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Volume-selective NMR spectroscopy with self-refocusing pulses
JOURNAL
OF MAGNETIC
RESONANCE
Volume-Selective LYNDON
87, I- 17 ( 1990)
NMR Spectroscopy with Self-Refocusing Pulses EMSLEY AND GEOFFREY BODENHAUSEN
Section de Chimie, Universite’de Lausanne, Rue de la Barre 2. CH 1005 Lausanne, Switzerland Received April 6, 1989; revised July 6, I989 If the 90” selective pulses that are normally used for volume localization are replaced with self-refocusing 270” pulses having a Gaussian time-domain amplitude profile, various pulse sequences can be made shorter, thus reducing the effects of transverse relaxation, flow, or diffusion. The sensitivity advantage is ascertained quantitatively for some of the commonly used sequences. The volume of interest defined in these experiments is determined by the offset dependence (spatial profile) of the conversion of transverse xy magnetization into z magnetization and vice versa. It is demonstrated that the gradients can be switched on and off within the intervals where the radiofrequency pulses are ap plied without affecting the spatial profile. We compare the shapes of the volumes of interest defined by (i) 90” Gaussian pulses, (ii) 90” sine pulses, and (iii) self-refocusing 270 Gaussian pulses. In the first two cases, refocusing or predefocusing is necessary, while in the latter case it is not. The comparisons lead us to propose three new methods, (i) selfrefocusing spatial and chemical-shift-encoded excitation (SR-SPACE), (ii) selfrefocusing sputially resolved spectroscopy (SR-SPARS), and (iii) self-refocusing selected-volume excitation using stimulated echoes (SR-VEST). o 1990 Academic press, hc.
Spatially localized NMR spectroscopy, although still in its infancy, has already been demonstrated to have enormous potential for applications ranging from in vivo clinical diagnosis ( Z,2) to materials science (3,4). Several pulse sequences for achieving spatial localization, such as spatial and chemical-shift-encoded excitation (SPACE) (5) (Fig. la), s&ally resolved spectroscopy (SPARS) (6, 7) (Fig. 1b), and selected-volume excitation using stimulated echoes (VEST) (8) (Fig. 2)) have been proposed. In these techniques, the selected magnetization spends a significant amount of time in the transverse plane of the rotating frame, so these methods are sensitive to short T, relaxation times or to flow effects, and therefore are beleaguered by low sensitivity. In the SPACE method of Fig. 1a (5)) the transverse magnetization is first excited selectively in a slice, and after refocusing it is returned to the z axis with a hard pulse which simultaneously excites transverse magnetization of the rest of the sample, which then dephases while the sequence is repeated with the field gradient applied along three orthogonal directions. Conversely the SPARS technique of Fig. 1b (6, 7) works by exciting transverse magnetization of the entire sample and returning only magnetization in a selected slice back to the z axis of the rotating frame. In both sequences the result is that magnetization of a particular volume of interest (VOI) remains along the z axis, and observation of the spectrum of the VOI is achieved by applying a nonselective 90” pulse at the end of the sequence followed by collection of 1
0022-2364190 $3.00 Copyright 0 1990 by Academic Press., Inc. All rights of reproduction in any form reserved.
2
EMSLEY 180;
a: SPACE
AND
BODENHAUSEN
180;
9qx
90;
180;
I-
-
9%
90;
9%
90$
RF lr
tP Gx
tr
tP
tr
tP
I /
\
GY
\
/
GZ
\
90;
b: SPARS
180;
-
90;
180,
wx
90;
180;
wx
wx
RF tr Gx
GY
GZ
tr
tP /
tr
tP
\
/
\
/
\
FIG. 1. Pulse and field gradient schemes suitable for (a) spatial and chemical-shift-encoded excitation (SPACE) and (b) spatially resolved spectroscopy (SPARS). The uppercase subscripts X, Y, and 2 refer to orthogonal spatial directions through the sample, while the lowercase subscripts x and y refer to the phases of the radiofrequency pulses in the rotating frame. If selective 90” Gaussian pulses of duration t, with 2.5% truncation are used I, = 0.55t,. In these schematics, it is assumed that the gradient pulses are switched on before the beginning of the RF pulses, and switched off after the end of the RF pulses.
the FID in the absence of external field gradients. The primary difference between SPARS and SPACE is that the gradients are temporarily switched off during the 180 pulses in the SPARS sequence, so as to circumvent the use of nonselective pulses in the presence of field gradients since this is impractical on whole-body imaging systems. While this reduces the RF power requirements, it is obvious that the overall duration of the experiment increases, bearing in mind the need for finite delays to switch the gradients. During these delays the magnetization resides in the transverse plane and will therefore decay through T2 relaxation. Thus, a sequence such as SPARS represents a compromise between antagonistic demands which is not entirely satisfactory. The VEST sequence of Fig. 2 (8) achieves volume localization by generating a stimulated echo ( 9, 10). Thus, as shown in the coherence transfer pathway diagram
VOLUME
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b
FIG. 2. (a) Pulse and field gradient echoes (VEST). (b) Coherence transfer
scheme suitable for selected-volume pathways for the stimulated echo.
excitation
using
stimulated
( 1 I ) of Fig. 2b, the first pulse generates transverse magnetization in a selected slice; the second pulse, during which a field gradient is applied along an orthogonal direction, then restores populations, but only along a selected column through the object. The final pulse, which is combined with a third orthogonal gradient, reconverts populations into transverse magnetization of the selected volume, which results in the formation of a stimulated echo. We have previously shown the dramatic advantages of using self-refocusing 270 Gaussian pulses in one- and two-dimensional high-resolution NMR spectroscopy (12). We also presented experimental evidence that the offset dependence of the phase and amplitude of the magnetization excited by a 270” Gaussian pulse with an inhomogeneous RF field is exactly as predicted by theory (12). We have also demonstrated the usefulness of 270” pulses in slice-selection techniques for magnetic resonance imaging ( 13) by showing that, if one wishes to convert M, into MXYmagnetization, there is no need for gradient reversal to form an echo after the pulse if selfrefocusing 270” Gaussian pulses are used instead of the conventional 90” pulses. In this paper we show how 270” Gaussian pulses can greatly improve volume-selective spectroscopic pulse sequences. SELECTIVE
EXCITATION AND
IN THE PRESENCE OF A MAGNETIC FIELD SELF-REFOCUSING 270” GAUSSIAN PULSES
GRADIENT,
It has long been known that the selective excitation of a narrow frequency band in an inhomogeneously broadened spectrum is a nontrivial matter since the finite length
4
EMSLEY
AND BODENHAUSEN
RF
Gradient a.
d. FIG. 3. Timing of gradients relevant for selective manipulations of magnetization. The RF pulse shown is a truncated Gaussian, but the gradient scenarios may also be combined with other pulse shapes. (a) Selective pulse to convert M, into MXy or vice versa in the presence of a field gradient without any compensation for phase errors; (b) selective excitation (to convert kf, into MX,) with refocusing of the transverse magnetization by reversal of the gradient to compensate for the phase dispersion which builds up during the pulse; (c) selective pulse to convert M+, into M, with compensation of phase errors by predefocusing of the transverse magnetization by reversing the gradient prior to the pulse; (d) same as (a), but the field gradient is switched on and off during gradient “ramping” times tp within the duration t, of the truncated Gaussian pulse.
of a selective RF pulse results in a dispersion ofthephases of the transverse magnetization vectors building up during the pulse. In the presence of a continuous distribution of offsets, there will be a distribution of effective fields spanning a continuous range of tilt angles with respect to the z axis of the rotating frame. Consequently, at the end of an RF pulse with a small on-resonance flip angle applied according to the scheme of Fig. 3a, the magnetization will have almost completely dephased, resulting in a nearly vanishing signal after removal of the gradient. Most selective pulses, when acting on z magnetization, fortunately give rise to transverse magnetization with a phase gradient that is approximately linear as a function of offset. It is possible to remove this phase dispersion by refocusing, either by using a nonselective 180” pulse or by reversing the magnetic field gradient as shown in Fig. 3b. Much effort has been put into the design of pulse shapes for selective conversion of A4, into MXYwith the aim of achieving a “top hat” response. Most of this research has quite naturally paid little attention to the phase response, assuming that refocusing of the magnetization would be possible after the pulse. However, if it is the purpose of the pulse selectively to return transverse A&.. magnetization to the z axis, such
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as in SPARS or VEST, the issue of the phase dispersion becomes more important because it is necessary to prepare the distribution of the phases of the magnetization vectors in the transverse plane in a suitable way before applying the pulse. In our previous discussions of the effects of 270” Gaussian pulses ( 12, 13) we only considered pulses acting on z magnetization. Fortunately, given the effect of a pulse on z magnetization, it is possible to show analytically what its effect will be on transverse magnetization. It can be proven (14, 15) that if a constant-phase pulse with a time-symmetric envelope is applied along the y axis concurrently with a time-symmetric amplitude-modulated field gradient producing a magnetic field along the z axis, then the magnetization experiences a total rotation Rtot about an axis that must lie in the yz plane. Thus for a given magnetization vector the problem is exactly analogous to the rotation produced under a constant-amplitude pulse in the presence of a constant field. It can easily be shown (IO) that if a pulse with a given effective flip angle Per and effective tilt angle Beffin the yz plane acts on initial magnetization MO aligned along the z axis the resultant magnetization is M: = M’sin &sin
Befi
M; = M”( 1 - cos &@)sin eerrcos Bee M: = M”(cos28,ff+
cos &.ffsin20,ff).
