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Pulsed-Field GradientNuclear MagneticResonance as a Tool forStudying TranslationalDiffusion: Part 1.Basic TheoryWILLIAM S. PRICE

Water Research Institute, Sengen 2-1-6, Tsukuba, Ibaraki 305, Japan

ABSTRACT: Translational diffusion is the most fundamental form of transport inchemical and biochemical systems. Pulsed-field gradient nuclear magnetic resonance pro-vides a convenient and noninvasive means for measuring translational motion. In thismethod the attenuation of the echo signal from a Hahn spin-echo pulse sequence contain-ing a magnetic field gradient pulse in each t period is used to measure the displacement ofthe observed spins. In the present article, the physical basis of this method is considered in

( )detail. Starting from the Bloch equations containing diffusion terms, the analytical equa-tion linking the echo attenuation to the diffusion of the spin for the case of unrestrictedisotropic diffusion is derived. When the motion of the spin occurs within a confinedgeometry or is anisotropic, such as in in vivo systems, the echo attenuation also yieldsinformation on the surrounding structure, but as the analytical approach becomes mathe-matically intractable, approximate or numerical means must be used to extract the motionalinformation. In this work, two common approximations are considered and their limitationsare examined. Measurements in anisotropic systems are also considered in some detail.Q 1997 John Wiley & Sons, Inc. Concepts Magn Reson 9: 299 ] 336, 1997

KEY WORDS: diffraction, diffusion, molecular dynamics, pulse field gradient, restricteddiffusion

Received July 18, 1996; revised May 5, 1997;accepted May 9, 1997.

Address for correspondence: Dr. William S. Price, WaterResearch Institute, Sengen 2-1-6, Tsukuba, Ibaraki 305, Japan.

Ž . Ž .Ph: 81-298 58 6186, FAX: 81-298 58 6144. e-mail: [email protected] 1997 John Wiley & Sons, Inc. CCC 1043-7347r97r050299-37

299

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INTRODUCTION

Self-diffusion is the random translational motionŽ .of molecules or ions driven by internal kinetic

Ženergy. Translational diffusion not to be con-.fused with spin diffusion or rotational diffusion

Ž .is the most fundamental form of transport 1]3and is responsible for all chemical reactions, sincethe reacting species must collide before they can

Ž .react 4 . Diffusion is also closely related tomolecular size, as can be seen from the Stokes]Einstein equation,

kTw xD s 1

f

where k is the Boltzmann constant, T is tempera-ture, and f is the friction coefficient. For thesimple case of a spherical particle with an effec-

Ž .tive hydrodynamic radius i.e., Stokes radius r inSa solution of viscosity h the friction factor is givenby

w xf s 6phr . 2S

Generally, however, molecular shapes are morecomplicated and may include contributions fromfactors such as hydration, and the friction factor

Ž .must be modified accordingly 4]9 . As a conse-quence, the diffusion also provides informationon the interactions and shape of the diffusingmolecule.

Because of its noninvasive nature, nuclearŽ .magnetic resonance NMR spectroscopy is a

unique tool for studying molecular dynamics inŽ .chemical and biological systems 10]19 . There

are two main ways in which NMR may be used tostudy self-diffusion coefficients, which are alsoknown as tracer-diffusion or intradiffusion coef-

Ž . Ž . Ž .ficients 11, 13, 20 Fig. 1 : a analysis of relax-w Ž .x Ž .ation data e.g., Refs. 21, 22 and b pulsed-field

Ž .gradient PFG NMR. However, the two methodsreport on motions in very different time scalesand thus, even though a translational diffusioncoefficient can be derived in both cases, the twoestimates will agree only under certain circum-

Ž .stances 23 since the relaxation method is in factsensitive to rotational diffusion, whereas the PFGmethod measures translational diffusion. Gener-ally, in experiments involving the solution state,relaxation measurements are sensitive to motionsoccurring in the picosecond to nanosecond timescale}that is, motion on the time scale of thereorientational correlation of the nucleus. While

in PFG measurements, motion is measured overthe millisecond to second time scale.

In the first method, relaxation data are ana-lyzed to determine the rotational correlation

Ž . Ž . Ž .time s t of a probe species 24 . t can thenc cbe related to the solution viscosity, and ulti-mately, to the translational diffusion coefficientŽ . Ž .Fig. 1 by using the Debye equation 25]27 ,

3 Ž . w xt s 4phr r 3kT 3c S

Ž w x.and the Stokes]Einstein equation i.e., Eq. 1 .However, a number of assumptions which, de-pending upon the system being studied, may ormay not be justified need to be made in perform-ing this analysis. First, the relaxation mechanismof the probe species needs to be known, and it isrequired that the intermolecular contributions tothe relaxation can be separated from the in-

Ž .tramolecular contributions 28 . Second, only ifthe molecule is spherical can its rotational dy-namics be properly characterized by a single cor-relation time. Third, depending on the size of theprobe molecules compared to the molecules ofthe bulk solution, they may not see the solutionas being continuous; as a consequence, one of thebasic requirements for the validity of the Debye

Ž .equation is violated 8, 9 . Thus, serious assump-tions are involved in applying this method tostudying biological systems when a small probespecies is used since the solution normally has a

Žlarge macromolecular component e.g., a largepart of the cytoplasm of red blood cells is com-

.posed of hemoglobin . The final problem with thismethod is that the Stokes radius of the probemolecule needs to be known and the determina-tion of this is not straightforward.

In the PFG method, the attenuation of a spin-echo signal resulting from the dephasing of thenuclear spins due to the combination of the trans-lational motion of the spins and the imposition ofspatially well-defined gradient pulses is used tomeasure motion. In contradistinction to the relax-ation method, no assumptions need to be made

Ž .regarding the relaxation mechanism s or in relat-ing t to the translational motion of the probecmolecule. However, to determine the ‘‘true’’ dif-fusion coefficient, D, as against an ‘‘apparent’’diffusion coefficient D the effects of structurala p pboundaries that affect the natural diffusion of theprobe species need to be considered. The mathe-matics required to model anything except for freediffusion or diffusion within simple geometriesbecomes rather complicated, and as a result, ana-

PULSED-FIELD GRADIENT NMR 301

Figure 1 Schematic representation of the relaxation and pulsed-field gradient methods fordetermining molecular dynamics. In our representation of the relaxation method, we haveassumed that the probe molecule is a sphere with an effective hydrodynamic radius of r .S

lytical solutions are generally not possible andnumerical solutions must be sought.

Ž .In practice, both B i.e., magnetic and B0 1w Ž .x Ž .i.e., radiofrequency rf gradients 15, 29, 30 canbe used, but in the present article we will concen-trate on B gradients, although it should be noted0

that the theoretical aspects are generally analo-gous. The application of B gradients to high-0

resolution NMR is now commonplace and pro-vides superior methods of water suppression,coherence selection, and quadrature detection,and methods for controlling the effects of radia-

Ž .tion damping 15, 19, 31]40 . Gradients also pro-vide the basis of spatial resolution in NMR mi-

wcroscopy and imaging i.e., magnetic resonanceŽ .x Ž .imaging MRI 41]44 , but the application of

B gradients to the study of molecular dynamics is0

less widespread. Gradients afford a powerful toolŽnot only for studying molecular diffusion under

y17 2 y1.favorable circumstances down to - 10 m s ,but also for providing structural information inthe range of 0.1]100 mm when the diffusion is

Ž .restricted e.g., diffusion in a cell on the NMRtime scale. The use of magnetic field gradients

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allows diffusion to be added to the standard NMRobservables of chemical shifts and relaxation timesŽi.e., longitudinal or T ; transverse or T ; and in1 2

.the rotating frame or T . Gradient-based diffu-1r

sion measurements have been found to have clini-Ž .cal utility in NMR imaging studies 45]49 espe-

cially in regard to study of cerebral ischaemiaw Ž . xe.g., Ref. 50 and references therein .

The aim of this article is to present an intro-duction to the PFG experiment and the theoreti-cal basis used for interpreting the data, to deter-mine the diffusion coefficient of a probe speciesand perhaps information on the geometry in whichit is diffusing. As it is not possible to provide acomprehensive review of the literature in a singlepaper, a number of pertinent references havebeen mentioned in the text that may be consultedfor more in-depth coverage of some aspects. Theanalysis of PFG NMR experiments is inherentlymathematical, and general books on mathemati-

w Ž .xcal methods e.g., Ref. 51 , mathematical func-w Ž .x wtions e.g., Ref. 52 , and integrals e.g., Ref.

Ž .x53 are useful references. Particular emphasis isplaced on developing a physical feeling for thePFG method. It should be noted that the theorypresented is quite general and applies equally toboth in ¨ i o and in ïtro samples. In the nextsection, the effects of a magnetic gradient onnuclear spins is discussed, followed by an intuitiveexplanation of how diffusion can be related to theattenuation of the echo signal in the PFG NMRexperiment. Finally, the concept of restricted dif-fusion is introduced. In the third section, themathematical background relating diffusion to theecho attenuation and the experimental parame-ters is considered in detail. First, an analyticalmacroscopic approach starting from the Blochequation is derived. The effects of flow superim-posed upon diffusion are also considered. Next,two common approximate methods, the Gaussian

Ž .phase distribution GPD approximation and theŽ .short gradient pulse SGP approximation, are

presented. To illustrate these approaches, equa-tions relating echo attenuation to the experimen-tal variables and the diffusion coefficient are de-rived for the case of free diffusion. The analogybetween PFG measurements and scattering is ex-plained. Finally, the concepts of ‘‘diffusivediffraction’’ and of imaging molecular motion areillustrated using diffusion within a rectangularbarrier pore as an example. In the final section,we consider the general relationships between theexperimental variables and echo attenuation inrestricted geometries and the validity of the GPD

and SGP approximations. The differences andsimilarities between the two approaches are eluci-dated pictorially using the example of diffusion ina sphere. PFG diffusion measurements in aniso-tropic systems, which commonly occur in liquidcrystal and in ¨ i o studies, are examined in thelast subsection of the article.

NUCLEAR SPINS, GRADIENTS, ANDDIFFUSION

Magnetic Gradients as Spatial Labels

All of the NMR theory needed for understandingthe effects of B gradients on nuclear spins has0the Larmor equation as the origin:

w xv s gB 40 0

Ž y1 .where v is the Larmor frequency radians s ,0Ž y1 y1. Ž .g is the gyromagnetic ratio rad T s , B T is0

the strength of the static magnetic field, and wehave neglected the small effect of the shieldingconstant. We consider B to be oriented in the0z-direction. Since B is spatially homogeneous, v0

w xis the same throughout the sample. Equation 4Ž .holds for a single quantum coherence i.e., n s 1 .

However, if in addition to B there is a spatially0Ž y1 .dependent magnetic field gradient g T m ,

and accounting for the possibility of more thansingle quantum coherence, v becomes spatiallydependent,

Ž . Ž Ž .. w xv n , r s n v q g g ? r 5e f f 0

where we define g by the grad of the gradientfield component parallel to B , i.e.,0

B B Bz z z w xg s =B s i q j q k 60 x y z

where i, j, and k are unit vectors of the laboratoryframe of reference. The important point is that ifa homogeneous gradient of known magnitude isimposed throughout the sample, the Larmor fre-quency becomes a spatial label with respect to thedirection of the gradient. In imaging systems,which typically can produce equally strong mag-netic field gradients in each of the x, y, and zdirections, it is possible to measure diffusion along

Žany of the x, y, or z-directions or combinations.thereof ; however, in normal NMR spectrome-

ters, it is more common to measure diffusion with


Žthe gradient oriented along the z-axis i.e., paral-.lel to B . For simplicity, in most of the present0

article we are concerned only with the case wherethe gradient is oriented along z, although someattention is paid to the use of gradients alongmore than one axis when we consider anisotropicdiffusion in the final subsection. In the case of asingle gradient oriented along z, the magnitudeof g is only a function of the position on thez-axis, g s g ? k, which we will henceforth referz

w xto simply as g. It can be seen from Eq. 5 thatŽ .successively higher homonuclear quantum tran-

sitions are more sensitive to the effects of thegradient, whereas zero quantum transitions areunaffected by the presence of the gradient. Forheteronuclear multiple quantum transitions, Eq.w x5 must be modified to account for the coherentspins.

In the case of a single quantum coherence, wew xcan see from Eq. 5 that for a single spin the

cumulative phase shift is given by

tŽ . Ž . Ž . w xf t s gB t q g g t9 z t9 dt9 7H0^ _ 0^ ` _static field

applied gradient

where the first term on the right-hand side corre-sponds to the phase shift due to the static field,and the second term represents the phase shiftdue to the effects of the gradient. Thus, from the

w xsecond term of Eq. 7 we can see that the degreeof dephasing due to the gradient pulse is propor-

Ž .tional to the type of nucleus i.e., g , the strengthŽ .of the gradient i.e., g , the duration of the gradi-

Ž .ent i.e., t , and the displacement of the spinalong the direction of the gradient. Although thegradient is normally applied in a pulse of constant

Ž . w xamplitude, we have written g as g t in Eq. 7 toemphasize that the gradient may itself be a func-

Ž .tion of time i.e., not merely a rectangular pulse .However, for simplicity, in the present article wewill concern ourselves only with constant ampli-tude pulses. In this case we can think of the‘‘area’’ or ‘‘dephasing strength’’ of the gradientpulse as equaling ggt.

