- Email: [email protected]
Exact solutions for selective-excitation pulses
JOURNAL OF MAGNETIC RESONANCE
94,3X-386
( 199 1)
Exact Solutions for Selective-Excitation
Pulses
J. W. CARISON Radiologic Imaging Laboratory, University of California, San Francisco, 400 Grandview Drive, South San Francisco, California 94080
Received January 24, 1991 Inverse scattering is used to solve the general problem of finding an applied RF field to produce a desired selective-excitation profile. The formalism is used to calculate the exact, closed-form solutions to the waveforms which produce a collection of refocusing slice profiles, the limit of which is a rectangular section. It is seen that this solution is also effective at producing a waveform which is effective as an excitation pulse or an inversion pUk.
0 1991 Academic F’res, Inc.
In the past few years there has been a steady interest in the general problem of creating RF pulses to produce a desired excitation profile. The need for these pulses grew with the development of imaging; the first reported pulses exploited the Fourier transform relationship between profile and input RF pulses ( 1) . Even though these arguments are strictly valid only for small-angle excitation, the pulses have shown to be adequate for larger-angle excitation and refocusing pulses. Improvement of this basic pulse shape has usually entailed numerical techniques to optimize the output profile (2, 3) or composite pulses (4) and results in improvement in profile of some other figure of merit. There has been some work on analytical solutions to the profiles for large-angle excitations (5, 6) but these have not been shown to be satisfactory refocusing pulses. This work describesa solution to the inverse problem for selective excitation. With some constraints on the desired slice profile for an imaging experiment, an amplitudemodulated RF pulse can be calculated which will produce that slice profile. A straightforward extension of this work allows it to be applied to frequency-selectivepulses in spectroscopy. SCATTERING
FORMALISM
The starting point for this analysis is the S&r&linger equation for a spin- 1 particle in a magnetic field in the interaction representation (rotating frame),
where B is the magnetic field in the rotating frame. The interaction representation wavefunction is related to the Schriidinger representation by a rotation around the static magnetic field:
[21 0022-236419 1 $3.00 Copyright 0 1991 by Academic Pnx, Inc. AU right.3of reprodwtion in any form reserved.
376
SOLUTIONS FOR SELECTIVE-EXCITATION
PULSES
377
A constant factor h in Eq. [l] has been can&M on both sides. Equation [I] is equivalent to the classical Bloch equation, neglecting relaxation effects. Take another time derivative of Eq. [l] and multiply by i to get
t31 The magnetic field consists of two terms: a static field along z proportional to the gradient strength and a time-varying component along the (rotating frame) x direction. The Pauli matrices satisfy (c.B)~ = (B)2 = B;(t) + (Gz)~.
]41 With these substitutions Eq. [ 31 has the form of a time-independent Schrodinger equation with a spin-dependent potential. The variable t now plays the role of a spatial coordinate in this new problem. The eigenvahte for the wave equation is the constant factor (yG~/2)~. The notation will be greatly simplified if the wave equation is expressedin dimensionlessvariables. We can assumethat ,therewill be a length parameter, ZO,in this problem; this turns out to be the slice thickness in a selective-excitation experiment. Define new time and field variables using the dimensionless quantities IrlGzot 2 ’
Bx Gz,’
[51
For the remainder of this paper, t and B, are taken to mean these dimensionless quantities. The potential can be diagonalized by a rotation around the y axis by 90”. Define
t61 The wave equation is
-a:x + I/x = k2x,
171
with v
=
3%
+
i&B,)
0
0 -(Bz - i&B,)
if31
and k = z/zo. The general properties of the solution to Eq. [l] are important later. It is assumed that, as t + &co, the RF field decreasesto zero. The solutions are then linear combinations of the up and down spinors and have the form
378
J. W. CARLSON
The asymptotic spinors are related by an SU( 2) rotation (l;)
= (-$*
t)(t)>
with i or12 + I /312 = 1. For RF polarized along the x axis in the rotating frame and B, = 0, an RF pulse yields rotation matrix with LY= cos(8/2), @ = -i sin(8/2). Four basic solutions to Eq. [I] are wavefunctions whose incoming or outgoing states are eigenstatesof oz. The first is
44+ (‘d” 1,
[Ill
as t + -cc. As t + +CXIthe solution approaches see J/I +
ikt
WI
‘1
( -B*epiktl . Similarly 1+5tr is the state which approaches a down spinor as t --) -cc . As t --, + cc the solution approaches [I31
The outgoing forms of Ic/rrrand &v are states which are eigenstatesof gz. These states are 1141
as t + -co. The outgoing form is
tiIII
+ i
zkt1 ,
Gw--I* ,-9, . (
i
[I51
The four basic solutions rewritten in terms of X: are
aeikt ‘11’ *
h
+
_ aeikt
pe-ikt + pe
-ikt
’
ast+--coand * ikt _ ace -a*eikt
flee-&t _
pe-ikt
1171
ast++cc.
