Molecular-Level Insights into the NMR Relaxivity of Gadobutrol Using Quantum and Classical Molecular Simulations

Abstract

MRI is an indispensable diagnostic tool in modern medicine; however, understanding the molecular-level processes governing NMR relaxation of water in the presence of MRI contrast agents remains a challenge, hindering the molecular-guided development of more effective contrast agents. By using quantum-based polarizable force fields, the first-of-its-kind molecular dynamics (MD) simulations of Gadobutrol are reported where the ¹H NMR longitudinal relaxivity r ₁ of the aqueous phase is determined without any adjustable parameters. The MD simulations of r ₁ dispersion (i.e., frequency dependence) show good agreement with measurements at frequencies of interest in clinical MRI. Importantly, the simulations reveal key insights into the molecular level processes leading to r ₁ dispersion by decomposing the NMR dipole–dipole autocorrelation function G(t) into a discrete set of molecular modes, analogous to the eigenmodes of a quantum harmonic oscillator. The molecular modes reveal important aspects of the underlying mechanisms governing r ₁, such as its multiexponential nature and the importance of the second eigenmodal decay. By simply analyzing the MD trajectories on a parameter-free approach, the Gadobutrol simulations show that the outer-shell water contributes ∼50% of the total relaxivity r ₁ compared to the inner-shell water, in contrast to simulations of (nonchelated) gadolinium-aqua where the outer shell contributes only ∼15% of r ₁. The deviation between simulations and measurements of r ₁ below clinical MRI frequencies is used to determine the low-frequency electron-spin relaxation time for Gadobutrol, in good agreement with independent studies.

1. Introduction

MRI (magnetic resonance imaging) contrast agents based on paramagnetic species are often used to shorten the ¹H NMR (nuclear magnetic resonance) relaxation times of nearby water molecules in bodily tissues, thereby improving image contrast. , Gadolinium-based contrast agents (GBCAs) are commonly used to achieve this contrast enhancement in clinical practice despite growing concerns about their safety. Studies have shown that GBCAs can bioaccumulate in the brain, bones, and kidneys, even in patients with normal renal function. − This poses significant health risks. The development of safer MRI contrast agents remains a major challenge, with an urgent need for new approaches to study paramagnetic ion hydration, chelation, and NMR relaxation, which could lead to the next generation of more sensitive and safer MRI contrast agents. ,

There has been a significant amount of work to develop new MRI contrast agents with lower toxicity and higher contrast, − including strategies to develop manganese-based chelating agents, modulate proton exchange in contrast agents or modify side chains of gadolinium-based contrast agents. Despite many efforts, no groundbreaking new generation of contrast agents has yet met the safety and efficacy goals for clinical use due, in part, to a limited understanding of the molecular-level relationships among chemical structure, chelation chemical stability, and the molecular-scale processes underlying NMR relaxation. In fact, because the molecular-level processes governing relaxivity in MRI remain poorly understood, the modeling and interpretation of the physics of NMR relaxation still rely on severe assumptions. , The extended Solomon-Bloembergen-Morgan (SBM) model, which is the mainstay approach for interpreting relaxation in MRI, assumes unlike-spin dipole pairs are tumbling hard-spheres (no potential of interaction between the dipoles) anchored at a fixed distance. − This model has been demonstrated to be insufficient to accurately describe even simple Gd³⁺-aqua complexes in water. Yet another centerpiece model is the so-called Ayant, Belorizky, Hwang, and Freed (ABHF) model, which aims to account for translational diffusion effects in the outer sphere around the paramagnetic ion. , The ABHF model also includes severe approximations such as hard-spheres diffusing in a viscous continuum, ignoring the anisotropy of the molecules and intermolecular potential interactions (e.g., hydrogen bonds in water). Their limitations notwithstanding, these models have helpfully guided the interpretation of NMR relaxation in MRI contrast agents. However, more physically accurate approaches or models with fewer parameters could significantly improve how this technology is used and more effectively direct the search for new contrast agents.

In this work, quantum and molecular simulations are used to investigate the NMR relaxivity of Gadobutrol, a commonly used MRI contrast agent, properly accounting for the potential of interaction in the system as well as the fluid structure and dynamics. This represents the first-of-its-kind molecular dynamics simulations of this MRI contrast agent system using highly refined polarizable models. Gadobutrol is a second-generation GBCA, and its macrocyclic structure (DO3A-butrol) forms a stable complex with gadolinium and enhances its NMR relaxivity response compared to first-generation linear GBCAs, making it a preferred choice in many MRI applications. , The trihydroxybutyl group in Gadobutrol enhances the hydrophilicity of the complex, thus increasing the in vivo tolerance while lowering protein binding. At frequencies relevant to MRI and human body temperature, simulations of NMR relaxivity show good agreement with experiments, without any free parameters or assumed NMR relaxation models. Overall, this methodology enables an understanding of the inherent molecular-level properties and NMR relaxivity behavior of MRI contrasts, serving as a tool for the development of emerging “next-generation” GBCAs and alternative classes of MRI contrast agents based on paramagnetic species such as manganese and iron.

2. Methodology

2.1. AMOEBA Force Field Parameters

The AMOEBA polarizable force field is used to model gadolinium, water, and Gadobutrol. The AMOEBA model is intermediate in complexity between a fixed charged model and a fully quantum mechanical model that allows electron densities to distort or delocalize and for atomic orbitals of the heavy metal ion to respond to the field of the ligands. Earlier exploratory calculations established the unsuitability of the fixed charge model, a so-called classical, nonpolarizable force field, for modeling Gd­(III)-aqua, but a fully quantum mechanical model is also not feasible computationally. In the AMOEBA model, the physics of polarization is captured, but not the physics of charge transfer or subtle effects related to electron spin relaxation dynamics. , As the results below indicate, electron spin relaxation effects are likely muted in the ¹H NMR frequencies of interest in MRI practiced today (f ₀ ≳ 5 MHz). The physics of charge transfer is important in the context of metal ions, but the results support the idea that the polarizable model captures a substantial part of the physics of ¹H relaxivity r ₁ at MRI frequencies.

Within MD simulations, metal coordination bonds can be modeled in force fields via bonded methods or nonbonded methods, and in most cases the parameters are usually adjusted to ensure that the distances between the metal and electron-donor groups match experiments. , In the absence of experimental data on the structure of Gd­(III)-DO3A-butrol chelate in liquid water at human body temperature, the approach in this study follows a nonbonded method for the metal coordination and does not assume any a priori structure for metal coordination. Thus, we performed the force field parametrization of bare aqueous DO3A-butrol to ensure that all atomic charges and multipoles, as well as key energetic contributions of this chelating agent, were accurately described. This approach ensures the transferability of parameters, allowing the same molecular description of DO3A-butrol to be applied across different chemical environments, including in the presence of other metal ions. The nontrivial agreement obtained in computed r ₁ relaxivities for the Gd³⁺-DO3A-butrol complex (see below) indicates substantial validity for this approach.

The AMOEBA electrostatics parameters for DO3A-butrol were calculated using the Poltype2 package, , which performed different calculation steps automatically from quantum mechanics simulations to force field parameter estimation. Following the AMOEBA parametrization protocol, , the initial guess structure for DO3A-butrol was optimized using the Gaussian package at the MP2/6–311G­(1d,1p) level of theory. Vibrational frequency calculations were performed to confirm that the optimized structure was an actual minimum point. Figure (a) shows the optimized structure of DO3A-butrol obtained from quantum simulations, while Figure (b) shows the Gd³⁺-DO3A-butrol complex and the adopted number indexing for the oxygen and nitrogen atoms in the DO3A-butrol structure. Next, the electronic density of the optimized structure of DO3A-butrol was analyzed by the Gaussian Distributed Multipole Analysis package (GDMA), which computed initial partial charges, dipoles, and quadrupoles for each atom with respect to local frames defined by the local symmetry of the atom and its surrounding neighbors. , Finally, these dipole and quadrupole moments were refined using the Potential module of the Tinker package to accurately reproduce the electrostatic potential (ESP) surfaces calculated using quantum simulations at a higher level of theory (MP2/aug-cc-pVTZ), with a convergence threshold of 0.1.