Measuring Diffusion with Magnetic FieldGradients

w x Ž .From Eq. 5 it is apparent that a well-definedmagnetic field gradient can be used to label theposition of a spin, albeit indirectly, through theLarmor frequency. This provides the basis for

measuring diffusion. The most common approachŽ .is to use a simple modification 54]56 of the

Ž .Hahn spin-echo pulse sequence 57]59 , in whichequal rectangular gradient pulses of duration d

Žare inserted into each t period the ‘‘Stejskal and. Ž .Tanner sequence’’ or ‘‘PFG sequence’’ Fig. 2 .

Applying the magnetic field gradient in pulseswinstead of continuously i.e., steady gradient ex-

Ž .xperiment 57, 58 circumvents a number of ex-Ž . Ž .perimental limitations 54 : a Since the gradient

is off during acquisition, the line width is notbroadened by the gradient, and thus the methodis suitable for measuring the diffusion coefficient

Ž .of more than one species simultaneously. b Therf power does not have to be increased to cope

Ž .with a gradient-broadened spectrum. c Smallerdiffusion coefficients can be measured since it is

Ž .possible to use larger gradients. d The time overwhich diffusion is measured is well defined be-cause the gradient is applied in pulses; this is ofparticular importance to studies of restricted dif-

Ž . Ž .fusion see the next section . e As the gradientŽ .is applied in pulses it is normally possible to

separate the effects of diffusion from spin]spinrelaxation; this will be explained below. Gener-ally, the applied gradient pulses are much strongerthan any background gradients that may be pre-

Žsent; as a result, the background gradients e.g.,due to differences in susceptibility in the sample,

.inhomogeneities in the main magnetic field, etc.will be neglected in the analysis given below.

We will now qualitatively explain how thismethod works. The mechanism is shown schemat-ically in Fig. 2. Imagine that we have an ensemble

Žof diffusing spins at thermal equilibrium i.e., the.net magnetization is oriented along the z-axis . A

pr2 rf pulse is applied which rotates the macro-scopic magnetization from the z-axis into the x]y

Ž .plane i.e., perpendicular to the static field . Dur-ing the first t period at time t , a gradient pulse1of duration d and magnitude g is applied so thatat the end of the first t period, spin i experiencesa phase shift,

t qd1Ž . Ž . w xf t s gB t q gg z t dt 8Hi 0 i^ _ t1^ ` _static fieldapplied gradient

where the first term is the phase shift due to themain field, and the second one due to the gradi-

w xent. Different from Eq. 7 we have taken g outw xof the integral in Eq. 8 since we are considering

a constant amplitude gradient.

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At the end of the first t period, a p rf pulse isapplied that has the effect of reversing the sign of

Ž .the precession i.e., the sign of the phase angleor, equivalently, the sign of the applied gradientsand static field. At time t q D, a second gradient1pulse of equal magnitude and duration is applied

Žn.b., the p pulse has the effect of changing thesign of the first gradient pulse; this leads to theidea of an ‘‘effective’’ field gradient; see The

.Macroscopic Approach . If the spins have notundergone any translational motion with respectto the z-axis, the effects of the two applied gradi-


ent pulses cancel and all spins refocus. However,if the spins have moved, the degree of dephasingdue to the applied gradient is proportional to the

Ždisplacement in the direction of the gradient i.e.,. Žthe z-direction in the period D i.e., the duration

.between the leading edges of the gradient pulses .Thus, at the end of the echo sequence, the totalphase shift of spin i relative to being located atz s 0 is given by

t qd1Ž . Ž .f 2t s gB t q gg z t dtHi 0 i½ 5t1^ ` _

first t periodt qDqd1 Ž .y gB t q gg z t9 dt9H0 i½ 5

t qD1^ ` _second t period

t qd t qDqd1 1Ž . Ž .s gg z t dt y z t9 dt9H Hi i½ 5t t qD1 1

w x9

We should recall that in NMR we are concernedŽwith an ensemble of nuclei with different spatial

.starting and finishing positions , and thus, theŽ .normalized intensity i.e., an attenuation of the

w Ž .xecho signal at t s 2t i.e., S 2t is given byŽ .58, 60 ,

ìfŽ . Ž . Ž . w xS 2t s S 2t P f , 2t e df. 10Hgs0

y`

Ž . Žwhere S 2t is the signal i.e., resultant mag-gs0.netic moment in the absence of a field gradient.

Ž .If we consider only the real component of S 2t ,and recalling De Moivre’s theorem,

if w xe s cos f q i sin f 11

we have

`Ž . Ž . Ž . w xS 2t s S 2t P f , 2t cos f df 12Hgs0

y`

Ž . Ž .where P f, 2t is the relative phase-distributionfunction. For the PFG sequence, some authors

Ž .write P f, D , since it is the gradient pulses andthe separation between them that constitute the

Ž .Figure 2 A schematic representation of how the Stejskal and Tanner or PFG pulsesequence measures diffusion and flow. This is a Hahn spin-echo pulse sequence with arectangular gradient pulse of duration d and magnitude g inserted into each t delay. Theseparation between the leading edges of the gradient pulses is denoted by D. The applied

Ž .gradient is generally along the z-axis the direction of the static field . The second half of theŽ . Ž .echo is digitized denoted by dots and used as the free induction decay FID . In this

schematic description, we assume that we start the pulse sequence with a sample consistingŽ .of four in-phase spins really an ensemble! and we consider only the precession due to the

Ž .gradient i.e., we use a rotating reference frame rotating at v . We assume that the center0Ž .of the gradient coincides with the center of the sample i.e., z s 0 . Accordingly, the spins

above and below this point acquire phase shifts owing to the gradient pulses, but in oppositesenses. In the absence of diffusion, the effect of the first gradient pulse, denoted by the

Žcurved arrows in the first phase diagram, is to create a magnetization helix i.e., the solid. Ž .ellipses in the center phase diagram with a pitch of 2 pr gd g . Although we have

represented the gradient pulses as having a finite width it is easier to consider them in theŽ .limit of d ª 0 i.e., the short gradient pulse limit . The p pulse reverses the sign of the

Ž .phase angle i.e., the dotted ellipses in the center phase diagram , and thus, after the secondgradient pulse, the helix is unwound and all spins are in phase, which gives a maximum echosignal. In the presence of diffusion, the winding and unwinding of the helix are scrambled bythe diffusion process, resulting in a distribution of phases, although it is not easily seen sinceour sample consists of only four spins. Larger diffusion would be reflected by poorerrefocusing of the spins, and consequently by a smaller echo signal. The effects of restrictionupon the diffusion process will also contribute to this loss of phase coherence. In the

Ž .absence of any background gradients, diffusion in the periods before e.g., 0 ª t and after1Ž .i.e., t q D q d the gradient pulses does not affect the signal attenuation. In the presence1

Ž .of flow imagine that the outflowing spins are replaced by inflowing spins along theŽ .direction of the gradient in the yz direction with velocity n in the present example and

neglecting diffusion, all the spins receive the same change in phase. The greater the flow is,the larger is the net phase change. If both diffusion and flow processes are present, then thewhole diffusion-induced phase distribution receives a net phase shift.

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‘‘active’’ part of the sequence. By definition,Ž .P f, 2t must be a normalized function, and so

`Ž . w xP f , 2t df s 1. 13H

y`

Below, we will consider the further derivationw xof Eq. 12 in the context of the GPD approxima-

Ž .tion see The GPD Approximation . However, forw x w xthe present, Eqs. 9 and 12 provide a very clear

conceptual idea as to how the PFG method works.w xFrom Eq. 9 , it can be seen that the phase shift

due to the static field cancels. In the absence ofdiffusion, the phase shifts due to the two gradient

Žpulses or, conversely, in the presence of diffusion.but with g s 0 will also cancel; thus, f s 0 fori

w xall i, and as cos f s 1 in Eq. 12 , a maximumŽsignal will be recorded see the first series of.phase diagrams in Fig. 2 . However, if we have

Ž .diffusion, then the displacement function z t isitime dependent and the phase shifts accumulatedby an individual nucleus due to the action of thegradient pulses in the first and second t periodsŽduring the gradient pulses to be precise see Eq.w x9 ; n.b., we neglect the effects of background

.gradients do not cancel. The degree of miscancel-Ž .lation i.e., larger phase shift increases with

Žincreasing displacement due to diffusion i.e., ran-.dom motion along the gradient axis. These ran-

dom phase shifts resulting from the diffusion areaveraged over the whole ensemble of nuclei that

contribute to the NMR signal. Hence, the ob-served NMR signal is not phase shifted but atten-uated, and the greater the diffusion is, the larger

Žis the attenuation of the echo signal see the.second series of phase diagrams in Fig. 2 . Simi-

larly, as the gradient strength is increased in thepresence of diffusion the echo signal attenuates.In Fig. 3 some experimental 13C-NMR PFG spec-tra of 13CCl are presented to illustrate the loss4of echo signal intensity due to diffusion. Net flow,on the other hand, causes a net phase shift of the

Žecho signal see the third series of phase dia-.grams in Fig. 2 and the end of this subsection

instead of the diffusion-induced ‘‘blurring’’ of thephases which results in a diminution of the echosignal.

It is important to understand the differencebetween gradient echoes and spin echoes. Inmeasuring diffusion, we generally choose to use

Žthe PFG pulse sequence i.e., a spin-echo se-.quence instead of a gradient-echo pulse se-Žquence i.e., the PFG pulse sequence without the

p pulse and with the second gradient pulse hav-.ing an opposite polarity to the first pulse . The

reason is that as well as refocusing the sign of thephase angle accumulated during the first t pe-riod, the p pulse has the effect of refocusingchemical shifts and the frequency dispersion dueto the residual B inhomogeneity and susceptibil-0ity effects in heterogeneous samples, etc. A gradi-

Figure 3 13C-PFG NMR spectra of a sample of 13CCl . The spectra were acquired at 303 K4with D s 100 ms, d s 4 ms, and g ranging from 0 to 0.45 T my1 in 0.05-T my1 increments.The spectra are presented in phase-sensitive mode with a line broadening of 5 Hz. As theintensity of the gradient increases, the echo intensity decreases due to the effects ofdiffusion.


ent echo, on the other hand, refocuses only thephase dispersion resulting from the gradientpulses. It is because of the additional propertiesof spin echoes that diffusion measurements arealmost invariably performed using spin-echo]based sequences.

In our discussion above, we did not considerthe relaxation process that occurs during the echosequence. Thus, in the absence of diffusionandror the absence of gradients, we would havethe signal at t s 2t equal to

2tŽ . Ž . w xS 2t s S 0 exp y 14gs0 ž /T2

Ž .where S 0 is the signal without attenuation dueto relaxation]that is, the signal that would beobserved immediately after the pr2 pulse. Weassume here that the observed signal originates

Žfrom a single species i.e., the observed signalresults from one population with a single relax-

.ation time . In the presence of diffusion andgradient pulses, the attenuation due to relaxationand the attenuation due to diffusion and theapplied gradient pulses are independent, and sowe can write,

2tŽ . Ž . Ž .S 2t s S 0 exp y f d , g , D , Dž / ^ ` _T2

attenuation due to^ ` _diffusionattenuation due to

relaxation

w x15

Ž .where f d, g, D, D is a function that representsŽthe attenuation due to diffusion e.g., compare

w x w x.Eq. 15 with Eq. 10 . Thus, if the PFG measure-ment is performed whilst keeping t constant, it ispossible to separate the contributions. Hence, by

w x w xdividing Eq. 15 by Eq. 14 we normalize out theattenuation due to relaxation, leaving only theattenuation due to diffusion,

Ž .S 2tŽ . w xE s s f d , g , D , D . 16Ž .S 2t gs0

ŽIn the steady-gradient experiment i.e, D s d s.t , however, since the gradients are on for all of

the sequence, only g can be altered indepen-dently of t. Recall the well-known diffusion termin the expression for the intensity of the Hahn

Ž .spin-echo sequence 57]59 ,

Ž . Ž . Ž . Ž 2 2 3 .S 2t sS 0 exp y2trT exp y2g D g t r3 ,2 ^ ` _^ ` _attenuation dueattenuation due

to diffusionto relaxation

w x17

whereas in the PFG experiment, we can alter d,D, or g independently of t and still perform thisnormalization. This is a very important distinctionbetween the steady-gradient diffusion experimentand the PFG experiment.

Although the effects of relaxation are normal-ized out, since we use E as our experimental

Ž .measure, the time scale of the experiment i.e, Dis limited by the relaxation time of the probespecies. As D increases, so must t, and eventually

Žthe signal will become too small to measure seew x.Eq. 14 . The smallest value of D will be limited

by the performance of the gradient system. Inpractice, D is normally between 1 ms and 1 s. dmust be smaller than D and is typically in therange of 0]10 ms. The magnitude of g is machinedependent, and currently the largest gradientpulses on commercially available equipment areof the order of 20 T my1.

Ž .We now need to equate the attenuation E ofthe echo signal to the experimental variables; that

Ž .is, we need to derive f d, g, D, D . The methodsfor doing this will be presented in the next sec-tion. However, we first need to digress a little andconsider diffusion itself.

