204 Bulletin of Magnetic Resonance

MRI Measurements of Paramagnetic Tracer Ion DiffusionBruce J. Balcom*, Alan E. Fischer, T. Adrian Carpenter and Laurence D. Hall

Herchel Smith Laboratory for Medicinal ChemistryUniversity of Cambridge School of Clinical Medicine

Cambridge CB2 2PZ, U. K.* Current Address: MRI Centre, Department of Physics

The University of New BrunswickP.O. Box 4400

Fredericton, New Brunswick, Canada, E3B 5A3

Contents

I. Introduction

II. Experimental

III. Theory

IV. Results

V. Conclusion

VI. Acknowledgments

VII. References

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I. Introduction

Molecular diffusion is of fundamental importancein determining the mass transport properties of fluidsaturated porous media (1). Measurement of thediffusion coefficient in such a system is a powerfulmethod to probe its microscopic structure (1). Wehave recently developed a quantitative MRI methodto extract the mutual diffusion coefficient of param-agnetic tracers as they penetrate aqueous gels (2,3).An introduction to the method, and its application,is the purpose of this article.

Quantitative measurements of a molecular prop-erty such as concentration can be difficult to quan-tify in an MRI image due to the large number ofcontrast generating mechanisms present in a sample.These mechanisms, principally local variations indynamics of the system, alter the underlying NMRrelaxation times T\ and T2. In addition the protondensity, for 1H imaging, and its spatial variation di-rectly influence the observed image. Traditionally,quantitative NMR imaging required acquisition of

a series of NMR images with different acquisitionparameters in order to determine the local protondensity and relaxation times (4). This approachcan be very time consuming and may still result ina relaxation time image which is not interpretablein terms of the property of interest. To analyzethe diffusion equation one must quantitatively de-termine the spatial variation of concentration of atracer, in the medium of interest, at a fixed time.Alternatively one can track a chosen concentrationas it moves through the sample as a function of time(5). Regardless of the approach chosen, analyticalinformation on concentration as a function of po-sition is required in what may be a rather compli-cated molecular system. The null point techniqueexploited in our method localizes a chosen para-magnetic concentration directly, and in particularminimizes the influence of T^ on image intensity.

In order to more fully understand and explorethe method we begin with a simple phantom and se-quence then move to a more complicated phantom

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a)Reservoir

Gel

b)

*• *

Reservoir

Gel

Figure 1: Sample geometry for MRI null pointimaging experiments, (a) One-dimensional sample.Paramagnetic tracers diffuse into the gel from thereservoir, (b) Two-dimensional sample. The para-magnetic tracer diffuses radially from the axial voidinto the gel.

and sequence. We take as validation of the morecomplicated sequence, which is the more generallyuseful, reproduction of the diffusion coefficient mea-sured with the simple phantom and sequence. Wevalidate the results of the simple system by compari-son to results in the literature. We believe a system-atic approach such as this is necessary, in general,to validate quantitative information from MRI mea-surements. As many authors have discovered, it isrelatively easy to misinterpret MRI data.

Several other groups have recently reported diffu-sion coefficient measurements of paramagnetic trac-ers in porous media. (6,7,8) The methods employedby these groups are more restricted in their appli-cation, principally because of the influence of T<i onthe observed signal.

II. Experimental

All MRI measurements employed an Oxford Re-search Systems Biopsec 1 console interfaced to anOxford Instruments 2T, 31 cm horizontal bore su-perconducting magnet. The gradient set was 20 cmin diameter, homebuilt, and based on Helmholtz-Golay coils. Gradients were driven by duplexedsets of Crown 7570 amplifiers to a maximum of 12kHz/cm. The radio frequency probe was a split

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ring resonator (9) driven by an Amplifier Research150LA amplifier. All measurements were made atambient temperature.

One-dimensional experiments were performed oncylinders of polyacrylamide gel and agarose pre-formed in perspex tubes. Sharp edges at the inter-face with the tracer solution were ensured by cuttingthe gel with a sharp blade or wire. Silica gel sam-ples were prepared by stirring a mixture of silica andwater, contained in the diffusion vessel, then allow-ing the mixture to settle under gravity. A detaileddescription of sample preparation may be found inBalcom et al. (2) Diffusion experiments commencedwith the addition of a finite volume of the param-agnetic agent in solution added to the surface of thegel, Figure la. The volume of solution added to thegel was sufficient to prevent depletion of the reser-voir concentration over the course of the experiment(2).

