MAGNETICRESONANCE I N MEDICINE18,28-38 ( 1991)
A Reduced Power Selective Adiabatic Spin-Echo Pulse Sequence STEVENCONOLLY, GARYGLOVER, * DWIGHTNISHIMURA, AND ALBERTMACOVSKI Department ofElectrica1 Engineering, Stanford University, Durand 120, Stanford, California 94305: and *GE Medical Systems Group Received December 19, 1989, revised April 6 , 1990 We introduce a selective adiabatic pulse sequence suitable for generating selective spinechoes for both MR imaging and spectroscopy. The technique is simple; one uses the echo generated by any pair of identical selectiveadiabatic inversion pulses. The nonlinear phase across the slice is compensated perfectly by the second T pulse. This compensation is immune to RF inhomogeneity and nonlinearity. For imaging applications, we concentrate on a reduced-power version of the pulse sequence in which time is traded off variably for RF amplitude in the presence of a time-varying gradient. This technique, known as variablerate excitation, mildly degrades the off-resonant slice profile when applied to amplitudemodulated pulses. We present theoretical explanations and experimental results that show that the variable-rate adiabatic pulses are immune to off-resonant degradation of the rnagnitude normally encountered in MR imaging. 0 1991 Academic Press, Inc. INTRODUCTION
Selective adiabatic 7r pulses have been applied to inverting slices in MR imaging for several years ( 1) . Adiabatic fast passage ( AFP) pulses’ immunity to RF variations ( 2 - 4 ) makes the pulses attractive for selective spin-echo experiments in MR imaging and spectroscopy. Conventional AFP 7r pulses present two problems with respect to spin-echo generation for MR imaging: they deposit relatively high power in the patient, and they leave a nonlinear phase variation across the excited slice. Recent research has seen the development of compensated 7r pulses that mitigate these problems. Kunz ( 5 ) compensated the phase of the 7r pulse with the 7r/2 pulse. Although this solution is reasonable in terms of RF power, the phase compensation varies with any RF field variations because the 7r/2 pulse is not adiabatic. We constructed a selective spin-echo pulse by prefacing an inversion pulse with a selective 27r pulse ( 6 ) .However, this 37r pulse deposits four times the power of a conventional inversion pulse, which precludes clinical use. Ugurbil et al. introduced four AFP spinecho pulses in ( 7), but none of these pulses is suitable for selective spin-echo generation in multislice MR imaging. In this paper, we introduce a spin-echo pulse sequence that addresses both the power and phase problems. The pulse sequence generates a selective spin-echo with full immunity to RF variations while depositing the same power as a conventional ir pulse. Furthermore, our pulse sequence is suitable for multislice magnetic resonance imaging. 0740-3194/9 I $3.00 Copynghl 0 1991 by Academic Press, Inc. All nghts of reproduction in any form reserved.
28
SELECTIVE ADIABATIC PULSE SEQUENCE
29
The technique is simple; one uses the second echo generated by any identical pair of selective adiabatic inversion pulses. The nonlinear phase across the slice is compensated perfectly by the second a pulse. This compensation is immune to both RF inhomogeneity and nonlinearity. We concentrate on a reduced-power version of the pulse sequence in which time is traded off variably for RF amplitude in the presence of a time-varying gradient. This technique, known as Variable-rate selective excitation or VERSE ( 8 ) , degrades the off-resonant slice profile when applied to amplitudemodulated pulses. We present both theoretical and experimental proof of the fact that VERSE adiabatic pulses are immune to off-resonant degradation of the magnitude normally encountered in MR imaging. METHOD
The even echoes generated by any series of identical selective AFP a pulses are collected. Unlike the first echo, the slice profile of the second echo shows no phase variation across the excited slice. The second a pulse cancels the frequency-dependent phase variation perfectly, even in the presence of RF amplitude variations. This is important because, as explained in the next section, the phase variation from each of the a pulses is sensitive to amplitude variations. If the amplitude variation remains constant for the delay between the two a pulses then the compensation is perfect. To reduce the power of the pulse sequence, we use the method of variable-rate selective excitation (8, 9 ) . In this method, one dilates the time base of both the excitation waveforms-the RF and gradient. To preserve the on-resonant slice profile, we now show that if the instantaneous rate is halved at any time during the pulse, then the amplitude of both the RF and the gradient must be halved at that time as well.
Continuous-Time VERSE Because most AFP pulses are designed with analytic waveforms, it helps to consider a continuous-time version of variable-rate excitation, which has been studied in (10, 11). Let us define M ( t , z ) and B ( t , z ) to be the magnetization vector and the excitation vector respectively on t E (0, T ) .The Bloch equation is M(t)=
M ( t )X y B ( t , z),
where B ( t , z ) = ( B , J t ) ,B,,(t), G z ) ,and the dot denotes the time derivative. Further, let us define a time-dilation waveform, ~ ( t )and , let P ( t , z ) = M ( T ( t ) , z ) be the "dilated" magnetization vector. Of course, time must increase, so +( t ) > 0. We restrict the variable-rate waveform to be of the same duration as the constant-rate pulse: T( T ) = T . This implies that P( T , z ) = M ( T (T ) ,z ) = M ( T , z ) , i.e., identical slice profiles reached at different schedules. The dilated Bloch equation for P is P(t) = + ( t ) M ( T ( t ) ) =
M ( T )X ~ + B ( T )
=
P ( t )X yi(t)B(T(t)).
30 1.8
0.21
a
CONOLLY ET AL. ,
,
,
0:l
0:2
013
,
,
0:4
0:5
,
,
0:6 017
Normalizedtime
,
,
018
0'9
,
0.25
1
-O"'0
b
,
.
.
0.1 '
0.2 '
0.'3
.
0.4 '
I
,
,
,
.
0.5 '
0.'6
0.7 '
0.' 8
0.9 '
Normalizedlime
FIG. 1. (a) The constant-rate and variable-rate gradient waveforms. (b) The corresponding RF waveforms. Note that the VERSE RF pulse has less power.
