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Analytical Expressions for the DANTE Pulse Sequence
JOURNAL OF MAGNETIC RESONANCE,
Series A 117, 103–108 (1995)
Analytical Expressions for the DANTE Pulse Sequence D. CANET,* J. BRONDEAU,*
AND
C. ROUMESTAND†
*Laboratoire de Me´thodologie RMN [URA CNRS 406 (LESOC), FU CNRS E008 (INCM) ], Universite´ Henri Poincare´, Nancy I, BP 239, 54506 Vandoeuvre les Nancy Cedex, France; and †Centre de Biochimie Structurale, CNRS-UMR 995, INSERM-U414, Universite´ de Montpellier I, Faculte´ de Pharmacie, 15 Avenue Charles Flahault, 34060 Montpellier Cedex 1, France Received February 8, 1995; revised April 18, 1995
s 2x Å s 2y Å s 2z Å 1 (identity)
Since it was first published (1), the DANTE pulse sequence and its numerous variants (2) have been used mainly for the purpose of selective excitation. The basic DANTE train can be represented as
sxsy Å isz
sysx Å 0isz
sysz Å isx
szsy Å 0isx
[( a )x 0 t]m ,
szsx Å isy
sxsz Å 0isy .
[1]
where ( a )x stands for a radiofrequency pulse of flip angle a applied along the x axis of the rotating frame and t is a precession interval which, together with the repetition of the elementary sequence m times, ensures frequency selectivity. It has been often claimed that a DANTE sequence, which is made of hard pulses, is equivalent to a rectangular soft pulse of appropriate length and amplitude. This may be questioned mathematically and further checked by numerical simulations. Obtaining the selectivity profile is in any event useful for the evaluation of sequences which may involve one or more DANTE trains of different characteristics. The performance of a given DANTE pulse train can actually be obtained through the analytical expressions which are derived below. To the best of our knowledge, such a derivation does not exist in the literature and we feel it could be useful for (i) speeding up numerical calculations, and (ii) gaining some insight into the intrinsic features of the DANTE sequence. The derivation is based on the use of Pauli matrices and rests on two simplifying assumptions: (i) relaxation is disregarded throughout the whole sequence, and (ii) RF pulses are supposed strong enough so that off-resonance effects can be ignored. Let us first recall how Pauli matrices can be used to handle rotations: an arbitrary rotation by an angle d around a direction specified by its unitary vector n of components (nx , ny , nz ) can be represented by Rn ( d ) Å cos( d /2) / i sin( d /2) sn
[2]
[3]
Following a procedure used in an earlier publication (3), we shall first attempt to find the characteristics of a single rotation equivalent to the elementary sequence [( a )x 0 t], recognizing that the t interval amounts to a rotation about z by an angle b Å 2pnt which, of course, depends on the precession frequency (in the rotating frame) of the magnetization considered. Hence [( a )x 0 t] can be represented by Rz ( 0 b )Rx ( 0 a ),
[4]
where the sense of nutation and precession (for a positive gyromagnetic ratio) are accounted by the signs of the angles appearing in the rotation operators. Using the general representation outlined in [2], we can write Rz ( 0 b )Rx ( 0 a ) Å [cos( b /2) 0 i sin( b /2) sz ] 1 [cos( a /2) 0 i sin( a /2) sx ] Å cos( a /2)cos( b /2) 0 i sin( a /2)cos( b /2) sx 0 i sin( a /2)sin( b /2) sy
[5]
0 i cos( a /2)sin( b /2) sz .
The single rotation by an angle dd around a direction defined by the unitary vector n (nx , ny , nz ) follows immediately from the identification of [2] and [5]
with cos( dd /2) Å cos( a /2)cos( b /2)
sn Å nxsx / nysy / nzsz ,
where sx , sy , and sz are the three Pauli matrices which will be used here only through the relationships
nx Å 0 [sin( a /2)cos( b /2)]/sin( dd /2) ny Å 0 [sin( a /2)sin( b /2)]/sin( dd /2)
103
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NOTES
tended to more than two rotations and therefore could be applied to a more complicated pulse train. In order to gain some insight into the rotation process induced by a simple DANTE sequence, let us denote by z * the direction defined by n and let us define the frame (x *, y *, z * ) such that z lies in the x *z * plane (Fig. 1). This frame is defined with respect to the (x, y, z) frame by the angles ud and fd . It turns out that cos( ud ) is simply nz , hence cos( ud) Å 0 [cos( a /2)sin( pnt )]/sin( dd /2) with cos( dd /2) Å cos( a /2)cos( pnt ),
[7]
whereas tan( fd ) is seen to be equal to ny /nx ; therefore, fd Å pnt.