One-dimensional MRI experiments (profiles)used the pulse sequence illustrated in Figure 2a. Atypical experiment employed an 8000 Hz/cm readgradient applied parallel to the long axis of the sam-ple. With a spectral width of 42 kHz, this gave afield of view of 5.2 cm and a spatial resolution of 0.2mm for 256 time domain points. The echo time r ewas 10 ms For a reservoir concentration, Co,of 15mM copper sulfate, the null delay r^ for a trackedconcentration of 2.0 mM was 325 ms. A repetitiontime of 40 s between each of 128 profiles gave anexperiment time of approximately 90 min.

Two-dimensional experiments were performed onaqueous gels of polyacrylamide and agarose pre-formed as right cylinders in a perspex former. In-sertion of a 5 mm NMR tube into the gel precursorsolution gave a well defined axial void in the gelupon setting of the gel and removal of the NMRtube, Figure lb. Radial diffusion commenced uponlongitudinal flow of paramagnetic agent through thisvoid. Radial diffusion in a thin slice from the middleof the phantom, Figure lb, was followed as a func-tion of time. A sufficient flow rate was maintainedto prevent depletion of the concentration of para-magnetic agent in the void during the experiment(3).

Two-dimensional MRI experiments employed afast inversion recovery imaging sequence illustratedin Figure 2b. In a typical experiment with spec-tral width 42 kHz, a read gradient of 10,800 Hz/cm

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a)Tx/Rx

180° 90° 180°

I—I I AGread

b)Tx/Rx

Gread

Jslice

Gphase

Figure 2: Null point imaging sequences, (a) One-dimensional inversion recovery imaging sequence.The read gradient is collinear with the long axis ofthe sample, (b) Two-dimensional FIR imaging se-quence. This sequence is related to (a), but withthe addition of a slice selective 90° pulse, a phaseencode gradient to encode the second spatial dimen-sion, and a delay W between transients which is lessthan 5 times T\.

gave a field of view of 3.7 cm and spatial resolutionof 0.15 mm for 256 time domain points. Phase en-coding was carried out in 128 discrete steps. The256x128 data matrix was zero filled in the phaseencode direction to yield a symmetric image matrixwith field of view and resolution equivalent in eachdimension. The echo time r e was 10 ms and slicethickness was 0.9 cm. For a reservoir concentration,Co of 15.0 mM copper sulfate a delay W 0.46 s andTd 0.10 s nulled 4.0 mM copper sulfate. With twostep phase cycling, image acquisition required 150s. Successive images were typically acquired every180 s.

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III. Theory

The key feature which underpins the methodis the linear relationship which exists between theinverse 7\ of water protons in an aqueous mediumand paramagnetic concentration, eqn. 1 (10).

bC (1)

The concentration of the paramagnetic tracer isC. The intercept of such a plot, a, is media depen-dent while the slope, 6, is dependent on the param-agnetic contrast agent employed. While the valuesof both a and b can vary widely, in our experiencethe linear relationship of eqn. 1 is operative in awide range of porous media, with a large numberof paramagnetic tracers. For copper sulfate in 10%polyacrylamide gel, the intercept a is 0.509 s"1 andthe slope b is 0.814 s"1 mM"1.

Based on the linear relationship of eqn. 1, an in-version recovery MRI imaging sequence can be em-ployed to localize a chosen concentration C. Theanalytical expression for the signal intensity at apoint in an MRI image, for an inversion recoveryimaging sequence is given by eqn. 2.

S = p exp(-r e /T2)( l - 2exp (2)

where p is proton density, re is the echo time andTd is the delay after a 180° inversion pulse. Exam-ination of eqn. 2 shows that the signal will be zerowhen Td = In 2 I \ , irrespective of the other vari-ables in the equation. This general relation makesthe method very robust and in particular avoids thetroublesome behaviour of T2 which can cloud theinterpretation of image intensity. We note in pass-ing that while the null point is independent of theunderlying sample T2 the null is only observable ifT2 is finite. Simulations (2) have shown that a T2as short as 10 ms still yields a well resolved nullpoint. The null point method is frequently used inspectroscopy to give a rapid estimate of T\ (11).

