A B1-Gradient method for the detection of slow coherent motion

JOURNAL

OF MAGNETIC

RESONANCE

91, 128-135

( 1991)

A &Gradient Method for the Detection of Slow Coherent Motion D. *Centre

BOURGEOIS*

AND

M.

DECORPS*

,t

d’Etudes Nuclkaires de Grenoble, LBIO/RMBM, 85 X, 38041 Grenoble Cedex, France,. j-Groupe d ‘Applications de la RMN d la Neurobiologie, Laboratoire de Neurobiophysique (INSERM lJ318). Universite’ Joseph Fourier, H6pital Albert Michallon, Pavillon B, BP 217X, 38043 Grenoble Cedex, France Received

November

21, 1989; revised

May

and

30, 1990

Since Hahn’s experiment ( 1)) NMR has proved to be a powerful tool for studying motion at a microscopic or macroscopic scale and a number of methods have been proposed (2, 3). The application of such methods to the study of water transport in biological systems, for example, in vegetables, requires that the technique be able to determine flow in the presence of a large amount of stationary water. With this aim, some techniques avoiding the use of a pulsed static field gradient have been proposed ( 4-6 ) . Recently the use of linear RF gradients to detect motion was demonstrated ( 7, 8). However, these methods produce a decrease in the signal amplitude when motion occurs. In this study we introduce a pulse sequence which uses RF gradients to determine flow velocity and gives, in a single experiment, a null signal for stationary samples. Consider the pulse sequence OX--~-~-,-ACQUIRE,

[II

where 8, is the flip angle at position r of a pulse applied during 7 along the x axis of the rotating frame, and T is a free precession delay. With the assumption that the pulse duration 7 is long enough, RF inhomogeneities produce a complete dispersion of the magnetization vectors in the yz plane. In the absence of motion and if T = 0, all the magnetization vectors refocus along the z axis at the end of the sequence: the sequence generates a rotary echo along the z axis (9). If T is chosen so that T %TT , a complete dispersion of the transverse magnetization occurs during delay T, but it is easy to show that a rotary echo still occurs along the z axis just after the LX pulse ( 7). It should be noted that in this case the pulse sequence is strictly equivalent to 8,-T&-ACQUIRE, [21 which is easier to implement than sequence [l] because critical adjustments of the phase and the amplitude of pulses are avoided. This pulse sequence is shown in Fig. 1. Karczmar et al. ( 7) have shown that the amplitude of the z component of the rotary echo decreases as the mean velocity increases. To observe this component they had 0022-2364191

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Copyright 0 1991 by Academic Pms, Inc. All rights of reproduction in any form reserved.

128

129

NOTES

ROTARY ECHO

HAHN ECHO

FIG. 1. Pulse sequence [ 21. Two identical long RF pulses are applied during T, using the inhomogeneous RF field produced by a flat coil. They am separated by a free precession delay T. The length of T must be enough that a complete dispersion of the magnetization occurs in the yz plane. Data may be acquired either during a rotary echo which occurs immediately aRer the second pulse or during a Hahn echo which occurs at a time T after this pulse. These ethos occur only in the presence of motion.

to apply a readout pulse immediately after the last gradient pulse. We show in this Note that, in the presence of motion, the rotary echo occurs along an axis which differs from the z axis, and consequently that sequences [l] or [ 21 directly produce a signal whose amplitude is a function of the velocity component along the gradient direction and is zero for stationary fluids. In addition, we show that the amplitude of the Hahn echo produced by sequences [l] or [ 21 is also a function of the velocity and goes to zero in the absence of motion. In the presence of RF inhomogeneities, the rotation angle O(r ) of the magnetization M( r)d I/ of a sample element at position r, produced by the pulses during T, is locationdependent, e(r) = yBl(rb,

t31

where B, (r) is the amplitude of the transverse rotating component of the RF field at location r, and y is the gyromagnetic ratio. We assume that motion occurring during the pulses is negligible. Because of motion during delay T ( T ti T), a given sample element at r during the first pulse will be at r’ during the second. Neglecting offset effects during the pulse durations but taking into account B. inhomogeneity and relaxation effects occuring during the T delay, we can write the magnetization M(r’, T + 7)d V just before the second pulse as

MZ(r’,

MJr’,

T + 7) = MO(r)sin

M,(r’,

T + T) = MO(r)sin B(r)cos cp(r, r’)exp(-T/T2)

T+ T) = MO(r)(l

B(r)sin cp(r, r’)exp(-T/T*)

- exp(-T/T,))

+ MO(r)cosB(r)exp(-T/T,),

141 I-s1 [6]

where Mo( r)d I/is the initial equilibrium magnetization from the volume element d V at r, (o(r, r’ ) is the local phase shift during the free precession delay, and T, , T, are the relaxation times. The magnetization after the second pulse (t = T + 27) is then given by M,(r’,

T + 27) = M,(r’,

T + T)

171

MJr’,

T + 27) = M,(r’,

T + T)sin O(r’) + M,.(r’,

T + 7)cos fI(r’)

I81

MAr’,

T + 27) = M,(r’,

T + 7)cos 6(r’) - M,(r’,

T + T)sin t9(r’).

