JOURNAL OF MAGNETIC RESONANCE, Series A 118, 78–83 (1996)Article No. 0011

Rotating-Frame-Imaging Technique for Spatially Resolved Diffusionand Flow Studies in the Fringe Field of RF Probe Coils

BERND SIMON, RAINER KIMMICH, AND HERBERT KOSTLER

Universitat Ulm, Sektion Kernresonanz, 89069 Ulm, Germany

Received March 6, 1995; revised September 19, 1995

A method for rotating-frame NMR diffusometry in the fringe- in context with the steady gradients used in the superconfield B1 gradients of solenoid RF coils in standard probeheads is fringe-field (SFF) technique (3) .described. A 5 mm RF solenoid coil, for instance, produces B1 Instead of using B0 gradients for the preparation of a zgradients of up to 3.3 T/m. The gradients can be enhanced by magnetization grid, it is also feasible to employ gradients ofso-called flux concentrators so that values comparable to those the RF amplitude B1 (5, 7–10) . The techniques describedcommon in laboratory-frame PGSE experiments should be feasi-

in Refs. (7–10) are based on the formation and the recordble. The principle of the technique is to produce a z magnetizationof rotary echoes. Reference (5) suggests directly imaginggrid with the aid of a B1 gradient pulse. The wavenumber of thethe z magnetization grid by rotating-frame imaging (11)grid is varied by the length of the preparation pulse. After a certainwithout formation of distinct rotary echoes. The imagingdiffusion delay, the grid is rendered as an image using a rapidpart of this procedure can be performed employing the samerotating-frame-imaging technique. The B1 gradient need not be

constant. Diffusion coefficients are rather evaluated locally based B1 gradients as used for the preparation of the grid. Obvi-on the local B1 gradients. Single-transient diffusion experiments ously, there is a complete analogy to the real-space represen-in toroid resonators, and the employment of the B1 gradients pro- tation of laboratory-frame diffusometry experiments.duced by the skin effect of conducting materials are suggested. The appealing advantage of the magnetization grid rotat-q 1996 Academic Press, Inc. ing-frame imaging (MAGROFI) diffusometry experiments

is that no constant B1 gradient is required. As the z magneti-zation grid is rendered as an image, it is only the local B1

INTRODUCTION gradient which is relevant for the evaluation of diffusionconstants or flow velocities. In the experiment, the variation

The standard method of NMR diffusometry (1, 2) is to of the B1 gradient across the sample reveals itself by changesmeasure the stimulated-echo attenuation by strong pulsed or of the grid ‘‘wave’’ length. Thus, the coil geometry can besteady gradients of the external magnetic-flux density B0 . In optimized for strong B1 gradients regardless of any homoge-our previous work (3, 4) , we have shown that this sort of neity requirements.experiment can be favorably carried out in the fringe-field The purpose of the present paper is to elaborate the MA-gradients of superconducting magnets. We now demonstrate GROFI diffusometry principle and to present experimentalthat, in a sense, there is a rotating-frame analogue referring results obtained with an improved version of this technique.to the fringe-field gradients of the RF field of the probe The spatial selectivity inherent to MAGROFI will be demon-solenoids in standard NMR probeheads (5) . strated in particular.

According to the real-space interpretation of B0 gradientexperiments (4, 6) , the procedure begins with the production

METHODof a z magnetization grid by the first two RF pulses. In asubsequent interval, the z magnetization grid will be leveledmore or less by diffusion. A third RF pulse serves the conver- Figure 1 shows the RF pulse sequence. A B1 gradient

pulse serving the preparation of the z magnetization grid ission of the grid information into an observable signal. In thewavenumber (or k) space representation, the signal appears followed by a rotating-frame-imaging sequence applied after

a delay t. This interval is assumed to be much longer thanas the stimulated-echo attenuated more or less due to transla-tional displacement. If a reading gradient is applied during all other intervals so that it may be considered as the only

period during which diffusion effects (apart from spin–lat-the acquisition of the echo signal, the residual z magnetiza-tion grid can be directly imaged with the aid of the Fourier tice relaxation) are relevant. Therefore, it may be referred

to as ‘‘diffusion interval.’’ The transverse magnetization alsotransformation from the k space to its conjugated counter-part, the real space. This view suggests itself in particular produced by the first pulse need not be considered further

781064-1858/96 $12.00Copyright q 1996 by Academic Press, Inc.All rights of reproduction in any form reserved.

