Theories describing transverse relaxation rate (1/T2) enhancement due to the presence of superparamagnetic (SPM) particles agree well with experimental and Monte Carlo (MC) simulation data¹ under the condition that the particles are monodisperse. Furthermore, they also assume that the magnetization phase of all particles is the same. However, it is well known that some synthesis techniques (in particular chemical precipitation) produce particles with a wide range of size and possibly magnetization values.² We present a distributed relaxation model that takes into account the spread in both particle size and magnetization. We test the model using MC simulation under both the Motional Averaging (MAR) and Static Dephasing Regimes (SDR).

Distributed Model: For a spherical particle with radius R, the MAR relaxation rate¹ is given by $$$(1/T2)^{MAR}=(\frac{\text{16}}{\text{45}})(\omega_{r})^{2}f\tau_{D}$$$ where f is the particle volume fraction, $$$\omega_{r}=\sqrt{8}\mu_{0}\gamma M/3$$$, is the rms Larmor frequency experienced by the proton at the surface of the particle of radius R and magnetization M, and $$$\tau_{D}=R^{2}/D$$$ is the diffusion characteristic time and D is the diffusion coefficient. For the SDR regime $$$(1/T2)^{SDR}=(\frac{2\pi\sqrt{3}}{9})\omega_{r}f$$$.

For a polydisperse sample any theoretical estimation of 1/T2 should, therefore, take into account the individual relaxation rate 1/T2_i of each group (i) containing N_i particles with the uniform size R_i and specific volume fraction f_i defined by $$$f_{i}=(\frac{4\pi}{3})N_{i}R_i^3/V$$$. Obviously, $$$\sum_i f_i=f$$$, and V is the accessible volume.

For the MAR regime (1/T2) can then be given by:

$$(1/T2)_{Dist}^{MAR}=\sum_i(1/T2)_i^{MAR}=(\frac{16}{45})\sum_i\omega_{r,i}^2f_{i}\tau_{D,i}$$

$$=(\frac{64\pi}{135VD})(\sqrt{8}\mu_{0}\gamma/3)^{2}\sum_iM_i^2N_iR_i^5 ..............[1]$$

τ_D,i and ω_r,i are the individual group diffusion correlation time ($$$\tau_{D,i}=R_i^2/D$$$) and frequency ($$$\omega_{r,i}=\sqrt{8}\mu_{0}\gamma M_{i}/3$$$), respectively. In the above distributed model, originally suggested by³, 1/T2 will depend on the average of the fifth power of the radius $$$(<R_i^5>)$$$. The SDR distributed 1/T2 value is then given by:

$$(1/T2)_{Dist}^{SDR}=\sum_i(1/T2)_i^{SDR}=(\frac{2\pi\sqrt{3}}{9})\sum_i\omega_{r,i}f_{i}$$

$$=(\frac{8\pi^2\sqrt{3}}{27V})(\sqrt{8}\mu_{0}\gamma/3)\sum_iM_iN_iR_i^3 ..............[2]$$

A comparison between the model and MC simulation is shown in Figure 1. For each data point a distribution of particles of different radii were chosen such that the MAR condition (ω_r τ_D < 1) is satisfied. For example, one distribution of particles included one each of radii 100, 200, and 300 nm abbreviated as [(1,100), (1,200), (1,300)]. The combinations of particle numbers/sizes are shown in Table 1 (Figure 2). The distributed model predicts the values successfully. When the particle size distribution is widened to cover both motional regimes the model (that combines both equations 1 and 2 above) fails (Figure 3). The SDR limit 1/T2 value exhibited by the large radii particles dominate over those of the smaller particles. Combinations of particle size included radii smaller and larger than the value R=600 nm satisfying (ω_r τ_D = 1). We introduced variations in the particles’ magnetization values (in addition to their size). Physically, these could ensue due to different magnetization phases or indeed different chemical compositions of the particles. The first data point in Figure 4 shows 1/T2 values for two particles of different size and equal magnetization: [(1,100; 1.0), (1,200; 1.0)]. The magnetization value = 1.0 corresponds to ω_r =2.36×10⁴ rad/s. The value of the magnetization of the small particle was increased by 5% and 10% in the other two data points: [(1,100; 1.05), (1,200; 1.0)] and [(1,100; 1.10), (1,200; 1.0)]. The model agrees well with MC simulation within the MAR regime. Figure 5 confirms that the model-based estimates (using $$$<R_i^5>$$$) yield larger values⁴ than estimates using the fifth power of the mean radius value ($$$<R_i>^5$$$). The latter is what is commonly used when the size distribution is not available. This work addresses the important issue of polydispersity of the nanoparticles^4,5. We recall that relaxometric parameters depend on the fifth power of the radius while magnetometric parameters depend on the third power of the radius so small errors in the determination of particle sizes can significantly affect the determination of relaxation rates. We tested a distributed relaxation model using MC simulation for both motional regimes. The model ascertains that relaxation data can be averaged according to the population of sizes. This intrinsically assumes that the spins are able to sample “average” all particles’ sizes within the measurement time. This is only true for the fast diffusion regime and explains why the model works well under the MAR conditions and not under the larger particles scale. Furthermore, we have successfully tested averaging according to magnetization, an important issue when considering surface/core nanoparticle structures which often results in samples with a range of M values. Current work is investigating the effects of echo times, D, and larger values of both M and its dispersity, in addition to combining both regimes.