J. Phys. Chem. 1996, 100, 8093-8100
8093
Rotational Dynamics of MRI Paramagnetic Contrast Agents in Viscous Media John W. Chen,*,†,‡ Robert B. Clarkson,‡,§ and R. Linn Belford†,‡ Department of Chemistry, Department of Veterinary Clinical Medicine, and College of Medicine, UniVersity of Illinois at Urbana-Champaign, Illinois 61801 ReceiVed: October 27, 1995; In Final Form: February 26, 1996X
Electron paramagnetic resonance spectra of model magnetic resonance imaging contrast agents vanadyl ethylenediaminetetraacetate and vanadyl diethylenetriaminepentaacetate in 46% w/v sucrose solutions have been acquired from near physiological temperature to near freezing to study their rotational dynamics in a viscous environment. The sucrose solution simulates environments that may arise in ViVo and also when the contrast agents are bound either covalently or noncovalently (e.g., via hydrogen bonds) to another molecule either by chance or by design to improve the relaxivity and the specificity. We report that an isotropic Brownian model, which described the rotational dynamics of these agents in water, does not fully portray the rotational dynamics in this environment, especially in the intermediate or slow tumbling regime. However, an axial anisotropic rotational model simulated the experimental spectra well and can account for the trends exhibited by the data. The dynamics results suggest that the motion deviates substantially from StokesEinstein behavior, and possible physical models to account for the behavior are discussed.
1. Introduction The addition of dilute paramagnetic complexes to a solvent, such as water, increases the relaxation rate of proton nuclei in the solvent. The motions of the paramagnetic complexes, which create a local fluctuating magnetic field, contribute to an increase of the proton relaxation efficiency. The strength of the dipolar interaction between the paramagnetic ion (i.e., the unpaired electron spins) and the proton nuclei determines the efficacy of this proton relaxation process, which is strongly influenced by the rotational motion of the paramagnetic complex. Hence, the agent’s rotational dynamics is extremely important in determining the proton relaxation rate, and an understanding of this behavior in different environments will aid in the rational development of effective contrast agents. We recently developed a method to study the rotational motion of paramagnetic contrast agents by substituting the magnetically anisotropic vanadyl ion (VO2+) for the virtually isotropic gadolinium ion (Gd3+) and have shown that in the case of small molecular weight chelates such as ethylenediaminetetraacetate (EDTA) and diethylenetriaminepentaacetate (DTPA) in low-viscosity solvents, the rotational motion obeys Brownian isotropic diffusion dynamics and that the vanadyl and the gadolinium analogs display nearly the same effective dipolar distance between the solvent protons and the metal ion.1,2 However, contrast agents in medical diagnostic use are not in Vitro agents, but rather in ViVo agents that must be injected into a subject (human patient) to be effective. Living systems present a wide range of environments that can alter the dynamics of the nominally isotropically-tumbling agents. For example, the blood contains numerous cofactors and serum proteins that may bind the agents; the intracellular matrix (cytosol) contains numerous proteins (such as keratin, collagen, and enzymes) as well as other structures and compounds that not only will increase the viscosity of the solvent but also may bind to the * To whom correspondence should be addressed. † Department of Chemistry. ‡ College of Medicine. § Department of Veterinary Clinical Medicine. X Abstract published in AdVance ACS Abstracts, April 15, 1996.
S0022-3654(95)03210-2 CCC: $12.00
Figure 1. Chelates used in this work: (a) EDTA ) ethylenediaminetetraacetate, (b) DTPA ) diethylenetriaminepentaacetate.
