JOURNAL OF MAGNETIC RESONANCE, Series A 117, 311–316 (1995)

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A Robust, Single-Shot Method for Measuring DiffusionCoefficients Using the ‘‘Burst’’ Sequence

SIMON J. DORAN* AND MICHEL DECORPS

INSERM U 438—RMN Bioclinique, Pavillon B, Centre Hospitalier Universitaire, BP 217 X, 38043 Grenoble Cedex, France

Received May 4, 1995; revised August 14, 1995

NMR methods have long been used for the measurement However, the method is not directly applicable to the Burstexperiment, since the system is far from being in dynamicof diffusion coefficients. Most previous studies have been

based on the pulsed-gradient spin-echo (PGSE) method of equilibrium after 64 pulses.A more fruitful approach is to recognize that the excitationStejskal and Tanner (1) , but a number of faster new methods

have recently been suggested (2, 3) . This paper describes a is a DANTE pulse train, to use the original small-angleapproximation of Morris and Freeman (9) and to modify it innovel use of a pulse sequence called Burst (4–7) , originally

proposed in an imaging context, to measure bulk diffusion order to analyze the effects of diffusion. For these theoreticalresults to be applicable experimentally, much smaller totalcoefficients in a single shot. The Burst sequence permits

high-speed imaging by creating a train of echoes from a (on-resonance) flip angles are used here than was the casein the original imaging studies (4) . For quantitative diffusionseries of low-flip-angle radiofrequency pulses applied under

a gradient. Rather than phase-encoding the echoes as in (4– measurement applications, the advantages of being able eas-ily to describe both the form of the signal decay and the7) , we use their intrinsic sensitivity to diffusion in the se-

quence of Fig. 1. The small-angle approximation (8, 9) is distribution of the excited magnetization justify using theselower angle pulses, despite the fact that the experimentalused as a basic framework to describe theoretically the trans-

verse magnetization created, and diffusion is introduced into signal-to-noise ratio is reduced.Consider a spin-echo experiment, with an excitation pulsethe model to obtain an expression for the echo attenuation.

The theory is verified experimentally and accurate bulk dif- defined by the arbitrary function B1( t) , applied along the/x axis of the rotating frame and under a gradient G , whichfusion measurements made for DMSO, water, and acetoni-

trile. Using a sample of CuSO4-doped water, it is shown that we shall call the excitation gradient. For simplicity, we shallignore T2 relaxation effects—Morris and Freeman (9) showa modification of the basic model allows the measurement

of D even in the presence of a short T2 . how these may be included—and, in the first instance, diffu-sion, too. Let the transverse magnetization, initially zero, beThe implications of diffusion for the Burst sequence have

not previously been explored thoroughly. Two basic methods represented by the complex variable M/ Å Mx = / iMy = . Thelinear response theory described by Ernst et al. in (8) leadshave been used in the past to attempt to calculate the effect

of random molecular diffusion on the magnetization excited to a familiar relation between the transverse magnetizationcreated by a low-flip-angle pulse and the Fourier transformby a multiple pulse sequence. The first, which Kaiser and

co-workers (10) term the ‘‘partition method’’ and Hennig of the excitation pulse shape—see also Jakob et al. (17)for the particular application to the Burst sequence. If theand Hodapp (4) the ‘‘extended phase graph algorithm’’

(11) , has its origins in the work of Carr and Purcell (12) nonzero part of the pulse is confined to the interval 0tp £t £ 0, then the magnetization in the transverse plane at theand Woessner (13) . The technique has not yet been applied

in the presence of diffusion to the calculation of echo ampli- end of the pulse istudes for a Burst sequence. In the second method, Kaiser etal. start with the Bloch equations (14) , as modified by Tor- M/(x , 0) Å igM0(x) *

0

0tp

B1( t *)exp(/igGxt *)dt *. [1]rey (15) to include the effects of diffusion, and express theevolution of transverse magnetization as a Fourier integral.

By suitable definition of the function B1( t) , we shall replacethese limits by the more conventional {` in what follows.* Current address: Department of Physics, University of Surrey,

Guildford, Surrey, GU2 5XH, United Kingdom. At a time T after the end of the excitation pulse train, the

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which one would detect in a standard one-pulse spin-echosequence in the presence of a gradient and provided that theconditions outlined below are satisfied, each echo will be afaithful reproduction of M0 . (This fact is important in im-aging applications of the sequence, where each echo pro-vides the data for a separate phase-encoding step.) Notethat the signal is ‘‘played back’’ in the reverse order of theexcitation.

