Theoretical description of hyperpolarization formation in the SABRE-relay method

SABRE (Signal Amplification By Reversible Exchange) has become a widely used method for hyper-polarizing nuclear spins, thereby enhancing their Nuclear Magnetic Resonance (NMR) signals by orders of magnitude. In SABRE experiments, the non-equilibrium spin order is transferred from parahydrogen to a substrate in a transient organometallic complex. The applicability of SABRE is expanded by the methodology of SABRE-relay in which polarization can be relayed to a second substrate either by direct chemical exchange of hyperpolarized nuclei or by polarization transfer between two substrates in a second organometallic complex. To understand the mechanism of the polarization transfer and study the transfer efficiency, we propose a theoretical approach to SABRE-relay, which can treat both spin dynamics and chemical kinetics as well as the interplay between them. The approach is based on a set of equations for the spin density matrices of the spin systems involved (i.e., SABRE substrates and complexes), which can be solved numerically. Using this method, we perform a detailed study of polarization formation and analyze in detail the dependence of the attainable polarization level on various chemical kinetic and spin dynamic parameters. We foresee the applications of the present approach for optimizing SABRE-relay experiments with the ultimate goal of achieving maximal NMR signal enhancements for substrates of interest.