## 1 Introduction  
The ability to synthesize nanoparticles and functionalize them have opened many areas for scientific and technical applications in recent years. Nanoscale magnetic particles possess unique physical, and particularly magnetic properties extremely different from their bulk counterparts. In moving from ferromagnetism to superparamagnetism, many critical parameters need to be quantified based on not only their nano size but also considering the new interactions and modified environment. Finite size effects [1] (e.g., canting and disordered surfaces) influence many magnetic parameters such as anisotropy energy (\(E_{a}\)) [2; 3; 4], Neel (\(\tau_{N}\)) and Brownian (\(\tau_{B}\)) relaxation times, and blocking temperature (\(T_{B}\)) [5]. Understanding not only the individual evolution of these parameters with scale, magnetic field, temperature, but also their interaction, mutual and with the surrounding environment, is of paramount importance to the design and application of intelligent and multifunctional magnetic nanoparticles (MNP). In biomedicine, MNPs and in particular superparamagnetic iron oxides (SPIO) have a wide range of applications such as magnetic particle imaging (MPI) [6], drug delivery [7], magnetic hyperthermia (MH) [8; 9; 10], MRI [11], cell tracking [12; 13], magnetic biosensors [14; 15], regenerative medicine and tissue engineering [16; 17], etc. When used as MRI contrast agents (CA) SPIOs can reduce both \(T_{1}\) and \(T_{2}\) relaxation times and hence enhance tissue contrast [18; 19; 20; 21]. Modern synthesis techniques of nanoparticles control their physical and chemical properties to great precision in order to control and predict interactions and functionality of the MNP within tissues such as their biocompatibility and toxicity, spatial distribution, and the relaxation enhancement per unit concentration (i.e., relaxivity). Transverse relaxation enhancement is controlled mainly by the MNP size and magnetization (or susceptibility [2; 3]) which create magnetic field inhomogeneities that are sampled by the diffusing spins. Generally, SPIO induced relaxation of water protons is due to two outer sphere elements. The first is the above mentioned process of spins moving through magnetic field inhomogeneities modulated by the translational correlation time. This is known as Curie (or susceptibility) relaxation because it is the nonfluctuating average electronic spin of the MNP which relaxes water protons. It is a dipolar interaction and modulated by reorientation and exchange. The second element is due to the fluctuations or relaxation of the superparamagnetic magnetic moment which is characterized by global correlation time (\(\tau_{C}^{-1}=\tau_{N}^{-1}+\tau_{B}^{-1}\)) and its relation to the proton's precessional frequency [2]. Crystal anisotropy influences this process through Neel's time constant (\(\tau_{N}=\tau_{0}(E_{a})\exp(E_{a}/kT)\)) where \(E_{a}\) is the anisotropy energy, \(\tau_{0}\) is the preexponential factor, and \(kT\) is thermal energy. At high magnetic fields (typical of clinical MRI scanners) and large MNP the first relaxation mechanism is dominant because the magnetic moment is locked along the external field and Neel relaxation is not possible. For small particles (few nm), both mechanisms contribute because crystal anisotropy is small leading to the appearance of low field dispersion in the Nuclear Magnetic Resonance Dispersion (NMRD) profile [2; 3]. It is known that spins diffusing around magnetic material, such as blood vessels or magnetic nanoparticles, accumulate phase due to the presence of magnetic field gradients created by the susceptibility difference between these materials and the surrounding background [22; 23]. It is the non-uniform nature of the magnetic fields that create phase dispersion leading to transverse magnetization decay.