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79ROTATING-FRAME IMAGING IN RF FRINGE FIELD

where v1 Å v1(r) Å gB1(r) is the RF amplitude at theposition r in angular-frequency units, and tp Å t*1 / t1 .

At the end of the diffusion interval, the longitudinal mag-netization at position r has changed to

Mz(r , tp / t) Å M0F1 0 (1 0 »cos[v1tp / w] …)

FIG. 1. B1 gradient pulse sequence of the MAGROFI technique as 1 expH0 t

T1JG . [2]applied in the present study. The preparation pulse producing the z magneti-

zation grid is incremented in steps Dt1 starting from a width t*1 . The subse-quent diffusion interval allows for displacements by self-diffusion or flowto be detected in the experiment. A further B1 gradient pulse compensates The cosine term represents the grid profile after the delaythe increments of the preparation pulse. The z magnetization grid is then

t. The brackets indicate the (local) ensemble average overrendered as a one-dimensional image with the aid of the rapid rotating-the phase shift w Å w(r , t) arising by translational displace-frame-imaging sequence consisting of a train of read B1 gradient pulses.

The dots indicate the dwell-time intervals in which the signal data points ments along the gradient direction.are acquired ‘‘stroboscopically.’’ We distinguish phase shifts wcoh caused by coherent flow

from phase shifts winc due to incoherent displacements bydiffusion characterized by an even distribution function.Equation [2] reads then

because it either vanishes during t owing to transverse relax-ation or is spoiled by B0 inhomogeneities. (In the experi- Mz(r , tp / t) Å M0H1 0 [1 0 cos(v1tp / wcoh )ments described in the following, the B0 inhomogeneity wascharacterized by a FID decay time T*2 Å 0.02 s comparedwith a diffusion delay t § 0.1 s.)

1 »exp( iwinc ) …]expS0 t

T1DJ . [3]In a typical diffusion experiment, the signal is recorded

as a function of the grid wavelength, that is, dependenton the preparation RF pulse width or—equivalently—

Based on the B1 gradient,dependent on the pulse amplitude (5 ) . After the diffusioninterval t, the grid is to be imaged. It turned out to be

G1 Å G1(r) Å ÇB1(r) , [4]favorable not to image the actual grid as it is effectiveduring the diffusion interval, but to stretch the grid wave-

a grid wave vectorlength just for imaging purposes so that the required spa-tial resolution is feasible and independent of the actualgrid wavelength. This is achieved by applying a pulse kp Å kp(r) Å gtpG1(r) [5]compensating the variable part of the preparation pulsebefore the grid is imaged using the rapid rotating-frame- may be introduced. The wavelength of the grid thus isimaging method (12, 13 ) . d Å 2p /kp .

The diffusion coefficient is determined from the t1 depen- For the sake of simplicity, we assume that the gradientdence of the amplitude of the stretched z magnetization grid can be approximated by locally constant values in the experi-which is rendered as an image by a sine Fourier transforma- mental length scale of diffusion and flow. However, overtion of the signal S Å S( t2) , ‘‘stroboscopically’’ acquired longer distances within the sample, the magnitude as wellin the gaps of the rapid rotating-frame-imaging pulse train. as the direction of the gradient may vary. That is, each

In the following treatment, exactly resonant spins are con- particle probes a certain spatial range about its starting posi-sidered for simplicity. Later, it will be argued that off-reso- tion r within which the gradient is approached by the lowest-nance effects are minor and can normally be neglected. Dif- order term of the spatial Taylor expansion. The local phasefusion and relaxation during the pulses is neglected. As we shifts can then be expressed asare dealing with an imaging experiment, we may refer tolocal quantities of the observables and experimental parame- wcoh Å wcoh (r , t) Å t£k(r)kp(r) [6]ters. The local magnetization immediately after the prepara-

winc Å winc (r) Å kp(r)xk(r) , [7]tion interval, that is, at the initial positions r of the spin-bearing particles, is given by

where £k and xk are the velocity and diffusive-displacementcomponents along the local gradient direction, respectively.