If, on the other hand, the same pulse acts on MO aligned along the -x axis the resultant components are M: = -MOcos &ff Ml. = -M’sin M: = M’sin
&&OS 8,* &esin ~9~~.
[21
Equations [I] and [ 21 show clearly that the M, response of a pulse acting on -x magnetization is the same as the M, response of a pulse acting on z magnetization. Therefore the slice profile selected by a 270” Gaussian pulse (or for that matter any pulse which fulfills the conditions set out above) will be the same regardless of whether it is acting on magnetization initially along the M, or M, axis. Figures 4a and 4c show the M, profiles for 90” Gaussian- and 90” sine-shaped pulses acting on initial magnetization aligned along the -x axis. The simulations also describe the M, profiles that are obtained if the same pulses are acting initially on M,, and correspond to the naive scheme of Fig. 3a. If the required transformation is of M, into M, the profile can be greatly improved by the refocusing scheme of Fig. 3b. If the required transformation is of M, into M,, refocusing is not possible, but a viable alternative is predefocusing. This consists of applying a field gradient to create a phase dispersion of the transverse magnetization before applying the RF pulse, according to the scheme of Fig. 3c. It will be realized from the above symmetry arguments that the phase dispersion created in this way must be equal but opposite to the phase dispersion which would be created by a given pulse in going from M, to Mxy, so that the pulse will now achieve the transformation of Mxy into M,. Figures 4b and 4d show the profiles which can be obtained in this way with 90” Gaussian- and 90“ sincshaped pulses. Obviously this procedure requires that the magnetization spend more
6
EMSLEY
AND
BODENHAUSEN GRADIENT
PULSE
H b
Ar
-2
-1
0
1
2
FIG. 4. Profiles of the M, magnetization created by y pulses acting on initial M,. It is assumed for simplicity that the gradients can be switched instantaneously. The abscissas represent the distance r from the center of the slice in a direction normal to the applied field gradient. (a) and (b) show the profiles with and without predefocusing (see obtained with a 90” Gaussian pulse (duration t,, 2.5% truncation) Fig. 3 for the precise timing of the gradients). (c) and (d) show the profiles for a 90’ sine pulse with six zero crossings, with the same overall duration f, as the 90’ Gaussian pulse. The peak RF amplitude is 2.6 times that of the 90’ Gaussian pulse. (e) shows the profile obtained with a 270” Gaussian pulse (duration t,, 2.5% truncation) withouf predefocusing. The peak RF amplitude B y is 1.15 times that of the 90” sine pulse. The excellent selectivity in (e) is obtained with a pulse/gradient sequence having an overallduration of only 65% of the sequences required to obtain the profiles of(b) or (d). In (b), (d), and (e) we indicate the slice thickness of these profiles; for (b) this is defined by the point where the excitation falls below 0.1 Ma, and for both (d) and (e) it is defined by the first zero points. The relative Ar values in (b), (d), and (e) are 1.75:3.1: I. To obtain the Same Ar value as in (d), the 90” and 270” Gaussians must be shorter by a factor 0.56 and 0.32, and their amplitudes scaled by 1.77 and 3. I, respectively.
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time in the transverse plane, thus reducing the sensitivity of the experiment if T2 relaxation times are short, since the overall duration of the pulse and gradient sequence is extended by about 55% when predefocusing is combined with either a 90” Gaussian pulse or a 90” sine pulse. In the context of volume localization it is useful to visualize the way in which the one-dimensional profiles of Fig. 4 (which correspond to selected planes) will combine to form the selected volume. Figure 5 shows the effect of applying two successive “building blocks” with gradients applied along orthogonal spatial directions. It is apparent that any imperfections are compounded, and thus we can see how predefocusing is absolutely necessary to obtain a useful VOI if 90” pulses are used. Figure 4e shows the profile of a 270” Gaussian pulse acting on x magnetization according to the scheme of Fig. 3a without any refocusing or predefocusing step. All the profiles in Fig. 4 were calculated for pulses of the same duration and assuming instantaneous gradient switching. Horizontal axes are scaled in arbitrary units which, if the field gradient is linear, will be directly proportional to distance. Thus we compare the behavior of the magnetization under two Gaussian pulses differing only in a three fold increase in ~;lax for the 270” pulse in comparison to the 90” pulse; they have the same overall duration. Note how the 270” Gaussian is more selective than the refocused 90” Gaussian, and how they are both much more selective than the 90” sine pulse. Note also that the peak RF amplitude of the 270” Gaussian is o&y 1.5% larger than the peak of the 90” sine pulse with six zero crossings. In order to obtain the same slice thickness Ar, defined in Fig. 4, for each pulse, the 90” sine pulse must have a peak amplitude 1.5 times that of the 90” Gaussian, and the 270” Gaussian a peak amplitude 3 times that of the 90” sine. The relative durations tp of pulses having the same Ar must be 0.56: 1.0:0.35, respectively, for the 90” Gaussian, 90” sine, and 270” Gaussian. For example, a 1.5 ms 270” Gaussian needs only a Bi field of 1 kHz to obtain a breadth of excitation corresponding to the entire ‘H chemical-shift range ofloppmat 1.5T(=640Hz). As mentioned elsewhere ( 23), the magnitude of the observable signal and the profile of a 270” Gaussian pulse are very sensitive to the truncation level of the Gaussian function. We find that the optimum truncation is around 0.025 BF. We have previously demonstrated ( 12) the self-refocusing effect of 270” Gaussian pulses using the conventional “grapefruit diagrams,” where the trajectories are shown on the unit sphere, but we present here what we find is a more enlightening approach. Thus, we consider the evolution of the phases C#J = arctan ( MY/M,) of a family of magnetization vectors at different offsets as a function of time, as shown for a 90” Gaussian pulse in Fig. 6a. Notice how the trajectories have a virtualfocus (16) at tf = 0.45t,, where tp is the overall duration ofthe pulse. For a Gaussian-shaped pulse with the same overall duration, but having a 270” flip angle, the focus of the same magnetization vectors occurs at around 0.85t,, as shown in Fig. 6b. It is this shift of the focus of the magnetization which accounts for the vastly reduced phase dispersion obtained at the end of a self-refocusing 270” Gaussian pulse, and this in turn leads to the MZ profile of Fig. 4e for the 270” Gaussian. The slice thickness, shown in Fig. 4e and expressed in terms of reduced frequency ( 12)) is Ar = 0.65. The contour plot in Fig. 5c of the twodimensional selectivity of the 270” Gaussian pulse shows an almost square Vol.
EMSLEY
AND
BODENHAUSEN
a
X
c
I
I X
FIG. 5. Two-dimensional spatial selectivity of 90” and 270” Gaussian pulses. The plots represent the M, profiles obtained as a function of the spatial axes X and Y if two of the “building blocks” shown on the left are applied in succession, with gradients directed along the X and Y directions. (a) shows the profile obtained by a 90’ Gaussian with predefocusing of the transverse magnetization, as used in SPARS experiments. (b) shows how attempting to dispense with the predefocusing step causes the resulting 1!4~ response to be distorted beyond recognition. By simply increasing the RF amplitude to obtain a 270” Gaussian pulse, however, we can transform the chaotic response of(b) into the excellent distribution of(c) because of the self-refocusing effect of 270” Gaussian pulses. Contour levels were plotted at ?5, 10,25, 50,75, and 90% of M” in each case.
