Introduction

NMR, considered as one of the most potent analytical methods, traditionally requires dedicated personnel and delicate equipment thanks to the use of superconducting magnets, sizable electronics, and intricate probe and antenna placements1. Only recently, owing to the advancement of permanent-magnet design2, integrated electronics3,4, and antenna miniaturization5, portable NMR systems6 have emerged as a viable surrogate. Thanks to the reductions in footprint, maintenance needs and price tag, the miniaturized sensor assemblies have extended their uses beyond conventional NMR laboratories to point-of-care medical diagnostics7,8, subterranean explorations9, flow metering10, fluid authentication11, and artefact preservation12. In those “field” applications, it is highly desirable that the machinery operates autonomously and self-optimizes based on properties of the samples under investigation.

The needs for optimizing NMR spectroscopy become more pressing when considering that the quantities of interest, such as relaxation times (T1 and T2), diffusion coefficient, J-coupling, and chemical shift oftentimes span a large numerical range up to several orders of magnitude13. Consequently, a fit-for-all-purpose pulse sequence (PS) often does not exist. Misuses of pulse sequence could result in either a prolonged experiment time or a loss of measurement accuracy. Previous efforts on measurement optimization have been focusing on samples of simple compositions (containing single or double fluid species) and/or 1D functional forms of forward models14,15,16.

Another challenge for autonomous optimization at the embedded sensing devices stems from the limited computing infrastructure, where microprocessors of merely tens of MHz CPU clock-rate and fast memories of tens of KB to a few MB are available17. The so-called “constrained devices” may connect to a gateway or cloud platform of much greater computing throughput, but often the connection is slow and intermittent18,19. In those scenarios, realtime optimizations need to be executed in its entirety at the sensory nodes of meager resources.

We wondered whether it would be possible to optimize multidimensional NMR relaxometry that measures NMR relaxation times20 of complex fluids, in realtime, on a mobile sensor generally regarded too “dumb” to perform such tasks. Instead of optimizing one sequence to all probable samples15,16, we utilized a suite of sequences that were individually optimized for samples with distinctive ranges of relaxation properties.

More specifically, we used the inversion-recovery-CPMG (IRCPMG) pulse sequence for T1 − T2 correlation spectroscopy. The forward model that describes the signal evolution as a function of experimental parameters is21:

$$S({\tau }_{1},{\tau }_{2})=\iint (1-\theta {e}^{-{\tau }_{1}/{T}_{1}}){e}^{-{\tau }_{2}/{T}_{2}}f({T}_{1},{T}_{2})d{T}_{1}d{T}_{2}+\epsilon ,$$
(1)

where \(\epsilon \) is the experimental noise, θ is a calibration coefficient of the instrument, τ1 and τ2 are prescribed parameters that the spectrometer traverses through, with S(τ1, τ2) the corresponding recorded signals. Inversion methods22 are applied to obtain the sample {T1, T2} correlation spectrum, f(T1, T2). As real-life samples often have broadly distributed f(T1, T2), we made no assumptions other than T1 ≥ T223 on the mathematical constructs of the spectra.

Three classes of fluids were considered in the sequence design. As shown in Fig. 1, class A contained components that were longer than 0.1 s in both T1 and T2 dimensions; class B embodied high T1/T2 ratios, where T1 had components longer than 0.1 s while T2 spanned [1 ms, 0.1 s]; and class C had relatively short T1 and T2 that each spanned [1 ms, 0.1 s]. Accordingly, Table 1 shows the optimal sequences for the respective fluid classes (i.e. sequence α for fluid A, β for B, and γ for C). In particular, both sequences α and β had τ1 up to 10 s, capable of measuring T1 up to 2 s, while sequence γ had the maximal τ1 of 1 s that sufficed to measure T1 up to 0.2 s. Meanwhile, sequence α had the maximal τ2 = Ne × te of 10 s, capable of measuring T2 up to 2 s, in contrast to sequences β and γ with the maximal τ2 of 0.6 s. The shorter echo spacing, te, used in sequences β and γ than in sequence α could further help resolve fast T2 components (Fig. S1).

Figure 1
figure 1

The three sample classes: (A) have T1 and T2 components that are longer than 0.1 s; (B) have components of large T1/T2 ratio; and (C) only have short T1 and T2 components up to 0.1 s. The IRCPMG pulse sequence is shown in (D), where te is echo spacing, τ1 is T1 encoding time that includes N1 steps of iteration, τ2 is echo time, WT is polarization time, and NA is the number of scans; the vertical bars represent RF pulses and the in-between triangles represent echo signals.

