#### Designing an Efficient Contrast Agent  
CA interaction with the \({}^{1}\)H protons of water is governed by dipolar interactions; therefore, the distance between the electronic and the nuclear spin defines the interaction strength and the relaxation speed; the shorter the distance, the faster the relaxation. Hence, in a metal ion, the relaxivity will depend on the \({}^{1}\)H proton interaction with the inner, second, and outer spheres of the lanthanide ion (Equations (1) and (2)) [10]. \[\frac{1}{T_{i,\ p}}=\left(\frac{1}{T_{i,\ p}}\right)^{IS}+\left(\frac{1}{T_{i, \ p}}\right)^{OS} \tag{1}\]  
\[r_{i}=r_{i}^{IS}+r_{i}^{2 nd}+r_{i}^{OS},\ i=1,2 \tag{2}\]  
Following the general Solomon-Bloembergen-Morgan (SBM) theory, the inner sphere contribution to the rate relaxation improvement results from the chemical exchange of \({}^{1}\)H water protons closest to the lanthanide ion with the solution, in which one area is much more populated than the other. The relaxation of these \({}^{1}\)H protons is governed by dipole-dipole and contact interactions, also called scalars; however, the latter is virtually negligible at frequencies above 10 MHz, since the correlation times of these contributions tend to zero at high frequencies [10]. Therefore, by applying the SBM equations of the paramagnetic relaxation theory, these terms are obtained (Equations (3)-(6)) [20]. \[\left(\frac{1}{T_{1}}\right)^{IS}=\frac{cq}{55.5}\frac{1}{T_{1m}+\tau_{m}}=P_ {m}\frac{1}{T_{1m}+\tau_{m}} \tag{3}\]  
\[\left(\frac{1}{T_{2}}\right)^{IS}=\frac{P_{m}}{\tau_{m}}\frac{T_{2m}^{-2}+ \tau_{m}^{-1}T_{2m}^{-1}+\Delta\omega_{m}^{2}}{\left(\tau_{m}^{-1}+T_{2m}^{-1 }\right)^{2}+\Delta\omega_{m}^{2}} \tag{4}\]\[\frac{1}{\tau_{ci}}=\frac{1}{\tau_{R}}+\frac{1}{T_{ie}}+\frac{1}{\tau_{m}},\textit{ where }i=1,\,2 \tag{5}\]  
\[\frac{1}{\tau_{sci}}=\frac{1}{T_{ie}}+\frac{1}{\tau_{m}},\textit{where }i=1,\,2 \tag{6}\]  
These equations condense the inner sphere relaxivity calculations, where c is the molar concentration of the CA and q is the number of water molecules bonded to the lanthanide center, called the hydration number, and depends on the ion, the structure, and the size of the complex. \(\frac{1}{T_{1m}}\) y \(\frac{1}{T_{2m}}\) are the relaxation times of the \({}^{1}\)H water proton closest to the metal ion. The correlation time for the magnetic fluctuation (\(\tau_{c}\)) is the shorter term of the rotational correlation time (\(\tau_{R}\)), \(T_{1e}\) or \(T_{2e}\) are the electronic relaxation time, \(\tau_{m}\) is the water residency time, and \(\tau_{sci}\) is the scalar correlation time [10]. In Equation (3), it is clear that if the variable \(q\) increases, the relaxivity will increase with it. However, a rise in this variable may reduce the thermodynamic stability and kinetic inertness. The \(\tau_{m}\) is one of the most important variables in the SBM equations given that decreasing its value increases the number of molecules relaxed per second. Thus, enhancing the rate of the \({}^{1}\)H proton's chemical exchange with other water molecules in the bulk improves the relaxivity of the CA. Zero Field Splitting (ZFS) describes the multiple degenerated micro-energy levels of a molecule or ion in the presence of one or more unpaired electrons due to the Zeeman Effect produced by the mutual magnetic dipole-dipole interaction of each unpaired electron. Therefore, using several Gd\({}^{3+}\) close to each other may enhance the magnetic fluctuations due to the spin density in the complex, improving the \({}^{1}\)H proton relaxation rate [21]. The theoretical calculations using the SBM equations within the Redfield limit show the relationship between molecular weight and relaxivity. Nevertheless, this augmentation reaches a limit after a few kDa [22]. Although weight is directly related with \(\tau_{R}\), there are more factors involved such as rigidity, which, regarding the macromolecule structure of the CA, can be under fast local or global correlation times (\(\tau_{i}\), \(\tau_{M}\)).