Numerical design of composite radiofrequency pulses

JOURNAL

OF MAGNETIC

RESONANCE

70, t l-20 ( 1986)

Numerical Design of Composite Radiofrequency Pulses* DAVID J. LURIE Department of Bit-Medical Physics and Bio-Engineering, University of Aberdeen, Foresterhill, Aberdeen AB9 220, United Kingdom Received December 11, 1985; revised May 28, 1986 A numerical procedure is presented for the design of composite pulses according to specific experimental requirements. Removing arbitrary restrictions on the pulse parameters allows the design of efficient composite pulses with fewer components and shorter overall length than before. New dual-compensating composite inversion pulseswith three to seven components are presented and their tolerance to setting-up errors is discussed. o 1986 Academic Press. Inc.

Pulse excitation is used extensively in NMR spectroscopyand imaging. Many pulse sequencese m p loy 90 and 180’ pulses which are intended to flip the magnetization uniformly throughout the sample volume. Square radiofrequency pulses, while being easyto implement, are inefficient in the presenceof radiofiequency field inhomogeneity and resonanceoffset. In the former casethe radiofrequency magnetic field strength is reduced from its n o m inal value, giving rise to changes in the flip angle throughout the sample. Additionally, the pulse length may have been m isset, altering the flip angle by an equal proportion everywhere. Resonanceoffset arises from chemical shift and from spatial inhomogeneity of the static magnetic field, causing the effective magnetic field to be tilted away from the equatorial plane in the rotating frame, again altering the flip angle. W h ile 90” “readout” pulses are relatively insensitive to these effects, the efficiency of a 180” inversion pulse may be seriously impaired. In recent years methods of compensating these effects have been developed which involve replacing the conventional pulse with a “composite pulse,” a cluster of two or more square pulses whose lengths (flip angles)and phasesare chosen to correct for the pulse imperfections (1-11). The design of composite pulses was at first intuitive, assisted by computer simulation of the paths of magnetization vectors during the sequenceof pulses (1-5). More elaborate techniques have been developed, using recursive expansion (6, 7), “reversed nutation pulses” (a), averageHamiltonian theory (!g), Magnus expansions(IO), and fixed-point analyses(I I), which go some way toward designing composite pulse sequencesaccording to specific performance requirements. In many casesthe allowed values of the component pulses’phase (and often length also) have been restricted, frequently to m u ltiples of 7r/2 radians. Such restrictions are necessarywhen intuitive design techniques are e m p loyed and may correspond to the * First presented at the Fourth Annual Meeting, Society of Magnetic Resonance in Medicine, London, UK, August 1985. II

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

12

DAVID

J. LURIE

only available phase values on some pulse programmers; however, they inevitably impair the flexibility of composite pulse design. More recent design techniques have allowed the restrictions on either pulse length or phase to be relaxed, resulting in composite pulses with improved immunity to pulse imperfections (9-11). Another design method, originally suggested by Levitt (2), is to allow a computer program to find the combination of component pulse lengths and phases which will best satisfy a given experimental requirement, for example, to achieve greater than 99% inversion efficiency over a stated range of resonance offset and radiofrequency field strength values. A similar approach has been used successfully in the design of selective radiofrequency pulses for NMR imaging (12). In this paper a systematic design procedure for composite pulses will be described in which all arbitrary restrictions on the component pulses are lifted, allowing the design of efficient pulses with fewer components than before. New composite inversion pulses will be presented and their tolerance to setting-up errors discussed. The effect of reintroducing phase-angle restrictions will be indicated. Throughout this paper a composite pulse’s components will be denoted by a(4), where (Yis the nominal flip angle and 4 is the phase angle (in degrees). For example, the familiar three-component inversion pulse 90°(X)1800( Y)9O”(X) (I), often written as X2YX, will be denoted 90”(0”)180”(90”)90”(0”). In the familiar rotating frame, the magnetization vector m precesses about the effective magnetic field, which is the sum of the radiofrequency magnetic field Bi , lying in the transverse plane and the resonance offset field AB, which is perpendicular to it. If relaxation is neglected, the length of m remains unchanged. A radiofrequency pulse with nominal tIip angle a! is normalized such that yB?t, = a

