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Enhancement of double-quantum-filtered signals using optimized tip angle RF pulses
JOURNAL
OF MAGNETIC
90,606-611
RESONANCE
( 1990)
Enhancementof Double-Quantum-Filtered Signals Using Optimized Tip Angle RF Pulses J. F. SHEN AND P. S. ALLEN* Department of Applied Sciences in Medicine, University of Alberta, Edmonton, Alberta, Canada T6G 2G3 Received May 22, 1990; revised July 17, 1990
In addition to their exploitation in high-resolution NMR, multiple-quantum filters provide a promising tool for spectral editing in in vivo spectroscopy. In particular, their use has been extensively explored for the purpose of observing the proton spectrum of metabolites with coupled spins such as lactate ( l-10). One of the more challenging difficulties faced when working in vivo is the low concentration of many metabolites of interest. As a result, any means of increasing the signal strength is of value. This Communication describes how the double-quantum-filtered signal can be enhanced for some double-quantum filters by up to 69%. This enhancement can be brought about by changing the tip angles of some of the pulses within the filter sequence from the conventional 90” to an optimized value that yields maximum conversion between anti-phase coherence (AC) and doublequantum coherence (DQC). Conversion pulses with tip angles other than 90” have been used in two-dimensional INADEQUATE experiments to achieve quadrature detection in the double-quantum frequency domain (11, 12).
Figure 1 illustrates variants of two typical double-quantum-filter formats which use the gradients labeled G and 2G for filtering. The unlabeled gradients occurring in these two sequences are trimming gradients used effectively to increase field inhomogeneities. When the tip angles of the pulses labeled rc/and 6 in Fig. 1 are both 90”, Fig. 1a reduces to the double-quantum filter proposed by Sotak et al. (4) and Fig. 1b reduces to that used to integrate double-quantum filtering and volume selection when all three 90” pulses are made slice selective ( 7). However, the optimal value for 1c/and 6 is not necessarily 90”, because the conversions from DQC to AC by the $ pulse and from AC to DQC by the 6 pulse have efficiencies that are a function of the tip angle of the II/ and 6 pulses. This functionality arises from the phase factor that must be incorporated into the analysis in order to account for field inhomogeneities. For a spin system with two weakly coupled spins I and S, for example, the doublequantum coherence before the filter gradient G can be represented in projection operator formalism ( 7, 13) by cz+s+ + c*z_s-,
[II
where c is a scalar coefficient and c* is its complex conjugate, which together take account of the differences between the two sequences in Fig. 1. If we adopt the con* To whom correspondence should be addressed. 0022-2364190 $3.00 Copyright 0 1990 by Academic Press. Inc. AN rights of reproduction in any form reserved.
606
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FIG. I. Double-quantum-filtering sequences (a) with and (b) without 180” refocusing pulses. The filter gradients are labeled G and 2G. Unlabeled gradients are trimming gradients.
vention of positive rotation in the right-handed sense (namely, clockwise rotation when looking in the direction of the relevant rotation axis) ( 13)) the DQC immediately after the gradient G becomes + c*I-S-ei24,
cI+S+e-i2”
Dl
where the phase factor 6 is spatially distributed and is given by C#J =
s
G-rdt,
and where we have assumed a short gradient period so that chemical-shift dephasing can be neglected during the gradient pulse. For either of the sequences in Fig. 1, a read pulse (labeled $) of arbitrary tip angle 0, applied along the x axis of the rotating frame, will give rise to the following transformation to AC of the first DQC component of Eq. [ 21: I, S+e-i24 -
ox
0 i(l+S, + S+Z,)e-i2%in 0 cos2~ + i(l-S;
6 + S_Z,)e-‘24sin 0 sin2 2
+ z-order, zero- and double-quantum
coherences.
