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Phase cycling of composite refocusing pulses to eliminate dispersive refocusing magnetization
JOURNAL
OF MAGNETIC
65, 348-354 (1985)
RESONANCE
PhaseCycling of Composite RefocusingPulsesto Eliminate Dispersive RefocusingMagnetization H.P. HETHERINGTONANDD.L.ROTHMAN Department of Molecular Biophysics and Biochemistry, Yale University, New Haven, Connecticut 06511 Received June 25, 1985
Several schemes have been proposed for the construction of composite ?r pulses designed to compensate for Hi inhomogeneity (Z-4). However, when these composite r pulses are used as refocusing pulses within a spin-echo sequence with a coil possessing poor H, homogeneity they produce phase and intensity errors in the acquired spectrum. Bendall and Pegg (5) have described the cycling required to eliminate these artifacts for a single refocusing pulse or a series of single refocusing pulses. EXORCYCLE as described by Bodenhausen et al. (6) eliminates these artifacts when the composite pulse train is used as an inversion pulse but can fail when the pulse train is used to refocus transverse magnetization (7, 8). With the wide application of spin-echo sequences to in vivo surface-coil NMR and NMR imaging, where H1 inhomogeneity and off-resonance effects are severe, a general scheme of cycling a train of pulses with varying phase and pulse angle to refocus transverse magnetization would be of general utility. In this work we (1) describe the source of the phase errors that must be eliminated, (2) derive the transformations required to eliminate all such errors from the matrix describing the pulse train using the matrix methodology described by Bendall and Pegg (5), (3) interpret these transformations in terms of the required pulse-train cycling constructions, and (4) compare the proposed cycling to that described by Bendall and Pegg (5) and to EXORCYCLE (6) for two refocusing trains. The effect of a train of y1pulses of arbitrary angle and phase can be described in the matrix formalism of Bendall and Pegg (5), by the operator P, where, P = I-J
(-~ir,)(-LYir,>(eil,>((Yir,>(~~~*)
PI
where i = y1- j + 1, & is the phase of the pulse, (Y~is the tilt angle due to off-resonance effects, Bi is the effective pulse angle which is given by ei = [(gf,>2 + (+(J2]“27p.
l-21
7p is the duration of the pulse in seconds, and I=, I,, and 1, are rotation matrices. The operator P can be expressed in this formalism as a 3 X 3 matrix Pll
PI2
PI3
P31
P32
P33
[31
0022-2364185$3.00 Copyright 0 1985 by Academic Press, Inc. All rights of reproduction in any form reserved.
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The effect of the pulse train P on an initial spin state, I’, is given by If = PI0
[41
or
If the train of pulses described by the operator P is to be used as a refocusing pulse it is necessary to determine the elements pii of P which give rise to phase anomalies. This can be done, by analyzing the effect of the sequence &-T-P-r-AQ. The effect of the excitation pulse ox and the evolution period 7 is to convert the initial 1, magnetization to I&OS(B) - Z+in(@cos(w7) + I&n(B)sin(wT). 161 The magnetization time 7 is given by
after application of the train P and evolution for an additional
[[(pi I + p2Jsin(oT)cos(oT) +
Z(p13COS(w7)
-
- (p12cos2(w~) + p2,sin2(or))]sin(e)
p23Sin(07))COS(e)llr,[[(p11sin2(w7)
+ (p2, - pi2)sin(07)cos(w7)]sin(0)
+
-
[(p*3COS(WT)
~22COS*(o7))
+ pi3sin(wT))]cos(0)]ly.