111

where BP is the nominal radiofrequency magnetic field strength in the rotating frame, tp is the duration of the pulse, and y is the gyromagnetic ratio. Considering a pulse intended to invert z magnetization, a convenient index of its efficiency is the quantity -MJ&, where Ma is the equilibrium magnetization. Its performance in the presence of pulse imperfections may be indicated by a diagram showing contours of -M,IMo as a function of normalized resonance offset A B/BP and radiofrequency field strength B,/BP . An efficiency plot for a simple 180” pulse is shown in Fig. 1. The 99% contour encloses only a small elipticrd region, indicating the conventional pulse’s poor tolerance to both types of error. A composite pulse with N components is described by 2N parameters, being the flip angle 0 and phase 4 of each component pulse. They may be written as a 2Ndimensional vector x = (al, . . . , (YN,$1, . . . , &). The response of the magnetization in the presence of resonance offset and radiofrequency field inhomogeneity is also a function of these parameters and may be written S(x, a, b) where a = AB/Bp and b = B,/BP. The overall efficiency of a composite pulse may be character&d by a function F(x), defined as F(X) = Z: Z Wo(l(Ui, bj) - 8% Ui, bj)12 i

where

(Ui,

PI

j

bj) represents a point on the AB/Bp - B,/BP plane at which the ideal or

DESIGN OF COMPOSITE PULSES

13

-

1.4

1.2-

8.8 8.6 -I

e

-6.5

8.5

ABiB; FIG. I. Calculated inversion efficiency for a simple 180” pulse as a function of normalized radiofrequency magnetic field strength (I&/&‘) and normalized resonance offset (AB/BP).

desired response is l(ai, bj). The relative contribution to F(x) from each point is determined by the weighting factors Wij. M inimizing F(x) with respect to the parameters x will yield the optimum composite pulse for the stated requirements. Altering the grid (ai, bj) and the weighting factors w, will “mold” the efficiency contours so that a composite pulse can be designed according to the prevailing experimental imperf&ions. In the present study, a computer program uses a geometrical model of the spin system to calculate the effect of applying an arbitrary sequence of radiofrequency pulses in the presence of resonance offset and Bi inhomogeneity. Function m inimization is carried out by commercially available routines (23) which explore the multidimensional surface of F(x) in search of the global m inimum corresponding to the “best” set of parameter values. M inimization routines using simplex (14) and quasiNewton (15) search methods have been used, the latter tending to converge more rapidly due to the relatively “well-behaved” nature of F(x). As an illustration of the method, consider the design of a Wee-component composite pulse, intended to invert longitudinal magnetization in the presence of +40% B, inhomogeneity. The grid (ai, bj) is set up to cover the range -0.05 to 0.05 in normalized resonance offset and 0.6 to 1.4 in normalized radiofrequency field strength. A good starting point for the optimization is the composite inversion pulse 9O”(O”)18O”(9O”)9O”(O”)(I) which atfords some degree of insensitivity to B, inhomogeneity (Fig. 2). M inimizing the function F,(X)

= 2 i

Z (OX& j

ai, bj))*

[31

(with the proviso that A4, < 0) yields the optimum composite pulse 88”(0”)352”( 110”)88”(0”). Figure 3 shows the efficiency diagram for this pulse, together with

14

DAVID

J. LURIE

6.6 -

-I

-8.5

0

8.5

I

I

AB/E( FIG. 2. Cakxdated inversion efficiency for the three-component composite suggested by Levitt and Freeman (I).

p& 90°(00)1800(~o)~o(0”),

the grid of points (ai, bj); it is evident that the contours have been stretched to enclose the grid. The inversion-efficiency response compares favorably with the four-component composite pulse 180”(0”)180”( 105”)180”(210”)360”(59”) suggested by Tycko et al. (20). The ability of a composite pulse to conform with a given set of requirements is strongly dependent on the number of component pulses, N, and is not known a priori.

1.4-

1.2 -

4 BY

-I

-6.5

e

a.5

FIG. 3. Calculati inversion efficiency for the optimized three-component Also shown is the grid of points at which the efficiency is optimized.

I

pulse

880(00)3520(1 10”)88°(O”).