In introducing molecular diffusion effects into the small-angle model, we continue to make the assumption of a linearresponse of the spin system to the pulse train. This basicFIG. 1. Pulse-sequence diagram describing the basic spin-echo Bursthypothesis is equivalent to saying that the transverse magne-pulse sequence used for measuring diffusion coefficients. A read gradienttization from each RF pulse evolves under diffusion in theis switched on and the sample excited by means of a train of very low

(typically õ17) RF pulses. The 1807 pulse and the second gradient lobe same way as that created in a normal, single-pulse spin-refocus a series of echoes, one corresponding to each pulse, and the delays echo experiment; that is, the attenuation of the l th echoare arranged so that the spin and gradient echoes coincide. The delays (corresponding to the (n 0 l 0 1)th RF pulse) isshown are explained in the main text.

Al Å exp[0D(gG)2d 2l (Dl 0 dl /3)] , [5]

gradient is switched off and then a 180 7x pulse is applied at where dl and Dl play the same role as the normal Stejskal–t Å TE/2. Later, at time TE 0 T , the gradient is turned back Tanner (1) parameters d and D. They are defined in Fig. 1on at the same amplitude as before and the magnetization for the particular sequence used here.in the transverse plane may be written as As long as the following conditions are satisfied, we may

regard the l th echo as the product uniquely of the (n 0 l 0M/(x , t ú TE 0 T ) 1)th pulse, and its diffusion behavior is completely indepen-

dent from that of the other echoes. Condition 1: The elemen-Å exp[0igGx( t 0 TE)]M*/(x , 0)tary pulses must each have a low flip angle (u) and the totalflip angle on resonance nu ! p /2. This situation should beÅ 0igM0(x)exp[0igGx( t 0 TE)]compared with that where higher pulse angles are used (4)

1 *`

0`

B*1 ( t *)exp(0igGxt *)dt *, [2] and a phase-graph interpretation is used to calculate signalintensities. Later, we give a more precise interpretation ofthe nu ! p /2 condition, showing that for accurate measure-

where the complex conjugation indicates the effect of thement of diffusion coefficients, nu ( 307. Condition 2: In

1807 pulse.order for echoes not to overlap, M0(k) must be a rapidly

In a Burst experiment, B1 may be modeled by a string ofdecaying function. This will depend on the sample size,

n d functions at intervals t: B1 Å 0(u /g) ( nlÅ1 d[ t / ( l 0 the excitation/read gradient, the bandwidth and duration of

1)t] . This leads to a transverse magnetization acquisition, and the value of t. In all the experiments pre-sented here, the condition is well satisfied, but it must be

M/(x , t ú TE 0 T ) emphasized that if very high spatial frequencies are present(e.g., small capillary sample, sample with very sharp edges) ,Å iuM0(x) ∑

n

lÅ1

exp{0igGx[ t 0 TE 0 ( l 0 1)t)]}. [3]problems may occur.

Once the data have been acquired, the normalized loga-rithms of the echo amplitudes are plotted against the quan-The signal, which is proportional to the integral of the mag-tity d 2

l (Dl 0 dl /3) and a straight line is obtained, whosenetization in the transverse plane, isslope gives the diffusion coefficient. An alternative methodof analysis of the data, which allows the effects of T2 decay

S( t ú TE 0 T ) } iu ∑n

lÅ1

MH 0(kl) , [4] during the echo train to be incorporated, is to rearrange thelinear relation above into an expression of the form Al Åexp(0Pl 3 0 Ql 2 0 Rl / S) . Using the relations dl Å lt /where kl Å gGxtl , tl Å t 0 TE 0 ( l 0 1)t and M0 isd1 and Dl Å lt / d2 (see Fig. 1) , Eq. [5] yields

the Fourier transform of M0 , M0(k) Å *`

0`M0(x)

exp(0ikx)dx . P Å D(gG)2( 23t

3) ; Q Å D(gG)2(d1 / d2)t 2 ;This means that the signal is the sum of a series of echoes,

separated by intervals t. Each echo is simply the signal R Å D(gG)2(2d1d2) . [6]

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FIG. 2. Decay of signal during the echo train in a basic spin-echo Burst experiment, with four different gradient strengths, for (a) water and (b)olive oil. Each point on the y axis is the average of 10 points at the center of the profile obtained by Fourier transforming the corresponding Burst echoand is normalized to the value for the first echo. Very similar results are found by taking simply the peak (modulus) value of each echo.

