1978
J. Phys. Chem. B 2001, 105, 1978-1983
Mechanisms of the Intermolecular Nuclear Magnetic Relaxation Dispersion of the (CH3)4N+ Protons in Gd3+ Heavy-Water Solutions. Interest for the Theory of Magnetic Resonance Imaging S. Rast,† E. Belorizky,‡ P. H. Fries,*,† and J. P. Travers§ Laboratoire de Reconnaissance Ionique, SerVice de Chimie Inorganique et Biologique (UMR 5046), De´ partement de Recherche Fondamentale sur la Matie` re Condense´ e, CEA-Grenoble, F-38054 Grenoble Ce´ dex 9, France, Laboratoire de Spectrome´ trie Physique, CNRS-UMR 5588, UniVersite´ Joseph Fourier, BP 87, F-38402 Saint-Martin d’He` res Ce´ dex, France, and Laboratoire Physique des Me´ taux Synthe´ tiques, SerVice des Interfaces et des Mate´ riaux Mole´ culaires et Macromole´ culaires, De´ partement de Recherche Fondamentale sur la Matie` re Condense´ e, CEA-Grenoble, F-38054 Grenoble Ce´ dex 9, France ReceiVed: October 6, 2000; In Final Form: December 29, 2000
The relative motion of a tetramethylammonium (CH3)4N+ ion with respect to a Gd3+ octoaqua complex, together with the quantum dynamics of the electronic spin of this lanthanide, is probed by the nuclear magnetic relaxation dispersion (NMRD) of the (CH3)4N+ proton spins. The measured proton resonance frequencies range between 10 and 800 MHz. A pronounced maximum is observed at around 90 MHz. This behavior is interpreted by assuming that the relative diffusion of (CH3)4N+ and Gd(D2O)83+ accounts for their repulsive potential of mean force, calculated with the help of the hypernetted chain approximation for two charged hard spheres in discrete, polar, and polarizable water, and by using a detailed picture of the Gd3+ electronic relaxation, based on an independent electronic paramagnetic resonance study. The standard dipolar nuclear relaxation formalism of Solomon-Bloembergen, valid for the above frequencies, leads to overall good agreement with the experimental data without any adjustable parameters. NMRD experiments using probe solutes of well-known spatial dynamics with respect to a Gd3+ complex, can be combined with the SolomonBloembergen theory to provide an indirect estimate of the longitudinal electronic relaxation time of this complex. This knowledge is useful in the theory of magnetic resonance imaging relaxivity.
1. Introduction In medical magnetic resonance imaging (MRI), the intensity of the NMR signal of protons of spin I ) 1/2, mainly due to water, is measured for spatially encoded volume elements in the body. The image contrast results primarily from the different relaxation rates of water protons in different tissues.1 This contrast can be enhanced by paramagnetic ions carrying electronic spins S that increase the relaxation rates (relaxivity) of the protons of the surrounding water through their dipoledipole interactions. For this purpose, extensive studies have recently been performed using paramagnetic agents such as trivalent lanthanide complexes and mainly monomeric and dimeric Gd3+ (S ) 7/2) complexes2 that can be injected in the form of a stable poly(aminocarboxylate) complex with an acceptable level of toxicity. These contrast agents have a high relaxivity in particular because of the large value of the Gd3+ electronic magnetic moment. A detailed knowledge of the mechanisms that produce relaxivity is essential to the design of new drugs, and the field dependence of nuclear relaxation rates, commonly called nuclear magnetic relaxation dispersion (NMRD), is a privileged technique provid* Author for correspondence. E-mail:
[email protected]. † Service de Chimie Inorganique et Biologique (UMR 5046), De ´ partement de Recherche Fondamentale sur la Matie`re Condense´e, CEA-Grenoble. ‡ Universite ´ Joseph Fourier. § Service des Interfaces et des Mate ´ riaux Mole´culaires et Macromole´culaires, De´partement de Recherche Fondamentale sur la Matie`re Condense´e, CEA-Grenoble.
