Prediction of proton relaxation rates from measurements of deuterium relaxation in aqueous systems

JOURNAL

OF MAGNETIC

RESONANCE

83,246-25

1 ( 1989)

Prediction of Proton Relaxation Rates from Measurements of Deuterium Relaxation in AqueousSystems J. C. GORE, M. S. BROWN, J. ZHONG, Departments

ofDiagnostic Radiology and *Molecular School ofh4edicine, 333 Cedar Street,

AND I. M. ARMITAGE

Biophysics New Haven,

*

and Biochemistry, Yale University Connecticut 06510

Received June 27,1988; revised October 26,1988 Spin-lattice relaxation for deuterium in aqueous solutions of macromolecules is dominated by quadrupolar interactions so that the rotational correlation time for DzO and HDO may be determined by measuring the deuteron spin-lattice relaxation time T, This correlation time may then be used to predict the dipole-dipole contribution to proton relaxation from rotational intramolecular water interactions. By subtracting these calculated values from the directly measured proton relaxation rates, the contribution to proton relaxation due to translation and intermolecular dipolar effects may be found. In solutions of polyethylene glycol the translational contribution was estimated to be a constant 40% ofthe overall relaxation rate when the deuteron relaxation rate varied fourfold, consistent with the view that the translation correlation time varies in direct proportion with the rotational time. It is proposed that deuteron relaxation measurements thus offer a novel method for estimating the hydrodynamic contribution to relaxation due to water-water interactions only in solutions of macromolecules and tissues, separate from the effects of cross relaxation, and these measurements are thus useful for understanding the relative importance of different proton relaxation mechanisms in such media. 0 1989 Academic Press, Inc.

The relative importance of different mechanisms that may contribute to proton relaxation in aqueous solutions of proteins or tissues has not been accurately quantified although the mechanisms have a fundamental role in determining contrast in NMR images, and potentially as sensitive probes of tissue structure. In pure water, proton relaxation is dominated by the intramolecular dipole-dipole interactions in which the dipolar field from each proton within each molecule is modulated by the rotational correlation time tR of the water molecule ( I ) . Self-diffusion of water molecules also introduces a translational contribution due to intermolecular interactions that has been estimated to account for 35% of the observed relaxation rate (2). In heterogeneous systems such as protein solutions and tissues, other mechanisms may contribute to the water relaxation, particularly intermolecular cross relaxation between water protons and macromolecular protons. For a better understanding of the factors that determine relaxation in complex media it would be useful to be able to separately quantitate the magnitude of each process. In the presence of macromolecules the effective average correlation times for the translation or rotation of water molecules are lengthened so the intramolecular water (rotational) and intermolecular water-water (translational) dipole terms are increased. Such hydrodynamic effects should be distinguished from direct cross relaxation at the macromolecular 0022-2364189

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246

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surface, and in this report we describe a novel method for estimating the effects of solute on the bulk water molecule motions only. By replacing water protons with deuterons and measuring the deuterium relaxation rate, the rotational correlation time tR can be directly estimated. Deuterium relaxes by quadrupolar relaxation only and does not exchange magnetization with macromolecules in solution. Using the value of tR derived, the proton relaxation rate can be calculated from the appropriate dipole-dipole equations and this value may then be compared with the value obtained experimentally. This concept has been evaluated by studying solutions of polyethylene glycol (PEG), and the results demonstrate that the intramolecular water relaxation accounts for a constant 60% of the total rate, and the calculated rate contribution from translation varies in direct proportion to the rotational term. THEORY

Deuterium has The proton dipole ation rate in pure relaxation rate for

spin-l and thus possesses both dipole and quadrupole moments. magnetogyric ratio is about 6.5 that of deuterium, and the relaxD20 due to dipolar interactions is expected to be lower than the protons in pure H20 by a factor

if small changes in viscosity and density are ignored ( 2 ) . The contribution to the relaxation rate of deuterons in HDO due to D-H dipolar interactions would still be less than the rate for protons in H20 (3) by a factor

