Note on the Error Analysis of the Generalized Rankannihilation Method

ABSTRACT

The generalized rank annihilation method (GRAM) is a method for curveresolution and quantitation that uses two data matrices simultaneously, i.e., onefor the unknown and one for the calibration sample. Requirements have beenderived that ensure the unique resolution of the analyte of interest in thepresence of unknown interferences if the data matrices are free fromexperimental error. In this paper, it is shown that the same requirements allowfor correct determination of bias and variance in the quantitative results obtainedby GRAM if the data matrices are not free from experimental error.     

Simultaneous spectrofluorometric determination of piroxicam and pyridoxine using generalized rank annihilation method.

The application of generalized rank annihilation method (GRAM) to the analysis of fluorescence excitation-emission matrices of mixtures of piroxicam and pyridoxine is described. The input of GRAM consists of two bilinear data matrices, i.e. one for unknown and one for the calibration sample. The excitation wavelength range was from 290 to 340 nm and the emission was recorded from 370 to 560 nm. Piroxicam and pyridoxine were determined in the concentration ranges 0.33 - 4.00 microg ml(-1) and 0.66 - 8.00 microg ml(-1), respectively. To check the accuracy of the proposed method, several binary synthetic mixtures and one real sample were analyzed successfully. No matrix effect was observed in mixture analysis, so a single external calibration sample was used for each analyte. The ability of GRAM to quantify the studied compounds and the comparability of GRAM results were evaluated by comparing them with those of PLS regression as a standard first-order calibration.     

Time shift correction in second-order liquid chromatographic data with iterative target transformation factor analysis

When the generalized rank annihilation method (GRAM) is applied to liquid chromatographic data with diode-array detection, an important problem is the time shift of the peak of the analyte in the test sample. This problem leads to erroneous predictions. This time shift can be corrected if a time window is selected so that the chromatographic profile of the analyte in the test sample is trilinear with the peak of the analyte in the calibration sample. In this paper we present a new method to determine when this condition is met. This method is based on the curve resolution with iterative target transformation factor analysis (ITTFA). The calibration and test matrices are independently decomposed into profiles and spectra, and aligned before GRAM is applied. Here we study two situations: first, when the calibration matrix has one analyte and second, when it has two analytes. When the calibration matrix has two analytes, we selectively determine the time window for the analyte to be quantified. There were considerably fewer prediction errors after correction.     

Outlier detection for the Generalized Rank Annihilation Method in HPLC-DAD analysis

The Generalized Rank Annihilation Method (GRAM) is a second-order calibration method that is used in chromatography to quantify analytes that coelute with interferences. For a correct quantification, the peak of the analyte in the standard and in the test sample must be aligned and have the same shape (i.e., have a trilinear structure). Variations in retention time and shape between the two peaks may cause the test sample to behave as an outlier and produce an incorrect prediction. This situation cannot be detected by checking the coincidence of the recovered spectrum with the known spectrum of the analyte because the spectral domain is not affected. It cannot be detected either by checking if the recovered profile is correct (i.e., unimodal and positive). Several plots are presented to detect such outliers. The first plot compares the particular elution profiles in the standard and in the test sample that are recovered by least-squares regression from the spectra estimated by GRAM. The calculated elution profiles from both peaks should coincide. A second plot uses the elution profiles and spectra calculated by GRAM to define the vector space spanned by the interferences. The measured peaks in the standard and in the test sample are projected onto the space that is orthogonal to the space spanned by the interferences. These projections are proportional (up to the noise) if data are trilinear. The proportionality is checked graphically from the first singular vector of the projected peaks, or from the plot of the orthogonal signal versus the net sensitivity. The use of these graphs is shown for simulated data and for the determination of 4-nitrophenol in river water samples with liquid chromatography/UV-Vis detection.     

Second-order standard addition for deconvolution and quantification of fatty acids of fish oil using GC-MS

In the present work two second-order calibration methods, generalized rank annihilation method (GRAM) and multivariate curve resolution-alternating least square (MCR-ALS) have been applied on standard addition data matrices obtained by gas chromatography-mass spectrometry (GC-MS) to characterize and quantify four unsaturated fatty acids cis-9-hexadecenoic acid (C16:1ω7c), cis-9-octadecenoic acid (C18:1ω9c), cis-11-eicosenoic acid (C20:1ω9) and cis-13-docosenoic acid (C22:1ω9) in fish oil considering matrix interferences. With these methods, the area does not need to be directly measured and predictions are more accurate. Because of non-trilinear conditions of GC-MS data matrices, at first MCR-ALS and GRAM have been used on uncorrected data matrices. In comparison to MCR-ALS, biased and imprecise concentrations (%R.S.D. = 27.3) were obtained using GRAM without correcting the retention time-shift. As trilinearity is the essential requirement for implementing GRAM, the data need to be corrected. Multivariate rank alignment objectively corrects the run-to-run retention time variations between sample GC-MS data matrix and a standard addition GC-MS data matrix. Then, two second-order algorithms have been compared with each other. The above algorithms provided similar mean predictions, pure concentrations and spectral profiles. The results validated using standard mass spectra of target compounds. In addition, some of the quantification results were compared with the concentration values obtained using the selected mass chromatograms. As in the case of strong peak-overlap and the matrix effect, the classical univariate method of determination of the area of the peaks of the analytes will fail, the “second-order advantage” has solved this problem successfully.     

