A calculational technique for custom tailoring of rf pulses in selective excitation magnetic resonance experiments

JOURNAL

OF MAGNETIC

RESONANCE

67, 551-555 (1986)

A Calculational Techniquefor Custom Tailoring of rf Pulsesin SelectiveExcitation Magnetic ResonanceExperiments J.W.CARLSON Radiologic

Imaging

Laboratory. University South San Francisco,

of California, 400 Grandview California 94080

Drive,

Received September 19, 1985; revised December 3, 1985

A continuing problem in selective excitation NMR imaging is the creation of new pulse sequences which produce desired magnetization profiles. Typically in an imaging sequence, one would like to be able to produce an rf pulse which acts like a 90 or 180” pulse for magnetization within a given range of the static magnetic field and does not affect the outlying regions. The starting approximation in selective excitation is to produce an rf pulse with a Fourier content that matches the desired profile. The limitations of this approximation generally become important for rotations through large flip angles. Furthermore, the effects of truncation of the pulse cause other noticeable deviations in the proftle. Recent attempt to improve on this approximation have relied on techniques such as windowing (I), continuous phase modulation (2), or composite pulses (3). This paper will describe a method for calculating improved amplitude modulated rf pulses. The procedure has the advantage of demanding neither phase modulation nor increasing sequence complexity. In fact, since this technique alters only the amplitude modulation of a single pulse, it can be used in conjunction with composite pulse techniques. Phase modulation may also be included through a straightforward extension of the technique. The method is illustrated with a numerical calculation of the relatively simple problem of a selective 90” pulse. A discussion of more complex examples and experimental results will be presented in a future publication. The basis of this approach is an expression for the change in magnetization of a sample after an excitation pulse due to a small change of an existing pulse. Of course, this could be calculated directly by brute force by solving the Bloch equations for a given rf pulse and for a perturbed rf pulse. But in the interests of computational speed and accuracy and for the purposes of additional insight, this paper will use the analytic expression for the effect of a perturbation of a pulse. We imagine that the amplitude of the rfpulse is specified by a sequence of piecewise constant steps over small time intervals At. If we ignore relaxation effects for the duration of the rf pulse, magnetization is rotated by the rf. Represent this rotation by the SO( 3) matrix 0,. The solution to the Bloch equations in the rotating frame is o(tk) = n ,+T-Biu. ick

551

0022-2364186 $3.00 Copyright 0 1986 by Academic Press, Inc. All rights of reproduction in any form reserved.

552

NOTES

where Bi is the magnetic field during the ith time interval, the z component of B is the value of static field, and the transverse components are the rf amplitude. For amplitude modulated pulses, the rf will be assumed to be in the x direction in the rotating frame. T are 3 X 3 Hermitian matrices which are the generators of SO(3) (4):

It is understood that the product is time ordered: it is evaluated with the exponential factors of later times placed to the left of terms evaluated at earlier times. The first order rf pulse is a sine pulse which extends to five zero crossings on either side of the peak. In calculations presented here the rf amplitude is discretized for 5 12 equal time intervals. For each time interval, the orthogonal matrix representing the rotation in a constant rf field is calculated exactly. The net rotation is the matrix product of the 5 12 matrices. The effect of changing the rf pulse amplitude Bi - Bi + 6Bi is oij(tk)

-

o&k)

+

aoij(tk)

To first order in 6B and At o(tk) + aott,) = n eirT.(Bi+~Bi)Al ixk

=

o(tk)

+

j

2 ick

{ fl bjai

ei-tTe3At)yT.

sBi&{n

&YT-B@f} lb

Taking the limit At - 0 gives the first-order change of an element of Oij: tk

dt’OTQt’)T,,O,(t’) . ySB(t’). s -cc The term OTTO is just the expression for the adjoint action of the orthogonal transformation on the generators of SO(3): 6OJtk) = jOif

This expression is just the first term in the Magnus expansion (5). In the absence of an initial rf pulse, then this expression for the change in the magnetization is equivalent to the approximation that the flip angle is proportional to the Fourier content of the rf pulse. However, it is important to emphasize that this expansion is also useful for calculating perturbations of a given rf pulse. The rest of this discussion will be specialized to a particular example for concreteness. Assume that the rf is purely amplitude modulated and extends in time from -T to T. A linear gradient produces a static field in the z direction. The rf field lies in the x direction in the rotating frame.