FIG. 1. Definition of a frame (x *, y *, z * ) where z * is the effective rotation axis, with respect to the normal rotating frame ( x, y, z).
nz Å 0 [cos( a /2)sin( b /2)]/sin( dd /2).
[6]
Thus, the effect of the DANTE train, involving m cycles, amounts to a rotation by an angle mdd about the axis defined by n. It can be mentioned that this approach could be ex-
[8]
From these considerations, we can visualize how a DANTE sequence works. Let us first consider the two limiting cases: (i) the on-resonance situation ( n Å 0) which implies that ud Å p /2 regardless of the value chosen for a; the effective rotation axis is along x and magnetization nutates about x. (ii) A largely off-resonance situation; we assume further small flip angles (as this is usually the case so that cos( a /2) and sin( a /2) can be approximated by 1 and 0, respectively). From the first relation in [6], it appears that dd É 0 b (the sign arises from the fact that the effective rotation angle dd must be negative for a positive frequency). Hence, nx , ny É 0, whereas nz É 1; ud is therefore close to zero and the
FIG. 2. Effective rotation axis (z * ) and effective rotation angles for a magnetization starting from equilibrium as a function of the precession frequency n (in the rotating frame). (Top) DANTE sequence, (Bottom) soft pulse.
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effective rotation axis is close to the z axis. For magnetization initially at equilibrium, there is no further rotation. These two limiting cases assess the selectivity features of a DANTE sequence. For an intermediate situation, and still for small a values, magnetization initially at equilibrium is rotated by an angle mdd É 02mpnt about the effective rotation axis (defined by the angles ud and fd ). This rotation is responsible for the well-known side lobes in DANTE selectivity profiles. The three situations that have just been discussed are depicted in Fig. 2 and compared with magnetization evolution under the application of a soft pulse. At this stage, it may be instructive to look at the selectivity profile of an inverting DANTE pulse train employed in the DANTE-Z procedure (4). In a general way, the z component of magnetization, after application of a DANTE sequence to the equilibrium magnetization M0 , is given by Mz Å M0[cos 2 ( ud ) / sin 2 ( ud )cos(mdd )].
1 1 / ( n1 / n ) 2
q
(cos 2ud cos 2fd / sin 2fd )cos mdd
[9]
mdd 2 / cos ud sin mdd
sin 2ud sin 2fd sin 2
mdd 2 0 sin ud sin fd sin mdd sin 2ud cos fd sin 2
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[10]
where ts is the soft pulse duration. The obvious difference with respect to the DANTE pulse train is the orientation of the effective rotation axis which lies here in the xz plane. In order to determine whether this difference will change the final result, we must examine the relevant rotation matrices. For that goal, we shall resort to the following general equation which describes the transformation of an arbitrary vector M under a rotation by an angle c about an axis defined by its unitary vector n
where r and Ú stand for scalar and vector products, respectively. The elements of the rotation matrix are then easily deduced from each of the components of M *, expressed as a linear combination of M components: M *x Å R11 Mx / R12 My / R13 Mz M *y Å R21 Mx / R22 My / R23 Mz M *z Å R31 Mx / R32 My / R33 Mz . The complete rotation matrix Rd corresponding to a DANTE experiment is obtained by this procedure using c Å mdd and nx Å cos fd sin ud , ny Å sin fd sin ud , nz Å cos ud (see Fig. 1, n being along the z * axis):
mdd 2 0 cos ud sin mdd
sin 2ud sin 2fd sin 2
/ sin 2ud cos 2fd
Rd Å
q
ds Å 02p n 21 / n 2 ts ,
M * Å (Mrn)n / cos c[M 0 (Mrn)] / sin cnÚM,