SOLUTIONS FOR SELECTIVE-EXCITATION
PULSES
379
Look first at the upper component of Xrr. If we interpret a term eikt as a wave moving in the pseudo-spatial coordinate t to the “right” and ewik’as a wave moving to the “left” then XI: describes a solution which is a left-moving wave with amplitude 1 / fi as t --f --COand a superposition of a left-moving wave with amplitude a/ A6 and a right-moving wave with amplitude ,!3/1(z as t --, +co . This is exactly the form of a one-dimensional quantum mechanical scattering problem with the potential v+ = -(BZ + id,B,),
1181
We can describe the scattered solutions in terms of the transmission coefficient for a wave of unit amplitude incident from the right,
T:(k) = A)
t191
and the reflection coefficient, R:(k) = p . The solution Xl can be interpreted as a wave incident from the right with momentum -k. X& and X& are solutions incident from the left with momenta +k and -k, respectively. Similarly the lower components of X describe scattered solutions in a one-dimensional problem: the lower components scatter in the complex potential V- = -(Bz - id,B,).
[211
In summary, the reflection and transmission coefficients are T:(k) = T:(k) = T;(k) = T;(k) = A T,f(-k) R:(k)
= T:(-k)
= T;(-k)
= T;(--k)
[221 = $
~231
= -I?:*(-k)
= -R;(k)
= J&*(-k)
= f
[241
R:(k) = -I?:*(-k)
= -R;(k)
= I?;*(-k)
= 5.
1251
Earlier work ( 7) has used the scattering formulation of the Bloch equations starting from a classical viewpoint. However, the derivation of the scattering problem and relationship between the asymptotic states and the slice profile is more easily seen using the quantum mechanical starting point. WAVEFORMS FOR REFOCUSING PULSES
This work centers on the creation of selective refocusing pulses. The need for this arises since it is generally seen that sine pulses are adequate as excitation pulses but less than optimum as refocusing pulses. The asymptotic effects of RF applied to a spin system are describable purely in terms of the parameters (Yand @ .For a spin-echo pulse sequence, the echo amplitude is proportional to
380
J. W. CARLSON
<&%>
[261
where the subscripts denote the number of the RF pulses (see the Appendix). The problem of inverting an excitation profile is therefore equivalent to an inverse scattering problem in one-dimensional mechanics. This problem can be solved in closed form for special cases when the reflection coefficient is the ratio of polynomials in k (8). A convenient form for the refocusing profile is proportional to 1 (1 + (k)2”)2 ’
v71
Recall that for RF polarized along x and on resonance, /3 is pure imaginary and proportional to sin( d/2). This gives p = -i sin(0/2) 1 + k2” - ’ In the limits of n + co and 0 + a this describes a rectangular profile of height 1 from z = -z. to z = +zo and 0 otherwise. First consider the simplest case, corresponding to n = 1. We must construct an (Y which satisfies the normalization constraint ] (Y1’+ I,6(2 = 1; cx must be an analytic function of k (9). A tentative form for a! is k2 + cos(fl/2) + 2ik sin(0/4) 1 +k2
w91
This satisfies the normalization constraint for real k. However, this is not quite satisfactory; CYis regular in the upper half k plane (9) and we assume that the potentials in Eqs. [ 18 ] and [ 2 l] have no bound states. This implies that (Ycan have neither zeros nor poles in the upper half plane. Equation [ 291 has a simple pole at k = i and a zero at k = ilh cos(8/4 + 7r/4). Modify Eq. [29] by multiplying by k+i\lZcos(B/4+?r/4) k - ifi cos(8/4 + r/4)
[301
The zeros and poles are reflected through the real line to the lower half plane and the normalization condition is preserved for k on the real line. Substituting this form of (Yand @into Eqs. [ 231 and [ 241 gives the reflection coefficients for the problem R:
= -R;
=
-i(k + i)sin(6/2) (k - i)(k + ifi cos(0/4 + ?r/4))(k + ifi cos(8/4 - 7r/4)) *
[311
To calculate the potential in the scattering problem we must solve the Marchenko equations
+t’)Af(t, t”)=0, t
rf(t +t’+)AF(t, t’+)st -m
dt”rl+(t” + t’)Al+(t, t”) = 0,
where r$ is the Fourier transform of R$,
t > t’,
1321 [331
SOLUTIONS
FOR SELECTIVE-EXCITATIOhi
PULSES
rf(t) =s$ s
eikfR:(k),
and
r?(t) =
381 t341
dk -lk*Rl’(k). iGe
The potential is V’(t)
= -2&4F(t,
t) = 2d,Af(t, t).
[361
The evaluation of rf and rf is a straightforward contour integration. First consider r: for t > 0. Close the contour in the upper half plane; there is a simple pole at k = i and r:(t)
=
-2i sin(8/2)e-’ ( 1 + 112cos(8/4 + 7r/4))( 1 + ti cos(0/4 - 7r/4)) .
[371
Ansatz a form for A:( t, t’) for t’ > t: A:(t, t’) = p+(t)e-“.
1381
Substitute this into Eq. [ 371 to yield an equation for p: p’(t)
=
2i sin(8/4)e-’ 1 + cos(6/4) + i sin(8/4)e-*’ ’
1391
The RF waveform can be calculated directly from A :’ through the relation 2id,B, = v- - vf = 2a,A;(t, t) - 2a,A:(t,
t),
[401
so (assuming both B, and A, go to zero for large I) B,(t) = +i(A:(t,
t) - A;(t,
t)).