NOTES

If we restrict the perturbation of the change in the rf is

553

of the rf pulse to be symmetric in time, then the form

s&(t) = cn

a, cos(yy).

The expression for the change in 0, gives the derivative of the SO(3) matrix with respect to a,: doij -

lyOik(T)

aa,

+

(032021

-

&O3~)(Tzhj}COS

(yE>].

The orthogonal matrices in this integral are given by the solution to the Bloch equations with the sine modulated rf pulse. One should evaluate the desired change in the orthogonal matrix, AO,, at a set of values in the static field. The intention is to calculate the value of the coefficients, a,,, which cancel the nonideal behavior to first order. This is done by solving the linear equations:

aOi, [aa, 1

a,, = AO,.

For the case of a 90” flip, we are interested in the final value of 023. The matrix of the linear equations is indexed by the value of the static field and the period of the perturbing cosine. Since the Fourier content of the truncated cosine is centered at (2n + l)n/2T, the derivative is peaked at values of the static field of (2n + l)a/2yT. One should evaluate 023 at a set of values in the static field separated by r/yT. This entire procedure can be iterated to find additional improvement. Figure 1 shows the computed profile for a sine pulse truncated at 10 zero crossings. (The t-f pulse was applied, then the gradient reversed and the magnetization allowed to process until the y component of the magnetization is a maximum (6).) The sine pulse is particularly susceptable to errors caused by truncation (7). The sidelobes outside the selected slice are primarily due to truncation effects and largely disappear with longer pulses. Two iterations of the improvements result in the reduction of the sidelobes as shown in Fig. 2. The amplitude of the unperturbed sine pulse and the improved pulse are shown in Fig. 3. The differences in the pulses are small. It initially appears to be simply a windowing of the sine pulse. However, it is not a windowing: the location of the zero crossings is different in the two curves. Unlike windowing techniques, this approach does not appreciably broaden the overall profile. (As an aside, this was done by taking the perturbations to be of the form cos(y$)-@sine(:) where the constant /3 is chosen so that the perturbations have no effect on the center of the slice. This allows for concentrating on eliminating sidelobes without destroying

554

NOTES

FIG. 1. Profile of component of magnetization following a sine shaped rf pulse and gradient reversal refocusing.

the center.) Further iterations result in increased residual y magnetization at offset fields between the sampled value. Consequently, the ideal number of iterations can only be determined on a case by case basis.

FIG.2. Profile with improved rf waveform.

555

NOTES

FIG. 3. Shape of sine pulse (dotted line) and improved rf waveform (solid line).

The improved rf pulse does not lead to an increase in the x component of the magnetization, nor does it produce residual y magnetization at very large values of the offset field. The requirement of which element of the SO( 3) matrix should be improved depends on the needs of the experiment and on details of the sequence. In the examples studied so far, the calculations show improvements even with the restriction of amplitude modulation. An extension of this technique can be applied to situations that allow for asymmetric pulses or phase modulation by a simple extension. ACKNOWLEDGEMENT This work is supported in part by Diasonics, MRI, Inc. REFERENCES I. P. R. LITHER, Philos. Trans. R. Sot. London Imag. 2, 169 (1983).

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IEEE

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2. M. S. SILVER, R. I. JOSEPH, AND D. I. HOULT, J. Magn. Reson. 59, 347 (1984). 3. J. FRAHM AND W. HANICKE, J. Magn. Reson. 60,320 (1984); D. G. NISHIMURA, Med. Phys. 12,413 (1985); A. J. SHAKA AND R. FREEMAN, J. Magn. Reson. 59, 169 (1984); R. TYCKO AND A. PINES, Chem. Phys. Lett. 111,462 (1984); R. TYCKO, Phys. Rev. Left. 51,775 (1983).

4. R. N. CAHN, “Semi-Simple Lie Algebras and Their Representations,” Benjamin/Cummings, Park, Calif., 1984. 5. W. MAGNUS, Commun. Pure Appl. Math. 7, 649 (1954). 6. D. I. HOULT, J. Magn. Reson. 26, 165 (1977). 7. W. S. WARREN,

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