This latter expression may be obtained by a direct calculation or from the complete rotation matrix (see Eq. [11]). Because ud goes slowly toward zero when frequency is increased (see Fig. 2), Mz can be seen to tend toward M0 by lower values (evidently without any overshoot) with some oscillations due to the factor cos(mdd ). From the approximation given above, the period of these oscillations is 1/mt in the frequency domain. A typical Mz profile illustrating these features is shown in Fig. 3. It emphasizes the advantages of resorting to the z component of magnetization which exhibits much fewer oscillations than transverse components. We turn now to the case of a soft pulse. One has for the angle us between the effective rotation axis z * and z
cos us Å
with n1 Å gB1 /2p, where B1 is the RF field amplitude used for the soft pulse. The angle by which magnetization rotates about z * is simply given by
(cos 2ud sin 2fd / cos 2fd )cos mdd 2
2
/ sin ud sin fd
mdd 2 / sin ud cos fd sin mdd sin 2ud sin fd sin 2
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mdd 2 / sin ud sin fd sin mdd
sin 2ud cos fd sin 2
md d 2 0 sin ud cos fd sin mdd sin 2ud sin fd sin 2
sin 2ud cos mdd / cos 2ud
.
[11]
106
NOTES
In the case of a soft pulse, the matrix Rs is established with c Å ds , nx Å sin us , ny Å 0, and nz Å cos us :
cos 2us cos ds / sin 2us 0cos us sin ds Rs Å
cos us sin ds sin 2us sin 2
ds 2
sin 2us sin 2
ds 2
cos ds
0sin us sin ds
sin us sin ds
sin 2us cos ds / cos 2us
In order to assess a possible equivalence between a DANTE sequence and a soft pulse, the rotation matrices [11] and [12] must be compared. Concerning the left-most and the central columns (which correspond to rotations applied to initial magnetization along x and y, respectively), we failed to find any mathematical relationship which would make the two methods equivalent. Conversely, such an equivalence can be found for the third column which corresponds to a rotation applied to magnetization initially along z. Let us first consider the z component of the resulting magnetization and denote by M dz and M sz the relevant quantities in the case of a DANTE sequence and of a soft pulse, respectively, M dz Å M0 (sin 2ud cos mdd / cos 2ud ) M sz Å M0 (sin 2us cos ds / cos 2us ), where M0 is the equilibrium magnetization supposed to lie initially along z. These two quantities are equivalent provided that ud and us are identical and that ds Å mdd . The equivalence is less obvious as far as transverse components
.
[12]
are concerned. However, forming Mx / iMy in both cases, we arrive at (Mx / iMy ) d Å M0 exp(ifd ) 1
S S
sin 2ud sin 2
(Mx / iMy ) s Å M0 sin 2us sin 2
D
mdd 0 i sin ud sin mdd 2
D
ds 0 i sin us sin ds . 2
Disregarding the phase factor exp(ifd ) (which amounts to a frequency-dependent phase correction), we can observe that both expressions are identical again with ud å us and ds å mdd . We are thus left with the conclusion that a DANTE sequence and a soft pulse can be regarded as equivalent provided the initial magnetization lies along the z axis. Yet, there is a small discrepancy regarding the phase factor involved in transverse magnetization; nevertheless, the modulus of transverse magnetization is identical in both cases. The strict equivalence for the longitudinal component as well as the dephasing which affects transverse components has been verified by pure numerical simulations (not based on the above analytical expressions) shown in Fig. 4. We must still worry about the meaning of the two equalities ud å us and ds å mdd . The latter relationship can be interpreted easily at zero frequency. Since in that case dd Å a, whereas ds Å 2pn1ts , this amounts to setting the global flip angle at the same value 2pn1ts Å 0ma.
[13]
Now considering the off-resonance situation for which dd can be approximated by 2pnt, we can write q
2p n 21 / n 2 ts Å 2pmnt. FIG. 3. The frequency profile (from 0200 to /200 Hz) of the longitudinal magnetization resulting from a DANTE inverting sequence ( a Å 4.57, t Å 1 ms, m Å 40), showing side lobes of period equal to 1/mt (in frequency units).
