MAGNETIC RESONANCE IN MEDICINE 19,20-30 ( 1991)

Analysis of the Higher-Order Echoes in SSFP

Department of Eleclrical Science, Koreu Advanced Institute of Science, P.O. Box 150, Cheongyangni, Seoul, Korea Received December 22, 1989; revised March 23, 1990 In the steady-state free precession ( SSFP) with an applied linear gradient, the transverse magnetizations are periodically distributed. Fourier analysis of this periodic distribution leads to the understanding of many interesting phenomena in SSFP. It is found that there are many other higher-order echoes in SSFP in addition to the previously known echoes. By deriving general description of the transverse magnetization of SSFP and by expanding the echo time independent term with Fourier series, the higher-order echoes including the two previously known FISP and CE-FAST are understood and explained. These higherorder echoes are studied in detail by both computer simulation and experiments and their results are reported. 0 1991 Academic Press, Inc. INTRODUCTION

Since steady-state free precession (SSFP) was first described by Carr ( I ) in 1958, several groups (2- 7) have analyzed the behavior of the magnetization and the resulting image contrasts. Major interests have been, however, the two components of SSFP signals: namely the signals obtained just after and before the rf pulses, i.e., M i known as FISP (8) and M ; known as CE-FAST ( 9 ) , the latter believed to be weighted by T 2 .In practice, simultaneous acquisition of M , f and M , has become more popular since each of these signals usually exhibits different contrasts ( 5 - 7 ) . Furthermore, by adjustment of the two signals i.e., by positioning the two signals exactly to the same position or making them coincide, an inhomogeneity map can also be measured from the interference pattern of the two signals ( 10 ) . The other echoes in SSFP in addition to the previously known two echoes were found by Zur et al. ( 1I ) and Mizumoto et al. ( 12). The former expanded the transverse magnetization just after rf pulse by Fourier series and has shown that each Fourier coefficient represents an echo when their complex exponential factor becomes unity. The existance of these various echoes was used to derive motion insensitive SSFP imaging technique ( I 1 ). In the latter they have also separated the various echoes by expanding the power series, and experimental images were obtained with multiple CE-FAST images ( 12). In this paper, we have also expanded the echo time independent term only in the Fourier series among the transverse magnetization. This Fourier expansion then provides a simple and easy way of analyzing the higher-order echoes with both their existance and characteristics. * Department of Radiological Sciences, University of California, Irvine, CA 927 17. 0740-3 194/9 1 $3.00 Copynghl 0 1991 by Academic Press, Inc All nghlr of reproduction in any form reserved.

20

HIGHER-ORDER ECHOES IN SSFP

21

In the steady-state free precession with an applied gradient, the pattern of the transverse magnetization appears more complex than the one usually observed in the spinecho sequence and the distribution pattern observed is the function of several parameters such as the repetition time ( TR), TI, T2, and flip angle ( a ) .As discussed in the following, the transverse magnetization can be divided into two components, namely the echo time dependent and the other with echo time independent term, respectively. The latter echo time independent term, h ( # ) ,is a term common to all the signals obtained, i.e., M i , M,, or inhomogeneity map. h ( 8 ) also depends on TR, T I , T2, flip angle (a),and total precession angle (6).Term h ( 0 ) is also conjugate symmetric and periodic, and thereby can be expanded by Fourier series, i.e., Ho, H,, ,Hk2,H+,, . . . . Here Hodenotes the average value of h(O), and H,, and Hk2 are the first and second harmonics of h( 19)and so on. Fourier coefficients of h( 0) are real function. In the SSFP imaging, a large linear gradient is usually applied after the gradient refocusing, and thereby dephases the spurious signals in a voxel so that the signal in a voxel becomes the average signal of the precession angle from 0 to 27r. Thus the obtained signal can be expressed by a combination of the Fourier coefficients. For example M: in SSFP is expressed by E( TE)Ho, where, E( TE) = exp(-TE/T2). Since there are other higher-order harmonics such as H-,, H P 2 ,. . . , H2, H,, one can also expect that there are other higher-order echoes in SSFP; these are discussed in the next section. THEORY