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Table 1 The three optimal pulse sequences.
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Ideally, sequences shall always be applied to the intended samples under study; but in continuous measurements on samples of changing properties, occasions do arise in which a sequence is suboptimally applied. As sensors generally couldn’t foresee temporal progression of sample properties, any combinations of fluid class and pulse sequence are practically probable. Here we show applications of each sequence to three exemplary fluids, namely dodecane (fluid A), emulsified fluid (fluid B), and glycerol (fluid C), in Fig. 2. In the 3 by 3 panels, the diagonal time-domain images were acquired by the respective optimal sequences, while the measurements that corresponded to the off-diagonal ones were either inefficient or erroneous (Fig. S1). The key challenge was to discern the fluid class from live time-domain images, regardless of the sequences in use, and apply the intended one in the subsequent runs. In practice, we used three trained ECOC (error-correcting output codes) learners24, a class of supervised learning models, for the realtime multiclass classification and inference task.

Figure 2
figure 2

Top: T1 − T2 spectra of dodecane (fluid A), an emulsified fluid (fluid B), and glycerol (fluid C), acquired by sequences α, β, and γ respectively, and inverted using the Fast Laplace Inversion algorithm. Bottom: 9 log-log plots of time-domain patterns of NMR T1 − T2 measurements on the three fluids. Columns are by the fluid types, and rows by the applied sequences. Each pattern is normalized by the first acquisition point of the longest τ1 time at the top left corner, and contour lines, from bottom to top of darkening colors, represent −0.9, −0.5, −0.1, 0.1, 0.5, and 0.9. Measurements are optimal within the yellow envelope, inefficient within the blue, and erroneous within the red.

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Supervised learning requires large quantities of labeled datasets for model training. The training sets can consist of either prior measurements on samples of known properties or forward-modeled simulations. We opted for the later approach thanks to the well-defined functional form of Eq. 1. Specifically, we approximated probable T1 − T2 distributions by a large ensemble of synthetic distributions, each consisting of three components of randomly generated \(\{{\tilde{T}}_{1},{\tilde{T}}_{2}\}\) pairs. Since the measurement volume was a constant and filled with fluids of similar proton density, we assigned each component by a randomly-generated weighting coefficient, \(\tilde{\mu }\), that sums to unity. The time-domain data, ST, for model training were generated as:

$${S}_{T}({\tau }_{1},{N}_{e},{t}_{e})=\mathop{\sum }\limits_{n=1}^{n=3}\,{\tilde{\mu }}_{n}(1-\theta {e}^{-{\tau }_{1}/{\tilde{T}}_{1,n}}){e}^{-{N}_{e}\cdot {t}_{e}/{\tilde{T}}_{2,n}}+{\epsilon }_{T},$$
(2)

where \({\sum }_{n=1}^{n=3}{\tilde{\mu }}_{n}=1\). After calibrating the experimental setup, we set θ to 1.85 and \({\epsilon }_{T}\) to a Gaussian noise with zero mean and 0.012 standard deviation.

To generate \(\{{\tilde{T}}_{1},{\tilde{T}}_{2}\}\) pairs, we stochastically sampled in a two-dimensional space, where each dimension consisted of 100 logarithmically distributed numbers from 1 ms to 2 s and all pairs satisfied the relation \({\tilde{T}}_{1}\ge {\tilde{T}}_{2}\). In total, 30,000 \({\tilde{T}}_{1}-{\tilde{T}}_{2}\) distributions were created. Subsequently, we sifted the sampled distributions, one by one, through a set of classification criteria, and labeled them accordingly (Fig. S3). A given distribution was labeled class A if the longest \({\tilde{T}}_{2}\) > 0.1 s and its associated weighting coefficient ≥0.05, labeled class B if the longest \({\tilde{T}}_{2}\) < 0.1 s, its associated \({\tilde{T}}_{1}\) > 0.15 s, and its associated weighting coefficient ≥0.05, and labeled C if the longest \({\tilde{T}}_{1}\) < 0.1 s. As a result, 11,663 were assigned to class A, 11,835 to B, and 1449 to C.

To further reduce the size of training datasets, we exploited the separable structures of the functional form, and applied singular value decomposition (SVD) on T1 and T2 kernels independently22. Consequently, for any given sample, the size of compressed datasets was 1.57 KB when acquired by sequence α, 1.34 KB when acquired by β, and 1.25 KB when acquired by γ. As elaborated in the supplementary information, a near 1000-fold reduction in memory usage was achieved with the SVD compression.

Subsequently, we trained three ECOC classifiers, an ensemble method for multiclass classification problem24; each classifier is used while running the corresponding pulse sequence. The ECOC models encoded the binary classification results from three linear support vector machines (SVM) into a coding design matrix25, using the “one-versus-one” strategy that distinguished a pair of labeled time-domain patterns in the training set while ignoring the third fluid class. For sequences α, β, and γ, the ECOC classifiers had the respective size of 4.7, 4.1 and 3.8 KB. In total, less than 13 KB of fast memory was required to store the models.

In the inference stage of classifying a new 2D NMR dataset, we utilized the loss-weighted decoding scheme26 to aggregate predictions of the binary learners, in which the weighted “hinge” error functions25 over all binary losses were minimized. After running a pulse sequence, the number of floating-point calculations for classifying the generated data, after normalization and SVD compression, is fewer than 700. More details of the model training, validation and inference can be found in the supplementary information.

