

To demonstrate the efficiency of the latter method, westart with the AM driving function (F1(t)) of an HS4 pulse.This choice is based on the ability of HSn pulses (for n>2)to perform inversions with significantly reduced peakpower.34 Additionally, with n=4 the AM shape of the pulseresembles that of a FOCI pulse (C-shaped52) of the same

duration [see Fig, 7(a)]. For the time dependent gradient ofGOIA, a function based on an HS2 AM shape,

F3(t)=1.00.9 sech 2tTp12 (20)was chosen because of its smoothness and continuity of itsderivatives. Other functions could be used, provided thathigher gradient intensity is maintained at the times when thepulse is building the borders of the inversion profile and thatlower gradient intensity occurs during the central region ofthe pulse, as shown in Fig. 7(c). Of course, experimental

requirements and hardware limitations (e.g. gradient slewrate) may restrict the choices for the F3(t) function. With thepresent choice ofF1(t) and F3(t) functions, eq. (19) was usedto calculate the driving function for the frequency modula-tion (F2(t)) shown in Fig. 7(b). By comparingB1(t) functions[Fig. 7(a)], it can be seen that the GOIA pulse usesconsiderably less peak power (approximately half) toperform essentially with the same transformation as a FOCI

pulse with the same duration and bandwidth. As can be seenfrom Fig. 7(c), the GOIA pulse also reduces demands ongradient slew rate. To allow a performance comparisonbetween GOIA and FOCI pulses, Figs 7(d) and 7(e) showplots of the power spectral density (PSD) and inversionprofile (Mz/M0) for a GOIA pulse based on HS4 and a FOCIpulse of the same duration. Table 2 compares the B1

intensity (B0

1 (99%)) and peak and average power required bythese pulses. For reference, these performance parametersare normalized to those of a square pulse of the sameduration although its spectral composition is obviouslydifferent.

The ability of the pulses to distribute RF energyuniformly over the requested range of isochromats isapparent from Fig. 8 which shows Mz/M0 profiles as afunction of and 1 intensity. As the 1 amplitudeincreases, the FOCI pulse inverts the isochromats at theborders earlier than the central portion of the profile [Fig.8(a)]. With the GOIA pulse, a relatively uniform response isobtained for all 1 amplitudes [Fig. 8(b)].

Practical aspects on the use of adiabatic pulses

The previous sections represent an attempt to explain theresponse of a spin system to adiabatic pulses and to proposemethods to generate their modulation functions. Here weconsider some practical aspects regarding experimentationsof adiabatic pulses.

When properly designed, adiabatic pulses do not neces-sarily deliver more RF power than conventional pulses.The common misconception that adiabatic pulses requirehigh RF power may arise in part from the fact that certainadiabatic pulses allow much wider bandwidths than

conventional pulses of the same duration. With OIAinversion pulses, the amount of power delivered per unitspectral width (or power spectral density, PSD) needs tobe no greater than that required by conventional pulses inorder to perform the same transformation on the spinsystem.34 Since the energy of OIA pulses is distributedincrementally in time, the peak power can be even lowerthan that required by conventional pulses. Hence, these

Figure 6. Amplitude modulation functions for FOCI (C-shaped) and aGOIA pulse. The solid line represents the AM function generated

according to the direct GOIA method, starting with the FM and gradientmodulation functions of the FOCI pulse. The inversion profile produced bythis GOIA pulse is comparable to that of the original FOCI pulse usingapproximately the same B1 amplitude, but the GOIA pulse clearly deliversless RF energy.




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adiabatic pulses can be used generously in experimentsformerly designed to accomplish the same tasks withconventional pulses provided that care is taken to avoidwastefully exceeding the thresholdB1 intensity needed to

accomplish the desired transformation. The ability to modulate the pulse frequency is not afeature of some NMR spectrometers. Most often, adia-batic pulses are implemented with phase instead offrequency modulation since digital phase shifters withhigh resolution (


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must be satisfied. In other words, the number of digitizedpulse samples must be at least equal to the numericalvalue of R. This requirement also applies to morecomplex adiabatic pulses such as BIR-1 and BIR-4.

