Analysis of 2D NMR relaxation data using Chisholm approximations

To analyze 2D NMR relaxation data based on a discrete delta-like relaxation map we extended the Padé-Laplace method to two dimensions. We approximate the forward Laplace image of the time domain signal by a Chisholm approximation, i.e. a rational polynomial in two dimensions. The poles and residues of this approximation correspond to the relaxation rates and weighting factors of the underlying relaxation map. In this work we explain the principle ideas of our algorithm and demonstrate its applicability. Therefore we compare the inversion results of the Chisholm approximation and Tikhonov regularization method as a function of SNR when the investigated signal is based on a given discrete relaxation map. Our algorithm proved to be reliable for SNRs larger than 50 and is able to compete with the Tikhonov regularization method. Furthermore we show that our method is also able to detect the simulated relaxation compartments of narrow Gaussian distributions with widths less or equal than 0.05s^-1. Finally we investigate the resolution limit with experimental data. For a SNR of 750 the Chisholm approximation method was able to resolve two relaxation compartments in 8 of 10 cases when both compartments differ by a factor of 1.7.