The steady-state magnetization that develops after a long series of identical rf pulses depends on several parameters such as TR, T I , T 2 ,flip angle a, and the precession angle 8. For a given set of rf pulses which would result in a spin precession around the x axis, the steady-state transverse magnetization at the time t after an rf pulse has been derived previously ( 4 , 13-15), and summarized as (see Fig. 1 )

M,(t)

=

MJt)

= Mo( 1

Mo( 1 - E,)sin (Y E(t)[sin O1

+ E2sin(19- Sl)]/D,

El)sin a E(t)[cos 0, - E2cos(0 - Sl)]/D,

[I1 where t is the time variable between pulses (0 < t < TR),O1 is the precession angle in the transverse plane during the time t, and -

D

=

( 1 - E~COS a ) (1 - E ~ C O 0) S- (El - cos a)(E2 - cos B)E2,

El

=

exP(-TR/TI),

E2

=

exp(-TR/T2),

E( t ) = exp( -t/ T 2 ) .

[21 The total precession angle in the transverse plane between rf pulses ( 0 ) can be divided into two components, namely the precession angle due to the inhomogeneity (4)and the precession angle due to the linear gradient ( # ) ,respectively. The precession angles during the time t due to the inhomogeneity and gradient are denoted by and #, , respectively and their relations are given as (see also Fig. 1 ) 81

=

41 + # I .

[31

22

KIM A N D CHO 2nd rf ( or (n+l)th )

1st rf (ornth)

h

I

I

I

I I

I I

I I

I I

I I

Time

F

Precession angle

I I

due to inhomogeneity

t

L

-

@

I

-@-@I-

I I

Precession angle due to gradient

Total precession angle

TR 4

-I I I I

w1

-I -

I,

el -e-el+ w1

@,- I I

w - w1I-

I I I

I I I I

I

I

FIG. I. Definitions of various precession angles and related time periods. As noted and J., denote precession angles due to the inhomogeneity and gradient during the time interval t , respectively. O1 then denotes total precession angle, i.e., sum of 6 ,and

+ +,

+ J/.Using the complex quantity M x y ( t )

Note also that 6' equals # i.e., 6' = # = My(t ) iMx(t ) , Eq. [ 11 can be written as

+

M , , ( t ) = Mo( 1 - El)sin a { l - Ezexp(-i6'))E(t)exp(i6',)/D. [41 From Eq. [ 41, it is easy to see that there are two characteristically different terms, namely the t-dependent and t-independent transverse magnetization components, respectively. The latter t-independent term which is noted as h( 6') is given by

h ( 6 ) = Mo( 1 - E,)sin a(1 - E2exp(-i6'))/D.

PI

Equation [4], therefore, can be rewritten as M x y ( t )= h(6')E(t)exp(id1). h(6') is a function of TR,T I , T 2 ,a , and 8, and is also a periodic and conjugate symmetric function. The term h(6')can be expanded by a Fourier series ( l l ) ,i.e., n=m

h(B) =

2

Hnexp(in6'),

n=-m

where Hn is the Fourier coefficient, i.e., 1

H n = 2-7T J

2a 0

h(8)exp(-inB)dB,

n = 0, f l , t-2,. . . .

171

Typical form of h( 6') and H,, are illustrated in Figs. 2 and 3 for various flip angles, i.e., a = lo", 20", 40", and 60" (parameters used for the computation are TR = 40 ms, TI = 770 ms, and T2= 460 ms, respectively). Since h( 6') is conjugate symmetric,

23

HIGHER-ORDER ECHOES IN SSFP

O0

ISO'

360-

FIG.2. Plots of the echo time independent term h( 0 ) as a function of the precession angle 0 from 0 to 271 for various flip angles, i.e., 01 = lo", 20", 30", and 60", respectively. ( a ) Re[h(0)] is real part of h ( 0 ) and ( b ) Im[h(B)] is the imaginary part of I t ( @ )Parameters . used for computation are TR of 40 ms, T , of I10 ms, and T z of 460 ms.