CONCLUDING REMARKS

In this review, adiabatic pulses were analyzed using aclassical description suitable for isolated spins. For morecomplex spin interactions such as scalar or dipolar coupling,it will be necessary to invoke quantum mechanics formal-ism to obtain accurate descriptions of the effects of

adiabatic pulses on the spin systems.57,58 Likewise, ouroptimizations of modulation functions were guided by theadiabatic condition, but other approaches, such as recursiveexpansion10, 15 and computer optimizations,59 may yieldfurther improvements. Finally, for the sake of brevity, it wasnot possible to consider all the many different types ofadiabatic pulses that have been reported and their potentialuses. Instead, we chose to focus on general principles and tolimit our discussion to some of the more common pulses inuse today.

Acknowledgements

This work was supported by NIH Grants RR08079 and CA64338 and theBrazilian agency FAPESP.

REFERENCES

1. Purcell, E. M., Torrey, H. C. and Pound, R. V. Resonanceabsorption by nuclear magnetic moments in a solid. Phys.Rev. 69, 37 (1946).

2. Bloch, F., Hansen, W. W. and Packard, M. Nuclear induction.

Phys. Rev. 69, 127 (1946).3. Ernst, R. R. and Anderson, W. A. Application of Fouriertransformation spectroscopy to magnetic resonance. Rev.Sci. Instrum. l37, 93101 (1966).

4. Starcuk, Z., Bartusek, K. and Starcuk, Z. Heteronuclearbroadband spin-flip decoupling with adiabatic pulses. J.Magn. Reson. A 107, 2431 (1994).

5. Bendall, M. R. Broadband and narrowband spin decouplingusing adiabatic spin flips. J. Magn. Reson. A 112, 126129(1995).

6. Kupce, E. and Freeman, R. Adiabatic pulses for widebandinversion and broadband decoupling. J. Magn. Reson. A115, 273276 (1995).

7. Fu, R. and Bodenhausen, G. Ultra-broadband decoupling. J.Magn. Reson. A 117, 324325 (1995).

8. Hwang, T.-L., Garwood, M., Tanns, A. and van Zijl, P. C. M.Reduction of sideband intensities in adiabatic decoupling

using modulation generated through adiabatic R-variation(MGAR). J. Magn. Reson. A 121, 221226 (1996).

9. Levitt, M. H. and Freeman, R. NMR population inversionusing a composite pulse. J. Magn. Reson. 33, 473476(1979).

10. Levitt, M. H. and Ernst, R. R. Composite pulses constructedby a recursive expansion procedure. J. Magn. Reson. 55,247 (1983).

11. Ugurbil, K., Garwood, M. and Bendall, M. R. Amplitude and

frequency modulated pulses to achieve 90 plane rotationswith inhomogeneous B1 fields. J. Magn. Reson. 72,177185 (1987).

12. Bendall, R. M., Ugurbil, K., Garwood, M. and Pegg, D. T.Adiabatic refocusing pulse which compensates for variableRF power and off-resonance effects. Magn. Reson. Med. 4,493499 (1987).

13. Conolly, S., Nishimura, D. and Macovski, A. A selectiveadiabatic spin-echo pulse. J. Magn. Reson. 83, 324334(1989).

14. Staewen, R. S., Johnson, A. J., Ross, B. D., Parrish, T.,Merkle, H. and Garwood, M. 3-D FLASH imaging using asingle surface coil and a new adiabatic pulse, BIR-4. Invest.Radiol. 25, 559567 (1990).

15. Garwood, M. and Ke, Y. Symmetric pulses to inducearbitrary flip angles with compensation for RF inhomoge-neity and resonance offsets. J. Magn. Reson. 94, 511525

(1991).16. de Graaf, R. A., Nicolay, K. and Garwood, M. Single-shot, B1-

insensitive slice selection with a gradient-modulatedadiabatic pulse, BISS-8. Magn. Reson. Med. 35, 652657(1966).

Figure 8. Contour plots of Mz/M0 vs 1 intensities and frequency offsets for (a) FOCIand (b) GOIA pulses with 20 kHz bandwidth and Tp=6 ms. The maximum value in the1 scale was chosen based on the minimum B1 intensity necessary for FOCI pulses toperform 99% inversion of the magnetization at =0. Evidence that the FOCI pulse

performs complete inversion of the isochromats at the borders earlier than in thecentral portion of the profile is shown in (a). A relatively uniform response is obtainedwith the GOIA pulse for a wide range of B1 intensities as shown in (b).