A potential drawback to volume localization techniques is that if the AI, profile generated by the selective pulse does not have a desirable box-like shape, one may face breakthrough from regions outside the VOI. Our simulations show that in the
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WITH SELF-REFOCUSING
PULSES
time
‘P
time
‘P
9
270” Gaussian
0
.a
FIG. 6. The evolution of the phases of a family of magnetization vectors under 90” and 270” Gaussian pulses. The horizontal time axis extends between points where the Gaussians are truncated. The 90” pulse has a virtual focus at tr = 0.45t,, and the 270’ Gaussian has a virtual focus in the vicinity of tr = OX%,. Note the reduced phase dispersion at the end of the 270” Gaussian pulse. The vectors are at reduced frequency intervals ( 12) of 0.058, and the pulses both have 2.5% truncation. The phases (excursion from the +x axis toward the +y axis) start at cp = 0 for the 90” pulse, and rp = - rr, i.e., at the -x axis, for the 270” pulse.
case where the spin density of the object under investigation, or the density of a particular type of spin in the object, is represented by a continuum, the contribution to the signal from outside the VOI (as defined by the slice thickness Ar, see Fig. 4e) after a self-refocusing 270” Gaussian pulse is only about 0.02% of the signal from within the
10
EMSLEY
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VOI. It is also interesting to note that of this negligible breakthrough is from within a distance Ar around the VOI. COMPENSATING
FOR
THE
EFFECT
OF
FINITE
GRADIENT
signal, over 90%
SWITCHING
TIMES
At first sight it would seem that a major obstacle to the implementation of selfrefocusing 270” Gaussian pulses is that the gradients cannot be switched on or off instantaneously. If we consider the conventional predefocusing sequence of Fig. 3c we may note that it has a certain degree of freedom to “tweek” the length of the predefocusing delay so as to compensate for finite gradient switching times. In contrast, a self-refocusing 270” pulse applied according to Fig. 3a does not have that degree of freedom. In fact, assuming that the magnetization is precisely aligned along the x axis before the gradient is activated, substantial phase errors will build up in the interval tp, thereby causing distortions in the resulting profile. Thus the simulations of the profiles calculated for Figs. 4e and 5c would seem to be nothing but pie in the sky, since they ignore the effects of this delay by assuming instantaneous gradient switching. The situation might, however, be improved if the gradient is switched within the duration of the RF pulse, in the manner of Fig. 3d. It is realistic to assume that the switching delay tg could be as much as 20% of the RF pulse length t,. Figure 7b shows the same M, profile as Fig. 4e, obtained with a 270” Gaussian pulse and instantaneous gradient switching. Figure 7d shows how the profile is distorted if we use the same RF pulse, but a field gradient that is linearly ramped up from t = 0 to 0.2t, and similarly ramped down from t = 0.8 to 1.Ot,,. Clearly, the distortions are severe, and it becomes necessary to devise a way of compensating for these effects. Toward this end we consider the consequences of gradient switching during the pulse on the time dependence of the instantaneous tilt angle 0. One of the intrinsic properties of a Gaussian pulse in the absence of gradient switching is that the effective field is aligned close to the z axis during the early part of the pulse, gradually moves toward the transverse plane as B, increases, and then moves back toward the z axis in the latter part of the pulse. This leads to so-called “teardrop” trajectories ( Z7) for magnetization vectors having large offsets. The behavior of the tilt angle t9(t) (excursion with respect to the z axis) is plotted in Fig. 7a for a 270” Gaussian pulse in the presence of a constant field. This results in the undistorted profile of Fig. 7b. If we now consider the behavior of 0( t) for a pulse in which the gradient is slowly ramped up, we can appreciate that at the very beginning of the RF pulse, in stark contrast to the previous situation, there is no z component of the effective field but there is a nonvanishing B,; therefore 0 (t = 0) is 7r/ 2. Figure 7c shows how the behavior of 8( t) is distorted throughout the gradient switching period. Clearly this will lead to severe distortion of the trajectories, so that the degraded profile of Fig. 7d is not surprising. To eliminate the effect of gradient switching on the excitation profile it is necessary to match the behavior of 0( t) during a sequence where tg Z 0 as far as possible to the behavior in the idealized case of Fig. 7a. The most obvious method would be simply to weight the pulse shape with the profile of the gradient, as 0 will remain constant if a reduction in the gradient amplitude is accompanied by a corresponding reduction in the RF amplitude. However, the trajectories will still be distorted by the ensuing
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Mm a
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11
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b
2
-1
0
1
2
d H
FIG. 7. Time dependence of the tilt angle 0 of the effective field for a vector at a reduced offset ( 12) of 0.058 during a 270” Gaussian pulse with 2.5% truncation. Ifthe field gradient is assumed to be instantaneously switched on and off at t = 0 and t,, one obtains 0(t) as shown in (a), and the nearly ideal M,(r) profile of(b). If the same RF pulse is used in a sequence where the gradient is being linearly ramped up from t = 0 to 0.22, and similarly ramped down from t = 0.8 to t, (t,/t, = 20%), the behavior of 0(t) is that of(c) with the degraded M,(r) profile of(d). If the same gradient timing with t.J t, = 20% is used with a 270” Gaussian pulse having 0.28% truncation, one obtains the time dependence of 0 shown in (e), and the resulting M,( r) profile is that of(f), which, apart from a scaling factor of I .2, is restored to the nearly ideal profileof(
reduction in the magnitude of the effective field I& during gradient ramping. A more satisfactory approach is to modify the pulse envelope with the gradient envelope and also allow time for a larger rotation at the beginning and end of the pulse where & is reduced. This can be achieved very simply by reducing the truncation of the Gaussian function. Figure 7e shows 19(t) for a 270” Gaussian pulse with 0.28% truncation, as opposed to 2.5% in Fig. 7c, and it can be seen that it is very similar to the ideal behavior of Fig. 7a, save for the fact that 0 tends to 7r/2 at the very beginning and end of the pulse. This small deviation is not important as Beff is so low at these points as to have little effect, and indeed we see that the profile of Fig. 7f is exactly the same as that of Fig. 7b, apart from a trivial scaling of the slice thickness caused by the slightly higher RF
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amplitude required for a pulse with lower truncation. The scaling factor is about 1.2 for the case where tp/tp = 0.2, and the peak RF amplitude must be increased by a factor 1.24. Thus by simply reducing the truncation level of a self-refocusing 270 Gaussian pulse it is possible to compensate almost completely for the effects of gradient switching. The simulations of Figs. 7c-7f are all for a situation where the gradient switching time t, is 0.2t,. To restore a “pseudo 2.5%” profile one should employ truncation levels of 1.05,0.28, and 0.03% for t.Jtp = 0.1,0.2, and 0.3, respectively. Above tJt, = 0.35 it is unfortunately impossible to restore a satisfactory profile simply by reducing the level of truncation. APPLICATIONS
OF SELF-REFOCUSING TO SPECTROSCOPIC
270” GAUSSIAN SEQUENCES
PULSES
From the above discussion it becomes clear that 270” Gaussian pulses without refocusing have a VOI that is at least as well defined as that of a 90” Gaussian with refocusing. Furthermore, sequences employing 270” Gaussian pulses are significantly less sensitive to flow or T, relaxation effects. In SPARS or SPACE type experiments, we suggest that self-refocusing 270” Gaussian pulses be used in place of the 90” selective pulses. In the SPACE experiment of Fig. la the refocusing gradient and the nonselective 180” pulse (which is a very demanding feature of the experiment) can be dispensed with, and the sequence becomes the (self-refocusing) SR-SPACE sequence of Fig. 8a. Similarly, in the SPARS experiment of Fig. 1b the need for the predefocusing gradient and the nonselective 180” pulse is obviated, and the sequence simplifies to that of our self-refocusing SPARS sequence (SR-SPARS) of Fig. 8b. These modifications should both improve the z magnetization profile and enhance the sensitivity by reducing the time spent in the transverse plane. If self-refocusing 270” Gaussian pulses are used to convert A4, into MXY, there is a small MY component that is antisymmetric with respect to offset, which may perturb slice selection in inhomogeneous objects (13). The self-refocusing SPARS experiment is not affected by this problem, because at each stage of the experiment the unwanted MY components are rendered inobservable by dephasing. In the original VEST sequence of Fig. 2 the relevant magnetization is in the transverse plane during the delay 7, which must be at least as long as the time required for the gradient pulses. In our SR-VEST method of Fig. 9, where 270” Gaussian pulses replace the usual 90” pulses, there is no need for refocusing or predefocusing gradient pulses, so that T, may be essentially zero and the sequence becomes largely insensitive to T2 _In the r2 delay the selected magnetization is aligned along the z axis; since this delay may also be zero in the SR experiment, the sequence is also largely insensitive to T, . The price to pay for these advantages is that it may be necessary to use more elaborate phase cycles. In order to obtain proper localization in the SR-VEST experiment, the first pulse must be alternated in phase together with the receiver, so as to ensure suppression of the free induction decay after the third pulse. Also, in order to suppress pathways other than the one leading to the stimulated echo (see coherence transfer pathways of Fig. 2b) one may insert a r2 interval between the second and the third pulses without signal losses due to T2 relaxation, and extend the GY and Gz
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GX GY
3
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k
90”
b: SR-SPARS
-
270’
270”
270”
RF
Gx
FIG. 8. Pulse and gradient scheme There is no need for any refocusing verse magnetization decays.
suitable for (a) self-refocusing SPACE and (b) self-refocusing gradients, and hence there are no unnecessary delays where
SPARS. the trans-
gradients into this r2 interval in order to dephase unwanted coherences (8). Alternatively, one may employ a four-step phase cycle. EFFECTS
OF
RF INHOMOGENEITY
We have shown both theoretically and by experiment (12, 13) that the shape of the magnetization profile generated by 270” Gaussian pulses is fairly tolerant to RF inhomogeneity. Nevertheless, if the sequences of Figs. 8 or 9 were to be blindly implemented with all RF pulses having the same phase, the total flip angles experienced by the magnetization in the VOI would be 1170” and 8 lo”, respectively, and the effects of RF inhomogeneity would be cumulative. However, if the SR-VEST sequence is modified so that the phases of the three pulses in Fig. 9 are +y, y, and -y, the total flip angle is reduced to an acceptable level of +270”. This prescription is compatible with the phase cycling required for suppression of the free induction decay following the last pulse, but is incompatible with the full phase-cycling requirements for unam-
14
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SR-VEST
2700
AND
270”
BODENHAUSEN
270”
::: GyY Gz FIG, 9. Pulse and gradient scheme sary delays where the magnetization
suitable for self-refocusing decays through transverse
VEST (SR-VEST). relaxation.
