Bayesian approach to peak deconvolution and library search for high resolution gas chromatography - Mass spectrometry.

A novel probabilistic Bayesian strategy is proposed to resolve highly coeluting peaks in high-resolution GC-MS (Orbitrap) data. Opposed to a deterministic approach, we propose to solve the problem probabilistically, using a complete pipeline. First, the retention time(s) for a (probabilistic) number of compounds for each mass channel are estimated. The statistical dependency between m/z channels was implied by including penalties in the model objective function. Second, Bayesian Information Criterion (BIC) is used as Occam's razor for the probabilistic assessment of the number of components. Third, a probabilistic set of resolved spectra, and their associated retention times are estimated. Finally, a probabilistic library search is proposed, computing the spectral match with a high resolution library. More specifically, a correlative measure was used that included the uncertainties in the least square fitting, as well as the probability for different proposals for the number of compounds in the mixture. The method was tested on simulated high resolution data, as well as on a set of pesticides injected in a GC-Orbitrap with high coelution. The proposed pipeline was able to detect accurately the retention times and the spectra of the peaks. For our case, with extremely high coelution situation, 5 out of the 7 existing compounds under the selected region of interest, were correctly assessed. Finally, the comparison with the classical methods of deconvolution (i.e., MCR and AMDIS) indicates a better performance of the proposed algorithm in terms of the number of correctly resolved compounds.

The use of δ(2)H and δ(18)O isotopic analyses combined with chemometrics as a traceability tool for the geographical origin of bell peppers.

Two approaches were investigated to discriminate between bell peppers of different geographic origins. Firstly, δ(18)O fruit water and corresponding source water were analyzed and correlated to the regional GNIP (Global Network of Isotopes in Precipitation) values. The water and GNIP data showed good correlation with the pepper data, with constant isotope fractionation of about -4. Secondly, compound-specific stable hydrogen isotope data was used for classification. Using n-alkane fingerprinting data, both linear discriminant analysis (LDA) and a likelihood-based classification, using the kernel-density smoothed data, were developed to discriminate between peppers from different origins. Both methods were evaluated using the δ(2)H values and n-alkanes relative composition as variables. Misclassification rates were calculated using a Monte-Carlo 5-fold cross-validation procedure. Comparable overall classification performance was achieved, however, the two methods showed sensitivity to different samples. The combined values of δ(2)H IRMS, and complimentary information regarding the relative abundance of four main alkanes in bell pepper fruit water, has proven effective for geographic origin discrimination. Evaluation of the rarity of observing particular ranges for these characteristics could be used to make quantitative assertions regarding geographic origin of bell peppers and, therefore, have a role in verifying compliance with labeling of geographical origin.

Probabilistic peak detection for first-order chromatographic data.

We present a novel algorithm for probabilistic peak detection in first-order chromatographic data. Unlike conventional methods that deliver a binary answer pertaining to the expected presence or absence of a chromatographic peak, our method calculates the probability of a point being affected by such a peak. The algorithm makes use of chromatographic information (i.e. the expected width of a single peak and the standard deviation of baseline noise). As prior information of the existence of a peak in a chromatographic run, we make use of the statistical overlap theory. We formulate an exhaustive set of mutually exclusive hypotheses concerning presence or absence of different peak configurations. These models are evaluated by fitting a segment of chromatographic data by least-squares. The evaluation of these competing hypotheses can be performed as a Bayesian inferential task. We outline the potential advantages of adopting this approach for peak detection and provide several examples of both improved performance and increased flexibility afforded by our approach.

Comprehensive study on the optimization of online two-dimensional liquid chromatographic systems considering losses in theoretical peak capacity in first- and second-dimensions: a Pareto-optimality approach.

A method to optimize different objectives (total analysis time, total peak capacity, and total dilution) has been applied to comprehensive two-dimensional liquid chromatography. The approach is based on Pareto-optimality, and it yields optimal parameters (column particle sizes, column diameters, and modulation times). Losses in the peak capacities in the first dimension (due to undersampling) and in the second dimension (due to high injection volumes) have been taken into account. The first effect (detection band broadening) reduces the original peak capacity by about a half, the second effect can reduce the total peak capacity by an additional half. Thus, the total loss in peak capacity may be 75% of its theoretical value. Analytical expressions to calculate these effects under gradient and isocratic conditions are derived. Of particular interest is the study of the optimal modulation times, which corresponded to between 2 and 3 two-dimensional runs per one-dimensional peak. The effects of using gradient or isocratic elution, conventional (40 MPa) or "ultra-high" (100 MPa) pressures, and focusing between the first and second dimensions were also studied. A trade-off between total peak capacity, total analysis time, and total dilution can be established.