ing insight into microscopic structural and dynamical properties of the investigated systems. For instance, this technique has shown that relaxivity can be improved by using Gd3+ complexes with slow rotational motion.1 In this paper, we focus our attention on the NMRD of the protons of (CH3)4N+ in D2O solutions of Gd3+ octoaqua ions. For this complex, extensive experimental data are available concerning its square antiprism structure,3 its electron paramagnetic resonance (EPR) line widths,1 the relaxivity of its water protons, 17O-enriched water NMR measurements,1 and the relaxation of protons and 7Li in probe cations such as (CH3)4N+ and Li(H2 O)n+.4 Of particular interest is the work of Dinesen and Bryant4 who gave the NMRD profiles of the (CH3)4N+ protons in the presence of Gd3+ aqua ions for magnetic field strengths ranging between 2.5 × 10-4 T (νI ) 10.6 kHz) and 7 T (νI ) 297.5 MHz). They used a field cycling sequence with a homemade instrument and observed a maximum at around 50 MHz. They tried to qualitatively explain their results by considering that the time fluctuations of the intermolecular dipolar coupling inducing the proton relaxation primarily arise from the short Gd3+ electronic relaxation times. They assumed that the latter are governed by the correlation time τv of the rapid internal fluctuations of the zero-field splitting (ZFS) tensor. Moreover, they noted that the various NMR dispersion curves obtained through various assumptions seemed to converge in the highfield limit.
10.1021/jp003686q CCC: $20.00 © 2001 American Chemical Society Published on Web 02/17/2001
(CH3)4N+ Protons in Gd3+ Heavy-Water Solutions However, in previous studies,5,6 we have shown that, in the high-frequency range 400 MHz e νI e 800 MHz, the ratio τ/T1e between the translational correlation time of the (CH3)4N+ and [Gd(H2O)8]3+ cations, τ ≈ 3 × 10-10 s, and the longitudinal electronic relaxation time of the Gd3+ ion, T1e J 10-8 s, is much lower than ωIτ J 0.4. Consequently, the longitudinal electronic relaxation of Gd3+ in the aqua complex [Gd(D2O)8]3+ does not interfere in the fluctuations of the interspin dipolar coupling in the high-field region. Our values of the intermolecular relaxation rate of the (CH3)4N+ protons could be interpreted with good accuracy within a purely diffusional model without any adjustable parameters. As the field decreases, ωIτ decreases, whereas τ/T1e increases,7 so that τ/T1e cannot be neglected with respect to ωIτ.5 Thus, the NMRD of protons requires a detailed knowledge of the electronic relaxation of the Gd3+ ion. For this purpose, we have undertaken a careful analysis of the EPR line widths of the [Gd(H2O)8]3+ complex performed by the Merbach group at various temperatures and fields.1 We have shown7 that, contrary to the usual assumption, the electronic relaxation is not only due to the effects of the transient zero-field splitting,4,8 but is also strongly influenced by the static crystal field effect in the molecular frame. This static crystal field is modulated by the random Brownian rotation of the complex characterized by a rotational correlation time τR ) 1/DR, where DR is the rotational diffusion constant of the paramagnetic complex. Typically, τR was estimated to be 1.95 × 10-10 s at T ) 298 K. It should be noted that the rotational correlation time that modulates the intramolecular dipolar relaxation is τ2 ) τR/6 ) 32 ps at T ) 298 K. This value is in reasonable agreement with the estimation of 22 ps derived from the Stokes-Einstein formula7 and with previous NMR determinations1,9 ranging between 29 and 45 ps. At the X band (νS ) 9.5 GHz), the static ZFS contribution to the EPR line width is by far dominant, whereas at the 2 mmband (νS ) 150 GHz), the transient ZFS contribution is about 5 times larger than the static one. We were able to interpret the transverse electronic relaxation with fitted crystal field parameters, correlation times, and activation energies, in good agreement with their expected values from the underlying processes. Our model allowed us to deduce the field dependence of the electronic relaxation functions. The longitudinal electronic relaxation function is well approximated using a single relaxation time, whereas the transverse electronic relaxation function is a superposition of four decreasing exponentials. With all of the above