1.

(a) Representation of the optimized structure of the ligand DO3A-butrol at the MP2/6–311G­(1d,1p) level of theory. (b) Molecular representation of the Gd³⁺-DO3A-butrol complex and the corresponding indexing adopted to identify different oxygen and nitrogen atoms in the structure.

The van der Waals, polarizability, bond, and angle parameters for the model were assigned by Poltype2 by identifying transferable bonded and nonbonded parameters available in the AMOEBA database. Dihedral parameters were estimated for the specific case of DO3A-butrol. This was performed by isolating the different dihedral types starting from the QM-optimized geometry of DO3A-butrol, and varying the dihedral angles along the conformational energy surface to generate configurations using the Minimize module of Tinker, as well as the Analyze module for computing the corresponding energies. , These configurations were later optimized by using quantum simulations, and their energies were determined at the ωB97X-D/6–311+G* level of theory. Finally, the AMOEBA dihedral parameters were obtained by fitting to best reproduce the QM torsion energy profiles. All of the AMOEBA parameters obtained for DO3A-butrol, as well as AMOEBA parameters used for water and gadolinium­(III), are available in Supporting Information.

2.2. Molecular Simulation Details

The longitudinal NMR relaxation of the Gd³⁺-DO3A-butrol complex in water was investigated at 37°C (human body temperature). The molecular structures and the simulation box were assembled using the Packmol package, with 2006 molecules of water and one Gd³⁺-DO3A-butrol complex at an initial random conformation. Simulations were conducted in triplicate using the GPU-accelerated OpenMM-7.5.1 package in cubic simulation boxes with periodic boundary conditions. In accordance with the AMOEBA force field guidelines, the real space electrostatic potential was truncated at 0.9 nm, van der Waals interactions were treated with a cutoff of 1.2 nm, and a mutual polarization convergence tolerance of 10^–5 Debye was applied. Long-range electrostatic interactions were handled via the particle-mesh Ewald (PME) method with a force error tolerance set to 10^–5. A time step of 0.5 fs was employed following recommendations to ensure proper energy equipartition in the system.

The simulation protocol started in the isothermal–isobaric ensemble (NpT) at the desired temperature (37°C) and atmospheric pressure (1 atm) using a Monte Carlo isotropic barostat with a frequency of 25 fs. Within a 1 ns time frame, the density of the systems stabilized, and the dimensions of the simulation boxes settled to approximately 40 Å. The final configuration was used as an input for the following steps in the canonical ensemble (NVT) using three chains of Nosé-Hoover thermostats with a collision frequency parameter of 10 ps^–1. In order to allow the Gd³⁺-DO3A-butrol complex to probe different chelation conformations until the most stable equilibrium chelated structure is obtained, a Simulated Annealing procedure was adopted starting with a 0.5 ns temperature rising step from 37 to 97°C (right before the liquid–vapor phase transition), followed by a 5.0 ns step at 97°C, then a 0.5 ns step of temperature dropping from 97 to 37°C, and finalized by a 2.0 ns step of equilibration at 37°C. The subsequent production step was carried out for 15 ns, and the atomic coordinates were stored every 0.05 ps. The analysis of the data was performed with the last 13.1072 ns (131072 or 2¹⁷ frames) of the production set.

The structure of the Gd³⁺-DO3A-butrol complex was analyzed in terms of the radial distribution function g _j (r) and the coordination number n _j (r) around the paramagnetic ion of any atom type j (either oxygen, hydrogen, or nitrogen). The radial distribution function was calculated as ,

$$g_j(r)=\frac{⟨{\rho }_j(r)⟩}{⟨{\rho }_j⟩}$$ 1

where ⟨ρ _j (r)⟩ is the average density of particles j at a distance r from the central paramagnetic ion, and ⟨ρ _j ⟩ is average density of particles j in the entire simulation box. The radial integration of g _j (r) leads to the coordination number as a function of distance r as in ,

$$n_j(r)={\int }_0^{r}4\pi r^2{g}_j(r)\mathrm{d}r$$ 2

The Brownian dynamics of the Gd³⁺-DO3A-butrol complex and the water molecules in the outer shell around the complex can be characterized by the self-diffusion coefficient D _i , where i is any molecule of interest in the system. Because finite-size effects of the simulation box are known to play a role in the calculation of mass-transport properties, the self-diffusion coefficients are obtained through Einstein’s relation followed by the Yeh and Hummer analytical correction to determined the self-diffusivity D _i at the thermodynamic limit, i.e., −

$$D_i^{\infty }=\underset{t\to \infty }{\lim }\frac{1}{6t}⟨\mathrm{\Delta }{r}_i^2(t)⟩+\frac{k_{\mathrm{B}}T}{6\pi \eta }\frac{\xi }{L}$$ 3

where t is time, and ⟨Δr _i (t)⟩ is the mean square displacement of molecules i, k _B is the Boltzmann constant, T is temperature, η is the shear viscosity of the system, ξ = 2.837297 for cubic simulation boxes, and L is length of the box. It is important to highlight that the Yeh and Hummer correction is critical to determine the proper diffusivity of the solvent in the simulation box (i.e., water), but solutes close to the infinite dilution limit are barely subjected to finite-size effects, and therefore, this correction is not used in such cases.

2.3. ¹H NMR Relaxivity

The NMR relaxation dipole–dipole autocorrelation function, G(t), between the paramagnetic ion Gd³⁺ and the ¹H nuclear magnetic dipole from the water molecules can be determined by analyzing the dynamics of thermal fluctuations around equilibrium. Assuming that the system is isotropic, and adopting an arbitrary direction for the hypothetical external magnetic field B ₀, the NMR dipole–dipole autocorrelation function, G _i (t), can be determined due to fluctuating magnetic fields around the i-th ¹H in water (with spin I = 1/2) in the presence of N _S electron dipoles (with spin S) by −

$$G_i(t)=\frac{1}{4}{\left(\frac{{\mu }_0}{4\pi }\right)}^2{\mathrm{\hslash }}^2{\gamma }_I^2{\gamma }_S^{2}S(S+1)\times \frac{1}{N_S}\sum _{s=1}^{N_S}{⟨\frac{(3{\cos }^2{\theta }_{is}(t+t^{\prime })-1)}{r_{is}^3(t+t^{\prime })}\frac{(3{\cos }^2{\theta }_{is}(t^{\prime })-1)}{r_{is}^3(t^{\prime })}⟩}_{t^{\prime }}$$ 4

where μ₀ is the vacuum permittivity, ℏ is the reduced Planck’s constant (ℏ = h/2π), γ _I and γ _S are the gyromagnetic ratios for particles with spin I and S, respectively, t′ is the lag-time, and θ _is is the polar angle between the dipoles with respect to the external magnetic field B ₀. The current simulation uses a single Gd³⁺ ion (N _S = 1) in the simulation box with a spin of S = 7/2. To improve statistical sampling, these autocorrelation functions were calculated over 7 different (and equidistant) directions for the magnetic field B ₀, important to compute the angles in eq ; over the three independent simulations, this represents a total of 21 autocorrelation functions, which were averaged to the final NMR relaxation autocorrelation function. , This approach is preferable over using a simplified version of eq in which isotropicity is already assumed, because the former has better statistics and provides autocorrelation functions with less noise.