F91

130

NOTES

If a coil producing a homogeneous RF field is used as receiver, the signal is directly proportional to the transverse component of k, where .& is the sum of magnetization vectors at the different points of the sample. After the second pulse, this total magnetization is given by JZ,(r’, &Jr’,

T + 27) =

s

MJr’,

T + 27) =

T + 27) = -

s

M,(r’,

T + 7)dV

T + r)cos B(r’)dV+

X (1 - exp(-T/T1)dV+ &Jr’,

s

MJr’,

s

M,,(r)cosB(r)sin

T + T)sin B(r’)dV+

S

X (1 - exp(-T/T,)dV+

MO(r)sin @r’)

s

Mo(r)cos

&r’)exp(-T/T,)dV

s

[ll]

MO(r)cos &r’)

8(r)cos B(r’)exp(-T/Tr)dV.

[12]

If we assume that Tz < T, MJr’, T + 7) and M,,(r’, T + T) vanish, and therefore &( r’ , T + 27) and the first term in [ 111 and [ 121 also vanishes. Furthermore the RF field inhomogeneities and r are assumed to be large enough for a complete dispersion of d(r) and tl(r’) to occur over the sample volume. Therefore (sin O(r)), (cos fYr)), (sinto( etr’))), and (cos( 6( r) + 0( r ’ ))) go to zero. Then the use of the expansions sin x cos y = f [ sin(x - y) + cos(x + y)] and cos x cos y = $ [ cos(x - y) + cos( x + y)] yields &,(r’, .&Jr’,

T+ 27) = i

.M,,(r’, T + 27) = i

T + 27) = 0

S S

[I31

&&(r)sin(e(r’)

- B(r))exp(-T/Tr)dV

[I41

M,,(r)cos(O(r’)

- e(r))exp(-T/T1)dV.

iI51

From this expression it appears that an echo occurs in the yz plane of the rotating frame. In the absence of motion e(r) = 0( r ’ ), J)1, = 0, the echo occurs along the z axis. In the presence of coherent motion JI1, # 0, and the echo occurs along an axis of the yz plane which differs from z and generates a signal. To properly evaluate the difference 0( r’ ) - 0(r) and since our aim is to measure very low velocities we have to take into account motion due to self-diffusion. Let us suppose that the RF gradient is uniformly generated along the Xaxis of the laboratory frame. VX and rx are the components along X of the velocity and the position. Since we choose r 4 T, we have &r’)

- d(r) = vdx,

[I61

where gl is the Bi field gradient and Zx = P-‘~- rx. The distribution of values lX is gaussian, with a mean VxT and a standard deviation u = ( 1:) rJ2 = (2TD) 112,where D is the diffusion constant (10). Finally the magnetization can be written as

131

NOTES

J12, = i Jil, = i

Mo(r)sin(yYxgrT-r)exp(-y2T2g:TD)exp(-T/T,)dV

[171

M0(r)cos(yYXg,T~)exp(-y2~2g:TD)exp(-T/T,)dV.

II181

s s

For a stationary sample, ‘V, = 0 and then J12, = 0, whatever the diffusion effects In this sense the technique is insensitive to incoherent motion and to the presence of stationary fluids. For a uniform sample with a uniform velocity, Eq. [ 17 ] reduces to Jn,, = ~.Mosin(yVXglT~)exp(-y2~2g~TD)exp(-T/T,), where & is the total For laminar flow = 2VX( 1 - r*/R*), velocity. The integral found:

magnetization. through a tube of radius R, the velocity is given by VX( v) where r is the distance from the tube axis and K is the mean [ 17 ] can be solved analytically and the following result can be

with A = f exp(-y2T2gjTD)exp(-T/T,) and a = yg,rT.