Hence, the following excitation vectors produce identical slices: + ( t ) B ( 7 ( t ) ,Z ) B(t, z ) . This is the continuous-time variable-rate theorem. To restate it more concretely, the RF pulse B I( t ) in the presence of the static gradient G excites the same slice onresonance as the RF pulse +( t ) B ,( T( t ) ) in the presence of the time-varying gradient +(t )G .' The flexibilityafforded by the time-dilation waveform, 7( t ) , has proven useful for power reduction ( 8 ) and for excitation during the ramping of the gradient ( 12). Figure 1 shows plots of a sinc pulse and a VERSE facsimile. Note that the time-varying gradient, which is proportional to +( t ) , is low at the middle of the pulse and high at the edges. Using this strategy, one distributes the area of a pulse more uniformly over the duration of the pulse and thereby reduces the power of the pulse. Figure 2 shows the variable-rate AFP a pulse ( a sech/tanh pulse (3))used in our experiments and its time-varying slice-select gradient. The VERSE a pulse has 57% of the average power, and only 16% of the peak power relative to the constant-rate pulse. Of course, for off-resonance spins one cannot perform the time dilation properly because one cannot vary chemical shift, Bo inhomogeneity, etc. For amplitude-modulated pulses the degree of off-resonant slice degradation increases with the degree of power reduction. In the theory section we investigate the off-resonant behavior of VERSE adiabatic pulses. THEORY
In this section we analyze the nonlinear phase variation from an adiabatic a pulse and show that identical a pulses compensate this phase. We then analyze the offresonant mechanics for AFP variable-rate pulses.
Phase Variation and compensation Suppose one applies a selective P pulse immediately following a nonselective r / 2 pulse. After the first pulse, the magnetization vectors point in the y direction. The P More generally, the RF pulse E l ( t ) in the presence of the vector gradient G (t ) excites the same slice onresonance as the RF pulse i ( t ) B , ( ~ ( tin) )the presence of the gradient + ( t ) G ( ~ ( t ) ) .
SELECTIVE ADIABATIC PULSE SEQUENCE I
31
'
C
0 .T) 2 UJ
Pm U
K
.. . . ...
. .
. .
Gradient
0
0.1
0.2
0.3
0.4
0.5
0.6 0.7
0.8 0.9
Normalizedtime
FIG. 2. The variable-rate sech pulse and corresponding gradient waveform used in the experimental section.
pulse is equivalent, in a frame rotating with the RF excitation vector, to an excitation vector that sweeps from the north pole to the south pole in the xz plane. By the adiabatic theorem, spins initially perpendicular to the adiabatic sweep vector remain perpendicular. Clearly then, a spin pointing along the x axis would finally point along the -x axis after such a pulse, whereas a spin pointing along the y axis would finally be returned to the y axis. However, in addition to this phase conjugation, the perpendicular spins experience a position-dependent phase variation while following the excitation sweep. This phase variation has been studied in ( 5 , 7, 1 3 ) . As an example, consider a generic pulse with RF amplitude B 1 ( t )and frequency sweep f i ( t ) . This pulse, at amplitude A , and in a gradient field y Gz has the effective excitation trajectory yBedA, 2 , t ) = ( ~ - 4 B l ( t0, ) , rGz
+ Q(t)).
Because the perpendicular spins precess about the Betffield at the Larmor frequency, their spatial phase variation is given by
IN-4, z ) l
=
y
soT
IIBeff(A, z, 7)IldT.
Note that the phase of these spins will be sensitive to amplitude variations of the excitation vector. Any robust phase-compensation scheme must take this phase sensitivity into account. To compensate the phase of the first P pulse, we use an identical ?r pulse. It should be obvious that the second P pulse will produce the same magnitude of spatial phase variation. We now argue that because the spins' phase has been conjugated by the first r pulse, the second pulse cancels rather than doubles the phase. In effect, the spins are rotated counterclockwisefrom above by the first T pulse, and counterclockwise from below by the second P pulse. This compensation mechanism is completely analogous to the B,tf flip of ( I d ) , in that the instantaneous direction of the sweep vector
32
CONOLLY ET AL.
switches from -z to + z from the end of the first pulse to the beginning of the second pulse. A similar compensation mechanism was also introduced for designing a “compensated” 2a pulse in ( 1 3 ) . In that work, two selective a pulses were run with no delay in between, so that only the in-slice region would see a zero-degree flip. Of course, the spatial phase variation is completely compensated only in the in-slice region, because the out-of-slice spins are never flipped by the ?r pulses. If the local RF amplitude is constant during the delay between the pulses, then the phase is compensated with immunity to spatial RF variations. It is important to understand that this phase compensation depends only on the symmetry of the two T pulses’ sweep diagrams. Hence, two identical VERSE T pulses in the presence of a static off-resonance field will also compensate perfectly, provided that the sweeps are adiabatic. This symmetry property forms the theoretical foundation for this paper. Another way to understand this pulse sequence is more algebraic. Any pulse produces a rotation. We know that an adiabatic a pulse rotates spins a radians about some axis in the xy plane. The phase variation left by a a pulse indicates that this rotation axis depends on the spin’s position. An identical a pulse repeats this rotation. At each position, then, the rotations add because their axes of rotation are identical. Hence, every spin is rotated 2a radians about some axis in the x y plane. This 2a rotation has no net effect, of course, so no spatial dependence is observed. One need only ensure that proper delays are added to allow for observation of the second echo.
Of-Resonant Mechanics The constant-rate sweep in the FM frame takes the generic form yBen(A, Z , t ) = (./ABi(t), 0,
+ TGZ + Qt(t)),
where B1( t ) / Q t( ) could be sech/tanh or cos/sin or any other pair of waveforms that performs an adiabatic inversion, and Aw represents the local departure from resonance. The variable-rate sweep is ( y A i ( t ) B l ( ~0, ) ,AU
+ i ( t ) [ y G z+ Q ( T ) ] ) .
For concreteness, let us assume Q(0)= Qo and Q( T ) = - 00.The sweep inverts magnetization if the sign of the z component changes from the beginning to the end of the sweep. This happens where sign { Aw
+ i(0 ) (yGz + Qo)} = - sign { Aw + i( T ) (yGz - no)}.
