Initially optimize the complexes ([Gd(DOTA)(H₂O)]^-, [Gd(DTPA)(H₂O)]^2- and [Gd(DTPA-BMA)(H₂O)]) ^12,13 and the hybrids δ-FeOOH(100).[Gd(DTPA)(H₂O)]^2- and δ-FeOOH(100).[Gd(DTPA-BMA)(H₂O)]), in the gaussian 09 program ¹⁴ , using the semi-empirical Parameterization Method 6 (PM6) ^15,16 .

After optimization, we made the molecular dynamics simulations (MD) for the complexes of Gd(III) using the program developed by van Duin and col. (REAX-FF) ¹⁷ , which is part of ADF-BAND program package. For the simulations was used the force field NiCH. For the MD simulation the box size was fixed at 8000 ų and was held at a temperature 310.65 K (physiologic temperature) throughout the simulation. Studies have shown that this temperature is adequate to simulate this type of model. For these simulations a 500 ps thermalization face (for system stabilization) and an additional 2.0 ns period are required, the box was built by the density of liquid water (ρ=0.996 g cm^-3) ¹⁸ .

After the MD simulation it is necessary to try to reduce the number of conformations for the later quantum calculations (decrease the computational cost). For this, we selected the uncorrelated configurations of the Gd(III) complexes, Scilab 2.7 ¹⁹ program was used. The method was developed and applied for the first time by the Canuto’s group ²⁰ . This method uses the statistical interval obtained from the energy autocorrelation, the interval between uncorrelated configurations, or the correlation step s, is calculated by integration from zero to infinity of C(n), Eq. 1. The interval between uncorrelated configurations, or the correlation step τ (the molecular rotational correlation time in Eq. 2) is calculated by integration from zero to infinity of C(n). The theory shows that separate the settings by 2τ, or larger intervals, are considered uncorrelated.

(1)

(2)

With uncorrelated structures we did the constant calculations of hyperfine coupling (A_iso) for the complexes with water molecules.

The hyperfine coupling constant (A_iso) calculations were carried out in the program Gaussian 09, with uncorrelated structures from MD simulation of Gd³⁺ complexes and with the lowest energy structure of the hybrid. For the Gd³⁺ complexes, the simulation was performed using the functional PBE1PBE ²¹ and basis set EPR-III for the H and O atoms, 6-31G for the C and N atoms, MWB53 for the Gd atom. For the hybrid compounds was also used the above-mentioned base function and we added the lanl2dz for the Fe atom.

The geometry of the complex was fully optimized using the method PM6, the geometry according mounted as shown in Fig. 2 and the bond distances from the metal coordination environment are listed in Tab. 1 ⁷ .

From the results of Tab. 1, it is possible to observe that our calculations were able to reproduce reasonably well the distances between the Gd^III and the ligand, observed with the experimental results performed by x-ray.

We observed for the complex that [Gd(DOTA)(H₂O)]^-, the inner sphere water molecule has a bond distance around 2.45 Å, what satisfies our theoretical value 2.56 Å. For the complexes [Gd(DPTA)(H₂O)]^- and [Gd(DTPA-BMA)(H₂O)] water molecules in the inner sphere have a connection distance between 2.49 Å, and 2.44 Å, which satisfies the theoretical values 2.52 Å and 2.46 Å, respectively. This can be attributed, at least in part, to the fact that the implicit solvation model (which uses the dielectric constant of the medium) cannot explain some specific interactions between the complex and the solvent, for example, the hydrogen bonds. Indeed, it has been shown that continuous dielectric solvent models are often inadequate to investigate solutes that concentrate on the charge density with strong local solute-solvent interactions ⁷ . Thus, to try to overcome this deficiency, we performed calculations of geometry optimization using only one coordinated water molecule with Gd. Table 1 shows the distances of the complex bonds compared with the experimental values.

Figure 2. Figure 2. Structure of Gd(III) complexes.

Table 1. Table 1. Distance values of experimental and theoretical bond for the complex.

MD calculations provide thousands of conformations, so it is possible to perform quantum calculations of all these conformations. Thus, methods to select the main structures of MD have been studied. Currently, one method that has been highly effective is statistical inefficiency ^18-21 . With this in mind, in the present work we use statistically different structures for quantum mechanics calculations, the method uses the energy correlation function of MD simulations ^22,23 . It is important to mention that this method was developed and studied deeply by the Coutinho and Canuto group ²³ . The Canuto and Coutinho group showed that the statistical interval, C(n), is particularly important for a Marovian process, where C(n) follows an exponential deterioration ²² . In this way, uncorrelated configurations, τ, is calculated by integrating zero to infinity of C(n). Configurations separated by 2τ, or larger intervals, are considered uncorrelated ^23-25 . Figure 3 shows exponential decay.

