Two dimensional IR-FID-CPMG acquisition and adaptation of a maximum entropy reconstruction
Introduction
More and more companies have in recent years proposed new compact Nuclear Magnetic Resonance (NMR) systems, which are very affordable, versatile, high performance and easy-to-use instruments. These low-resolution (or Time Domain) NMR spectrometers (TD-NMR) are particularly interesting for process control and formulation in food industries and are conventionally used to measure the solid fat content of fat blends used for butters, margarines and oils on the basis of well-known official methods from the American Oil Chemists’ Society (AOCS) [1], [2], [3]. Low-field spectrometers can also be equipped with linear magnetic field gradient units, allowing diffusion measurements, which can be very useful for instance in characterizing emulsions. The analytical potential of these spectrometers is however much wider. Based on longitudinal (T1) and transverse (T2) relaxation time measurements [4], [5], [6], benchtop NMR equipment can also be used to obtain quantitative information about the composition and the liquid/solid state of multiphasic compounds in products other than food, such as semi-clathrate hydrates, cements, wood or rocks in oil and gas production [7], [8]. NMR relaxation times are not only sensitive to the physical state of molecules (for example liquid or solid) [9], but are also sensitive to the liquid/solid interface in porous materials [10], [11], [12], [13], [14], [15] and to chemical and diffusional exchanges of protons between molecules and compartments [16]. These phenomena result in distributions of relaxation times related to the distribution and mobility of molecules in samples subjected to various dipolar interactions. Analysis of solid samples is currently performed by the acquisition of Free Induction Decay (FID) or solid echo signals. Applications for these sequences were demonstrated in the early 1970s when several low-field NMR methods were introduced, allowing rapid and precise determination of the solid fat content (SFC) in fatty products (see references in [9]). Later, with the improvement of the electronic and computational specifications of TD-NMR spectrometers, coupled FID-CPMG data (Fig. 1a) could be acquired [17] and extended to other food products [18], [19].
Extraction of the relaxation parameters from a decaying signal measured using CPMG, FID, or the classical inversion recovery (IR) sequence corresponds to a numerical inversion of the Laplace transform. Such signal processing task is known to be a large-scale inverse problem. One of the approaches to analyzing multi-exponential data consisted of a Non-Negative Least Squares (NNLS) fitting procedure [20], under suitable regularization constraints. This can be implemented, for instance, by the CONTIN software package [21]. Methods based on the inverse Laplace transform were developed later, but these showed numerical instability in the sense that a small noise in the data can cause considerable changes in the resulting distribution (see references in [22], [23], [24], [25]). However, the method used by Mariette et al., consisting of comparing Marquardt and Maximum Entropy Method (MEM) fitting of CPMG signals, was more successful [19]. Only a few publications have referred to the curve-fitting of combined FID-CPMG data [17]. Most publications have tended to use predefined functions generally based on NNLS algorithms and separate fitting of FID and CPMG signals because of the different mobility regimes of molecules described by Gaussian, Pake, and Lorentzian (exponential) functions [17], [19], [26]. For the same reason, no method based on MEM has been developed until now to adjust combined FID-CPMG data. In addition to this unidimensional approach, that is easily applied in the food and petroleum industries, two-dimensional (2D) cross-correlation relaxation is one of the major developments in low-field NMR in the last fifteen years. As in high-field NMR spectroscopy, the principle of the signal acquisition is based on the evolution of spin systems while applying two or more independent variables, such as time, instead of chemical shift or other specific magnetic interaction. In relaxometry, spins can be differentiated or correlated due to their different T1 and T2 relaxation times, characteristic of molecular species and specific dynamics depending on the molecule environment and dipolar interactions. Research has already benefited from this two-dimensional technique for many applications, from cement-based materials and rock to food and biological products, such as plants and wood (see references in [27]). However, these studies would not have been possible without the technical advances in computer hardware and the development of fast and efficient algorithms for two-dimensional inverse Laplace transformations. The first reconstruction of two dimensional relaxation distributions was successfully performed using NNLS and imposing positivity constraints on the solution amplitudes [28]. As a commonly used regularization method, the maximum entropy method (MEM) has shown satisfactory results for the reconstruction of 2D T1–T2 distributions [29], [30], [31], [32]. However, none of these reports have involved measurement of the solid phase in samples. NMR experiments to date have mostly been based on T1-weighted CPMG acquisitions (Fig. 1b), although several two-dimensional hybrid pulse sequences based on the multi-window analysis of T1-weighted FID signals have been developed [33], [34]. There are two main reasons to explain the lack of simultaneous studies of solid and liquid phases using a two-dimensional method: neither the sequence based on T1-weighted FID-CPMG (Fig. 1c), nor the 2D reconstruction method, which needs the combination of various analytical functions, has ever been implemented. Moreover, applications of T1–T2 experiments have focused principally on samples with a small or no solid fraction.
We propose here an adaptation of the existing 2D reconstruction algorithm [29] in order to process T1-weighted FID-CPMG datasets. We have therefore introduced a direct model based on a combination of the standard CPMG decreasing exponentials and Abragam’s FID function in the reconstruction method [35], [36], [37]. We show that these developments lead to an accurate reconstruction of the IR-FID-CPMG relaxation time distributions using the Maximum Entropy Method in two dimensions (MEM2D) and provide more relevant information about the overall composition of complex samples as compared to the distribution obtained from CPMG data alone.
Section snippets
Experimental section
1H NMR measurements were performed using a Time-Domain spectrometer (Minispec MQ20 Bruker, Germany) operating at a resonance frequency of 20 MHz and equipped with a 10 mm 1H high-low variable temperature probe head (PH H20-10-33/RH1/GY3 with a dead-time of 11.6 μs, Bruker SA, Wissembourg, France). Magnetic field tuning, homogeneity of the magnet, detection angles, receiver gain and pulse lengths were checked for each sample. The NMR system was regulated using dried air, cooled in a dedicated
Reconstruction of T1–T2 distributions from IR-CPMG acquisition
Conventional T1–T2 NMR experiments are performed with a CPMG sequence after the inversion recovery (IR) block, as illustrated in Fig. 1b. In this pulse sequence, the recovery time τ1 and the half-echo time τ2 are independent time variables giving rise to the acquisition of the two-dimensional array Y(τ1, τ2). The 2D NMR signal measured Y(τ1, τ2) depends on the continuous distribution S(T1, T2) of the T1 and T2 relaxation times according to the equation:
Results and discussion
The new sequence IR-FID-CPMG illustrated in Fig. 1c and the algorithm based on Eqs. (2), (12) were tested on model samples constituted of classical food ingredients, i.e., water, fats, and starch. The aim was to mimic physically compartmentalization of water or to create a multiphasic system where water and others constituents were not influenced by each other, or in other terms, did not interact. Two model samples were therefore prepared based on the physical separation of these ingredients,
Conclusions
We have presented here an adaptation of an efficient inversion method based on maximum entropy regularization in order to reconstruct 2D T1–T2 correlation maps obtained using the new IR-FID-CPMG sequence. We demonstrated that the insertion of an adapted Abragam’s FID function into the 2D Laplace inversion algorithm was remarkably effective for quantitative analysis of IR-FID-CPMG data. In this processing step, the sine function angle value (ω) can be adjusted or determined from the NNLS fitting
Acknowledgments
This work was performed using the NMR facilities of the PRISM Research Platform (Rennes, France). The authors thank the Regional Council of Brittany of France for their financial support.
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