[7]
Analysis of Eq. [7] shows that all frequency-dependent terms of Z, magnetization can be eliminated if PI3 = ~23 = 0, PI2 = pzl, and p1 i = -~22. These criteria lead to the refocusing event, (pilsin2(w~) - pz2cos2(w7))sin(0)l,, orp,,sin(@, along the y axis, i.e., absorption phase magnetization. However, these same criteria lead to the refocusing event, -(p12cos2(w~) + p2,sin2(w7))sin(0)1,, or -pi2sin(0) for Z, magnetization, which will contribute a dispersive component to the acquired signal. Since the relative magnitudes of the dispersive and absorption components are functions of pulse angle, the net phase of the acquired signal will vary throughout the sample if a coil having an inhomogenous Hi is used. This in turn will result in phase distortions if the observed resonances are inhomogenously distributed throughout the sample volume as in many in vivo systems. At this point it sh(ould be noted that this refocusing dispersive phase magnetization is not eliminated by EXORCYCLE, which eliminates only nonrefocusing, frequency-dependent magnetization (6). To eliminate this component we must derive a cycling such that p12 = p21 = 0. Therefore the matrix representing the net effect of the operator P must be of the form,
PI
Hence, we must find a set of transformations,
Tj, such that
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where P is the operator describing the net effect of the cycling, Uj = f 1 and Z?(B)is a function of the pulse angle 0. Before deriving the specific set of transformations ajZJ-it is useful to note the effect of several operations 01, 02, 03, and 04 which will form the basis from which the transformations Tj are constructed; these are
01P = 1;:
;;
it]
= PT
O,P = [ jjil
g*
?J
03P = [ ;Yf,
4
;l+]
= ~-~zy~P~Tzy~
08
-jT;:
fjj
= ~-T/2zz~P~d2zz~.
= [ ;I2
= ~-~zz~P~Tzz~
[lOI
The elimination of the elements p13 and pz3 from the resultant matrix can be achieved by the addition of P + OZP = P + (-7rZz)P(7rZz)
El11
which in matrix form is given by
We can eliminate all other anomalies if we can construct a train such that its matrix representation is given by 2
[ 1 P22
PI2
*
P21
Pll
*
.
.
.
[I31
and subtract it from Eq. [ 121, by inverting the receiver phase during the acquisition. The operation 040301P yields the matrix
which is equivalent to the matrix specified in Eq. [ 131. The terms describing the creation of transverse magnetization from residual longitudinal magnetization, in this case ~~32 and -Pan, can be eliminated by a subsequent operation of 02 as specified in Eq. [ 111. Thus the full set of transformations ajq as described by Eq. [9] is given by ajTj = (a,Z, a202, a 3 0 4 0 3 0
1,
a4 0 2 0 4 0 3 0 1 >
where Z is the identity operator, al = u2 = +l and a3 = a4 = -1.
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The operators 02, O3 and O4 can be interpreted as coordinate rotations, which can be represented as a simple linear transformation such that their effect on the pulsetrain operator P is given by
R-'PR = R-u1 (-cbil,)(-aiZ,)(eiZ~)(~iZ~)(~ir,)lR
1161
where R and R-' are the specific rotation matrices. However, Eq. [ 161 is equivalent to
Thus, performing the transformation on the operator describing the pulse train P is equivalent to performing the same transformation on each and every pulse within the train. Therefore, the operation 02P is equivalent to
I-J [(-~~,)(-~~i~,)(-~ir,)(eil,)(air,)(4~~~~~~~~~1
1181
which can be simplified to n
[(-(r
+
~i)Z,)(-cyiz,)(eiz~)(~iZ,><(~
+
4iYz)l
[I91
or is equivalent to adding 7~to the phase of each pulse in the train. Since the effect of the transpose operator on the actual pulse train cannot be expressed as a coordinate rotation, we need to find a representation of O,P, (PT), in terms of the angles 4i, (Y~,and Bi and their respective operators Z,, I,, and Z,. We start by noting that
pT =
WI
For the product of matrices X1 * * * X,, * - - Xn-IXn)T
(X1X2
=
x:x:-*
- - * XTXT
Pll
such that PT is equivalent to
II [(-4jz=)(-ol,r,)(e,l,)(4jZ=)lT.