15

DESIGN OF COMPOSITE PULSES

Therefore, the “molding” of efficiency contours must proceed in stages,the grid over which the overall efficiency is guaged being expanded stepwise until the desired performance is achieved. Although function m inimization with a small number of parameters (M < 20) on a mainframe computer is not computationally demanding, it is always desirable to make M as small as possible. This reduces the number of iterations of the m inimization routine and, more importantly, increasesthe likelihood of locating the global m inimum of F(x). A reduction of M may be achieved by placing constraints on the composite pulse. For the purpose of optimizing inversion efficiency, the absolute phase is unimportant so that one may set 4i = 0 without loss of flexibility. A further reduction in. the number of parameters will result by constraining the sequence of pulses to be palindromic. It is usually desirable that the efficiency contours be symmetrical with respect to positive and negative resonance offsets; Freeman et al. have shown that, with a three-component composite inversion pulse, such a response can only be obtained if the first and last pulses have identical length and phase (2). This result has been confirmed empirically in the present study: if the grid (ai, bj) is symmetrical with respect to positive and negative resonance offsets than the optimization of an unconstrained composite pulse with an odd number of components will always yield a symmetrical sequenceof pulses. With these restrictions, a three-component composite pulse is described by only three parameters--a,, (Ye,and 42. Composite inversion pulses with up to seven components have been designed using the above procedure, with the objective of achieving simultaneous compensation of resonance offset and radiofrequency field inhomogeneity errors. Table 1 summarizes the composite pulses obtained by optimization with no constraints on the allowed values of pulse length or phase. Increasing the number of components improves the achievable performance, although even the three-component pulse gives some degree of dual compensation (Fig. 4).

TABLE 1

OptimizedDual-Compensating Composite Inversion Pulses Region where inversion efficiency is greater than 97% Number of c:omponents 1 3 4 5 6 7

Sequence 180=(0”) 79”(O”)276”(106”)79°(O”) 64”(O”)146”(185”)320°(3100)77”(192’) 63”(O”)140”(148”)340°(2400)1400(148”)63”(0”) 52”(O”)94”(139”)66”( 196”)323”(251°)1430(1590)63”(10”) 54”(O”)135”(163”)177”(2950)381“(1 l”)177”(295”)135°(163”)54”(O”)

Total length compared with simple pulse

Resonance offset

Radiofrequency field strength

1.0 2.4

-0.12-0.12 -0.40-0.40

0.92-I .08 0.88-1.05

3.4

-0.40-0.40

0.85-1.05

4.1

-0.43-0.43

0.70-l. 15

4.1

-0.47-0.47

0.68-1.15

6.2

-0.50450

0.62-1.15

16

DAVID

J. LURIE

FIG. 4. Calculated inversion efficiency for the optimized dual-compensating pulses 79”(0”)276”( 106”)79”(0”) (above) and 63”(O”)140°(1480)3400(2400)1400(1480)630(Oo) (below).

These pulses compare well with composite pulses suggested by other workers. The five-component pulse gives a similar performance to the six-pulse dual-compensating sequence suggested by Shaka and Freeman (8) but its total duration is less by a factor of approximately two. The inversion-efficiency response of the optimized seven-component pulse is almost as good as that of a sequence obtained by Tycko et al. using an iterative phase-shift expansion scheme (II); however, the latter has 25 components and is more than four times as long. While maximum flexibility is obtained by allowing all of the variable parameters to take any value, the optimization procedure is easily adapted to account for instrumental limitations, for example, in the allowed values of phase. Figure 5 shows efficiency diagrams for five-component composite pulses optimized with the allowed phase values restricted to multiples of 10 and 22.5”, respectively. Even the more stringent restriction does not greatly impede the achievable performance. Note that

DESIGN OF COMPOSITE PULSES

17

1.2-

6.6

FIG. 5. Calculated inversion efficiency for five-component optimized pulses with restricted phase values. The upper dii is for the pulse 63”(O”)142”(150”)350”(250°)1420(1500)630(Oo) (phase restricted to multiples of 10’) while the lower diagram is for the pulse 62D(0”)1450(157.50)3580(2700)1450(157.5”)620(Oo) (phase restricted to multiples of 22.5”).

rounding the freely optimized phase values to the nearest allowed value does not necessarily give the optimum sequence of phase-restricted pulses. The above calculations of inversion efficiency assumethat it is possible to implement the optimized composite pulses exactly as stated. In practice, however, errors in the component pulses’length and phase will inevitably degrade the performance from the ideal case. The effect of errors in the pulse parameters was investigated by calculating efficiency contours for m isset composite pulses. Inaccurate pulse-length calibration causesa proportionate change in length of each component pulse from its optimum value. Figure 6 shows the effect of +5 and -5% changes in pulse length on the 97% efficiency contour for a five-component optimized inversion pulse. The result is to shift (and slightly distort) the efficiency contours along

18

DAVID

J. LLJRIE

-0.5

B

-

I.4

1.2.

Bl et

I-

I 8.6t

I

6.6

I -I

a.5

i 1

AEVB; FIG. 6. Calculated 97% inversion efficiency contour for a five-component optimized composite pulse (solid line). The dashed lines show the effect of plus and minus 5% errors in pulse-length calibration.

the radiofrequency field strength axis; provided one is not operating at the limits of the pulse’s performance then errors of this magnitude are acceptable. Uncertainty in the pulse length and phase was simulated by calculating efficiency contours for a composite pulse whose parameters had been altered by random amounts from their optimum values, the maximum deviation being fixed. Figure 7 shows that I 1 4 t

I

2

!