I31
The analogous transformation of 1-S may be obtained by taking the complex conjugate of Eq. [ 31. To understand the effect of the spatially distributed phase factors, it is convenient to regroup the AC so produced into two different components. The first, proportional to sin f3cos*( o/2), corresponds to the pathway Z+ + I+ (or S, --, S,). The second, proportional to sin 0 sin2( 8/2), corresponds to the pathway I+ --) Ii (or S, + S,). The subsequent rephasing filter gradient 2G, which is twice as strong (or long) as the dephasing filter gradient, refocuses one, but not both, of these two components. Which component is refocused is determined by the relative polarity of the two filter gradients. If the two gradients have the same polarity, it is the second component that is refocused. The amplitude of the AC so produced varies between 0 and 3fi/8 ( x0.65) depending on 0, with the maximum value occurring at 0 = 120”.
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This maximum value is about 30% greater than that occurring when 6 = 90”. If the two gradients have opposite polarity, the first component is refocused and the variation with 0 of the AC amplitude covers the same range as the previous case, but this time it maximizes at 6 = 60”. By comparison, when the filter gradients G and 2G are absent, a read pulse tip angle of 0 = 90” gives the maximum conversion of DQC to AC (i.e., amplitude = 1). The sequence in Fig. lb can be further enhanced over that possible for Fig. la by using an optimal tip angle for the preceding pulse (labeled 6) also. Since no 180” refocusing pulse is used between the first and the second pulses in Fig. 1b, the antiphase coherence before the second pulse is already dephased by field inhomogeneities and can be written as 2(Z,cos f$i + Z,sin &)S, + 2(S,cos 4s + S,sin $s)Z= = (Z+@+ + Z-ei@l)S, + (S+epih + S-e’@)Z,,
[4]
where $i and & are phase factors caused by field inhomogeneities and the trimming gradient, as well as by the chemical shifts of the I and S spins, respectively. The trimming gradient ensures that the dephasing is complete. An RF pulse with a tip angle 0 and phase x will give rise to the following transformation of the AC for the I spin:
ox
(Z+e-‘“I + Z_e’+~)S, ----
-
0
tJ
Z+e-‘+I + Z-eibl cos* -2 + (Z-e- I41+ Z+e’“*)sin* 2I I( ) X
&sin 6 + z-order and single-quantum
i =-[(Z+S+e~ib1-Z-S-ei~1)cos2~-(Z~S-e~.m~2 + z-order, zero-quantum
coherences
1
Z+S+ei”l)sin2 4 sin 0 and single-quantum
coherences.
[5]
The analogous transformation of the AC for the S spin may be obtained by interchanging the I and the S spins in Eq. [ 51. The result of the transformation of Eq. [ 41 can be separated into two components that are entirely analogous to those obtained from the transformation of Eq. [2]. In order to refocus the field inhomogeneities occurring in the first 1 / (2 J) period with the field inhomogeneities present during the second 1/( 2J) period in Fig. lb, the combined pathway for the 6 and + pulses has to be Z, + Z5 (or S, --* ST), because the field inhomogeneities producing 4, (or 4s) may not be reversed manually like the relative polarity of the filter gradients. Two overall pathways satisfy this requirement, Z, 5 Z+ A Z7 and Z+ 2 Z7 5 Z5. However, in order to form an echo, the effect of the filter gradients must also be refocused. Our preceding analysis on the read pulse showed that only one of the two pathways, Z+ A ZT and Z+ J$ ZT (same as Z+ A I,), refocuses the effect of the filter gradients at any one time. If the two filter gradients have the same polarity, the pathway Z, A Zt A IT leads to the refocusing of the effects of both the field inhomogeneities and the filter gradients. If the two filter gradients have opposite polarity, the pathway Z, L IT A IT will give rise to the refocusing.