results, we tried to interpret the experimental NMRD dispersion curve of Dinesesn and Bryant.4 We could reproduce the main features, in particular the existence of a maximum, but a satisfactory agreement could not be achieved. For this reason, we studied a dilute 0.1 M solution of (CH3)4N+ in D2O containing Gd3+ ions at a concentration of 3.08 × 10-3 M. We performed new measurements of the longitudinal relaxation time of the tetramethylammonium protons for an external field varying between 0.24 and 18.5 T (10 MHz e νI e 800 MHz). The experiments are briefly described in section 2, the theory is provided in section 3, and the results are discussed in section 4. 2. Experimental Section 2.1. Ionic Solution. Tetramethylammonium chloride (CH3)4N+Cl- (Aldrich) was recrystallized in ethanol and dried under vacuum for 24 h at 60 °C. A diamagnetic 0.1 M stock solution of (CH3)4N+Cl- was made in pure heavy water (Eurisotop, 99.8 atom %D, sealed under argon). A 3.08 × 10-3 M concentration
J. Phys. Chem. B, Vol. 105, No. 10, 2001 1979 of [Gd(D2O)8]3+ paramagnetic species was obtained by dissolving a weighed quantity of Gd(NO3)3(H2O)6 salt (Aldrich). To eliminate the paramagnetic oxygen impurities, nitrogen was bubbled through the sample for half an hour. 2.2. NMR Measurements. The proton spin-lattice relaxation time, T1, of the tetramethylammonium cations, was measured using the standard inversion recovery sequence (π - τ - π/2). For Larmor frequencies ranging from 9.9 to 95 MHz, experiments were carried out on a Bruker CXP 100 spectrometer in a temperature-regulated room atmosphere. Whatever the frequency, the measured sample temperature was found to be in the range of 23.0-25.4 °C. In fact, the observed magnetic relaxation is not trivial. The free induction decay (FID) clearly results from the superimposition of two components with different T2 values: (i) a short T2 signal that can be attributed to the residual protons of the HOD molecules in strong interaction with the Gd3+ ions and (ii) a long T2 signal, coming from the tetramethylammonium protons. Similarly, the longitudinal relaxation itself could not be accounted for by a single exponential. A detailed analysis showed that the HOD protons were relaxing much more rapidly than those of tetramethylammonium. As an example, at 90 MHz, the short and the long T1 values were estimated to be less than 35 ms and equal to 315 ms, respectively. Therefore, particular experimental conditions were used for the measurements. The T1 of tetramethylammonium protons were measured (i) by using delays, τ, longer than 170 ms in the inversion recovery sequence and (ii) by starting the acquisition of the FIDs only after the decrease of the short T2 component. On the other hand, at 200 MHz and above,5,6 the measurements were performed on high-resolution Bruker spectrometers, where the (CH3)4N+ protons display a NMR signal well separated from that of the HOD molecules and have a spinlattice relaxation time T1 that can be recorded straightforwardly. In these cases, the temperature was well calibrated at 25 °C. 3. Theory of the Intermolecular Dipolar Relaxation The intermolecular relaxation rate Reln of the spins I ) 1/2 of the (CH3)4N+ methyl protons due to their dipolar coupling with the Gd3+ electronic spins S ) 7/2 is defined as
Reln ) R1 - R10 ) 1/T1 - 1/T10
(1)
where T1 and T10 are the measured longitudinal relaxation times of the protons in the para- and diamagnetic solutions, respectively. 3.1. The Solomon Model. If the quantum motion of the electronic spin S of the hydrated Gd3+ ion could be approximated as a free Larmor precession during the translational correlation time τ, Reln would be simply given by the Solomon expression5,10
RelnSolomon )
NS τ 8π2 2 7 γI (gSµB)2S(S + 1) 3 hj 2(ωIτ) + hj 2(ωSτ) 5 3 πb (2)
[
]
In this equation, NS is the number density of the paramagnetic ions, γI is the proton gyromagnetic ratio, and gS ) 2 is the Lande´ factor of the Gd3+ ion with S ) 7/2. The quantities ωI ) -γIB0 and ωS ) gSµBB0/p are the nuclear and electronic angular frequencies, respectively. b is the minimal distance of approach of the centers of the two ions (CH3)4N+ and Gd(D2O)83+ considered as hard spheres. τ ) b2/D is the correlation time of the relative translational motion of these ions, D being their
1980 J. Phys. Chem. B, Vol. 105, No. 10, 2001
Rast et al.