The autocorrelation function G(t) at t = 0, defined as the amplitude G(0), is an important property of the system related to the second moment (i.e., strength) of the dipole–dipole interactions. According to eq , G(0) depends on the distance r between the Gd³⁺ and ¹H according to the functional form 1/r ⁶, i.e.,

$$G(0)=\frac{1}{5}{\left(\frac{{\mu }_0}{4\pi }\right)}^2{\mathrm{\hslash }}^2{\gamma }_I^2{\gamma }_S^{2}S(S+1)\sum _{i=1}^{N_I}{⟨\frac{1}{r_{is}^6(t^{\prime })}⟩}_{t^{\prime }}$$ 5

Given that dipoles in the liquid undergo several different motions over a wide range of time scales, such processes can be thought as different frequencies of atomic motion. Thus, the Fourier analysis of the NMR relaxation autocorrelation function decomposes the response into a spectrum of signals over multiple frequencies, also known as the spectral density function, J _i (ω). This spectral density function can be calculated through the two-sided even Fourier transform of G _i (t) by

$$J_i(\omega )=2{\int }_0^{\infty }{G}_i(t)\cos (\omega t)\mathrm{d}t$$ 6

where J _i (ω) is the spectral density function for the i-th ¹H in water. Given that a continuous spectrum of atomic motion frequencies (bond and angle stretching, rotation, translation, diffusion, etc.) is sampled during the simulations, J _i (ω) can be recovered over the entire range of frequencies of interest, enabling NMR dispersion analysis. Because of quantum selection rules, NMR relaxation times will inherently depend on multiples (0, 1, and 2) of such frequencies in the spectral density function J _i (ω).

For the case of ¹H (I = 1/2 and Larmor frequency ω₀ = γ _I B ₀ = 2πf ₀) in water interacting with the paramagnetic ion Gd³⁺ (S = 7/2 and Larmor frequency ω _e = 658ω₀), because ω _e ≫ ω₀, the longitudinal relaxation time T ₁ is given in terms of the spectral density function by

$$\frac{1}{T_{1i}}=J_i({\omega }_0)+\frac{7}{3}{J}_i({\omega }_e)$$ 7

where T _1i is the longitudinal relaxation time for the i-th ¹H in water.

The expressions for the total NMR relaxation autocorrelation function G(t), total spectral density function J(ω), and average (i.e., measured) longitudinal relaxation time T ₁ over all the N _I number of ¹H nuclei in water are given by

$$G(t)=\sum _{i=1}^{N_I}{G}_i(t)$$ 8

$$J(\omega )=\sum _{i=1}^{N_I}{J}_i(\omega )$$ 9

$$\frac{1}{T_1}=\frac{1}{N_I}\sum _{i=1}^{N_I}\frac{1}{T_{1i}}$$ 10

A more convenient way to express longitudinal relaxation is to write it in terms of its relaxivity rate r ₁, given by

$$r_1=\frac{1}{[{\mathrm{G}\mathrm{d}}^{3+}]}\frac{1}{T_1}=\frac{1}{[\mathrm{H}]}\sum _{i=1}^{N_I}\frac{1}{T_{1i}}$$ 11

where [Gd³⁺] is the molar concentration of gadolinium ions (which is equal to the molar concentration of Gadobutrol), [H] is the molar concentration of ¹H atoms in water (where [H] = 2­[H₂O]), and [Gd³⁺] = [H]/N _I for the simulation box. At 37 °C and 1 atm, the molar concentration of pure water is [H₂O] ≈ 55,138 mM, and thus [H] ≈ 110,276 mM.

At this point, observe that the “fast-exchange” regime can be considered because the relaxation happens much faster than the residence time τ _m of water molecules around the paramagnetic ion. , This implies that the relaxivity rate r ₁ is independent of τ _m , and it can thus be compared with NMR relaxation measurements of the observed relaxation rate R ₁ by

$$R_1=[{\mathrm{G}\mathrm{d}}^{3+}]r_1+R_{1,DI}$$ 12

where [Gd³⁺] is the molar concentration of Gadobutrol in the experiment, and R _1,DI is the relaxation rate of deionized water where [Gd³⁺] = 0.

2.4. Recovery of Molecular Eigenmodes

From theoretical developments, it is known that magnetic dipoles undergoing NMR relaxation under normal diffusion lead to a multiexponential decay in the NMR relaxation dipole–dipole autocorrelation function. Therefore, G(t) from molecular simulations can be written as

$$G(t)=\sum _{k=1}^{n}P({\mu }_k)\exp (-{\mu }_{k}t)$$ 13

where n is the number of exponential decays, and P(μ _k ) is the amplitude (probability) for an eigenmode with decay-rate μ _k . The goal is to find the underlying amplitudes and decay rates in the G(t) from molecular simulations, keeping in mind the limitations due to simulation noise and time resolution constraints (i.e., time decays shorter than the sampling frequency of the trajectory cannot be captured).

Observe that the Laplace transform of G(t), as a multiexponential function, is given by

$$L(p)={\int }_0^{\infty }\exp (-pt)G(t)\mathrm{d}t$$ 14

$$L(p)=\sum _{k=1}^n\frac{P({\mu }_k)}{p-{\mu }_k}$$ 15

The values of P(μ _k ) and μ _k in eq will be obtained by rewriting L(p) in terms of its Padé approximant of order [n – 1, n]­

$$L(p)=\frac{A_{n-1}(p)}{B_n(p)}$$ 16

in which A _n–1(p) is a polynomial of order n – 1 and B _n (p) is a polynomial of order n. Next, eq is expanded in a Taylor series around any arbitrary point p ₀ as

$$d_0+d_1(p-p_0)+...+d_{2n-1}{(p-p_0)}^{2n-1}=\frac{a_0+a_1(p-p_0)+...+a_{n-1}{(p-p_0)}^{n-1}}{1+b_1(p-p_0)+...+b_n{(p-p_0)}^n}$$ 17

where a good choice for p ₀ is the inverse of the half decay time of G(t). Notice that the Taylor series coefficients d ₀, d ₁, ..., d _2n–1 are given by

$$d_k=\frac{1}{k!}{\left(\frac{d^{(k)}L}{d{p}^{(k)}}\right)}_{p=p_0}$$ 18

and can be calculated using numerical integration of the derivatives of eq by

$$d_k=\frac{1}{2}\frac{\mathrm{\Delta }t}{k!}[{(-t_0)}^k\exp (-p_0{t}_0)G(t_0)+{(-t_{M-1})}^k\exp (-p_0{t}_{M-1})G(t_{M-1})]+\frac{\mathrm{\Delta }t}{k!}\sum _{n=1}^{M-2}{(-t_n)}^k\exp (-p_0{t}_n)G(t_n)$$ 19

Once all coefficients d ₀, d ₁, ..., d _2n–1 are determined, the coefficients from the Taylor expansion of A _n–1(p) and B _n (p) can be calculate by substituting the values of d ₀, d ₁, ..., d _2n–1 into eq as in

$$[d_0+d_1(p-p_0)+...+d_{2n-1}{(p-p_0)}^{2n-1}][1+b_1(p-p_0)+...+b_n{(p-p_0)}^n]=a_0+a_1(p-p_0)+...+a_{n-1}{(p-p_0)}^{n-1}$$ 20

and solving the corresponding linear system formed by ensuring that all the terms with like-powers of p on both sides of the equation agree. Finally, by performing a partial fraction decomposition of the polynomial ratio $\frac{A_{n-1}(p)}{B_n(p)}$ using MATLAB, a sum of fractions is obtained as in eq from where the values of P(μ _k ), μ _k , and the number of poles N that are physical can be identified. , Because the number of poles present in the signal is not initially known, N is determined by trying different orders of Padé approximants starting from n = 1 up to point n = N + 1 when no more new physical poles are identified. The final physical poles and corresponding amplitudes from this inversion process are usually refined using a standard Particle Swarm Optimization (PSO) procedure to improve the adjustment to the simulation data.