agents. The extracellular matrix is filled with a viscous, hydrophilic ground substance (glycosaminoglycans) that increases dramatically the viscosity of the environment and may slow or even immobilize the contrast agent. Finally, other types of glycosaminoglycans are found in the skin, cornea, synovial fluid of joints, vitreous humor of the eye, and umbilical cordswhich may be targets of future magnetic resonance imaging (MRI) contrast agents. In all these cases, even small molecular weight agents may no longer tumble isotropically. Therefore, in this study, we set out to model some characteristics of the above environments, i.e., increased viscosity and noncovalent bonding. To accomplish this, we added sucrose to the experimental solutions. Sucrose is ideal for this study because its viscosity behavior is well understood and the sugar may form hydrogen bonds with the agents. It also can serve as a crude model for glycosaminoglycans (which are essentially repeated units of acid and amino sugars) that are found nearly ubiquitously in the body. The chelates studied in this report are shown in Figure 1. 2. Materials and Methods 2.1. Chemical Preparation. The stock vanadyl sulfate (Aldrich) solutions were prepared from deionized water and purged with argon to prevent oxidation of the vanadyl ion (pH ) 3). Sample solutions were prepared by combining in a 1:1.2 © 1996 American Chemical Society
8094 J. Phys. Chem., Vol. 100, No. 20, 1996
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ratio the stock vanadyl sulfate solution and the powdered chelates. Sodium bicarbonate (5% solution) was used to raise the pH to physiological pH ()7.4). N-(2-hydroxyethyl)piperazine-N′-2-ethanesulfonic acid (HEPES) buffer was added to prevent the formation of vanadyl hydroxide3 and to buffer the solutions. An appropriate amount of sucrose (Sigma) was added to the solutions to make 46% w/v sucrose solutions. The final solutions were again purged with argon. The concentrations of the solutions were determined by plasma emission spectroscopy on a Perkin-Elmer 2000. 2.2. Spectroscopy. 2.2.1. UltraViolet and Visible Spectroscopy. UV-vis spectroscopy was performed on a HewlettPackard 8452 Diode Array Spectrometer. All solutions exhibited the characteristic UV-vis spectra of a vanadyl ion chelated by EDTA and DTPA,3 signifying that chelation is complete. 2.2.2. Electron Paramagnetic Resonance (EPR) spectroscopy. Variable-temperature EPR spectra were taken with a Varian X-band spectrometer (12 in. magnet) with a TE102 cavity. The sample was held in a 1 mm i.d. quartz tube, and its temperature was regulated by flowing gaseous nitrogen precooled by liquid nitrogen through a Varian variable-temperature controller. A standard field calibration utilizing a DTM-141 Digital Teslameter was also performed on the X-band spectrometer. 2.3. Computation. All computations were performed on an IBM RS/6000 Model 3AT, except for the powder pattern analysis, which used a Gateway 2000 486/DX2-66V. The programs mentioned here may be obtained from the anonymous ftp server at the Illinois EPR Research Center.4 Powder pattern computation to derive A and g employed the SIMPOW program.1,5 FIT (version 2.21), an automatic nonlinear least-squares fitting program capable of single-parameter (parabolic interpolation)6 and multiparameter (simplex)7 optimizations, was developed and used in the analysis of the motionally-modulated spectra. The simulation engine of FIT was based on the EPRLF programs developed by Budil et al. The EPRLF programs used the stochastic Liouville equations that include nonsecular contributions in the spin Hamiltonian to simulate the motionally-modulated EPR spectra. Briefly, the simulation uses the result that the absorption intensity of an unsaturated magnetic resonance spectrum is10,11
I(ω - ω0) )
(π1)〈ν|[(Γ - iL) + i(ω - ω )I] 0
-1
|ν〉
(1)
where L is the Liouville superoperator associated with the spin Hamiltonian and Γ is the diffusion superoperator used to model the rotational motion. The matrix elements of Γ relevant to our system, where the complex undergoes Brownian diffusion and does not have Heisenberg spin exchange, are11,12
〈L1M1K1ps1qs1pI1qI1|Γ|L2M2K2ps2qs2pI2qI2〉 ) δL1,L2δM1,M2δK1,K2δps1,ps2δqs1,qs2δpI1,pI2δqI1,qI2{R⊥L1(L1 + 1) + R|K21 - R⊥K21} (2) where R| and R⊥ are the parallel and perpendicular components of the rotational diffusion tensor, ps ) ms - m′s, qs ) ms + m′s, and similarly for pI and qI (I and S being the nuclear spin and the electron spin, respectively). For a complex undergoing isotropic Brownian diffusion, R| is equal to R⊥. The expression (Γ - iL) is known as the stochastic Liouville superoperator. The simulation program takes as input the rigid-limit A- and g-matrices and a starting vector consisting of the rotational diffusion tensor and, if applicable, the diffusion tilt angle (see below). The program then constructs the stochastic Liouville
TABLE 1: Rigid-Limit A- and g-Matrices for VO(EDTA) and VO(DTPA) in Sucrose complex (in 46% sucrose) VO(EDTA) VO(DTPA)
Axxa (MHz)
Ayya (MHz)
Azza (MHz)
gxx
gyy
gzz
-184.3 -173.2 -501.5 1.981 1.979 1.944 -188.6 -175.5 -508.3 1.981 1.978 1.943
a The values of A can be converted from megahertz to gauss with MHz/Gauss ) 〈giso〉β/h.