Free and Restricted Diffusion

In the PFG experiment, we probe the particle’smotion by taking a measurement at time t s t1and a second measurement at time t s t q D.1The key point is that in the PFG experiment theecho attenuation gives information on the dis-

Žplacement along the gradient axis the z-axis in.the present case that has occurred during the

period D, which can then be related to the diffu-sion coefficient but it does not, at least directly,give us information on how the particle movedbetween the initial and the final positions. Specif-ically, it gives information on the self-correlation

Ž . Ž .function 61 , P r , r , t }that is, the conditional0 1probability of finding a particle initially at a posi-

Ž .tion r , at a position r after a time t. P r , r , t0 1 0 1is given by the solution of the diffusion equation.Hence we need to examine the diffusion equation

Ž .and how we can obtain P r , r , t from it.0 1

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In terms of the concentration in number ofŽ .particles per unit volume, c r, t , the flux of a

particle is given by Fick’s first law of diffusion tow Ž .xbe for example, see Ref. 2, 62 ,

Ž . Ž . w xJ r, t s yD=c r, t . 18

Ž .The minus sign indicates that in isotropic mediathe direction of flow is from larger to smallerconcentration. Because of the conservation ofmass, the continuity theorem applies, and thus,

Ž .c r, tŽ . w xs y= ? J r, t . 19

t

w x Ž .In other words, Eq. 19 states that c r, t rt isthe difference between the influx and efflux from

w xthe point located at r. Combining Eqs. 18 andw x19 we arrive at Fick’s second law of diffusionw Ž .xe.g., Refs. 4, 51, 62 ,

Ž .c r, t2 Ž . w xs D= c r, t 20

t

So far in our mathematical descriptions ofdiffusion, we have, perhaps simplistically, as-sumed that the diffusion process is isotropic andcan therefore be described by the isotropic diffu-

Ž .sion coefficient D i.e., a scalar . More generallythe diffusion process is represented by a carte-

Ž .sian, or rank two, tensor i.e., a 3 = 3 matrixŽ . Ž51 , D D where a and b take each of theab

.Cartesian directions ; thus, written more gener-w xally, Eq. 18 can be written as

Ž . Ž . w xJ r, t s yD=c r, t , 21

which is shorthand for

Ž .c x , t

xD D Dx x x y x zŽ .J x , t Ž .c y , tD D DŽ .J y , t s y .y x y y y z

yŽ . D D DJ z , t z x z y z z Ž .c z , t

z

w x22

ŽWe note that the diagonal elements of D, e.g.,.a s b scale concentration gradients and fluxes

in the same direction, the off-diagonal elementsŽ .e.g., a / b couple fluxes and concentration gra-

dients in orthogonal directions, and similarly, Eq.w x20 becomes

Ž .c r, tŽ . w xs = ? D=c r, t . 23

t

For simplicity in most of what follows, we areconcerned only with isotropic diffusion. However,in the section on anisotropic diffusion, we willconsider in detail the significance of anisotropicdiffusion in PFG diffusion measurements.

In the case of self-diffusion, there is no netconcentration gradient, and instead we are con-

Ž .cerned with the total probability, P r , t of find-1ing a particle at position r at time t. This is given1by

Ž . Ž . Ž . w xP r , t s r r P r , r , t dr 24H1 0 0 1 0

Ž . Žwhere r r is the particle density the formal0definition of the particle density is considered in

. Ž . Ž .detail below , and thus, r r P r , r , t is the0 0 1probability of starting from r and moving to r in0 1time t. The integration over r accounts for all0possible starting positions. Similar to concentra-

Ž .tion, P r , t describes the probability of finding1a particle in a certain place at a certain time.Ž .P r , t is a sort of ensemble-averaged probability1

concentration for a single particle, and it is thusreasonable to assume that it obeys the diffusion

Ž .equation 41 . Because the spatial derivatives inFick’s laws refer to r , we can rewrite Fick’s laws1

Ž .in terms of P r , r , t with the initial condition,0 1

Ž . Ž . w xP r , r , 0 s d r y r 250 1 1 0

Ž w xn.b., d in Eq. 25 is the Dirac delta function, not.the length of the gradient pulse . Thus, if in Eq.

w x Ž . Ž .18 P r , r , t is substituted for c r, t , J be-0 1comes the conditional probability flux. Similarly,

Ž . w xin terms of P r , r , t Eq. 20 becomes0 1

Ž .P r , r , t0 1 2 Ž . w xs D= P r , r , t . 260 1t

w xIn the case of anisotropic diffusion, Eq. 26 canw x Ž .be changed similarly to Eq. 23 . P r , r , t is0 1

commonly termed the Green’s function or diffu-Ž .sion propagator 55, 63 .


Ž .For the case of three-dimensional diffusionŽin an isotropic and homogeneous medium i.e.,

. Ž .boundary condition P ª 0 as r ª ` , P r , r , t1 0 1w xcan determined from Eq. 26 using Fourier trans-

Ž . Ž .forms 64 and is given by 62

2Ž .r y r1 0y3r2Ž . Ž .P r , r , t s 4pDt exp y .0 1 ž /4Dt

w x27

w xEquation 27 states that the radial distributionfunction of the spins in an infinitely large systemwith regard to an arbitrary reference time is

w x Ž .Gaussian. We note from Eq. 27 that P r , r , t0 1does not depend on the initial position, r , but0depends only on the net displacement, r y r1 0Žthe vector r y r moved during time t is often1 0

.referred to as the dynamic displacement R . ThisŽ .reflects the Markovian nature 10, 65 of Brown-

w xian motion. The solution of Eq. 26 becomesmuch more complicated when the displacement

Žof the particle is affected by its boundaries e.g.,. Ž .diffusion in a sphere and P r , r , t is no longer0 1

w xGaussian. The solutions of Eq. 26 for manycases of interest can be found in the literatureŽ .62, 66 . It should be noted that the mathematicsof heat conduction, after making the appropriatechanges of notation, is identical to that for de-

Ž .scribing diffusion 62 .Ž .It is an appropriate stage to consider the r r ,0

the probability that a spin starts at r , in some0Ž .detail. Formally, r r is given by0

Ž . Ž . w xr r s lim P r , r , t dr 28H0 0 1 1tª`

and thus is independent of r , because after infi-0nite time the finishing position of a particle in thesystem will be independent of the starting posi-

Ž . w x Žtion. P r , r , t as given by Eq. 27 i.e., free0 1.diffusion approaches 0 as t ª `, but the ‘‘effec-

Ž .tive volume’’ i.e., the integral over r becomes1Ž .proportionally larger; consequently, r r stays0

Ž .constant 1 is a convenient choice . In the case ofŽ .an enclosed geometry, r r is given by the in-0

verse of the volume. Also, by definition we mustw Ž .xhave e.g., ref. 65

Ž . w xr r dr s 1. 29H 0 0

The mean-squared displacement is given byw Ž .xe.g., Ref. 10

2Ž .r y r² :1 0

` 2Ž . Ž . Ž .s r y r r r P r , r , t dr dr .H 1 0 0 0 1 0 1y`

w x30

Ž . w xUsing P r , r , t as given by Eq. 27 , we can0 1calculate the mean-squared displacement of free

w xdiffusion. To do this we rewrite Eq. 27 in Carte-Ž .sian form i.e., r s x i q y j q zk

2Ž .x y x1 0y3r2Ž . Ž .P r , r , t s 4pDt exp y0 1 ž /4Dt

2Ž .y y y1 0= exp yž /4Dt

2Ž .z y z1 0 w x= exp y 31ž /4Dt

w x Ž .and using Eq. 30 and noting that r r s 1 for0the case of free diffusion. We can evaluate Eq.w x w30 using the standard integral e.g., Eq. 3.462 8.

Ž .xin Ref. 53 ,

` 22 ym x q2n xx e dxHy`

21 p n 2n rms 1 q 2 e( ž /2m m m

w < < x w xarg n - p , Re m ) 0 32

where in our case x s x y x , y y y , z y z ,1 0 1 0 1 0Ž .y1 Žm s 4Dt and n s 0, and thus we obtain n.b.

.for free diffusion

2Ž . w xr y r s nDt 33² :1 0

where n s 2, 4, or 6 for one, two, or three dimen-w xsions, respectively. Equation 30 presents a rela-

tionship between the molecular displacement dueto diffusion and the diffusion equation. Specifi-cally for free diffusion, it states that the mean-squared displacement changes linearly with time.

When we use the PFG method to measurediffusion in free solution and in the absence of

Ž .exchange, the length of time we choose i.e., D isŽirrelevant and we get the same result i.e., from

w xEq. 33 the mean-squared displacement scales.linearly with time . This is, of course, assuming

Ž .that the relaxation time s of the species in ques-

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tion is sufficiently long so that we still get ameasurable signal and that the measurement isunaffected by eddy currents or other experimen-

w Ž .xtal complications see, for example, ref. 19 .However, in the case of a species diffusing withina confined space we must be careful to properlyaccount for the effects of the restricting geometryon the motion of the species. If a particle is

Ždiffusing within a restricted geometry sometimes.referred to as a ‘‘pore’’ , the displacement along

the z-axis will be a function of D, the diffusioncoefficient, and the size and shape of the restrict-ing geometry. Consequently, if the boundary ef-fects are not properly accounted for and we ana-lyze the data using the model for free diffusionŽ .see the next section , we will measure an appar-

Ž .ent diffusion coefficient D and not the truea p pdiffusion coefficient. We illustrate this effect laterin this section.

Before further considering the problem of re-stricted diffusion, it is appropriate to briefly con-sider what constitutes the true diffusion coeffi-

Ž .cient. In a pure liquid e.g., water the true diffu-sion coefficient corresponds to the bulk diffusioncoefficient. However, the situation is rather more

Žcomplex in a macromolecular solution e.g., cellcytoplasm, polymer solutions, protein solutions,

. Ž .etc. where the probe molecule e.g., water has toskirt around the larger ‘‘obstructing’’ moleculesŽ .e.g., proteins, organelles as well as perhaps in-

wteracting with protein hydration shells e.g., Ref.Ž .x67 . These effects operate on a time scale muchsmaller than the smallest experimentally availableD and consequently are well averaged on the timescale of D. For example, if we consider a reason-ably small value of D of 5 ms and that at 298 Kwater has a diffusion coefficient of about 2.3 =

y9 2 y1 Ž . w x10 m s 68, 69 , then from Eq. 30 the meandisplacement of a water molecule during D isabout 5 mm. The true diffusion coefficient will bean average bulk diffusion coefficient consisting ofall of the interactions that affect the probemolecule diffusion. The situation can be furthercomplicated by the effects of exchange through

Ž .cell membranes 70 . The different time scales ofthe averaging processes is one of the major rea-sons that diffusion probed by using relaxationstudies and diffusion measured using PFG NMRare essentially different things with the relaxationbased measurements probing motion on the timescale of the correlation time of the probe molecule

Ž .and not on the much longer time scale of DŽ .21, 71 . We mention in passing that in a polymersolution if the diffusion of the polymer itself is

studied there can be additional complications ow-ing to the entanglement of the polymer moleculesw Ž . xe.g., Ref 19 and references therein .

We now explain the concept of restricted dif-fusion and how it relates to PFG NMR diffusionmeasurements. Consider two cases where we havea particle with the same diffusion coefficient; in

Žone case the particle is freely diffusing i.e., an.isotropic homogeneous system , while in the other

case it is confined to a reflecting sphere of radiusŽ .R Fig. 4 . By ‘‘reflecting’’ we mean that the spin

is neither transported through the boundary norrelaxed by the contact with the boundary. From

w xEq. 30 , we can define the dimensionless variableŽ .i.e., n s 1, t s D ,

2 w xj s DDrR , 34

which is useful in characterizing restricted diffu-sion as will be seen below. In the case of freelydiffusing particles, the diffusion coefficient deter-mined will be independent of D and the displace-ment measured in the z-direction will reflect thetrue diffusion coefficient, since the mean-squared

Ždisplacement scales linearly with time see Eq.w x.33 . However, for the particle confined to thesphere, the situation is entirely different. Forshort values of D such that the diffusing particlehas not diffused far enough to feel the effect of

Ž .the boundary i.e., j < 1 , the measured diffu-sion coefficient will be the same as that observedfor the freely diffusing species. As D becomes

Ž .finite i.e., j f 1 , a certain fraction of the parti-Žcles i.e., in a real NMR experiment there is an

.ensemble of diffusing species will feel the effectsof the boundary and the mean squared displace-ment along the z-axis will not scale linearly with

ŽD; thus, the measured diffusion coefficient i.e.,. Ž .D will appear to be observation time depen-a p p

dent. At very long D, the maximum distance thatthe confined particle can travel is limited by theboundaries, and thus the measured mean-squareddisplacement and diffusion coefficient becomesindependent of D. Thus, for short values of D themeasured displacement of a particle in a restrict-ing geometry observed via the signal attenuationin the PFG experiment is sensitive to the diffu-sion of the particle. At long D the signal attenua-tion becomes sensitive to the shape and dimen-sions of the restricting geometry. The relation-ships between the experimental parameters arefurther examined in the next section. If the re-stricting geometry is spherically symmetric, thenthere will be no-orientational dependence with


Figure 4 In the PFG experiment, we use the first gradient pulse to label the startingposition of the diffusing species and the second gradient pulse, at a time D later, to probe itsfinishing position with respect to the gradient direction. The important point is that we donot know what happens between these two times. In this diagram we schematically representwhat happens when we measure the diffusion coefficient of a species when it is undergoingfree diffusion or restricted diffusion in a sphere of radius R. r denotes the starting position0` vŽ . Ž . Ž ., and r denotes the position at a time D later. The length of the arrows i.e., R1

denote the measured displacement in the direction of the gradient which is normally in the zdirection and is taken to be up the page in the present diagram. We consider three relevant

Ž . Ž 2.time scales for the measurement of the effects of the restricted diffusion; i j s DDrRŽ .< 1 the short time limit ; the particle does not diffuse far enough during D to feel the

effects of restriction. Measurements performed within this time scale lead to the trueŽ . Ž .diffusion coefficient i.e., D . ii j f 1; some of the particles feel the effects of restriction

Ž .and the diffusion coefficient measured within this time scale will be apparent i.e., D andappbe a function of D. The fraction of particles that feel the effects of the boundary will be

Ž . Ž .dependent on the surface-to-volume ratio S rV. iii j ) 1 the long time limit ; allgeoparticles feel the effects of restriction. In this time scale, the displacement of the particle is

Ž .independent of D and depends only on R. Thus, restriction causes a measuring-time -de-pendent diffusion coefficient in which at D the displacement is limited by the embeddinggeometry.

respect to the gradient direction of the measureddisplacement. However, particularly in in ¨ i o

Ž .systems e.g., muscle cells , where the restrictinggeometry is normally not spherically symmetric,and in liquid crystals where the diffusion can beanisotropic, the observed signal attenuation willhave an orientational dependence. This aspectwill be considered subsequently.