Our prototype diffusion experiment is one-dimensional free diffusion into a semi-infinite rod,from a constant concentration reservoir (5). Thediffusion equation has a particularly simple solutionwith these boundary conditions, eqn. 3.

(3)

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ouu

1.00

0.80 -

0.60 -

0.40 -

0.20 -

0.000.00 0.10 0.20 0.30 0.40 0.50 0.60

Displacement / (cm)

Figure 3: Simulated diffusion experiment. Free dif-fusion from a constant concentration reservoir into asemi-infinite rod. The simulation assumes D = 3.0x 10~6 cm2 s"1 and times of 200, 1000, 2500 and5000 s. C/Co as a function of x was calculated witheqn. 3.

The concentration C(x, t), normalized by the reser-voir concentration Co, is a function of distance xfrom the reservoir interface, and time t after ini-tiation of the diffusion experiment. The diffusioncoefficient is D.

Figure 3 shows the solution to eqn. 2 at fourdifferent experimental times for a system with D =3.0 x 10~6 cm2 s"1. If the diffusing species is para-magnetic there will exist a corresponding variationof T\ within the sample. The spatial variation of T\is related, by eqn. 1, to the spatial variation in con-centration of the paramagnetic contrast agent. Fig-ure 4 simulates the result of a one-dimensional MRIimage in a 10% polyacrylamide gel with a reservoirconcentration of 15 mM where the delay r^ was cho-sen to null 2 mM copper sulfate. The simulation wasbased on the distribution of copper sulfate shown inFigure 3 with the spatial variation of T\ governedby eqn. 1. The image intensity as a function of po-sition was calculated by eqn. 3, with the first twoterms set equal to one.

IV. Results

Two representative one-dimensional diffusion im-ages are shown in Figure 5. The reservoir (rightside of profile) was 15 mM copper sulfate and the

1.00 -

0.80 -

_ 0.60 -cD)i/5 0.40 -

0.20

0.000.00 0.10 0.20 0.30 0.40 0.50 0.60

Displacement / (cm)

Figure 4: Simulated one-dimensional inversion re-covery null point imaging experiment. The tempo-ral and spatial variation of copper sulfate for eachof the four curves is given in Figure 2. The delay r^was chosen to null 2.0 mM copper sulfate, the reser-voir concentration was 15.0 mM (C/Co = 0.13). Theplot is signal magnitude according to eqn. 2. Notethe symmetry of the signal about the null point.

nulled concentration was 2 mM. The two experimen-tal profiles shown are three and forty one minutesafter addition of the copper sulfate solution to thediffusion cell. Diffusion takes place from right toleft in the profile, from the reservoir into the bulk ofthe gel. The resulting profiles have several charac-teristic features (2). The null point is well resolved,the signal is symmetric about the null point andthe trough about the null point broadens with time.All three features were predicted by the simulationshown in Figure 4. The first observation proves weare able to zero a chosen concentration and the re-sulting null point is clearly identifiable. The spatialvariation of T\ determines how well resolved the nullpoint will be. In practise we are able to work with alarge range of paramagnetic tracer concentrations.The symmetry of the null point and the broadenedtrough are related to the spatial derivative of theparamagnetic tracer concentration. Theory showsthat the spatial derivative of the signal is directlyproportional to the spatial derivative of paramag-netic concentration at the null point. Because ourimages are magnitude images, we loose the sign ofthe signal (which will change at the null point) andthe symmetrical response is therefore merely a re-flection of a constant slope through the zero point.

208 Bulletin of Magnetic Resonance

The spatial derivative of concentration, for a chosenconcentration, increases with time as seen in Fig-ure 3. The trough will therefore broaden with timebecause the spatial derivative of concentration is de-creasing.