1411
We can repeat the calculation to solve for A;. From 13q. [ 251, this gives the same solution with 0 replaced by -0. The result is 4 sin(fI/4)[ 1 + cos(0/4)]eP2’ = 2 sech[2t - In tan (e/8)]. Bx(t) = [ 1 + cos(19/4)]* + sin2(0/4)ee4’
[42]
A calculation of the RF field for t < 0 follows using the same procedure using A f . The result of the calculation is a symmetric waveform II,.. -t) = B,.. t) . For n greater than 1 the calculation is similar but with a minor complication. Following Eq. [ 27 ] we can take a tentative form of LYto be k*” + cos(B/2) + 2ik”sin(f3/4) l+k*” -*
[431
This has poles at locations k” = fi
t441
and zeros at k” = ?ifi
cos(O/4 +- a/4).
[451
382
J. W. CARLSOK
We want to move all poles and zeros in the upper half plane to a point reflected through the real line. This gives fi
(k
+
ATj)(
k
-t
37-j)
1461
a= (k
j=l
+
Tj)"
'
where A and B are the positive, real solutions to .+I’ = fi cos(d/4 - a/4), 3” = $ COS(~/~ + r/4) an d 7j are the 2nth roots of - 1 in the upper half plane, Tj = er(2J-‘)r’2n. As before construct RF and evaluate a contour integral to give rf . There is a simple pole in the upper half plane in the contour integral at each value of Tj, 1 G j G n. The result for r:(t) is r:(t) = 5
iYijeiTJ’,
> 0,
t
[471
j=l
where %j is the residue of the pole at 7j: -2i sin(8/2) C7j + T*) ii 7jt l + &)t l + fB) psl (Tj - Tp)(Tj + + .B7p) * p+i Ansatz a solution to the Marchenko equation [ 321 of the form yjf
=
[481
ATp)(Tj
A:(t,
t’) = i
pf(t)e”j”.
[491
j=l
Substitution of this into Eq. [30] gives a comphcated algebraic equation. We can group the terms according to the time dependence in t’; in order for the equation to be satisfied we must individually cancel each term. This gives a series of algebraic equations of the form [501 The solution for A:( t, t) is therefore ei(Tj+Tk)t
A:(t, t) = -i 2 i,k
3.
-1 1
yj.ei(TI+Tk)f
(Tj+Tk)
’
9
J
[511
which can be rewritten as L,i(rj+Tk)l
sjk
+ crj
+
7kl
Bj
.
[521
As before, the solution for A ; has the same form with Bj replaced by - -Bj. This gives B,(t) = -ia,ln
det( 6jk + ( 6?i(rj+Tk)‘/( + Tk)) Bj) det( 6jk - ( f?i’7J+Tk”/( + Tk)) %j) Tj
[531 ’ The determinant evaluation for Eq. [ 521 is straightforward but cumbersome. The determinant can be written as a sum of n + 1 groups of terms: the first term is 1, the second are terms linear in one 32, etc., up to a final term proportional to the product of all 3’s. The general expression is Tj
SOLUTIONS FOR SELECTIVE-EXCITATION
383
PULSES
ei(lj+Tk)t
det Sj, + (Tj
d- 7k)
16jl
1 x
e2i(~jl+~~2)~~j,~j2
+
. . .
+
25
lsj,-ejyz..
X
$i(rJ,+Th+.
+-
-
47jl7j2
C7jl
+
lj2
I2
-
2 “Tj,7j2 * ’ . Tj,
. ‘+3”)‘j$&.
1 2%,7** * ‘7,
. . xjm
+
. . .
[54]
There is a corresponding expression for the derivative of the determinant. The final expression for B results from the ratio of the derivative and the determinant m inus the corresponding terms with the signs on all powers of .% reversed. At this point we can see the difficulty of the exact solutions. Equation [ 531 is the complete solution, but it is difficult to extract an intuitive analysis from the properties. Using Eq. [ 541, one can write the explicit form for n = ;! and II = 3. Beyond that, however, the explicit expression become extremely long; the number of terms in both the numerator and the denominator of & grows as 2”. The solution for it = 1 was a nonoscillating exponential; for n > 1 the poles move off the imaginary axis, and the complex exponentials e2iT’contain oscillations as well as exponential damping. Furthermore, as n increases,a pair of poles approaches the red axis. This implies that the overall exponential attenuation of the waveform becomes less significant. Nonetheless it is quite easy to generate waveforms by a direct numerical evaluation of Eq. [ 541 and the corresponding expression for the derivative. The calculation is straightforward and consumes an insignificant amount of computer time. Some typical waveforms for 6 = r and for it = 1 to II = 4 are shown in Fig. 1. The IZ = 1 waveform has a discontinuous first derivative at the origin; all others are continuous. And as expected, the temporal extent of the waveform increaseswith n, as does the RF power required. Numerically, we find that power increaseslinearly with n and peak amplitude as the square root of n . The pulse area is, of course, independent of n. The results of a numerical simulation of an NMR puke sequence are shown in Fig. 2. These are the slice profiles in a spin-echo sequence using a nonselective 90’ pulse followed by one of the selective refocusing pulses of Fig. 1. The degree of selectivity increases for the higher-order pulses. Extension of these pulse shapes to higher n is straightforward and continues to yield increasingly selective pulses. OTHER PULSES
The motivation for this calculation was the creation of selective-excitation pulses. However, the calculation is general and is not necessarily lim ited to this pulse type. The contribution of the first RF pulse to the spin-echo amplitude is proportional to the product of a/3. From the normalization condition we can write the magnitude of this as Id-T-m.
