Closed‐form solution for multisite 2D relaxation exchange NMR

Abstract

Recently, Landeghem et al. (1) reviewed T 2 -T 2 relaxation exchange spectroscopy (REXSY), which is an attractive means for studying exchange in porous media and biological tissue. This detailed review of REXSY data analysis will be a valuable resource for many wishing to perform such experiments. However, for the sake of thoroughness, we wish to point out one key omission that may be of interest to the readers of this journal.In their review, the authors note ''there is no published analytical solution for the case of relaxation and exchange between more than two sites'' for the T 2 -T 2 spectral peak amplitudes; therefore, ''only numerical simulations that take relaxation and exchange into account at all time periods of the 2D exchange experiment can support the interpretation of experimental data.'' However, a recent article by Dortch et al. (2) proposed and validated such an expression:[1]where P(t m ) is a general N-pool expression for the peak amplitudes derived from 2D inverse Laplace transform (ILT) of T 2 -T 2 data. Other terms in this equation are defined as follows: U L 2 is a matrix whose columns are the eigenvectors of L 2 ; L 1,2 ¼ ÀR 1,2 þ K, where R 1,2 is a diagonal matrix of R 1 s or R 2 s for each pool and K is a matrix of exchange rates; M 0 is a vector of equilibrium magnetizations; t m is the mixing time; 1 1 Â N is a 1 Â N vector of ones; represents element-wise matrix multiplication; and the superscript T represents the matrix transpose. This expression can be expressed analytically for two-pool systems, although in practice we typically compute it numerically. However, unlike numerical 2D ILT methods, this is a very stable and computationally inexpensive task.