There
are no unneces-
biguous selection of the coherence transfer pathways, and in the case where RF inhomogeneity is a problem, suppression of other pathways must be achieved by applying gradients in the 72 interval. In the SR-SPARS sequence of Fig. 8b on the other hand, where each building block of the sequence contains two pulses, we suggest that the following phases be used: 90;,270",
90",270",,
90",,270",
90;,.
This reduces the total flip angle in the VOI to 90” and since there are no phase-cycling requirements in SR-SPARS there should be no obstacle to the implementation of these phases. EFFECTS
OF
TRANSVERSE
RELAXATION
It is possible to quantify the effects of relaxation on the original sequences of Figs. 1 and 2 and our self-refocusing sequences of Figs. 8 and 9 by including a simple monoexponential T2 damping in the simulations of the Bloch equations. We neglect the effect of longitudinal T, relaxation. For clarity we compare only Gaussian pulses having the same duration between truncation points, so that we do not take into account the fact that, to obtain the same slice thickness, a 270’ Gaussian must be 37% shorter than the corresponding 90” Gaussian, which will clearly benefit the relative performance of the 270” Gaussian in the presence of relaxation. Our simulations show that if the sequence with gradient reversal of Fig. 3b is used on a system with a T2 relaxation time equal to the length t, of the (truncated) 90” Gaussian pulse, the signal intensity will be attenuated to a mere 30%. In contrast, if the sequence of Fig. 3a is used with the same relaxation time and a self-refocusing 270” pulse, as much as 60% of the signal remains. Thus the elimination of the gradient reversal step yields an improvement in sensitivity by a factor of 2. Although we have included a monoexponential damping in the Bloch equations, the decay of the magnetization at a given offset during the pulse is generally multiexponential. Also, for a given pulse length, each offset will have a different trajectory, and therefore each offset will relax in a slightly different way to another, which leads to a slight distortion of the resulting
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PULSES
s/s,
0.0
1.0
2.0
3.0
4.0
5.0
VT2
$/T2 FIG. 10. Plots of the relative signal intensity S/S,, against the dimensionless ratio tP/ T2, where lr, is the duration of the Gaussian pulses, and TZ the transverse relaxation time. (a) Dash-dotted line: 90’ Gaussian pulse with the scenario of Fig. 5a; short dashes: 270” Gaussian pulse with the self-refocusing scenario of Fig. 5c. The dimensionless decay constants (“relaxation coefficients”) of these curves are k = 1.1 and 0.43, respectively. In (b) we show the attenuation of several magnetization vectors at different offsets undergoing the scenario of Fig. 5c as a function of tP/ T,, in order to demonstrate the multiexponential behavior of the effective relaxation. The vectors are at reduced offsets ( 12) of 0 (solid line), 0. I5 (long dashes), 0.23 (medium dashes), 0.35 (dash dots), 0.40 (dots), and 0.42 (short dashes).
magnetization profile because of transverse relaxation. Fortunately, for a 270” Gaussian pulse this distortion is so small as to be negligible. A useful way to quantitate the effects of transverse relaxation on various pulse sequences is to plot the signal intensity as a function of the dimensionless ratio tP/ T2 (see Fig. 10). Including t, in the numerator simply normalizes the resulting curves (to first order) with respect to selectivity. Relaxation leads to a distortion of the magnetization trajectories which is dependent on tP/ T,, and therefore each offset experi-
16
EMSLEY
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ences a multiexponential decay, not only during the pulse, but also as a function of the ratio tP/ Tz as demonstrated in Fig. lob. It can be seen from the plot of Fig. 10a that the attenuation of the integrated signal over the slice is almostmonoexponential for Gaussian pulses applied according to the schemes of Fig. 3. This suggests that the decay of a given experiment can be described approximately by (S/So>; e exp 1 -kit,/
Tz).
[31
The dimensionless decay constant ki, which we call the “relaxation coefficient,” serves as a good measure of the sensitivity of a sequence to transverse relaxation. Indeed it can be given a physical interpretation in that a value of k, = 1 would indicate that the magnetization is attenuated by the same amount as if it were simply residing in the transverse plane for the duration t,. The smaller the value of k, the less sensitive the sequence is to relaxation, and the better the performance. The relaxation coefficients of the curves obtained in Fig. 1Oa are ki = 1.1 for the sequences with a 90 Gaussian pulse of Figs. 3b or 3c and ki = 0.43 for sequences involving a 270” Gaussian pulse according to Figs. 3a or 3d. It is a simple matter to extend this analysis to the three-dimensional localized spectroscopic sequences considered in this paper, since all such sequences essentially consist of three consecutive building blocks. Therefore (S/&h
x exp { -3kitpl
Tz >2
[41
and the relaxation coefficients of the three-dimensional experiments are thus simply three times those for the one-dimensional building blocks; i.e., 3ki = 3.3 for the normal SPARS and VEST sequences, and 3ki = 1.3 for the corresponding SR sequences. This clearly shows how the self-refocusing sequences are much less sensitive to relaxation than the gradient refocused equivalents. We have not explicitly considered the effect of relaxation on the stimulated echo sequences including 7, delays longer than the time required for gradient switching, but it is clear that increasing pi simply increases the sensitivity to relaxation and this can be described by modifying Eq. [ 41 to (S/So),i-exp{(-3kit,+27,)/T~},
[51
with7, = 7, - 2t,, where t, is the time required for gradient predefocusing or refocusing, and the factor 2 in Eq. [ 51 is introduced to account for the fact that the magnetization is in the transverse plane during two T, delays if acquisition is delayed until the top of the stimulated echo. CONCLUSIONS
We have shown how self-refocusing 270” Gaussian pulses can dramatically reduce the sensitivity of localized spectroscopic pulse sequences to T2 relaxation by significantly reducing their duration. We have also shown that the volume of interest generated by 270” Gaussian pulses is well defined and that the experiments should be insensitive to breakthrough signal from surrounding tissue. The new self-refocusing sequences of Figs. 8 and 9 will be less sensitive to the effects of flow or diffusion in the sample. We see no reason why these sequences should not replace the original
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sequences of Figs. 1 and 2, especially since the new sequences are more straightforward to implement than their original counterparts. We believe that self-refocusing 270” Gaussian pulses will be of particular importance to sequences such as SPARS and VEST which are designed for implementation on whole-body NMR imaging systems, and for that reason we have highlighted those sequences here. However, the principle of substituting self-refocusing 270” Gaussian pulses for 90” Gaussian pulses will be valid in many of the various pulse sequences which have been proposed for spatially localized spectroscopy. ACKNOWLEDGMENTS Many routines.
thanks go to Urs Eggenberger and Peter Pftindler for assistance This work was supported in part by the Swiss National Science
with the high-resolution Foundation.
graphics
REFERENCES I.
K.
L. BEHAR, J. A. DEN HOLLANDER, M. E. STROMSKI, PETROFF, AND J. W. PRITCHARD, Proc. Natl. Acad. Sci.
T. OGINO,
R. G. SCHULMAN,
0.
USA 80,4945 ( 1983). 2. H. BLUM, B. CHANCE, AND G. P. BLJZBY, Surg. Gynecol. Obstet. 164( 5), 409 ( 1987). 3. E. W. MCFARLAND, L. J. NEWRINGER, AND M. J. KUSHMERICK, Magn. Reson. Imaging (1988). 4. D. G. CORY,
Sot., Div.
A. M. REICHWEIN,
Polym.
5. D. M. DODDRELL,
Magn.
Reson.
Chem.)
J. W. M. VAN OS, AND
Polym.
Prepr.
(Am.
Chem.
J. M. BULSING,
J. FIELD,
M. G. IRVING,
AND H. BADDELEY,
J.
( 1986).
6. P. R. LUYTEN, A. J. H. MARIEN, B. SIJTSMA, AND (1986). 7. D. J. JENSEN, J. L. DELAYRE, AND P. A. NARAYANA,
J. A. DEN HOLLANDER,
J. Magn.
Reson.
69,552
J. Magn.
R. FREEMAN, J. FRIEDRICH, AND Wu, C. BAUER, R. FREEMAN, T. FRENKIEL,
XI-LI, J. Magn. Reson. 79,561 ( 1988). J. KEELER, AND A. J. SHAKA, J. Magn.