We should also mention that ‘‘obstruction’’ canbe thought of as a type of restricted diffusion.The change in the diffusion coefficient due toobstruction is a source of information on theshape of the obstructing particle, e.g., Refs.Ž .72]74 . The mathematical description of ob-struction effects is very difficult, since the diffu-sion path of the probe molecule can be very

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complicated; also the obstructing particles are notŽ .necessarily distributed in space in a totally or-dered or totally random way.

CORRELATING SIGNAL ATTENUATIONWITH DIFFUSION

Introduction

We will now discuss the mathematical formula-tions necessary to relate the signal attenuation tothe diffusion coefficient and boundary conditionsin the PFG experiment. Starting from the Blochequations modified to include the diffusion of

Ž .magnetization 75, 76 it is possible to derive thenecessary relationships analytically for free diffu-sion, as we shall show below. However, in thecase of restricted diffusion this macroscopicapproachbecomes mathematically intractable.Thus, in general case one is forced to use differ-ent approximations to find formulae relating E tothe diffusion coefficient, boundary, and experi-mental conditions. There are two common ap-proximations, namely: the GPD approximationand the SGP approximation. However, even usingthese approximations, analytic solutions are gen-erally not possible and numerical methods mustbe used. In this section, we will only consider thecase of free diffusion and describe the macro-scopic approach and the SGP and GPD approxi-mations in this case. It is assumed that the gradi-ent pulses are rectangular. Detailed discussion ofthe signal attenuation of spins undergoing re-stricted diffusion will be deferred until later inthis section.

The Macroscopic Approach

Bloch Equations Including the Effects of Diffusion.The Bloch equations for the macroscopic nuclear

Ž .magnetization, M r, t s M q M q M , includ-x y zing the diffusion of magnetization, are given byŽ .75, 76 ,

Ž .M r, t M i q M jx yŽ .s gM = B r, t yt T2

Ž .M y M kz 0 2 w xy q D= M. 35T1

In the case of anisotropic diffusion, the last termw xin Eq. 35 would be replaced by = ? D ? =M. If we

Ž .now take as is usually the case B to be oriented0

along the z-axis and that this is superposed by agradient g vanishing at the origin which is parallel

Žto B we assume that the inhomogeneities caused0.by g are much smaller than B , and thus we can0

write

B s 0, B s 0,x y

Ž . w xB s B q g ? r sB q g x q g y q g z 36z 0 0 x y z

w x w xIf Eq. 36 is then substituted into Eq. 35 , notingthat

Ž . Ž .M = B s M B y M B q M B y M B yy z z y z x x zx

Ž . w xq M B y M B 37x y y x z

Ž .and defining the complex transverse magnetiza-tion as

w xm s M q iM 38x y

we obtain

m2Ž .s yiv m y ig g ? r m y mrT q D= m.0 2tw x39

The Stejskal and Tanner Pulse Sequence in theAbsence of Diffusion. In the absence of diffusionŽ .i.e., D s 0 , m relaxes exponentially with a timeconstant T , and thus we set2

yi v 0 ty tr T2 w xm s ce 40

where c represents the amplitude of the precess-ing magnetization unaffected by the effects of

w x w xrelaxation. If we substitute Eq. 40 into 39 , weobtain

c2Ž . w xs yig g ? r c q D= c. 41

t

w xIn the absence of diffusion, Eq. 41 is a first-orderordinary differential equation with solution

Ž . Ž . w xc r , t s S exp yigr ? F 42

where S is a constant and

tŽ . Ž . w xF t s g t9 dt9. 43H0

Now, if we consider the case of the PFG pulsesequence, then during the period from the pr2

Ž w x.pulse to the p pulse, we have i.e., Eq. 42

Ž . Ž . w xc r , t s S exp yigr ? F , 44


and S corresponds to the value of c immediatelyŽafter the pr2 pulse. After the p pulse neglect-

ing the phase angle of the p pulse, which is of no.consequence here , we have

Ž . Ž Ž .. w xc r , t s S exp yigr ? F y 2f , 45

where

Ž . w xf s F t . 46

w xThus, from Eq. 45 we can see that the effect ofthe p pulse is to set back the phase of c by twicethe amount that it had advanced up until the p

Žpulse see the first series of phase diagrams in. w x w xFig. 2 . Equations 44 and 45 can then be com-

bined into

Ž . Ž Ž Ž . .. w xc r , t s S exp yigr ? F y 2 H t y t f 47

Ž .where H t is the Heaviside step function. Wew xnote here that Eq. 47 is valid for the Hahn

spin-echo pulse sequence.

The Stejskal and Tanner Pulse Sequence in thePresence of Diffusion. In the previous section, we

w xconsidered the solution to Eq. 41 in the absenceof the diffusion. In this section, we derive a

w xsolution to Eq. 41 including the effects of diffu-w xsion. We assume a solution to Eq. 41 , including

w xthe diffusion term, to be of the form of Eq. 47w Ž .xbut allow S to be a function of t i.e., S t . By

w x w xsubstituting Eq. 47 into Eq. 41 , we obtain

Ž .dS t 22 w Ž . x Ž . w xs yg D F y 2 H t y t f S t 48dt

w xNow we integrate Eq. 48 from t s 0 to t s 2t

Ž .S 2tŽ Ž ..ln sln E 2tŽ .S 0

t 2 t 22 2 2 w xs y g DF dt q y g D F y 2f dtH H0 t

2t 2 t2 2 2s yg D F dt y 4f F dt q 4 f tH H½ 50 t

w x49

w xThe application of Eq. 49 to the calculation ofthe echo attenuation resulting from the effects ofdiffusion and the application of gradients is quitestraightforward but rather tedious. If we apply thegradient pulses as shown in the pulse sequencesin Fig. 2 and neglect the effects of any back-

Ž .ground gradients, then we can define g t and theŽ .effective field gradient, g t , as in Table 1. It ise ff

important to note that the lower limit of integra-w xtion in Eq. 43 refers to the start of the sequence.

Ž .For example, using the above definition of g t ,Ž .F t for t q D - t F t q D q d is calculated as1 1

follows,

t t qd1 1Ž .F t s 0 dt q g dtH H0 t1

t qD t1q 0 dt q g dtH Ht qd t qD1 1

Ž . w xs g t q d y t y D . 501

An example of the use of the symbolic algebraŽ . w xpackage Maple 77 to calculate Eq. 49 is given

in the Appendix, and from this we obtain theŽ .result 54

Ž . 2 2 2 Ž . w xln E s yg g Dd D y dr3 . 51

The term dr3 accounts for the finite width of thew xgradient pulse. Equation 51 is not a function of

t , and thus the placement of the gradient pulses1in the sequence is of no consequence; for exam-ple, there is no requirement that the gradientpulses be symmetrically placed around the ppulse. If instead we had imposed a steady gradi-

Žent throughout the echo sequence i.e., D s d s.t , we would have reproduced the well-known

diffusion term in the expression for the intensityŽ w x.of the Hahn spin-echo sequence see Eq. 17 , as

expected.w xIt is instructive to consider Eq. 51 in some

detail. Let us suppose that we do an experimentat, say 298 K on a sample containing a smallmolecule, such as water, which has a diffusion

y9 2 y1 Ž .coefficient of about 2.3 = 10 m s 68, 69and a protein with a diffusion coefficient of 1 =10y10 m2 sy1. From our discussion above and also

w xEq. 51 , it is apparent that after we have cali-

( ) ( )Table 1 g t and g t for the Stejskal andeff

Tanner Pulse Sequence

Subinterval of PulseŽ . Ž .Sequence g t g te ff

0 - t F t 0 01t - t F t q d g yg1 1t q d - t F t q D 0 01 1t q D - t F t q D q d g g1 1t q D q d - t F 2t 0 01

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brated the gradient and decided upon which nu-cleus we shall use to probe diffusion, we are leftwith three experimental variables to choose fromŽ .i.e., d, D, or g . Increasing any of these threeparameters will lead to increased signal attenua-tion, and is thus a means of measuring diffusion;for example, in Fig. 3 we altered d. While we arefree to choose which parameter we wish to vary,the relaxation characteristics of the sample andtechnical reasons may limit our choice. Somesimulated ‘‘typical experimental results’’ for aPFG experiment on the water]protein solutionare plotted in Fig. 5. This simple plot conveys awealth of information. We have chosen to plot E

2 2 2Ž .on a log scale versus g g d D y dr3 , and thusw xfrom Eq. 51 we see that each data set is a

straight line with a slope given by yD, where Dis the respective diffusion coefficient of the speciesin question. Of course, we could have just plottedour data against the experimental variable, but by

2 2 2Ž .using g g D D y dr3 as the abscissa, data ac-quired using different experimental conditions aremore easily compared. It can be clearly seen that

Žas the diffusion coefficient decreases i.e., larger.molecule andror more viscous solution , the slope

decreases which experimentally is reflected byless attenuation.

In all of the discussion above, the gradientpulses have been taken to be rectangular. This ismore out of technical and mathematical conve-nience than necessity. It should be mentionedthat in the PEG experiment the gradient pulsesdo not have to be rectangular, and in fact tominimize the generation of eddy currents, it maybe preferable to have nonrectangular pulses. Us-

w xing Eq. 49 , the effects of arbitrarily shaped gra-Ž .dient pulses can be considered 78 and the com-

putations can be conveniently performed bysimple modification of the Maple worksheet givenin the Appendix. Another commonly used and

w xtotally equivalent means of solving Eq. 41 is tow xsubstitute Eq. 44 , but where S is a function of t,

w xinto Eq. 41 directly and solving for S to obtain

t2 2Ž Ž .. w xln E t s yg D F dt. 52H0

w xIn evaluating Eq. 52 , the applied field gradientmust be replaced by the effective field gradient,g , such that the sign of the gradient is changede ff

Figure 5 A plot of the simulated echo attenuation for determining the diffusion coefficientŽ . Ž . w xof water }} and protein ] ] ] . The simulations were performed using Eq. 51 with

g s g1H s 2.6571 = 108 rad Ty1sy1, g s 0.2 T my1, D s 100 ms, and d ranging from 0 to

10 ms. The diffusion coefficient of water and the protein were taken to be 2.33 = 10y9 and1 = 10y10 m2sy1, respectively. As the diffusion coefficient increases, the slope of the line

Ž .increases. If ln E is plotted on a linear scale versus the same abscissa the slope is given byŽ w x.yD see Eq. 51 .


w xevery time a p pulse is applied. Thus, if Eq. 52is used to evaluate the PFG pulse sequence, g e ffis used as defined above and the final results is, as

w xbefore, given by Eq. 51 . Sometimes, especially inw xclinically oriented literature, Eq. 52 is written as

Ž Ž .. w xln E t s ybD 53

where the ‘‘gradient’’ or ‘‘diffusion weighting’’factor b is defined by

t2 2 w xb s g F dt. 54H0

Of course, for the Stejskal and Tanner sequence,2 2 2Žin the case of free diffusion, b s g g d D y

.dr3 .Although only isotropic diffusion was consid-

ered in this section, and therefore we used ascalar diffusion coefficient, D, the derivationscould equally well have been performed for aniso-tropic diffusion using the diffusion tensor D. Thisis considered in detail later.

The Stejskal and Tanner Pulse Sequence in thew xPresence of Diffusion and Flow. If Eq. 35 is sup-

Žplemented with a term reflecting flow i.e.,.y=vM where v is the velocity of the medium in

which the spins are in and similar analysis iscarried out as above, then we get, assuming flow

Ž .along the direction of gradient, 55

Ž . 2 2 2 Ž .ln E s yg g Dd D y dr3 q igd gD¨ .^ ` _ ^` _attenuation net phase change

w x55

We note that whereas diffusion results in a loss ofŽecho intensity, flow causes a net phase shift n.b.,

.the complex ‘‘i’’ . This is depicted in the thirdseries of phase diagrams in Fig. 2.