[551
384
J. W. CARLSON
FIG. 1. Amplitude-modulation envelope of the input RF pulses to produce a refocusing pulse (@ = r) with n = 1 (solid line), n = 2 (dotted line), n = 3 (dashed line), and n = 4 (dot-dashed line). The peak amplitude and power required increases with n, as does the apparent time extent of the pulse.
If we evaluate the RF pulse with 8 = r, the limiting form of this profile in the limit of n + cc is a rectangular section. This may not be the squarest profile possible for an excitation pulse, but does exhibit the good selectivity. The basic shape of the waveforms for 90” pulses is substantially different from that of the corresponding 180” pulse of the same order. Not only is the amplitude different, but the location and apparent number of zero crossings vary in the two pulses. The z magnetization after an RF pulse for spins aligned with the magnetic field is proportional to ] LYI 2 - I p I ‘. In the limit of n + cc the slice profile for the refocusing pulse becomes highly selective. The refocusing pulses generated using the techniques of this paper should therefore also be suitable as inversion pulses. A separate issue with this formalism is the distortion of the input waveforms. Inverse scattering problems with potentials of the form of Eq. [ 181 or [ 2 1] are closely connected with solutions to the modified Korteweg-deVries equation ( 10). If we evolve the input waveform in a new variable s according to the nonlinear equation d,B,+ 6B$3,B,+d:Bx
= 0,
[561
then the magnitudes of the reflection and transmission coefficients are unchanged. CONCLUSIONS
This method of solution for RF waveforms may be applied to a wide variety of selective-excitation problems. Closed-form solutions are obtainable for many general problems involving protiles which can be written as ratios of polynomials in z. Some common forms, such as Gaussians, are not in that form, but they may be approximated to any desired accuracy as a Pade approximate.
SOLUTIONS FOR SELECTIVE-EXCITATION
385
PULSES
1c
-I
FIG. 2. Calculated slice profiles for the four input RF pubes of Fig. 1. Transverse magnetization is “created’ with a nonselective 90” pulse. After a free precession, the signal is refocused using the RF pubes of Fig. 1. The slice profile is measured by summing the transverse magnetization at the echo time. The parameters are chosen to give a total slice thickness of 1 cm. These curves correspond to n = 1 to n = 4 and give a signal profile proportional to ( 1 + k2”)-2. The curves are labeled the same as in Fig. 1.
The method of calculating RF waveforms for more general profiles is a straightforward procedure. We can extend the methods of inverse scattering to calculate the waveform for any excitation or. refocusing profile provided it can be written as the square of a ratio of polynomials in 2 (for a refocusing pulse) or a product of polynomials that satisfies the normalization condition (for an excitation pulse). One need only construct the two quantities a and /3 and locate the poles of /3/a in the upper half plane and evaluate the residues. The waveform follows directly from Eq. [ 531. One can extend the technique to generatepulses with different shapesfor imaging or specific nulls for spectroscopic peak suppression. APPENDIX: DERIVATION
OF EQ. [261
Start with a spinor initially prepared in an eigenstate of u, with eigenvalue + 1. The action of an excitation RF pulse acts on this spinor as an SU (2) rotation of the state. For an amount of time, to, the state evolves in some magnetic field along the z axis. After this precession, a second RF pulse again rotates the state, followed again by free precession for time t. The spinor after these four operations is
386
J. W. CARLSON
The expectation value of the complex transverse magnetization is
(mx+ in?,)= (a*(t)b*(t)) (i i)(E::i)y
1‘421
which is -(la,
I2 -
IPI
l’)~*Ge
2 ,irB(f+fo) + cu:fi1f12*2eirfNb-0. j-fBf- a1* 8,* (Y*E
[A31
The signal from a macroscopic number of spins is an average of the expectation values weighted by the local magnetic field. We can imagine that the local field varies linearly over the sample and perform the averaging. After doing so we can identify the first term as the FID of the second pulse, the second term as the FID from the first pulse, and the third term as the spin echo. We must also repeat the calculation with the state prepared in a state with eigenvalue - 1 and average the results with a Boltzmann factor. The spin echo of the down state is the same as before with an overall minus sign. The result is that the spin echo is proportional to
ACKNOWLEDGMENT This work was supported by Toshiba America MRI, Inc. REFERENCES B. L. TOMLINSON AND H. D. W. HILL, J. Chem. Phys. 59, 1775 (1973). S. CONNOLY, D. NISHIMURA, AND A. Mocovs~~, IEEE Trans. Med. Imaging MI-S, 106 ( 1984). J. B. MURDOCH, A. H. LENT, AND M. R. KRITZER, J. firagn. Reson. 74,226 ( 1987). S. WIMPWS, J. Magn. Reson. 83,509 ( 1989). F. HIOE, Whys. Rev. A 30,210O (1984). M. S. SILVER, R. 1. JOSEPH, AND D. I. HOULT, J. Magn. Reson. 59,347 ( 1984). A. HASENFELD, J. Magm. Reson. 72,509 (1987). 8. I. KAY, Commun. Pure Appl. Math. 13, 371 ( 1960). 9. K. CHADAN AND P. C. SABATIER, “Inverse Problems in Quantum Scattering Theory,” 2nd ed., p. 330, Springer-Verlag, New York, 1989. 10. G. L. LAMB, JR ., “Elements of Sol&on Theory,” pp. 134.- 142, Wiley, New York, 1980. I. 2. 3. 4. 5. 6. 7.