superoperator, computes the matrix elements of the superoperator, and tridiagonalizes it with the Lanczos algorithm. The tridiagonal matrix is then solved for eq 1 to obtain the simulated spectrum. The simulated and experimental spectra are compared, and the starting vector is automatically adjusted according to the nonlinear least-squares routines mentioned above to produce the best-fit spectrum. Please consult Chen et al.1 and references contained therein for more details on the SIMPOW and the EPRLF programs. 3. Results and Discussion Table 1 shows the A- and g-matrices derived from rigidlimit spectra of the complexes. These values were used as part of the initial starting vector for the motional simulations. Note that the values are very similar to each other. In simulating the spectra, we first employed an isotropic Brownian diffusion model that was successfully applied to describe small-chelate contrast agents in saline solution.1 In these simulations, only the rotational diffusion coefficient (DR) was varied; DR is related to the rotational correlation time (τR) by the equation
τR )
1 6DR
(3)
Figures 2 and 3 show representative spectra and their bestfit simulations using the isotropic Brownian diffusion model. The simulations matched the experimental spectra quite well at high temperatures (near fast limit) and at low temperatures (near rigid limit). However, in the intermediate-tumbling regime, this simple model did not adequately describe the more complex dynamics exhibited by the model agents; the fits were rather poor, as exemplified by Figures 2b,c and 3b,c. It was not surprising that this model did not fit the slow-motional experimental spectra well. Campbell and Freed showed that for aquo vanadyl ion in sucrose, the isotropic Brownian diffusion model was not a reasonable description for the motion.8 Although they reported that isotropic jump diffusion and isotropic free diffusion models fit the spectra better than does the isotropic Brownian diffusion model, none of these models could reproduce all the line shapes and match the intensity of the peaks. Therefore, it seemed likely to us that the motion in the slow-tumbling regime was not isotropic rigid rotation. The anisotropic tumbling in our experiments can arise in two ways. First, the isotropic model requires a spherical tumbling complex. Being fairly large and flexible, it is unlikely that the agents are truly spherical. In low-viscosity solvents such as water, small deviations from spherical symmetry will be averaged out by the rapid rotation of the complex. However, as we slow the motion down by increasing the viscosity, the motion may no longer be capable of completely averaging out the deviations. In this case, there would be minimally two unequal axes of rotation (axial symmetry), and the complex could be described as either a football (prolate ellipsoid) or a disk (oblate ellipsoid). Anisotropic tumbling also could arise from the interaction between the sucrose molecules and the model agents. Hydrogen bonding is a likely interaction since the chelates possess many
Rotational Dynamics of MRI Contrast Agents
Figure 2. Some representative VO(EDTA) (in 46% sucrose) EPR spectra and their isotropic best fits: (a) τR ) 0.261 ns (309 K), (b) τR ) 0.433 ns (293 K), (c) τR ) 1.97 ns (286 K), and (d) τR ) 4.31 ns (271 K). In the intermediate-tumbling regime, as shown in b and c, the isotropic model does not fit the spectra well.
electron-rich atoms that may interact with the electron-deficient hydrogen atoms on sucrose, much like the second-sphere interaction between water and the agents. In this case there also will be two axes of rotation: one about the agent and the other about the overall solvent-agent complex. This model is applicable to other systems where the agents may be bound to macromolecules such as dendrimers9 or albumin. It is likely that both scenarios play a role in modulating the dynamics of the agents and may not be mutually exclusive. The increase in viscosity may slow the motion sufficiently to allow the solvent molecules (sucrose-water) to bind to the agents for a long enough time to alter the motion from nearly isotropic to very anisotropic. Therefore, to better simulate these spectra, we used an anisotropic model composed of parallel (τ|) and perpendicular (τ⊥) motions and a diffusion tilt angle (Φ) between
J. Phys. Chem., Vol. 100, No. 20, 1996 8095
Figure 3. Some representative VO(DTPA) (in 46% sucrose) EPR spectra and their isotropic best fits: (a) τR ) 0.333 ns (325 K), (b) τR ) 0.540 ns (306 K), (c) τR ) 0.149 ns (298 K), (d) τR ) 5.142 ns (275 K). In the intermediate-tumbling regime, as shown in b and c, the isotropic model does not fit the spectra well.
the magnetic and the diffusion frames. Note that the two motions can correspond to axial rotation either of an asymmetric complex (Figure 4) or of a paramagnetic complex bound to a macromolecule. In the latter case, τ| is the motion of the complex while τ⊥ is the motion of the macromolecule (Figure 5). This model therefore describes both scenarios above, and they are in fact computationally equivalent.11 To implement this model, three parameters must be covaried (τ|, τ⊥, Φ). Because multiparameter fittings are inherently less robust than one-parameter fittings (like that used in the isotropic model), each simulation was restarted several times with the starting vector taken to be the tentative best-fit values increased or decreased by an order of magnitude to minimize trapping in
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Figure 4. Axial rotational model for an asymmetric molecule. Zm is the magnetic z-axis, taken in this illustration to be along the VO bond. Zr is the rotational z-axis. Φ is the angle between th two frames.