12/12

17. Hwang, T.-L., van Zijl, P. C. M. and Garwood, M. Broadbandadiabatic refocusing without phase distortion. J. Magn.Reson. 24, 250254 (1997).

18. Norris, D. G. and Haase, A. Variable excitation angle AFPpulses. Magn. Reson. Med. 9, 435 (1989).

19. Abragam, A. The Principles of Nuclear Magnetism. Clar-endon Press, 1978.

20. Solomon, I. Rotary spin echoes. Phys. Rev. Lett. 2, 301305(1959).

21. Ross, B. D., Merkle, H., Hendrich, K., Staewen, R. S. andGarwood, M. Spatially localized in vivo 1H magneticresonance spectroscopy of an intracerebral rat glioma.Magn. Reson. Med. 23, 96108 (1992).

22. Schupp, D. G., Merkle, H., Ellermann, J. M., Ke, Y. andGarwood, M. Localized detection of glioma glycolysisusing edited 1H MRS. Magn. Reson. Med. 30, 1827(1993).

23. Inglis, B. A., Sales, K. D. and Williams, S. C. R. BIRIANI, anew composite adiabatic pulse for water-suppressed pro-ton NMR spectroscopy. J. Magn. Reson. B 105, 6166(1994).

24. de Graaf, R. A., Luo, Y., Terpstra, M., Merkle, H. andGarwood, M. A new localization method using an adiabaticpulse, BIR-4. J. Magn. Reson. B106, 245252 (1995).

25. Garwood, M. and Merkle, H. Heteronuclear spectral editingwith adiabatic pulses. J. Magn. Reson. 94, 180185 (1991).

26. de Graaf, R. A., Luo, Y., Terpstra, M. and Garwood, M.Spectral editing with adiabatic pulses. J. Magn. Reson. B109, 184193 (1995).

27. Garwood, M., Nease, B., Ke, Y., de Graaf, R. A. and Merkle,H. Simultaneous compensation for B1 inhomogeneity andresonance offsets by a multiple-quantum NMR sequenceusing adiabatic pulses. J. Magn. Reson. A 112, 272274(1995).

28. Merkle, J. Wei, H., Garwood, M. and Ugurbil, K. B1-Insensitive heteronuclear adiabatic polarization transfer forsignal enhancement. J. Magn. Reson. 99, 480494 (1992).

29. Kim, S.-G. and Garwood, M. Double DEPT using adiabaticpulses. Indirect heteronuclear T1 measurement with B1insensitivity. J. Magn. Reson. 99, 660667 (1992).

30. van Zijl, P. C. M., Hwang, T.-L., ONeil Johnson, M. andGarwood, M. Optimized excitation and automation for

high-resolution NMR using B1-insensitive rotation pulses.J. Am. Chem. Soc. 118, 55105511 (1996).

31. Silver, M. S., Joseph, R. I. and Hoult, D. I. Highly selective /2 and pulse generation. J. Magn. Reson. 59, 347351(1984).

32. Baum, J., Tycko, R. and Pines, A. Broadband and adiabaticinversion of a two-level system by phase-modulatedpulses. Phys. Rev. A 32, 3435 (1985).

33. Kupce, E. and Freeman, R. Optimized adiabatic pulses forwideband spin inversion. J. Magn. Reson. A 118, 229303(1996).

34. Tanns, A. and Garwood, M. Improved performance offrequency-swept pulses using offset-independent adiaba-ticity. J. Magn. Reson. A 120, 133137 (1966).

35. Dunant, J. J. and Delayre, J., US Patent 3,975,765 (1976).36. Baum, J., Tycko, R. and Pines, A. Broadband population

inversion by phase modulated pulses. J. Chem. Phys. 79,46434647 (1983).37. Hardy, C. J., Edelstein, W. A. and Vatis, D. Efficient adiabatic

fast passage for NMR population inversion in the presenceof radiofrequency field inhomogeneity and frequencyoffsets. J. Magn. Reson. 66, 470482 (1986).

38. Kunz, D. Use of frequency-modulated radiofrequencypulses in MR imaging experiments. Magn. Reson. Med. 3,377 (1986).

39. Bendall, M. R. and Pegg, D. T. Uniform sample excitationwith surface coils for in vivo spectroscopy by adiabaticrapid half passage. J. Magn. Reson. 67, 376381 (1986).