B, 1BP 06

i 8.6

i

I -I

-

a AWB;

6.5

FIG. 7. Calculated 97% inversion e5ciency contour for a five-component optimized composite pulse (solid line) and for the same pulse a&ctcd by random errors in its components’ lengths and phases (dashed line). The “errors” were randomly generated and lay between +2” and -2”.

DESIGN OF COMPOSITE PULSES

19

the effect of random errors is to distort the efficiency contours; however, a single simulation is insufficient to estimate the tolerable uncertainty in the pulse parameters. This was done by accumulating calculated efficiency data for many composite pulses, each being affected by a different set of random errors, in order to determine the region where a given efficiency level was guaranteed. Figure 8 shows the 97% efficiency contour for a five-component composite pulse together with the “guaranteed’ contour corresponding to -t2” uncertainty in the pulse parameters. Errors of this magnitude do not appreciably degrade the composite pulse’s performance. In conclusion, a numerical optimization procedure has been developed and applied to the design of composite inversion pulses. The method is easily extended to the design of other types of pulse, for example, composite 90” pulses. Any aspect of the magnetization’s response can be optimized-for example, “phase distortion” (10) is easily accounted for by introducing a phase-error penalty into F(x). Although the optimization method is empirical and as such does not provide insight into the mechanism of error compensation, its advantage over other techniques is its flexibility and generality, allowing the design of composite pulses according to specific experimental requirements. While iterative expansion schemes generate efficient pulses, their disadvantage is that each iteration necessitatesan increase in the number of component pulses, so that the final composite pulse may have 25 or more components (II). Numerical optimization allows the composite pulse’s performance to be optimized while constraining the number of components and/or the total pulse length. Best results are obtained by removing all restrictions on the pulse parameters; however, experimental lim itations are easily accounted for. Restricting the allowed phase values to multiples of 7r/8 radians, for example, does not seriously degrade the achievable

FIG. 8. Calculated 97% inversion efficiency contour for a five-component optimized composite pulse (solid line). Regions within the dashed contour are guaranteed to have inversion efficiency above 97% when the uncertainty in the pulse parameters is +2”.

20

DAVID

J. LURJE

performance. Composite pulses designed in this way should prove useful in highresolution spectroscopy and especially in NMR imaging where dual compensation is frequently required, but where the overall pulse duration must be kept to a minimum to avoid excessive power deposition. ACKNOWLEDGMENTS The author was supported by an MRC Research Fellowship for the duration of this work under Grant SPG 7509376. I thank Professor J. R. Mallard for his guidance and support. Computational facilities were provided by the University of Aberdeen Computer Centre. REFERENCES 1. M. H. LEVITT AND R. FREEMAN, J. Magn. Reson. 33,473 (1979). 2. R. FREEMAN, S. P. KEMPSELL, AND M. H. LEVIIT, J. Magn. Reson. 38,453 (1980). 3. M. H. LEVITT AND R. FREEMAN, J. Magn. Reson. 43,65 (1981). 4. M. H. LEVI-IT, J. Magn. Reson. 48,234 (1982). 5. M. H. LEVITT, J. Magn. Reson. 50,95 (1982). 6. M. H. LE~IIT AND R. R. ERNST, J. Magn. Reson. 55,247 (1983). 7. A. J. SHAIU AND R. FREEMAN, .I. Magn. Reson. 59, 169 (1984). 8. A. J. SHAKA AND R. FREEMAN, J. Magn. Reson. 55,487 (1983). 9. R. WCKO, Phys. Rev. Lett. 51,775 (1983). 10. R. TYCKO, H. M. CHO, E. SCHNEIDER, AND A. PINES, .I. Magn. Reson. 61,90 (1985). II. R. Tyc~o, A. PINES, AND J. GUCKENHEIMER, J. Chem. Phys. 83,2775 (1985). 12. D. J. LURIE, Magn. Reson. Zmaging3,235 (1985). 13. Numerical Algorithms Group. Routines E04CCF and E04JBF. NAG Fortran Library Manual Mark 10 (1982). (Numerical Algorithms Group, Maytield House, 256 Banbury Road, Oxford, United Kingdom.) 14. J. A. NELDER AND R. MEAD, Comput. J. 7,308 (1965). 15. P. E. GILL AND W. MURRAY, J. Inst. Math. Appl. 9, 91 (1972).