A fast Monte Carlo sampler for NMR T2 inversion

Abstract

The inversion of noisy NMR T2 echo data into a T2 spectrum is widely recognized as an inherently non-unique process [1]. One approach to quantifying this uncertainty is to use Monte Carlo sampling. Uncorrelated measurement noise combine with the non-negativity constraint on T2 spectral values to yield spectra following a non-negative normal distribution. There are two published samplers for truncated normal distributions [2], of which nonnegative normal samples are a subset, but we show that these converge too slowly to be practical for the T2 spectral inversion problem. This is because they are based on Gibbs’ samplers that update the spectral estimate just one T2 component at a time. When all of the spectral elements are fixed but one, that one has little room for change without violating the noise constraints on the data. Thus each spectral sample can only be slightly different from the preceding sample, indicating a high degree of statistical correlation and slow convergence. Our solution is to simultaneously update two neighboring spectral components at a time, allowing changes due to one spectral component to be offset by changes in its neighbor. Central to this improvement is a fast 2D slice sampler for non-negative normal distributions. This improves convergence by more than two orders of magnitude. Such speedup allows routine Monte Carlo inversion of 1D NMR spectra, and opens the door for the inversion of 2D NMR spectra.