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TABLE 1 Conversion Efficiencies of the S and # Pulses Conversion efficiency at a tip angle of 8 Relative polarity of filter gradients
For 6
Optimal angle (degrees)
For rl/
Conversion efficiency at
For 8
For $
0 = 90”
0 = Optimal angle
Same
c9 sin 6 cos* 2
sinesin*-
e 2
60
120
0.5
g
Opposite
0 sin 0 sin* 2
sinOcos*-
6 2
120
60
0.5
3Jj c=z0.65 8
8
zz 0.65
The results for the conversion from DQC to AC by the + pulse and from AC to DQC by the 6 pulse are summarized in Table 1, where the conversion efficiency occurring in the table is defined as the coefficient of the refocused component, and where the optimal angle is defined as the tip angle of the conversion pulse that yields the maximum conversion efficiency. Although a two-spin system has been used in our analysis, the results are also valid for any product operators involving only two spin operators. For instance, for an AX3 spin system with scalar coupling constant J between the A and the X spins, the double-quantum coherence immediately before the read pulse for both sequences shown in Fig. 1 is proportional to (7, 10) + C*I-Sjeei2’),
$ (CI+Sj+e-‘*’
]=I
[61
where ZI correspond to the A spins and S’, correspond to the X spins and where the mutual complex conjugates c and c* take account of the differences in the two sequences in Fig. 1. We have also neglected, in Eq. [ 61, the more complex terms involving four spins, which only contribute to the A quartet signal. Since each individual term in Eq. [ 6 ] involves only two spins, our previous analysis predicts a 30% increase in the doublet signal if the tip angle of the read pulse is changed from 90” to the optimal value. The antiphase coherence arising from the X spins in an AX3 system immediately before the second pulse in Fig. lb is given by 3
C
j=l
(SjxCOS
4
+
&sin
4)Zz,
171
where the phase factor 4 is caused by field inhomogeneities, trimming gradients, and chemical shifts. Once again each individual term in Eq. [ 71 contains only two spins; a further increase of about 30% in the detectable doublet signal is expected if the tip angle of the second pulse is changed from 90” to the optimal value. Using optimal tip angles for both the second and the third pulses in sequence Fig. lb will yield a total increase in the doublet signal of about 69%. These predictions have been verified by experiments with the AX3 spin system of lactate dissolved in D20. Figure 2 and Fig. 3 show the doublet spectra acquired using
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1.31
(a)
v=90”
I v=60”
FIG. 2. Doublet components of spectra obtained from a sample of lithium lactate in DzO using the pulse sequence shown in Fig. la. The two filter gradients have the same polarity in a and have opposite polarity in b.
the sequences of Fig. la and Fig. lb, respectively. The number above each doublet is its amplitude measurement. In order to refocus the effects of field inhomogeneities in the sequence of Fig. lb, it was necessary to introduce minor adjustments into the timing which would not be anticipated for the case of a perfectly uniform static magnetic field ( 7). Because the manifestation of the field inhomogeneities is dependent on which pathways are chosen to be exploited, the timing adjustments are also pathway dependent. When the filter gradients have the same polarity, the interval between the first two pulses needs to be shortened by 27, where T is the interval between the second and the third pulse in Fig. lb. When opposite polarity filter gradients are used, the same shortening, 27, should be applied to the interval between the last pulse and the start of acquisition. In practice, the actual adjustments made were not exactly 27 and this probably reflects the difference in eddy current fields during the corresponding intervals. This change in timing could result in partial loss of signal due to incomplete conversion between in-phase and anti-phase coherences. However, without the timing adjustment, the effects of field inhomogeneities are not fully refocused and this could have led to a larger loss of signal and to distortions of the spectra. All data were acquired on a Bruker 100 MHz spectrometer with a 40 cm bore magnet. The measured value of 1/( 2J) for lactate was 72 ms. To improve the homogeneity of the B1 field, a 20 mm diameter Helmholtz coil was used to accommodate a 5 mm diameter NMR tube filled to a height of about 10 mm with lactate solution. Homemade 100 mm diameter gradient coils were used to reduce eddy current effects.
611
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(4
1.00
1.31
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FIG. 3. Doublet components of spectra obtained from a sample of lithium lactate in D20 using the pulse sequence shown in Fig. lb. The two filter gradients have the same polarity in a and have opposite polarity in b.
