Measurement of Spin–Lattice Relaxation Times and Concentrations in Systems with Chemical Exchange Using the One-Pulse Sequence: Breakdown of the Ernst Model for Partial Saturation in Nuclear Magnetic Resonance Spectroscopy

Abstract

A fundamental problem in Fourier transform NMR spectroscopy is the calculation of observed resonance amplitudes for a repetitively pulsed sample, as first analyzed by Ernst and Anderson in 1966. Applications include determination of spin–lattice relaxation times (T1's) by progressive saturation and correction for partial saturation in order to determine the concentrations of the chemical constituents of a spectrum. Accordingly, the Ernst and Anderson formalism has been used in innumerable studies of chemical and, more recently, physiological systems. However, that formalism implicitly assumes that no chemical exchange occurs. Here, we present an analysis of N sites in an arbitrary chemical exchange network, explicitly focusing on the intermediate exchange rate regime in which the spin–lattice relaxation rates and the chemical exchange rates are comparable in magnitude. As a special case of particular importance, detailed results are provided for a system with three sites undergoing mutual exchange. Specific properties of the N-site network are then detailed. We find that (i) the Ernst and Anderson analysis describing the response of a system to repetitive pulsing is inapplicable to systems with chemical exchange and can result in large errors in T1 and concentration measurements; (ii) T1's for systems with arbitrary exchange networks may still be correctly determined from a one-pulse experiment using the Ernst formula, provided that a short interpulse delay time and a large flip angle are used; (iii) chemical concentrations for exchanging systems may be correctly determined from a one-pulse experiment either by using a short interpulse delay time with a large flip angle, as for measuring T1's, and correcting for partial saturation by use of the Ernst formula, or directly by using a long interpulse delay time to avoid saturation; (iv) there is a significant signal-to-noise penalty for performing one-pulse experiments under conditions which permit accurate measurements of T1's and chemical concentrations. The present results are analogous to but are much more general than those that we have previously derived for systems with two exchanging sites. These considerations have implications for the design and interpretation of one-pulse experiments for all systems exhibiting chemical exchange in the intermediate exchange regime, including virtually all physiologic samples.