94,3X-386
( 199 1)
Exact Solutions for Selective-Excitation
Pulses
J. W. CARISON Radiologic Imaging Laboratory, University of California, San Francisco, 400 Grandview Drive, South San Francisco, California 94080
Received January 24, 1991 Inverse scattering is used to solve the general problem of finding an applied RF field to produce a desired selective-excitation profile. The formalism is used to calculate the exact, closed-form solutions to the waveforms which produce a collection of refocusing slice profiles, the limit of which is a rectangular section. It is seen that this solution is also effective at producing a waveform which is effective as an excitation pulse or an inversion pUk.
0 1991 Academic F’res, Inc.
In the past few years there has been a steady interest in the general problem of creating RF pulses to produce a desired excitation profile. The need for these pulses grew with the development of imaging; the first reported pulses exploited the Fourier transform relationship between profile and input RF pulses ( 1) . Even though these arguments are strictly valid only for small-angle excitation, the pulses have shown to be adequate for larger-angle excitation and refocusing pulses. Improvement of this basic pulse shape has usually entailed numerical techniques to optimize the output profile (2, 3) or composite pulses (4) and results in improvement in profile of some other figure of merit. There has been some work on analytical solutions to the profiles for large-angle excitations (5, 6) but these have not been shown to be satisfactory refocusing pulses. This work describesa solution to the inverse problem for selective excitation. With some constraints on the desired slice profile for an imaging experiment, an amplitudemodulated RF pulse can be calculated which will produce that slice profile. A straightforward extension of this work allows it to be applied to frequency-selectivepulses in spectroscopy. SCATTERING
FORMALISM
The starting point for this analysis is the S&r&linger equation for a spin- 1 particle in a magnetic field in the interaction representation (rotating frame),
where B is the magnetic field in the rotating frame. The interaction representation wavefunction is related to the Schriidinger representation by a rotation around the static magnetic field:
[21 0022-236419 1 $3.00 Copyright 0 1991 by Academic Pnx, Inc. AU right.3of reprodwtion in any form reserved.
376
SOLUTIONS FOR SELECTIVE-EXCITATION
PULSES
377
A constant factor h in Eq. [l] has been can&M on both sides. Equation [I] is equivalent to the classical Bloch equation, neglecting relaxation effects. Take another time derivative of Eq. [l] and multiply by i to get
t31 The magnetic field consists of two terms: a static field along z proportional to the gradient strength and a time-varying component along the (rotating frame) x direction. The Pauli matrices satisfy (c.B)~ = (B)2 = B;(t) + (Gz)~.
]41 With these substitutions Eq. [ 31 has the form of a time-independent Schrodinger equation with a spin-dependent potential. The variable t now plays the role of a spatial coordinate in this new problem. The eigenvahte for the wave equation is the constant factor (yG~/2)~. The notation will be greatly simplified if the wave equation is expressedin dimensionlessvariables. We can assumethat ,therewill be a length parameter, ZO,in this problem; this turns out to be the slice thickness in a selective-excitation experiment. Define new time and field variables using the dimensionless quantities IrlGzot 2 ’
Bx Gz,’
[51
For the remainder of this paper, t and B, are taken to mean these dimensionless quantities. The potential can be diagonalized by a rotation around the y axis by 90”. Define
t61 The wave equation is
-a:x + I/x = k2x,
171
with v
=
3%
+
i&B,)
0
0 -(Bz - i&B,)
if31
and k = z/zo. The general properties of the solution to Eq. [l] are important later. It is assumed that, as t + &co, the RF field decreasesto zero. The solutions are then linear combinations of the up and down spinors and have the form
378
J. W. CARLSON
The asymptotic spinors are related by an SU( 2) rotation (l;)
= (-$*
t)(t)>
with i or12 + I /312 = 1. For RF polarized along the x axis in the rotating frame and B, = 0, an RF pulse yields rotation matrix with LY= cos(8/2), @ = -i sin(8/2). Four basic solutions to Eq. [I] are wavefunctions whose incoming or outgoing states are eigenstatesof oz. The first is
44+ (‘d” 1,
[Ill
as t + -cc. As t + +CXIthe solution approaches see J/I +
ikt
WI
‘1
( -B*epiktl . Similarly 1+5tr is the state which approaches a down spinor as t --) -cc . As t --, + cc the solution approaches [I31
The outgoing forms of Ic/rrrand &v are states which are eigenstatesof gz. These states are 1141
as t + -co. The outgoing form is
tiIII
+ i
zkt1 ,
Gw--I* ,-9, . (
i
[I51
The four basic solutions rewritten in terms of X: are
aeikt ‘11’ *
h
+
_ aeikt
pe-ikt + pe
-ikt
’
ast+--coand * ikt _ ace -a*eikt
flee-&t _
pe-ikt
1171
ast++cc.