All gradient pulse lengths were 2 ms and T was set to 7 ms to allow a 5 ms settling time after the filter gradient and before the read pulse. ACKNOWLEDGMENTS This work has been supported by an equipment grant from the Alberta Heritage Foundation for Medical Research (AHFMR) and an operating grant from the Medical Research Council of Canada. A Medical Scientist Award (PSA) from AHFMR is also gratefully acknowledged. REFERENCES 1. C. L. DUMOULIN, J. Magn. Reson. 64, 38 (1985). 2. C. L. DUMOULIN AND D. VATIS, Mugn. Reson. Med. 3,282 ( 1986). 3. C. H. SOTAK AND D. M. FREEMAN, J. Magn. Reson. 77, 382 (1988). 4. C. H. SOTAK, D. M. FREEMAN, AND R. E. HURD, .I. Magn. Reson. 78,355 ( 1988). 5. G. C. MCKINNON AND P. BOSIGER, Magn. Reson. Med. 6,334 ( 1988). 6. G. C. MCKINNON AND P. WSIGER, Magn. Resort Med. 8,355 (1988). 7. A. KNUTTEL AND R. KIMMICH, Magn. Reson. Med. 10,404 ( 1989). 8. R. E. HURD AND D. M. FREEMAN, Proc. Natl. Acad. Sci. USA 86,4402 ( 1989). 9. W. NOSEL, L. A. TRIMBLE, J. F. SHEN, AND P. S. ALLEN, Magn. Resort. Med. 11, 398 ( 1989). 10. L. A. TRIMBLE, J. F. SHEN, A. H. WILMAN, AND P. S. ALLEN, J. Mugn. Reson. 86, 191 ( 1990). 11. T. H. MARECI AND R. FREEMAN, J. Mugn. Reson. 48, 158 ( 1982). 12. D. PIVETEAU, M. A. DELSUC, E. GUITTET, AND J. Y. LALLEMAND, Magn. Reson. Chem. 23, 127 (1985). 13. R. R. ERNST, G. BODENHAUSEN, AND A. WOKAUN, “Principles of Nuclear Magnetic Resonance in One and Two Dimensions,” p. 32, Clarendon Press, Oxford, 1987.
OF MAGNETIC
90,606-611
RESONANCE
( 1990)
Enhancementof Double-Quantum-Filtered Signals Using Optimized Tip Angle RF Pulses J. F. SHEN AND P. S. ALLEN* Department of Applied Sciences in Medicine, University of Alberta, Edmonton, Alberta, Canada T6G 2G3 Received May 22, 1990; revised July 17, 1990
In addition to their exploitation in high-resolution NMR, multiple-quantum filters provide a promising tool for spectral editing in in vivo spectroscopy. In particular, their use has been extensively explored for the purpose of observing the proton spectrum of metabolites with coupled spins such as lactate ( l-10). One of the more challenging difficulties faced when working in vivo is the low concentration of many metabolites of interest. As a result, any means of increasing the signal strength is of value. This Communication describes how the double-quantum-filtered signal can be enhanced for some double-quantum filters by up to 69%. This enhancement can be brought about by changing the tip angles of some of the pulses within the filter sequence from the conventional 90” to an optimized value that yields maximum conversion between anti-phase coherence (AC) and doublequantum coherence (DQC). Conversion pulses with tip angles other than 90” have been used in two-dimensional INADEQUATE experiments to achieve quadrature detection in the double-quantum frequency domain (11, 12).
Figure 1 illustrates variants of two typical double-quantum-filter formats which use the gradients labeled G and 2G for filtering. The unlabeled gradients occurring in these two sequences are trimming gradients used effectively to increase field inhomogeneities. When the tip angles of the pulses labeled rc/and 6 in Fig. 1 are both 90”, Fig. 1a reduces to the double-quantum filter proposed by Sotak et al. (4) and Fig. 1b reduces to that used to integrate double-quantum filtering and volume selection when all three 90” pulses are made slice selective ( 7). However, the optimal value for 1c/and 6 is not necessarily 90”, because the conversions from DQC to AC by the $ pulse and from AC to DQC by the 6 pulse have efficiencies that are a function of the tip angle of the II/ and 6 pulses. This functionality arises from the phase factor that must be incorporated into the analysis in order to account for field inhomogeneities. For a spin system with two weakly coupled spins I and S, for example, the doublequantum coherence before the filter gradient G can be represented in projection operator formalism ( 7, 13) by cz+s+ + c*z_s-,
[II
where c is a scalar coefficient and c* is its complex conjugate, which together take account of the differences between the two sequences in Fig. 1. If we adopt the con* To whom correspondence should be addressed. 0022-2364190 $3.00 Copyright 0 1990 by Academic Press. Inc. AN rights of reproduction in any form reserved.