H,, is a real function. In Figs. 2( a) and 2(b), plots of real and imaginary part of h( 0 ) for various flip angles, i.e., a = lo", 20", 40°, and 60", are shown, respectively. And their Fourier coefficients are plotted in Fig. 3. Note here that the high frequency components of h ( 0 ) become larger as the flip angle cy becomes smaller. In the imaging condition, a large linear gradient is usually applied such that each voxel has widely spread transverse precession angles ranging from 0 to 27r. Since those transverse precession angles are uniformly distributed over the entire voxel, the signal obtained is the average signal in a voxel(16), i.e.,

or

24

KIM AND CHO

Hn 03

02

0. I

0.0

-0. I

-0.2

I

-0 3 -8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

n

8

FIG.3. Plots of Fourier coefficients H , for various flip angles, i.e., 01 = lo", 20", 30", and 60", respectively. The parameters are the same as those given in Fig. 2. Because h(B) is conjugate symmetric, H, is real function. As noted, the larger signal of the higher-order echoes are obtained at small flip angles.

In typical imaging sequences, the gradients are positive, negative, or zero. The precession angle 4b1 due to gradient during the time duration t , therefore, can be negative, positive, or zero so that the precession angle 4b1 can be the integer multiples of $. Therefore, one can expect that MXy( t ) would contain a large number of terms, each of which corresponds to the Fourier coefficients of h( 0); i.e., many higher-order echoes which correspond to the Fourier coefficients of h( 8) are produced if a suitable adjustment of the reading gradient is made. In Fig. 4, an illustrative example is shown in which the signals are generated at various points which correspond to their respective precession angle (developed by the gradient strength between the rf pulse and the echo center). Note here that the precession angle is again integer multiples of the total precession angle +. For example, = -24b at t = 7:, = -$ at t = r : , = 0 at t = T O + , = at t = 24b at t = r ; , and = 34b at t = 7;. Thus if is m ( m is an integer) = 70, multiples of $, 0, in Eq. [ 31 can be written as

+

=

41 + m#

=

-

m4

+ m8.

[91

rf

I I

I

gradient I I

I I

/

I I I I

I

\

I I I 1 I

I I

/i I

I I

I I

signal

1 I

I I

I I I

I I I

I I I I

I I I I

c--------r 2

:

I

4

c--------- 7,;

/

I

I

I

I

-t

I I

I

FIG.4. Illustration of echoes expected to be generated by the zeroth-, first-, and second-order Fourier coefficients. From the top, the applied rfpulse, reading gradient, and the various generated echoes are shown.

FIG.5. Echoes obtained by the computer simulation and experiments. Parameters used for the simulation and experiment are TR = 40 ms, TI = 770 ms, TZ= 460 ms, (Y = lo", and the diameter of spherical object = 35 mm, respectively. 25

26

KIM AND CHO

By substituting 19, in Eq. [ 91 into Eq. [ 81, MXy( t ) can be rewritten as

or =

1 E(t)exp{i(41 - m 4 ) } 27r

-J

2 , ~ n=m

2

0

Hnexp{i(n + m)19)d19.

fl=~a)

After some computation it can be shown that the Gxy( t ) can be reduced to Mxy(t)

=

E(t)fLnexp{i(41 - m 4 ) ) .

[I11

In Fig. 5, a computer-simulated result of Eq. [ 101 or [ I 11 and its corresponding experimental result of higher-order echoes up to the second order for both M + and M - are shown. In this case a total of 10 echoes are visible. Note, however, that the conventionally observed echoes M i (FISP) and M i (CE-FAST) are also a part of those echoes. Additionally observed higher-order echoes are the groups ( M T , M: , M T , M y ) and (MTt, Mi., My?,M;.) and the latter group is the echoes with gradientreversed cases of the former. Above simulations and experiments were performed with T, of 40 ms, TI of 770 ms, T2of 460 ms, LY of lo", and an applied reading gradient of 0.2 G/cm, respectively. In the real NMR imagings, the final images are obtained by taking the absolute values of the complex image data which is given by [I21

I M x y ( t ) l = E(t)lH-,I.

