## 2 Methods  
Theoretical models quantify \(R_{2}\) and \(R_{2}\)* in terms of a dynamic parameter known as the diffusion or translational correlation time (\(\tau_{D}\)) which is given by \(\tau_{D}=R^{2}/D\). \(R\) is the radius of the particle and \(D\) is the diffusion coefficient of the sample investigated. Two limiting regimes exist based on the value of the product (\(\omega_{r}\)\(\tau_{D}\)) where \(\omega_{r}\) is the Larmor frequency experienced by the proton at the equatorial line of the particle surface (radius = \(R\), \(\theta=\pi/2\)) of a magnetic particle of radius \(R\) and magnetization \(M\), and is given by \(\omega_{r}=\mu_{0}\gamma\)M/3. For small spherical particles satisfying (\(\omega_{r}\)\(\tau_{D}<<1\)), the MAR (also known as diffusional narrowing regime) relaxation rate [24; 25; 26] is given by (\(T_{2}=T_{2}^{*}\))  
\[R_{2}^{*}=R_{2}=\left(\frac{16}{45}\right)(\omega_{r})^{2}f\tau_{D} \tag{1}\]  
where \(f\) is the particle volume fraction. The water spins are very efficient in sampling and averaging the magnetic field variations introduced by the small MNP as long as they are not interrupted by refocusing RF pulses or MNP boundaries. Both \(R_{2}\) and \(R_{2}\)* are equal and independent of the echo time. For larger particles in SDR satisfying (\(\omega_{r}\)\(\tau_{D}\gg 1\)), relaxation is independent of both \(R\) and \(D\) and reaches a limiting value given by [23; 27]  
\[R_{2}^{*}=\left(\frac{2\pi}{3\sqrt{3}}\right)\omega_{r}f \tag{2}\]  
These two limiting cases are shown in Figure 1 along with the theoretical models given by Equations (1) and (2) above. The data shows the ability of MC technique to accurately predict relaxation rates. Details of the MC simulation are explained below. For spin-echo (SE) sequences (large particles) and as the echo-time (\(\tau_{CP}\)) is reduced the refocusing RF pulses become more effective and partial refocusing occurs and \(R_{2}\) decreases compared to \(R_{2}^{*}\). This is the echo-limited regime (ELR) [32; 33; 34] valid for \(\tau_{D}>2\tau_{CP}\), the relaxation time is given by  
\[R_{2}=\frac{7.2\ f\ D\ x^{1/3}\ (1.52+f\chi)^{5/3}}{4R^{2}} \tag{3}\]  
where \(x=\ \omega_{r}\tau_{CP}\) and \(\tau_{CP}\) is half the echo-time in a single Hahn echo or half the 180 RF pulses spacing in a Carr-Purcell sequence. All the above models assume uniform size and magnetization distributions, spherical particle shape, and low volume fractions. The dependence of \(R_{2}\) on \(\tau_{CP}\) is shown in Figure 1.  
Figure 1: \(R_{2}\) and \(R_{2}\)* values are plotted vs. the diffusion correlation coefficient (\(\tau_{D}\)) for three different echo-times \(\tau_{CP}\) = 0.02, 0.05, and 2.0 ms. The agreement of Monte Carlo (MC) values with the asymptotic models in the two limiting regimes is evident for all three echo-times. The echo-limited regime (ELR) dependence on echo-time is also shown (circles) for large particle radii (\(\tau_{D}\) > 5 \(\times\)\(10^{-4}\) s). We will evaluate these models under the condition of large particle volume fraction against \(R_{2}\) and \(R_{2}\)* values obtained using Monte Carlo simulations as these techniques have demonstrated agreement with both theoretical models and experimental findings by many previous studies [22; 35; 36]. Spins sample the inhomogeneous magnetic fields randomly due to Brownian motion and accumulate phase shift (\(\Delta\phi\)) in time (\(\Delta t\)) according to the relation  
\[\Delta\phi=\gamma B_{z}\left(\overrightarrow{r}\right)\Delta t \tag{4}\]  
\[B_{z}(r,\phi)=\frac{\mu_{0}M}{3}(\frac{R}{r})^{3}\Big{(}3 cos^{2}\phi-1\Big{)} \tag{5}\]  
where \(B_{z}\) is the magnetic field component parallel to the static field \(B_{0}\) and is the sum of contributions from the all MNPs according to their size and magnetization. The time step \(\Delta t\) is chosen such that the step length of each displacement (given by [PERSON]'s relation \(\Delta h=\sqrt{2D\ \Delta t}\) for one-dimensional displacement) does not exceed half the smallest particle radius.