This paper demonstrates how the multi-linear PARAFAC model can with advantage be used to decompose 2D diffusion¿relaxation correlation NMR spectra prior to 2D-Laplace inversion to the T2¿D domain. The decomposition is advantageous for better interpretation of the complex correlation maps as well as for the quantification of extracted T2¿D components. To demonstrate the new method seventeen mixtures of wheat flour, starch, gluten, oil and water were prepared and measured with a 300 MHz nuclear magnetic resonance (NMR) spectrometer using a pulsed gradient stimulated echo (PGSTE) pulse sequence followed by a Carr¿Purcell¿Meiboom¿Gill (CPMG) pulse echo train. By varying the gradient strength, 2D diffusion¿relaxation data were recorded for each sample. From these double exponentially decaying relaxation data the PARAFAC algorithm extracted two unique diffusion¿relaxation components, explaining 99.8% of the variation in the data set. These two components were subsequently transformed to the T2¿D domain using 2D-inverse Laplace transformation and quantitatively assigned to the oil and water components of the samples. The oil component was one distinct distribution with peak intensity at D = 3 × 10¿12 m2 s¿1 and T2 = 180 ms. The water component consisted of two broad populations of water molecules with diffusion coefficients and relaxation times centered around correlation pairs: D = 10¿9 m2 s¿1, T2 = 10 ms and D = 3 × 10¿13 m2 s¿1, T2 = 13 ms. Small spurious peaks observed in the inverse Laplace transformation of original complex data were effectively filtered by the PARAFAC decomposition and thus considered artefacts from the complex Laplace transformation. The oil-to-water ratio determined by PARAFAC followed by 2D-Laplace inversion was perfectly correlated with known oil-to-water ratio of the samples. The new method of using PARAFAC prior to the 2D-Laplace inversion proved to have superior potential in analysis of diffusion¿relaxation spectra, as it improves not only the interpretation, but also the quantification.
Tonning, E., Polders, D., Callaghan, P. T., & Engelsen, S. B. (2007). A novel improved method for analysis of 2D diffusion-relaxation data-2D PARAFAC-Laplace decomposition. Journal of Magnetic Resonance, 188(1), 10-23. https://doi.org/10.1016/j.jmr.2007.05.018