This gives a maximum

transverse magnetization

for

Vxmax = 1.165/a, and the value of this magnetization

Furthermore the observed at time T absence of coherent RF inhomogeneities motion. Indeed the M,(r”,

is

flow-dependent modulation of the magnetization may also be after the second pulse. At this time a Hahn echo occurs. In the motion, the amplitude of this echo is equal to zero if T and the are large enough. This is not true in the presence of coherent transverse magnetization at time T after the second pulse is

2T + 27) = exp(-T/T2){MX(r’,

T+ 27)cos cp(r’, r”) + M,(r’,

A&(r”,

2T + 27) = exp(-T/T2)(M,,(r’,

T+ 2r)sin cp(r’, r”)}

[25]

T + 27)cos cp(r’, r”) - M,(r’,

T + 2r)sin cp(r’. r”)},

[26]

where r” is the location at t = 2T + 27, and cp(r’, r” ) is the local phase shift associated with the free precession during the second T delay. As we observe only very low

132

NOTES

velocities, we can consider &, to be the same at r, r’, and r”. Then we have cp(r, r) = cp(r, r’ ) = cp(r’, r”). Moreover, due to &, inhomogeneities over the whole sample and if T is long enough, a complete dispersion of angles (o occurs over the sample volume. Then after elementary contributions are summed, all terms vanish but one, which may be written as .&Jr”,

2T + 27) =

s

A&(r)sin

B(r)cos O(r’)cos2y?(r, r)exp(-2T/T2)dV

[27]

or &Jr”,

2T+

27) = $J)Zo(r)sin(y~~g,T7)exp

-(y2r2g:TD)exp(-2T/T2)dV.

[281 It should be noted that the occurrence of this Hahn echo can be simply demonstrated by noting that the first pulse generates a transverse magnetization proportional to sin 0(r). This magnetization is refocused by the second pulse 0( r ’ ) . It is known ( I1 ) that the echo amplitude is proportional to sin 6(r) sin2 [ 19(r’ )/2]. Using the assumption of a complete dispersion of pulse angles, it follows that this amplitude is proportional to (sin 8(r)sin2[6(r’)/2]) = $ sin[B(r) - d(r’)] = $ sin(yVxglT7). In order to assessthe technique, we used the inhomogeneous RF field produced by a 4 cm diameter flat coil along its axis X. water circulates through a glass tube along the coil axis (Fig. 2). The flow is controlled by a high-precision peristaltic pump. Since the gradients produced by a flat coil on the two sides are of opposite signs, the signals produced due to the effect of flow cancel out. It follows that the emitter coil cannot be used as receiver. Therefore to collect the NMR signal we used a saddleshaped coil producing a RF field perpendicular to the flat coil axis. Setting the center of the saddle-shaped coil 1 cm (half the radius of the flat coil) away from the center of the flat coil allows us to obtain signals originating only from a region of the sample which experiences an approximately linear RF gradient. Capacitive coupling between both coils was minimized by using electrically balanced matching schemes. Inductive coupling was geometrically minimized. The isolation between the coils is then better than 50 dB. Experiments were carried out with a 2.35 T horizontal superconducting magnet and a Bruker CXP spectrometer. The B1 gradient at the center of the receiver ONE TURN CIRCULAR (Emitter coil)

COIL

FIG. 2. Experimental setting used with the pulse sequence of Fig. 1. A 4 cm diameter flat coil producing a roughly linear RF gradient along its axis is used for emission. A saddle-shaped coil (2 cm diameter, 2 cm length) is used for reception with its axis perpendicular to the flat coil axis. The center of the saddle-shaped coil is placed at a distance of 1 cm from the center of the flat coil.

133

NOTES

coil was calculated from the measurement of the a/2 pulse at the center of the coil and found to be equal to 7.6 kHz/cm. Peaks are averaged over 30 scans to minimize the effects of instabilities that may be caused by fluctuations in output power or RF phase. Figure 3 shows the amplitude of the Hahn echo as a function of the flow through a 5 mm inner-diameter glass tube. The pulse sequence parameters (7 = 3 ms: T = 200 ms) were adjusted so that the condition yVXgl Tr 6 a/ 2 was satisfied. The linearity of the response clearly appears and Fig. 3 shows that velocities lower than 100 pm/s are detectable. The residual signal for stationary water is due both to instrumental instabilities (small inequalities of the two pulses) and to slow fluid motion, which could be due to thermal convection. Acquiring the signal produced by the Hahn echo could allow one to spatially encode the transverse magnetization and then to obtain angiographies. However, when T2 6 T1, this advantage is offset by the fact that 7 must be of the order of T2 or shorter, rather than of the order of T, . Furthermore the signal from the Hahn echo is halved in comparison to the signal immediately after the second pulse. The signal obtained just after the second pulse, from the rotary echo, is plotted in Fig. 4 as a function of the velocity of the flowing water in an 8 mm inner-diameter tube for different sets of parameters (set A: 7 = 1 ms, T = 700 ms; set B: 7 = 3 ms, T = 150 ms; set C: 7 = 4 ms, T = 700 ms). The signal intensity is normalized to the intensity of the signal produced by a 90” one-pulse experiment carried out by using the saddle coil as emitter and receiver. Therefore, if the RF field is assumed to be homogeneous, this normalized signal is equal to J&,/J&. If a laminar profile is assumed, Eq. [ 201 can be used, and the theoretical curves can be fitted to the experimental data by adjusting parameters A and a. As shown in Table I, there is a good agreement between the values of A and a obtained from the experimental data and those determined from Eq. [21] and [22] (with T, = 4s, D = 2.5 X lo-” cm2/s, yg1/27r = 7.6 kHz/cm ) . This agreement has been obtained despite the poor linearity of the RF gradient over the receiver coil size. This is probably because, due to the field distribution of the receiver coil which is gaussian rather than homogeneous (4, 12), most of the signal arises from the center of the receiver coil, a region where the gradient is ap8-