If we choose to normalize the signal to that of the echo l Å 1, fects described in the theory above. In Fig. 2b, the resultsfor olive oil are invariant under different gradients. Olivethen S Å P / Q / R. A multicomponent, nonlinear, least-

squares fit can then be used to determine the value of the diffu- oil is an example of a slowly diffusing liquid; with oursystem, we were able to place an upper limit on the diffusionsion coefficient. The effect of T2 may be represented simply as

an extra decay exp(02lt 0 d2) and thus as an addition to the coefficient of 4 1 1008 cm2 s01 . In this case, signal decayis due to T2 . We believe that the small ‘‘bumps’’ in theterms R and S. (In addition, since Q Å P3(d1 / d2)/2t, we

can reduce by one the number of parameters fitted.) decay may be due to J-modulation effects.Figure 3 demonstrates the application of the theory aboveThe experiments described in this paper were performed

on a 2.35 T, horizontal, 40 cm room-temperature bore, to the measurement of diffusion coefficients. The sequenceused is the same as for the previous figure. In Fig. 3a, theBruker superconducting magnet, using an MSL console.

Gradients and probe (a birdcage coil for high RF homogene- logarithm of the normalized data is plotted against the Stej-skal–Tanner parameter d 2(D 0 d /3) for three liquids withity) were built in-house. The exact flip angle for each pulse

length was found by making a composite 1807 from a series different diffusion coefficients. In each case, the excitation/read gradient was 36 mT/m. The values of the relevantof these elementary pulses separated by 5 ms. For our ampli-

fier (Kalmus Engineering International Ltd., Model parameters in Fig. 1 were t Å 640 ms, d1 Å 0.95 ms, d2 Å2.69 ms, and the length of the elementary pulse of the Burst162LPS), with the attenuation settings used, the linear rela-

tion between flip angle and pulse length was well verified, DANTE excitation was again 1 ms (0.257) . The echo ampli-tudes were normalized to the value for the first echo and abut with a nonzero intercept: u Å mtpulse / c , with m Å

0.757 /ms and c Å 00.57. linear least-squares fit was performed on points with ln(A /A0) ú 02.5 in order to exclude echoes below the noiseFigure 2 shows results obtained using 64 pulses, each of

length 1 ms (0.257) , occurring at intervals of 640 ms. The threshold. The aim of this study was to test any deviationsfrom the theoretical predictions above, as far as possible inacquisition bandwidth was 100 kHz, so that with 64 points

per echo, the total acquisition time was 40.96 ms. Delays the absence of noise, and so 32 signal averages were per-formed. However, almost identical results for the diffusionwere arranged so that the spin echo corresponding to each

pulse coincided exactly with the gradient echo. The excita- coefficient were obtained with a single shot.The straightness of the lines demonstrates that Eq. [5] istion and read gradients were in all cases equal, so that echoes

occurred every 640 ms, and results for four different gradient very well obeyed. The measured diffusion coefficients forDMSO, water and acetonitrile are tabulated in Table 1. Forstrengths from 9 to 36 mT/m are shown. Echo amplitude is

plotted against echo number. In Fig. 2a, the results for water water, where we have access to literature data (20) at thetemperature used here (277C), there is excellent agreement(D É 2.2 1 1005 cm2 s01) show a clear decay of the echo

amplitude along the train, which varies dramatically with with our results. For DMSO and acetonitrile, the measure-ments obtained here are somewhat higher than, but consis-gradient strength. Since the T2 of water is much longer than

the acquisition period, the decay is due to the diffusion ef- tent with, previous results at 217C (19) .

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FIG. 3. (a) Straight-line fits obtained by plotting Burst diffusion data using the standard Stejskal–Tanner formula; (b) results for CuSO4(aq) modelincluding the effects of T2 , as described in the text (solid line) , and straight-line fit to d 2(D 0 d /3) , as in (a) , ignoring the effects of T2 (dotted line) .

Figure 3b shows the results of a similar experiment on a is negligible for later points. ( iii ) Gradient calibration: Insample of water, doped with CuSO4 to reduce its T2 to 44 our case, this was accurate to {3%, leading to a possiblems. The data are plotted using a nonlogarithmic y axis. Equa- systematic error in D of up to 6%, since a G 2 appears in thetion [6] was used to fit the curve, which yielded D Å 2.47 formula. ( iv) Temperature control: Not possessing a thermo-1 1005 cm2 s01 and T2 Å 43.4 ms. Shown as a dotted line statting facility for our magnetic bore, the temperature offor comparison is a curve calculated for the same diffusion the sample could not be stabilized to a suitable value forcoefficient, but with a model ignoring the effects of T2 . exact comparison with given theoretical results. The method