In Eq. [ 121, TT effect is neglected. From Figs. 1 and 4 and Eq. [ 121, it can be shown that at the time o f t = TO' and m = 0, Eq. [ 121 can be reduced to the well known zeroth-order echo M i (FISP) i.e.,

IMxy(70')l = ~(70f)lff~I. ~ 3 1 In Eq. [ 131, again the signal loss due to T: is neglected. The zeroth-order signal M i is the E(~i)-weightedmagnitude of the average signal HO.Other examples at the time o f t = T: with m = -1, t = 7: with m = -2, and t = 7; with m = 1 are given by

lMxv(7~)l = E(T:)IH~I,

t

=

7:

and m

=

-I,

1141

lMxy(7;)l = E ( ~ : ) I H ~ I ,

t

=

7;

and m

=

-2,

~ 5 1

lM,,(~i)l = E ( T ~ ) ~ H1, - , t = 7; and m = 1. [161 As is known, I ax,,( 70) I in Eq. [ 161 which is equivalent to M , is the well known T2-weighted echo signal. This again can be interpreted as a signal due to the first harmonic K1weighted by E(70). Computer simulations are performed for the various combinations of T Iand T2 values to examine the characteristics of each echo. For the simulation study, three different materials are selected, namely ( 1 ) T I = 800 ms and T 2 = 70 ms, ( 2 ) T I = 800 ms and T 2 = 140 ms, and ( 3 ) T I = 1500 ms and T 2 = 200 ms, respectively.

27

HIGHER-ORDER ECHOES IN SSFT

In Figs. 6 (a) and 6 (b), the signal strengths as a function of rf flip angle are plotted to show the flip angle-dependent signal strength. In this simulation the susceptibility effect is assumed to be negligible. Simulation shows that the higher-order echoes have in general smaller signal strengths and also have a tendency to show their signal maxima at smaller flip angles. For example, in the case of the TI = 800 ms and T 2 = 70 ms combination, it has signal maxima at a! = 23" for M i , a! = 35" for M i , a! = 11O for M T , a! = 19" for M y , a! = 8" for M:, and a! = 15" for M Y , respectively. In the real imaging condition, the signal loss due to TT in a voxel must be considered. As is known, TT is mainly generated by local field gradients, such as the main magnetic field inhomogeneities, inhomogeneities due to spatially dependent eddy current, and differences in magnetic susceptibilities due to heterogeneous tissues or at air / tissue interfaces. Local field gradients induce the signal loss by the mutual cancellation of individual signal components, especially in the selection direction. As implied in the term exp{ i( 41 - rn6)) in Eq. [ 1 11, if the order of echoes becomes larger, the signal loss due to TT increases. For example, the zeroth-order echo experiences the susceptibility effect only during the echo time. The first-order echo, however, experiences the susceptibility effect throughout the periods which include both the echo time duration and a repetition time. The second-order echo suffers even further experiencing an additional susceptibility effect for an added repetition time in addition to that of the first echo; therefore the susceptibility artifact appears even more pronounced. The existence of multi-echoes can also be understood by drawing the phase diagram which describes the evolution of the phase change of the spin system along the time axis ( I 7-19). Figure 7 illustrates the phase diagram in SSFP sequence. Let us first consider the case of rn = 0. In this case after the application of the first rf pulse, spins experience the transverse precessions due to the gradients and reach a substate mag-

0 164

0 164

0 123

0 123

0 082

0 a82

0 041

0

a 0"

a

b

FIG.6. Plot of signals as a function of flip angle m. ( a ) Plot of M ; , M:, and M: for the three different combinations of T I and T 2 values. (b) Plot of M ; , M ; , and M i with same T I and T2 combinations. Combinations are ( 1 ) T , = 800 ms and T , = 70 ms; ( 2 ) T I = 800 ms and T , = 140 ms; and ( 3 ) T I = 1500 ms and T 2 = 200 ms, respectively.