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If the RF field amplitude n1 is sufficiently low (as normally implied in the case of a soft pulse), ( n1 / n ) 2 is negligible with respect to 1 and we arrive at
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NOTES
FIG. 4. Frequency profiles (from 0200 to 200 Hz) of Mx , My , Mz components obtained under the application of a DANTE 907 sequence (full curves) and of an ‘‘equivalent’’ soft pulse (dashed curves). Characteristics of the DANTE pulse train: a Å 4.57, t Å 1 ms, m Å 20; soft pulse duration: 20 ms. A perfect equivalence can be noted for the longitudinal component, whereas differences observed for transverse components are ascribed to a phase factor (see text).
ts Å mt.
[14]
This latter identity demonstrates that the same selectivity is achieved through a DANTE sequence or a soft pulse provided that their global duration are identical. Relationships [13] and [14] can be shown to be approximately consistent with the condition cos dd Å cos us . As mentioned before, cos us É 1 0 (1/2)( n1 / n ) 2 . On the other hand, cos ud can be expressed as cos ud Å 0tan( pnt )/tan( dd /2) (see Eq. q [7]). From relations [10] and [14], dd /2 Å pnt 1 / ( n1 / n ) 2 and can be expanded as a function of ( n1 / n ) 2 , yielding cos ud É 1 0 (1/2)( n1 / n ) 2 1/ sin c(2pnt ). It thus appears that the equivalence between us and ud is verified by n significantly smaller than 1/4 t. Discrepancies between a soft pulse and the corresponding DANTE sequence may be discussed in the light of modifications proposed by Wu et al. (5), especially that concerning a symmetrized DANTE sequence which can be schematized as [( t /2) 0 ( a )x 0 ( t /2)]n .
[15]
Carrying out similar calculations as for the original DANTE
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sequence yields the same expression for dd , whereas the components of the unitary vector defining the effective rotation axis become nx Å 0sin( a /2)/sin( dd /2) ny Å 0 nz Å 0 [cos( a /2)sin( b /2)]/sin( dd /2).
[16]
Apart the fact that nz and ud (see [6] and [7]) are unmodified, the striking feature is that the y component is zero. Therefore, as for the soft pulse, the effective rotation axis lies in the xz plane. Moreover, the rotation matrices for the soft pulse and for the symmetrized DANTE sequence share exactly the same structure with ds substituted by dd and us substituted by ud (see [12]). As a matter of fact, the phase shift, alluded to above in the case of the original DANTE sequence, totally disappears, and even if tiny numerical differences still exist, they are almost invisible in the plotted profiles (not shown). As a conclusion, it can be stressed that the analytical expressions for the original DANTE sequence indicate an approximate mathematical equivalence with a soft pulse pro-
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NOTES
vided that magnetization is initially along the z axis. However, both procedures become quasi-identical, for any initial situation, when a symmetrized DANTE sequence is used (see [15]). This equivalence rests on the choice of identical flip angles at zero frequency and of identical overall durations. Finally, it can be mentioned that the whole rotation matrix (given by [11]) could be useful to optimize experiments which involve concatenated DANTE pulse trains (6). ACKNOWLEDGMENT It is a pleasure to thank Professor Ray Freeman who pointed out that the symmetrized DANTE removes the apparent dephasing with respect to the equivalent soft pulse.
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REFERENCES
1. G. Bodenhausen, R. Freeman, and G. A. Morris, J. Magn. Reson. 23, 171 (1976). 2. R. Freeman, Chem. Rev. 91, 1397 (1991). 3. J. Brondeau and D. Canet, J. Magn. Reson. 47, 159 (1982). 4. D. Boudot, D. Canet, J. Brondeau, and J. C. Boubel, J. Magn. Reson. 83, 428 (1989). 5. X. L. Wu, P. Xu, J. Friedrich, and R. Freeman, J. Magn. Reson. 81, 206 (1989). 6. C. Roumestand, D. Canet, N. Mahieu, and F. Toma, J. Magn. Reson. A 106, 168 (1994).