relative translational diffusion constant. The values of all of the geometrical and dynamical parameters describing the interacting ions are given in refs 5 and 6. The reduced spectral density hj2 is a dimensionless function that depends on the effective interaction between the ions and on their translational and rotational dynamics. More precisely, it is related to the relative motion of the interacting spins I and S through the relation
hj 2(ωτ) )
πb3 j (ω) NSτ 2
(3)
where j(ω) is the Fourier transform of the time correlation function g2(t) of the random functions r-3Y2q(θ,φ) of the vector b(r,θ,φ), r which joins the position of the proton spin I to that of the electronic spin S. We have
g2(t) ) 〈r0-3Y2q(θ0,φ0)r-3Y2q(θ,φ)*〉
(4)
The reduced spectral density hj2 depends on the pair distribution gIS(R) of the two ions carrying the I and S spins. This distribution is related to the ion-ion potential of mean force (PMF)11-13 wFC IS (R) through the relation
[
]
wFC IS (R) gIS(R) ) exp kT
(5)
The superscipt FC indicates that the interacting ion pair is surrounded by a finite concentration (FC) of ions. The finiteconcentration PMF wFC IS (R) is derived from the infinite-dilution (R) using the simple Debye-Hu¨ckel screening forlimit wID IS malism, which was shown to be reliable up to moderate ionic strengths.11 wID IS (R) is computed within the reference hypernetted chain (RHNC) approximation12,13 for two hard spherical ions solvated by discrete, polar, and polarizable molecules modeling the behavior of water.11-13 The model of liquid solution considered here is a mixture of hard spheres with embedded electric multipoles. In this case, the many-body spatial correlations between the particles can be better taken into account by using reference closures in the integral equation theory that gives the liquid structure.14,15 However, note that the simpler hypernetted chain (HNC) approximation is often very accurate for fluids of molecules of anisotropic shapes.16,17 The numerical procedure for computing hj2 from gIS(R) is described in ref 18. In expression 2, we omitted the Curie contribution,19 which was shown to be negligible for Gd3+ ions.5 3.2. The Effects of the Electronic Relaxation. In the laboratory (L) frame, the electronic Hamiltonian He of the Gd3+ ion (S ) 7/2, gS ) 2) is the sum of a time-independent Zeeman Hamiltonian H0 ) ωSSz and a fluctuating part with two contributions: a first contribution H (L) T (t) corresponding to the usual transient zero-field splitting (ZFS) and a second contribution H (L) 1 (t) due to the random reorientation in the laboratory frame of the mean crystal field, which is static in the molecular (M) frame.7 Thus (L) He ) H0 + H (L) T (t) + H 1 (t)
(6)
We need to calculate the quantum correlation functions of the components of B S, which in the high-temperature limit are the ensemble averages
kzz(t) ) k+-(t) )
1 1 tr S U+(t)SzUe ) tr S S (t) (7a) 2S + 1 e z e 2S + 1 e z z
1 1 tr S U+(t)S-Ue ) tr S S (t) 2S + 1 e + e 2S + 1 e + -
(7b)
where tre is the trace in the subspace of the electronic spin states, Ue is the evolution operator in this space, and the bar stands for the ensemble average over the other degrees of freedom of the complex. When only the Zeeman Hamiltonian acts on the spin S, kzz and k+- have constant and periodic behaviors, respectively. However, the existence of small fluctuating perturbations is responsible for the time decay of these two functions. In a preceding paper,7 we showed that, for the hydrated Gd3+ complex, kzz(t) has a quasi mono-exponential decay, characterized by a single electronic relaxation time T1e, the variation of which versus the applied magnetic field B0 was displayed in Figure 4 of ref 7 for T ) 298.15 K. T1e monotonically increases with the field from ∼2.5 × 10-10 s to 1.0 × 10-8 s for B0 ) 0.34 T (X band) and B0 ) 8 T, respectively. Thus
1 kzz(t) ) S(S + 1)e-t/T1e 3
(8)
This result was obtained from eq 7a, using the Redfield theory, by solving the set of 2S + 1 ) 8 differential equations describing the time evolution of the components ZM(t) ) 〈M|Sz|M〉 in terms of an 8 × 8 relaxation matrix, the elements of which are (L) functions of H (L) T (t), H 1 (t), and the EPR angular frequency ωS. The calculation showed that, among the eight exponentials associated with the eigenvalues of this relaxation matrix, one of them was by far dominant, leading to the simple relation in eq 8. On the other hand, the decay of the transverse relaxation function k+-(t) ) k/-+(t) was shown by a similar procedure to behave as a weighted sum of four decreasing exponentials 4 2 iωSt k+-(t) ) S(S + 1)e ( wie-t/T2ie) 3 i)1
∑
(9)
The weights wi (∑i wi ) 1), which strongly depend on the field below B0 ) 4 T, and the characteristic times T2ie were computed at T ) 298.15 K7 for all of the fields of interest for this NMRD study. As for kzz(t), this result was derived by solving the set of seven differential equations describing the time evolution of the seven matrix elements X+ M(t) ) 〈M + 1|S+|M〉 in terms of a 7 × 7 relaxation matrix. Among the seven exponentially decreasing solutions of these differential equations, three of them have vanishing weights wi, but for the Gd3+ aqua complex considered here, none of the four remaining weights can be neglected, except at very low field below ∼0.1 T. It is quite reasonable to suppose that the spatial molecular diffusion and the motion of the electronic spin are uncorrelated. Then, the correlation functions CRβ(t), which are relevant to the nucleus-electron dipole-dipole interaction, are simply the products
CRβ(t) ) g2(t)kRβ(t)
(10)
of the spatial correlation function g2(t) of the interspin position18,20,21 and the correlation functions of the spin components kzz, k+-, or k-+. We are now in a position to give the general expression for the theoretical intermolecular dipolar nuclear-electron relax-
(CH3)4N+ Protons in Gd3+ Heavy-Water Solutions
J. Phys. Chem. B, Vol. 105, No. 10, 2001 1981
Figure 1. NMRD curves describing the field dependence of the intermolecular spin-lattice relaxation rate Reln of the (CH3)4N+ (0.1 M) protons in the presence of Gd3+ ions (3.08 mM) in D2O at 25 °C. The experimental data (b) below B0 ) 2.23 T were corrected to the reference temperature of 25 °C (see text). Model i (continuous line) includes the effects of the ion-ion repulsion and of the quenching by the electronic relaxation derived from the EPR study;7 model ii (longdashed line) includes the effects of the ion-ion repulsion only; model iii (dashed line) includes the effects of the quenching by the electronic relaxation only.