For ordinary diffusion, we have (1, 2)Mz(r , tp) Å M0cos(v1tp) , [1]

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80 SIMON, KIMMICH, AND KOSTLER

»exp( iwinc ) … Å exp(0k 2pDt) , [8]

1 exp(0k 2pDt)expS0 t

T1D

where D Å D(r) is the local diffusion coefficient. The localz magnetization at the end of the diffusion interval thus is

/ 12

cos[k*1 x / t£kkp]Mz(r , tp / t)

1 exp(0k 2pDt)expS0 t

T1D ,Å M0H1 0 (1 0 cos[v1tp / t£kkp] [11]

where k1 Å gG1t1 , kp Å gG1tp , and k*1 Å gG1t*1 . The onlyquantities on the right-hand side of Eq. [11] which are not1 exp(0k 2

pDt))expS0 t

T1DJ ,

predetermined by the pulse program settings are the local[9]

B1 gradient G1 , the diffusion coefficient D , the local velocitywhere v1 Å v1(r) , £k Å £k(r) , kp Å kp(r) , D Å D(r) , and component £k , and the spin–lattice relaxation time T1 .T1 Å T1(r) in principle are local quantities. The modulated z magnetization is then rendered as an

The compensation pulse with the rotating-frame direction image using any suitable NMR imaging method. Particularlyof B1 assumed along the x direction of that frame converts favorable for our purpose is the standard rotating-framethe local magnetization into zeugmatography technique (11) or a multipulse variant of

it (12, 13) . All unknown parameters of Eq. [11] can in prin-Mz (r , tp / t / t1) ciple conveniently be fitted to the local image data, that is,

to data proportional to Mz(x , t*1 , t, t1) . The situation isÅ M0H1 0 (1 0 cos[v1tp / t£kkp]exp(0k 2

pDt)) particularly simple in the absence of flow (£k Å 0), forknown spin–lattice relaxation times, and for temporally andspatially constant diffusion coefficients.

1 expS0 t

T1DJcos(v1t1)

TEST EXPERIMENTS

Å M0cos(v1t1)S1 0 expS0 t

T1DD All experiments were carried out with a Bruker Biospec

BMT 47/40 tomograph operating at 200 MHz. The sampleswere placed in the fringe RF field of a three-turn solenoidprobe coil with a diameter of 5 mm. The inner diameter of/ M0

2cos[v1( t*1 / 2t1) / t£kkp]exp(0k 2

pDt)the sample tubes was 4 mm. In a first step, the rapid rotating-frame imaging pulse train is optimized by measuring the

1 expS0 t

T1D / M0

2cos[v1t*1 / t£kkp] profiles of water samples without preparation and compensa-

tion pulses. The flip angle of the read pulses is restrictedby the Nyquist sampling theorem to be smaller than 1807everywhere within the sample.1 exp(0k 2

pDt)expS0 t

T1D , [10]

The minimum read-pulse delay is given by the dead timeof the probehead and the receiver. On the one hand, the totalwhere t*1 Å tp 0 t1 .duration of the pulse train must be long enough to ensureThe B1 gradient may locally be approached by a constantsufficient spatial resolution, and on the other, it should bevalue G1 . Assume further that the gradient is aligned alongshort enough to permit transverse-relaxation effects to bethe x axis of the laboratory frame. Equation [10] may thenneglected. In our experiments, the read-pulse width was Dt2be rewritten in the formÅ 4 ms. The number of read pulses was 128.