WI
If the identity in [21] is applied to [22] PT is given by
where the product is taken over the variable j, such that the time order of the pulses has been reversed (see Eq. [l]). However, since (BjZJT = -OjZx, (ajZy)’ = -air, and (4jZz)T = -4jZ2, PT is equivalent to n
(-4jZ=>(-oiir,>(-ejZ~)(~jZ~)(4jZ~).
[241
Subsequent operation of 03 and O4 on 0,) PT, yields (-?r12z,>(-~z~)PT(?rly)(~/21z)
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which by using the property 4jIzrIy = ~~~(-~jI~), reduces to II (-Cai2 -
~j)IZ)(-~j~~)(-?T~y)(-eiTx)(nly)(oljly)((n/2
Noting that (-~I~)(-tijr,)(~I~)
-
$j)L).
P61
= (ejr,>, Eq. [26] is equivalent to
which represents a train of pulses with the opposite time order as that described by P with a phase given by 7r/2 - $j. Thus the complete cycling to eliminate all nonrefocusing and dispersive refocusing magnetization of a train of n pulses, P, ((Or, &, a,) - - - (e,, &, (w,)) where 8, 4 and (Y represent the tip angle, phase, and off-resonance tilt angle, is given by (1)
(04,
41,d
(2)
((6,
41
(3) wn, a/2
- *ml, f
r, -
4 hi,
dbz,
%J)
* * - vA2,
42
4.
* *@1,
(4) ((en, 3~12 - h, 4.
R +.X +
r,
a/2
R +.X
a,)> -
41,
al))
R-X
- -(e,, 3d2 - h, 4)
R-,
P81
where R+, and R-, indicate the phase of the receiver. Similarly it can be shown that the dispersive refocusing magnetization can be acquired independently of the absorption refocusing component using the following cycling: (1)
Wl,
41,
4’
* *@2,
(2) ue,, h, 4. (3) ((e,, -4,,
67,
R 4-x
4)
- - (en, db + 75 4) 4.
(4) ((en, T - 4,, 4.
- m,
-h,
4)
. +4, K - h, 4)
R +.X R-x R-,.
WI
Application of EXORCYCLE to the well known composite ?r pulse X-2 Y-X, yields the four cycling elements, X-2 Y-x, --$2 Y-x, Y-2X-Y and The cycling -- Y-2X-Y. described in Eq. [28] yields X-2 Y-X, X-2 Y-X, Y-2X-Y and Y-2X-Y, which is different from that of EXORCYCLE. As shown in Fig. Ic, when the above composite ?r pulse is used as a refocusing pulse, dispersive magnetization is obtained with EXORCYCLE, while only absorption phase magnetization, Fig. 1a, is obtained if the cycling described in Eq. [2X] is used. As shown in Fig. lb the dispersive refocusing component can be acquired if the cycling described in Eq. [29] is used. Recently Hore (9) has described the use of frequency-selective pulses which when used in spin-echo sequences can have significant advantages in the suppression of large solvent resonances and the alteration ofj modulation for in vivo NMR studies (10-12). To validate the applicability of the described cycling to an arbitrary semiselective pulse, we must show that the transformations Tj yield a physically real transformation when chemical-shift evolution periods are included within the trains. We start by noting that a semiselective pulse P’, can be written as P’ = J-J (w7il;)(-Qzilr)(-Lyi~y)(eiJx)((yi~~)(~i~=)
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F[G. I. The result of the sequence (r/2)r~-P-~-AQ applied on resonance to a sample of Hz0 in a 5 mm tube in a standard 10 mm ‘H probe using the composite pulse P, OJS,&, and the cyclings (a) described in Eq. [28], (b) described in Eq. [29], and (c) EXORCYCLE. The ‘H resonance of Hz0 is plotted as a function OFthe applied pulse angle 8. Spectra were acquired with a Bruker WB-360 spectrometer.
where i = IZ - j + 1, o is the offset in radian hertz, and 7i refers to the duration of the ith evolution period following the ith pulse. The transformation 02P’ corresponds to =
02p
I-J
[311
(-?rI,)(~7i)(-cpir,)(-cuiZ~)(e,i,)((Yi~~)(~)iJz)(~Z~).