From the simulation MD, as can be seen in Fig. 3, the correlation time of the complex coordinated with water molecules ([Gd(DOTA)(H₂O)]^-, [Gd(DTPA)(H₂O)]^2- and [Gd(DTPA-BMA)(H₂O)] were 4.09, 6.01 and 6.53 ps, respectively. According to the calculations of statistical inefficiency for the complex [Gd(DOTA)(H₂O)]^- 244 structures were uncorrelated, for the [Gd(DTPA)(H₂O)]^2- 164 structures were uncorrelated and for the complex [Gd(DTPA-BMA)(H₂O)] 153 structures were uncorrelated. We observed that the complex [Gd(DTPA-BMA)(H₂O)] has a larger correlation time relative to other complexes, thus has a smaller number of uncorrelated structures.

Figure 3. Figure 3. Graphic of the auto-correlation function for the time in picoseconds. a) ([Gd(DOTA)(H2O)]-, b) [Gd(DTPA)(H2O)]2-, c) [Gd(DTPA-BMA)(H2O)]. The blue curve is the correction and the red curve the adjustment done.

In recent decades, the MRI has emerged as a powerful diagnostic tool that uses longitudinal relaxation times (T₁) and transverse (T₂) of the atoms ¹H and ¹⁷O of water molecules to obtain tissue images. The value T₁ is related to the return time magnetization to the longitudinal axis and it is influenced by the interaction of spins with the network (environment). The value of T₂ refers to the reduction of magnetization in the transverse plane and it is influenced by the spin-spin (dipole-dipole) interaction. The dipolar magnetic interactions between protons of water with other local interactions, are able to gradually restore the original orientation of the magnetization vector along the main magnetic field ²⁶ , that way, to evaluate the influence of contrast agents on T₁ and T₂ times it is necessary that the compound be paramagnetic. Thus, the Eqs. 3 and 4 represent the relaxation time T₁ and T₂, respectively.

(3)

(4)

Observing Eqs. 1 and 2, we have that the longitudinal relaxation time (T₁) depends on several parameters, such as: the electron spin (S), the electronic (ge) and proton g factors (gN), the Bohr magneton (β), the nuclear magneton (β.), the hyperfine coupling constant (A), the ion-nucleus distance (r), and the Larmor frequencies for the proton (𝜔_𝐼) and electron spins (𝜔_𝑆), 𝜏𝑒 is the correlation time that characterizes the time of internal rotational correlation of molecules. In the Eq. 2, besides the constants already mentioned we also have 𝜏𝑐 , which is the correlation time characterized by the rate of change of the ion interactions between metal and neighboring hydrogens. In these equations it is important to highlight the hyperfine coupling constant, which is the most sensitive parameter and what our calculations were performed ²¹ .

We evaluate the constant values of hyperfine coupling to ¹H e ¹⁷O, and was chosen the Aiso parameters to evaluate the effects of structures, because the Aiso values are more sensitive to geometric parameters of structures, thereby facilitating the observation of a variation of the parameters ²⁷ . Initially we will start to analyze the Aiso coupling constant of the complex [Gd(DOTA)(H₂O)]^- water molecules coordinated with. According to Tab. 2, we note that for the structure in equilibrium A𝑖𝑠𝑜 eq (PBE1PBE(H₂O)//PBE1PBE(H₂O)) obtained Aiso values equal to 0.53 MHz for the ¹H and 0.87 MHz for the ¹⁷O. It was also made calculations with the implicit solvent and explicit A𝑖𝑠𝑜 eq (PBE1PBE (H₂O)/PCM//PBE1PBE(H₂O)). The values were 0.33 MHz and 0.82 MHz for the ¹H and ¹⁷O, respectively, the result indicate that the implicit solvent does not influence significantly our system and it shows that the amount of water molecules are allowed sufficient to realistically simulate our system. Thus, analyzing the calculations now uncorrelated with the values of MD A𝑖𝑠𝑜 300K (MD(H₂O)//MD(H₂O)) we have 0.92 MHz for the ¹H and 0.72 MHz for the ¹⁷O. By analyzing these results, it is observed that the thermal effects influence the system, making the Aiso values closer to the experimental. This increase in Aiso values is to be expected since thermal effects are important in the system

Table 2. Table 2. Values of Aiso of the Water in the presence of [Gd(DOTA)(H2O)]-.