For simplicity, let us consider the symmetric case: i(0) = i( T ) .Using the restriction i( t ) > 0, we obtain the inverted region -
80
+
< ~ G z Aw/;(O)
< Qo.
Note that VERSE does not alter the slice thickness. If we define the time-varying gradient g ( t ) = Gi(t ) , then off-resonant slice is shifted a distance Aw/yg(O). Hence, off-resonance fields cause a simple shift of the slice. We have observed secondary effects that arise from inadequate fulfillment of the adiabatic condition ( 1 5 ) especially at the edges of sweep. It is helpful to analyze this effect in the undilated time frame, where the static off-resonance field appears modulated.
SELECTIVE ADIABATIC PULSE SEQUENCE
33
The identity P( t ) = M( T ( t ) )can be written as M( t ) = P( T - ’ ( t ) ) ,where F1( t ) is the inverse of the function T( t ) . The existence of ~ - ‘ ( tis) guaranteed by the condition + ( t )> 0. The dilated Bloch equation with an off-resonance field A u is
+
P = P X [ ; ( t ) y B ( ~ ) Auk]. Returning to the original time frame, we have M ( t ) = T1’(t)P(7-’(t)) = T-’(t)P(T-’(t)) X
+
[ i ( ~ - ’ ( t ) ) ~ B ( tAuk] )
where the last line follows from the relation -dT-I(
t)-
1
dt +(7-I ( t ) )’ which can be derived by implicit differentiation of the identity 7-’( T( t ) ) = t . Hence, variable-rate excitation in an off-resonance field Aw is equivalent to constant-rate excitation in the presence of the time-varying inhomogeneity Am/+( T - ’ ) . A qualitative analysis of the shape of this function illuminatesthe secondary off-resonant degradation. Figure 3a shows a typical gradient waveform used for power reduction in variablerate excitation. Figure 3b shows the associated dilation waveform T ( t ) and its inverse 7-’( t ) .Note that the slope of T ( t ) is small at the center of the pulse, where one wishes to reduce the amplitude the most. Figure 3c shows the shape of the inhomogeneity modulation waveform 1/i(T - ’ ) . Note that the “DC” shift is 1/ i (0) = 0.7, consistent with the analysis above and with the initial amplitude of the gradient. The peak shift is inversely proportional to the gradient modulation at the center of the pulse: 1/+( .5 ) x 1.75 in the figure. Hence, the “AC” modulation has amplitude A = 1/+( .5) - 1 / +(O). For the example above, A = 1. That is, the degree of variation between the inverse gradient at minimum and peak of the gradient determines the amplitude of off-resonance field modulation; the more power reduction, the more modulation. The constant-rate sweep suffers a modulation to its z component that is maximal at the center of the pulse and of magnitude AAw. If this z modulation is smaller than the bandwidth of the pulse, then the sweep trajectory remains adiabatic. As a rule of thumb, we should expect to see a degradation in the adiabatic character of the sweep for AAw greater than about Do. This rule is useful because it specifies the maximum allowable power reduction by VERSE for a given pulse bandwidth and a desired spectral bandwidth. Figure 4a shows three sech/tanh sweep diagrams ( 1 3 )in the undilated time frame for the gradient waveform shown in Fig. 3a, for which A = 1. Note that the sweep diagram becomes badly warped for A u / O o > 1. The sweep diagrams indicate that the degradation of adiabatic character should be asymmetric. Indeed, a comparison of the “adiabaticity ratio,” 11 Be*11 /8,as a function of time indicates that the side of the slice shifted toward the origin (the right side if shifted to the left) should suffer a preferential loss of adiabaticity. See Fig. 4b for these plots. In the next section we present experiments that confirm this prediction.
34
CONOLLY ET AL.
1 . 5 , .
+-
.
.
.
,
,
,
,
,
I
1.4-
0.9-
1.3-
0.8-
5 1.2.-
._ 6 0.6-
1~
0.5-
z.-s
a
0.7-
1.1 -
0.9-
g 0.8 0.7 0.6 0.5i
a
'
0.1
.
0.2
'
0.3
'
0.4
.
0.5
.
0.6
'
0.7
'
0.8
'
0.9
I
1
b
Normalizedtime
0.6;
'
0.1
C
1
0.2
'
0.3
'
0.4
1
0.5
Normalized time
0.6
07
'
I
0.8 0.9
Normalizedtime
FIG.3. ( a ) A typical gradient waveform used for power reduction in a variable-rate excitation format. ) ~ - ' ( tfor ) the gradient waveform shown in Fig. 3a. (c) A plot of the off-resonant ( b ) Plots of ~ ( tand modulation function I /+(T-') for the gradient shown in Fig. 3a.
2
b 15
x componentof excitation vector
FIG.4.( a ) Sweep diagrams for sech/tanh pulses with the off-resonant modulation function from Fig. 3c added to the frequency sweep. The plots are parameterized by the ratio r = Aw/Qo, where Q, = 1. (b) Plots of the adiabaticity ratio as a function of time on the right side and left side of the slice. The shift is toward the right. Note that the left side suffers a preferential loss of adiabaticity.
35
SELECTIVE ADIABATIC PULSE SEQUENCE
RFI
RFQ
Gz
0
5
10
15
20
25
30
35
40
45
50
Time (ms) FIG. 5. Timing diagram for experimental test of the spin-echo pulse sequence. Readout is in the sliceselect direction, and the r / 2 pulse is nonselective. The r pulses are 5.376 ms in duration and have a bandwidth of 2.6 kHz.
EXPERIMENTAL RESULTS
Figure 5 shows the pulse sequence used to test the spin-echo pulse sequence on our 1.5-T General Electric Signa system. The echo was read out in the direction of slice selection. Both the first and second echoes were collected for comparison. The ~ / 2 pulse was nonselective. The duration of the adiabatic K pulse was 5.376 ms and the bandwidth was 2.6 kHz. Above a threshold amplitude, the response remained fixed. Figure 6 compares phantom results for the first and second echoes on-resonance. Note the flatness of the phase profile of the second echo, and that the magnitude of the first echo is smaller than the second echo. This signal loss manifests the dependence
': Second echo
i
U
._
b-'5\ Normalized position
-0.8 -0.6 -0.4 -0.2
0
012 014
0:6
018
!