AP: Mag Res
Series A 117, 103–108 (1995)
Analytical Expressions for the DANTE Pulse Sequence D. CANET,* J. BRONDEAU,*
AND
C. ROUMESTAND†
*Laboratoire de Me´thodologie RMN [URA CNRS 406 (LESOC), FU CNRS E008 (INCM) ], Universite´ Henri Poincare´, Nancy I, BP 239, 54506 Vandoeuvre les Nancy Cedex, France; and †Centre de Biochimie Structurale, CNRS-UMR 995, INSERM-U414, Universite´ de Montpellier I, Faculte´ de Pharmacie, 15 Avenue Charles Flahault, 34060 Montpellier Cedex 1, France Received February 8, 1995; revised April 18, 1995
s 2x Å s 2y Å s 2z Å 1 (identity)
Since it was first published (1), the DANTE pulse sequence and its numerous variants (2) have been used mainly for the purpose of selective excitation. The basic DANTE train can be represented as
sxsy Å isz
sysx Å 0isz
sysz Å isx
szsy Å 0isx
[( a )x 0 t]m ,
szsx Å isy
sxsz Å 0isy .
[1]
where ( a )x stands for a radiofrequency pulse of flip angle a applied along the x axis of the rotating frame and t is a precession interval which, together with the repetition of the elementary sequence m times, ensures frequency selectivity. It has been often claimed that a DANTE sequence, which is made of hard pulses, is equivalent to a rectangular soft pulse of appropriate length and amplitude. This may be questioned mathematically and further checked by numerical simulations. Obtaining the selectivity profile is in any event useful for the evaluation of sequences which may involve one or more DANTE trains of different characteristics. The performance of a given DANTE pulse train can actually be obtained through the analytical expressions which are derived below. To the best of our knowledge, such a derivation does not exist in the literature and we feel it could be useful for (i) speeding up numerical calculations, and (ii) gaining some insight into the intrinsic features of the DANTE sequence. The derivation is based on the use of Pauli matrices and rests on two simplifying assumptions: (i) relaxation is disregarded throughout the whole sequence, and (ii) RF pulses are supposed strong enough so that off-resonance effects can be ignored. Let us first recall how Pauli matrices can be used to handle rotations: an arbitrary rotation by an angle d around a direction specified by its unitary vector n of components (nx , ny , nz ) can be represented by Rn ( d ) Å cos( d /2) / i sin( d /2) sn
[2]
[3]
Following a procedure used in an earlier publication (3), we shall first attempt to find the characteristics of a single rotation equivalent to the elementary sequence [( a )x 0 t], recognizing that the t interval amounts to a rotation about z by an angle b Å 2pnt which, of course, depends on the precession frequency (in the rotating frame) of the magnetization considered. Hence [( a )x 0 t] can be represented by Rz ( 0 b )Rx ( 0 a ),
[4]
where the sense of nutation and precession (for a positive gyromagnetic ratio) are accounted by the signs of the angles appearing in the rotation operators. Using the general representation outlined in [2], we can write Rz ( 0 b )Rx ( 0 a ) Å [cos( b /2) 0 i sin( b /2) sz ] 1 [cos( a /2) 0 i sin( a /2) sx ] Å cos( a /2)cos( b /2) 0 i sin( a /2)cos( b /2) sx 0 i sin( a /2)sin( b /2) sy
[5]
0 i cos( a /2)sin( b /2) sz .
The single rotation by an angle dd around a direction defined by the unitary vector n (nx , ny , nz ) follows immediately from the identification of [2] and [5]
with cos( dd /2) Å cos( a /2)cos( b /2)
sn Å nxsx / nysy / nzsz ,
where sx , sy , and sz are the three Pauli matrices which will be used here only through the relationships
nx Å 0 [sin( a /2)cos( b /2)]/sin( dd /2) ny Å 0 [sin( a /2)sin( b /2)]/sin( dd /2)
103
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1064-1858/95 $12.00 Copyright q 1995 by Academic Press, Inc. All rights of reproduction in any form reserved.
