Exchange-dependent relaxation in the rotating frame for slow and intermediate exchange -- modeling off-resonant spin-lock and chemical exchange saturation transfer.

Chemical exchange observed by NMR saturation transfer (CEST) and spin-lock (SL) experiments provide an MRI contrast by indirect detection of exchanging protons. The determination of the relative concentrations and exchange rates is commonly achieved by numerical integration of the Bloch-McConnell equations. We derive an analytical solution of the Bloch-McConnell equations that describes the magnetization of coupled spin populations under radiofrequency irradiation. As CEST and off-resonant SL are equivalent, their steady-state magnetization and dynamics can be predicted by the same single eigenvalue: the longitudinal relaxation rate in the rotating frame R1ρ . For the case of slowly exchanging systems, e.g. amide protons, the saturation of the small proton pool is affected by transverse relaxation (R2b ). It turns out, that R2b is also significant for intermediate exchange, such as amine- or hydroxyl-exchange or paramagnetic CEST agents, if pools are only partially saturated. We propose a solution for R1ρ that includes R2 of the exchanging pool by extending existing approaches, and verify it by numerical simulations. With the appropriate projection factors, we obtain an analytical solution for CEST and SL for nonzero R2 of the exchanging pool, exchange rates in the range 1-10(4) Hz, B1 from 0.1 to 20 μT and arbitrary chemical shift differences between the exchanging pools, whilst considering the dilution by direct water saturation across the entire Z-spectra. This allows the optimization of irradiation parameters and the quantification of pH-dependent exchange rates and metabolite concentrations. In addition, we propose evaluation methods that correct for concomitant direct saturation effects. It is shown that existing theoretical treatments for CEST are special cases of this approach.