Normalized position
FIG. 6. ( a) Comparison of the experimental spin-echo magnitude slice profile for the first and second echoes. Note the signal loss of the first echo due to RF inhomogeneity. ( b ) Comparison of the experimental spin-echo phase slice profile for the first and second echoes. Note that the phase of the second echo is flat.
36
CONOLLY ET AL.
of the phase variation on RF field strength. That is, the magnitude is decreased by projection through the distribution of phase angles in the plane of the slice. The magnitude slice profile on resonance and 1 kHz off resonance are shown in Fig. 7a, and the corresponding phase profile is shown in Fig. 7b. The RF transmitter was used to create the off-resonance condition. Note that 1 kHz off-resonance is close to the condition AAw = O o , since 00 = 1.3 kHz and that it far exceeds the normal spectral bandwidth needed for imaging at 1.5 T, where the water-fat separation is about 250 Hz. The off-resonant slice is probably acceptable for imaging. Also note that the asymmetry of the off-resonant slice profile agrees with the qualitative sweepdiagram analysis from above. DISCUSSION
One drawback with the sequence as described is that the minimum echo-time (TE ) is greater than that for standard spin-echo imaging, making the imaging of short-T2 species difficult. It is worth noting that a small modification remedies this problem for single-echo experiments. The first a pulse can be started immediately after the conclusion of the 1 / 2 pulse, because one does not need the full echo from the first a pulse. The refocusing and phase-encode can then be impressed during the half-echo after the first a pulse. During the a pulses, the spins do not experience much T2decay because they move quickly around the plane perpendicular to the sweep vector as they are being swept about the x axis. Using this scheme, the spins remain in the transverse plane for roughly the same amount of time as for a standard spin-echo experiment. This modified pulse sequence is shown in Fig. 8a. To demonstrate the feasibility of the pulse sequence, we include an axial head image of a healthy volunteer in Fig. 8b, obtained on a 1.5-T GE Signa system. There are many clinical procedures that could benefit from near-perfect spin-echoes. A multiecho version of this technique might prove helpful for quantitative T 2determination. Majumdar et al. (16) have shown that RF pulse imperfections undermine measurements of T2. However, it seems difficult to generalize the short first-echo experiment described above to a multiecho experiment.
3 c
P a
Ij
-10
-12
Position in arb. units
,
,
,
,
,
Position in arb. units
,
,
,
\, I
FIG. 7. ( a ) Comparison of the experimental spin-echo magnitude profile on-resonance and 1 kHz offresonance. (b) Comparison of the experimental spin-echo phase profile on-resonance and 1 lcHz off-resonance.
SELECTIVE ADIABATIC PULSE SEQUENCE
37
daql===l ub
20.0 25.0 30.0 35.0 40.0 45.0 50.0 55.0 60.0 65.0
a
Time (ms)
FIG. 8. ( a ) Pulse sequence that reduces the time spent in the transverse plane to about the same time as in a standard 2D FT experiment. (b) Experimental image of a healthy volunteer obtained with the pulse sequence shown in Fig. 8a on a 1.5-T GE Signa system. No effort was made to optimize the contrast; this image is intended to demonstrate only the viability of the proposed pulse sequence on a current MR scanner.
Another important result ofthis paper is that all spatially selective adiabatic T pulses should now be applied in a variable-rate format because the only drawback-offresonant slice degradation-is negligible. SUMMARY
The even echoes from any sequence of identical selective adiabatic r pulses are exactly phase compensated with immunity to spatial RF variations. The variable-rate version of the spin-echo sequence deposits far less power in the patient and produces no degradation of the off-resonant slice profile. This selective spin-echo pulse sequence deposits relatively low power, it offers immunity to RF inhomogeneity and nonlinearity, and it produces an excellent slice profile that would enable contiguous slice acquisition. ACKNOWLEDGMENTS This work was supported in part by National Institutes of Health Grants HL-39478; HL-39297, HV38045, and HL-34962. In addition, the authors thank members of the GE Medical Systems Group for their help and support. Finally, the authors appreciate the comments from our second reviewer. REFERENCES 1. J. BAUM,R. TYCKO,AND A. PINES,J. Chem. Phys. 79,4643 ( 1983). 2. F. HIOE,Phys. Rev. A 30, 2100 (1984). 3. M. SILVER,R. JOSEPH,AND D. HOULT,Phys. Rev.A 31,2753 ( 1985). 4. J. BAUM,R. TYCKO,AND A. PINES,Phys. Rev.A 32, 3435 ( 1985). 5. D. KUNZ,Magn. Reson. Med. 4, 129 (1987).
38
CONOLLY ET AL.
S. CONOLLY, D. NISHIMURA, AND A. MACOVSKI, J. Magn. Reson. 83, 324 ( 1989). K. U ~ U R B I M. L , GARWOOD, A. RATH, AND M. R. BENDALL,J. Magn. Reson. 78,472 ( 1988). S. CONOLLY, D. NISHIMURA, A. MACOVSKI, AND G. GLOVER,J. Magn. Reson. 78,440 ( 1988). S. CONOLLY, U.S. Patent 4,760,336 (1988). 10. S. CONOLLY,“Magnetic Resonance Selective Excitation,” Ph.D. thesis, Stanford University, 1989. 11. D. LEWIS,B. TsuI, P. MORAN,AND D. SALONER, in “Society of Magnetic Resonance in Medicine 8th Annual Meeting,” p. 27, 1989. 12. J. PAULY,S. CONOLLY, D. NISHIMURA, AND A. MACOVSKI, J. Magn. Reson., in press. 13. S. CONOLLY, D. NISHIMURA, AND A. MACOVSKI, J. Mugn. Reson. 83, 549 ( 1989). 14. M. BENDALL, M. GARWOOD, K. UdURBIL, AND D. PEGC, Magn. Reson. Med. 4,493 ( 1987). 15. A. ABRAGAM, “The Principles of Nuclear Magnetism,” Oxford Univ. Press, London, 1961. 16. S. MAJUMDAR, S. OPRHANOUDAKIS, A. GMITRO,M. ODONNELL, AND J. GORE,M a p . Reson. Med. 6. 7. 8. 9.