To determine the diffusion coefficient, a time se-ries of profiles, usually 128, are acquired and thenull point position plotted against the square rootof time for representative points, Figure 6. Theslope of the resulting plot yields, after simple ma-nipulation the diffusion coefficient D while the in-tercept is the position of the interface between geland reservoir (2). A series of experiments in poly-acrylamide have determined (2) that D for coppersulfate is 3.3 x 10~6 cm2 s"1. In 10% agarose D forthe same paramagnetic probe is 4.8 x 10~6 cm2 s~1

while in silica gel D is 3.5 x 10~6 cm2 s"1. Uncer-tainty in the measurement of D is generally less than10%. Many systems amenable to study by the nullpoint imaging method are difficult to probe exper-imentally via other techniques. Reasonable agree-ment exists, however, between measures of D bythe null point method and measurements on relatedsystems in the literature (2). As expected the diffu-sion coefficient of the paramagnetic in a gel will beslightly less than the free solution value. Copper sul-fate diffuses in solution (14 mM, 25°C) at a rate of6.5 x 10~6 cm2 s^1 (12). Very close agreement existsbetween our measure of D for copper sulfate diffu-sion in agarose and a value of D reported for coppersulfate diffusion in an agar gel, 5 x 10~6cm2s (13).

The one-dimensional imaging method outlinedabove is sufficient to study many model systemswhere one knows the sample has axial symmetry.Clearly however if one wishes to examine more re-alistic systems, one cannot assume or impose ax-ial symmetry. We have therefore developed a two-dimensional analogue to the null point imagingmethod. Eqn. 2 implicitly assumed that the timebetween successive transients was 5 times T\. If werequire 256 transients to build up a two-dimensionalimage of a thin slice, the overall acquisition timecan become prohibitively long with this restriction.Even though diffusion in liquids is recognized to bea slow process, the null point will move a distance onthe order of one pixel (0.2 mm) unless the acquisi-tion is less than two minutes. If the null point movesmuch more than this a characteristic motion artifactis present in the image (3). Fast Inversion Recov-

41 minutes

3 minutes

Figure 5: Experimental one-dimensional null pointimages of copper diffusing into a 10% polyacry-lamide gel. The reservoir is on the right of theprofile. Field of view is 5.2 cm. The nulled concen-tration was 2.0 mM copper sulfate with a reservoirconcentration of 15 mM. Note the symmetry of thenull point and the trough which broadens as timeincreases.

ery (FIR) is a well known method of acceleratingTi measurements (3,15). In FIR the interproceduredelay W is reduced to less than 5 * Ti. Eqn. 2 nolonger applies for FIR imaging because longitudinalmagnetization is not completely recovered betweensuccessive acquisitions. We must take into accountthe finite time W shown in Figure 2b. Figure 2b isthe FIR sequence used to image the null point in astandard porous media paramagnetic ion diffusionexperiment. For the FIR sequence the nulled Tidepends on both the time between acquisitions Wand the inversion delay T&. The relation between anulled Ti and the parameters W and r^ is given byeqn. 4.

Td = Tiln[2 - (4)

Eqn. 4 reduces to the familiar expression T̂ = In 27i when W » T\.

In order to test this sequence we developed a ra-dially symmetric diffusion phantom where the diffu-sion equation no longer has the simple form of eqn.3; but is instead described by eqn. 5 (5)

rooC(r,t)/C0 = 1 + (2/TT) / exp(-Dtu2) Q du

JoJ0{ru) Y0(lu) - J0(lu) Y0(ru)

Q = (5)u(J02(lu) + Y0

2(lu))Where C{r, t) is the concentration of paramagneticmolecule at a given radial point at a given time, Co

Vol. 18, No. 3/4 209

Eo

ca>E<DU<na.10

2.240 60

time1 2! (sec1 2

80 1 0 0

Figure 6: Plot of null point position, from a timeseries of one-dimensional null point images, versusthe square root of time. The uncertainty in eachnull point is ± 1/2 pixel (0.1 mm). The diffusioncoefficient, derived from the slope of this plot (usingeqn. 3 with C/Co = 0.13), is 3.5 x 106 cm2 s"1.

