R1ρ relaxation for two-site chemical exchange: General approximations and some exact solutions

Analytic expressions for the nuclear magnetic spin relaxation rate constant for magnetization spin-locked in the rotating reference frame under an applied radiofrequency field, R1ρ, are obtained for two-site chemical exchange. The theoretical approach is motivated by Laguerre's method and obtains R1ρ as the root of a (p1, q1) Padé approximant. The general formula for R1ρ obtained by this approach is substantially simpler than existing expressions and is equally or slightly more accurate, in most cases. In addition, particular solutions for the R1ρ rate constant are presented for two special cases: equal populations of the two exchanging sites, or placement of the radiofrequency carrier at the average resonance frequency of the two sites. The solutions are exact when the R1 and R2 relaxation rate constants are identical, and nearly exact under realistic experimental conditions.