AP: Mag Res
104
NOTES
tended to more than two rotations and therefore could be applied to a more complicated pulse train. In order to gain some insight into the rotation process induced by a simple DANTE sequence, let us denote by z * the direction defined by n and let us define the frame (x *, y *, z * ) such that z lies in the x *z * plane (Fig. 1). This frame is defined with respect to the (x, y, z) frame by the angles ud and fd . It turns out that cos( ud ) is simply nz , hence cos( ud) Å 0 [cos( a /2)sin( pnt )]/sin( dd /2) with cos( dd /2) Å cos( a /2)cos( pnt ),
[7]
whereas tan( fd ) is seen to be equal to ny /nx ; therefore, fd Å pnt.
FIG. 1. Definition of a frame (x *, y *, z * ) where z * is the effective rotation axis, with respect to the normal rotating frame ( x, y, z).
nz Å 0 [cos( a /2)sin( b /2)]/sin( dd /2).
[6]
Thus, the effect of the DANTE train, involving m cycles, amounts to a rotation by an angle mdd about the axis defined by n. It can be mentioned that this approach could be ex-
[8]
From these considerations, we can visualize how a DANTE sequence works. Let us first consider the two limiting cases: (i) the on-resonance situation ( n Å 0) which implies that ud Å p /2 regardless of the value chosen for a; the effective rotation axis is along x and magnetization nutates about x. (ii) A largely off-resonance situation; we assume further small flip angles (as this is usually the case so that cos( a /2) and sin( a /2) can be approximated by 1 and 0, respectively). From the first relation in [6], it appears that dd É 0 b (the sign arises from the fact that the effective rotation angle dd must be negative for a positive frequency). Hence, nx , ny É 0, whereas nz É 1; ud is therefore close to zero and the
FIG. 2. Effective rotation axis (z * ) and effective rotation angles for a magnetization starting from equilibrium as a function of the precession frequency n (in the rotating frame). (Top) DANTE sequence, (Bottom) soft pulse.
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105
NOTES
effective rotation axis is close to the z axis. For magnetization initially at equilibrium, there is no further rotation. These two limiting cases assess the selectivity features of a DANTE sequence. For an intermediate situation, and still for small a values, magnetization initially at equilibrium is rotated by an angle mdd É 02mpnt about the effective rotation axis (defined by the angles ud and fd ). This rotation is responsible for the well-known side lobes in DANTE selectivity profiles. The three situations that have just been discussed are depicted in Fig. 2 and compared with magnetization evolution under the application of a soft pulse. At this stage, it may be instructive to look at the selectivity profile of an inverting DANTE pulse train employed in the DANTE-Z procedure (4). In a general way, the z component of magnetization, after application of a DANTE sequence to the equilibrium magnetization M0 , is given by Mz Å M0[cos 2 ( ud ) / sin 2 ( ud )cos(mdd )].
1 1 / ( n1 / n ) 2
q
(cos 2ud cos 2fd / sin 2fd )cos mdd
[9]
mdd 2 / cos ud sin mdd
sin 2ud sin 2fd sin 2
mdd 2 0 sin ud sin fd sin mdd sin 2ud cos fd sin 2
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[10]
where ts is the soft pulse duration. The obvious difference with respect to the DANTE pulse train is the orientation of the effective rotation axis which lies here in the xz plane. In order to determine whether this difference will change the final result, we must examine the relevant rotation matrices. For that goal, we shall resort to the following general equation which describes the transformation of an arbitrary vector M under a rotation by an angle c about an axis defined by its unitary vector n
where r and Ú stand for scalar and vector products, respectively. The elements of the rotation matrix are then easily deduced from each of the components of M *, expressed as a linear combination of M components: M *x Å R11 Mx / R12 My / R13 Mz M *y Å R21 Mx / R22 My / R23 Mz M *z Å R31 Mx / R32 My / R33 Mz . The complete rotation matrix Rd corresponding to a DANTE experiment is obtained by this procedure using c Å mdd and nx Å cos fd sin ud , ny Å sin fd sin ud , nz Å cos ud (see Fig. 1, n being along the z * axis):