Since it is usually more convenient to write characteristic times rather than rates for different NMR relaxation decays, and by recognizing that μ _k = 1/τ _k , eq can be rewritten by

$$G(t)=\sum _{k=1}^{n}P({\tau }_k)\exp (-\frac{t}{{\tau }_k})$$ 21

Notice that the combination of all n physical poles identified in the system should satisfy

$$G(0)=\sum _{k=1}^{n}P({\tau }_k)$$ 22

from which an average characteristic time ⟨τ⟩ can be defined by

$$⟨\tau ⟩=\frac{1}{G(0)}\sum _{k=1}^{n}P({\tau }_k){\tau }_k$$ 23

Finally, the spectral density is then determined (using eq ) as such

$$J(\omega )=\sum _{k=1}^{n}P({\tau }_k)\frac{2{\tau }_k}{1+{(\omega {\tau }_k)}^2}$$ 24

2.5. NMR Relaxation Measurements

The Gadobutrol complex, i.e., [Gd­(DO3A-butrol)­H₂O], was obtained from the commercial Gadavist ^Ⓡ solution used in clinical MRI. A solution of the concentrated contrast agent was prepared in deionized water with an equivalent concentration of [Gd³⁺] = 1.0 mM, along with a [Gd³⁺] = 0 sample to determine R _1,DI for pure deionized water. Observe that the concentration of Gd³⁺ is the same concentration of the [Gd­(DO3A-butrol)­H₂O] complex since its stoichiometry relationship is 1:1.

The ¹H longitudinal relaxation rate R ₁ dispersions of the Gd³⁺-DO3A-butrol was measured across a frequency range from 30 kHz to 35 MHz (¹H Larmor frequency) using a Spinmaster FFC2000 1T Relaxometer (Stelar s.r.l., Mede (PV), Italy). Standard prepolarized (PP) and nonpolarized (NP) sequences were used to determine R ₁. , The polarization and acquisition magnetic fields were fixed at 15 and 16.3 MHz, respectively, for all measurements. A switching time of 3 ms was employed with a field slew rate of 13 MHz/ms. Additionally, R ₁ was measured at fixed magnetic fields of 90 MHz, using a 2.1 T permanent magnet; 300 MHz, with a Varian NMR spectrometer at 7.0 T; and 400 MHz, with a Bruker spectrometer at 9.4 T. Relaxation profiles were recorded at 37°C, with experimental uncertainties typically below 6%, covered by the size of the data points in the graphs.

3. Results and Discussion

3.1. Inner-Shell Structure

The structure of the hydration inner-shell of Gd³⁺-DO3A-butrol was investigated at human body temperature (37°C). Figure shows the radial distribution function and the radial coordination number of oxygen and hydrogen atoms from water around the gadolinium ion for both Gd³⁺-DO3A-butrol and Gd³⁺-aqua complexes (for comparison purposes). Observe that, while the Gd³⁺-aqua complex displays a q = 8 ↔ 9 water molecules coordination number in the hydration inner-shell, the Gd³⁺-DO3A-butrol complex displays only q = 1 water molecule coordinating the ion in its inner shell, which is in agreement with experimental measurements for Gadobutrol. , Because the inner shell of Gd³⁺-DO3A-butrol is not radially symmetric with respect to the central ion due to the steric effects caused by the chelating agent (DO3A-butrol), a proximal radial distribution was calculated: A cone with vertex angle of 30° centered at gadolinium with axis pointing toward the center of an oxygen atom of a water molecule in the inner shell is used to sample the inner-shell water in all cases. Figure shows that the inner shell of Gadobutrol is shifted by ∼0.2 Å compared to the inner shell of Gd³⁺-aqua, and the former is also broader by about ∼0.15 Å and presents a smaller amplitude for the same coordination number (i.e., same area according to the integration in eq ).

2.

Radial distribution function g(r) and number coordination function n(r) for (a) Gd³⁺-O _w from water molecules and (b) Gd³⁺-H _w from water molecules. The continuous lines () represent g(r), while the dashed lines (---) represent n(r).

3.

Radial distribution function g(r) and number coordination function n(r) for Gd³⁺-O _w from water molecules within a conical region starting from the center of Gd³⁺ and pointing toward one O _w atom with an angle of angle of 30°. The continuous lines () represent g(r), while the dashed lines (---) represent n(r).

These findings about the structure of the inner shell were corroborated by quantum simulations of the Gd³⁺-aqua and Gd³⁺-DO3A-butrol complexes at the meta-GGA level of theory with a continuum solvent model for water at 37°C. Table summarizes the distances of the inner shell for the studied complexes obtained by both quantum and molecular dynamics simulations, showing agreement between these two approaches. Quantum calculations were performed with the uM06 functional, which has good accuracy for describing transition metal chemistry, including for systems involving noncovalent interactions such as chelation. Relativistic effects in the inner-core electrons of gadolinium were accounted for by effective core potentials (ECP) with the def2-TZVPP basis set, while the remaining atoms in the system were simulated with an aug-cc-pVTZ basis set.

1. Average Distances (Å) Obtained via Molecular Dynamics (MD) and Quantum Mechanics (QM) Simulations for Gd³⁺ with Respect to Oxygen and Hydrogen Atoms in Water in the Inner Shell of the Ion, for Both Gd³⁺-Aqua and Gd³⁺-DO3A-Butrol at 37°C .

	Gd3+-aqua		Gd3+-DO3A-butrol
	MD	QM	MD	QM
Gd3+-O w (inner shell)	2.45 ± 0.05	2.47	2.64 ± 0.03	2.59
Gd3+-H w (inner shell)	3.04 ± 0.05	3.13	3.25 ± 0.03	3.19

^aQuantum simulations were performed at the uM06/aug-cc-pVTZ level of theory and with the def2-TZVPP+ECP basis set for the Gd­(III) ion.

^bAverage between the [Gd­(H₂O)₈]³⁺ and [Gd­(H₂O)₉]³⁺ complexes.

3.2. Aqueous Structure

The structure of the Gd³⁺-DO3A-butrol complex in aqueous media was also investigated. It is known that the crystalline structure of the dimer [Gd­(DO3A-butrol)]₂ is dehydrated and presents the Gd³⁺ coordinated by 8 electron-donor groups, namely the four nitrogen atoms (N₍₁₎, N₍₂₎, N₍₃₎, and N₍₄₎), the three negatively charged carboxylate oxygen atoms (O₍₁₎, O₍₃₎, and O₍₅₎), and one of the hydroxyalkyl oxygen atoms (O₍₇₎). However, differently from the solid structure, the average aqueous Gadobutrol complex does not present all eight electron-donor groups of DO3A-butrol chelating the paramagnetic ion. Figure presents the radial distribution function of all of the electron-donor groups in the DO3A-butrol structure with respect to the Gd³⁺ ion at equilibrium in aqueous media and human body temperature (37°C).

4.