Using the property that (-‘ITI,)(wTiZz) = (wT~Z~)(--XZ~),Eq. [3 l] becomes IT
(@TiL)(-(di
+
?r)Z=)(-aiZ~)(eil,)(air,)((~i
+
~V.J
~321
or a simple phase shift of ?r radians to the phase of each pulse in the train leaving the evolution periods unchanged. The transformation 0,P’ yields OIP’ = J-j
[331
(-~jZ~)(-~jz~)(-ejZ~)(otil,)(~jZ=)(-W7ilz).
The operation of 0403 on Eq. [33] yields fl (-(r/2
-
~j)zz)(-oLir,)(Bjz~)(oljl,)((~/2
-
djY.z)(WTiZz)
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or a reversal of the time order of the sequence with the phase of each pulse changed to 7r/2 - $j. Comparison of this cycling scheme with EXORCYCLE reveals that they are equivalent for the semiselective pulses described by Hore (9). However, the cycling will not be equivalent for an arbitrary train. While EXORCYCLE eliminates all nonrefocusing magnetization produced by the refocusing trains, phase errors can still occur with inhomogenous coils unless the dispersive refocusing components are eliminated. The elimination of these dispersive refocusing components can be achieved if the cycling described in Eq. [28] is used. The elimination of these phase errors will allow the application of composite pulse refocusing trains to surface coils where the Hi compensation afforded by these trains could significantly increase the sensitivity and allow for the quantitative application ofj-modulation-based double-resonance techniques (13). In addition the dependence of the relative intensities of the absorptive and dispersive phase refocusing magnetization on pulse angle may allow the cycling in Eq. [28] to be used in the design of Hi-selective pulses as described by Bendall and co-workers (5,14) and recently applied with composite pulses (2, 15). ACKNOWLEDGMENTS We thank M. J. Avison, K. L. Raheb, M. R. Bendall, T. Jue, and R. G. Shulman for helpful discussions. This work was supported by Public Health Service Grant GM 30267, and National Institutes of Health Grant AM 27121. REFERENCES 1. M. H. LEVITT AND R. FREEMAN, .I. Magn. Reson. 43,65 (1981). 2. A. J. SHAKA AND R. FREEMAN, J. Magn. Reson. 59, 169 ( 1984). 3. M. H. LEVITT AND R. R. ERNST, J. Magn. Reson. 55,247 (1983). 4. R. TyCKO, Phys. Lett. Rev. 51, 775 (1983). 5. M. R. BENDALL AND D. T. PEoo, Magn. Reson. Med. 2, 9 1 (1985). 6. G. BODENHAUSEN, R. FREEMAN, AND D. L. TURNER, J. Magn. Reson. 27,5 11 (1977). 7. M. R. BENDALL AND D. T. PEGG, .I Magn. Resort. 63, in press. 8. R. T~CKO, H. M. CHO, E. SCHNEIDER, AND A. PINES, J. Mugn. Reson. 61,90 (1985). 9. P. J. HORE, J. Mugn. Reson. 55,283 (1983). 10. K. M. BRINDLE, R. PORTELJS, AND 1. D. CAMPBELL, J. Magn. Reson. 56,543 (1984). II. H. P. HETHERINGTON, M. J. AVISON, AND R. G. SHULMAN, Proc. Natl. Acad. Sci.
USA 82, 3 115
(1985). 12. T. JUE, F. ARIAS-MENWZA,
N. C. GONNELLA,
G. I. SHULMAN,
AND R. G. SHULMAN,
Proc. Natl.
Acad. Sci. USA, in press. 13. D. L. ROTHMAN,
K. L. BEHAR,
H. P. HETHERINGTON,
AND R. G. SHULMAN,
USA 81,633O (1984). 14. M. R. BENDALL AND R. E. GORDON, J. Mugn. Reson. 53,365 (1983). 15. R. TYCKO AND A. PINES, J. Mugn. Reson. 60, 156 (1984).