Figure 5. Model for a (nearly) symmetric paramagnetic complex bound noncovalently or covalently to a macromolecule. Zm is the magnetic z-axis, taken in this illustration to be along the VO bond. Zr is the rotational z-axis, taken here to be along the bond formed between the agent and the macromolecule. Φ is the angle between the two frames.
a local minimum. The final converged solutions do not deviate by more than 5% from each other. Figures 6 and 7 show the best fits under this model for the same experimental spectra as those in Figures 2 and 3. The simulated spectra now fit the experimental spectra quite closely. Even near the slow and fast limits, the anisotropic model gave better fits than those derived from the isotropic model. However, in light of the failure of the isotropic model to adequately fit the slow-motional spectra, it was surprising that it could fit the spectra near these limits. The fact that both the isotropic and the anisotropic models fit the low- and hightemperature spectra nearly equally well could be interpreted to mean that we may no longer be able to deconvolute the tumbling motion into distinct modes at these limits. However, this seems unlikely, as the anisotropic model does fit the spectra better by utilizing fairly large anisotropies (N ) (τ⊥/τ|) > 10) in the motions (see Tables 2 and 3). Looking at the two scenarios described above, one can see why the isotropic model is capable of describing the contrast agents’ dynamics near the fast limit. At higher temperatures, the agents will tumble faster on account of the increased thermal energy. The viscosity is also decreased at higher temperatures. These changes will allow the agents to eventually tumble rapidly and average out the deviation from spherical symmetry (fast limit). Therefore, we observe that the isotropic model generates better and better fits as the temperature increases. Near freezing temperatures, all the motions become so slow that no significant rotational averaging occurs. Consequently,
Figure 6. Some representative VO(EDTA) (in 46% sucrose) EPR spectra and their anisotropic best fits: (a) τ⊥ ) 2.2 ns, τ| ) 79 ps, Φ ) 56°, N ) 28 (309 K), (b) τ⊥ ) 3.9 ns, τ| ) 0.19 ns, Φ ) 66°, N ) 20 (293 K), (c) τ⊥ ) 8.3 ns, τ| ) 0.30 ns, Φ ) 71°, N ) 36, (286 K), and (d) τ⊥ ) 24 ns, τ| ) 0.49 ns, Φ ) 34°, N ) 50 (271 K).
motion no longer contributes much to the spectra. The resultant spectra are thus mainly determined by the Zeeman and hyperfine interactions. As a result, the isotropic model becomes adequate to describe the spectra. It is worth noting that one can compute an “average” τR with the formula12
τRave ) xτ| × τ⊥
(4)
This numerical value can sometimes be useful in characterizing complex motions, and it can be compared to the results from the isotropic model to more clearly see differences in the two approaches. Tables 2 and 3 summarize our results. The isotropic and the anisotropic results are shown side by side for
Rotational Dynamics of MRI Contrast Agents
J. Phys. Chem., Vol. 100, No. 20, 1996 8097 TABLE 2: Best-Fit Results of the Anisotropic and Isotropic Rotational Models for VO(EDTA)a T (K) τ⊥ (ns) τ| (ns) τRave (ns) Φ(°) N
R2
τR (ns)
R2
309 300
2.2 2.2
0.079 0.094
0.41 0.46
56 59
28 0.99 24 0.98
0.26 0.29
0.98 0.97
293 289 286 281
3.9 6.1 8.3 12
0.19 0.26 0.30 0.35
0.85 1.3 1.6 2.1
66 69 69 71
20 23 28 36
0.43 1.7 2.0 2.5
0.92 0.86 0.89 0.93
276 271
26 24
0.38 0.49
3.1 3.4
43 34
69 0.98 50 0.98
3.2 4.3
0.95 0.96
0.97 0.97 0.98 0.98
a The horizontal spaces divide the data into the three regions discussed in the text.
TABLE 3: Best-Fit Results of the Anisotropic and Isotropic Rotational Models for VO(DTPA)a T (K) τ⊥ (ns) τ| (ns) τRave (ns) Φ(°) N
R2
τR (ns)
R2
325
1.7
0.12
0.45
59
15 0.99
0.33
0.98
310 306 298
2.3 3.9 5.4
0.20 0.24 0.32
0.68 0.97 1.3
65 66 69
11 0.97 16 0.97 17 0.98
0.48 0.54 1.5
0.92 0.91 0.91
0.40 0.54
3.0 3.1
41 28
57 0.98 33 0.99
3.0 5.2
0.95 0.98
287 275
23 18
a The horizontal spaces divide the data into the three regions discussed in the text.
Figure 8. τ⊥ vs η/T for both VO(EDTA) and VO(DTPA). The inset identifies the three regions discussed in the text for VO(EDTA).