3,397 (1986).
A Reduced Power Selective Adiabatic Spin-Echo Pulse Sequence STEVENCONOLLY, GARYGLOVER, * DWIGHTNISHIMURA, AND ALBERTMACOVSKI Department ofElectrica1 Engineering, Stanford University, Durand 120, Stanford, California 94305: and *GE Medical Systems Group Received December 19, 1989, revised April 6 , 1990 We introduce a selective adiabatic pulse sequence suitable for generating selective spinechoes for both MR imaging and spectroscopy. The technique is simple; one uses the echo generated by any pair of identical selectiveadiabatic inversion pulses. The nonlinear phase across the slice is compensated perfectly by the second T pulse. This compensation is immune to RF inhomogeneity and nonlinearity. For imaging applications, we concentrate on a reduced-power version of the pulse sequence in which time is traded off variably for RF amplitude in the presence of a time-varying gradient. This technique, known as variablerate excitation, mildly degrades the off-resonant slice profile when applied to amplitudemodulated pulses. We present theoretical explanations and experimental results that show that the variable-rate adiabatic pulses are immune to off-resonant degradation of the rnagnitude normally encountered in MR imaging. 0 1991 Academic Press, Inc. INTRODUCTION
Selective adiabatic 7r pulses have been applied to inverting slices in MR imaging for several years ( 1) . Adiabatic fast passage ( AFP) pulses’ immunity to RF variations ( 2 - 4 ) makes the pulses attractive for selective spin-echo experiments in MR imaging and spectroscopy. Conventional AFP 7r pulses present two problems with respect to spin-echo generation for MR imaging: they deposit relatively high power in the patient, and they leave a nonlinear phase variation across the excited slice. Recent research has seen the development of compensated 7r pulses that mitigate these problems. Kunz ( 5 ) compensated the phase of the 7r pulse with the 7r/2 pulse. Although this solution is reasonable in terms of RF power, the phase compensation varies with any RF field variations because the 7r/2 pulse is not adiabatic. We constructed a selective spin-echo pulse by prefacing an inversion pulse with a selective 27r pulse ( 6 ) .However, this 37r pulse deposits four times the power of a conventional inversion pulse, which precludes clinical use. Ugurbil et al. introduced four AFP spinecho pulses in ( 7), but none of these pulses is suitable for selective spin-echo generation in multislice MR imaging. In this paper, we introduce a spin-echo pulse sequence that addresses both the power and phase problems. The pulse sequence generates a selective spin-echo with full immunity to RF variations while depositing the same power as a conventional ir pulse. Furthermore, our pulse sequence is suitable for multislice magnetic resonance imaging. 0740-3194/9 I $3.00 Copynghl 0 1991 by Academic Press, Inc. All nghts of reproduction in any form reserved.
28
SELECTIVE ADIABATIC PULSE SEQUENCE
29
The technique is simple; one uses the second echo generated by any identical pair of selective adiabatic inversion pulses. The nonlinear phase across the slice is compensated perfectly by the second a pulse. This compensation is immune to both RF inhomogeneity and nonlinearity. We concentrate on a reduced-power version of the pulse sequence in which time is traded off variably for RF amplitude in the presence of a time-varying gradient. This technique, known as Variable-rate selective excitation or VERSE ( 8 ) , degrades the off-resonant slice profile when applied to amplitudemodulated pulses. We present both theoretical and experimental proof of the fact that VERSE adiabatic pulses are immune to off-resonant degradation of the magnitude normally encountered in MR imaging. METHOD
The even echoes generated by any series of identical selective AFP a pulses are collected. Unlike the first echo, the slice profile of the second echo shows no phase variation across the excited slice. The second a pulse cancels the frequency-dependent phase variation perfectly, even in the presence of RF amplitude variations. This is important because, as explained in the next section, the phase variation from each of the a pulses is sensitive to amplitude variations. If the amplitude variation remains constant for the delay between the two a pulses then the compensation is perfect. To reduce the power of the pulse sequence, we use the method of variable-rate selective excitation (8, 9 ) . In this method, one dilates the time base of both the excitation waveforms-the RF and gradient. To preserve the on-resonant slice profile, we now show that if the instantaneous rate is halved at any time during the pulse, then the amplitude of both the RF and the gradient must be halved at that time as well.
Continuous-Time VERSE Because most AFP pulses are designed with analytic waveforms, it helps to consider a continuous-time version of variable-rate excitation, which has been studied in (10, 11). Let us define M ( t , z ) and B ( t , z ) to be the magnetization vector and the excitation vector respectively on t E (0, T ) .The Bloch equation is M(t)=
M ( t )X y B ( t , z),
where B ( t , z ) = ( B , J t ) ,B,,(t), G z ) ,and the dot denotes the time derivative. Further, let us define a time-dilation waveform, ~ ( t )and , let P ( t , z ) = M ( T ( t ) , z ) be the "dilated" magnetization vector. Of course, time must increase, so +( t ) > 0. We restrict the variable-rate waveform to be of the same duration as the constant-rate pulse: T( T ) = T . This implies that P( T , z ) = M ( T (T ) ,z ) = M ( T , z ) , i.e., identical slice profiles reached at different schedules. The dilated Bloch equation for P is P(t) = + ( t ) M ( T ( t ) ) =
M ( T )X ~ + B ( T )
=
P ( t )X yi(t)B(T(t)).
30 1.8
0.21
a
CONOLLY ET AL. ,
,
,
0:l
0:2
013
,
,
0:4
0:5
,
,
0:6 017
Normalizedtime
,
,
018
0'9
,
0.25
1
-O"'0
b
,
.
.
0.1 '
0.2 '
0.'3
.
0.4 '
I
,
,
,
.
0.5 '
0.'6
0.7 '
0.' 8
0.9 '
Normalizedlime
FIG. 1. (a) The constant-rate and variable-rate gradient waveforms. (b) The corresponding RF waveforms. Note that the VERSE RF pulse has less power.