-I

Signal (Arbitrary

units)

64-

Micron/s I 0

200

400

I

600

800

FIG. 3. Signal amplitude of the Hahn echo versus the velocity of circulating water through CT = 3 ms, T = 200 ms).

a glass tube

134

NOTES

f

Signal

FIG. 4. The signal intensity, acquired immediately after the second pulse, plotted versus the velocity for different sets of parameters. Plot A: ~=1ms,T=7OOms;plotB:7=3ms,T=150ms;plotC:r=4 ms, T = 700 ms. The theoretical curves were calculated for laminar flow with Eq. [20] by adjusting the values of the parameters A and a (Table 1). The vertical scale is given in units of the signal obtained from a 90” one-pulse experiment carried out with the saddle coil used as emitter and receiver.

proximately linear. However, the agreement is better for sets of parameters A and B than for set C. The reason could be that the influence of diffusion is negligible for sets A and B ( y %*g: TD < 1) but not for set C; the signal attenuation due to diffusion depends on g: and therefore is very sensitive to variations in the RF gradient. Errors introduced by the nonlinearity of the B, field gradient could be strongly reduced by decreasing the receiver coil size. Another source of error is the calibration of the pump, which was found not to be reproducible, yielding some uncertainty in measurements of slow velocities. The main disadvantage of the method is closely related to its very high sensitivity, since instabilities in the RF power produce signal in the absence of motion and are likely to severely degrade the results. Thanks to its insensitivity to stationary fluids, this method, by allowing measurements of velocities of the order of 100 pm/s, can TABLE 1 Experimental” and Calculatedb Values of Parameters A and u (Eqs. [21] and [22]) for Three Sets of Pulse Sequence Parameters

A =rp

&JC a,,, (s mm’) hc (s m-‘)

Set A

Set B

setc

0.43 0.40 2950 3340

0.44 0.45 1780 2150

0.15 0.22 13,250 13,370

0 Obtained by fitting the theoretical curves obtained with Eq. [20] to the experimental data (Fig. 4). b Obtained from Eqs. [2 l] and [22] with T, = 4 s, D = 2.5 X 10m5cm*/s, and yg,/2s = 7.6 kHz/cm and r=1ms,T=7OOms(setA);r=3ms,T=150ms (set B); 7 = 4 ms, T = 700 ms (set C).

be useful for flow studies in biological systems, particularly in plants. Finally the technique may be easily combined with standard imaging protocols. REFERENCES I. E. L. HAHN, Phys. Rev. 80, 580 ( 2. H. VAN As AND T. J. SCHAAFSMA,

3. 4. 5 6. 7

1950). in “An Introduction to Biomedical Nuclear Magnetic Resonance” (S. B. Petersen, R. N. Muller, and P. A. Rinch, Eds.), pp. 68-95, Thieme Verlag. Stuttgart. W. Germany, 1985. STILBS, Prog. NMR Specfrosc. 19, 1 ( 1987). A. HEMMINGA, P. A. DE JAGER, AND A. SONNEVELD, J. Magn. Reson. 27, 359 ( 1977). A. HEMMINGA AND P. A. DE JAGER, J. Magn. Reson. 37, 1 ( 1980). VAN As AND T. J. SCHAAFSMA, J. Magn. Reson. 74,526 ( 1987).

P. M. M. H. G. S. KARCZMAR, D. B. TWIEG, T. 3. LAWRY, G. B. MATSON, AND M. W. WEINER, Magn. Rcsw~ Med. 7, 11 I ( 1988). 8 D. CANET, B. DITER, A. BELMAJDOUB, J. BRONDEAU, J. C. BOUBEL, AND K. ELBAYED. J. Magn. Reson. 81, 1 (1989). 9. J. SOLOMON, Phys. Rev. Lets. 2, 301 (1959). IO. H. Y. CARR AND E. M. PURCELL, Phys. Rev. 93,630 ( 1954 1. Ii. D. E. WOESSNER, J. Chem. Phys. 34,2057 ( 1961). I? D. 1. HOULT, J. Magn. Reson. 21, 337 (1976).