We performed realtime optimizations of continuous NMR experiments with the trained ECOC classifiers, as illustrated in Fig. 3. The mobile NMR sensor27, shown in Fig. 3B, was miniaturized largely due to the use of an NMR ASIC (Application Specific Integrated Circuit)3. The NMR probe, embodied in a Halbach-array magnet, was made of a solenoid coil wound around a polymer capillary, interrogating fluid samples of 17 μL in volume. During operation, the spectrometer executed a selected pulse sequence that was downloaded from the laptop, on which the acquired data were input to ECOC classifiers for realtime inference.

Figure 3
figure 3

Realtime optimizations of continuous NMR T1 − T2 experiments with ECOC classifiers. (A) The optimization workflow where S(X,Y) is the raw time-domain pattern from fluid of class X acquired by sequence Y and Sc(X,Y) is the pattern after normalization and SVD compression; (B) A photo of the miniaturized NMR system; (C) A series of NMR experiments on sequentially displaced fluid samples. Sample 1 to 6 are mixtures of glycerol and water with different volume ratios: 0:100 (1), 10:90 (2), 20:80 (3), 50:50 (4), 70:30 (5), and 100:0 (6); sample 7 is a well-gelled emulsified fluid. The vertical dashed lines mark the time stamps of sample injection. In the upper panel, the crosses signify applied sequences, with the red and blue circles highlighting inaccurate and inefficient measurements, respectively. In the lower panel, the circles and crosses signify calculated means of T1 and T2 times from the obtained 2D datasest; (D) Continuous NMR measurements on an emulsified fluid under static conditions, where y axis on the left indicates the sequence in use, and on the right indicates the calculated volume fraction of bulk base oil (any components with T2 > 0.1 s). The light gray band highlights the transition of optimal sequences. The inset shows the T1 − T2 correlation spectra of the fluid at the two marked time stamps.

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Figure 3C shows a series of experiments on sequentially displaced samples. The samples were six water/glycerol mixes of varying volume fractions and one emulsified fluid. As the volume fraction of glycerol increases, the relaxation times of the mixtures shorten from T2 = T1 = 2 s of pure water to T2 = T1/2 = 15 ms of pure glycerol; meanwhile, the T1/T2 ratio also inflates thanks to the elevated fluid viscosity. We started with sample 1 of pure water while applying sequence γ; the classifier correctly identified the fluid as type A and instructed to use sequence α for the subsequent run. Thereafter, sequence α was optimally applied for samples 1, 2 and 3. Sample 4 had T1/T2 slightly above 1 with T2 = 0.1 s. Consequently, the optimization routine signified the sample as type B. Sequence β was properly applied till sample 5 was loaded, which was classified as fluid C. Subsequently, sequence γ was applied on sample 5 and 6 of rather short relaxation times. Finally, we displaced glycerol by a well-gelled emulsion sample, which the ECOC classifier correctly deduced as a class B fluid; the spectrometer subsequently applied sequence β for the rest of the experiments.

In addition to physical displacement, a given sample could also evolve over time. For example, the emulsion fluids, which are multiphasic mixes of oil, brine, organoclays, and naturally-mined barite particles, could experience phase separation under static conditions. As the emulsion collapsed and solid particles sedimented, the emancipated oil exhibited a characteristic T2 time much longer than the original fluid, calling for a different optimal sequence.

Experimentally, we performed the optimization routine for continuously monitoring an emulsion sample under static conditions. As shown in Fig. 3D, the initial well-gelled emulsion fell in the class B fluid, to which sequence β was optimally applied. At about 50,000 s after the experiment commenced, signals of bulk oil started to appear with a T2 at ca. 0.5 s. As volume fraction of free oil increased, the fluid gradually morphed to class A, with the corresponding optimal sequence α. Notably, A transition window presented at ca. 60,000 s, where the free fluid content was still marginal while the inference results hinged partially on noise realizations of each measurement.

In practice, it is important to ensure that each sequence has sensitivities over the entire numerical domain of T1 − T2 spectra under consideration. Failure to meet the requirement could cause misclassification and thereby erroneous results. For example, the maximum τ1 in sequence γ, which is optimized for samples of fast relaxation times, should be designed so that the signal decays significantly with the maximal T1. Mathematically, it should satisfy 1 − exp(−τ1,max/T1,max) ≫ σ, where σ2 is the variance of the Gaussian noise. The relation is indeed held in the work, as τ1,max = 1 s for sequence γ, T1,max = 2 s, and σ2 = 0.0122.

Although we focus on relaxometry, the method can be extended to other types of NMR measurements of increasing complexities, such as multidimensional spectroscopy and MRI, at the core of which are forward models of similar mathematical constructs (exponential, sine and cosine functions). In conjunction with minimal requirements on computing resources, the demonstrated approach may further NMR methods to a substantially broadened usage in a wide range of field applications.