Reson.
67, 148
( 1986).
8. J. GRANOT, J. Magn. Reson. 70,488 ( 1986). 9. E. L. HAHN, Phys. Rev. 80,580 ( 1950). IO. R. R. ERNST, G. BODENHAUSEN, AND A. WOKAUN, “Principles of Nuclear Magnetic One and Two Dimensions,” Clarendon, Oxford, 1987. II. G. BODENHAUSEN, H. KOGLER, AND R. R. ERNST, J. Magn. Reson. 58,370 ( 1984). 12. L. EMSLEY ANDG. BODENHAUSEN, J. Magn. Reson. 82,211( 1989). 13. L. EMSLEY ANDG. BODENHAUSEN, Magn. Reson. Med. 10,273 ( 1989). 14. C. COUNSELL, M. H. LEVIS, ANDR. R. ERNST, J. Magn. Reson. 63,133 (1985). 1.5. J. T. NGO AND P. MORRIS, J. Magn. Reson. 75, 122 ( 1987). 16. 17.
6( 5), 507
29( 1 ), 92 ( 1988).
W. M. BROOKS,
68,367
W. S. VEEMAN,
A. C.
Reson.
Resonance
58,442
( 1984).
in
OF MAGNETIC
RESONANCE
Volume-Selective LYNDON
87, I- 17 ( 1990)
NMR Spectroscopy with Self-Refocusing Pulses EMSLEY AND GEOFFREY BODENHAUSEN
Section de Chimie, Universite’de Lausanne, Rue de la Barre 2. CH 1005 Lausanne, Switzerland Received April 6, 1989; revised July 6, I989 If the 90” selective pulses that are normally used for volume localization are replaced with self-refocusing 270” pulses having a Gaussian time-domain amplitude profile, various pulse sequences can be made shorter, thus reducing the effects of transverse relaxation, flow, or diffusion. The sensitivity advantage is ascertained quantitatively for some of the commonly used sequences. The volume of interest defined in these experiments is determined by the offset dependence (spatial profile) of the conversion of transverse xy magnetization into z magnetization and vice versa. It is demonstrated that the gradients can be switched on and off within the intervals where the radiofrequency pulses are ap plied without affecting the spatial profile. We compare the shapes of the volumes of interest defined by (i) 90” Gaussian pulses, (ii) 90” sine pulses, and (iii) self-refocusing 270 Gaussian pulses. In the first two cases, refocusing or predefocusing is necessary, while in the latter case it is not. The comparisons lead us to propose three new methods, (i) selfrefocusing spatial and chemical-shift-encoded excitation (SR-SPACE), (ii) selfrefocusing sputially resolved spectroscopy (SR-SPARS), and (iii) self-refocusing selected-volume excitation using stimulated echoes (SR-VEST). o 1990 Academic press, hc.
Spatially localized NMR spectroscopy, although still in its infancy, has already been demonstrated to have enormous potential for applications ranging from in vivo clinical diagnosis ( Z,2) to materials science (3,4). Several pulse sequences for achieving spatial localization, such as spatial and chemical-shift-encoded excitation (SPACE) (5) (Fig. la), s&ally resolved spectroscopy (SPARS) (6, 7) (Fig. 1b), and selected-volume excitation using stimulated echoes (VEST) (8) (Fig. 2)) have been proposed. In these techniques, the selected magnetization spends a significant amount of time in the transverse plane of the rotating frame, so these methods are sensitive to short T, relaxation times or to flow effects, and therefore are beleaguered by low sensitivity. In the SPACE method of Fig. 1a (5)) the transverse magnetization is first excited selectively in a slice, and after refocusing it is returned to the z axis with a hard pulse which simultaneously excites transverse magnetization of the rest of the sample, which then dephases while the sequence is repeated with the field gradient applied along three orthogonal directions. Conversely the SPARS technique of Fig. 1b (6, 7) works by exciting transverse magnetization of the entire sample and returning only magnetization in a selected slice back to the z axis of the rotating frame. In both sequences the result is that magnetization of a particular volume of interest (VOI) remains along the z axis, and observation of the spectrum of the VOI is achieved by applying a nonselective 90” pulse at the end of the sequence followed by collection of 1
0022-2364190 $3.00 Copyright 0 1990 by Academic Press., Inc. All rights of reproduction in any form reserved.
2
EMSLEY 180;
a: SPACE
AND
BODENHAUSEN
180;
9qx
90;
180;
I-
-
9%
90;
9%
90$
RF lr
tP Gx
tr
tP
tr
tP
I /
\
GY
\
/
GZ
\
90;
b: SPARS
180;
-
90;
180,
wx
90;
180;
wx
wx
RF tr Gx
GY
GZ
tr
tP /
tr
tP
\
/
\
/
\
FIG. 1. Pulse and field gradient schemes suitable for (a) spatial and chemical-shift-encoded excitation (SPACE) and (b) spatially resolved spectroscopy (SPARS). The uppercase subscripts X, Y, and 2 refer to orthogonal spatial directions through the sample, while the lowercase subscripts x and y refer to the phases of the radiofrequency pulses in the rotating frame. If selective 90” Gaussian pulses of duration t, with 2.5% truncation are used I, = 0.55t,. In these schematics, it is assumed that the gradient pulses are switched on before the beginning of the RF pulses, and switched off after the end of the RF pulses.
the FID in the absence of external field gradients. The primary difference between SPARS and SPACE is that the gradients are temporarily switched off during the 180 pulses in the SPARS sequence, so as to circumvent the use of nonselective pulses in the presence of field gradients since this is impractical on whole-body imaging systems. While this reduces the RF power requirements, it is obvious that the overall duration of the experiment increases, bearing in mind the need for finite delays to switch the gradients. During these delays the magnetization resides in the transverse plane and will therefore decay through T2 relaxation. Thus, a sequence such as SPARS represents a compromise between antagonistic demands which is not entirely satisfactory. The VEST sequence of Fig. 2 (8) achieves volume localization by generating a stimulated echo ( 9, 10). Thus, as shown in the coherence transfer pathway diagram
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b
FIG. 2. (a) Pulse and field gradient echoes (VEST). (b) Coherence transfer
scheme suitable for selected-volume pathways for the stimulated echo.
excitation
using
stimulated
( 1 I ) of Fig. 2b, the first pulse generates transverse magnetization in a selected slice; the second pulse, during which a field gradient is applied along an orthogonal direction, then restores populations, but only along a selected column through the object. The final pulse, which is combined with a third orthogonal gradient, reconverts populations into transverse magnetization of the selected volume, which results in the formation of a stimulated echo. We have previously shown the dramatic advantages of using self-refocusing 270 Gaussian pulses in one- and two-dimensional high-resolution NMR spectroscopy (12). We also presented experimental evidence that the offset dependence of the phase and amplitude of the magnetization excited by a 270” Gaussian pulse with an inhomogeneous RF field is exactly as predicted by theory (12). We have also demonstrated the usefulness of 270” pulses in slice-selection techniques for magnetic resonance imaging ( 13) by showing that, if one wishes to convert M, into MXYmagnetization, there is no need for gradient reversal to form an echo after the pulse if selfrefocusing 270” Gaussian pulses are used instead of the conventional 90” pulses. In this paper we show how 270” Gaussian pulses can greatly improve volume-selective spectroscopic pulse sequences. SELECTIVE
EXCITATION AND
IN THE PRESENCE OF A MAGNETIC FIELD SELF-REFOCUSING 270” GAUSSIAN PULSES
GRADIENT,
It has long been known that the selective excitation of a narrow frequency band in an inhomogeneously broadened spectrum is a nontrivial matter since the finite length
4
EMSLEY
AND BODENHAUSEN
RF
Gradient a.
d. FIG. 3. Timing of gradients relevant for selective manipulations of magnetization. The RF pulse shown is a truncated Gaussian, but the gradient scenarios may also be combined with other pulse shapes. (a) Selective pulse to convert M, into MXy or vice versa in the presence of a field gradient without any compensation for phase errors; (b) selective excitation (to convert kf, into MX,) with refocusing of the transverse magnetization by reversal of the gradient to compensate for the phase dispersion which builds up during the pulse; (c) selective pulse to convert M+, into M, with compensation of phase errors by predefocusing of the transverse magnetization by reversing the gradient prior to the pulse; (d) same as (a), but the field gradient is switched on and off during gradient “ramping” times tp within the duration t, of the truncated Gaussian pulse.