SOLUTIONS FOR SELECTIVE-EXCITATION
PULSES
379
Look first at the upper component of Xrr. If we interpret a term eikt as a wave moving in the pseudo-spatial coordinate t to the “right” and ewik’as a wave moving to the “left” then XI: describes a solution which is a left-moving wave with amplitude 1 / fi as t --f --COand a superposition of a left-moving wave with amplitude a/ A6 and a right-moving wave with amplitude ,!3/1(z as t --, +co . This is exactly the form of a one-dimensional quantum mechanical scattering problem with the potential v+ = -(BZ + id,B,),
1181
We can describe the scattered solutions in terms of the transmission coefficient for a wave of unit amplitude incident from the right,
T:(k) = A)
t191
and the reflection coefficient, R:(k) = p . The solution Xl can be interpreted as a wave incident from the right with momentum -k. X& and X& are solutions incident from the left with momenta +k and -k, respectively. Similarly the lower components of X describe scattered solutions in a one-dimensional problem: the lower components scatter in the complex potential V- = -(Bz - id,B,).
[211
In summary, the reflection and transmission coefficients are T:(k) = T:(k) = T;(k) = T;(k) = A T,f(-k) R:(k)
= T:(-k)
= T;(-k)
= T;(--k)
[221 = $
~231
= -I?:*(-k)
= -R;(k)
= J&*(-k)
= f
[241
R:(k) = -I?:*(-k)
= -R;(k)
= I?;*(-k)
= 5.
1251
Earlier work ( 7) has used the scattering formulation of the Bloch equations starting from a classical viewpoint. However, the derivation of the scattering problem and relationship between the asymptotic states and the slice profile is more easily seen using the quantum mechanical starting point. WAVEFORMS FOR REFOCUSING PULSES
This work centers on the creation of selective refocusing pulses. The need for this arises since it is generally seen that sine pulses are adequate as excitation pulses but less than optimum as refocusing pulses. The asymptotic effects of RF applied to a spin system are describable purely in terms of the parameters (Yand @ .For a spin-echo pulse sequence, the echo amplitude is proportional to
380
J. W. CARLSON
<&%>
[261
where the subscripts denote the number of the RF pulses (see the Appendix). The problem of inverting an excitation profile is therefore equivalent to an inverse scattering problem in one-dimensional mechanics. This problem can be solved in closed form for special cases when the reflection coefficient is the ratio of polynomials in k (8). A convenient form for the refocusing profile is proportional to 1 (1 + (k)2”)2 ’
v71
Recall that for RF polarized along x and on resonance, /3 is pure imaginary and proportional to sin( d/2). This gives p = -i sin(0/2) 1 + k2” - ’ In the limits of n + co and 0 + a this describes a rectangular profile of height 1 from z = -z. to z = +zo and 0 otherwise. First consider the simplest case, corresponding to n = 1. We must construct an (Y which satisfies the normalization constraint ] (Y1’+ I,6(2 = 1; cx must be an analytic function of k (9). A tentative form for a! is k2 + cos(fl/2) + 2ik sin(0/4) 1 +k2
w91
This satisfies the normalization constraint for real k. However, this is not quite satisfactory; CYis regular in the upper half k plane (9) and we assume that the potentials in Eqs. [ 18 ] and [ 2 l] have no bound states. This implies that (Ycan have neither zeros nor poles in the upper half plane. Equation [ 291 has a simple pole at k = i and a zero at k = ilh cos(8/4 + 7r/4). Modify Eq. [29] by multiplying by k+i\lZcos(B/4+?r/4) k - ifi cos(8/4 + r/4)
[301
The zeros and poles are reflected through the real line to the lower half plane and the normalization condition is preserved for k on the real line. Substituting this form of (Yand @into Eqs. [ 231 and [ 241 gives the reflection coefficients for the problem R:
= -R;
=
-i(k + i)sin(6/2) (k - i)(k + ifi cos(0/4 + ?r/4))(k + ifi cos(8/4 - 7r/4)) *
[311
To calculate the potential in the scattering problem we must solve the Marchenko equations
+t’)Af(t, t”)=0, t
rf(t +t’+)AF(t, t’+)st -m
dt”rl+(t” + t’)Al+(t, t”) = 0,
where r$ is the Fourier transform of R$,
t > t’,
1321 [331
SOLUTIONS
FOR SELECTIVE-EXCITATIOhi
PULSES
rf(t) =s$ s
eikfR:(k),
and
r?(t) =
381 t341
dk -lk*Rl’(k). iGe
The potential is V’(t)
= -2&4F(t,
t) = 2d,Af(t, t).
[361
The evaluation of rf and rf is a straightforward contour integration. First consider r: for t > 0. Close the contour in the upper half plane; there is a simple pole at k = i and r:(t)
=
-2i sin(8/2)e-’ ( 1 + 112cos(8/4 + 7r/4))( 1 + ti cos(0/4 - 7r/4)) .
[371
Ansatz a form for A:( t, t’) for t’ > t: A:(t, t’) = p+(t)e-“.
1381
Substitute this into Eq. [ 371 to yield an equation for p: p’(t)
=
2i sin(8/4)e-’ 1 + cos(6/4) + i sin(8/4)e-*’ ’
1391
The RF waveform can be calculated directly from A :’ through the relation 2id,B, = v- - vf = 2a,A;(t, t) - 2a,A:(t,
t),
[401
so (assuming both B, and A, go to zero for large I) B,(t) = +i(A:(t,
t) - A;(t,
t)).
