## 2 Methods
This is very important in order to yield adequate sampling of the magnetic field spatial inhomogeneity and consistent results. Our voxel is chosen to be cubic and its volume is calculated according to the chosen values of the volume fraction for any particle size distribution (i.e., voxel or sample volume \(V=\ q^{3}=\ \frac{4\pi}{3}\sum_{i}R_{i}^{3}/f\)). The boundary conditions at the surface of the cube are periodic where a diffusing spin is re-entered at the opposite face of the cubic voxel if the size of the next random walk jump would take it outside the voxel. The surface of the spherical MNP is impermeable and therefore spins are not allowed to either bypass the particle nor to take a jump that would leave the spin inside it. Details of the MC simulation are similar to those used in references mentioned above. The spin performs a random walk (jumps \(\Delta h\) in each Cartesian direction) and accumulates transverse phase according to Equation (4). This process continues for duration of the total MRI experiment \(TE\). For simulating \(T_{2}\) decay, the phase is inverted due to the application of the refocusing RF pulses. The number of spin jumps during \(\tau_{CP}\) is governed by the value (\(\tau_{CP}/\Delta t\)). The value of the transverse magnetization is given by  
\[M_{xy}=\ \frac{1}{N}\sqrt{\left[\sum_{i=1}^{N}\cos(\phi_{i})\right]^{2}+ \left[\sum_{i=1}^{N}\sin(\phi_{i})\right]^{2}} \tag{6}\]  
where \(N\) is the number of spins (\(N=4000\)). The simulated magnetization is fitted to a monoexponential curve from which \(R_{2}\) and \(R_{2}\)* are extracted. The parameters' values chosen in this study are as follows: \(\omega_{r}=2.36\ \times 10^{4}\) rad/s, \(D=2.5\ \times 10^{-9}\) m\({}^{2}\)/s, particle's radii: 1040-3000 nm for SDR and 69-200 nm for MAR, volume fraction (\(f\)) \(1\ \times 10^{-6}\)-\(2\ \times 10^{-1}\), the number of spin echoes is eight. This process is repeated for 40 spatial distributions (i.e., particle's locations) of the MNP and averaged together. For each limiting motional regime we used two particle size arrangements: variable and fixed size arrangements. For MAR Table 1 shows the two size arrangements below. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **Arrangement 1** & & **Volume** & & **Arrangement 2** & \\ \hline
**No. Particles** & \(q\) **(nm)** & **R (nm)** & **Fraction** & **No. Particles** & \(q\) **(nm)** & **R (nm)** \\ \hline
1 & \(1.89\ \times 10^{-5}\) & 200 & \(5.0\ \times 10^{-6}\) & 5 & \(3.22\ \times 10^{-5}\) & 200 \\
4 & \(1.50\ \times 10^{-5}\) & 126 & \(1.0\ \times 10^{-5}\) & 5 & \(2.56\times 10^{-5}\) & 200 \\
8 & \(8.75\ \times 10^{-6}\) & 100 & \(5.0\ \times 10^{-5}\) & 5 & \(1.50\times 10^{-5}\) & 200 \\
12 & \(6.95\ \times 10^{-6}\) & 87.4 & \(1.0\ \times 10^{-4}\) & 5 & \(1.19\ \times 10^{-5}\) & 200 \\
16 & \(4.06\ \times 10^{-6}\) & 79.4 & \(5.0\ \times 10^{-4}\) & 5 & \(6.95\ \times 10^{-6}\) & 200 \\
20 & \(3.22\ \times 10^{-6}\) & 73.7 & \(1.0\ \times 10^{-3}\) & 5 & \(5.51\ \times 10^{-6}\) & 200 \\
24 & \(1.89\ \times 10^{-6}\) & 69.3 & \(5.0\ \times 10^{-3}\) & 5 & \(3.22\ \times 10^{-6}\) & 200 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Number of particles and their radii along with sample space side (_q_) in the first three columns. Similar data is shown for fixed particle size arrangement in the last three columns. The center column contains the volume fraction values common to both arrangements.