Effects of diffusion in magnetically inhomogeneous media on rotating frame spin-lattice relaxation

In an aqueous medium containing magnetic inhomogeneities, diffusion amongst the intrinsic susceptibility gradients contributes to the relaxation rate R_1ρ of water protons to a degree that depends on the magnitude of the local field variations ΔB_z, the geometry of the perturbers inducing these fields, and the rate of diffusion of water, D. This contribution can be reduced by using stronger locking fields, leading to a dispersion in R_1ρ that can be analyzed to derive quantitative characteristics of the material. A theoretical expression was recently derived to describe these effects for the case of sinusoidal local field variations of a well-defined spatial frequency q. To evaluate the degree to which this dispersion may be extended to more realistic field patterns, finite difference Bloch-McConnell simulations were performed with a variety of three-dimensional structures to reveal how simple geometries affect the dispersion of spin-locking measurements. Dispersions were fit to the recently derived expression to obtain an estimate of the correlation time of the field variations experienced by the spins, and from this the mean squared gradient and an effective spatial frequency were obtained to describe the fields. This effective spatial frequency was shown to vary directly with the second moment of the spatial frequency power spectrum of the ΔB_z field, which is a measure of the average spatial dimension of the field variations. These results suggest the theory may be more generally applied to more complex media to derive useful descriptors of the nature of field inhomogeneities. The simulation results also confirm that such diffusion effects disperse over a range of locking fields of lower amplitude than typical chemical exchange effects, and should be detectable in a variety of magnetically inhomogeneous media including regions of dense microvasculature within biological tissues.