of a selective RF pulse results in a dispersion ofthephases of the transverse magnetization vectors building up during the pulse. In the presence of a continuous distribution of offsets, there will be a distribution of effective fields spanning a continuous range of tilt angles with respect to the z axis of the rotating frame. Consequently, at the end of an RF pulse with a small on-resonance flip angle applied according to the scheme of Fig. 3a, the magnetization will have almost completely dephased, resulting in a nearly vanishing signal after removal of the gradient. Most selective pulses, when acting on z magnetization, fortunately give rise to transverse magnetization with a phase gradient that is approximately linear as a function of offset. It is possible to remove this phase dispersion by refocusing, either by using a nonselective 180” pulse or by reversing the magnetic field gradient as shown in Fig. 3b. Much effort has been put into the design of pulse shapes for selective conversion of A4, into MXYwith the aim of achieving a “top hat” response. Most of this research has quite naturally paid little attention to the phase response, assuming that refocusing of the magnetization would be possible after the pulse. However, if it is the purpose of the pulse selectively to return transverse A&.. magnetization to the z axis, such
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as in SPARS or VEST, the issue of the phase dispersion becomes more important because it is necessary to prepare the distribution of the phases of the magnetization vectors in the transverse plane in a suitable way before applying the pulse. In our previous discussions of the effects of 270” Gaussian pulses ( 12, 13) we only considered pulses acting on z magnetization. Fortunately, given the effect of a pulse on z magnetization, it is possible to show analytically what its effect will be on transverse magnetization. It can be proven (14, 15) that if a constant-phase pulse with a time-symmetric envelope is applied along the y axis concurrently with a time-symmetric amplitude-modulated field gradient producing a magnetic field along the z axis, then the magnetization experiences a total rotation Rtot about an axis that must lie in the yz plane. Thus for a given magnetization vector the problem is exactly analogous to the rotation produced under a constant-amplitude pulse in the presence of a constant field. It can easily be shown (IO) that if a pulse with a given effective flip angle Per and effective tilt angle Beffin the yz plane acts on initial magnetization MO aligned along the z axis the resultant magnetization is M: = M’sin &sin
Befi
M; = M”( 1 - cos &@)sin eerrcos Bee M: = M”(cos28,ff+
cos &.ffsin20,ff).
If, on the other hand, the same pulse acts on MO aligned along the -x axis the resultant components are M: = -MOcos &ff Ml. = -M’sin M: = M’sin
&&OS 8,* &esin ~9~~.
[21
Equations [I] and [ 21 show clearly that the M, response of a pulse acting on -x magnetization is the same as the M, response of a pulse acting on z magnetization. Therefore the slice profile selected by a 270” Gaussian pulse (or for that matter any pulse which fulfills the conditions set out above) will be the same regardless of whether it is acting on magnetization initially along the M, or M, axis. Figures 4a and 4c show the M, profiles for 90” Gaussian- and 90” sine-shaped pulses acting on initial magnetization aligned along the -x axis. The simulations also describe the M, profiles that are obtained if the same pulses are acting initially on M,, and correspond to the naive scheme of Fig. 3a. If the required transformation is of M, into M, the profile can be greatly improved by the refocusing scheme of Fig. 3b. If the required transformation is of M, into M,, refocusing is not possible, but a viable alternative is predefocusing. This consists of applying a field gradient to create a phase dispersion of the transverse magnetization before applying the RF pulse, according to the scheme of Fig. 3c. It will be realized from the above symmetry arguments that the phase dispersion created in this way must be equal but opposite to the phase dispersion which would be created by a given pulse in going from M, to Mxy, so that the pulse will now achieve the transformation of Mxy into M,. Figures 4b and 4d show the profiles which can be obtained in this way with 90” Gaussian- and 90“ sincshaped pulses. Obviously this procedure requires that the magnetization spend more
6
EMSLEY
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PULSE
H b
Ar
-2
-1
0
1
2
FIG. 4. Profiles of the M, magnetization created by y pulses acting on initial M,. It is assumed for simplicity that the gradients can be switched instantaneously. The abscissas represent the distance r from the center of the slice in a direction normal to the applied field gradient. (a) and (b) show the profiles with and without predefocusing (see obtained with a 90” Gaussian pulse (duration t,, 2.5% truncation) Fig. 3 for the precise timing of the gradients). (c) and (d) show the profiles for a 90’ sine pulse with six zero crossings, with the same overall duration f, as the 90’ Gaussian pulse. The peak RF amplitude is 2.6 times that of the 90’ Gaussian pulse. (e) shows the profile obtained with a 270” Gaussian pulse (duration t,, 2.5% truncation) withouf predefocusing. The peak RF amplitude B y is 1.15 times that of the 90” sine pulse. The excellent selectivity in (e) is obtained with a pulse/gradient sequence having an overallduration of only 65% of the sequences required to obtain the profiles of(b) or (d). In (b), (d), and (e) we indicate the slice thickness of these profiles; for (b) this is defined by the point where the excitation falls below 0.1 Ma, and for both (d) and (e) it is defined by the first zero points. The relative Ar values in (b), (d), and (e) are 1.75:3.1: I. To obtain the Same Ar value as in (d), the 90” and 270” Gaussians must be shorter by a factor 0.56 and 0.32, and their amplitudes scaled by 1.77 and 3. I, respectively.
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time in the transverse plane, thus reducing the sensitivity of the experiment if T2 relaxation times are short, since the overall duration of the pulse and gradient sequence is extended by about 55% when predefocusing is combined with either a 90” Gaussian pulse or a 90” sine pulse. In the context of volume localization it is useful to visualize the way in which the one-dimensional profiles of Fig. 4 (which correspond to selected planes) will combine to form the selected volume. Figure 5 shows the effect of applying two successive “building blocks” with gradients applied along orthogonal spatial directions. It is apparent that any imperfections are compounded, and thus we can see how predefocusing is absolutely necessary to obtain a useful VOI if 90” pulses are used. Figure 4e shows the profile of a 270” Gaussian pulse acting on x magnetization according to the scheme of Fig. 3a without any refocusing or predefocusing step. All the profiles in Fig. 4 were calculated for pulses of the same duration and assuming instantaneous gradient switching. Horizontal axes are scaled in arbitrary units which, if the field gradient is linear, will be directly proportional to distance. Thus we compare the behavior of the magnetization under two Gaussian pulses differing only in a three fold increase in ~;lax for the 270” pulse in comparison to the 90” pulse; they have the same overall duration. Note how the 270” Gaussian is more selective than the refocused 90” Gaussian, and how they are both much more selective than the 90” sine pulse. Note also that the peak RF amplitude of the 270” Gaussian is o&y 1.5% larger than the peak of the 90” sine pulse with six zero crossings. In order to obtain the same slice thickness Ar, defined in Fig. 4, for each pulse, the 90” sine pulse must have a peak amplitude 1.5 times that of the 90” Gaussian, and the 270” Gaussian a peak amplitude 3 times that of the 90” sine. The relative durations tp of pulses having the same Ar must be 0.56: 1.0:0.35, respectively, for the 90” Gaussian, 90” sine, and 270” Gaussian. For example, a 1.5 ms 270” Gaussian needs only a Bi field of 1 kHz to obtain a breadth of excitation corresponding to the entire ‘H chemical-shift range ofloppmat 1.5T(=640Hz). As mentioned elsewhere ( 23), the magnitude of the observable signal and the profile of a 270” Gaussian pulse are very sensitive to the truncation level of the Gaussian function. We find that the optimum truncation is around 0.025 BF. We have previously demonstrated ( 12) the self-refocusing effect of 270” Gaussian pulses using the conventional “grapefruit diagrams,” where the trajectories are shown on the unit sphere, but we present here what we find is a more enlightening approach. Thus, we consider the evolution of the phases C#J = arctan ( MY/M,) of a family of magnetization vectors at different offsets as a function of time, as shown for a 90” Gaussian pulse in Fig. 6a. Notice how the trajectories have a virtualfocus (16) at tf = 0.45t,, where tp is the overall duration ofthe pulse. For a Gaussian-shaped pulse with the same overall duration, but having a 270” flip angle, the focus of the same magnetization vectors occurs at around 0.85t,, as shown in Fig. 6b. It is this shift of the focus of the magnetization which accounts for the vastly reduced phase dispersion obtained at the end of a self-refocusing 270” Gaussian pulse, and this in turn leads to the MZ profile of Fig. 4e for the 270” Gaussian. The slice thickness, shown in Fig. 4e and expressed in terms of reduced frequency ( 12)) is Ar = 0.65. The contour plot in Fig. 5c of the twodimensional selectivity of the 270” Gaussian pulse shows an almost square Vol.
EMSLEY
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BODENHAUSEN
a
X
c
I
I X
FIG. 5. Two-dimensional spatial selectivity of 90” and 270” Gaussian pulses. The plots represent the M, profiles obtained as a function of the spatial axes X and Y if two of the “building blocks” shown on the left are applied in succession, with gradients directed along the X and Y directions. (a) shows the profile obtained by a 90’ Gaussian with predefocusing of the transverse magnetization, as used in SPARS experiments. (b) shows how attempting to dispense with the predefocusing step causes the resulting 1!4~ response to be distorted beyond recognition. By simply increasing the RF amplitude to obtain a 270” Gaussian pulse, however, we can transform the chaotic response of(b) into the excellent distribution of(c) because of the self-refocusing effect of 270” Gaussian pulses. Contour levels were plotted at ?5, 10,25, 50,75, and 90% of M” in each case.