These examples show that dipolar relaxation is not important for deuterium. As noted by Abragam ( 1)) the coupling of the nuclear quadrupole moment with the fluctuating electric fields that exist in the liquid state is almost always the main relaxation mechanism for deuterons. This quadrupolar relaxation mechanism accounts for the measured deuterium relaxation time value r, = 0.41 s in pure D20. The fluctuating electric fields are modulated by the rotation of water molecules, the same motions which modulate the proton intramolecular dipole-dipole coupling (3). The spin-lattice relaxation time r, for nuclei with a quadrupole moment, in the extreme narrowing limit, is given by 1 3 2Z+3 -=-. T1 40 Z2(2Z- 1) where Z is the spin of the nucleus, 7 is the asymmetry factor, tR is the rotational correlation time, and QCCis the quadrupole coupling constant. For deuterium, Z = 1 and 17 is of order 0.1 so to a good approximation we can write

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ET

AL.

The quadrupole coupling constant is 230 kHz or 1.445

X

lo6 rad/s (4). This gives

[II which is the expression used to calculate the The protons in HZ0 relax mainly via (modulated by rotational tumbling) and (modulated by translation). The spin-lattice by +-(rot) I

= :%I(]+

water rotational correlation time. the intramolecular dipole interaction the intermolecular dipole mechanism relaxation time due to rotation is given l)fR,

where b is the distance between the spins in the molecule. Using the value b = 1.58 A (2) and for spin-i, we obtain +-(rot)=5+5279X I

10”XtR.

[21

Thus we can predict that the rotational rate for protons is 0.0705 the rate of deuterium in solution. The total proton relaxation rate will be given by 1 -=+-(rot)++(trans). TI 1 1 Various models have been analyzed in order to derive expressions for the translation term. Torrey (5) considered the random walks of hard spheres, and this model was refined by later workers such as Hubbard (6) and Zeidler ( 7). However, these models are at best only qualitative and the expressions derived rely on a knowledge of several parameters, such as the coefficient of self-diffusion, that may vary in solutions of macromolecules. In consequence, in order to determine the intermolecular contribution to proton relaxation, we make use of Eqs. [ 21 and [ 31, The total relaxation rate for protons may be measured directly and the translation term found by subtracting the intramolecular contributions. The ratio of translational and rotational contributions can thus be estimated for different concentrations of solute. These concepts have been evaluated in a simple model system as described below. METHODS

Polyethylene glycol (MW 8000, Sigma Chemical Co.) was dissolved in doubly distilled water to form solutions of concentration O-2 g/ml. Equivalent solutions were made using D20-enriched water ( 5- 10% D20). For the more concentrated solutions it was necessary to warm the mixtures to dissolve the polymer completely, but the polymer then remained in solution at room temperature. Relaxation studies were performed on a Bruker CXP-200 spectrometer of field strength 4.7 T operating at 200 MHz for protons and 30 MHz for deuterons. Inversion-recovery pulse sequences with 15-20 delay intervals were used for the determination of the spin-lattice relaxation times T, for both deuterium and water protons. The relaxation curves were fit to a single exponential curve using a three-parameter fitting routine. In all cases the relaxation curves showed no evidence of multiexponential behavior. Samples were

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run both with and without degassing to determine whether there were any effects of dissolved oxygen. No measurable differences outside of normal experimental errors were found in either the proton relaxation or the deuteron relaxation. Measurements of the bulk viscosity of the polymer solutions in water were performed using Ostwald viscometers and a pycnometer. RESULTS

Table 1 gives the measured deuterium relaxation times and water proton relaxation times for PEG solutions of several different concentrations. Table 1 also shows the rotational correlation times calculated using Eq. [ 1 ] and the predicted proton spinlattice relaxation rates due to rotation obtained using Eq. [ 2 1. Subtraction of the predicted rotational rates from the measured total rates gives the estimated translational rates, and Table 1 shows these along with the calculated ratios of the translational to rotational contributions. The mean ratio is 0.67 ( z!zO.12). Figure 1 is a plot of measured proton relaxation rates vs calculated proton rates assuming a constant ratio of rotational to translational contributions of 0.67. The figure illustrates good agreement between the predicted and the observed values with this assumption. Figure 2 shows the nonlinear relationship of the measured D20 relaxation times vs the bulk viscosity of the PEG solution. CONCLUSIONS