28

KIM AND CHO 1st rf ( or nth)

2nd rf ( or (n+l)th )

a rf

Gradient

I I I I I I I I I

FIG.7. Phase diagram representing the evolution of the spins in the SSFP sequence with an arbitrary flip angle a.The labels on the vertical axes in the right correspond to the spin states while the dashed horizontal lines correspond to the z values. When the gradient is applied, the substates F , , F,, . . . , F: , F; , . . . are affected but not the Z , , Z,, . . . , Z ;, Z ;, . . . . Signals are produced at the crossing with the horizontal baseline M, or My between the rf pulses where the spin system is in a state of pure transverse magnetization. As an example, the substate F, just after rf pulse generates M: signal and the substate F2 generates M: signal and so on. On the other hand the substate F: generates M ; while the substate F ; generates M ; and so on.

netization F1 at the end of gradient pulse, i.e., right before the second rf pulse. This state, i.e., F , then has transverse precession angle of 8. The second rf pulse then leads to the inversion of the magnetization to its complex conjugate FT and at the same time leads to the generation of z magnetization, i.e., Z , and Z 7 . The latter appears to act as a storage function of the phase information of the spin system ( I 7, 18). These states, i.e., 2, and Z 7 lead to the subsequent generation of the stimulated echoes. Since the phase information stored in Z l and 2 7 is identical to that of F , and F 7 , and do not change with time; i.e., Z l and Z 7 remain in the same horizontal (dashed) lines as shown in Fig. 7. In a similar way further evolution of the spin system in the SSFP sequence leads to the subsequent states F2, FF , . . . , F,, FE , Z,, Z ;, . . . , Z,,, and 2 ; . These various substates experience transverse precessions which correspond to . . . , -38, -28, -8, 0, 8, 28, 38 (see Eq. [lo]). If appropriate gradient is applied as shown

HIGHER-ORDER ECHOES IN SSF'P

(4

(el

29

(9

FIG.8. Images of a human volunteer's head corresponding to the six experimentally observable echoes, namely MT, ( a) , MT (b), Mo+( c ) , M i ( d ) , M ; (e), and My, (f), respectively. The repetition time used was 40 ms with an rf flip angle ( a )of 20".

in Fig. 7, these various transverse states can be refocused to generate NMR signals. For example, the spins which experience the transverse precession of angle 19 can be refocused by applying a suitable gradient (strength and time period) that corresponds to - B and the transverse magnetization just after an rf pulse corresponds to the case of m = 0 in Eq. [ 121 which leads to the signal amplitude of Ho. Similarly F, substate corresponds to the case of rn = -n which leads to H , and so on. Signals are then produced at the crossing points with the horizontal baseline M, or M y ,i.e., the substate F , generates signal M: and so on. EXPERIMENTAL RESULTS AND DISCUSSIONS

For the experimental proof of the existence of the higher-order echoes, a ping-pong ball having diameter of 35 mm was measured using the KAIS 2.0-T superconducting NMR system. When the phantom was filled with 0.25 m M Gd-DTPA solution, T I and T , values were 770 and 460 ms, respectively. Relatively long T , allowed us to conveniently observe the higher-order echoes. Pulse sequence was operated with TR of 40 ms with flip angle of 10". As shown in Fig. 5(c) the experimentally observed signals with the applied gradient show the expected performance. First row (a) is the applied reading gradient, second (b), and third row (c) are the real part of the signals obtained by computer simulation and experiments, respectively. With the given reading gradient 10 echoes are produced, namely, M:!, M:!, M i , M:, M i , M,, M ; , M ; , My!, and M ; f . Note here that (M:., Mi., M;?, My.) are the corresponding echo signals of ( M : , M i , M ; , M ; ) when the gradient is reversed. As expected the