Towards unsupervised analysis of second-order chromatographic data: automated selection of number of components in multivariate curve-resolution methods.

A method to apply multivariate curve-resolution unattendedly is presented. The algorithm is suitable to perform deconvolution of two-way data (e.g. retrieving the individual elution profiles and spectra of co-eluting compounds from signals obtained from a chromatograph equipped with multiple-channel detection: LC-DAD or GC-MS). The method is especially adequate to achieve the advantages of deconvolution approaches when huge amounts of data are present and manual application of multivariate techniques is too time-consuming. The philosophy of the algorithm is to mimic the reactions of an expert user when applying the orthogonal projection approach--multivariate curve-resolution techniques. Basically, the method establishes a way to check the number of significant components in the data matrix. The performance of the method was superior to the Malinowski F-test. The algorithm was tested with HPLC-DAD signals.

A comparative study of the performance of acetonitrile and methanol in the multi-linear gradient separation of proteic primary amino acids.

The performance of the separation of proteic primary amino acids using multi-linear gradients of acetonitrile and methanol was studied under an experimental-design basis, using an Inertsil ODS-3 column and pre-column derivatization with o-phthaldialdehyde (OPA) and N-acetyl-l-cysteine (NAC). Elution strength, peak properties, resolution, and analysis time, were examined. The optimal separation was established through modeling, using information obtained from isocratic data. By optimizing the separation with gradients of increasing complexity, acceptable resolution was possible, being glycine/threonine the critical pair. Multi-criteria decision-making (Derringer desirabilities) was applied to balance resolution and analysis time. The more favorable peak distribution for methanol gradients allowed a larger reduction of analysis times, keeping satisfactory resolution, but its smaller elution strength forces the use of concentrations significantly larger. Methanol is, however, less toxic, and the final cost is similar for both solvents.

Robust interpretive optimisation in high-performance liquid chromatography considering uncertainties in peak position.

In the context of interpretive chromatographic optimisation, robustness is usually calculated by introducing deliberated shifts in the nominal optimal conditions and evaluating their effects on the monitored objective function, mimicking thus the experimental procedures used in method validation. However, such strategy ignores a major source of error: the uncertainties associated to the modelling step, that may give rise to deceiving results when conditions that were expected to yield baseline separation are reproduced in the chromatograph. Two approaches, based on the peak purity concept, are here proposed to evaluate the robustness of the objective function under the perspective of measurement errors and modelling. The first approach implements these uncertainties as an extra band broadening for each chromatographic peak. The second one implements them as peak fluctuations in simulated replicated assays, which gives rise to a distribution of peak purities, easily computed through Monte-Carlo simulations. Both approaches predict satisfactorily a decreased separation capability, with respect to the conventional approach, for those situations where the uncertainties in peak position make the objective function critical. The first approach is less optimistic and formally less rigorous than the second one, but its computation is simpler. It can be used to map the critical resolution regions, to be comprehensively appraised further by the slower, although more rigorous, Monte-Carlo approach.

Automatic program for peak detection and deconvolution of multi-overlapped chromatographic signals part I: peak detection.

A series of two papers describing a procedure for automated peak deconvolution is presented. The goal is to develop a package of routines that can be used by non-experienced users. Part I (this paper) concerns peak detection, whereas Part II is dedicated to the deconvolution itself. In this first part, the most interesting features of the peak detection algorithms, which precede the deconvolution step, are outlined. High-order derivatives provide valuable information to assess the number of underlying compounds under a given peak cluster. A smoothing technique was found essential to compute properly the derivatives, since the noise is amplified when differences are calculated. The Savitsky-Golay smoother was applied in combination with the Durbin-Watson criterion to automate the window size selection. This strategy removed the noise without loosing valuable information. In some cases, it was found preferable to split the chromatogram in different elution regions, and apply the Durbin-Watson test and the Savitsky-Golay smoother to each region, separately. The derivatives allowed obtaining estimates of both peak parameters and the corresponding ranges for each eluting compound to be used in the deconvolution. An algorithm oriented to compare peaks from different chromatograms is also presented to perform deconvolution, using information from several related chromatograms.

Automatic program for peak detection and deconvolution of multi-overlapped chromatographic signals part II: peak model and deconvolution algorithms.

