The traditional peak integration method for quantitative NMR spectroscopy is inherently limited by its ability to resolve overlapping peaks, susceptibility to noise and phase errors; these obstacles become especially challenging with decreasing field strengths. The need to accurately analyse complex mixtures, including those data acquired with medium-field benchtop instruments, motivates the development of more robust model-based approaches for quantitative NMR [1].
Parametric model-based methods allow one to automate the analysis of NMR data by fitting them with specifically shaped peaks, e.g. Lorentzians. In the multitude of proposed techniques, the methods of Bayesian statistics stand out as particularly effective for the purpose of parameter estimation [2, 3, 4]. They smoothly incorporate any prior knowledge about the studied system into the model and provide a principled way to characterize the uncertainty of results.
Here, we develop a general model for an NMR signal and test it using simulations as well as experimental data. Our model takes into account the effects of chemical shifts, relaxation, phasing, and baseline distortions. Furthermore, it allows us to compensate for lineshape imperfections caused by inhomogeneity of the external magnetic field or other factors. This is an important feature for analysing data obtained from medium-field spectrometers or during reaction and process monitoring. In the chosen examples, where peaks are well separated and the SNR is high (> 40 dB), the conventional peak integration in the Fourier domain readily achieves almost perfect results. Yet we show that even in this challenging competition, our model-based estimation performs as well, and usually slightly better. Our approach achieves an accuracy of at least 0.01 mol/mol for the concentrations of all species. Most importantly, it is successful even when it is practically impossible to determine the correct phasing of the spectrum due to high levels of noise, and integration methods fail.