ation rate Reln due to electronic spins, the components of which have correlation functions that decay according to eqs 8 and 9. Define the spectral density j2(ω,1/Tie) and the associated dimensionless reduced quantity hj2(ωτ,τ/Tie) by
j2(ω,1/Tie) )
NSτ 1 hj 2(ωτ,τ/Tie) ) 2π πb3
The new expression of 2, reads
RelnSB )
Reln,
∫-∞+∞g2(t)e-iωt e-t/T
ie
dt (11)
which should be used instead of eq
[
NSτ 8π2 2 S(S + 1) hj 2(ωIτ,τ/T1e) + γ1 (gSµB)2 5 πb3 7
4
∑ wihj 2(ωSτ,τ/T2ie) 3 i)1
]
(12)
The generalized Solomon-Bloembergen (SB) eq 12 is the starting point for calculating the NMRD of the tetramethylammonium protons in our solution. The reduced spectral density hj2(ωτ,τ/Tie), defined by eq 11, can be rewritten as
hj 2(ωτ,τ/Tie) )
πb3 b3 j2(ω,1/Tie) ) Re[g˜ 2(σ)iω+1/Tie)] NSτ NSτ (13)
where g˜ 2(σ) is the Laplace transform of the time correlation function g2(t) of the relative microdynamics of the (CH3)4N+/ Gd3+ pair. In general, this Laplace transform is computed numerically, but it can be obtained analytically for some simple models of intermolecular dynamics.13,18,20,21 Results and Discussion In Figure 1, the experimental NMRD data are compared to the theoretical predictions of three different models. Model i incorporates the influence of the Coulomb repulsion of the PMF of eq 5 on the ion-ion relative diffusion. It also accounts for the effects of the electronic relaxation, discussed in subsection
3.2, by using the generalized Solomon-Bloembergen eq 12. Moreover, one or two water molecules of the coordination shell of some Gd3+ cations can be replaced by a dissolved NO3anion. The new GdNO32+ complexes have a charge of +2 instead of +3 for the fully hydrated Gd3+ complex. These new complexes are then less repulsive for the (CH3)4N+ probe. As shown by Vigouroux et al.,5 this results in a measurable enhancement of the relaxation of the (CH3)4N+ protons that go nearer to the Gd3+ paramagnetic centers. At the present Gd(NO3)3 concentration of 3.08 × 10-3 M, the obtained association equilibrium constant Kapp ) 11.8 L mol-1 leads to the formation of nearly 10% of GdNO32+ complexes. Here, we calculated that this coordination effect can be taken into account for all magnetic fields B0, simply by multiplying the theoretical intermolecular relaxation rate Reln, calculated for the (CH3)4N+/ Gd3+ pair, by an average enhancement factor of 1.035. As we are interested in the subtle and rather weak effects of the electronic relaxation on Reln, the neglect of a systematic influence, even small, would not be justified a priori. Model ii accounts for the repulsive ion-ion PMF as in model i, but neglects all of the effects of the electronic relaxation by using the Solomon eq 2. Finally, model iii describes the ion-ion relative diffusion as that of neutral hard spheres in a viscous continuum and assumes that the spins are located at the ion centers. This is the reference ABHF dynamical model that was proposed independently by Ayant and Belorizky20 and then by Hwang and Freed21 and which has an analytical solution. The effects of the electronic relaxation are also incorporated by using the generalized SB eq 12 as in model i. With the most realistic model i, the overall agreement is good for all of the investigated fields, and in particular, the position of the maximum at B0 ≈ 2.2 T is well predicted. The theoretical predictions are excellent at high fields >4 T, where the electronic relaxation effects vanish so that the Solomon eq 2 applies.6 This justifies the model of interionic dynamics. The theoretical values are slightly lower than the experimental data at low fields, but the discrepancy never exceeds ∼15%. This is quite remarkable because there are no adjustable parameters and because the NMR experimental accuracy is of the order of 5%. Note that, for 0.23 T < B0 < 2.23 T, we corrected the measured relaxation rate Reln for the temperature fluctuations of the sample (varying between 23.4 and 25.1 °C) to the reference temperature of 25 °C. This was done by considering that, at these frequencies, the terms hj2(ωsτ,τ/T2ie) are negligible (ωSτ g 12), so that