The GPD Approximation

In this section, the first of the two common ap-proximations used to relate the echo-signal atten-uation to the diffusion coefficient and the experi-mental variables is introduced. In the second sec-tion it was shown that the echo-signal attenuation

Ž w x.could be defined as i.e., Eq. 12

`Ž . Ž . Ž .S 2t s S 2t P f , 2t cos f dfHgs0

y`

Ž w x. Ž .with f being defined by Eq. 9 f 2t si� t1qd Ž . t1qD qd Ž . 4gg H z t dt y H z t9 dt9 . We need tot i t qD i1 1

Ž . w xderive P f, 2t from Eq. 9 . We begin by notingŽ .that z t is described by the one-dimensionali

diffusion equation which is a Gaussian for theŽcase of unbounded diffusion i.e., the one-dimen-

w x w xsional version of Eq. 27 , i.e., start with Eq. 31and integrate over the x and y coordinates using

w x .Eq. 64 below ,

yz 2y1r2Ž . Ž . w xP 0, z , t s 4pDt exp . 56ž /4Dt

Now as the probability density for the integral ofw Ž .xa variable in the present case z t , which itselfi

has a Gaussian probability density, is Gaussianw Ž .xe.g., see Ref. 65 , we have

yf2y1r22Ž . Ž ² : . w xP f , 2t s 2p f exp 57a¨ 2ž /² :2 f a¨

² 2:where f is the mean-squared phase change att s 2t, which is given by

² 2:f a¨

2t qd t qDqd1 12 2 Ž . Ž .s g g z t dt y z t dt .H Hi i½ 5¦ ;t t qD1 1 a¨

w x58

To avoid confusion with t, we use t and t as oura bw xdummy variables of integration, and thus, Eq. 58

becomes

t qd t qd1 12 2 2² :f s g g dt dta¨ H H a b½t t1 1

t qd t qDqd1 1y2 dt dtH H a bt t qD1 1

t qDqd t qDqd1 1q dt dtH H a b 5t qD t qD1 1

² Ž . Ž .: w x= z t z t . 59aä b

w xFrom Eq. 59 , we see that the computation of² 2:f can be separated into two pieces: a spatialpart given by the mean-squared displacement in

² Ž . Ž .:the direction of the gradient, z t z t , and aaä bŽ .temporal part i.e., the time integrals . Thus, we

² Ž . Ž .:first need to calculate z t z t . We need toaä b² Ž . Ž .:express z t z t as the products of theaä b

probability of each motion times the correspond-

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ing displacement in the direction of the gradient,Ž .which can be written most generally as 79

² Ž . Ž .:z t z t aä b

Ž . Ž . Ž . Ž .s r yr r yr r r P r , r , tHHH 1 0 2 0 0 0 1 az z

= Ž . w xP r , r , t y t dr dr dr . 601 2 b a 0 1 2

w xIt should be noted that Eq. 60 holds only whenw xt ) t . We will now evaluate Eq. 60 for theb a

Ž . Ž .particularly simple case of P r , r , t as given0 1w x Ž .by Eq. 27 i.e., free diffusion . Since we are only

interested in motion in one dimension, we cansimplify our task by using the one-dimensional

w xversion of Eq. 27 ,

2Ž .z y z1 0y1r2Ž . Ž .P z , z , t s 4pDt exp y0 1 ž /4Dt

w x61

w xand making obvious changes to Eq. 60 thusobtain

² Ž . Ž .:z t z t aä b

` ` `Ž .Ž .Ž .s r z z y z z y zH H H 0 1 0 2 0

y` y` y`

y1r2Ž .= 4pDta

=

2Ž .z y z1 0 y1r2Ž Ž ..exp y 4pD t y tb až /4Dta

=

2Ž .z y z2 1exp y dz dz dz .0 1 2Ž .ž /4D t y tb a

Now we let Z s z y z and Z s z y z , and1 1 0 2 2 0thus,

`Ž .r z dzH 0 0

y`

=`

2Z1y1r2Ž .Z 4pDt exp y dZH 1 a 1ž /4Dty` a

=` y1r2Ž Ž ..Z 4pD t y tH 2 b a

y`

=

2Ž .Z y Z2 1exp y dZ .2Ž .ž /4D t y tb a

w xBy noting Eq. 29 , we can remove the integralover z , and making the substitution ZX s Z y0 2 2Z , we then get1

`2Z1y1r2Ž .Z 4pDt exp y dZH 1 a 1ž /4Dty` a

=` y1r2XŽ .Ž Ž ..Z q Z 4pD t y tH 2 1 b a

y`

=ZX 2

2 X w xexp y dZ . 622Ž .ž /4D t y tb a

X w xWe now consider the integral over Z in Eq. 62 ,2which we rewrite as

y1r2Ž Ž ..4pD t y tb a

`X 2Z2

= Z exp y dZ9H1½ Ž .ž /4D t y ty` b a

`X 2Z2X w xq Z exp y dZ9 . 63H 2 5Ž .ž /4D t y ty` b a

w xThe first integral in Eq. 63 can be evaluated withwthe standard integral e.g., integral 3.323 2. in Ref.

Ž .x53 ,

'` p2 2 2 2yp x " q x q r4 p w xe dx s e . 64H < <py`

X Ž Ž ..y1r2by setting x s Z , p s 4D t y t and2 a bŽ Ž ..1r2q s 0, to give 4pD t y t . The second inte-b a

w xgral in Eq. 63 can be evaluated using the stan-w Ž .xdard integral Eq. 3.462 6. in Ref. 53 ,

` q p2 2 2yp x q2 q x q r p w xxe dx s e Re p ) 0 .H (p py`

w x65

X Ž .In our case, x s Z , p s 4D t y t and q s 0,2 b aw xand so this integral equals 0. Hence, Eq. 63

w xreduces to simply Z and now Eq. 62 becomes1

` y1r22² Ž . Ž .: Ž .z t z t s Z 4pDta¨ Ha b 1 ay`

Z 21 w x= exp y dZ . 661ž /4Dta


Finally, noting the standard integral given in Eq.w x32 , we obtain the final result of

² Ž . Ž .: w xz t z t s 2 Dt 67aä b a

which is of course equal to the mean-squareddisplacement for the one-dimensional diffusion

Ž w x.equation see Eq. 30 .We now come to a subtle point in evaluating

w x w xEqs. 59 and 60 . In performing the integrals, weneed to consider the range of integration when

² Ž . Ž .:inputting z t z t ; that is, we have to inter-aä bw xchange t for t in Eq. 60 depending on whethera b

t ) t or t - t . This can be understood bya b a bw xnoting that the exponentials in Eq. 60 must have

Ž w x.negative exponents recall the validity of Eq. 60 ;Ž .hence, we get 60, 80

² Ž . Ž .: ² 2 Ž .:z t z t s z t s 2 Dt if t - ta¨ aä b a a a b

² Ž . Ž .: ² 2 Ž .:z t z t s z t s 2 Dt if t ) t .a¨ aä b b b a b

w x68

w xContinuing on with our derivation of Eq. 58 , wehave

t qd t1 a2 2 2² :f s g g 2 Dt dta¨ H H b b½ t t1 1

t qd1q 2 Dt dt dtH a b ata

t qd t qDqd1 1y2 2 Dt dt dtH H a b at t qD1 1

t qDqd t1 aq 2 Dt dtH H b bt qD t qD1 1

t qDqd1q 2 Dt dt dtH a b a 5ta

2 2 2 Ž . w xs g g 2d D y dr3 69

w xIf we evaluate Eq. 12 using the distribution ofw xphases as given in Eq. 57 , we find that the echo

attenuation is given by

Ž ² 2: . w xE s exp y f r2 . 70

w x w xFinally, if we substitute Eq. 69 into Eq. 70 , weŽ w x.get our final result i.e., Eq. 51 as before,

Ž . 2 2 2 Ž .ln E s yg g Dd D y dr3 .

As expected, if we modify the limits of integra-w xtion in Eq. 59 , then the GPD approximation can

be used to calculate the diffusion term in theexpression for the intensity of the Hahn spin-echo

Ž w x. Ž .sequence see Eq. 17 60, 80 .We have shown in this section that since the

Ž ² 2:.mean-squared phase change i.e., f can becalculated exactly for unrestricted diffusion, theGPD approximation gives the same result as the

Ž w x.macroscopic approach i.e., Eq. 51 . Subse-quently, the validity of the GPD approximation

w xrepresented by Eq. 57 and its ramifications whenthe diffusion is bounded will be considered fur-ther.

SGP Approximation

To understand the SGP approximation, we startw xback at Eq. 7 but ignore the effects of motion

Žduring the gradient pulse rigorously, one as-sumes that the gradient pulse is like a delta

< <function, that is, d ª 0 and g ª `, while their.product remains finite . Experimentally, this con-

dition is approximated by keeping d < D. Hence,the effect of a gradient pulse of duration d on aspin at position r is given by, neglecting the effectof the static field,

Ž . w xf r s gdg ? r. 71

The scalar product arises because only motionparallel to the direction of the gradient will causea change in the phase of the spin. Hence, if weconsider the phase change of a spin which was atposition r during the first gradient pulse and at0position r during the second, then the change in1phase in moving from r to r is given by0 1

Ž . Ž . w xDf r y r s gdg ? r y r . 721 0 1 0

Ž w xn.b., the D in Eq. 72 represents the difference.in f, not the duration between gradient pulses .

Now we need to consider the probability of a spinŽ .starting at r i.e., the starting spin density at0

Ž .t s 0, r r , 0 , which is usually taken as being0Ž . Žequal to the equilibrium spin density r r see0

w x.Eq. 28 . This assumption requires that insignifi-cant relaxation occur between the first rf pulseŽ .i.e., excitation and the first gradient pulse. Inpractice this is normally the case. Now the proba-

Žbility of moving from r to r in time D i.e., the0 1.separation between the gradient pulses is, of

Ž .course, given by P r , r , t as before. Thus, the0 1

PRICE318

probability of a spin starting from r and moving0Ž w x.to r in time D is given by recall Eq. 241

Ž . Ž . w xr r P r , r , D . 730 0 1

The NMR signal is proportional to the vectorsum of the transverse components of the magne-tization, and so the signal from one spin is givenby

Ž . Ž . igdg?Žr 1yr 0 . w xr r P r , r , D e . 740 0 1

But in NMR, the signal results from the ensembleof spins, and thus we must integrate over allpossible starting and finishing positions, and fi-

Ž .nally we arrive at our result 55, 56

Ž . Ž . Ž .E g, D s r r P r , r , DHH 0 0 1

= igdg?Žr 1yr 0 . w xe dr dr . 750 1

Thus, the total signal is a superposition of signalsŽ .transverse magnetizations , in which each phaseterm is weighted by the probability for a spin tobegin at r and move to r during D.0 1

w xAs an example, let us rederive Eq. 51 usingŽ w x.the SGP approximation i.e., Eq. 75 . From Eq.

w x w x75 and 27 , we have

1 2yŽr yr . r4 DD1 0Ž . Ž .E g, D s r r eHH 0 3r2Ž .4pDD

igdg?Žr 1yr 0 . w x= e dr dr . 761 0

ŽIn the present case, we set g s g we will dropz.the subscript z, however , and R s r y r . Us-1 0

Žing spherical polar coordinates i.e., R is theradius and u and w are the polar and azimu-

.thal angles, respectively , we note that dR sR2 sin u dR du dw; also, since u is the angle be-tween R and g we have

`1 2p 2yR r4 DD 2Ž .E g, D s dw e RH H3r2Ž . 0 04pDD

=p

igd g R cos u w xe sin u du dR 77H0

As there is no w dependence, it can be integratedout

`2p 2yR r4 DD 2e RH3r2Ž . 04pDD

=p

igd g R cos u w xe sin u du dR 78H0

then, by noting that d cos u s ysin u du, we get

`2p 12yR r4 DD 2 igd g Rue R e du dR.H H3r2Ž . 0 y14pDD

w x79

The integral over u is then performed and evalu-w xated using Eq. 11 , resulting in

`4p 2yR r4 DD Ž .R e sin gd gR dRH3r2Ž . 0gd g 4pDD

w x80

wWe note from a table of standard integrals e.g.,Ž . x53 Eq. 3.952 1. that

'` a p2 2 2 2yp x ya r4 pŽ . w xxe sin ax dx s e . 81H 34 p0

Ž .y1r2In our case, x s R, p s 4DD , and a s gd g,and thus we obtain the final result

Ž . Ž 2 2 2 . w xE g , D s exp yg g Dd D . 82

w xThis is the same as Eq. 51 , but in the limit ofd ª 0; thus the dr3 term which accounts for thefinite width of the gradient pulse is absent. Whenwe consider restricted diffusion, the evaluation of

w xEq. 75 proceeds in exactly the same manner,Ž .except that we must substitute the relevant r r0

Ž .and P r , r , t . However, as the confining geom-0 1etry becomes more complicated, so does themathematical complexity.

The Analogy between PFG Measurements and Scat-tering. Returning back to the derivation of the

w xSGP approximation, and in particular, Eq. 72 ,Žbecause we are using a constant commonly

.termed ‘‘linear’’ gradient, what is important isnot the actual starting and finishing positions ofthe spin but the net displacement between thetwo points in the direction of the gradient. Asbefore, we can write R s r y r and, analo-1 0

w xgously to Eq. 73 , the probability that a particlethat starts at r displacing a distance R during D0

Ž w x.is given by recall Eq. 24

Ž . Ž . w xr r P r , r q R, D . 830 0 0


w xNow, if we integrate Eq. 83 over all possiblestarting positions, we obtain the ‘‘average propa-

Ž .gator.’’ This is the probability, P R, D , that amolecule at any starting position will displace by

Ž .R during the period D 12, 63

Ž . Ž . Ž . w xP R, D s r r P r , r q R, D dr . 84H 0 0 0 0

w x w xUsing Eq. 84 , Eq. 75 can be rewritten as

igdg?RŽ . Ž . w xE q, D s P R, D e dR. 85Hw xThus, from Eq. 85 we can see that PFG NMR is

Ž .sensitive to the average propagator P R, D . It isconvenient to include the effects of the gradientinto the analysis by defining the parameter, q, byŽ .81

y1Ž . w xq s 2p gdg, 86

y1 Žwhere q has units of m n.b., some authors use. w xk s gdg , and thus we can rewrite Eq. 85 as

i2 p q?RŽ . Ž . w xE q, D s P R, D e dR. 87H

Physical insight can be gained by noting fromw xEq. 87 that there is a Fourier relationship

Ž . Ž . Ž . Ž .82, 83 between E q, D and P R, D 63, 84, 85 ;Ž .that is, the Fourier transform of E q, D with

Ž . Ž .respect to q returns an image of P R, D . P R, DŽ .will be equivalent to P r , r , t only when0 1

Ž .P r , r , t is independent of the starting position0 1Žr . While this is true for free diffusion see Eq.0

w x .27 and the discussion thereafter , it is not truein the case of restricted diffusion or diffusion inmacroscopically heterogeneous systems. We canthink of PFG diffusion measurements as q-space

w ximaging. From Eq. 86 , we see that we can tra-verse q-space by either changing d or g, and we

Žcan change the direction by altering g i.e., the.direction of the gradient . Thus, PFG diffusion

measurements are analogous to normal NMRŽ . Ž .imaging also termed k-space imaging 41, 42 ,

except that normal NMR imaging returns theŽ .spin density r r . Or, to be more mathematically0

succinct, in PFG diffusion measurements q-spaceis conjugate to R, while in normal imaging k-spaceis conjugate to r .0

w xThus, Eq. 75 is analogous to the scatteringfunction which applies in neutron scattering andq corresponds to the scattering wave vector

Ž .41, 86, 87 . However, there are major differencesin the temporal and spatial time scales of each

Ž .type of experiment. Further, E q, D is measuredin the time domain of D in PFG experiments andin the frequency domain for neutron scatteringexperiments.