is the centre void concentration of paramagnetic, Dis the diffusion coefficient, t the time after fillingthe centre reservoir, I the diameter of the internalvoid, and r the radial distance from the center ofthe cylinder. The functions Jo and YQ are Besselfunctions of the first kind and u is an integrationvariable. Figure 7 shows a series of images of a thinslice from this radial diffusion phantom. The nullpoint in each image is 4 mM and was chosen bysetting the sequence parameters W and r^ accord-ing to eqn. 4. The images demonstrate that the nullpoint is once again well resolved and that diffusion issymmetric about the internal void as one would ex-pect. As with the one-dimensional experiment, thenull point may be localized from such an image andindeed tracked as it moves with time. The analysisrequired to extract D from this cylindrical system isconsiderably more complicated than our treatmentof eqn. 3. Non-linear regression after a transforma-tion of variables does, however, permit extractionof the diffusion coefficient (3). This diffusion coeffi-cient should be the same as that determined in theone-dimensional experiment for an identical para-magnetic tracer and medium. The diffusion coeffi-cient of copper sulfate in 10% polyacrylamide deter-mined by the FIR imaging method is 3.4 x 10 6 cm2

Figure 7: Experimental two-dimensional null pointimages acquired using the FIR sequence of Figure2b. The image shows a thin cross section of thesample in Figure lb. Gel samples are 2.6 cm in di-ameter. The nulled concentration was 4 mM coppersulfate with a reservoir concentration of 15 mM cop-per sulfate. Experimental times are a) 20, b) 45, c)90 and d) 150 minutes

s"1, which is within experimental error of our earlierone-dimensional measurement (2,3). The null pointobserved in the two-dimensional experiment has adifferent characteristic shape (note the bright sec-ondary ring in Figure 7). This effect occurs due topartial saturation of the long T\ spins at the periph-ery of the sample and is considered in more detail inreference (3). Nevertheless the null point can be ob-served and tracked for a wide variety of ratios CjCo

as in the one-dimensional experiment.

V. ConclusionWe have developed one and two-dimensional null

point imaging methods which permit the analyti-cal determination of the mutual diffusion coefficientof paramagnetic contrast agents in fluid saturatedporous media. This measurement is distinctly dif-ferent from the more familiar NMR pulsed field gra-dient measurement of solvent self-diffusion. The dif-fusing molecule can be any one of a large number of

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paramagnetic transition metals or stable free radi-cals or a larger molecule with the paramagnetic in-dicator chemically bound - for example a proteinlabelled with nitroxyl free radicals. The methodhas been applied to a variety of model gel systemsand we predict it will find increasing use in otherfluid saturated porous media such as food systemsor plants.

VI. Acknowledgments

We wish to thank Herchel Smith for an endow-ment (LDH, TAC) and studentship (AEF); NSERCof Canada for a post-doctoral fellowship (BJB).

Bulletin of Magnetic Resonance

VII. References

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2B. J. Balcom, A. E. Fischer, T. A. Carpenter,L. D. Hall, J. Am. Chem. Soc. 115, 3300 (1993).

3A. E. Fischer, B. J. Balcom, E. J. Fordham, T.A. Carpenter, L. D. Hall, J. Phys. D28, 384 (1995).

4J. Attard, L. Hall, N. Herrod, S. Duce, PhysicsWorld July, 41 (1991).

5J. Crank, The Mathematics of Diffusion; Ox-ford Science: Oxford, (1989).

6G. Guillot, G. Kassab, J. P. Hulin, P. Rigord J.Phys. D 24, 763 (1991).

7Z. Pearl, M. Margaritz, P. Bendel, J. Magn. Re-son. 95, 597 (1991).

8T. Asakura, M. Demura, H. Ogawa, K. Mat-sushita, M. Imanari, Macromolecules 24, 620(1991).

9L. D. Hall, T. Marcus, C. Neale, B. Powell, J.Sallos, S. L. Talagala, J. Magn. Reson. 62, 525(1985).

10A. Abragam, The Principles of Nuclear Mag-netism; Oxford Science: Oxford, (1989).

U J . K. M. Sanders, B. K. Hunter, Modern NMRSpectroscopy; Oxford University Press: Oxford,(1989).

12W. G. Eversole, H. M. Kindswater, J. D. Pe-terson, J. Phys. Chem. 46, 370 (1942).

13R. Lee, F. R. Meeks, J. Colloid Interface Sci.35, 584 (1971).

14H. Hansum, J. Magn. Reson. 45, 461 (1981).