Hence, the following excitation vectors produce identical slices: + ( t ) B ( 7 ( t ) ,Z ) B(t, z ) . This is the continuous-time variable-rate theorem. To restate it more concretely, the RF pulse B I( t ) in the presence of the static gradient G excites the same slice onresonance as the RF pulse +( t ) B ,( T( t ) ) in the presence of the time-varying gradient +(t )G .' The flexibilityafforded by the time-dilation waveform, 7( t ) , has proven useful for power reduction ( 8 ) and for excitation during the ramping of the gradient ( 12). Figure 1 shows plots of a sinc pulse and a VERSE facsimile. Note that the time-varying gradient, which is proportional to +( t ) , is low at the middle of the pulse and high at the edges. Using this strategy, one distributes the area of a pulse more uniformly over the duration of the pulse and thereby reduces the power of the pulse. Figure 2 shows the variable-rate AFP a pulse ( a sech/tanh pulse (3))used in our experiments and its time-varying slice-select gradient. The VERSE a pulse has 57% of the average power, and only 16% of the peak power relative to the constant-rate pulse. Of course, for off-resonance spins one cannot perform the time dilation properly because one cannot vary chemical shift, Bo inhomogeneity, etc. For amplitude-modulated pulses the degree of off-resonant slice degradation increases with the degree of power reduction. In the theory section we investigate the off-resonant behavior of VERSE adiabatic pulses. THEORY
In this section we analyze the nonlinear phase variation from an adiabatic a pulse and show that identical a pulses compensate this phase. We then analyze the offresonant mechanics for AFP variable-rate pulses.
Phase Variation and compensation Suppose one applies a selective P pulse immediately following a nonselective r / 2 pulse. After the first pulse, the magnetization vectors point in the y direction. The P More generally, the RF pulse E l ( t ) in the presence of the vector gradient G (t ) excites the same slice onresonance as the RF pulse i ( t ) B , ( ~ ( tin) )the presence of the gradient + ( t ) G ( ~ ( t ) ) .
SELECTIVE ADIABATIC PULSE SEQUENCE I
31
'
C
0 .T) 2 UJ
Pm U
K
.. . . ...
. .
. .
Gradient
0
0.1
0.2
0.3
0.4
0.5
0.6 0.7
0.8 0.9
Normalizedtime
FIG. 2. The variable-rate sech pulse and corresponding gradient waveform used in the experimental section.
pulse is equivalent, in a frame rotating with the RF excitation vector, to an excitation vector that sweeps from the north pole to the south pole in the xz plane. By the adiabatic theorem, spins initially perpendicular to the adiabatic sweep vector remain perpendicular. Clearly then, a spin pointing along the x axis would finally point along the -x axis after such a pulse, whereas a spin pointing along the y axis would finally be returned to the y axis. However, in addition to this phase conjugation, the perpendicular spins experience a position-dependent phase variation while following the excitation sweep. This phase variation has been studied in ( 5 , 7, 1 3 ) . As an example, consider a generic pulse with RF amplitude B 1 ( t )and frequency sweep f i ( t ) . This pulse, at amplitude A , and in a gradient field y Gz has the effective excitation trajectory yBedA, 2 , t ) = ( ~ - 4 B l ( t0, ) , rGz
+ Q(t)).
Because the perpendicular spins precess about the Betffield at the Larmor frequency, their spatial phase variation is given by
IN-4, z ) l
=
y
soT
IIBeff(A, z, 7)IldT.
Note that the phase of these spins will be sensitive to amplitude variations of the excitation vector. Any robust phase-compensation scheme must take this phase sensitivity into account. To compensate the phase of the first P pulse, we use an identical ?r pulse. It should be obvious that the second P pulse will produce the same magnitude of spatial phase variation. We now argue that because the spins' phase has been conjugated by the first r pulse, the second pulse cancels rather than doubles the phase. In effect, the spins are rotated counterclockwisefrom above by the first T pulse, and counterclockwise from below by the second P pulse. This compensation mechanism is completely analogous to the B,tf flip of ( I d ) , in that the instantaneous direction of the sweep vector
32
CONOLLY ET AL.
switches from -z to + z from the end of the first pulse to the beginning of the second pulse. A similar compensation mechanism was also introduced for designing a “compensated” 2a pulse in ( 1 3 ) . In that work, two selective a pulses were run with no delay in between, so that only the in-slice region would see a zero-degree flip. Of course, the spatial phase variation is completely compensated only in the in-slice region, because the out-of-slice spins are never flipped by the ?r pulses. If the local RF amplitude is constant during the delay between the pulses, then the phase is compensated with immunity to spatial RF variations. It is important to understand that this phase compensation depends only on the symmetry of the two T pulses’ sweep diagrams. Hence, two identical VERSE T pulses in the presence of a static off-resonance field will also compensate perfectly, provided that the sweeps are adiabatic. This symmetry property forms the theoretical foundation for this paper. Another way to understand this pulse sequence is more algebraic. Any pulse produces a rotation. We know that an adiabatic a pulse rotates spins a radians about some axis in the xy plane. The phase variation left by a a pulse indicates that this rotation axis depends on the spin’s position. An identical a pulse repeats this rotation. At each position, then, the rotations add because their axes of rotation are identical. Hence, every spin is rotated 2a radians about some axis in the x y plane. This 2a rotation has no net effect, of course, so no spatial dependence is observed. One need only ensure that proper delays are added to allow for observation of the second echo.
Of-Resonant Mechanics The constant-rate sweep in the FM frame takes the generic form yBen(A, Z , t ) = (./ABi(t), 0,
+ TGZ + Qt(t)),
where B1( t ) / Q t( ) could be sech/tanh or cos/sin or any other pair of waveforms that performs an adiabatic inversion, and Aw represents the local departure from resonance. The variable-rate sweep is ( y A i ( t ) B l ( ~0, ) ,AU
+ i ( t ) [ y G z+ Q ( T ) ] ) .
For concreteness, let us assume Q(0)= Qo and Q( T ) = - 00.The sweep inverts magnetization if the sign of the z component changes from the beginning to the end of the sweep. This happens where sign { Aw
+ i(0 ) (yGz + Qo)} = - sign { Aw + i( T ) (yGz - no)}.
For simplicity, let us consider the symmetric case: i(0) = i( T ) .Using the restriction i( t ) > 0, we obtain the inverted region -
80
+
< ~ G z Aw/;(O)
< Qo.