Figure 7. Some representative VO(DTPA) (in 46% sucrose) EPR spectra and their anisotropic best fits: (a) τ⊥ ) 1.7 ns, τ| ) 0.12 ns, Φ ) 59°, N ) 15 (325 K), (b) τ⊥ ) 3.9 ns, τ| ) 0.24 ns, Φ ) 66°, N ) 16 (306 K), (c) τ⊥ ) 5.4 ns, τ| ) 0.32 ns, Φ ) 69°, N ) 17 (298 K), and (d) τ⊥ ) 18 ns, τ| ) 0.54 ns, Φ ) 28°, N ) 33 (275 K).
comparison. We have also computed the average τR from the anisotropic data. Note that the average τR values computed are within a factor of 2 of the isotropic τR values. Since the anisotropic simulations are orders-of-magnitude more expensive computationally than are the isotropic simulations (on our RS/ 6000, an isotropic simulation takes a few minutes while an anisotropic simulation may take hours to days), the isotropic model, although physically incorrect, may nevertheless be used to formulate a starting point for the anisotropic fitting. To further cut computational times, a basis set pruning following Vasavada et al.13 was done for each complex and for each model. The basis sets that yielded an error tolerance e10-4 were selected as the minimal basis sets for the simulations.
Figure 9. τ| vs η/T for both VO(EDTA) and VO(DTPA). The inset identifies the three regions discussed in the text for VO(EDTA).
Figures 8 and 9 show the plots of the anisotropic rotation data against viscosity and temperature. It is obvious that the Stokes-Einstein relationship is not obeyed for the entire temperature/viscosity range studied for either the perpendicular or the parallel motion. Both VO(EDTA) and VO(DTPA)
8098 J. Phys. Chem., Vol. 100, No. 20, 1996 exhibit nearly the same behavior for both motions. The figures suggest dividing the motional behavior into roughly three regions, indicated by the vertical lines shown in the insets for the VO(EDTA) data. Interestingly, the rotational correlation times are not the only indication of these three regions. They can also be discerned by looking at the R2 values of the isotropic fits. Regions I and III correspond to the data points that the isotropic model fits adequately, while region II corresponds to the data points that only the anisotropic model can explain. The anisotropic data also suggest these different regions. If one looks at the diffusion tilt angles (Φ), region III corresponds to a sharp drop in Φ. Region I is similarly indicated by a drop in Φ. Region II, on the other hand, shows a relatively constant Φ (≈70). The anisotropy parameter (N ) τ⊥/τ|) also suggests the three regions. Region II shows a monotonically increasing anisotropy, whereas regions I and III contradict this trend by an increase in N and a decrease in N, respectively. All these reveal that there are three regimes of motional behavior in our data. The three regions are also marked by the horizontal spaces in Tables 2 and 3. Although not central to our report of the existence and modeling of anisotropic motion in contrast agents, it is very interesting and challenging to propose and discuss possible models to explain the trends shown by our data. However, we realize that much of the theory of the condensed phase, especially of complex fluids like our system, remain unknown. Therefore, we strive in this discussion to develop an intuitive and physical picture of the dynamics. In doing so we must make simplifications and assumptions. However, it is our hope that our discussion will spur more discourse on the subject and further our understanding of fluid dynamics, especially that of contrast agents. Our solvent is fairly complex. It is composed of about 50% water and 50% sucrose molecules. Because of the existence of numerous hydroxyl groups, sucrose is hydrophilic. Therefore, water will interact chemically (i.e., form hydrogen bonds) with sucrose. This was shown by Einstein in his pioneering work on hydrodynamics, in which he found that the hydrodynamic radius of sugar in water is larger than that of the sugar itself.14 There are also possible interactions between water molecules and between sucrose molecules. But since we are considering this system from the contrast agent’s point of view and because the solvents are all capable of forming hydrogen bonds to the agent, we shall treat the solvent as a homogeneous environment. Therefore, from the agent’s point of view, the solvent offers frictional resistance to rotation and, more importantly, the ability to alter the agent’s dynamics dramatically via hydrogen bonding with the agents. We will, however, return to solvent-solvent interactions later when we discuss region III. We will now visit a contrast agent molecule, which we will assume to be an oblate molecule (e.g., with the ratio of the axes ≈1:2) surrounded by our solvent molecules defined