30

KIM AND CHO

experimentallyobtained higher-order echoes are somewhat smaller than the computer simulated ones due to T: effect. Subsequently human imaging was performed and images corresponding to the six echoes were obtained on the same run, namely M:., M:, M i , M ; , M y , and MTr, respectively. The second-order echoes such as Mzr, M: , M ; , and MTrwere too small to be useful for the imaging and therefore discarded. In Fig. 8 corresponding images of M:f, M:, M ; , M,, M ; , M;. are shown with susceptibility artifact correction. The susceptibility artifact was observed in both M: and M ; . Since image degradations due to local gradients are relatively large in the conventional images due to the incomplete refocusing, resulting images are severely distorted (20). In practice, susceptibility artifact is avoided by use of multiple step imaging, i.e., 16 steps of different refocusing gradients are applied for each imaging and the final image is then obtained by summing the 16 images. Since each image now has a different susceptibility artifact the sum of these images results in a susceptibility compensated image. As is known, the M$ and M: images are TI-weightedimages while M , and M ; are the T2-weighted images, respectively. In conclusion, we have successfully extended and observed the higher-order echoes in the region of M + in addition to the previously observed higher-order echoes in the domain of M - by Mizumoto et al, ( 1 2 ) , and thereby completed observation of the entire transverse magnetizations. These extended echoes are both theoretically studied and experimentally confirmed. REFERENCES 1. H. Y. CARR,Phys. Rev. 112, 1693 (1958). 2. R. B. BUXTON,J. Magn. Reson. 83, 576 (1989). 3. P. VAN DER MEULEN,J. P. GROEN,A. M. C. TINUS,AND G. BRUNTINK,Magn. Reson. Imaging 6, 335 (1988). 4. Y. ZUR,S. STOKAR,AND P. BENDEL,Magn. Reson. Med. 6, 175 (1988). 5 . S. Y. LEEAND Z. H. CHO, Magn. Reson. Med. 8, 142 (1988). 6. T. W. REDPATHAND R. A. JONES, Magn. Reson. Med. 6,224 ( 1988). 7. H. BUNDER,H. FISCHER,R. GRAUMANN, AND M. DEIMLIG,Magn. Reson. Mrd. 7, 35 ( 1988). 8. A. OPPELT,R. GRAUMANN, H. BARFUSS,H. FISHER,W. HARTL,AND W. SCHAJOR,Electromedica 54, 15 (1986). 9. M. L. GYNGELL,N. D. PALMER, A N D L. M. EASTWOOD, in “Abstracts, SOC.Magn. Reson. Med.,” p. 666, 1986. 10. S. Y. LEE,C. Y. RIM,J. B. RA, AND Z. H. CHO, in “Abstracts, SOC.Magn. Reson. Med.,” p. 959, 1988. 11. Y. ZUR,M. L. WOOD, AND L. NEURINGER, in “Abstracts, SOC.Magn. Reson. Med.,” p. 1128, 1989. 12. C . T. MIZLIMOTO, E. YOSHITOME.K. YAMAGUCHI, AND T. MATOZAKI,in “Abstracts, SOC. Magn. Reson. Med.,” p. 833, 1989. 13. R. FREEMAN AND H. D. W. HILL,J. Magn. Reson. 4, 366 ( 1971). 14. R. R. ERNSTAND W. A. ANDERSON, Rev. Sci. Instrum. 31, 93 ( 1966). 15. S. MATSUI,M. KURODA,AND H. KOHNO,J. Magn. Reson. 62, 12 (1985). 16. K. SEKIHARA, IEEE MI-6, 157 (1987). 17. E. L. HAHN,Phys. Rev. 80, 580 (1950). 18. D. E. WOSSENGER,J. Chem. Phys. 34,2057 ( 1961). 19. J. HENNIG,J. Magn. Reson. 78, 397 (1988). 20. J. FRAHM,K. D. MERBOLDT,AND W. HANICKE,Magn. Reson. Med. 6,474 ( 1988).