‘‘Diffusi©e Diffraction’’ and Imaging Molecular Mo-tion. Consider a spin trapped within a fully en-

Žclosed pore e.g., a spin diffusing between parallel. Ž .planes . In the long-time limit i.e., D ª ` , all

species lose memory of their starting positionŽi.e., they become independent of their starting

.position and, therefore, the diffusional process ,and so

Ž . Ž . w xP r , r , ` s r r 880 1 1

and the average propagator becomes

Ž . Ž . Ž . w xP R, ` s r r r r q R dr . 89H 0 0 0

Ž .Thus, P R, ` is the autocorrelation function ofŽ . Žthe molecular density r r or the convolution of0

. w xthe density with itself . From Eq. 87 and usingŽ . Žthe Wiener]Kintchine theorem 82, 88 i.e., the

Fourier transform of a time autocorrelation func-.tion is the frequency power spectrum , we find

Ž . Ž .that E q, ` is the power spectrum of r r0Ž .41, 85, 89 ,

i2 p q?RŽ . Ž .E q, ` s P R, ` e dRH

Ž . Ž . i2 p q?Rs r r r r q R dr e dRHH 0 0 0

Ž . Ž . i2 p q?Žr 1yr 0 .s r r r r dr e dRHH 0 1 0

Ž . yi 2 p q?r 0 Ž . i2 p q?r1s r r e dr r r e drH H0 0 1 1

Ž . Ž .s S* q S q

< Ž . < 2 w xs S q 90

Ž . Ž .where S q is the Fourier transform of r r .1w xAlternately, Eq. 90 can easily be derived directly

w x w xfrom Eqs. 75 and 88 .This is the origin of diffraction-like effects in

PFG diffusion studies. In quasielastic neutron< Ž . < 2scattering S q is known as the elastic incoher-

ent structure factor, whereas in scattering theoryit is referred to as the form factor of the confining

Ž . Ž .volume 86 . S q is analogous to the signal mea-

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Ž .sured in conventional NMR imaging 41]43 .However, whereas conventional imaging returnsthe phase-sensitive spatial spectrum of the re-

Ž .stricting pore, E q, ` measures the power spec-< Ž . < 2 Ž .trum, S q . Thus, E q, ` is sensitive to average

features in local structure, not the motional char-Ž .acteristics. Further, because E q, ` measures the

Ž .power spectrum of S q , Fourier inversion cannotbe used to obtain a direct image of the pore.However, the q-space imaging has the potentialto give much higher resolution than conventionalk-space imaging, since the entire signal from thesample is available to contribute to each pixel in

Ž . Ž .R-space i.e., R, the dynamic displacement 85Ž .rather than from a volume element i.e., voxel as

in conventional k-space imaging. Thus, the reso-lution achievable in q-space imaging is limitedonly by the magnitude of q.

We will illustrate the diffraction effect withrecourse to diffusion in between parallel platesŽ .see the inset to Fig. 6 with a separation of 2 RŽ .n.b. not R, the dynamic displacement . For thisgeometry, the analysis linking the experimentalvariables and the diffusion of the particle is per-formed in a manner entirely analogous to that

already presented for free diffusion earlier, ex-cept that the mathematics is more tedious. Briefly,

w xthe solution to Eq. 26 for this geometry with thew x Ž .initial condition of Eq. 25 is given by 66

` 2 2n p DtŽ .P z , z , t s 1 q 2 exp yÝ0 1 2ž /Ž .2 Rns1

=np z np z0 1 w xcos cos . 91ž / ž /2 R 2 R

w x w xIf Eq. 91 is substituted into Eq. 75 , except thatwe now write the equation in terms of q, we get

Ž .the SGP solution 56 ,

w Ž Ž ..x2 1 y cos 2p q 2 RŽ .E q , D s 2Ž Ž ..2p q 2 R

` 2 2n p DD2Ž Ž ..q 4 2p q 2 R exp yÝ 2ž /Ž .2 Rns1

=

nŽ . Ž Ž ..1 y y1 cos 2p q 2 Rw x. 9222 2Ž Ž .. Ž .2p q 2 R y np

Ž . w xFigure 6 A plot of E q, ` versus q calculated using Eq. 93 for two values of theŽ . Ž . Ž .interplanar spacing i.e., slit width; 2 R , 2 R s 26 mm ] ] ] and 30 mm }} . The

diffractive minima are clearly R dependent, and in the case of planes, the minima occurŽ .when q s nr2 R n s 1, 2, 3 . . . . Generally, when there is only one characteristic distance,

Žit is more convenient to plot the abscissa in terms of the dimensionless parameter qR see.Fig. 8 .


Ž .In the long time limit j 4 1 i.e., D ª ` , Eq.w x90 becomes

w Ž Ž ..x2 1 y cos 2p q 2 RŽ .E q , ` s 2Ž Ž ..2p q 2 R

< Ž Ž .. < 2 w xs sinc pq 2R 93

Ž . Ž . Ž .where sinc x s sin x rx. E q, ` versus q isplotted for two values of R in Fig. 6. From Eq.w x93 , it is easy to see that diffractive minima result

Ž . Ž .when q s nr2 R n s 1, 2, 3 . . . and E q, ` s 0,Ž .since sin np s 0. The diffraction-like effects in

the echo-attenuation curves clearly demonstratethat there is a direct analogy between the PFGdiffusion measurement of a spin undergoing re-stricted diffusion in an enclosed pore and optical

Ž .diffraction by a single slit 89]92 . Also, theR-dependence of the attenuation curve also showsthat structural information about the enclosinggeometry can be obtained from the characteris-tics of the diffraction pattern. It is important tonote that the above discussion regards diffusiveŽ . Ž .i.e., q-space diffraction, not Mansfield k-space

Ž . Ž .diffraction 87, 93 . Mansfield k-space diffrac-tion depends on the relative positions at fixed

wtime n.b., normal imaging returns an image ofŽ .xr r , whereas q-space diffraction depends on the

relative displacements from the molecular originduring D.

The effects of diffraction and the experimentalconditions that will lead to their observation arefurther considered in the following section.

PFG MEASUREMENTS IN RESTRICTEDGEOMETRIES

( )General Relationships between E q, D ,q and D , and Displacement

The above discussion and considerations, espe-w x w xcially Eqs. 85 and 88 , reveal some pertinent

Ž Ž .y1 .points about the roles of D and q s 2p gdgin PFG diffusion measurements in systems inwhich the diffusion is restricted by barriers on aspatial scale of R. When the condition qR < 1 is

Ž .met, the behavior of E q, D is dominated by thediffusive motion of the spin. If the condition jŽ 2 .s DDrR - 1 is met, then the measured diffu-

Ž .sion coefficient i.e., D will tend to that of thea p pŽ .bulk solution Fig. 4 . As j increases, the effects

of the restricting geometry will become increas-

ingly important. When j is ) 1, structural infor-mation can be obtained directly from the PFG

Žsignal i.e., diffraction effects, if the restricting.geometry has local order by varying q such that

qR G 1.In analyzing the PFG dependencies for re-

stricted geometries, one often implies an analogyto the free diffusion case and defines an apparentdiffusion coefficient by

w Ž .x1 ln E q , DŽ . w xD D s y lim . 94a p p 2D qqª0

w xThe origin of Eq. 94 can be easily understood byw xsubstituting Eq. 82 , written in terms of q for

Ž .E q, D . Considerable insight can be gained byŽ .performing a Taylor expansion 51 with q s 0

Ž . w xi.e., Maclaurin’s series of Eq. 85 for the case ofshort D

Ž y1 .E q < R , D

n n i2 p q?R` Ž .i2p qR eŽ .f P R, D dRÝH 2n! qns0

w x95

For simplicity we assume that the gradient isŽ .directed along z n.b. Z s z y z , and thus1 0

w x Ž .from Eq. 95 we obtain 14, 94, 95 ,

Ž y1 .E q < R , D

2 2Ž . Ž .i 2p q Z 2p q ZŽ .f P Z, D 1 q yH 1! 2!

3 43 4Ž . Ž .i 2p q Z 2p q Zy q ??? dZ

3! 4!

2 2Ž . ² Ž .:2p q z Ds 1 y

2!4 4Ž . ² Ž .:2p q z D

w xq ??? 964!

w xIn deriving the last step, we used Eq. 30 ; also we` Ž .note, by definition, that H P Z, D dZ s 1. Asy`

can be seen, all the odd orders vanish. From Eq.w x Ž .96 , we see that the initial decay of E q, D withrespect to q gives the mean-squared displacement

PRICE322

² 2Ž .: w xz D . Further, using Eq. 33 , the apparenttime-dependent diffusion coefficient can be ob-tained, i.e.,

Ž . ² 2 Ž .: Ž . w xD D s z D 2D 97a p p

Ž .from the low q limit of E q, D . As might beexpected, the time dependence of the apparent

Ždiffusion coefficient over observation time i.e.,. ² 2Ž .:1r2

D , such that D is finite but that z D is lessthan the distance between the confinements,provides information on the surface-to-volume ra-

w Ž .tio of the confining geometry see case ii in Fig.x Ž .4 . Mitra and coworkers 96]98 derived the rela-

tionship

4 Sg eo 1r2 1r2Ž .D D s D 1 y D D q ???a p p ' V3d p

w x98

where d is the number of spatial dimensions andS rV is the surface-to-volume ratio. Thus, Dg eo a p p

deviates from D approximately linearly with D1r2.These surface effects have been nicely illustrated

Ž .in a recent review by Callaghan and Coy 14 .

The Validity of the Different Approaches

In the third section we presented three ap-proaches for calculating the effects of diffusionon the signal attenuation in the PFG experiment.The macroscopic approach provides analytical so-lutions but is mathematically tractable only forfree diffusion and for diffusion superimposedupon flow. We then presented the GPD and SGPapproximations. It should be noted that these are

Žnot the only approximations available e.g., Refs.Ž .x99, 100 . In the case of free diffusion, the GPDapproximation is valid and gives the same resultas the macroscopic approach, while the SGP ap-proximation gives the same result but in the limitof d ª 0 as g ª `. However, in chemical and

Žbiochemical systems e.g., cells, micelles, zeolites,.etc. , it is often the case that the diffusion of the

probe species is restricted on the time scale of DŽ .see Free and Restricted Diffusion , and to ana-lyze the experimental data, the SGP or GPDapproximation is normally used if the system ismathematically tractable. If the system is toocomplicated numerical methods must be resortedto.

It is important to understand the implicationsof the approximations involved in the GPD and

w Ž .xSGP approaches e.g., Refs. 94, 101]104 . Thevalidity of the GPD approach is determined by

w xthe validity of Eq. 57 . From the above discussionit was seen that the Gaussian phase approxima-tion is justified in the case of free diffusion.Consequently, it will also hold in the case ofrestricted diffusion when D is so short that very

Žfew of the spins are affected by the boundary i.e.,.j < 1; see earlier text and Fig. 2 , since the

Žpropagator describing the restricted diffusion i.e.,Ž ..P r , r , t will reduce to that of the free diffu-0 1

Ž w x. Ž .sion case i.e., Eq. 27 . Similarly, Neuman 79showed that when D becomes so long that theprobability of being at any position at the end ofD is independent of the starting position, thechange in phase becomes independent of thephase distribution. From the central limit theo-

Ž .rem 65 , the distribution of the sums of thephase changes becomes Gaussian. However, theenforcing of a Gaussian phase condition is asevere approximation and, as a consequence can-

Ž .not yield interference effects 105 .In many experiments, particularly where the

sample has a short T relaxation time, which2limits the value of D in the PFG pulse sequence,it may be impossible to comply closely enoughwith the requirements for SGP approximation. Inthis case, the GPD is useful, since it accounts forthe finite length of the gradient pulse. However,the GPD approximation is exact only in the limit

Ž .of free diffusion i.e., R ª ` ; that is, where thephase distribution is Gaussian, while the SGPequation is only strictly valid for infinitely small d.

Ž .Balinov et al. 101 used computer simulations ofBrownian motion to test the validity of the GPDand SGP approximations. They found that theGPD approximation solution for diffusion within

Ž w x .a reflecting sphere see Eq. 99 below simulatedthe data very well in the limit of j - 1, fairly wellfor j f 1, and well for j ) 1. In contrast, theSGP solution described the data well for largevalues of j and small gradient strengths. Theresults showed that at j f 1 the long time limit of

Žthe SGP equation i.e., the attenuation has be-.come independent of D is already applicable

Ž . Ž . w Ž .x101 . Blees 102 and others e.g., 94, 105 ,using numerical simulations, considered the ef-fects of finite d on the SGP approximation sol-

Ž w x.ution Eq. 92 for spin diffusing between re-Ž .flecting planes Fig. 6 . They found that as the

duration of the gradient pulse becomes finite,the diffraction minima shift toward higher q. Thehigher-order minima were more affected than thefirst minimum.