A potential drawback to volume localization techniques is that if the AI, profile generated by the selective pulse does not have a desirable box-like shape, one may face breakthrough from regions outside the VOI. Our simulations show that in the
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time
‘P
time
‘P
9
270” Gaussian
0
.a
FIG. 6. The evolution of the phases of a family of magnetization vectors under 90” and 270” Gaussian pulses. The horizontal time axis extends between points where the Gaussians are truncated. The 90” pulse has a virtual focus at tr = 0.45t,, and the 270’ Gaussian has a virtual focus in the vicinity of tr = OX%,. Note the reduced phase dispersion at the end of the 270” Gaussian pulse. The vectors are at reduced frequency intervals ( 12) of 0.058, and the pulses both have 2.5% truncation. The phases (excursion from the +x axis toward the +y axis) start at cp = 0 for the 90” pulse, and rp = - rr, i.e., at the -x axis, for the 270” pulse.
case where the spin density of the object under investigation, or the density of a particular type of spin in the object, is represented by a continuum, the contribution to the signal from outside the VOI (as defined by the slice thickness Ar, see Fig. 4e) after a self-refocusing 270” Gaussian pulse is only about 0.02% of the signal from within the
10
EMSLEY
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VOI. It is also interesting to note that of this negligible breakthrough is from within a distance Ar around the VOI. COMPENSATING
FOR
THE
EFFECT
OF
FINITE
GRADIENT
signal, over 90%
SWITCHING
TIMES
At first sight it would seem that a major obstacle to the implementation of selfrefocusing 270” Gaussian pulses is that the gradients cannot be switched on or off instantaneously. If we consider the conventional predefocusing sequence of Fig. 3c we may note that it has a certain degree of freedom to “tweek” the length of the predefocusing delay so as to compensate for finite gradient switching times. In contrast, a self-refocusing 270” pulse applied according to Fig. 3a does not have that degree of freedom. In fact, assuming that the magnetization is precisely aligned along the x axis before the gradient is activated, substantial phase errors will build up in the interval tp, thereby causing distortions in the resulting profile. Thus the simulations of the profiles calculated for Figs. 4e and 5c would seem to be nothing but pie in the sky, since they ignore the effects of this delay by assuming instantaneous gradient switching. The situation might, however, be improved if the gradient is switched within the duration of the RF pulse, in the manner of Fig. 3d. It is realistic to assume that the switching delay tg could be as much as 20% of the RF pulse length t,. Figure 7b shows the same M, profile as Fig. 4e, obtained with a 270” Gaussian pulse and instantaneous gradient switching. Figure 7d shows how the profile is distorted if we use the same RF pulse, but a field gradient that is linearly ramped up from t = 0 to 0.2t, and similarly ramped down from t = 0.8 to 1.Ot,,. Clearly, the distortions are severe, and it becomes necessary to devise a way of compensating for these effects. Toward this end we consider the consequences of gradient switching during the pulse on the time dependence of the instantaneous tilt angle 0. One of the intrinsic properties of a Gaussian pulse in the absence of gradient switching is that the effective field is aligned close to the z axis during the early part of the pulse, gradually moves toward the transverse plane as B, increases, and then moves back toward the z axis in the latter part of the pulse. This leads to so-called “teardrop” trajectories ( Z7) for magnetization vectors having large offsets. The behavior of the tilt angle t9(t) (excursion with respect to the z axis) is plotted in Fig. 7a for a 270” Gaussian pulse in the presence of a constant field. This results in the undistorted profile of Fig. 7b. If we now consider the behavior of 0( t) for a pulse in which the gradient is slowly ramped up, we can appreciate that at the very beginning of the RF pulse, in stark contrast to the previous situation, there is no z component of the effective field but there is a nonvanishing B,; therefore 0 (t = 0) is 7r/ 2. Figure 7c shows how the behavior of 8( t) is distorted throughout the gradient switching period. Clearly this will lead to severe distortion of the trajectories, so that the degraded profile of Fig. 7d is not surprising. To eliminate the effect of gradient switching on the excitation profile it is necessary to match the behavior of 0( t) during a sequence where tg Z 0 as far as possible to the behavior in the idealized case of Fig. 7a. The most obvious method would be simply to weight the pulse shape with the profile of the gradient, as 0 will remain constant if a reduction in the gradient amplitude is accompanied by a corresponding reduction in the RF amplitude. However, the trajectories will still be distorted by the ensuing
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b
2
-1
0
1
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d H
FIG. 7. Time dependence of the tilt angle 0 of the effective field for a vector at a reduced offset ( 12) of 0.058 during a 270” Gaussian pulse with 2.5% truncation. Ifthe field gradient is assumed to be instantaneously switched on and off at t = 0 and t,, one obtains 0(t) as shown in (a), and the nearly ideal M,(r) profile of(b). If the same RF pulse is used in a sequence where the gradient is being linearly ramped up from t = 0 to 0.22, and similarly ramped down from t = 0.8 to t, (t,/t, = 20%), the behavior of 0(t) is that of(c) with the degraded M,(r) profile of(d). If the same gradient timing with t.J t, = 20% is used with a 270” Gaussian pulse having 0.28% truncation, one obtains the time dependence of 0 shown in (e), and the resulting M,( r) profile is that of(f), which, apart from a scaling factor of I .2, is restored to the nearly ideal profileof(
reduction in the magnitude of the effective field I& during gradient ramping. A more satisfactory approach is to modify the pulse envelope with the gradient envelope and also allow time for a larger rotation at the beginning and end of the pulse where & is reduced. This can be achieved very simply by reducing the truncation of the Gaussian function. Figure 7e shows 19(t) for a 270” Gaussian pulse with 0.28% truncation, as opposed to 2.5% in Fig. 7c, and it can be seen that it is very similar to the ideal behavior of Fig. 7a, save for the fact that 0 tends to 7r/2 at the very beginning and end of the pulse. This small deviation is not important as Beff is so low at these points as to have little effect, and indeed we see that the profile of Fig. 7f is exactly the same as that of Fig. 7b, apart from a trivial scaling of the slice thickness caused by the slightly higher RF
12
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AND
BODENHAUSEN
amplitude required for a pulse with lower truncation. The scaling factor is about 1.2 for the case where tp/tp = 0.2, and the peak RF amplitude must be increased by a factor 1.24. Thus by simply reducing the truncation level of a self-refocusing 270 Gaussian pulse it is possible to compensate almost completely for the effects of gradient switching. The simulations of Figs. 7c-7f are all for a situation where the gradient switching time t, is 0.2t,. To restore a “pseudo 2.5%” profile one should employ truncation levels of 1.05,0.28, and 0.03% for t.Jtp = 0.1,0.2, and 0.3, respectively. Above tJt, = 0.35 it is unfortunately impossible to restore a satisfactory profile simply by reducing the level of truncation. APPLICATIONS
OF SELF-REFOCUSING TO SPECTROSCOPIC
270” GAUSSIAN SEQUENCES
PULSES
From the above discussion it becomes clear that 270” Gaussian pulses without refocusing have a VOI that is at least as well defined as that of a 90” Gaussian with refocusing. Furthermore, sequences employing 270” Gaussian pulses are significantly less sensitive to flow or T, relaxation effects. In SPARS or SPACE type experiments, we suggest that self-refocusing 270” Gaussian pulses be used in place of the 90” selective pulses. In the SPACE experiment of Fig. la the refocusing gradient and the nonselective 180” pulse (which is a very demanding feature of the experiment) can be dispensed with, and the sequence becomes the (self-refocusing) SR-SPACE sequence of Fig. 8a. Similarly, in the SPARS experiment of Fig. 1b the need for the predefocusing gradient and the nonselective 180” pulse is obviated, and the sequence simplifies to that of our self-refocusing SPARS sequence (SR-SPARS) of Fig. 8b. These modifications should both improve the z magnetization profile and enhance the sensitivity by reducing the time spent in the transverse plane. If self-refocusing 270” Gaussian pulses are used to convert A4, into MXY, there is a small MY component that is antisymmetric with respect to offset, which may perturb slice selection in inhomogeneous objects (13). The self-refocusing SPARS experiment is not affected by this problem, because at each stage of the experiment the unwanted MY components are rendered inobservable by dephasing. In the original VEST sequence of Fig. 2 the relevant magnetization is in the transverse plane during the delay 7, which must be at least as long as the time required for the gradient pulses. In our SR-VEST method of Fig. 9, where 270” Gaussian pulses replace the usual 90” pulses, there is no need for refocusing or predefocusing gradient pulses, so that T, may be essentially zero and the sequence becomes largely insensitive to T2 _In the r2 delay the selected magnetization is aligned along the z axis; since this delay may also be zero in the SR experiment, the sequence is also largely insensitive to T, . The price to pay for these advantages is that it may be necessary to use more elaborate phase cycles. In order to obtain proper localization in the SR-VEST experiment, the first pulse must be alternated in phase together with the receiver, so as to ensure suppression of the free induction decay after the third pulse. Also, in order to suppress pathways other than the one leading to the stimulated echo (see coherence transfer pathways of Fig. 2b) one may insert a r2 interval between the second and the third pulses without signal losses due to T2 relaxation, and extend the GY and Gz
VOLUME
SELECTION
WITH
SELF-REFOCUSING
PULSES
a: SR-SPACE
GX GY
3
GZ
k
90”
b: SR-SPARS
-
270’
270”
270”
RF
Gx
FIG. 8. Pulse and gradient scheme There is no need for any refocusing verse magnetization decays.