Several interlinked algorithms for peak deconvolution by non-linear regression are presented. These procedures, together with the peak detection methods outlined in Part I, have allowed the implementation of an automatic method able to process multi-overlapped signals, requiring little user interaction. A criterion based on the evaluation of the multivariate selectivity of the chromatographic signal is used to auto-select the most efficient deconvolution procedure for each chromatographic situation. In this way, non-optimal local solutions are avoided in cases of high overlap, and short computation times are obtained in situations of high resolution. A new algorithm, fitting both the original signal and the second derivatives is proved to avoid local optima in intermediate coelution situations. This allows achieving the global optimum without the need of background knowledge by the user. A previously reported peak model, a Gaussian with a polynomial standard deviation whose complexity can be modulated to enhance the fitting quality, was applied. However, the original formulation was modified to account baseline outside the peak region. Also, the optimal model complexity was auto-selected via error propagation theory. The method is able to process simultaneously several related chromatograms. The software was tested with both simulated and experimental chromatograms obtained with monolithic silica columns.

Limits of multi-linear gradient optimisation in reversed-phase liquid chromatography.

The concept of limiting peak purity was applied to quantify the degree of completion of the separation capability of a chromatographic system using multi-linear gradients. The objective was to check whether the complexity of a gradient program deserves be increased to enhance resolution by inserting more linear segments, or on the contrary, no significant improvements can be expected under more complex gradients. A set of 19 isoindole derivatives of primary amino acids was selected to test the performance of isocratic, single linear and multi-linear gradients. Accurate simulated chromatograms were obtained via numerical integration of the general equation of gradient elution, using pre-established start and end conditions of the gradient program. The overall peak purity was selected as objective function. Good--although not baseline--resolution was achieved with an optimal trilinear gradient. Excellent agreement between experimental and predicted optimal chromatograms was found. With the proposed approach, a degree of completion of the separation capability of the chromatographic system of 21.2, 49.7, 81.5 and 88.5% was accomplished with optimal gradients with one, two, three and four segments, respectively. More complex gradients did not enhance the latter figure significantly. Also, multi-linear gradients gave rise to more benefits than complementary gradients.

Estimation of significant solvent concentration ranges and its application to the enhancement of the accuracy of gradient predictions.

The solvent concentration range actually useful for gradient predictions is significantly narrower than the total range scanned in a gradient run. This range, called "solvent informative range" (SIR), if known with the highest accuracy, allows to predict gradient retention times (t(g) with minimal error. The small size of the SIR supports the application of the linear solvent strength theory (LSST). Furthermore, LSST allows a closed-form solution to the integral required to predict gradient retention times, which eliminates numerical integration, needed with other retention models. A methodology that calculates the SIR by applying error analysis, and uses it to improve the accuracy in the prediction of t(g) from isocratic experiments, is proposed. The importance of those mobile-phase compositions that do not contribute significantly to the prediction of t(g) is selectively attenuated within the prediction algorithm, relying the predictions more heavily on the SIR. As a result, t(g) was found to be predicted with similar accuracy using isocratic training data with regard to predictions based on gradient training data. The approach is useful for all situations where the chromatographer is able to provide predictions of retention at constant solvent concentration, and wish to predict the retention in gradient mode.

Error analysis and performance of different retention models in the transference of data from/to isocratic/gradient elution.

The transferability of retention data among isocratic and gradient RPLC elution modes is studied. For this purpose, 16 beta-blockers were chromatographed under both isocratic and gradient elution with acetonitrile-water mobile phases. Taking into account the elution mode where the experimental data come from, and the mode where the retention should be predicted, the following combinations are possible: isocratic predictions from (i) isocratic or (ii) gradient experimental designs; and gradient predictions from (iii) isocratic or (iv) gradient data. Each of these possibilities was checked using three retention models that relate the logarithm of the retention factor: (a) linearly and (b) quadratically with the volume fraction of organic solvent, and (c) linearly with a normalised mobile phase polarity parameter. The study was carried out under two different perspectives: a straightforward examination of the prediction errors and the analysis of the uncertainties derived from the variance-covariance matrix of the fitted models. The best combinations of prediction mode and model were: (i)-(b), (ii)-(c), (iii)-(b), and (iv)-(a) or (c).

Enhanced calculation of optimal gradient programs in reversed-phase liquid chromatography.