mdd 2 0 cos ud sin mdd
sin 2ud sin 2fd sin 2
/ sin 2ud cos 2fd
Rd Å
q
ds Å 02p n 21 / n 2 ts ,
M * Å (Mrn)n / cos c[M 0 (Mrn)] / sin cnÚM,
This latter expression may be obtained by a direct calculation or from the complete rotation matrix (see Eq. [11]). Because ud goes slowly toward zero when frequency is increased (see Fig. 2), Mz can be seen to tend toward M0 by lower values (evidently without any overshoot) with some oscillations due to the factor cos(mdd ). From the approximation given above, the period of these oscillations is 1/mt in the frequency domain. A typical Mz profile illustrating these features is shown in Fig. 3. It emphasizes the advantages of resorting to the z component of magnetization which exhibits much fewer oscillations than transverse components. We turn now to the case of a soft pulse. One has for the angle us between the effective rotation axis z * and z
cos us Å
with n1 Å gB1 /2p, where B1 is the RF field amplitude used for the soft pulse. The angle by which magnetization rotates about z * is simply given by
(cos 2ud sin 2fd / cos 2fd )cos mdd 2
2
/ sin ud sin fd
mdd 2 / sin ud cos fd sin mdd sin 2ud sin fd sin 2
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mdd 2 / sin ud sin fd sin mdd
sin 2ud cos fd sin 2
md d 2 0 sin ud cos fd sin mdd sin 2ud sin fd sin 2
sin 2ud cos mdd / cos 2ud
.
[11]
106
NOTES
In the case of a soft pulse, the matrix Rs is established with c Å ds , nx Å sin us , ny Å 0, and nz Å cos us :
cos 2us cos ds / sin 2us 0cos us sin ds Rs Å
cos us sin ds sin 2us sin 2
ds 2
sin 2us sin 2
ds 2
cos ds
0sin us sin ds
sin us sin ds
sin 2us cos ds / cos 2us
In order to assess a possible equivalence between a DANTE sequence and a soft pulse, the rotation matrices [11] and [12] must be compared. Concerning the left-most and the central columns (which correspond to rotations applied to initial magnetization along x and y, respectively), we failed to find any mathematical relationship which would make the two methods equivalent. Conversely, such an equivalence can be found for the third column which corresponds to a rotation applied to magnetization initially along z. Let us first consider the z component of the resulting magnetization and denote by M dz and M sz the relevant quantities in the case of a DANTE sequence and of a soft pulse, respectively, M dz Å M0 (sin 2ud cos mdd / cos 2ud ) M sz Å M0 (sin 2us cos ds / cos 2us ), where M0 is the equilibrium magnetization supposed to lie initially along z. These two quantities are equivalent provided that ud and us are identical and that ds Å mdd . The equivalence is less obvious as far as transverse components
.
[12]
are concerned. However, forming Mx / iMy in both cases, we arrive at (Mx / iMy ) d Å M0 exp(ifd ) 1
S S
sin 2ud sin 2
(Mx / iMy ) s Å M0 sin 2us sin 2
D
mdd 0 i sin ud sin mdd 2
D
ds 0 i sin us sin ds . 2
Disregarding the phase factor exp(ifd ) (which amounts to a frequency-dependent phase correction), we can observe that both expressions are identical again with ud å us and ds å mdd . We are thus left with the conclusion that a DANTE sequence and a soft pulse can be regarded as equivalent provided the initial magnetization lies along the z axis. Yet, there is a small discrepancy regarding the phase factor involved in transverse magnetization; nevertheless, the modulus of transverse magnetization is identical in both cases. The strict equivalence for the longitudinal component as well as the dephasing which affects transverse components has been verified by pure numerical simulations (not based on the above analytical expressions) shown in Fig. 4. We must still worry about the meaning of the two equalities ud å us and ds å mdd . The latter relationship can be interpreted easily at zero frequency. Since in that case dd Å a, whereas ds Å 2pn1ts , this amounts to setting the global flip angle at the same value 2pn1ts Å 0ma.
[13]
Now considering the off-resonance situation for which dd can be approximated by 2pnt, we can write q
2p n 21 / n 2 ts Å 2pmnt. FIG. 3. The frequency profile (from 0200 to /200 Hz) of the longitudinal magnetization resulting from a DANTE inverting sequence ( a Å 4.57, t Å 1 ms, m Å 40), showing side lobes of period equal to 1/mt (in frequency units).