above. It has a small diffusion tilt angle. At an arbitrarily high temperature where viscosity is low and thermal energy is high, the agent tumbles rapidly and appears, on the EPR time scale, spherical because the rapid rotation averages out the deviation from spherical symmetry and because solvent molecules cannot form hydrogen bonds with a lifetime long enough to be detectable on the EPR time scale (i.e., the average bond time is less than the inverse electronic Larmor frequency, or for X-band, ωs-1 ≈ 18 ps). This is the fast-tumbling limit (not reached in our study). Now we slowly lower the temperature. We begin to see that the contrast agent is really tumbling about two axes, one faster and one slightly slower, and that the faster axis and the magnetic
Chen et al. z-axis are not coincident, though the difference is small. About the slower rotation axis of the agent, we begin to see solvent molecules forming detectable hydrogen bonds. However, the solvent molecules have much less success at forming detectable bonds about the faster rotation axis. This will increase the anisotropy in the motion. As we continue to decrease the temperature, some solvent molecules succeed in forming detectable hydrogen bonds about the faster axis as well as about the slower axis. This will serve to decrease the anisotropy. (E.g., assume that the axes are of lengths 1 and 2 and that each hydrogen-bonded solvent layer adds 0.5 to the lengths. We have, initially, N ) 2 (2/1) with no solvent-agent bonds; subsequent binding of solvent molecules to only the slower axis will increase N: N(2 + 0.5 + 0.5)/1 ) 3. Finally, when solvent molecules bind to both axes N decreases: N ) (2 + 0.5 + 0.5 + 0.5 + 0.5)/(1 + 0.5 + 0.5) ) 2). We see that the asymmetric addition of the solvent molecules to certain favorable sites (e.g., negatively charged or electron-rich atoms) further shifts the diffusion z-axis away from the magnetic axis. The beginning of detectable anisotropic rotation corresponds to the onset of region I, and the rapid rotation allows the isotropic model to still be a reasonable approximation. Due to the different tumbling rates about the two axes, there will be slightly more solvent molecules bound about the slower axis. As we decrease temperature at some point the slower axis will be much more accessible to the solvent molecules than the faster axis. Many more solvent molecules will be bound about the slower axis, and we observe a dramatic decrease in the rotational rate about the slower axis. Meanwhile, the faster axis still forms hydrogen bonds with the solvents, though not as many solvent molecules can bind to it. This can be verified by examining the slopes in region II of the two motions in Figures 8 and 9. Anisotropy will now increase dramatically, and the isotropic model no longer can explain our observations. This is region II. The diffusion z-axis continues to shift away from the magnetic z-axis, though the angle appears to be stabilizing as we reach the maximum number of solvent molecules that can solvate the agent. Toward the lower temperature end of region II, the decreasing amount of thermal energy will allow the solvent molecules to form more bonds with each other than with the contrast agent so as to form a more orderly structure that is energetically less expensive. As a result, since the faster axis requires more thermal energy to bond, increasingly more solvent molecules, once broken off of the faster axis, may not return. Since there were relatively few solvent molecules bonded to the faster axis from the beginning, we observe that the rotational rate about this axis still decreases due to decreasing thermal energy, but at a slower rate. However, the slower axis is still energetically favorable for the solvent molecules to bond to and thus continues to hold on to the solvent molecules. Near freezing, however, the solvent molecules will also leave the slower axis, and we see that they leave en masse to add to the growing crystalline structure. At this point, the solvent molecules are no longer bound to the agent but to each other. This is region III. The anisotropy is at a maximum just at the beginning of region III, just as the solvent molecules start to freeze. The diffusion tilt angle consequently decreases as the diffusion z-axis returns toward the original position. Since the motion of solvent molecules impedes the agent’s motion, the formation of rigid solvent structures will in effect enable freer movement of the contrast agent. Since the slower axis motion is the motion most coupled to (or hindered by) the solvent, we observe a speeding up in the slower axis motion while the faster motion remains unperturbed.