Reln is proportional to τjh2(ωIτ,τ/T1e), the temperature dependence of which is primarily that of τ ) b2/D ≈ η(T)/T, η(T) being the D2O viscosity. The induced correction remains smaller than 5%. Now, turn to a physical explanation of the maximum at B0 ) 2.2 T of the NMRD profile. As seen above, in eq 12, the terms hj2(ωSτ,τ/T2ie) are always small for the investigated frequency range, and the field dependence of Reln is that of hj2(ωIτ,τ/T1e) given by eq 13. When B0 increases from 0.23 to 2.2 T, ωIτ increases from 0.02 to 0.18, while τ/T1e decreases from 1.5 to 0.17, as T1e is roughly proportional7 to B0 in this field domain. The relevant spectral density hj2(ωIτ,τ/T1e) mainly depends on the decreasing argument τ/T1e. As hj2 is a decreasing function of its arguments, we obtain the observed increase in Reln. On the other hand, when B0 exceeds 2.2 T, τ/T1e rapidly becomes negligible.7 Consequently, hj2(ωIτ,τ/T1e) ≈ hj2(ωIτ,0) decreases at high fields. To our knowledge, this is the first time that an electronic relaxation model of the Gd3+ ion, with convenient parameters for the underlying physical processes, is able to interpret the
1982 J. Phys. Chem. B, Vol. 105, No. 10, 2001
Figure 2. NMRD curves describing the field dependence of the proton intermolecular relaxation rate Reln. The experimental data (b) below B0 ) 2.23 T were corrected as in Figure 1. Model i (continuous line) assumes the longitudinal electronic relaxation rate 1/T1e derived from the EPR study;7 model i′ (long-dashed line) uses the adjusted 1/T1e values given by the empirical eq 14.
EPR line widths at various temperatures and fields and that the same set of parameters accounts, in a reasonably accurate way, for the effect of the fast electronic relaxation on the proton magnetic relaxation dispersion due to the translational encounters of the cation/Gd3+ pairs. For completeness, now consider the Reln predictions of models ii and iii, also represented in Figure 1. The smaller results of model i (continuous curve) with respect to the values of model ii (long-dashed curve) display the influence of the electronic relaxation on the NMRD profile for B0 < 4 T. At B0 ) 0.5 T, which is the MRI field of reference, the (CH3)4N+ proton relaxation rate Reln is decreased by ∼50% because of the short value of T1e with respect to the translational correlation time τ. The smaller values of model i (and of model ii) with respect to the predictions of model iii (dahed line) indicate the marked influence of the Coulomb repulsion on the spatial relative dynamics of the (CH3)4N+/Gd3+ ion pair. This electrostatic repulsion is in no way negligible, leading to a reduction of Reln by a factor of ∼2. Finally, at very low fields when B0 decreases below 0.1 T, the spectral density values hj2(ωSτ,τ/T2ie) at the electronic Larmor frequency ωS contribute more and more significantly to the intermolecular relaxation rate Reln given by eq 12. More precisely, as B0 varies from 0.1 to 0 T, ωSτ decreases from 5.3 to 0, while τ/T21e only slightly increases from 2.85 to 3.2, where T21e is the unique transverse relaxation time playing a role below 0.1 T, as w1 g 0.999 with T21e ) T1e (extreme narrowing case). Thus, for B0 ) 0, hj2(0,τ/T21e) ) hj2(0,τ/T1e). Therefore, a notable enhancement of Reln is expected in this region within the generalized Solomon-Bloembergen formalism used in this work. However, Dinesen and Bryant4 did not observe such an enhancement at a field as low as 2.5 × 10-4 T. This result can be explained by the breakdown of the validity of the Redfield theory for the electronic relaxation and, consequently, of eqs 9 and 12. We intend to develop a new theoretical approach to deal with this difficult open problem. In the present work, the values of the electronic relaxation time T1e of a Gd3+ complex were derived indirectly from a relaxation model fitted to independent EPR data. Then, we found it possible to interpret the proton NMRD profile of the (CH3)4N+ probe. Conversely, such NMRD experiments with probe solutes can be the starting point of an indirect method for estimating
Rast et al.