Radial distribution function g(r) and number coordination function n(r) for (a) Gd³⁺-N _i for all nitrogen atoms in DO3A-butrol (i = 1 – 4), and (b) Gd³⁺-O _i for all oxygen atoms in DO3A-butrol (i = 1 – 9), at 37C°. The continuous lines () represent g(r), while the dashed lines (---) represent n(r). See Figure (b) for indexing adopted to identify different oxygen and nitrogen atoms in the structure

To the extent that the relevant physics is captured in the MD simulations, all four nitrogen atoms (N₍₁₎, N₍₂₎, N₍₃₎, and N₍₄₎) are asymmetrically chelating the ion, but only two oxygen atoms (O₍₁₎ and O₍₇₎) of the DO3A-butrol ligand (along with a water molecule) chelate the Gd­(III) ion. Although one cannot claim that this is the most stable physical conformation of the complex in water, there are good reasons why this result is reasonable for the structure in solution: (1) there is strain in the chelation ring that is relieved when two of the oxygens move out and (2) that movement is also aided by favorable gain in hydration by the molecular solvent. The asymmetric chelation provided by the DO3A-butrol nitrogen atoms in an aqueous environment is due to the asymmetric nature of the chemical structure of Gadobutrol. Table shows that, within error bars, MD simulations present all three nitrogens connected to a carboxylate groups in DO3A-butrol (namely N₍₁₎, N₍₃₎, and N₍₄₎) chelating the Gd³⁺ at the same average distance of about ∼2.48 Å, and the nitrogen atom connected to the hydroxyalkyl group (namely N₍₂₎) chelating the central ion weakly and at a distance of about ∼3.09 Å. Unfortunately, no experimental data are available on the structure of Gadobutrol in liquid water at the human body temperature to directly support the results obtained from molecular dynamics simulations.

2. Average Distances (Å) Obtained via Molecular Dynamics (MD) and Quantum Mechanics (QM) at the Simulations for Gd³⁺ with Respect to Different Electron-Donor Groups in DO3A-Butrol and for the Water in the Inner Shell at 37°C .

	[Gd-(DO3A-butrol)(H2O)]			[Gd(DO3A-butrol)]2
	MD	QM (vacuum)	QM (implicit water)	Crystalline
Gd3+-O w (inner shell)	2.64 ± 0.03	2.51	2.47	
Gd3+-N(1)	2.48 ± 0.03	2.68	2.66	2.648–2.741
Gd3+-N(2)	3.09 ± 0.03	3.10	2.82
Gd3+-N(3)	2.45 ± 0.03	2.75	2.67
Gd3+-N(4)	2.51 ± 0.03	2.68	2.63
Gd3+-O(1)	2.26 ± 0.03	2.34	3.642	2.342–2.427
Gd3+-O(3)	5.50 ± 0.03	2.29	2.29
Gd3+-O(5)	4.10 ± 0.03	2.23	2.37
Gd3+-O(7)	2.48 ± 0.03	2.48	2.53	2.405

^aQuantum simulations were performed at the uM06/6-311G level of theory and with the def2-TZVPP+ECP basis set for the Gd­(III) ion. Experimental crystallographic value for [Gd­(DO3A-butrol)]₂ in the solid state are present for comparison.

^bSignifies that the atom is not chelating the Gd³⁺ ion in the complex.

The MD results for the structure of Gadobutrol were compared against quantum chemical calculations at the uM06/6–311G level of theory with the def2-TZVPP basis set and effective core potentials for Gd­(III), with and without implicit water solvent. Without any solvent, quantum chemical calculations show N4O4 chelation. However, with a surrounding water described as a continuous dielectric medium, stable structures where Gd­(III) is coordinated by four nitrogens and only three oxygens are found; these are elements of the same ring distortion found in the MD simulations. The lack of thermal effects in the quantum calculation (beyond the presence of a temperature dependent dielectric) is a likely reason the quantum chemical calculation does fully recapitulate the binding motif seen in the MD simulations. Table summarizes these results along with crystallographic data for the dimer of Gadobutrol dehydrated and in the solid state, for the purpose of comparison.

Figure represents the coordination structure obtained from molecular dynamics (MD) simulations of the two stable Gd³⁺-aqua complexes and the Gadobutrol complex in aqueous media at 37°C. Observe that the [Gd­(H₂O)₈]³⁺ complex presents a bicapped trigonal prism (BTP) structure, while the [Gd­(H₂O)₉]³⁺ presents a monocapped square antiprism (CSAP) structure. These results are in agreement with theoretical calculations for the most stable structure of ML₈ and ML₉ coordination complex. It was believed that Gadobutrol would present a distorted CSAP with eight electron-donor groups in the chelating agent and one solvent molecule, while the actual structure observed in the MD simulations is closer to a distorted capped octahedron, , with six electron-donor groups from the chelating agent and one solvent molecule. Similar distorted capped octahedron structures with seven ligand groups were observed in other chelating complexes, including with other lanthanide ions such as Dy³⁺. ,

5.

Average molecular structures obtained with MD simulations in aqueous media at 37°C. (a) The [Gd­(H₂O)₈]³⁺ complex presents a bicapped trigonal prism (BTP) structure, while (b) the [Gd­(H₂O)₉]³⁺ complex displays a monocapped square antiprism (CSAP) structure, and (c) the [Gd­(DO3A-butrol)­(H₂O)] complex presents 6 electron-donor groups from the chelating agent and one water molecule, forming a distorted capped octahedron structure.

3.3. Molecular Eigenmodes of NMR Relaxation

The NMR relaxation autocorrelation function for the Gadobutrol complex at human body temperature was investigated by using trajectories from MD simulations via eq and . Figure (a) shows the NMR dipole–dipole autocorrelation function G(t) for both Gd³⁺-DO3A-butrol and Gd³⁺-aqua complexes for comparison purposes.

6.

(a) NMR relaxation autocorrelation function over time for both Gd³⁺-aqua and Gd³⁺-DO3A-butrol complex obtained via MD simulations at 37°C, and (b) the corresponding normalized function over time normalized by ⟨τ⟩. The dashed lines (---) in both plots represent the corresponding NMR relaxation autocorrelation function reconstructed with Padé-Laplace inversion results according to eq , and the dotted line in plot (b) represents the prediction from the SBM model.

Figure (b) shows the NMR autocorrelation function normalized in amplitude by G(0) as a function of the corresponding time normalized by the average characteristic ⟨τ⟩. Notice that the Solomon-Bloembergen-Morgan (SBM) model, which assumes a single exponential time decay for the autocorrelation, is not able to capture the correct underlying relaxation for neither Gd³⁺-aqua nor Gd³⁺-DO3A-butrol, with Gadobutrol deviating more substantially from the single exponential decay approximation. In order to identify the underlying exponential decays in the NMR autocorrelation function according to eq for both Gd³⁺-aqua and Gd³⁺-DO3A-butrol at 37°C, the Padé-Laplace (PL) inversion technique was employed as described in Section . The reconstructed NMR autocorrelation function from the Padé-Laplace modes is presented in Figures (a) and (b) for both complexes, showing good agreement with the original MD simulation results in the region where the autocorrelation is not subject to much simulation noise (i.e., commonly at shorter times).

During the course of the simulation, several dynamical processes occur in the system, including the rotation of water molecules around their axis in the bulk and around the central paramagnetic ion, the rotation of water molecules around the inner shell, and the translation of water molecules from/toward the paramagnetic ion. The combination of all these and other processes govern how the relative distance between each hydrogen atom and the Gd³⁺ ion, as well as their angles with respect to the external magnetic field B ₀, change over time (i.e., r(t), θ­(t), and ϕ­(t)). The overall dynamic relative motion between each ¹H–Gd³⁺ pair can be described by a diffusion propagator function ρ­(r, t), which is related to G(t) through an integral relationship. This diffusion propagator contains all of the information on the relaxation dynamics and can be written as a summation of different eigenstates of the system, analogous to the case of the quantum harmonic oscillator and its corresponding possible eigenmodes. The discrete nature of the molecular eigenmodes was explored by Pinheiro et al. (2024), and it can be understood as independent possible dynamic contributions to the total NMR relaxation phenomena, given by the corresponding series of eigenfunctions that add up to restore the total diffusion propagator. Figure illustrates the eigenmodes decomposition of the dynamics of a ¹H–Gd³⁺ pair, and how these contributions correlate to a series of molecular eigenmodes with characteristic times τ _k and corresponding amplitudes P(τ _k ), for k = 1, 2, 3, ...