The resolution of a mixture of 16 beta-blockers under gradient elution was optimised using both isocratic and gradient training sets, with a reversed-phase column and acetonitrile-water eluents. Error theory was applied to measure the information extracted from different gradient experimental designs. This allows checking the expected accuracy when gradient predictions exceed the initial solvent concentrations tested in the training set. This work applies the results on modelling found in a previous study [J. Chromatogr. A 1018 (2003) 169] where the performance of several retention models was compared. Enhanced retention predictions were applied to the optimisation of gradient programs involving three factors (gradient slope, initial solvent composition and gradient curvature), using the peak purity criterion as resolution assessment. Peak shape parameters required in peak purity evaluation were modelled by adapting previous developments in isocratic mode. The mixture, which required prohibitive analysis times under isocratic elution, was almost baseline resolved in less than 35 min with linear gradients. Curvilinear gradients did not enhance this result significantly.

Net analyte signal as a deconvolution-oriented resolution criterion in the optimisation of chromatographic techniques.

The performance of two multivariate calibration measurements, multivariate selectivity (SEL(s)) and scalar net analyte signal (scalar NAS), as chromatographic objective functions (COFs), was investigated. Since both assessments are straightforwardly related to the quantification of analytes in the presence of interferents, they were expected to confer new features in the optimisation of compound resolution, not present in conventional assessments. These capabilities are especially interesting in situations of low resolution, where peak deconvolution becomes an attractive alternative. For comparison purposes, chromatographic resolution (R(s)) and peak purity (p(s)) were used as reference COFs. In order to correlate COFs with the probability of deconvolution error, an artificial peak crossing was used to generate 73 different peak arrangements, which were deconvolved using three different methods. SEL(s) exhibited the best correlation, which allowed predicting properly the risk of obtaining inaccurate deconvolutions. The optimisation of a poorly resolved mixture of 16 aromatic compounds by reversed-phase liquid chromatography with methanol-water and acetonitrile-water mobile phases was examined to investigate the differences in performance among the resolution criteria. In situations like these, SEL(s) tends to consider acceptable mobile phase compositions with partial coelution, which permits however the deconvolution with low errors. In contrast, p(s) selects compositions where the resolution of some compounds is sacrificed to enhance the separation of others. Scalar NAS was not so favourable as expected, since it depends on sampling frequency and peak widening. SEL(s) was not affected by these factors.

Peak deconvolution in one-dimensional chromatography using a two-way data approach.

A deconvolution methodology for overlapped chromatographic signals is proposed. Several single-wavelength chromatograms of binary mixtures, obtained in different runs at diverse concentration ratios of the individual components, were simultaneously processed (multi-batch approach), after being arranged as two-way data. The chromatograms were modelled as linear combinations of forced peak profiles according to a polynomially modified Gaussian equation. The fitting was performed with a previously reported hybrid genetic algorithm with local search, leaving all model parameters free. The approach yielded more accurate solutions than those found when each experimental chromatogram was fitted independently to the peak model (single-batch approach). The improvement was especially significant for those chromatograms where the peaks were severely affected by the tails of the preceding compounds. Peak shifts among chromatograms, which are a usual source of non-bilinearity, were modelled in a continuous domain instead of in a discrete way, which avoided some drawbacks associated with latent variable methods. An experimental design involving simulated chromatograms was applied to check the method performance. Five main factors affecting the deconvolution were examined: concentration pattern, chromatographic resolution, number of batches and replicates, and noise level, which were evaluated using first- and second-order figures of merit. The method was also tested on three real samples containing compounds showing different overlap. Four multi-batch deconvolution methods were considered differing in the nature of the processed information and kind of peak matching among chromatograms. In all cases, the multi-batch deconvolution yielded better performance than the single-batch approach.

Complementary mobile-phase optimisation for resolution enhancement in high-performance liquid chromatography.

An optimisation methodology in high-performance liquid chromatography (HPLC) is presented for the selection of two or more mobile phases having an optimal complementary resolution. The complementary mobile phases (CMPs) are selected in such a way that each one resolves optimally only some compounds in the mixture, while the remainder, resolved by the other mobile phase(s), can overlap among them. The methodology is based on the computation of a peak purity measurement for each solute, using an asymmetrical peak model for peak simulation. Two global resolution criteria (product of elementary resolutions and worst elementary resolution) and two methods for solving the problem (a systematic examination of all possible solute arrangements, and the use of genetic algorithms to expedite the calculation time) were used to find the optimal CMPs. The CMP optimisation methodology was applied to the resolution of a mixture of 10 diuretics and beta-blockers, which could not be resolved using a single mobile phase; virtual baseline resolution was achieved, however, with two CMPs.