Discrete and Continuous Spin–Spin Relaxation Rate Distributions Derived from CPMG NMR Response Curves: a Comparative Analysis Exemplified by Water in Meat

Abstract

The spin–spin relaxation rate distribution of water in a porcine longissimus dorsi muscle was derived from an inverse integral transformation of the proton CPMG (Carr–Purcell–Meiboom–Gill) NMR (nuclear magnetic resonance) signal at each hour during a 49 h drip period. This “continuous (C)” relaxation rate distribution was found to be excellently represented by an empirical peak function, characterized by three parameters: a peak width, an average relaxation rate, and a skewness parameter, which enable the distribution to be quantitatively defined. In addition, the same CPMG response was fitted to a sum of three single-exponential decay functions, denoted a “discrete (D)” relaxation rate model. The analysis shows that when the fraction of the slow relaxation component \(f_{2}^{\text{C}}\) from the continuous model is close to 5 %, which is a rather typical value, the mean relaxation rate \(\bar{R}_{22}^{\text{D}}\) from the discrete model becomes larger than the corresponding relaxation rate \(\bar{R}_{22}^{\text{C}}\) from the continuous model by nearly 25 % and \(f_{2}^{\text{D}}\) becomes larger than \(f_{2}^{\text{C}}\) by more than 75 %. Likewise, when \(f_{2}^{\text{C}}\) approaches 2.5 %, \(\bar{R}_{22}^{\text{D}}\) becomes larger than \(\bar{R}_{22}^{\text{C}}\) by more than 75 % and \(f_{2}^{\text{D}}\) becomes larger than \(f_{2}^{\text{C}}\) by more than a factor of 3 which are supported by model simulations. The relative quality and goodness of the two different relaxation rate models are discussed. Finally, the number of transients needed to obtain a preset error in relaxation rate and/or mole fraction was determined by model simulation.