7.

Diagram representing the decomposition of the NMR relaxation dynamics of ¹H around Gd³⁺ into multiple eigenmodes, each of which displays a characteristic time τ _k and an amplitude P(τ _k ), with k = 1, 2, 3, ... The combination of all these eigenmodes recovers the complete NMR relaxation autocorrelation function, according to eq .

Figure (a) shows that both studied complexes display three main exponential decays detectable within the numerical resolution of the MD simulations; other decays with shorter characteristic times and small amplitude may be present, but their overall influence is implicitly accounted for in the amplitude of the remaining captured decays. Each of these exponential decays represents one molecular eigenmode with a corresponding amplitude P(τ) and a characteristic time τ. Because the amplitude G(0) of the NMR autocorrelation function of Gd³⁺-aqua is much higher than the one from Gd³⁺-DO3A-butrol, the former presents eigenmodes with higher amplitudes P(τ). Notice, however, that the eigenmodes for Gd³⁺-DO3A-butrol (⟨τ⟩ = 49.1 ps) present higher characteristic times than the ones for Gd³⁺-aqua (⟨τ⟩ = 21.1 ps), revealing that the water molecules around Gadobutrol undergo slower molecular dynamics. The comparison of the ⟨τ⟩ values obtained for each case with experimental data is presented in Table , showing good agreement with independent measurements.

8.

(a) Padé-Laplace inversion modes of the NMR relaxation autocorrelation function according to eq for both Gd³⁺-aqua and Gd³⁺-DO3A-butrol complex obtained via MD simulations at 37°C, and (b) the corresponding normalized modes over the corresponding characteristic times normalized by ⟨τ⟩. Uncertainties for both τ and P(τ) are within less than 10%. The dashed lines (---) in plot (a) represent the ⟨τ⟩ for Gd³⁺-aqua and Gd³⁺-DO3A-butrol complexes.

3. Characteristic times and diffusivities obtained via MD simulations for Gd³⁺-aqua and Gd³⁺-DO3A-butrol in comparison with literature measurements. All quantities are at 37°C.

	Gd3+-aqua		Gd3+-DO3A-butrol
	This work	Literature	This work	Literature
τ R (ps)	21.1	25, 26.24, 17.37, 22.91	69.3	57 ± 2
N I,in (≡ 2q)	17	18, 16	2	2
r in (Å)	3.06	3.05, 3.20, 3.04	3.18	3.1
D Gd (μm2 ms–1)	0.539	0.630	0.385	
D W (μm2 ms–1)	2.94	3.04 ,	2.94	3.04 ,
d O (Å)	4.60	3.9	5.01	
τ D (ps)	60.8	33	75.5	60
τ D (ps)			80.3
T e0 (ps)	316	277.6	104	111 ± 6
τ m (ns)	1.10	1.95, 1.58, 1.52	>22.8	176 ± 21, 250

^aSignifies the approximation given by τ _R ∼ ⟨τ⟩

^bSignifies the case of bulk water without a contrast agent in the system.

It is known that the amplitudes of the molecular eigenmodes can be normalized by 4πP(τ _k )/⟨1/r ⁶⟩, but since G(0)∝⟨1/r ⁶⟩, a more practical way to normalize the amplitudes can be done by

$$P{({\tau }_k)}^{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}=\frac{P({\tau }_k)}{G(0)}$$ 25

Similarly, the characteristic times can be normalized as 6D*τ _k /⟨r ²⟩, where D* is the relative diffusion coefficient of hydrogen atoms with respect to the central paramagnetic ion. Since D* can be difficult to determine, and noting that ⟨τ⟩ ∝ D*τ _k /⟨r ²⟩, a more practical way to normalize the characteristic times can be done by

$${\tau }_k^{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}=\frac{{\tau }_k}{⟨\tau ⟩}$$ 26

Figure (b) shows the normalized molecular eigenmodes for both the Gd³⁺-aqua and Gd³⁺-DO3A-butrol complexes. This normalization allows for a comparison of the eigenmodes among different systems normalized by their corresponding dynamics (D*) and structural effects (⟨r ²⟩ and ⟨1/r ⁶⟩). Notice that the normalized characteristic times for each of the three eigenmodes on Gd³⁺-aqua and Gd³⁺-DO3A-butrol are similar, indicating that the underlying eigenmodes for both cases are similar. This implies that apart from being at different characteristic times and physical distances with respect to the central paramagnetic ion, similar mechanisms for the underlying r ₁ dispersion are present in both Gd³⁺-DO3A-butrol and Gd³⁺-aqua complexes.

The fact that the amplitudes of the eigenmodes differ between Gd³⁺-aqua and Gd³⁺-DO3A-butrol indicates that the relative importance of each underlying mechanism is different. In the case of Gd³⁺-DO3A-butrol, the second eigenmode τ₂ presents higher importance than the first τ₁ (i.e., P(τ₂)/P(τ₁) = 0.602) when compared to the case of Gd³⁺-aqua (P(τ₂)/P(τ₁) = 0.271). It is known that the n^th-eigenmode presents n – 1 nodes (i.e., region with zero transition probability) in the eigenfunction, which indicates that a mechanism involving a bimodal radial diffusion propagator plays a more important role in the case of Gd³⁺-DO3A-butrol than that for Gd³⁺-aqua. This may be related to the fact that the water residence time in the inner shell of Gd³⁺-DO3A-butrol is significantly higher than the residence time in the Gd³⁺-aqua complex at the same temperature.

3.4. NMR Relaxivity Dispersion

Given that MD simulation trajectories contain information about fluctuations in the NMR relaxivity at any given frequency of the external magnetic field B ₀, frequency-dependent longitudinal relaxation rates r ₁ were calculated for both Gd³⁺-aqua and Gd³⁺-DO3A-butrol complexes in aqueous media at 37°C. Figure (a) shows the total ¹H r ₁ dispersion (inner and outer shell) from MD simulations and measurements for the Gd³⁺-DO3A-butrol complex at human body temperature as well as other experimental data available in the literature for Gadobutrol under the same conditions. Figure (b) compares the r ₁ dispersion between Gd³⁺-DO3A-butrol and Gd³⁺-aqua at 37°C for both MD simulations and measurements. In both cases, for the Larmor frequencies of interest to clinical MRI (i.e., f ₀ ≃ 10 ↔ 300 MHz), good agreement between simulations and measurements is observed for both Gd³⁺-aqua and Gd³⁺-DO3A-butrol complexes.

9.

(a) NMR relaxivity r ₁ dispersion for Gd³⁺-DO3A-butrol at 37°C, with the continuous line representing MD simulations (calculated using the Padé-Laplace inversion) with the corresponded shaded areas representing uncertainties (calculated without Padé-Laplace noise attenuation) and the blue symbols representing experimental measurements for Gd³⁺-DO3A-butrol at 1.0 mM from (●) this work, (*) Laurent et al., 2016, (▲) Lohrke et al., 2022, and (◀) Hao et al., 2012. (b) Comparison of the r ₁ dispersion between both Gd³⁺-aqua and Gd³⁺-DO3A-butrol complexes at 37°C, where the continuous line are again the MD simulation results and red symbols represent measurements for Gd³⁺-aqua from Pinheiro et al., 2022 at concentrations of (●) 0.3 mM, (×) 1.0 mM, and (+) 2.0 mM.