Four possible sources of systematic error exist in the mea- chosen for monitoring the temperature was an alcohol ther-surements. Only points ( i) and (ii) below are specific to the mometer placed next to the sample inside the probe, mea-new Burst sequence and they contribute very small system- surement by thermocouple embedded in the sample havingatic errors compared with sources (iii ) and (iv) , which are been rejected because of the possibility of introducing noisecommon to all NMR diffusion measurements. ( i ) Noninfini- into the NMR measurements.tesimal pulse lengths: The slight deviations from linearity Figure 4 demonstrates the effect on measurements of dif-in Fig. 3a are due to the effects of the still finite pulse length, fusion coefficient of increasing the flip angle. The experi-but for a 0.257 flip angle, the error in the calculated diffusion ment in Fig. 4a, on a sample of water is the same as that incoefficient is negligible. ( ii ) Finite rise time of the gradients: Fig. 3, but using a gradient of 18 mT/m. Each line corre-This means that the delays d1 and d2 , and hence the gradient sponds to a different pulse length, ranging from 0.257 todephasing, are not accurately known. While in some cases 3.37. For u Å 0.257, behavior is almost exclusively due tothis may present a problem for the early echoes, the error diffusion and the results match closely the small-angle ap-

proximation. For u Å 1.17 and 1.77, agreement with themodel is still reasonable, but we begin to see a more rapiddecay of ln A /A0 at the start, which levels off and thenTABLE 1becomes more shallow than predicted for later echoes. Incor-

Measurement Literature rect values of D will be obtained by attempting a straightD (1005 cm2 s01) temperature (7C) value line fit to these data. For u Å 2.57, the effect is greatly

accentuated, but the signal decay is still monotonic. How-DMSO 0.83 { 0.04a 27 { 1 0.68 at 217Cb

ever, by the time u reaches 3.37 (nuÉ 1807) , all resemblanceWater 2.51 { 0.10a 27 { 1 2.50c

Acetonitrile 5.43 { 0.30a 27 { 1 4.11 at 217Cb to the linear model is lost.Figure 4b analyzes the same data by calculating the diffu-

a Systematic error estimate based on a maximum 3% error in gradient sion coefficient using the simple straight line fit and ignoringcalibration.the evident deviations from linearity shown in Fig. 4a. Withb Results from Ref. (19).

c Results from Ref. (20). 64 pulses, it is clear that if one requires a systematic error

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FIG. 4. (a) Deviation of measured signal intensity from small-angle approximation for increasing values of the elementary pulse angle of the Burstsequence; (b) variation in the measured value of the diffusion coefficient of water for different values of the elementary pulse angle—see main text fora more detailed description.

of under 5%, then u must be less than about 17. However, followed (for example, rapidly progressing polymerizationreactions) . ( ii ) The ease with which it is possible to generatethere are few situations when one needs to acquire as many

as 64 data points. Other data (not shown) indicate that if large numbers of echoes, and hence many diffusion points,makes the technique ideal for studying restricted diffusion.one reduces the number of points to 16, then the error re-

mains less than 1% for u õ 27. For still smaller numbers of (iii ) The method does not use pulsed diffusion gradientslike (1) or rapidly switched gradients as in (3) . The se-points, correspondingly higher flip angles are permitted. It

is possible that a more sophisticated analysis, based on the quence also has a number of potentially attractive featuresfor diffusion imaging. ( i) The diffusion attenuation of thetransition matrix approach of (11) with incorporation of the

Stejskal–Tanner equations, rather than on linear response signal occurs in parallel with the data acquisition and notseparately as in (20) . This is important, because, for a giventheory, could extend the range of applicability of the mea-

surement to higher flip angles. This would provide measure- T2 and maximum gradient, lower diffusion coefficients canbe measured. ( ii ) The method can be used for samples withments with higher signal-to-noise, but at the cost of sacrific-

ing one of the attractive features of the experiment, namely short T2 . ( iii ) Since a large number of diffusion points canbe acquired, a wide span of values of D can be measuredits simplicity.

Although we have shown results only for the bulk diffu- in a single experiment.sion coefficient of a uniform sample, the same pulse se-quence provides spatially resolved information, since the ACKNOWLEDGMENTSdiffusion gradient is also a read gradient. Simply by treating

S.J.D. acknowledges gratefully the support by a Poste Vert of the Frencheach point on the Fourier-transformed echo separately givesgovernment medical research organization INSERM. M.D. thanks the re-1-D diffusion coefficient profiles (data not shown). Additiongion Rhone Alpes for its support.of a second gradient, orthogonal to the diffusion gradient in

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