Proc. Natl. Acad. Sci.
OF MAGNETIC
65, 348-354 (1985)
RESONANCE
PhaseCycling of Composite RefocusingPulsesto Eliminate Dispersive RefocusingMagnetization H.P. HETHERINGTONANDD.L.ROTHMAN Department of Molecular Biophysics and Biochemistry, Yale University, New Haven, Connecticut 06511 Received June 25, 1985
Several schemes have been proposed for the construction of composite ?r pulses designed to compensate for Hi inhomogeneity (Z-4). However, when these composite r pulses are used as refocusing pulses within a spin-echo sequence with a coil possessing poor H, homogeneity they produce phase and intensity errors in the acquired spectrum. Bendall and Pegg (5) have described the cycling required to eliminate these artifacts for a single refocusing pulse or a series of single refocusing pulses. EXORCYCLE as described by Bodenhausen et al. (6) eliminates these artifacts when the composite pulse train is used as an inversion pulse but can fail when the pulse train is used to refocus transverse magnetization (7, 8). With the wide application of spin-echo sequences to in vivo surface-coil NMR and NMR imaging, where H1 inhomogeneity and off-resonance effects are severe, a general scheme of cycling a train of pulses with varying phase and pulse angle to refocus transverse magnetization would be of general utility. In this work we (1) describe the source of the phase errors that must be eliminated, (2) derive the transformations required to eliminate all such errors from the matrix describing the pulse train using the matrix methodology described by Bendall and Pegg (5), (3) interpret these transformations in terms of the required pulse-train cycling constructions, and (4) compare the proposed cycling to that described by Bendall and Pegg (5) and to EXORCYCLE (6) for two refocusing trains. The effect of a train of y1pulses of arbitrary angle and phase can be described in the matrix formalism of Bendall and Pegg (5), by the operator P, where, P = I-J
(-~ir,)(-LYir,>(eil,>((Yir,>(~~~*)
PI
where i = y1- j + 1, & is the phase of the pulse, (Y~is the tilt angle due to off-resonance effects, Bi is the effective pulse angle which is given by ei = [(gf,>2 + (+(J2]“27p.
l-21
7p is the duration of the pulse in seconds, and I=, I,, and 1, are rotation matrices. The operator P can be expressed in this formalism as a 3 X 3 matrix Pll
PI2
PI3
P31
P32
P33
[31
0022-2364185$3.00 Copyright 0 1985 by Academic Press, Inc. All rights of reproduction in any form reserved.
348
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The effect of the pulse train P on an initial spin state, I’, is given by If = PI0
[41
or
If the train of pulses described by the operator P is to be used as a refocusing pulse it is necessary to determine the elements pii of P which give rise to phase anomalies. This can be done, by analyzing the effect of the sequence &-T-P-r-AQ. The effect of the excitation pulse ox and the evolution period 7 is to convert the initial 1, magnetization to I&OS(B) - Z+in(@cos(w7) + I&n(B)sin(wT). 161 The magnetization time 7 is given by
after application of the train P and evolution for an additional
[[(pi I + p2Jsin(oT)cos(oT) +
Z(p13COS(w7)
-
- (p12cos2(w~) + p2,sin2(or))]sin(e)
p23Sin(07))COS(e)llr,[[(p11sin2(w7)
+ (p2, - pi2)sin(07)cos(w7)]sin(0)
+
-
[(p*3COS(WT)
~22COS*(o7))
+ pi3sin(wT))]cos(0)]ly.