At lower frequencies f ₀ ≲ 5 MHz, it is known that MD simulation results do not agree with measurements since the simulations do not account for electron-spin relaxation time, given that the employed semiclassical AMOEBA force field does not have any component to account for electron-spin effects. The mismatch at low frequencies was particularly large for the case of the Gd³⁺-DO3A-butrol complex, indicating that electron-spin relaxation is more important for Gadobutrol than for the Gd³⁺-aqua complex at 37°C. However, the fact that MD simulations capture all of the important physics of r ₁ at low frequencies except for the electron-spin relaxation allows the determination of the electron-spin correlation time for such complexes by making use of the mismatch between simulations and measurements for f ₀ ≲ 5 MHz.

The longitudinal and transverse electron-spin relaxation times are defined by T _1e and T _2e , respectively, and the low-frequency limit for these times are given by T _1e (0) = T _2e (0) = T _e0. By assuming that the rotational and translational-diffusion are uncorrelated with the electron-spin relaxation time T _e0 at low frequencies, the following relationship holds

$$r_1^{\prime }(0)=\frac{1}{[\mathrm{H}]}\frac{20}{3}G(0)⟨{\tau }^{\prime }⟩$$ 27

$$\frac{1}{⟨{\tau }^{\prime }⟩}=\frac{1}{⟨\tau ⟩}+\frac{1}{T_{e0}}$$ 28

where r ₁ (0) is the NMR relaxivity at f ₀ = 0 MHz, and the average characteristic time ⟨τ⟩ can be calculated by eq . Thus, by using simulation results (that do not have T _e0 effects) and measurements at f ₀ = 0 MHz, T _e0 can be determined through eqs and . It is found that T _e0 ≃ 316 ps for the Gd³⁺-aqua complex while T _e0 ≃ 104 ps for the Gd³⁺-DO3A-butrol complex, indicating that the low-frequency electron-spin relaxation time is ∼3 times shorter for Gadobutrol than for Gd³⁺-aqua. Comparison of these calculated values for T _e0 against independent studies is presented in Table , in which good agreement is found. Although electron spin relaxation does not play a role at frequencies relevant to clinical MRI as practiced today (f ₀ ≳ 5 MHz), recent improvements to use lower magnetic field strengths have produced cost-effective and more accessible imaging techniques, albeit with a lower signal-to-noise ratio (SNR). Thus, in the case of low-frequency fields where electron spin effects are important, the shorter electron spin relaxation time of Gadobutrol results in lower NMR relaxivity r ₁, leading to less efficient MRI contrast and, consequently, a worse contrast effect.

Note that the above analysis does not account for T _1e and T _2e dispersion. , However, as shown in Figure , the effects of electron-spin relaxation on r ₁ are negligible above f ₀ ≃ 5 MHz for both Gd³⁺-aqua and Gd³⁺-DO3A-butrol, and therefore T _1e,2e dispersion does not affect their r ₁ at frequencies of interest in clinical MRI. This is not always the case, as shown for other contrast agents , where T _1e,2e dispersion plays a role up to f ₀ ≃ 20 MHz. In principle, the current QM and MD simulation techniques can be extended to predict the T _1e,2e dispersion.

Finally, the translational-diffusion correlation time, τ _D , can also be determined from the MD simulation trajectories. τ _D is defined as the average time necessary for a molecule to diffuse through a diameter d _O , corresponding to the distance between the center of the paramagnetic ion and the second layer of oxygen atoms around it. The translational-diffusion correlation time can be estimated through

$${\tau }_D=\frac{d_O^2}{D_{\mathrm{W}}+D_{\mathrm{G}\mathrm{d}}}$$ 29

where D _W and D _Gd are the self-diffusivities of water and Gd³⁺ (either in the Gd³⁺-aqua or the Gd³⁺-DO3A-butrol complexes), respectively, calculated according to eq . For these calculations, d _O = 4.60 Å for the Gd³⁺-aqua complex and d _O = 5.01 Å for the Gd³⁺-DO3A-butrol complex were determined from the corresponding radial distribution functions. Table summarizes the self-diffusivities of water and the complexes under study, the corresponding translational-diffusion correlation times, and d _O . Notice that the translational-diffusion correlation time for the water molecules around Gd³⁺-DO3A-butrol is slightly longer than the corresponding case of Gd³⁺-aqua, indicating slightly slower translational dynamics for the case of Gadobutrol.

3.5. Inner- and Outer-Shell Contributions

The amplitude of G(t) at t = 0, defined as G(0) in eq , for Gd³⁺-DO3A-butrol is only about 5.6 times smaller than the amplitude for Gd³⁺-aqua, despite the fact that its inner shell displays 8.5 times fewer water molecules. This shows that the outer shell of Gadobutrol plays a role. At every instant in time t′ in eq , the simulation can naturally label inner-sphere water ¹H’s (N _I,in = 17 for Gd³⁺-aqua and N _I,in = 2 for Gadobutrol) from outer-sphere water ¹H’s (N _I,out = N _I – N _I,in), where N _I,in ≡ 2q. The simulation can then separate G(0) for inner versus outer-sphere ¹H’s such as

$$G(0)=G_{\mathrm{i}\mathrm{n}}(0)+G_{\mathrm{o}\mathrm{u}\mathrm{t}}(0)$$ 30

where the labeling of inner vs outer-sphere ¹H’s is updated at each time step t′. Indeed, the same water molecule is likely to be both inner- and outer-sphere water during the simulation run, as governed by the residence time τ _m for that particular water molecule. Despite the finite residence time, eq is exact given that the labeling of inner- versus outer-sphere ¹H is updated at each t′. Note that some early reports further separate outer-sphere water into second-sphere water plus all other waters, however this distinction has recently been dropped, as done in this report.

Table shows the relative contributions from G _in(0) versus G _out(0) for both Gd³⁺-aqua and Gd³⁺-DO3A-butrol complexes. Surprisingly, while the inner shell has a large contribution (∼84%) to the total G(0) for the Gd³⁺-aqua complex, the relative contribution of the inner shell for the Gd³⁺-DO3A-butrol complex is much smaller (∼44%). It is important to emphasize that although Gd³⁺-DO3A-butrol contains 8–9 times fewer inner-shell water molecules than Gd³⁺-aqua, its inner-shell contribution does not follow a simple 1/8 or 1/9 ratio. This discrepancy arises not only from the change in the number of water molecules, but also from significant differences in the Gd³⁺-¹H _w distances in both the inner and outer spheres between of the Gd³⁺-aqua and the Gd³⁺-DO3A-butrol complexes. According to eq , one can then use G _in (0) and N _I,in to determine the inner-shell separation r _in by

$$\frac{1}{r_{\mathrm{i}\mathrm{n}}^6}\simeq \frac{1}{N_{I,\mathrm{i}\mathrm{n}}}\sum _{i=1}^{N_{I,\mathrm{i}\mathrm{n}}}{⟨\frac{1}{r_{is}^6(t^{\prime })}⟩}_{t^{\prime }}$$ 31

yielding r _in ≃ 3.06 Å for Gd³⁺-aqua, and r _in ≃ 3.18 Å for Gd³⁺-DO3A-butrol, which is consistent with Table using $g_{{\mathrm{G}\mathrm{d}}^{3+}-{\mathrm{H}}_{\mathrm{w}}}(r)$ . Table lists r _in and N _I,in (≡ 2q) for convenience.

4. G _in(0) versus G _out(0) for Both Gd³⁺-Aqua and Gd³⁺-DO3A-Butrol Complexes at 37°C from MD Simulations, Including Absolute Values (in s^–2) and Relative Contributions to Total (in %).