606
COMMUNICATIONS
607
FIG. I. Double-quantum-filtering sequences (a) with and (b) without 180” refocusing pulses. The filter gradients are labeled G and 2G. Unlabeled gradients are trimming gradients.
vention of positive rotation in the right-handed sense (namely, clockwise rotation when looking in the direction of the relevant rotation axis) ( 13)) the DQC immediately after the gradient G becomes + c*I-S-ei24,
cI+S+e-i2”
Dl
where the phase factor 6 is spatially distributed and is given by C#J =
s
G-rdt,
and where we have assumed a short gradient period so that chemical-shift dephasing can be neglected during the gradient pulse. For either of the sequences in Fig. 1, a read pulse (labeled $) of arbitrary tip angle 0, applied along the x axis of the rotating frame, will give rise to the following transformation to AC of the first DQC component of Eq. [ 21: I, S+e-i24 -
ox
0 i(l+S, + S+Z,)e-i2%in 0 cos2~ + i(l-S;
6 + S_Z,)e-‘24sin 0 sin2 2
+ z-order, zero- and double-quantum
coherences.
I31
The analogous transformation of 1-S may be obtained by taking the complex conjugate of Eq. [ 31. To understand the effect of the spatially distributed phase factors, it is convenient to regroup the AC so produced into two different components. The first, proportional to sin f3cos*( o/2), corresponds to the pathway Z+ + I+ (or S, --, S,). The second, proportional to sin 0 sin2( 8/2), corresponds to the pathway I+ --) Ii (or S, + S,). The subsequent rephasing filter gradient 2G, which is twice as strong (or long) as the dephasing filter gradient, refocuses one, but not both, of these two components. Which component is refocused is determined by the relative polarity of the two filter gradients. If the two gradients have the same polarity, it is the second component that is refocused. The amplitude of the AC so produced varies between 0 and 3fi/8 ( x0.65) depending on 0, with the maximum value occurring at 0 = 120”.
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This maximum value is about 30% greater than that occurring when 6 = 90”. If the two gradients have opposite polarity, the first component is refocused and the variation with 0 of the AC amplitude covers the same range as the previous case, but this time it maximizes at 6 = 60”. By comparison, when the filter gradients G and 2G are absent, a read pulse tip angle of 0 = 90” gives the maximum conversion of DQC to AC (i.e., amplitude = 1). The sequence in Fig. lb can be further enhanced over that possible for Fig. la by using an optimal tip angle for the preceding pulse (labeled 6) also. Since no 180” refocusing pulse is used between the first and the second pulses in Fig. 1b, the antiphase coherence before the second pulse is already dephased by field inhomogeneities and can be written as 2(Z,cos f$i + Z,sin &)S, + 2(S,cos 4s + S,sin $s)Z= = (Z+@+ + Z-ei@l)S, + (S+epih + S-e’@)Z,,
[4]
where $i and & are phase factors caused by field inhomogeneities and the trimming gradient, as well as by the chemical shifts of the I and S spins, respectively. The trimming gradient ensures that the dephasing is complete. An RF pulse with a tip angle 0 and phase x will give rise to the following transformation of the AC for the I spin:
ox
(Z+e-‘“I + Z_e’+~)S, ----
-
0
tJ
Z+e-‘+I + Z-eibl cos* -2 + (Z-e- I41+ Z+e’“*)sin* 2I I( ) X
&sin 6 + z-order and single-quantum
i =-[(Z+S+e~ib1-Z-S-ei~1)cos2~-(Z~S-e~.m~2 + z-order, zero-quantum
coherences
1
Z+S+ei”l)sin2 4 sin 0 and single-quantum
coherences.