Note that VERSE does not alter the slice thickness. If we define the time-varying gradient g ( t ) = Gi(t ) , then off-resonant slice is shifted a distance Aw/yg(O). Hence, off-resonance fields cause a simple shift of the slice. We have observed secondary effects that arise from inadequate fulfillment of the adiabatic condition ( 1 5 ) especially at the edges of sweep. It is helpful to analyze this effect in the undilated time frame, where the static off-resonance field appears modulated.
SELECTIVE ADIABATIC PULSE SEQUENCE
33
The identity P( t ) = M( T ( t ) )can be written as M( t ) = P( T - ’ ( t ) ) ,where F1( t ) is the inverse of the function T( t ) . The existence of ~ - ‘ ( tis) guaranteed by the condition + ( t )> 0. The dilated Bloch equation with an off-resonance field A u is
+
P = P X [ ; ( t ) y B ( ~ ) Auk]. Returning to the original time frame, we have M ( t ) = T1’(t)P(7-’(t)) = T-’(t)P(T-’(t)) X
+
[ i ( ~ - ’ ( t ) ) ~ B ( tAuk] )
where the last line follows from the relation -dT-I(
t)-
1
dt +(7-I ( t ) )’ which can be derived by implicit differentiation of the identity 7-’( T( t ) ) = t . Hence, variable-rate excitation in an off-resonance field Aw is equivalent to constant-rate excitation in the presence of the time-varying inhomogeneity Am/+( T - ’ ) . A qualitative analysis of the shape of this function illuminatesthe secondary off-resonant degradation. Figure 3a shows a typical gradient waveform used for power reduction in variablerate excitation. Figure 3b shows the associated dilation waveform T ( t ) and its inverse 7-’( t ) .Note that the slope of T ( t ) is small at the center of the pulse, where one wishes to reduce the amplitude the most. Figure 3c shows the shape of the inhomogeneity modulation waveform 1/i(T - ’ ) . Note that the “DC” shift is 1/ i (0) = 0.7, consistent with the analysis above and with the initial amplitude of the gradient. The peak shift is inversely proportional to the gradient modulation at the center of the pulse: 1/+( .5 ) x 1.75 in the figure. Hence, the “AC” modulation has amplitude A = 1/+( .5) - 1 / +(O). For the example above, A = 1. That is, the degree of variation between the inverse gradient at minimum and peak of the gradient determines the amplitude of off-resonance field modulation; the more power reduction, the more modulation. The constant-rate sweep suffers a modulation to its z component that is maximal at the center of the pulse and of magnitude AAw. If this z modulation is smaller than the bandwidth of the pulse, then the sweep trajectory remains adiabatic. As a rule of thumb, we should expect to see a degradation in the adiabatic character of the sweep for AAw greater than about Do. This rule is useful because it specifies the maximum allowable power reduction by VERSE for a given pulse bandwidth and a desired spectral bandwidth. Figure 4a shows three sech/tanh sweep diagrams ( 1 3 )in the undilated time frame for the gradient waveform shown in Fig. 3a, for which A = 1. Note that the sweep diagram becomes badly warped for A u / O o > 1. The sweep diagrams indicate that the degradation of adiabatic character should be asymmetric. Indeed, a comparison of the “adiabaticity ratio,” 11 Be*11 /8,as a function of time indicates that the side of the slice shifted toward the origin (the right side if shifted to the left) should suffer a preferential loss of adiabaticity. See Fig. 4b for these plots. In the next section we present experiments that confirm this prediction.
34
CONOLLY ET AL.
1 . 5 , .
+-
.
.
.
,
,
,
,
,
I
1.4-
0.9-
1.3-
0.8-
5 1.2.-
._ 6 0.6-
1~
0.5-
z.-s
a
0.7-
1.1 -
0.9-
g 0.8 0.7 0.6 0.5i
a
'
0.1
.
0.2
'
0.3
'
0.4
.
0.5
.
0.6
'
0.7
'
0.8
'
0.9
I
1
b
Normalizedtime
0.6;
'
0.1
C
1
0.2
'
0.3
'
0.4
1
0.5
Normalized time
0.6
07
'
I
0.8 0.9
Normalizedtime
FIG.3. ( a ) A typical gradient waveform used for power reduction in a variable-rate excitation format. ) ~ - ' ( tfor ) the gradient waveform shown in Fig. 3a. (c) A plot of the off-resonant ( b ) Plots of ~ ( tand modulation function I /+(T-') for the gradient shown in Fig. 3a.
2
b 15
x componentof excitation vector
FIG.4.( a ) Sweep diagrams for sech/tanh pulses with the off-resonant modulation function from Fig. 3c added to the frequency sweep. The plots are parameterized by the ratio r = Aw/Qo, where Q, = 1. (b) Plots of the adiabaticity ratio as a function of time on the right side and left side of the slice. The shift is toward the right. Note that the left side suffers a preferential loss of adiabaticity.
35
SELECTIVE ADIABATIC PULSE SEQUENCE
RFI
RFQ
Gz
0
5
10
15
20
25
30
35
40
45
50
Time (ms) FIG. 5. Timing diagram for experimental test of the spin-echo pulse sequence. Readout is in the sliceselect direction, and the r / 2 pulse is nonselective. The r pulses are 5.376 ms in duration and have a bandwidth of 2.6 kHz.
EXPERIMENTAL RESULTS
Figure 5 shows the pulse sequence used to test the spin-echo pulse sequence on our 1.5-T General Electric Signa system. The echo was read out in the direction of slice selection. Both the first and second echoes were collected for comparison. The ~ / 2 pulse was nonselective. The duration of the adiabatic K pulse was 5.376 ms and the bandwidth was 2.6 kHz. Above a threshold amplitude, the response remained fixed. Figure 6 compares phantom results for the first and second echoes on-resonance. Note the flatness of the phase profile of the second echo, and that the magnitude of the first echo is smaller than the second echo. This signal loss manifests the dependence
': Second echo
i
U
._
b-'5\ Normalized position
-0.8 -0.6 -0.4 -0.2
0
012 014
0:6
018
!