Rotational Dynamics of MRI Contrast Agents Similar anomalous increases in one of the axial diffusion coefficients have been observed by several previous experimental studies on liquid crystals utilizing nitroxide probes.12,16,17 They found that the parallel rotational diffusion coefficient increased with decreasing temperature at the isotropic-nematic and then at the nematic-smectic transitions. While these studies utilized probes that resemble the solvent so as to study solvent diffusion characteristics, (thus are not strictly applicable to our problem where the solute is dissimilar to the solvent and may not undergo liquid crystalline changes), they reveal that in the presence of an ordering potential in the solvent (in our case, hydrogen bonding), anomalous diffusion characteristics can arise near a phase transition. In our study, we found evidence that when an agent undergoes anisotropic rotation in the very slow motional regime (i.e., near freezing) and when the motion of the agent is strongly coupled to or influenced by the solvent molecules, the perpendicular rotational diffusion coefficient increased with increasing viscosity near freezing. Allen also found a similar effect in the translational motion in his theoretical molecular dynamics study of the transition between the isotropic phase and the nematic phase.15 He observed that just above the transition the translational perpendicular diffusion coefficient (R⊥) for a very flat oblate ellipsoid increased while the translational parallel diffusion coefficient (R|) decreased monotonically as the density of the solvent increases (for a prolate ellipsoid, it is the parallel diffusion coefficient that showed the increase while the perpendicular diffusion coefficient decreased monotonically). He attributed this to the increasing order of the solvent environment that serves to offset the effect of the increase in friction,18 which is similar to the argument we invoke above to explain the decrease in τ⊥. While the contrast agent is unlikely to have a very flat oblate shape, the chemical interaction of the solvent molecules with each other and with the contrast agent could also make the solVated complex flat (oblate), as judged by the anisotropy parameter N in the results near phase transition. However, we note here that liquid crystalline arguments need not be used for our study; the transition from a liquid phase to solid phase without the intervening liquid crystalline phases will also provide the ordering solvent structure for our interacting (H-bond capable) solvent molecules at high viscosity. It should be noted that our results near the transition temperatures may be inaccurate since the dynamics may no longer be Browniansthe interaction between the solvent and the agent is no longer random but takes place along certain orientations. For example, the contrast agent could be held at a certain orientation for a brief period of time via interactions with a more oriented solvent structure and then “jump” to another orientation that also is favorable for interacting with the oriented solvent. A jump diffusion model can be used to model this, and information obtained from the simulation, such as the number of jump sites, can be utilized to infer solvent structure as well as contrast agent structure at the phase transition. It also is possible that the transition data may be better modeled with a solvent cage potential coupled to that of the contrast agent. In this case novel simulation programs developed by Freed’s group that take into account the cage potential can be used to better describe the data.19,20 However, the very high goodness-of-fit of the simpler model gives us confidence that the qualitative trend we observe near freezing is valid. In our simulations we have attempted to ascertain whether the powder pattern parameters, namely the g- and the Amatrices, are valid for motionally-modulated spectra. To accomplish this we fixed the motional parameters at the values
J. Phys. Chem., Vol. 100, No. 20, 1996 8099 found in Tables 2 and 3. We first varied the g diagonal matrix elements while holding the A diagonal matrix elements constant at the rigid-limit values. The starting values were the rigidlimit parameters reported in Table 1. The A-matrix then was varied with g-matrix elements held constant at the rigid-limit values. We then restarted the simulation again but held one set of matrix elements constant at the new values found by the simulation. All simulations were restarted to ensure that the global minimum was reached. As a check, we also used a starting vector composed of axially-symmetric matrix elements. Our results (not shown) indicate that varying these parameters only slightly improves the overall fitness of the simulation (about a 0.5% change in R2). Therefore, it appears that the rigid-limit parameters are valid throughout the dynamic range we studied. We also have studied residual line width (R′′) contributions, which may arise from unresolved hyperfine interactions, site inhomogeneities, and spin-rotational interaction. Again, we found that the goodness-of-fit changed only slightly (less than 0.2% improvement in R2). In most cases, the residual line widths are effectively zero. Only in region III do we observe a need for a residual line width contribution of about 2 G, probably caused by site inhomogeneities introduced by the freezing process. This is in accordance with the results found in our previous report1 as well as those found by Campbell and Freed.8 The lack of a significant residual line width contribution in slow-motional data reveals that the Brownian model is a good approximation for the dynamics experienced by the contrast agent in the slow-motional regime. 