[5]
The analogous transformation of the AC for the S spin may be obtained by interchanging the I and the S spins in Eq. [ 51. The result of the transformation of Eq. [ 41 can be separated into two components that are entirely analogous to those obtained from the transformation of Eq. [2]. In order to refocus the field inhomogeneities occurring in the first 1 / (2 J) period with the field inhomogeneities present during the second 1/( 2J) period in Fig. lb, the combined pathway for the 6 and + pulses has to be Z, + Z5 (or S, --* ST), because the field inhomogeneities producing 4, (or 4s) may not be reversed manually like the relative polarity of the filter gradients. Two overall pathways satisfy this requirement, Z, 5 Z+ A Z7 and Z+ 2 Z7 5 Z5. However, in order to form an echo, the effect of the filter gradients must also be refocused. Our preceding analysis on the read pulse showed that only one of the two pathways, Z+ A ZT and Z+ J$ ZT (same as Z+ A I,), refocuses the effect of the filter gradients at any one time. If the two filter gradients have the same polarity, the pathway Z, A Zt A IT leads to the refocusing of the effects of both the field inhomogeneities and the filter gradients. If the two filter gradients have opposite polarity, the pathway Z, L IT A IT will give rise to the refocusing.
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TABLE 1 Conversion Efficiencies of the S and # Pulses Conversion efficiency at a tip angle of 8 Relative polarity of filter gradients
For 6
Optimal angle (degrees)
For rl/
Conversion efficiency at
For 8
For $
0 = 90”
0 = Optimal angle
Same
c9 sin 6 cos* 2
sinesin*-
e 2
60
120
0.5
g
Opposite
0 sin 0 sin* 2
sinOcos*-
6 2
120
60
0.5
3Jj c=z0.65 8
8
zz 0.65
The results for the conversion from DQC to AC by the + pulse and from AC to DQC by the 6 pulse are summarized in Table 1, where the conversion efficiency occurring in the table is defined as the coefficient of the refocused component, and where the optimal angle is defined as the tip angle of the conversion pulse that yields the maximum conversion efficiency. Although a two-spin system has been used in our analysis, the results are also valid for any product operators involving only two spin operators. For instance, for an AX3 spin system with scalar coupling constant J between the A and the X spins, the double-quantum coherence immediately before the read pulse for both sequences shown in Fig. 1 is proportional to (7, 10) + C*I-Sjeei2’),
$ (CI+Sj+e-‘*’
]=I
[61
where ZI correspond to the A spins and S’, correspond to the X spins and where the mutual complex conjugates c and c* take account of the differences in the two sequences in Fig. 1. We have also neglected, in Eq. [ 61, the more complex terms involving four spins, which only contribute to the A quartet signal. Since each individual term in Eq. [ 6 ] involves only two spins, our previous analysis predicts a 30% increase in the doublet signal if the tip angle of the read pulse is changed from 90” to the optimal value. The antiphase coherence arising from the X spins in an AX3 system immediately before the second pulse in Fig. lb is given by 3
C
j=l
(SjxCOS
4
+
&sin
4)Zz,
171
where the phase factor 4 is caused by field inhomogeneities, trimming gradients, and chemical shifts. Once again each individual term in Eq. [ 71 contains only two spins; a further increase of about 30% in the detectable doublet signal is expected if the tip angle of the second pulse is changed from 90” to the optimal value. Using optimal tip angles for both the second and the third pulses in sequence Fig. lb will yield a total increase in the doublet signal of about 69%. These predictions have been verified by experiments with the AX3 spin system of lactate dissolved in D20. Figure 2 and Fig. 3 show the doublet spectra acquired using
COMMUNICATIONS
610
1.31
(a)
v=90”
I v=60”
FIG. 2. Doublet components of spectra obtained from a sample of lithium lactate in DzO using the pulse sequence shown in Fig. la. The two filter gradients have the same polarity in a and have opposite polarity in b.