[7]
Analysis of Eq. [7] shows that all frequency-dependent terms of Z, magnetization can be eliminated if PI3 = ~23 = 0, PI2 = pzl, and p1 i = -~22. These criteria lead to the refocusing event, (pilsin2(w~) - pz2cos2(w7))sin(0)l,, orp,,sin(@, along the y axis, i.e., absorption phase magnetization. However, these same criteria lead to the refocusing event, -(p12cos2(w~) + p2,sin2(w7))sin(0)1,, or -pi2sin(0) for Z, magnetization, which will contribute a dispersive component to the acquired signal. Since the relative magnitudes of the dispersive and absorption components are functions of pulse angle, the net phase of the acquired signal will vary throughout the sample if a coil having an inhomogenous Hi is used. This in turn will result in phase distortions if the observed resonances are inhomogenously distributed throughout the sample volume as in many in vivo systems. At this point it sh(ould be noted that this refocusing dispersive phase magnetization is not eliminated by EXORCYCLE, which eliminates only nonrefocusing, frequency-dependent magnetization (6). To eliminate this component we must derive a cycling such that p12 = p21 = 0. Therefore the matrix representing the net effect of the operator P must be of the form,
PI
Hence, we must find a set of transformations,
Tj, such that
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where P is the operator describing the net effect of the cycling, Uj = f 1 and Z?(B)is a function of the pulse angle 0. Before deriving the specific set of transformations ajZJ-it is useful to note the effect of several operations 01, 02, 03, and 04 which will form the basis from which the transformations Tj are constructed; these are
01P = 1;:
;;
it]
= PT
O,P = [ jjil
g*
?J
03P = [ ;Yf,
4
;l+]
= ~-~zy~P~Tzy~
08
-jT;:
fjj
= ~-T/2zz~P~d2zz~.
= [ ;I2
= ~-~zz~P~Tzz~
[lOI
The elimination of the elements p13 and pz3 from the resultant matrix can be achieved by the addition of P + OZP = P + (-7rZz)P(7rZz)
El11
which in matrix form is given by
We can eliminate all other anomalies if we can construct a train such that its matrix representation is given by 2
[ 1 P22
PI2
*
P21
Pll
*
.
.
.
[I31
and subtract it from Eq. [ 121, by inverting the receiver phase during the acquisition. The operation 040301P yields the matrix
which is equivalent to the matrix specified in Eq. [ 131. The terms describing the creation of transverse magnetization from residual longitudinal magnetization, in this case ~~32 and -Pan, can be eliminated by a subsequent operation of 02 as specified in Eq. [ 111. Thus the full set of transformations ajq as described by Eq. [9] is given by ajTj = (a,Z, a202, a 3 0 4 0 3 0
1,
a4 0 2 0 4 0 3 0 1 >
where Z is the identity operator, al = u2 = +l and a3 = a4 = -1.
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The operators 02, O3 and O4 can be interpreted as coordinate rotations, which can be represented as a simple linear transformation such that their effect on the pulsetrain operator P is given by
R-'PR = R-u1 (-cbil,)(-aiZ,)(eiZ~)(~iZ~)(~ir,)lR
1161
where R and R-' are the specific rotation matrices. However, Eq. [ 161 is equivalent to
Thus, performing the transformation on the operator describing the pulse train P is equivalent to performing the same transformation on each and every pulse within the train. Therefore, the operation 02P is equivalent to
I-J [(-~~,)(-~~i~,)(-~ir,)(eil,)(air,)(4~~~~~~~~~1
1181
which can be simplified to n
[(-(r
+
~i)Z,)(-cyiz,)(eiz~)(~iZ,><(~
+
4iYz)l
[I91
or is equivalent to adding 7~to the phase of each pulse in the train. Since the effect of the transpose operator on the actual pulse train cannot be expressed as a coordinate rotation, we need to find a representation of O,P, (PT), in terms of the angles 4i, (Y~,and Bi and their respective operators Z,, I,, and Z,. We start by noting that
pT =
WI
For the product of matrices X1 * * * X,, * - - Xn-IXn)T
(X1X2
=
x:x:-*
- - * XTXT
Pll
such that PT is equivalent to
II [(-4jz=)(-ol,r,)(e,l,)(4jZ=)lT.
WI
If the identity in [21] is applied to [22] PT is given by
where the product is taken over the variable j, such that the time order of the pulses has been reversed (see Eq. [l]). However, since (BjZJT = -OjZx, (ajZy)’ = -air, and (4jZz)T = -4jZ2, PT is equivalent to n
(-4jZ=>(-oiir,>(-ejZ~)(~jZ~)(4jZ~).
