Site-based description of R_1ρ relaxation in local reference frames

Nuclear magnetic spin relaxation in the presence of an applied radiofrequency field depends critically on chemical exchange processes that transfer nuclear spins between chemical or conformational environments with distinct resonance frequencies. Characterization of chemical exchange processes in R_1ρ relaxation dispersion, CEST, and DEST experiments provides powerful insights into chemical and conformational kinetics of biological macromolecules. The present work reformulates expressions for magnetization evolution and the R_1ρ relaxation rate constant by focussing on the orientations of the tilted rotating frames of reference for magnetization components in different sites, by treating the spin-locking field strength as a perturbation to free-precession evolution, and by applying the Homotopy Analysis and Laplace transform methods to approximate solutions to the Bloch-McConnell equations. The results provide an expression for R_1ρ that is invariant to the topology of the kinetic scheme, an approximate equation for evolution of spin-locked z-magnetization, and an approach for effective simplification of chemical exchange topologies for 2- and N-site chemical exchange processes. The theoretical approach also provides an accurate approximation for relaxation during a constant-amplitude radiofrequency field in the absence of exchange.