1411
We can repeat the calculation to solve for A;. From 13q. [ 251, this gives the same solution with 0 replaced by -0. The result is 4 sin(fI/4)[ 1 + cos(0/4)]eP2’ = 2 sech[2t - In tan (e/8)]. Bx(t) = [ 1 + cos(19/4)]* + sin2(0/4)ee4’
[42]
A calculation of the RF field for t < 0 follows using the same procedure using A f . The result of the calculation is a symmetric waveform II,.. -t) = B,.. t) . For n greater than 1 the calculation is similar but with a minor complication. Following Eq. [ 27 ] we can take a tentative form of LYto be k*” + cos(B/2) + 2ik”sin(f3/4) l+k*” -*
[431
This has poles at locations k” = fi
t441
and zeros at k” = ?ifi
cos(O/4 +- a/4).
[451
382
J. W. CARLSOK
We want to move all poles and zeros in the upper half plane to a point reflected through the real line. This gives fi
(k
+
ATj)(
k
-t
37-j)
1461
a= (k
j=l
+
Tj)"
'
where A and B are the positive, real solutions to .+I’ = fi cos(d/4 - a/4), 3” = $ COS(~/~ + r/4) an d 7j are the 2nth roots of - 1 in the upper half plane, Tj = er(2J-‘)r’2n. As before construct RF and evaluate a contour integral to give rf . There is a simple pole in the upper half plane in the contour integral at each value of Tj, 1 G j G n. The result for r:(t) is r:(t) = 5
iYijeiTJ’,
> 0,
t
[471
j=l
where %j is the residue of the pole at 7j: -2i sin(8/2) C7j + T*) ii 7jt l + &)t l + fB) psl (Tj - Tp)(Tj + + .B7p) * p+i Ansatz a solution to the Marchenko equation [ 321 of the form yjf
=
[481
ATp)(Tj
A:(t,
t’) = i
pf(t)e”j”.
[491
j=l
Substitution of this into Eq. [30] gives a comphcated algebraic equation. We can group the terms according to the time dependence in t’; in order for the equation to be satisfied we must individually cancel each term. This gives a series of algebraic equations of the form [501 The solution for A:( t, t) is therefore ei(Tj+Tk)t
A:(t, t) = -i 2 i,k
3.
-1 1
yj.ei(TI+Tk)f
(Tj+Tk)
’
9
J
[511
which can be rewritten as L,i(rj+Tk)l
sjk
+ crj
+
7kl
Bj
.
[521
As before, the solution for A ; has the same form with Bj replaced by - -Bj. This gives B,(t) = -ia,ln
det( 6jk + ( 6?i(rj+Tk)‘/( + Tk)) Bj) det( 6jk - ( f?i’7J+Tk”/( + Tk)) %j) Tj
[531 ’ The determinant evaluation for Eq. [ 521 is straightforward but cumbersome. The determinant can be written as a sum of n + 1 groups of terms: the first term is 1, the second are terms linear in one 32, etc., up to a final term proportional to the product of all 3’s. The general expression is Tj
SOLUTIONS FOR SELECTIVE-EXCITATION
383
PULSES
ei(lj+Tk)t
det Sj, + (Tj
d- 7k)
16jl
1 x
e2i(~jl+~~2)~~j,~j2
+
. . .
+
25
lsj,-ejyz..
X
$i(rJ,+Th+.
+-
-
47jl7j2
C7jl
+
lj2
I2
-
2 “Tj,7j2 * ’ . Tj,
. ‘+3”)‘j$&.
1 2%,7** * ‘7,
. . xjm
+
. . .
[54]
There is a corresponding expression for the derivative of the determinant. The final expression for B results from the ratio of the derivative and the determinant m inus the corresponding terms with the signs on all powers of .% reversed. At this point we can see the difficulty of the exact solutions. Equation [ 531 is the complete solution, but it is difficult to extract an intuitive analysis from the properties. Using Eq. [ 541, one can write the explicit form for n = ;! and II = 3. Beyond that, however, the explicit expression become extremely long; the number of terms in both the numerator and the denominator of & grows as 2”. The solution for it = 1 was a nonoscillating exponential; for n > 1 the poles move off the imaginary axis, and the complex exponentials e2iT’contain oscillations as well as exponential damping. Furthermore, as n increases,a pair of poles approaches the red axis. This implies that the overall exponential attenuation of the waveform becomes less significant. Nonetheless it is quite easy to generate waveforms by a direct numerical evaluation of Eq. [ 541 and the corresponding expression for the derivative. The calculation is straightforward and consumes an insignificant amount of computer time. Some typical waveforms for 6 = r and for it = 1 to II = 4 are shown in Fig. 1. The IZ = 1 waveform has a discontinuous first derivative at the origin; all others are continuous. And as expected, the temporal extent of the waveform increaseswith n, as does the RF power required. Numerically, we find that power increaseslinearly with n and peak amplitude as the square root of n . The pulse area is, of course, independent of n. The results of a numerical simulation of an NMR puke sequence are shown in Fig. 2. These are the slice profiles in a spin-echo sequence using a nonselective 90’ pulse followed by one of the selective refocusing pulses of Fig. 1. The degree of selectivity increases for the higher-order pulses. Extension of these pulse shapes to higher n is straightforward and continues to yield increasingly selective pulses. OTHER PULSES
The motivation for this calculation was the creation of selective-excitation pulses. However, the calculation is general and is not necessarily lim ited to this pulse type. The contribution of the first RF pulse to the spin-echo amplitude is proportional to the product of a/3. From the normalization condition we can write the magnitude of this as Id-T-m.