4. Conclusions The slow-motional results and the methodology developed here not only are useful to investigate anisotropic rotational dynamics of contrast agents in different environments but also could be used to augment nuclear magnetic relaxation dispersion studies where anisotropic rotational dynamics may arise and complicate the interpretation of the relaxation profile. The relaxation profile of a contrast agent undergoing anisotropic rotation is complex, and the conventional theory, the SolomonBloembergen equations,21,22 does not satisfactorily describe the experimental results. We point out here that Woessner has worked out theoretical modifications that may be applied to the Solomon-Bloembergen equations to account for anisotropic rotation.23,24 Finally, it has been shown that the relaxivity increases when the rotational dynamics is slowed, either by increasing the viscosity or by linking to a macromolecule.9,25 While at the viscosities studied in this report an isotropic rotational model may adequately describe the contrast agent’s rotational dynamics at physiological temperature, environments of different viscosity (from low, such as blood, to high, such as the extracellular matrix) exist in different body compartments that may induce anisotropic rotation in the agents. Moreover, in the case of macromolecular agents that may be developed to increase specificity, anisotropic rotation may play a significant role even in low-viscosity environments such as blood. Therefore, it is extremely important to the development of effective in ViVo contrast agents to understand how anisotropic rotation may modulate relaxation rates and how it may affect other physical processes. In this study, we found that in viscous solutions, anisotropic Brownian diffusion is the best model to describe the slowmotional regime of the contrast agents. Other motional models, such as a jump diffusion model or a solvent cage model, may be needed to better reflect the dynamics of the contrast agent near freezing. This model can be applied to study the motion of intracellular agents as well as extracellular agents where the
8100 J. Phys. Chem., Vol. 100, No. 20, 1996 viscosity is increased by the presence of proteins and factors (such as ground substance (glycosaminoglycans), RNA, keratin, collagen, etc.) and to study agents that noncovalently or covalently bond to macromolecules to increase the relaxivity and/or to increase the specificity. Acknowledgment. We thank Dr. David Schneider and Dr. David Budil for useful discussions and for supplying the EPRLF programs. We also thank the members of the Illinois EPR Research Center for their expert advice and helpful discussions. We thank Raymond Chen for laboratory assistance. J.W.C. thanks the University of Illinois College of Medicine for the Hazel I. Craig Fellowship and the U.S. Department of Education for the GAANN Fellowship in Chemistry. Most other facilities were provided by the Illinois EPR Research Center, an NIH Biomedical Technology Research Resource (P41-RR01811). Partial support for this work was provided by the NIH (GM42208). References and Notes (1) Chen, J. W.; Auteri, F. P.; Budil, D. E.; Belford, R. L.; Clarkson, R. B. J. Phys. Chem. 1994, 98, 13452. (2) Chen, J. W.; Auteri, F. P.; Belford, R. L.; Clarkson, R. B. Second Meeting of the Society of Magnetic Resonance, San Francisco, CA, 1994; Abstract 922. (3) Chasteen, N. D. Vanadyl(IV) EPR Spin Probes: Inorganic and Biochemical Aspects. Biological Magnetic Resonance; Berliner, L. J., Reuben, J., Eds.; Plenum Press: New York, 1981. (4) To obtain the software at the Illinois EPR Research Center’s ftp site use the following steps: 1. ftp ierc.scs.uiuc.edu; 2. at the login prompt type anonymous; 3. enter your e-mail address as the password; 4. type get README.1st to read what files are on the site and where they are located; 5. change directory to the desired directory with the command cd directory name; 6. issue the command get filename to get the desired file.
Chen et al. (5) Nilges, M. J. Ph.D. Thesis, University of Illinois, 1979. (6) Brent, R. P. Algorithms for Minimization without DeriVatiVes; Prentice-Hall: Englewood Cliffs, NJ, 1973; Chapter 5. (7) Nelder, J. A.; Mead, R. Comput. J. 1965, 7, 308. (8) Campbell, R. F.; J. H. Freed, J. H. J. Phys. Chem. 1980, 84, 2666. (9) Wiener, E. C.; Auteri, F. P.; Chen, J. W.; Brechbiel, M. W.; Gansow, O. A.; Schneider, D. J.; Belford, R. L.; Clarkson, R. B.; Lauterbur, P. C. J. Am. Chem. Soc., in press, (10) Freed, J. H. Theory of Slow Tumbling ESR Spectra for Nitroxides. Spin Labeling: Theory and Applications; Berliner, L. J., Ed.; Academic Press: New York, 1976; Vol. 1. (11) Schneider, D. J.; Freed, J. H. A User’s Guide to Slow-Motional ESR Lineshape Calculations. Biological Magnetic Resonance; Berliner, L. J., Reuben, J., Eds.; Academic Press: New York, 1989; Vol. 8. (12) Meirovitch, E.; Igner, E.; Igner, D.; Moro, G.; Freed, J. H. J. Chem. Phys. 1982, 77, 3915. (13) Vasavada, K. V.; Schneider, D. J.; Freed, J. H. J. Chem. Phys. 1987, 86, 647. (14) Einstein, A. InVestigations on the Theory of the Brownian MoVement; Dover Publications, Inc.: New York, 1956; translated by R. Fu¨rth and A. D. Cowper. (15) Allen, M. P. Phys. ReV. Lett. 1990, 65, 2881. (16) Polnaszek, C. F.; Freed, J. H. J. Phys. Chem. 1975, 79, 2283. (17) Lin, W.-J.; Freed, J. H. J. Phys. Chem. 1979, 83, 379. (18) Allen, M. P.; Evans, G. T.; Frenkel, D.; Mulder, B. M. AdV. Chem. Phys. 1993, 86, 1. (19) Polimeno, A.; Moro, G. J.; Freed, J. H. J. Chem. Phys. 1995, 102, 8094. (20) Earle, K. A.; Polimeno, N.; Moscicki, J.; Freed, J. H. 37th Rocky Mountain Conference on Analytical Chemistry, Denver, CO, 1995; Abstract 100. (21) Solomon, I. Phys. ReV. 1955, 99, 559. (22) Bloembergen, N. J. J. Chem. Phys. 1957, 27, 572. (23) Woessner, D. E. J. Chem. Phys. 1962, 36, 1. (24) Woessner, D. E. J. Chem. Phys. 1962, 37, 647. (25) Lauffer, R. B. Chem. ReV. 1987, 87, 901.
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