The set of data points recorded in the gaps of the read-pulse train may be called a ‘‘pseudo-FID.’’ The maximum,Mz(x , t*1 , t, t1)

M0 i.e., a ‘‘pseudo echo,’’ is reached after n read pulses if t*1Å nDt2 . Therefore, the width of the constant part of thepreparation pulse, t*1 , was adjusted so that the maximumÅ cos(k1x)F1 0 expS0 t

T1DG

signal arises in the middle of the read-pulse train. One tran-sient was acquired for each t1 increment. The real and imagi-nary parts of the pseudo-FID were zero-filled and Fourier/ 1

2cos[(kp / k1)x / t£kkp]

transformed separately. The magnetization grid images were

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81ROTATING-FRAME IMAGING IN RF FRINGE FIELD

FIG. 2. Typical magnetization grids recorded in a water sample at 247C as a function of t1 . Fixed experimental parameters are t*1 Å 220 ms and tÅ 0.1 s. The acquisition time t2 was subdivided into 128 intervals of 4 ms each. The value of t1 was incremented in 128 steps of 20 ms. The width ofthe read pulses was Dt2 Å 4 ms. The nutation frequency n1 is nonlinearly related to a spatial coordinate of the position within the sample. The positionis represented by the indicated numbers of the half-waves of the grid. The number 1 corresponds to the remote end of the sample where B1 is lowest.The integrals over the half-waves (shaded areas) defined between neighboring nodes have been evaluated as a mean measure of the local magnetization.

phase-corrected to account for the difference between the DISCUSSIONpulse and reference phases.

The MAGROFI technique is analogous to laboratory-Figure 2 shows four typical magnetization grids recordedframe NMR diffusometry in two respects. First, it corre-with water at room temperature. The grid amplitude decayssponds to the real-space version of pulsed or steady gradient-faster with increasing t1 close to the RF coil, where the RFspin-echo diffusometry using stimulated-echo pulse se-amplitude B1 and the RF gradients are higher. As the quantityquences. The principle is to prepare a z magnetization gridto be evaluated further, the integral over the half waves ofand to render it as an NMR image after a certain diffusionthe grid were formed (shaded areas in Fig. 2) . The positionsinterval so that the leveling effect of diffusion becomes visi-of the half waves are indicated by numbers. For each posi-

tion, the dependence of the integral values on the grid wave-number kp (Eq. [5]) was evaluated by fits to Eq. [11]. Basedon the known value of the water diffusion coefficient at247C, D Å 2.15 1 1009 m2/s (14) , the local B1 gradientswere calibrated separately. The result is plotted in Fig. 3.The maximal gradient was 3.3 T/m permitting a spatial reso-lution of 66 mm (11) . For comparison, the root-mean-squaredisplacement along the B1 gradient in the diffusion intervalwas 21 mm.

The method was tested with a series of organic liquids inone- or two-compartment samples. Figure 4 shows the re-sults for a two-compartment sample. The compartments werefilled with water and hexanol. The different diffusion coeffi-cients in the two compartments are clearly resolved. A listof diffusion coefficients measured with the MAGROFI tech-nique is given in Table 1 in comparison to literature data.

The liquids investigated in this work partially haveFIG. 3. Local B1 gradients in the fringe RF field of a three-turn probemultiline NMR spectra with appreciable chemical-shift dif-

coil with a diameter of 5 mm. The data are evaluated from the z magnetiza-ferences. Therefore, off-resonance effects must be consid-tion grids of Fig. 2 using a literature value for the water diffusion coefficientered. An estimation based on a treatment of the nutation ofat 247C, D Å 2.15 1 1009 m2/s (14) . The positions are represented by the

the magnetization about off-resonance effective fields numbers of the half-waves indicated in Fig. 2. The highest numbers refer(17, 18) shows that these effects can be neglected in all to positions closest to the outermost probe coil turn. The error bars illustrate

the statistical errors evaluated from the fits to the experimental data.cases dealt with here.