Figure 3. Comparison of the field dependence of the longitudinal electronic relaxation rates 1/T1e derived from the EPR study7 (continuous line) and given by the empirical eq 14 adjusted from the NMRD profile (long-dashed line).
T1e values that are too short to be directly measured by the presently available techniques. Indeed, at 0.5 T, which approximately corresponds to the reference MRI proton frequency of 20 MHz,22 and above, the theoretical relaxation rate R1e is given by eq 12 with hj2(ωSτ,τ/T2ie) ≈ 0. Then, if the relative motion of the Gd3+ complex and the NMR solute probe is accurately modeled, the values of T1e can be adjusted so that R1e gives the experimental NMRD data of this probe. Such an independent knowledge of T1e allows for a more reliable analysis23 of the NMRD profiles involving the H2O molecules and, thus, a better understanding of the subtle inner- and outersphere complexation dynamics that strongly influence the MRI relaxivity. For instance, in our system, the experimental NMRD profile can be almost perfectly obtained using model i′, which assumes the same interionic dynamics and transverse electronic relaxation times T1e as in model i but which leaves the longitudinal relaxation time T1e as an adjustable function of the field B0. As shown in Figure 2, we found that the empirical field dependence of 1/T1e
1 ) 6 × 109 T1e
1 s-1 B0 2 1+ 0.35
( )
(14)
where B0 is given in Tesla, allows for a convenient fit of the experimental NMRD data. At low fields, these NMRD values of 1/T1e are compared in Figure 3 to their EPR counterparts, which are used in model i and which were obtained indirectly from a model of the crystal field and dynamics of the Gd3+ aqua ion, as already discussed. The NMRD fitted values of 1/T1e are of the same order of magnitude as the EPR results but smaller. For B0 ) 0.5 T, the difference is only ∼10%. Should the theoretical proton relaxation rate increase by only 5%, which is within the model accuracy, the fitted NMRD 1/T1e value would become 10% larger, and the agreement would be perfect. On the other hand, the difference between the NMRD and EPR 1/T1e values reaches 50% above 1 T. At higher fields beyond 1.5 T, the Reln attenuation caused by the longitudinal electronic relaxation becomes too small to allow for a reliable NMRD determination of 1/T1e. The observed difference between the NMRD and EPR results can be easily attributed to the weaknesses of both methods. On one hand, the 1/T1e predictions derived from the
(CH3)4N+ Protons in Gd3+ Heavy-Water Solutions
J. Phys. Chem. B, Vol. 105, No. 10, 2001 1983 negligible with respect to R1e for the Gd3+ complexes, is omitted. The attenuation factor A represents the relative quenching of the Solomon process due to the short lifetimes of the Gd3+ electronic levels. Its variation with field for 0.23 T < B0 < 1.4 T is shown in Figure 4. It is 2 orders of magnitude larger than the attenuation factors of the other lanthanide Ln3+ ions, the electronic relaxation times of which are very short,10 being on the order of 1 ps. The above NMRD technique applied to probe solutes is a simple way of evaluating the attenuation factors of Gd3+ complexes, which should be as close as possible to unity to have optimum MRI performance.
Figure 4. Field dependence of the attenuation factor A (see text) of the relaxivity of the (CH3)4N+ (0.1 M) protons in the presence of Gd3+ ions (3.08 mM) in D2O at 25 °C. The experimental values of A (b) were derived from the measured intermolecular spin-lattice relaxation rates, which were corrected below B ) 2.23 T as in Figure 1. Model i (continuous line) assumes the longitudinal electronic relaxation time 1/T1e derived from the EPR study;7 model i′ (long-dashed line) uses the ajusted 1/T1e values given by the empirical eq 14 adjusted from the NMRD profile.