40. Ugurbil, K., Garwood, M. and Rath, A. Optimization ofmodulation functions to improve insensitivity of adiabaticpulses to variations in B1 magnitude. J. Magn. Reson. 80,448360 (1988).

41. Johnson, A. J., Ugurbil, K. and Garwood, M. Optimization ofmodulation functions to enhance B1 insensitivity and off-resonance performance of adiabatic pulses. Society ofMagnetic Resonance in Medicine, 8th Annual Meeting(1989).

42. Wang, Z. Theory of selective excitation by scaled frequency-amplitude sweep. J. Magn. Reson. 81, 617 (1989).

43. Johnson, A. J., Garwood, M. and Ugurbil, K. Slice selectionwith gradient-modulated adiabatic excitation despite thepresence of large B1 inhomogeneities. J. Magn. Reson. 81,653660 (1989).

44. Bhlen, J.-M., Rey, M. and Bodenhausen, G. Refocusingwith chirped pulses for broadband excitation withoutphase dispersion. J. Magn. Reson. 84, 191 (1989).

45. Conolly, S., Glover, G., Nishimura, D. and Macovski, A. Areduced power selective adiabatic spin-echo pulsesequence. Magn. Reson. Med. 18, 2838 (1991).

46. Ke, Y., Schupp, D. G. and Garwood, M. Adiabatic DANTEsequences for B1-insensitive narrowband inversion. J.Magn. Reson. 96, 663 (1992).

47. Skinner, T. E. and Robitaille, P. M. L. Adiabatic excitation

using sin2

amplitude and cos2

frequency modulationfunctions. J. Magn. Reson. A 103, 34 (1993).

48. Slotboom, J., Vogels, B. A. P. M., de Haan, J. G., Creyghton,J. H. N., Quack, G., Chamuleau, R. A. F. M. and Bov, W. M.M. J. Proton resonance spectroscopy study of the effects ofL-ornithine-L-aspartate on the development of encephalop-athy, using localization pulses with reduced specificabsorption rate. J. Magn. Reson. B105, 147156 (1994).

49. Kupce, E. and Freeman, R. Stretched adiabatic pulses forbroadband spin inversion. J. Magn. Reson. A 117, 246(1995).

50. de Graaf, R. A., Luo, Y., Garwood, M. and Nicolay, K. B1-insenstive, single-shot localization and water suppression.J. Magn. Reson. B113, 3545 (1996).

51. Rosenfeld, D., Panfil, S. L. and Zur, Y. Analytic solutions ofthe Bloch equation involving asymmetric amplitude and

frequency modulations. Phys. Rev. A 54, 24392443(1996).52. Ordidge, R. J., Wylezinska, M., Hugg, J. W., Butterworth, E.

and Franconi, F. Frequency offset corrected inversion(FOCI) pulses for use in localized spectroscopy. Magn.Reson. Med. 36, 562566 (1996).

53. Shen, J. Use of amplitude and frequency transformations togenerate adiabatic pulses of wide bandwidth and low RFpower deposition. J. Magn. Reson. Series B112, 131140(1996).

54. Rosenfeld, D., Panfil, S. L. and Zur, Y. Design of adiabaticpulses for fat-suppression using analytic solutions of theBloch equation. Magn. Reson. Med. 37, 793801 (1997).

55. Slichter, C. P. Principles of Magnetic Resonance, 3rd edn.Springer, New York, 1990.

56. Tanns, A., Garwood, M., Panepucci, H. and Bonagamba, T.J. Localized proton Spectroscopy with 3D-GMAX. 9thSociety of Magnetic Resonance in Medicine, New York, NY(1990).

57. Hediger, S., Meier, B. H., Kurur, N. D., Bodenhausen, G. andErnst, R. R. NMR cross polarization by adiabatic passagethrough the HartmannHahn condition (APHH). Chem.Phys. Lett. 233, 283288 (1994).

58. Capuan. S. Luca, F. D., Marinelli, L. and Maraviglia, B.Coherence-transfer process by adiabatic pulses. J. Magn.Reson. Series A 121, 17 (1996).

59. Poon, C. S. and Henkelman, R. M. Robust refocusing pulsesof limited power. J. Magn. Reson. A 116, 161180 (1995).



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