	G in(0)	G out(0)	G in(0)	G out(0)
	(s–2/1016)	(s–2/1016)	(%)	(%)
Gd3+-aqua	1.6189	0.3124	83.8	16.2
Gd3+-DO3A-butrol	0.1490	0.1919	43.7	56.3

While separating G _in(0) from G _out(0) is of interest, it says nothing about the difference in correlation times (i.e., the dynamics) for inner-sphere (τ_in) versus outer-sphere (τ_out) water. In MD simulations, the contribution of inner versus outer-sphere relaxivity cannot be readily separated from the total relaxivity (r ₁) in an analogous manner to G(0) given that a certain water molecule is likely to be both inner-sphere and outer-sphere water during the simulation trajectory. However, because the inner shell of the Gd³⁺-DO3A-butrol complex is composed by a single water molecule which remains in the inner shell for times much longer than the NMR relaxation autocorrelation time (fast exchange regime), the contribution from the inner-shell water molecule in the Gadobutrol complex to the ¹H r ₁ dispersion can be separated within the MD simulation trajectories, allowing the decomposition of inner and outer sphere contributions to the NMR dispersion of Gadobutrol in aqueous media at 37°C.

Figure (a) shows the ¹H r ₁ dispersion for water molecules around Gadobutrol with the separated contributions from the inner and outer-shell water molecules, and Figure (b) shows the relative contribution of inner and outer shells to the total ¹H r ₁ dispersion. These results were obtained from an 8 ns section of the MD trajectories in which the same water molecule remained in the inner shell of the Gd³⁺-DO3A-butrol complex. Observe that the contribution of the inner shell responds for only ∼55% for the total r ₁ dispersion, highlighting the importance of the outer shell for the relaxivity response of Gadobutrol. Figure S1 in the Supporting Information shows the outer-sphere water molecules around the Gd³⁺-DO3A-butrol complex that contribute to the vast majority of the total outer-sphere effect. Because the NMR relaxation signal decays with 1/r ⁶ of the relative distance between the paramagnetic ion and the ¹H nuclei, only the outer-sphere water molecules near the complex play a role in this term, where most of them interact with the complex’s hydrophilic side groups.

10.

Inner and outer shells (a) NMR relaxivity r ₁ and (b) relative contribution to the total NMR relaxivity r ₁ dispersion for Gd³⁺-DO3A-butrol at 37°C obtained with the Padé-Laplace inversion of the MD simulation results. The corresponded shaded areas represent the uncertainties, which were calculated using the MD simulation results across 7 independent B ₀ directions without inversion (therefore, without noise attenuation). Recall that MD calculations should be accurate compared to experimental data for frequencies f ₀ ≳ 5 MHz.

The contributions from inner and outer shells of Gadobutrol to the NMR relaxation autocorrelation function G(t) (Figure S2), as well as their corresponding eigenmodes decomposition (Figure S3), are presented in the Supporting Information. From this separation of the MD simulation results, the rotational diffusion time τ _R = 69.3 ps and the characteristic translational time τ _T = 35.7 ps were determined for Gadobutrol at 37°C. If the Hwang–Freed model is assumed for the outer shell, yet another estimate of the translational-diffusion correlation time for Gadobutrol can be determined by τ _D = 9/4τ _T = 80.3 ps. Table summarizes these quantities and shows the good agreement with estimates based on experimental measurements.

In the case of the Gd³⁺-aqua complex, the residence time was estimated as τ _m ∼ 1.1 ns, which is in relatively good agreement with estimates from measurements available in the literature. ,, However, the water residence time for the Gd³⁺-DO3A-butrol complex is much longer than the extent of the MD simulations, which in essence hinders the proper determination of τ _m for Gadobutrol from the simulations. For MD simulations with a total of 25 ns, water molecule residence times of at least 22.8 ns were observed, which sets a lower bound to τ _m from simulations; a proper determination of τ _m for Gadobutrol via MD simulations would require calculations at least 2 orders of magnitude longer than the performed ones, which is impractical for this approach.

4. Conclusions

The first-of-its-kind MD simulation of Gadobutrol, a widely used clinical MRI contrast agent, was performed and revealed important aspects of its structure in aqueous media and the underlying mechanism governing its NMR relaxation response at the human body temperature. The simulations assume no adjustable parameters, and the results present good agreement with measurements at Larmor frequencies of interest to clinical MRI. At low frequencies, the deviation between the measurements and MD simulation occurs because the simulations do not account for the electron-spin relaxation. This deviation is used to estimate the low-frequency electron-spin relaxation time for Gadobutrol, which is found to be ∼3 times shorter than nonchelated Gd³⁺-aqua at the same temperature. The approach presented here to study NMR relaxivity is not limited to the bare Gd­(III) ion and its chelated complexes but can be used to study contrast agents where the metal ion is encapsulated or within nanostructures, as well as manganese- and iron-based contrast agents.

To the extent of the physics captured within the polarizable force field, MD simulations show that the equilibrium structure of Gadobutrol in water has one solvent molecule in its inner shell (q = 1), and also that only six electron-donor groups of DO3A-butrol effectively chelate the paramagnetic ion, mostly due to dihedral tensions in the ring and solvation effects of the remaining electron-donor groups in aqueous media. The overall coordination number of Gd³⁺ in water at 37°C is seven (one water molecule and six electron-donor groups in DO3A-butrol), leading to distorted capped octahedron structures.

Significant differences between Gd³⁺-aqua and Gadobutrol are found with regards to the amplitude (∝ 1/r ⁶) of the NMR dipole–dipole autocorrelation function, which highlights the differences in the distance of the inner versus outer-sphere water to the Gd³⁺ ion in both cases. In fact, for the Gd³⁺-DO3A-butrol complex in aqueous media at 37°C, the ¹H r ₁ dispersion for water molecules from the inner shell responds for only ∼55% for the total r ₁ dispersion, highlighting the importance of the outer-shell water molecules (∼45%) for the total relaxivity response primarily due to water molecules hydrating the DO3A-butrol structure and in proximity to the paramagnetic ion. These results spotlight the significance of hydrophilic structures in the side groups of the macrocyclic agent, which implies that, given the importance of the outer sphere to the total NMR relaxation signal, (i) the design of newer contrast agents must not neglect the contribution of side groups’ interaction with the solvent, and (ii) the models used to describe outer sphere play a more important role than what was probably expected; thus, emphasis should also be given to accurately describe this contribution.

Finally, an analysis on the NMR eigenmodes of relaxation was presented for the Gd³⁺-DO3A-butrol complex, revealing that the underlying relaxation decays for Gadobutrol are slower (longer characteristic times) than the modes for Gd³⁺-aqua, but the normalized eigenmodes analysis suggest that similar mechanisms (despite being at different characteristic times and physical distance with respect to the central paramagnetic ion) are present in both Gd³⁺-DO3A-butrol and Gd³⁺-aqua complexes. It was also reported that in both Gd³⁺-aqua and Gd³⁺-DO3A-butrol complexes, P(τ₁) > P(τ₂), indicating that the major underlying contribution to relaxation displays a diffusion propagator eigenmode with no spatial nodes. However, Gd³⁺-DO3A-butrol presents a P(τ₂)/P(τ₁) ratio higher than the one for Gd³⁺-aqua, indicating that Gadobutrol displays an underlying relaxation mechanism where a diffusion propagator eigenmode with one spatial node plays an important role, possibly indicating the difficulty of water molecules to access/leave the inner shell of Gadobutrol due to the high residence times in the Gd³⁺-DO3A-butrol complex.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/cbmi.4c00080.

• AMOEBA parameters obtained for DO3A-butrol in this work, as well as the parameters used for water and gadolinium­(III); The inner and outer-shell contribution to the NMR relaxation autocorrelation function G(t) for the chelate, as well as their corresponding eigenmodes decomposition (PDF)

This manuscript has been authored by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for US government purposes. DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).

The authors declare no competing financial interest.