[241
Subsequent operation of 03 and O4 on 0,) PT, yields (-?r12z,>(-~z~)PT(?rly)(~/21z)
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which by using the property 4jIzrIy = ~~~(-~jI~), reduces to II (-Cai2 -
~j)IZ)(-~j~~)(-?T~y)(-eiTx)(nly)(oljly)((n/2
Noting that (-~I~)(-tijr,)(~I~)
-
$j)L).
P61
= (ejr,>, Eq. [26] is equivalent to
which represents a train of pulses with the opposite time order as that described by P with a phase given by 7r/2 - $j. Thus the complete cycling to eliminate all nonrefocusing and dispersive refocusing magnetization of a train of n pulses, P, ((Or, &, a,) - - - (e,, &, (w,)) where 8, 4 and (Y represent the tip angle, phase, and off-resonance tilt angle, is given by (1)
(04,
41,d
(2)
((6,
41
(3) wn, a/2
- *ml, f
r, -
4 hi,
dbz,
%J)
* * - vA2,
42
4.
* *@1,
(4) ((en, 3~12 - h, 4.
R +.X +
r,
a/2
R +.X
a,)> -
41,
al))
R-X
- -(e,, 3d2 - h, 4)
R-,
P81
where R+, and R-, indicate the phase of the receiver. Similarly it can be shown that the dispersive refocusing magnetization can be acquired independently of the absorption refocusing component using the following cycling: (1)
Wl,
41,
4’
* *@2,
(2) ue,, h, 4. (3) ((e,, -4,,
67,
R 4-x
4)
- - (en, db + 75 4) 4.
(4) ((en, T - 4,, 4.
- m,
-h,
4)
. +4, K - h, 4)
R +.X R-x R-,.
WI
Application of EXORCYCLE to the well known composite ?r pulse X-2 Y-X, yields the four cycling elements, X-2 Y-x, --$2 Y-x, Y-2X-Y and The cycling -- Y-2X-Y. described in Eq. [28] yields X-2 Y-X, X-2 Y-X, Y-2X-Y and Y-2X-Y, which is different from that of EXORCYCLE. As shown in Fig. Ic, when the above composite ?r pulse is used as a refocusing pulse, dispersive magnetization is obtained with EXORCYCLE, while only absorption phase magnetization, Fig. 1a, is obtained if the cycling described in Eq. [2X] is used. As shown in Fig. lb the dispersive refocusing component can be acquired if the cycling described in Eq. [29] is used. Recently Hore (9) has described the use of frequency-selective pulses which when used in spin-echo sequences can have significant advantages in the suppression of large solvent resonances and the alteration ofj modulation for in vivo NMR studies (10-12). To validate the applicability of the described cycling to an arbitrary semiselective pulse, we must show that the transformations Tj yield a physically real transformation when chemical-shift evolution periods are included within the trains. We start by noting that a semiselective pulse P’, can be written as P’ = J-J (w7il;)(-Qzilr)(-Lyi~y)(eiJx)((yi~~)(~i~=)
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I 150
1 165
F[G. I. The result of the sequence (r/2)r~-P-~-AQ applied on resonance to a sample of Hz0 in a 5 mm tube in a standard 10 mm ‘H probe using the composite pulse P, OJS,&, and the cyclings (a) described in Eq. [28], (b) described in Eq. [29], and (c) EXORCYCLE. The ‘H resonance of Hz0 is plotted as a function OFthe applied pulse angle 8. Spectra were acquired with a Bruker WB-360 spectrometer.
where i = IZ - j + 1, o is the offset in radian hertz, and 7i refers to the duration of the ith evolution period following the ith pulse. The transformation 02P’ corresponds to =
02p
I-J
[311
(-?rI,)(~7i)(-cpir,)(-cuiZ~)(e,i,)((Yi~~)(~)iJz)(~Z~).