Normalized position
FIG. 6. ( a) Comparison of the experimental spin-echo magnitude slice profile for the first and second echoes. Note the signal loss of the first echo due to RF inhomogeneity. ( b ) Comparison of the experimental spin-echo phase slice profile for the first and second echoes. Note that the phase of the second echo is flat.
36
CONOLLY ET AL.
of the phase variation on RF field strength. That is, the magnitude is decreased by projection through the distribution of phase angles in the plane of the slice. The magnitude slice profile on resonance and 1 kHz off resonance are shown in Fig. 7a, and the corresponding phase profile is shown in Fig. 7b. The RF transmitter was used to create the off-resonance condition. Note that 1 kHz off-resonance is close to the condition AAw = O o , since 00 = 1.3 kHz and that it far exceeds the normal spectral bandwidth needed for imaging at 1.5 T, where the water-fat separation is about 250 Hz. The off-resonant slice is probably acceptable for imaging. Also note that the asymmetry of the off-resonant slice profile agrees with the qualitative sweepdiagram analysis from above. DISCUSSION
One drawback with the sequence as described is that the minimum echo-time (TE ) is greater than that for standard spin-echo imaging, making the imaging of short-T2 species difficult. It is worth noting that a small modification remedies this problem for single-echo experiments. The first a pulse can be started immediately after the conclusion of the 1 / 2 pulse, because one does not need the full echo from the first a pulse. The refocusing and phase-encode can then be impressed during the half-echo after the first a pulse. During the a pulses, the spins do not experience much T2decay because they move quickly around the plane perpendicular to the sweep vector as they are being swept about the x axis. Using this scheme, the spins remain in the transverse plane for roughly the same amount of time as for a standard spin-echo experiment. This modified pulse sequence is shown in Fig. 8a. To demonstrate the feasibility of the pulse sequence, we include an axial head image of a healthy volunteer in Fig. 8b, obtained on a 1.5-T GE Signa system. There are many clinical procedures that could benefit from near-perfect spin-echoes. A multiecho version of this technique might prove helpful for quantitative T 2determination. Majumdar et al. (16) have shown that RF pulse imperfections undermine measurements of T2. However, it seems difficult to generalize the short first-echo experiment described above to a multiecho experiment.
3 c
P a
Ij
-10
-12
Position in arb. units
,
,
,
,
,
Position in arb. units
,
,
,
\, I
FIG. 7. ( a ) Comparison of the experimental spin-echo magnitude profile on-resonance and 1 kHz offresonance. (b) Comparison of the experimental spin-echo phase profile on-resonance and 1 lcHz off-resonance.
SELECTIVE ADIABATIC PULSE SEQUENCE
37
daql===l ub
20.0 25.0 30.0 35.0 40.0 45.0 50.0 55.0 60.0 65.0
a
Time (ms)
FIG. 8. ( a ) Pulse sequence that reduces the time spent in the transverse plane to about the same time as in a standard 2D FT experiment. (b) Experimental image of a healthy volunteer obtained with the pulse sequence shown in Fig. 8a on a 1.5-T GE Signa system. No effort was made to optimize the contrast; this image is intended to demonstrate only the viability of the proposed pulse sequence on a current MR scanner.
Another important result ofthis paper is that all spatially selective adiabatic T pulses should now be applied in a variable-rate format because the only drawback-offresonant slice degradation-is negligible. SUMMARY
The even echoes from any sequence of identical selective adiabatic r pulses are exactly phase compensated with immunity to spatial RF variations. The variable-rate version of the spin-echo sequence deposits far less power in the patient and produces no degradation of the off-resonant slice profile. This selective spin-echo pulse sequence deposits relatively low power, it offers immunity to RF inhomogeneity and nonlinearity, and it produces an excellent slice profile that would enable contiguous slice acquisition. ACKNOWLEDGMENTS This work was supported in part by National Institutes of Health Grants HL-39478; HL-39297, HV38045, and HL-34962. In addition, the authors thank members of the GE Medical Systems Group for their help and support. Finally, the authors appreciate the comments from our second reviewer. REFERENCES 1. J. BAUM,R. TYCKO,AND A. PINES,J. Chem. Phys. 79,4643 ( 1983). 2. F. HIOE,Phys. Rev. A 30, 2100 (1984). 3. M. SILVER,R. JOSEPH,AND D. HOULT,Phys. Rev.A 31,2753 ( 1985). 4. J. BAUM,R. TYCKO,AND A. PINES,Phys. Rev.A 32, 3435 ( 1985). 5. D. KUNZ,Magn. Reson. Med. 4, 129 (1987).
38
CONOLLY ET AL.
S. CONOLLY, D. NISHIMURA, AND A. MACOVSKI, J. Magn. Reson. 83, 324 ( 1989). K. U ~ U R B I M. L , GARWOOD, A. RATH, AND M. R. BENDALL,J. Magn. Reson. 78,472 ( 1988). S. CONOLLY, D. NISHIMURA, A. MACOVSKI, AND G. GLOVER,J. Magn. Reson. 78,440 ( 1988). S. CONOLLY, U.S. Patent 4,760,336 (1988). 10. S. CONOLLY,“Magnetic Resonance Selective Excitation,” Ph.D. thesis, Stanford University, 1989. 11. D. LEWIS,B. TsuI, P. MORAN,AND D. SALONER, in “Society of Magnetic Resonance in Medicine 8th Annual Meeting,” p. 27, 1989. 12. J. PAULY,S. CONOLLY, D. NISHIMURA, AND A. MACOVSKI, J. Magn. Reson., in press. 13. S. CONOLLY, D. NISHIMURA, AND A. MACOVSKI, J. Mugn. Reson. 83, 549 ( 1989). 14. M. BENDALL, M. GARWOOD, K. UdURBIL, AND D. PEGC, Magn. Reson. Med. 4,493 ( 1987). 15. A. ABRAGAM, “The Principles of Nuclear Magnetism,” Oxford Univ. Press, London, 1961. 16. S. MAJUMDAR, S. OPRHANOUDAKIS, A. GMITRO,M. ODONNELL, AND J. GORE,M a p . Reson. Med. 6. 7. 8. 9.
3,397 (1986).