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If the RF field amplitude n1 is sufficiently low (as normally implied in the case of a soft pulse), ( n1 / n ) 2 is negligible with respect to 1 and we arrive at
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NOTES
FIG. 4. Frequency profiles (from 0200 to 200 Hz) of Mx , My , Mz components obtained under the application of a DANTE 907 sequence (full curves) and of an ‘‘equivalent’’ soft pulse (dashed curves). Characteristics of the DANTE pulse train: a Å 4.57, t Å 1 ms, m Å 20; soft pulse duration: 20 ms. A perfect equivalence can be noted for the longitudinal component, whereas differences observed for transverse components are ascribed to a phase factor (see text).
ts Å mt.
[14]
This latter identity demonstrates that the same selectivity is achieved through a DANTE sequence or a soft pulse provided that their global duration are identical. Relationships [13] and [14] can be shown to be approximately consistent with the condition cos dd Å cos us . As mentioned before, cos us É 1 0 (1/2)( n1 / n ) 2 . On the other hand, cos ud can be expressed as cos ud Å 0tan( pnt )/tan( dd /2) (see Eq. q [7]). From relations [10] and [14], dd /2 Å pnt 1 / ( n1 / n ) 2 and can be expanded as a function of ( n1 / n ) 2 , yielding cos ud É 1 0 (1/2)( n1 / n ) 2 1/ sin c(2pnt ). It thus appears that the equivalence between us and ud is verified by n significantly smaller than 1/4 t. Discrepancies between a soft pulse and the corresponding DANTE sequence may be discussed in the light of modifications proposed by Wu et al. (5), especially that concerning a symmetrized DANTE sequence which can be schematized as [( t /2) 0 ( a )x 0 ( t /2)]n .
[15]
Carrying out similar calculations as for the original DANTE
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sequence yields the same expression for dd , whereas the components of the unitary vector defining the effective rotation axis become nx Å 0sin( a /2)/sin( dd /2) ny Å 0 nz Å 0 [cos( a /2)sin( b /2)]/sin( dd /2).
[16]
Apart the fact that nz and ud (see [6] and [7]) are unmodified, the striking feature is that the y component is zero. Therefore, as for the soft pulse, the effective rotation axis lies in the xz plane. Moreover, the rotation matrices for the soft pulse and for the symmetrized DANTE sequence share exactly the same structure with ds substituted by dd and us substituted by ud (see [12]). As a matter of fact, the phase shift, alluded to above in the case of the original DANTE sequence, totally disappears, and even if tiny numerical differences still exist, they are almost invisible in the plotted profiles (not shown). As a conclusion, it can be stressed that the analytical expressions for the original DANTE sequence indicate an approximate mathematical equivalence with a soft pulse pro-
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NOTES
vided that magnetization is initially along the z axis. However, both procedures become quasi-identical, for any initial situation, when a symmetrized DANTE sequence is used (see [15]). This equivalence rests on the choice of identical flip angles at zero frequency and of identical overall durations. Finally, it can be mentioned that the whole rotation matrix (given by [11]) could be useful to optimize experiments which involve concatenated DANTE pulse trains (6). ACKNOWLEDGMENT It is a pleasure to thank Professor Ray Freeman who pointed out that the symmetrized DANTE removes the apparent dephasing with respect to the equivalent soft pulse.
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REFERENCES
1. G. Bodenhausen, R. Freeman, and G. A. Morris, J. Magn. Reson. 23, 171 (1976). 2. R. Freeman, Chem. Rev. 91, 1397 (1991). 3. J. Brondeau and D. Canet, J. Magn. Reson. 47, 159 (1982). 4. D. Boudot, D. Canet, J. Brondeau, and J. C. Boubel, J. Magn. Reson. 83, 428 (1989). 5. X. L. Wu, P. Xu, J. Friedrich, and R. Freeman, J. Magn. Reson. 81, 206 (1989). 6. C. Roumestand, D. Canet, N. Mahieu, and F. Toma, J. Magn. Reson. A 106, 168 (1994).
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