the sequences of Fig. la and Fig. lb, respectively. The number above each doublet is its amplitude measurement. In order to refocus the effects of field inhomogeneities in the sequence of Fig. lb, it was necessary to introduce minor adjustments into the timing which would not be anticipated for the case of a perfectly uniform static magnetic field ( 7). Because the manifestation of the field inhomogeneities is dependent on which pathways are chosen to be exploited, the timing adjustments are also pathway dependent. When the filter gradients have the same polarity, the interval between the first two pulses needs to be shortened by 27, where T is the interval between the second and the third pulse in Fig. lb. When opposite polarity filter gradients are used, the same shortening, 27, should be applied to the interval between the last pulse and the start of acquisition. In practice, the actual adjustments made were not exactly 27 and this probably reflects the difference in eddy current fields during the corresponding intervals. This change in timing could result in partial loss of signal due to incomplete conversion between in-phase and anti-phase coherences. However, without the timing adjustment, the effects of field inhomogeneities are not fully refocused and this could have led to a larger loss of signal and to distortions of the spectra. All data were acquired on a Bruker 100 MHz spectrometer with a 40 cm bore magnet. The measured value of 1/( 2J) for lactate was 72 ms. To improve the homogeneity of the B1 field, a 20 mm diameter Helmholtz coil was used to accommodate a 5 mm diameter NMR tube filled to a height of about 10 mm with lactate solution. Homemade 100 mm diameter gradient coils were used to reduce eddy current effects.
611
COMMUNICATIONS 1.78
1.30
(4
1.00
1.31
V
I
1.72
FIG. 3. Doublet components of spectra obtained from a sample of lithium lactate in D20 using the pulse sequence shown in Fig. lb. The two filter gradients have the same polarity in a and have opposite polarity in b.
All gradient pulse lengths were 2 ms and T was set to 7 ms to allow a 5 ms settling time after the filter gradient and before the read pulse. ACKNOWLEDGMENTS This work has been supported by an equipment grant from the Alberta Heritage Foundation for Medical Research (AHFMR) and an operating grant from the Medical Research Council of Canada. A Medical Scientist Award (PSA) from AHFMR is also gratefully acknowledged. REFERENCES 1. C. L. DUMOULIN, J. Magn. Reson. 64, 38 (1985). 2. C. L. DUMOULIN AND D. VATIS, Mugn. Reson. Med. 3,282 ( 1986). 3. C. H. SOTAK AND D. M. FREEMAN, J. Magn. Reson. 77, 382 (1988). 4. C. H. SOTAK, D. M. FREEMAN, AND R. E. HURD, .I. Magn. Reson. 78,355 ( 1988). 5. G. C. MCKINNON AND P. BOSIGER, Magn. Reson. Med. 6,334 ( 1988). 6. G. C. MCKINNON AND P. WSIGER, Magn. Resort Med. 8,355 (1988). 7. A. KNUTTEL AND R. KIMMICH, Magn. Reson. Med. 10,404 ( 1989). 8. R. E. HURD AND D. M. FREEMAN, Proc. Natl. Acad. Sci. USA 86,4402 ( 1989). 9. W. NOSEL, L. A. TRIMBLE, J. F. SHEN, AND P. S. ALLEN, Magn. Resort. Med. 11, 398 ( 1989). 10. L. A. TRIMBLE, J. F. SHEN, A. H. WILMAN, AND P. S. ALLEN, J. Mugn. Reson. 86, 191 ( 1990). 11. T. H. MARECI AND R. FREEMAN, J. Mugn. Reson. 48, 158 ( 1982). 12. D. PIVETEAU, M. A. DELSUC, E. GUITTET, AND J. Y. LALLEMAND, Magn. Reson. Chem. 23, 127 (1985). 13. R. R. ERNST, G. BODENHAUSEN, AND A. WOKAUN, “Principles of Nuclear Magnetic Resonance in One and Two Dimensions,” p. 32, Clarendon Press, Oxford, 1987.