These measurements show that it is indeed feasible to use deuterium relaxation times as a probe to predict the effect of solute molecules on intramolecular waterwater proton relaxation rates. The theory used in this work is well known but has not been much used to formally demonstrate the compatibility of deuterium and proton relaxation measurements. No correction has been made for the different viscosities of D20 and H20. Pure D20 has a viscosity that is 23% greater than that of H20, but at 5- 10% deuteration, TABLE 1 Relaxation Rates for Protons and Deuterium at Different Concentrations of PEG Concentration PEG (g/ml) 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

T, for ‘H measured (4 0.41 + 0.02 0.37 kO.02 0.31 -t 0.02 0.27 kO.3 0.199 + 0.018 0.164 kO.015 0.141 kO.020 0.125 k 0.013 0.113 kO.028 0.105 f 0.073 0.09 15 + 0.0025

calculated (4

~/TI (rot) for ‘H calculated (s-l)

1/T, (total) for ‘H measured W’)

l/T, (tram) for ‘H calculated W’)

TI (rot) T, (trans)

3.11 3.45 4.12 4.73 6.42 7.79 9.06 10.22 11.30 12.16 13.96

0.172 0.191 0.228 0.262 0.355 0.43 I 0.500 0.565 0.625 0.676 0.792

0.275 + 0.006 0.334 + 0.006 0.408 + 0.010 0.490 f. 0.007 0.562 ?I 0.012 0.671 + 0.013 0.813 k 0.031 0.917 kO.016 0.980 + 0.045 1.099 -c 0.046 -

0.103 0.143 0.184 0.228 0.207 0.240 0.313 0.352 0.355 0.423 -

0.599 0.753 0.792 0.872 0.584 0.557 0.626 0.624 0.569 0.626 -

tRx 10’2

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GORE ET AL.

I

0

I

2

CALCULATED

3

4

T1 (WC)

FIG. 1. Plot of proton T, measured vs the calculated values obtained from deuterium relaxation studies and assuming T, (trans)/ T, (rot) = 1S. The regression line is T, (measured) = 0.96T, (calculated) + O.O76(R = 0.99).

only a l-2% change in viscosity would be expected. This can easily be accommodated into the calculation of proton intramolecular rates (8) but the results of Table 1 do not include this small correction. When incorporated, T, (rot)/ T, (trans) has an average value of 0.69 f 0.10. An important conclusion is that it is reasonable to assume equal effects of solute molecules on both the translational and the rotational contributions to dipolar effects, so that the ratio of their rates can be assumed approximately constant over a wide range of solute concentrations and viscosities. Thus a measure of the deuterium relaxation rate alone can be used to give a reasonable indication of the total hydrody-

.40 .

.35 if

'.30 F 6 6 2 x

-

. .

.25

-

.20 -

. .

.I5

-

. .

0 0

I 50

I 100 VISCOSITY

I 150

.

200

250

(centipoise)

FIG. 2. Plot of deuteron T, of D,O in polyethylene glycol solutions vs measured macroscopic viscosity.

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namic effects of macromolecules in solution, which would be a useful first step in quantifying the relative contributions of different mechanisms in complex systems. Rotational effects are 50% greater than translational effects, which thus constitute only 40% of the total rate. These results are in good agreement with other theoretical predictions (2). REFERENCES 1. A. ABRAGAM, “Principles of Nuclear Magnetism,” p. 298, Oxford Univ. Press, Oxford, 196 1. 2. A. CARRINGTON AND A. D. MCLACHLAN, “Introduction to Magnetic Resonance,” p. 193, Harper & Row, New York, 1967. 3. J. MCCONNELL, “The Theory of Nuclear Magnetic Relaxation in Liquids,” p. 145, Cambridge Univ. Press, Cambridge, 1987. 4. H. H. MANTSCH, H. SAITO, AND I. C. P. SMITH, Prog. NMR Spectrosc. 11,2 11 ( 1977). 5. H. C. TORREY, Phys. Rev. 92,962 ( 1953). 6. P. S. HUBBARD, Phys. Rev. 131,275 (1963). 7. M.D. ZEIDLER,MOI. Phys. 30,1441(1975). 8. J. ZHONG, J. C. GORE, AND I. M. ARMITAGE, Magn. Res. Med., in press.