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      where i is either 1 or 2, and R_i,obs is the observed relaxation rate which is defined as the reverse of the effective relaxation time T_i,eff. T_i is the relaxation time in the absence of any CA, c is the CA concentration, and r_i is the relaxivity, which is characteristic of CA. Higher the r_i values are, higher the effectiveness of the CA, meaning that a smaller concentration of CA leads to measurable effects. The effects of the CA on the two relaxivities are very often completely different. As such, CAs that reduce the T₁ values (high r₁) and increase the signal in T₁‐weighted images are called positive CAs, whereas CAs that reduce the T₂ values and subsequently reduce the intensity of the signal in T₂‐weighted images are called negative CAs.

      The mechanism by which CAs influence the relaxation times is related to the value of their magnetic moment and the time spent by the protons (mostly belonging to water molecules) in their proximity (retention time). Thus, paramagnetic ions with large magnetic moments (Mn²⁺, Gd³⁺, Dy³⁺) allow a close interaction of the ions with water molecules present in the first hydration sphere, leading to an increase of r₁ relaxivity values. These paramagnetic ions are usually administrated in the form of ion complexes that have the role to decrease their mobility and to increase their relaxivities. The phenomenon is known as Proton Relaxation Enhancement (PRE). These ions are known as T₁‐positive CAs.

      On the other hand, the advancement in the synthesis of different classes of MNPs with improved magnetic properties has represented a major discovery in the field of CAs. MNPs have huge values of their magnetic moments, with respect to paramagnetic ions, influencing the relaxation behavior of water molecules beyond the first hydration layer. Nevertheless, their coatings, used for improving their biocompatibility, impede a direct contact with water molecules. For these reasons, the MNPs are mostly used as negative T₂‐CAs. The first commercially available MNPs approved for clinical applications as CAs in MRI are Resovist and Feridex (Table 2.1). They consist of SPIONs of small magnetic cores (approximately ~10 nm) obtained by the coprecipitation method. Their r₂ values are lower than 100 s⁻¹ mM⁻¹. The relaxivity values are directly proportional to the square of MNPs magnetic moment, which in turn is proportional to their magnetization saturation (Ms) and volume. Therefore, MNPs with larger volumes and Ms values present several fold higher relaxivities. The use of thermal decomposition methods led to the synthesis of MNPs with improved crystallinity. This increases their magnetic moments as well as their relaxivities. However, monocrystalline MNPs with sizes higher than 20 nm are difficult to obtain because they involve laborious seed‐mediated growth synthesis methods. Moreover, the increase of MNP size can induce their transition into the so‐called “blocked state” in which they will have a nonzero remanent magnetization (Mr – induced magnetization remaining at zero magnetic field). A direct consequence of this nonzero Mr will be the occurrence of dipolar interaction and MNPs' self‐aggregation that make them less suitable for biomedical applications. For example, a 20 nm diameter appears to be an upper limit in the case of MNPs applications such as MRI CAs.

Another approach for improving the relaxivity properties of the MNPs is to include them in preformed aggregates or clusters of controlled structure, shape, and size. The existence of a cluster creates a stronger magnetic field gradient around it, thereby, changing the relaxation rate R₂ and the relaxivity r₂. However, the experimental results obtained for these types of structures show that the cluster size and MNPs composition influence the transverse relaxation rate and relaxivity in a nonmonotonous manner (Kostopoulou et al. 2014). With increasing the size and the number of clustered MNPs, the transverse relaxation rate increases, and this regime is called the “motional average regime.” Once the size limit is reached, the water molecules feel a constant magnetic field during their relaxation. This regime is called the “static dephasing regime” and determines the maximum relaxivity limit. If the MNPs' size is further increased, the relaxivity decreases and the measured relaxation rate depends on the echo time. This regime is called “echo limited regime.”

      Interestingly, in the case of longitudinal relaxivity, it has been observed that its value tends to decrease with increasing the size of the cluster. This effect is explained by the reduction of the surface accessible for water molecules as the size of the cluster increases.

In fact, the accessibility of water molecules to the magnetic core is a key factor in determining the effect of MNPs on relaxivities. This means, on the one hand, that the optimum coating allows for sufficient diffusion of water molecules to affect the relaxivity of protons and, on the other hand, it assures a rapid exchange for a maximum number of water molecules near the CAs (Laurent et al. 2008). The diffusion of water molecules around the CA as a function of the coating might be assessed by measuring the NMR dispersion (NMRD) profile. In this type of measurement, the relaxivity r₁ is measured as a function of the magnetic resonance frequency (which is determined by the value of the static magnetic field B₀). It also facilitates the acquisition of the frequency or the value of B₀ at which the CA is the most effective. A typical NMRD profile of magnetite MNPs is presented in Figure 2.1.