Analysing now the complex [Gd(DTPA)(H2O)]^2- , in Tab. 3, the A_iso values of equilibrium structure, A_𝑖𝑠𝑜 ^eq (PBE1PBE(H₂O)//PBE1PBE(H₂O)), was of 0.38 MHz for the ¹H and 0.85 MHz for the ¹⁷O. The calculations with the implicit solvent and explicit A_𝑖𝑠𝑜 ^eq (PBE1PBE (H₂O)/PCM// PBE1PBE (H₂O)), the values obtained were of 0.47 MHz for the ¹H and 0.80 MHz for the ¹⁷O, it was observed that the values of the explicit and implicit solvent next are the values only with explicit solvent, in other words, the water molecules placed as solvent were able to realistically represent our system. The calculations with uncorrelated structures of the MD, A_𝑖𝑠𝑜 ^300K (MD(H₂O)//MD(H₂O)), we have the values of 0.65 MHz for the ¹H and 0.75 for the ¹⁷O. Thus, the thermal effects were also shown to be important. In fact, the molecular dynamics calculations are important to simulate a more real system, thus, it is expected that the results are closer to the experimental ones.

Table 3. Table 3. Values of Aiso of the Water in the presence of [Gd(DTPA)(H2O)]2-.

Analyzing the last complex of work (Tab. 4), [Gd(DTPA-BMA)(H₂O)], the equilibrium structure, A_𝑖𝑠𝑜 eq (PBE1PBE(H₂O)//PBE1PBE(H₂O)), the values obtained were 0.33 MHz for the ¹H and 0.89 MHz for the ¹⁷O, and calculations with the implicit solvent and explicit A_𝑖𝑠𝑜 eq ( PBE1PBE (H₂O)/PCM// PBE1PBE (H₂O)), the values obtained were of 0.55 MHz for the ¹H and 0.75 MHz for the ¹⁷O. Calculations with uncorrelated structures of the MD, A_𝑖𝑠𝑜 300K (MD(H₂O)//MD(H₂O)), the values obtained were 0.95 MHz for the ¹H and 0.72 MHZ for the ¹⁷O. The thermal effects were important, the Aiso values were closer to the experimental. In Fig. 2 are shown the structures of Gd(III) complexes with different ligands. As noted, in both cases ([Gd(DOTA)(H₂O)]^- and [Gd(DTPA)(H₂O)]^2- ) in both cases ([Gd(DOTA)(H₂O)]- and [Gd(DTPA)(H₂O)]^2- ) the thermal effects were important. With the incessant movement of water molecules, more interactions can occur between the solvent and the solute and between solvent molecules (such as hydrogen bonds). These interactions are the main responsible for the considerable increase in Aiso values. The fact, thermal effects are important because they consider the movement of all solvent molecules, thus, this model is considered more realistic.

Table 4. Table 4. Values of Aiso of the Water in the presence of [Gd(DTPA-BMA)(H2O)].

As already mentioned, thermal effects are important. However, for our proposal of a new contrast agent this effect was neglected, in fact despite the importance of this effect, our objective is to verify if the hybrid compound can be used as CA. In this way, to reduce the computational cost, we perform calculations only with the balance structure. Thus, it was made A_iso calculations only with the lowest energy conformer of hybrids (δ-FeOOH(100).[Gd(DTPA)(H₂O)]^2-, δ-FeOOH(100).[Gd(DTPA-BMA)(H₂O)]). The values of A_iso for the hybrid compounds (Tab. 5) show that both significantly increase. For the first hybrids δ-FeOOH(100).[Gd(DTPA)(H₂O)]^2- values of 4.25 MHz and 5.30 MHz were obtained for the ¹H e ¹⁷O atoms, respectively. For the hybrid δ-FeOOH(100).[Gd(DTPA-BMA)(H₂O)] the values of A_iso were found to be 4.15 MHz and 5.15 MHz, respectively. Thus, it is noted that the hybrid compounds can be promising contrast agents for MRI since they showed a significant increase in the values of A_iso. Figure 4 shows the structures of hybrid compounds.

Table 5. Table 5. Values of Aiso of the water in the presence of hybrids.

Figure 4. Figure 4. Structures of hybrid compounds. a) δ-FeOOH(100).[Gd(DTPA)(H2O)]2- b) δ-FeOOH(100).[Gd(DTPA-BMA)(H2O)].