In the next section, we will illustrate someaspects of the above discussion by comparing theresults for a diffusion inside a sphere obtainedusing the GPD and SGP approximations. In the

Ž .present case, we consider the relatively simpleŽcase of spins within a reflecting sphere i.e., the

sphere is impermeable and collision with the sur-face of the sphere does not affect the relaxation

.of the spins .

An Example: Diffusion Within a Sphere

Although mathematical models have been de-rived for a number of complicated geometries for

w Žboth steady and PFG sequences e.g., 12,79,.x105]111 , here we will consider only the theoret-

ical solutions for the PFG experiment obtainedusing the SGP and GPD approximations for dif-fusion within reflecting spherical boundaries ofradius R. A reflecting sphere is a suitable firstapproximation for the diffusion of small molecules

Ž .such as metabolites inside cells 71 , or moleculesinside many porous systems. In agreement withthe earlier discussion concerning free and re-stricted diffusion, the solutions for the restrictinggeometries reduce to those for free diffusion in

Ž .the short time limit i.e., j - 1 , while in the longŽ .time limit i.e., j ) 1 the solutions become de-

pendent upon only the restricting geometry.The GPD approximation solution for a spin

diffusing within a reflecting sphere is calculatedanalogously to the case of free diffusion as given

Ž .previously. The solution is given by 106

Ž .E q , D

2g2 g 2

s exp y 2D�2 Ž . Ž .2a Dd y 2 q 2 L d y L D y dn

` Ž . Ž .q2 L D y L D q d= Ý 6 2 2Ž .a R a y 2 0n nns1

w x99

Ž . Ž 2 .where L t s exp ya Dt and a are the rootsn nof the equation

Ž . X Ž . Ž .a R J a R y 1r2 J a R s 0,n 3r2 n 3r2 n

where J is the Bessel function of the first kindw Ž .xe.g., Ref. 52 . Similarly, the SGP solution iscalculated in the same manner as the free-diffu-

sion example given previously. The solution isŽ .101

2wŽ . Ž . Ž .x9 2p qR cos 2p qR y sin 2p qRŽ .E q , D s 6Ž .2p qR

`22 XŽ . w Ž .xq 6 2p qR j 2p qRÝ n

ns0

Ž . 22n q 1 a nm=Ý 2 2a y n y nnmm

a2 DD 1nm=exp y 2 2ž / 2R 2 Ž .a y 2p qRnm

w x100

where a is the mth nonzero root of the equa-nmX Ž .tion j a s 0 and j is the spherical Besseln nm

w Ž .xfunction of the first kind e.g., Ref. 52 .As an example of the effects of restricted

diffusion, we have plotted some simulated datafor free diffusion and diffusion within a sphereŽ .using the GPD approximation in Fig. 7 using thesame experimental conditions and gradientstrength as in Fig. 5. To show the effects clearlywe have chosen to plot the data as a function ofD. As is readily apparent in the case of freediffusion, the mean-squared displacement scaleswith time, and as a result we obtain a straightline. However, in the case of diffusion within asphere, at very small values of D the results of thesimulation agree with that for free diffusion, butas D increases, there is a transition from freediffusion to surface effects as the boundaries sig-nificantly affect the motion of the diffusing spins,and the mean-squared displacement no longer

Žscales linearly with time i.e., the diffusion is no.longer purely Gaussian . At large values of D, the

motion becomes completely restricted, the dis-placement becomes time independent, and theattenuation curve plateaus out.

It is very interesting to compare the long-timeŽ . w x w xi.e., j ª ` behavior of Eqs. 99 and 100 . In

w x Ž .Eq. 99 , all the L t terms involving D disappear,leaving only an exponential function involving d

w x Ž .and D; thus, Eq. 99 becomes 79

2Ž . Ž . w xE q , ` s exp y 2p qR r5 , 101Ž .a monotonically decreasing function. However, in

w xthe cases of Eq. 100 , we have a totally differentsituation, as only the second term on the right-

w x whand side of Eq. 100 vanishes n.b., the

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Ž .Figure 7 A plot of simulated echo attenuation in the case of free diffusion }} andŽ . Ž w x.diffusion in a sphere ] ] ] based on the GPD approximation i.e., Eq. 99 versus D. The

parameters used in the simulation were d s 1 ms, D s 5 = 10y10 m2sy1, g s 1 T my1,R s 8 mm, and g s g

1H s 2.6571 = 108 rad Ty1sy1. The echo attenuation in the case ofŽ .diffusion in the sphere can be seen to go through three stages: i when j < 1, the diffusion

Ž .appears unrestricted and the result is the same as that of free diffusion, ii as D increasesŽ .the spins begin to feel the effects of the surface, and iii when j ) 1, the diffusion is fully

restricted and the attenuation curve plateaus out.

Ž 2 . x Žexp ya j term leaving the trigonometric i.e.,nm.periodic function

2wŽ . Ž . Ž .x9 2p qR cos 2p qR y sin 2p qRŽ .E q , ` s .6Ž .2p qR

w x102

Obviously, as q increases, the denominator of Eq.w x Ž w x102 increases such that Eq. 102 as a whole

.decreases , but the trigonometric functions inthe numerator result in the function having aninfinite series of maxima and minima. The min-ima occur when q takes a value such thatŽ . Ž . Ž .2p qR cos 2p qR y sin 2p qR s 0, for the firstminima, this occurs when q f 0.71rR. The simu-lated echo intensity calculated using both theGPD and SGP approximations versus qR is shownin Fig. 8. We can see that at small attenuationvalues, the GPD and SGP approximations agree

Ž .very well n.b., d g D , but at larger attenuationvalues, the SGP approximation gives diffractiveminima, whereas as expected, the GPD approxi-mation does not. In well-chosen systems wherethe signal-to-noise ratio is sufficient and the sam-

Žple geometry is monodisperse or at least not too

.polydisperse , it is possible to observe such min-w Ž .xima e.g., 105 . The diffractive minima are an

additional source of information and their posi-tion is R dependent.

Anisotropic Diffusion

Earlier, it was noted that isotropic diffusion isreally just a special case, and more generally, wemust consider anisotropic diffusion resulting fromeither the physical arrangement of the medium or

Ž .anisotropic i.e., nonspherical restriction. SuchŽsituations commonly arise in biological e.g., cells,

. wskeletal muscle and liquid crystals systems e.g.,Ž . xRefs. 112, 113 and references therein , thus the

diffusion process is represented by a CartesianŽ .tensor, D see Free and Restricted Diffusion . In

such systems, the echo-signal attenuation will havean orientational dependence with respectto the measuring gradient. For example, foranisotropic free diffusion, the g 2D term in Eq.w x51 must be replaced by g ? D ? g, where

w xg ? D ? g s D g g a , b s x , y , z 103ÝÝ ab a ba b


Figure 8 A plot of the simulated echo-attenuation data for diffusion within a sphereŽ . Ž .calculated using the SGP approximation }} and the GPD approximation ] ] ] versus

qR. The parameters used in the simulation were d s 1 ms, D s 100 ms, D s 1 =10y9 m2sy1, g s 1 T my1, R s 8 mm, and g s g

1H s 2.6571 = 108 rad Ty1sy1. Thew xminima in the SGP plot occur when q takes a value such that the numerator in Eq. 102

equates to 0.

Ž .and so we obtain 55

Ž . 2 2 Ž . w xln E s yg g ? D ? gd D y dr3 . 104

w xWe remark that Eq. 104 can be rewritten in aw xform similar to Eq. 53 , i.e.,

Ž . w xln E s y b D s yb:D 105ÝÝ ab aba b

where ‘‘:’’ is the generalized dot product and b isŽ .now a symmetric matrix given by 114

2t2 Ž Ž . Ž . .b s g F t9 y 2 H t y t fH0

=TŽ Ž . Ž . .F t9 y 2 H t y t f dt

2 2 Ž . w xs g g g d D y dr3 . 106a b

w xThe first line in Eq. 106 is a general definitionand the second line is the specific solution for thePFG sequence. Alternatively and totally equiva-

w xlently, we can rewrite Eq. 52 as

t2Ž Ž .. w xln E t s yg F ? D ? F dt. 107H0

w xFrom Eq. 104 , it can be seen that the directionin which the diffusion is measured is determinedby the gradient, and it is actually a diagonalelement of D, the diffusion tensor in the gradientframe, that is measured. Thus, the equation relat-ing the echo attenuation due to free diffusion

Ž w x.when measured using a z-gradient i.e., Eq. 51written in tensor notation is

Ž . 2 2 2 Ž .ln E s yg g D d D y dr3 s yb D .z z z z z z z

w x108

The diffusion tensor in the molecular frame, D9,Žcan be transformed to the laboratory i.e., gradi-

. Ž . went frame Fig. 9 by using rotation matrices e.g.,Ž .xRef. 115

y1 Ž . Ž . w xD ' R u, w ? D9 ? R u, w 109

Ž .where R u, w is the relevant rotation matrix andu and w are the polar and azimuthal angles be-tween the director and gradient frames, respec-tively. Thus, the off-diagonal elements of D willvanish only when the director and laboratoryframes of reference coincide. Thus, in the generalcase, both diagonal and off-diagonal elements of

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D9 will affect the measured echo attenuationŽ .114, 116 .

The situation with anisotropic restricted diffu-sion is more complicated, and we will illustratethis with reference to diffusion in a cylinder with

Ž .an arbitrary polar angle, u, between the symme-

try axis of the cylinder and the static magneticŽ .field which is also the direction of the gradient

Ž .Fig. 9 . Such a cylinder can be thought of, forexample, as a simplistic model of a muscle-fibercell. The SGP solution for this geometry is given

Ž .by 117

n42 2 2` ` ` Ž . Ž . Ž . Ž .2 K R 2p qR sin 2u 1 y y1 cos 2p qL cos u anm k mŽ .E q , D s Ý Ý Ý 2 22 2 22 2 2 2 2 2ns0 ks1 ms0 Ž . Ž . Ž . Ž .L npRrL y 2p qR cos u a y 2p qR sin u a y mk m k m

2 2a npk m2Xw Ž .x w x= J 2p qR sin u exp y q DD 110m ž / ž /½ 5R L

where L is the length of the cylinder, R is theradius of the cylinder, and a is the k th nonzerok m

X Ž .root of the equation J a s 0, where J is them k mBessel function of the first kind, and the constantK depends on the values of the indexes n andnmm according to

K s 1 if n s m s 0nm

K s 1 if n / m s 0 or m / 0 and n s 0nm

w xK s 1 if n , m / 0. 111nm

Now, the mathematical complexity is no concernto us here and the point that we wish to empha-size is the u dependence; thus, in contradistinc-tion to the cases of free diffusion and diffusionwith a sphere, in an anisotropic system the spin-echo attenuation is now a function of the direc-tion of the gradient. In fact, if we had a less

Ž .symmetric geometry e.g., an elliptic cylinder ,Ž .then the equation for E q, D should also be

dependent on the azimuthal angle, w. If we setw xu s 0, then the solution given by Eq. 110 re-

duces, as expected, to the solution for diffusionŽ w xbetween planes i.e., Eq. 92 and noting that

. w xL s 2 R . Similarly, if u s pr2, Eq. 110 reducesŽ .to the solution for diffusion in a cylinder 110 ,

Ž .E q , D` `

2Ž .s 2 2p qR Ý Ýks1 ms0

2 2X2 w Ž .x Ž .K a J 2p qR exp y a rR DD� 40 m k m m k m.222 2 2Ž . Ž .a y 2p qR a y mk m k m

w x112

The echo-attenuation curves for diffusion in acylinder versus D are plotted for three differentvalues of u in Fig. 10. The long time limiting

Ž .formula for the cylinder is given by 117

Ž .E q , `

22 w Ž .x w Ž .x8 R 1 y cos 2p qL cos u J 2p qR sin u1s .4 22Ž . Ž .2p qR L cos u sin u

w x113

When u s 0, this, of course, reduces to the longtime for diffusion between planes as given in Eq.w x Ž .93 n.b. L s 2 R , and when u s pr2, this re-

Ž .duces to 117

2w Ž .x2 J 2p qR1Ž . w xE q , ` s . 1142Ž .2p qR

The echo-attenuation curves for diffusion in acylinder versus qR are plotted for three differentvalues of u in Fig. 11.

