Understanding the detailed nature of transverse signal decay in the presence of magnetic perturbations plays an important role in many fields of MRI. It is well known that observed transverse relaxation times depend on several factors, such as the perturbation magnitude, the interplay between spin diffusion and the spatial scale of the perturbers, and the refocusing rate in a multi-echo spin echo (CPMG) sequence. In this study, we present a closed-form, analytical solution for the evolution of the net transverse magnetization from a medium using the so-called weak field approximation (WFA).¹ We compare this analytical solution to simulations and to an asymptotic assumption of mono-exponential decay when the medium is composed of randomly positioned spherical perturbers. We also explore the relaxation properties at high volume fractions, where the roles of “perturber” and “external medium” become reversed.²

The WFA applies to systems of spins in the motional narrowing regime and where the perturber boundaries are freely permeable.³ The formulation for how the field inhomogeneities reduce the signal magnitude at time T is given by:¹$$S(T)=\text{exp}\left\{-\frac{\gamma^2}{2}\int_0^T dt^\prime\sigma(t^\prime)\int_0^T dt\sigma(t)K\left( |t-t^\prime|\right)\right\}\qquad[1]$$where σ(t) is a spin flip function of magnitude 1 and which changes sign upon application of any 180° pulses and K(t) is the temporal correlation function. Jensen and Chandra derived an asymptotic form as T$$$\rightarrow$$$∞ for how the transverse relaxation rate would change (ΔR₂) assuming mono-exponential decay.¹

While knowledge of ΔR₂ may be sufficient for many applications, other applications may require a more detailed understanding of how the MR signal decays and refocuses prior to and after any 180° refocusing pulses. Using the approximation proposed in (1), we have derived a closed-form solution (CFS) to Eq. [1] that satisfies this requirement for an arbitrary number and timing of refocusing pulses (equation omitted for space). The perturbers considered here were randomly positioned spheres, for which G₀, the mean-square field offset, and r_c, a characteristic spatial length, are defined.¹

The CFS was compared against the asymptotic solution of Jensen and Chandra for multiple echo spacings (ΔTE). To determine the accuracy of the WFA as it applies to intravascular signal, we compared the CFS results to simulations of the transverse signal from multiple networks of randomly positioned spheres populated to volume fractions (ζ) between 3 – 75%. Two sets of distributions were considered, those where spheres were allowed to overlap, and those where they were not. The simulations were performed using the deterministic diffusion method in three-dimensions.⁴ To model red blood cells, sphere radii were set to 3 μm and the diffusion coefficient of water was set to 2.7 μm2/ms.⁵

Fig. 1 shows time series that were calculated using the CFS and compares them to the solutions obtained under the assumption of mono-exponential decay.¹ These were calculated at 3 T with ζ = 40% and susceptibility offset corresponding to a blood oxygenation of 60%.⁶ In the CPMG examples, ΔR₂ was underestimated by less than 2% if at least three echoes were fitted, however, this grew to 15% for the free induction decay (ΔTE = ∞).

Fig. 2 shows sample results of the CFS overlaid on the simulation results using ΔTE = 40 ms for ζ = 3% and 40% at multiple O₂ saturations. The root mean square error (RMSE) between the CFS and the simulations was less than 0.015 for all simulation settings. In accordance with the WFA, these results were from simulation networks where spheres were allowed to overlap. Fig. 3b shows the results when the number of perturbers remained the same as in Fig. 2b but they were not allowed to overlap. Here the CFS breaks down when using the true ζ; however, by scaling ζ in the CFS, the RMSE can be minimized (Fig. 3c). This scaling is well approximated by a cubic polynomial constrained to pass through 0 when ζ = 0 or 100% (Fig. 4), and it can be interpreted as the perturber and external medium reversing roles.²

The closed-form solution to the WFA presented here is in close agreement with simulations from sphere networks covering a wide range of volume fractions and field offsets. In the more realistic scenario of non-overlapping spheres, the volume fraction in the CFS must be scaled to accurately describe the simulations; in line with experiments⁷ and theory.² When compared against a commonly employed model of mono-exponential decay,¹ the two models agree. We are currently applying the CFS to ex vivo blood relaxometry, and it is foreseen that it could be applied in iron imaging and to calculate intravascular contributions in simulations of BOLD imaging.