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82 SIMON, KIMMICH, AND KOSTLER

component transverse to B0) planes within the sample shouldbe equigradient (magnitude of the gradient of the transverseB1 component) planes at the same time, so that the localgrid wavelengths displayed in the images are unambiguouslyrelated to distances in real space. This condition holds onlyclose to the coil axis. However, if the B1 gradients are cali-brated with the aid of the known diffusion coefficient of areference liquid, as was done in the present study, experi-mental errors on this basis are largely compensated.

The measurement of small diffusion coefficients with suf-ficient accuracy requires the design of a high-powerprobehead for strong B1 gradients. The gradients in the fringefield of solenoid coils or surface coils are larger for smallercoil diameters. The geometry of the RF coil can be effec-tively reduced with the aid of so-called flux concentrators

FIG. 4. Spatially resolved measurement of the diffusion coefficients in (19, 20) . In this way, coils with effective diameters less thana two-compartment sample filled with water and hexanol at 247C. Fixed 1 mm can be produced. The gradients, i.e., the wavenumbersexperimental parameters were t*1 Å 220 ms and t Å 0.2 s. The value of t1 of the z magnetization grids, become then comparable towas incremented in 128 steps of 20 ms each. The acquisition interval t2 was

those common in laboratory-frame techniques. Anti-Helm-subdivided into 128 intervals of 20 ms. The read pulse width was Dt2 Å 4ms. The dotted lines correspond to the conventionally measured literature holtz arrangements of flux concentrator coils may be particu-values for water at 247C, D Å 2.15 1 1009 m2/s (14) , and for hexanol at larly favorable in this context.217C, D Å 1.76 1 10010 m2/s (15) . Other probe geometries of interest are toroid cavity detec-

tors (21) or a coaxial resonators (22) . In these cases, theRF amplitude outside of the inner conductor is well definedand depends only on the distance from the axis. In a diffusionble (4, 5) . In both cases, the experimental parameter variedexperiment, a single preparation pulse thus produces a widefor evaluations of diffusion coefficients is the wavenumberrange of local wavenumbers of the z magnetization grid. Incharacterizing the grid.principle, a whole diffusion experiment can thus be per-The second analogy refers to variants of the techniquesformed in a single transient, if the grid amplitudes measuredemploying fringe-field B0 and B1 gradients of superconduct-at different positions are rescaled according to the distanceing magnets and probehead solenoid coils, respectively. Independence of the sensitivity (23) .both cases, the standard hardware of NMR spectrometers

Finally, it should be mentioned that strong RF field gradi-can be favorably used without further modification (3, 5) .ents can also be produced by the skin effect in conductingAs the techniques render one-dimensional images of the zmaterials. A corresponding rotating-frame-imaging experi-magnetization grid, there is no necessity to have constantment has been suggested in Ref. (24) . As an application,gradients.diffusion of hydrogen in palladium could be studied on thisIn practical circumstances, the gradient effective in thebasis. The room-temperature diffusion coefficient is reportedlength scale of diffusional displacements can be safely ap-to be on the order of 10011 m2/s (25) . Thus, a MAGROFIproximated by a locally constant value, which then may

vary from pixel to pixel. Note, however, that equifield (B1 experiment should be feasible.

TABLE 1Comparison of Diffusion Coefficients Measured in Various Liquids with the MAGROFI Technique (Fig. 1) with Literature Data

MAGROFI diffusometry Laboratory frame methods

Liquid u (7C) D (m2 s01) u (7C) D (m2 s01) Ref.

Acetone 24 (5.03 { 0.36) 1 1009 25 4.77 1 1009 16Ethanol 24 (1.05 { 0.06) 1 1009 25 1.01 1 1009 16Hexane 24 (3.84 { 0.22) 1 1009 25 4.21 1 1009 16Hexanol 24 (1.94 { 0.09) 1 10010 21 1.76 1 10010 15Tetradecane 24 (5.11 { 0.24) 1 10010 21 4.90 1 10010 15

Note. The experimental parameters are the same as given in the legend to Fig. 4.

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83ROTATING-FRAME IMAGING IN RF FRINGE FIELD

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