[551
384
J. W. CARLSON
FIG. 1. Amplitude-modulation envelope of the input RF pulses to produce a refocusing pulse (@ = r) with n = 1 (solid line), n = 2 (dotted line), n = 3 (dashed line), and n = 4 (dot-dashed line). The peak amplitude and power required increases with n, as does the apparent time extent of the pulse.
If we evaluate the RF pulse with 8 = r, the limiting form of this profile in the limit of n + cc is a rectangular section. This may not be the squarest profile possible for an excitation pulse, but does exhibit the good selectivity. The basic shape of the waveforms for 90” pulses is substantially different from that of the corresponding 180” pulse of the same order. Not only is the amplitude different, but the location and apparent number of zero crossings vary in the two pulses. The z magnetization after an RF pulse for spins aligned with the magnetic field is proportional to ] LYI 2 - I p I ‘. In the limit of n + cc the slice profile for the refocusing pulse becomes highly selective. The refocusing pulses generated using the techniques of this paper should therefore also be suitable as inversion pulses. A separate issue with this formalism is the distortion of the input waveforms. Inverse scattering problems with potentials of the form of Eq. [ 181 or [ 2 1] are closely connected with solutions to the modified Korteweg-deVries equation ( 10). If we evolve the input waveform in a new variable s according to the nonlinear equation d,B,+ 6B$3,B,+d:Bx
= 0,
[561
then the magnitudes of the reflection and transmission coefficients are unchanged. CONCLUSIONS
This method of solution for RF waveforms may be applied to a wide variety of selective-excitation problems. Closed-form solutions are obtainable for many general problems involving protiles which can be written as ratios of polynomials in z. Some common forms, such as Gaussians, are not in that form, but they may be approximated to any desired accuracy as a Pade approximate.
SOLUTIONS FOR SELECTIVE-EXCITATION
385
PULSES
1c
-I
FIG. 2. Calculated slice profiles for the four input RF pubes of Fig. 1. Transverse magnetization is “created’ with a nonselective 90” pulse. After a free precession, the signal is refocused using the RF pubes of Fig. 1. The slice profile is measured by summing the transverse magnetization at the echo time. The parameters are chosen to give a total slice thickness of 1 cm. These curves correspond to n = 1 to n = 4 and give a signal profile proportional to ( 1 + k2”)-2. The curves are labeled the same as in Fig. 1.
The method of calculating RF waveforms for more general profiles is a straightforward procedure. We can extend the methods of inverse scattering to calculate the waveform for any excitation or. refocusing profile provided it can be written as the square of a ratio of polynomials in 2 (for a refocusing pulse) or a product of polynomials that satisfies the normalization condition (for an excitation pulse). One need only construct the two quantities a and /3 and locate the poles of /3/a in the upper half plane and evaluate the residues. The waveform follows directly from Eq. [ 531. One can extend the technique to generatepulses with different shapesfor imaging or specific nulls for spectroscopic peak suppression. APPENDIX: DERIVATION
OF EQ. [261
Start with a spinor initially prepared in an eigenstate of u, with eigenvalue + 1. The action of an excitation RF pulse acts on this spinor as an SU (2) rotation of the state. For an amount of time, to, the state evolves in some magnetic field along the z axis. After this precession, a second RF pulse again rotates the state, followed again by free precession for time t. The spinor after these four operations is
386
J. W. CARLSON
The expectation value of the complex transverse magnetization is
(mx+ in?,)= (a*(t)b*(t)) (i i)(E::i)y
1‘421
which is -(la,
I2 -
IPI
l’)~*Ge
2 ,irB(f+fo) + cu:fi1f12*2eirfNb-0. j-fBf- a1* 8,* (Y*E
[A31
The signal from a macroscopic number of spins is an average of the expectation values weighted by the local magnetic field. We can imagine that the local field varies linearly over the sample and perform the averaging. After doing so we can identify the first term as the FID of the second pulse, the second term as the FID from the first pulse, and the third term as the spin echo. We must also repeat the calculation with the state prepared in a state with eigenvalue - 1 and average the results with a Boltzmann factor. The spin echo of the down state is the same as before with an overall minus sign. The result is that the spin echo is proportional to
ACKNOWLEDGMENT This work was supported by Toshiba America MRI, Inc. REFERENCES B. L. TOMLINSON AND H. D. W. HILL, J. Chem. Phys. 59, 1775 (1973). S. CONNOLY, D. NISHIMURA, AND A. Mocovs~~, IEEE Trans. Med. Imaging MI-S, 106 ( 1984). J. B. MURDOCH, A. H. LENT, AND M. R. KRITZER, J. firagn. Reson. 74,226 ( 1987). S. WIMPWS, J. Magn. Reson. 83,509 ( 1989). F. HIOE, Whys. Rev. A 30,210O (1984). M. S. SILVER, R. 1. JOSEPH, AND D. I. HOULT, J. Magn. Reson. 59,347 ( 1984). A. HASENFELD, J. Magm. Reson. 72,509 (1987). 8. I. KAY, Commun. Pure Appl. Math. 13, 371 ( 1960). 9. K. CHADAN AND P. C. SABATIER, “Inverse Problems in Quantum Scattering Theory,” 2nd ed., p. 330, Springer-Verlag, New York, 1989. 10. G. L. LAMB, JR ., “Elements of Sol&on Theory,” pp. 134.- 142, Wiley, New York, 1980. I. 2. 3. 4. 5. 6. 7.