Using the property that (-‘ITI,)(wTiZz) = (wT~Z~)(--XZ~),Eq. [3 l] becomes IT
(@TiL)(-(di
+
?r)Z=)(-aiZ~)(eil,)(air,)((~i
+
~V.J
~321
or a simple phase shift of ?r radians to the phase of each pulse in the train leaving the evolution periods unchanged. The transformation 0,P’ yields OIP’ = J-j
[331
(-~jZ~)(-~jz~)(-ejZ~)(otil,)(~jZ=)(-W7ilz).
The operation of 0403 on Eq. [33] yields fl (-(r/2
-
~j)zz)(-oLir,)(Bjz~)(oljl,)((~/2
-
djY.z)(WTiZz)
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or a reversal of the time order of the sequence with the phase of each pulse changed to 7r/2 - $j. Comparison of this cycling scheme with EXORCYCLE reveals that they are equivalent for the semiselective pulses described by Hore (9). However, the cycling will not be equivalent for an arbitrary train. While EXORCYCLE eliminates all nonrefocusing magnetization produced by the refocusing trains, phase errors can still occur with inhomogenous coils unless the dispersive refocusing components are eliminated. The elimination of these dispersive refocusing components can be achieved if the cycling described in Eq. [28] is used. The elimination of these phase errors will allow the application of composite pulse refocusing trains to surface coils where the Hi compensation afforded by these trains could significantly increase the sensitivity and allow for the quantitative application ofj-modulation-based double-resonance techniques (13). In addition the dependence of the relative intensities of the absorptive and dispersive phase refocusing magnetization on pulse angle may allow the cycling in Eq. [28] to be used in the design of Hi-selective pulses as described by Bendall and co-workers (5,14) and recently applied with composite pulses (2, 15). ACKNOWLEDGMENTS We thank M. J. Avison, K. L. Raheb, M. R. Bendall, T. Jue, and R. G. Shulman for helpful discussions. This work was supported by Public Health Service Grant GM 30267, and National Institutes of Health Grant AM 27121. REFERENCES 1. M. H. LEVITT AND R. FREEMAN, .I. Magn. Reson. 43,65 (1981). 2. A. J. SHAKA AND R. FREEMAN, J. Magn. Reson. 59, 169 ( 1984). 3. M. H. LEVITT AND R. R. ERNST, J. Magn. Reson. 55,247 (1983). 4. R. TyCKO, Phys. Lett. Rev. 51, 775 (1983). 5. M. R. BENDALL AND D. T. PEoo, Magn. Reson. Med. 2, 9 1 (1985). 6. G. BODENHAUSEN, R. FREEMAN, AND D. L. TURNER, J. Magn. Reson. 27,5 11 (1977). 7. M. R. BENDALL AND D. T. PEGG, .I Magn. Resort. 63, in press. 8. R. T~CKO, H. M. CHO, E. SCHNEIDER, AND A. PINES, J. Mugn. Reson. 61,90 (1985). 9. P. J. HORE, J. Mugn. Reson. 55,283 (1983). 10. K. M. BRINDLE, R. PORTELJS, AND 1. D. CAMPBELL, J. Magn. Reson. 56,543 (1984). II. H. P. HETHERINGTON, M. J. AVISON, AND R. G. SHULMAN, Proc. Natl. Acad. Sci.
USA 82, 3 115
(1985). 12. T. JUE, F. ARIAS-MENWZA,
N. C. GONNELLA,
G. I. SHULMAN,
AND R. G. SHULMAN,
Proc. Natl.
Acad. Sci. USA, in press. 13. D. L. ROTHMAN,
K. L. BEHAR,
H. P. HETHERINGTON,
AND R. G. SHULMAN,
USA 81,633O (1984). 14. M. R. BENDALL AND R. E. GORDON, J. Mugn. Reson. 53,365 (1983). 15. R. TYCKO AND A. PINES, J. Mugn. Reson. 60, 156 (1984).
Proc. Natl. Acad. Sci.