EPR spectra are obtained indirectly through an analysis of the line widths, which are due to several Gd3+ dynamical processes. As the contributions of these processes7 for the transverse electronic relaxation rates 1/T2ie are different from those for the longitudinal rate 1/T1e, the behavior of the latter is only approximately analyzed by EPR. In addition, at 1 T, the Reln attenuation caused by the longitudinal electronic relaxation is 25%. Then, an increase of only 5% of the theoretical proton relaxation rate would enhance the fitted NMRD 1/T1e value by 30% at 1 T. This would lead to a reduced difference of 25% with the EPR value, instead of ∼50%. NMRD experiments using probe solutes of well-known spatial dynamics with respect to a Gd3+ complex appear to be a promising indirect method for measuring T1e in the MRI field domain. Finally, as for the lanthanide Ln3+ ions other than Gd3+, one can introduce the attenuation factor (see eq 35 of ref 10)
A)
e Rln
ReSlnolomon
(15)
where RelnSolomon is the theoretical relaxation rate of model ii given by eq 2. In this equation, the Curie term, which is
Acknowledgment. We are indebted to Prof. A. Sacco for measuring the proton relaxation time T1 at 200 MHz and to D. Chapon for preparing NMR samples. We are grateful to Prof. A. E. Merbach for his interest in this research related to the EC-COST-D-8/18 action (Chemistry). References and Notes (1) Powell, D. H.; Dhubhghaill, O. M. N.; Pubanz, D.; Helm, L.; Lebedev, Ya. S.; Schlaepfer, W.; Merbach, A. E. J. Am. Chem. Soc. 1996, 118, 9333. (2) Lauffer, R. B. Chem. ReV. 1987, 87, 901. (3) Kowall, Th.; Foglia, F.; Helm, L.; Merbach, A. E. J. Phys. Chem. 1995, 99, 13078. (4) Dinesen, T. R. J.; Bryant, R. G. Chem. Phys. Lett. 1999, 303, 187. (5) Vigouroux, C.; Bardet, M.; Belorizky, E.; Fries, P. H.; Guillermo, A. Chem. Phys. Lett. 1998, 286, 93. (6) Favier, A.; Chapon, D.; Fries, P. H.; Rast, S.; Belorizky, E. Chem. Phys. Lett. 2000, 320, 49. (7) Rast, S.; Fries, P. H.; Belorizky, E. J. Chem. Phys. 2000, 113, 8724. (8) Clarkson, R. B.; Smirnov, A. I.; Smirnova, T. I.; Kang, H.; Belford, R. L.; Earle, K.; Freed, J. H. Mol. Phys. 1998, 95, 1325. (9) Koenig, S. H.; Epstein, M. J. Chem. Phys. 1975, 63, 2279. (10) Vigouroux, C.; Belorizky, E.; Fries, P. H. Eur. Phys. J. D 1999, 5, 243. (11) Sacco, A.; Belorizky, E.; Jeannin, M.; Gorecki, W.; Fries, P. H. J. Phys. II Fr. 1997, 7, 1299. (12) Jeannin, M.; Belorizky, E.; Fries, P. H.; Gorecki, W. J. Phys. II Fr. 1993, 3, 1511. (13) Fries, P. H.; Patey, G. N. J. Chem. Phys. 1984, 80, 6253. (14) Fries, P. H.; Patey, G. N. J. Chem. Phys. 1985, 82, 429. (15) Lomba, E.; Martı´n, C.; Lombardero, M. Mol. Phys. 1992, 77, 1005. (16) Fries, P. H.; Kunz, W.; Calmettes, P.; Turq, P. J. Chem. Phys. 1994, 101, 554. (17) Richardi, J.; Fries, P. H.; Fischer, R.; Rast, S.; Krienke, H. Mol. Phys. 1998, 93, 925. (18) Fries, P.; Belorizky, E. J. Phys. Fr. 1978, 39, 1263. (19) Gue´ron, M. J. Magn. Reson. 1975, 19, 58. (20) Ayant, Y.; Belorizky, E.; Alizon, J.; Gallice, J. J. Phys. Fr. 1975, 36, 991. (21) Hwang, L. P.; Freed, J. H. J. Chem. Phys. 1975, 63, 4017. (22) Aime, S.; Botta, M.; Fasano, M.; Terreno, E. Acc. Chem. Res. 1999, 32, 941. (23) Fries, P. H.; Rast, S.; Belorizky, E. Proceedings of the Table Ronde de la IXe Journe´ e Grenobloise de RMN; Centre Grenoblois de Re´sonance Magne´tique; Grenoble, France, 1999.