Clearly, the u dependence on the attenuationcurves and the diffraction patterns}or alter-nately, we can think of this as the orientation ofD with respect to the gradient}provides an addi-tional structural probe. If the restricted diffusioneffects are not accounted for and free diffusion is

w xassumed, and Eq. 104 , which is valid only forfree diffusion, is used to analyze the attenuationdata, then D9 is really an apparent diffusion ten-


sor, DX . When a single effective diffusion tensora p pis estimated for the entire sample, it is sometimesreferred to as ‘‘diffusion tensor MR spectroscopy,’’and when, as is commonly the case in imagingstudies, the estimation is performed for eachvoxel, it is referred to as ‘‘diffusion tensor MRimaging.’’ From the discussion on restricted dif-fusion above it should be clear that DX is per-a p p

X Ž .haps better written as D D , since it will bea p pobservation time dependent. For sufficiently short

Figure 9 Schematic diagram of diffusion in a cylinder.The cylinder is of length L and radius R with the

Ž .symmetry axis of the cylinder ] ] ] subtends an angleu with the direction of the gradient and static field,which is taken to be z in the present case. The labora-

Ž .tory or gradient frame is given by x, y, z , where zcoincides with the gradient direction. The director

Ž .frame for the cylinder is given by x9, x9, z9 , where z9coincides with the symmetry axis of the cylinder. If thiswere an elliptic cylinder for example, the director framewould be uniquely determined. Clearly, if u s 0, thetwo reference frames coincide. If u s 0 and a PFGdiffusion measurement is performed, the spin-echo at-tenuation will be described by diffusion between planar

Ž w x.boundaries i.e., Eq. 92 . Conversely, if u s pr2, thespin-echo attenuation will be described by diffusion

Ž w x.within a cylinder Eq. 112 .

values of D such that the diffusion of the probeX Ž .species is unaffected by the boundaries, D Da p p

would be isotropic, whereas for larger D, it be-comes increasingly anisotropic. This can be easilyvisualized from the divergence of the echo-signalattenuation plots for different values of u versusD given in Fig. 10. For the case of diffusion within

Ž . Xa cylinder Fig. 9 , D would be given bya p p

XD 0 0x xXX 0 D 0 w xD s 115y ya p p

X0 0 Dz z

where DX s DX owing to the axial symmetry ofx x y y

the cylinder. We remark that DX , DX , and DXx x y y z z

are, of course, eigenvalues of the matrix DX anda p p

are often termed the ‘‘principal diffusivities.’’ Ide-ally, it is possible to determine the dimensions ofthe restricting geometry from the restricted dis-placement}for example, in the case of the cylin-der, the long axis of the cylinder gives the largestapparent diffusion coefficient}but this requiresprior knowledge of the sample orientation so thatit is possible to align the director frame of refer-ence coincident with the gradient frame of refer-ence in the PFG experiment. Sometimes it isuseful to represent DX graphically by a diffusiona p p

Ž . Ž .ellipsoid 45 Fig. 12 , which can be constructedusing

22 2x9 y9 z9

q q s1.XX Xž / ž /ž /2 D D2 D D 2 D D' '' y yx x z z

w x116

w xEquation 116 can be derived starting from Eq.w x X27 , but substituting the D for the scalar D.a p p

w xWe note from Eq. 33 that the major axes of thew xellipsoids in Eq. 116 are the mean diffusionX 2 X'Ž ² : .distances i.e., x s 2 D D , etc. . As D' x x a p p

becomes more anisotropic the ellipsoid becomesmore prolate. For example, in muscle fiber, theeffective diffusion ellipsoid reflects the fiber ori-entation and the mean diffusion distances.

Generally, the relative alignment between thegradient and director frames is not known, thediffusion measurement returns a mixture ofthe different elements of the diffusion tensor, andthe orientational dependence becomes a problem.

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Figure 10 A plot of the simulated echo attenuation for PFG diffusion measurements in acylinder with the cylinder oriented at three different polar angles with respect to the

Ž . Ž . Ž .gradient, i.e., u s 0 }} , u s pr4 ]?]?] , and u s pr2 ] ] ] ] versus D calculatedw x w x w xusing Eqs. 92 , 110 , and 112 , respectively. Also shown is the result of a power distribution

Ž . w x w xof polar angles ?????? versus D calculated using Eqs. 110 and 119 . The parameters usedin the simulation were the same, as far as possible, as those used for the sphere in Fig. 7, i.e.,d s 1 ms, D s 5 = 10y10 m2sy1, g s 1 T my1, R s 8 mm, L s 24 mm, and g s g

1H s2.6571 = 108 rad Ty1sy1. The effects of the polar angle can be clearly seen on the

Žattenuation curves, and the curves go through three stages depending on D this is most. Ž .obvious in the u s pr2 case , similar to the results for diffusion in a sphere see Fig. 7 . As

would be expected, since it is an average over all possible polar angles, the powder averageecho-attenuation curve is between the limits of the attenuation curves for u s 0 andu s pr2. Because of the axial symmetry of the cylinder, it was unnecessary to average over

w xw. However, the normalization factor in Eq. 119 was changed appropriately.

For example, it makes it difficult to compare thediffusional characteristics of one sample to an-other. A solution is to determine D9 itself bymeasuring the diffusion coefficients in seven dif-

wferent directions i.e., except for the case ofŽ .charged moieties, D9 is symmetric 118 and so

xthere are only six independent elements . How-ever, because of experimental imprecision it isnormal to perform a much larger number ofmeasurements and determine D9 statisticallyŽ . w x114 . As seen in Eq. 108 , the use of a singlegradient direction in a diffusion measurementallows the diagonal elements of D to be probed.The off-diagonal elements can be probed by ap-plying gradients along various oblique directionsŽconsider the pulse sequence shown in Fig. 2, butwith the possibility of gradients along all three

.Cartesian directions . For example, if gradients

were applied along all three Cartesian directions,then the echo attenuation would be described byŽ w x.i.e., Eq. 105

Ž .ln E s y b D q b D q b Dx x x x y y y y z z z z

Ž . Ž .q b q b D q b q b Dx y y x x y x z z x x z

Ž . w xq b q b D . 117y z z y y z

As a further complication, the restricting ge-ometries may not all be uniformly aligned in the

w Ž .xsame direction e.g., brain white matter 119 orŽeven randomly aligned e.g., a suspension of red

.blood cells . The apparent D9 is then an averageof the different orientations. Measuring diffusionin three orthogonal directions so as to determinethe trace of the diffusion tensor has been pro-


Figure 11 Plot of the simulated echo attenuation for PFG diffusion in a cylinder with thecylinder oriented at three different polar angles with respect to the gradient, i.e., u s 0Ž . Ž . Ž . w x w x}} , u s pr4 ]?]?] , and u s pr2 ] ] ] versus qR calculated using Eq. 92 , 110 ,

w xand 112 , respectively. Also shown is the result of a powder distribution of polar anglesŽ . w x w x?????? versus qR calculated using Eqs. 110 and 119 . The parameters used in thesimulation were the same, as far as possible, as those used for the sphere in Fig. 8, i.e.,d s 1 ms, D s 1 = 10y9 m2sy1, g s 1 T my1, R s 8 mm, L s 24 mm, and g s g

1H s2.6571 = 108 rad Ty1sy1. The effects of the polar angle can be clearly seen on theattenuation curves and the position of the diffraction minima. The behavior of the diffrac-tive minima is quite different from that found for the sphere in Fig. 8.

posed as a means of overcoming the anisotropyŽ .problems 45, 120

1 1 X X X 1Ž . Ž . Ž .Tr D9 s D q D q D s Tr Dx x y y z z3 3 3

1 Ž . w xs D q D q D sD . 118x x y y z z a¨3

As the trace is invariant under rotations, theŽorientational dependence is removed i.e., from

w xEq. 118 , the trace of the diffusion tensor in thedirector or cell frame equals the trace of the

.diffusion tensor in the gradient frame . For com-pletion, we should note that where the diffusiontensor is observation time dependent, as it is inthe case of restricted diffusion, and under someexperimental circumstances, the measured tracecan differ from the true trace of the effectivediffusion tensor and the measured quantity is not

Ž .completely rotationally invariant 121 .Generally, however, diffusion is only measured

in one direction, and if the anisotropic system isnot oriented in one direction, it is necessary to

perform a powder average. In performing thepowder average, it is mathematically equivalent toconsider that there is only a single domain with adefined direction and that it is the field gradient

Ž .randomly oriented 122 ; thus,

Ž .E g, D p ow der

p1 2p Ž . w xs E g, D , u, w sin u du dw 119H H4p 0 0

Ž .where 1r4p sin u du dw is the probability of gbeing in the direction defined by u and w, and we

Ž .have written E g, D, u, w to emphasize the orien-tational dependence of the attenuation. The pow-der average of the echo attenuation due to diffu-

Ž w x.sion in a cylinder i.e., Eq. 110 is plotted againstD and qR in Figs. 10 and 11, respectively. Thesituation is much more complicated, though, if inthe time scale of D the diffusing molecules change

Ž .from one domain e.g., exchange between cells ,specified by a unique local director orientation,

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Ž .into another 123, 124 . Exact solutions to Eq.w x119 are known only for some very simple casesw Ž .x w xe.g., Refs. 41, 124, 125 and generally, Eq. 119

Ž .must be evaluated numerically 122, 124 .

CONCLUDING REMARKS

Pulsed-field gradient experiments provide astraightforward means of obtaining informationon the translational motion of nuclear spins.However, the interpretation of the data is compli-cated by the effects of restricting geometries andthe mathematical modeling required to accountfor this becomes nontrivial for anything but thesimplest of geometries. Generally, we have toresort to numerical methods andror approxima-tions to model diffusion within restricted geome-tries, and the type of approximation that wechoose should be consistent with our experimen-

tal conditions. For example, to use the SGP ap-proximation we must ensure that the conditiond < D holds.

In the present article we have presented theunderlying concepts of how PFGs may be used tomeasure diffusion. The mathematical modelingrequired to extract information from the attenua-tion of the echo signal on the diffusion processand structural information in restricting geome-tries was presented in some detail, and bothisotropic and anisotropic systems were consid-ered. However, the experimental aspects andcomplications were largely ignored. Further, wepresented only simple examples of restricting ge-ometries and have barely mentioned any of themany applications that PFG NMR can be appliedto such as measuring polymer dynamics, obtainingdiffusion and structural information in porousmedia with more complicated restricted geome-tries and measuring exchange.

w xFigure 12 Example of an effective diffusion ellipsoid calculated using Eq. 116 . Theparameters used in the simulation were D s 20 ms, D s 0.6 = 10y9 m2sy1, D s 1.2 =x x y y10y9 m2sy1, D s 1.5 = 10y9 m2sy1. The extremely anisotropic diffusion parameters werez zchosen to allow easy visualization of the ellipsoidal shape.


Experimental aspects of PFG NMR will bepresented in Part II of the series.

APPENDIX

Maple Worksheet for the Stejskal andTanner Equation

Define the integral used in determining F

( ) ( )> F:= g, ti ª int g, td = ti..t ;

tŽ .F [ g , ti ª g dtdHti

Define the time intervals and the relevant valueof g for each integral. Also calculate the value ofF for each interval remembering that it containsthe contribution from all of the intervals from thestart of the pulse sequence.

> l1:= 0;> g1:= 0;

( )> F1:= F g1, 11 ;

l1 [ 0g1 [ 0

F1 [ 0

> l2:= t1;> g2:= g;

( ) ( )> F2:= subs t = l2, F1 + F g2, l2 ;

l2 [ t1g2 [ g

F2 [ tg y t1 g

> l3:= t1 + delta;> g3:= 0;

( ) ( )> F3:= subs t = l3, F2 + F g3, l3 ;

l3 [ t1 q d

g3 [ 0

Ž .F3 [ t1 q d g y t1 g

> l4:= t1 + Delta;> g4:= g;

( ) ( )> F4:= subs t = l4, F3 + F g4, l4 ;

l4 [ t1 q D

g4 [ g

Ž .F4 [ t1 q d g y 2 t1 g q tg y gD

> l5:= t1 + Delta + delta;> g5:= 0;

( ) ( )> F5:= subs t = l5, F4 + F g5, l5 ;> l6:= 2* tau;

l5 [ t1 q D q d

g5 [ 0

Ž . Ž .F5 [ t1 q d g y 2 t1 g q tl q D q d g y gD

l6 [ 2t

w Ž .xDefine the function ‘‘ f ’’ s F tau

( )> f:= subs t = tau, F3 ;

Ž .f [ t1 q d g y t1 g

Define the integral of F between tau and 2* tau

( ) ( )> FINT:= int F3, t = tau..l4 + int F4, t = l4..l5( )+int F5, t = l5..l6 ;

1 2FINT [ ygdt1 y gdD q 3t gd y gd2

Define the integral of F n2 between 0 and 2* tau

( )> FSQINT := int F1^2, t = l1..l2( )+int F2^2, t = l2..l3( )+int F3^2, t = l3..l4( ) ( )+int F4^2, t = l4..l5 + int F5^2, t = l5..l6 ;

7 2 3 2 2FSQINT [ y g d y 3g d D3

q 8t g 2 d2 y 4 g 2 d2 t1

Define the function to give the Stejskal and Tan-ner relationship and simplify the result.

( ) ( (> ln E := simplify ygamma 2*D* FSQINT y) )4*f*FINT + 4*f^2*tau ;

1 2 2 2Ž . Ž .ln E [ g Dg d d y 3D3

ACKNOWLEDGMENT

Dr. A. V. Barzykin, Dr. K. Hayamizu, and Dr. P.van Gelderen are thanked for critically readingthe manuscript and their valuable suggestions.The author is also very grateful for the verydetailed comments provided by the referees.

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William S. Price received his B.Sc. andŽ .Ph.D. Biochemistry degrees from the

University of Sydney in 1986 and 1990,respectively. His Ph.D. studies were un-der the supervison of Professor Philip W.Kuchel and Dr. Bruce A. Cornell. He didpostdoctoral study at the Institute ofAtomic and Molecular Science in Taipei,

Ž .Taiwan 1990�1993 with Professor Lian-

Pin Hwang and at the National Institute of Material andŽ .Chemical Research in Tsukuba, Japan 1993�1995 with Dr.

Kikuko Hayamizu. In 1995 he joined the research staff at theWater Research Institute in Tsukuba, Japan and presentlyholds the position of Chief Scientist. His interests focus on theuse of NMR techniques such as Pulsed Field Gradient NMR,NMR microscopy, spin relaxation, and solid state 2 H NMR tostudy molecular dynamics in chemical and biochemical sys-tems.

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