suitable for (a) self-refocusing SPACE and (b) self-refocusing gradients, and hence there are no unnecessary delays where
SPARS. the trans-
gradients into this r2 interval in order to dephase unwanted coherences (8). Alternatively, one may employ a four-step phase cycle. EFFECTS
OF
RF INHOMOGENEITY
We have shown both theoretically and by experiment (12, 13) that the shape of the magnetization profile generated by 270” Gaussian pulses is fairly tolerant to RF inhomogeneity. Nevertheless, if the sequences of Figs. 8 or 9 were to be blindly implemented with all RF pulses having the same phase, the total flip angles experienced by the magnetization in the VOI would be 1170” and 8 lo”, respectively, and the effects of RF inhomogeneity would be cumulative. However, if the SR-VEST sequence is modified so that the phases of the three pulses in Fig. 9 are +y, y, and -y, the total flip angle is reduced to an acceptable level of +270”. This prescription is compatible with the phase cycling required for suppression of the free induction decay following the last pulse, but is incompatible with the full phase-cycling requirements for unam-
14
EMSLEY
SR-VEST
2700
AND
270”
BODENHAUSEN
270”
::: GyY Gz FIG, 9. Pulse and gradient scheme sary delays where the magnetization
suitable for self-refocusing decays through transverse
VEST (SR-VEST). relaxation.
There
are no unneces-
biguous selection of the coherence transfer pathways, and in the case where RF inhomogeneity is a problem, suppression of other pathways must be achieved by applying gradients in the 72 interval. In the SR-SPARS sequence of Fig. 8b on the other hand, where each building block of the sequence contains two pulses, we suggest that the following phases be used: 90;,270",
90",270",,
90",,270",
90;,.
This reduces the total flip angle in the VOI to 90” and since there are no phase-cycling requirements in SR-SPARS there should be no obstacle to the implementation of these phases. EFFECTS
OF
TRANSVERSE
RELAXATION
It is possible to quantify the effects of relaxation on the original sequences of Figs. 1 and 2 and our self-refocusing sequences of Figs. 8 and 9 by including a simple monoexponential T2 damping in the simulations of the Bloch equations. We neglect the effect of longitudinal T, relaxation. For clarity we compare only Gaussian pulses having the same duration between truncation points, so that we do not take into account the fact that, to obtain the same slice thickness, a 270’ Gaussian must be 37% shorter than the corresponding 90” Gaussian, which will clearly benefit the relative performance of the 270” Gaussian in the presence of relaxation. Our simulations show that if the sequence with gradient reversal of Fig. 3b is used on a system with a T2 relaxation time equal to the length t, of the (truncated) 90” Gaussian pulse, the signal intensity will be attenuated to a mere 30%. In contrast, if the sequence of Fig. 3a is used with the same relaxation time and a self-refocusing 270” pulse, as much as 60% of the signal remains. Thus the elimination of the gradient reversal step yields an improvement in sensitivity by a factor of 2. Although we have included a monoexponential damping in the Bloch equations, the decay of the magnetization at a given offset during the pulse is generally multiexponential. Also, for a given pulse length, each offset will have a different trajectory, and therefore each offset will relax in a slightly different way to another, which leads to a slight distortion of the resulting
VOLUME
SELECTION
WITH
SELF-REFOCUSING
15
PULSES
s/s,
0.0
1.0
2.0
3.0
4.0
5.0
VT2
$/T2 FIG. 10. Plots of the relative signal intensity S/S,, against the dimensionless ratio tP/ T2, where lr, is the duration of the Gaussian pulses, and TZ the transverse relaxation time. (a) Dash-dotted line: 90’ Gaussian pulse with the scenario of Fig. 5a; short dashes: 270” Gaussian pulse with the self-refocusing scenario of Fig. 5c. The dimensionless decay constants (“relaxation coefficients”) of these curves are k = 1.1 and 0.43, respectively. In (b) we show the attenuation of several magnetization vectors at different offsets undergoing the scenario of Fig. 5c as a function of tP/ T,, in order to demonstrate the multiexponential behavior of the effective relaxation. The vectors are at reduced offsets ( 12) of 0 (solid line), 0. I5 (long dashes), 0.23 (medium dashes), 0.35 (dash dots), 0.40 (dots), and 0.42 (short dashes).
magnetization profile because of transverse relaxation. Fortunately, for a 270” Gaussian pulse this distortion is so small as to be negligible. A useful way to quantitate the effects of transverse relaxation on various pulse sequences is to plot the signal intensity as a function of the dimensionless ratio tP/ T2 (see Fig. 10). Including t, in the numerator simply normalizes the resulting curves (to first order) with respect to selectivity. Relaxation leads to a distortion of the magnetization trajectories which is dependent on tP/ T,, and therefore each offset experi-
16
EMSLEY
AND
BODENHAUSEN
ences a multiexponential decay, not only during the pulse, but also as a function of the ratio tP/ Tz as demonstrated in Fig. lob. It can be seen from the plot of Fig. 10a that the attenuation of the integrated signal over the slice is almostmonoexponential for Gaussian pulses applied according to the schemes of Fig. 3. This suggests that the decay of a given experiment can be described approximately by (S/So>; e exp 1 -kit,/
Tz).
[31
The dimensionless decay constant ki, which we call the “relaxation coefficient,” serves as a good measure of the sensitivity of a sequence to transverse relaxation. Indeed it can be given a physical interpretation in that a value of k, = 1 would indicate that the magnetization is attenuated by the same amount as if it were simply residing in the transverse plane for the duration t,. The smaller the value of k, the less sensitive the sequence is to relaxation, and the better the performance. The relaxation coefficients of the curves obtained in Fig. 1Oa are ki = 1.1 for the sequences with a 90 Gaussian pulse of Figs. 3b or 3c and ki = 0.43 for sequences involving a 270” Gaussian pulse according to Figs. 3a or 3d. It is a simple matter to extend this analysis to the three-dimensional localized spectroscopic sequences considered in this paper, since all such sequences essentially consist of three consecutive building blocks. Therefore (S/&h
x exp { -3kitpl
Tz >2
[41
and the relaxation coefficients of the three-dimensional experiments are thus simply three times those for the one-dimensional building blocks; i.e., 3ki = 3.3 for the normal SPARS and VEST sequences, and 3ki = 1.3 for the corresponding SR sequences. This clearly shows how the self-refocusing sequences are much less sensitive to relaxation than the gradient refocused equivalents. We have not explicitly considered the effect of relaxation on the stimulated echo sequences including 7, delays longer than the time required for gradient switching, but it is clear that increasing pi simply increases the sensitivity to relaxation and this can be described by modifying Eq. [ 41 to (S/So),i-exp{(-3kit,+27,)/T~},
[51
with7, = 7, - 2t,, where t, is the time required for gradient predefocusing or refocusing, and the factor 2 in Eq. [ 51 is introduced to account for the fact that the magnetization is in the transverse plane during two T, delays if acquisition is delayed until the top of the stimulated echo. CONCLUSIONS
We have shown how self-refocusing 270” Gaussian pulses can dramatically reduce the sensitivity of localized spectroscopic pulse sequences to T2 relaxation by significantly reducing their duration. We have also shown that the volume of interest generated by 270” Gaussian pulses is well defined and that the experiments should be insensitive to breakthrough signal from surrounding tissue. The new self-refocusing sequences of Figs. 8 and 9 will be less sensitive to the effects of flow or diffusion in the sample. We see no reason why these sequences should not replace the original
VOLUME
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WITH
SELF-REFOCUSING
17
PULSES
sequences of Figs. 1 and 2, especially since the new sequences are more straightforward to implement than their original counterparts. We believe that self-refocusing 270” Gaussian pulses will be of particular importance to sequences such as SPARS and VEST which are designed for implementation on whole-body NMR imaging systems, and for that reason we have highlighted those sequences here. However, the principle of substituting self-refocusing 270” Gaussian pulses for 90” Gaussian pulses will be valid in many of the various pulse sequences which have been proposed for spatially localized spectroscopy. ACKNOWLEDGMENTS Many routines.
thanks go to Urs Eggenberger and Peter Pftindler for assistance This work was supported in part by the Swiss National Science
with the high-resolution Foundation.
graphics
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in