Quantification from highly drifted and overlapped chromatographic peaks using second-order calibration methods

For determining low levels of pesticides and phenolic compounds in river and wastewater samples by high performance liquid chromatography (HPLC), solid phase extraction (SPE) is commonly used before the chromatographic separation. This preconcentration step is not necessarily selective for the analytes of interest and it may retain other compounds of similar characteristics as well. In this case, we present, humic and fulvic acids caused a large baseline drift and overlapped the analytes to be quantified. The inaccurate determinations of the area of the peaks of these analytes made it difficult to quantify them with univariate calibration. Here we compare three second-order calibration algorithms (generalized rank annihilation method (GRAM), parallel factor analysis (PARAFAC) and multivariate curve resolution-alternating least squares (MCR-ALS)) which efficiently solve this problem. These methods use second-order data, i.e., a matrix of responses for each peak, which is easily obtained with a high performance liquid chromatography-diode array detector (HPLC-DAD). With these methods, the area does not need to be directly measured and predictions are more accurate. They also save time and resources because they can quantify analytes even if the peaks are not resolved. GRAM and PARAFAC require trilinear data. Biased and imprecise concentrations (relative standard deviation, %R.S.D. = 34) were obtained without correcting the time-shift. Hence, a time-shift correction algorithm to align the peaks was needed to obtain accurate predictions. MCR-ALS was the most robust to the time-shift. All three algorithms provided similar mean predictions, which were comparable to those obtained when sulfite was added to the samples. However, the predictions for the different replicates were more similar for the second-order algorithms (%R.S.D. = 3) than the ones obtained by univariate calibration after the sulfite addition (%R.S.D. = 13).     

An absorbance-based micro-fluidic sensor for diffusion coefficient and molar mass determinations

The H-Sensor reported herein is a micro-fluidic device compatible with flow injection analysis (FIA) and high performance liquid chromatography (HPLC). The device detects analytes at two separate off-chip absorbance flow cells, providing two simultaneous absorbance measurements. The ratio of these two absorbance signals contains analyte diffusion coefficient information. A theoretical model for the sensing mechanism is presented. The model relates the signal Ratio to analyte diffusion coefficient. The model is qualitatively evaluated by comparing theoretical and experimental signal Ratio values. Experimental signal Ratios were collected via FIA for a variety of analytes, including sodium azide, benzoic acid, amino acids, peptides, and proteins. Measuring absorbance at multiple wavelengths provides higher order data allowing the analyte signals from mixtures to be deconvolved via classical least squares (CLS). As a result of the H-Sensor providing two simultaneous signals as a function of time for each sample injection, two simulated second-order HPLC chromatograms were generated using experimental H-Sensor data. The chemometric deconvolution method referred to as the generalized rank annihilation method (GRAM) was used to demonstrate chromatographic and spectroscopic deconvolution. GRAM also provides the signal Ratio value, therefore simultaneously obtaining the analyte diffusion coefficient information during deconvolution. The two chromatograms successfully serve as the standard and unknown for the GRAM deconvolution. GRAM was evaluated on chromatograms at various chromatographic resolutions. GRAM was found to function to a chromatographic resolution at and above 0.25 with a percent quantitative error of less then 10%.     

Determination of the proteins in mixtures of meat,soymeal and rind from their chromatographic amino-acid pattern by the partial least-squares method

The simultaneous determination of muscle protein and rind protein in mixtures of muscle, soymeal and rind is described. Hydrolysates are derivatized with dansyl chloride and separated by reversed-phase liquid chromatography. The chromatographic pattern is subjected to multivariate evaluation. An indirect calibration method, partial least-squares modelling in latent variables (p.l.s.), is used. The rind protein is determined accurately regardless of sample composition. The accuracy of muscle protein determinations is lower when high concentrations of soymeal are present in the samples, because of the similarity in amino acid composition between soy protein and muscle. The method not only predicts the concentrations of the constituents but also calculates a measure of the similarity between the calibration set and each unknown sample. This gives an indication of the reliability of the predicted concentrations.     

Enhancing the Limit of Detection for Comprehensive Two‐Dimensional Gas Chromatography (GC×GC) using Bilinear Chemometric Analysis

The chemometric method referred to as the generalized rank annihilation method (GRAM) is used to improve the precision, accuracy, and resolution of comprehensive two‐dimensional gas chromatography (GC×GC) data. Because GC×GC signals follow a bilinear structure, GC×GC signals can be readily extracted from noise by chemometric techniques such as GRAM. This resulting improvement in signal‐to‐noise ratio (S/N) and detectability is referred to as bilinear signal enhancement. Here, GRAM uses bilinear signal enhancement on both resolved and unresolved GC×GC peaks that initially have a low S/N in the original GC×GC data. In this work, the chemometric method of GRAM is compared to two traditional peak integration methods for quantifying GC×GC analyte signals. One integration method uses a threshold to determine the signal of a peak of interest. With this integration method only those data points above the limit of detection and within a selected area are integrated to produce the total analyte signal for calibration and quantification. The other integration method evaluated did not employ a threshold, and simply summed all the data points in a selected region to obtain a total analyte signal. Substantial improvements in quantification precision, accuracy, and limit of detection are obtained by using GRAM, as compared to when either peak integration method is applied. In addition, the GRAM results are found to be more accurate than results obtained by peak integration, because GRAM more effectively corrects for the slight baseline offset remaining after the background subtraction of data. In the case of a 2.7‐ppm propylbenzene synthetic sample the quantification result with GRAM is 2.6 times more precise and 4.2 times more accurate than the integration method without a threshold, and 18 times more accurate than the integration method with a threshold. The limit of detection for propylbenzene was 0.6 ppm (parts per million by mass) using GRAM, without implementing any sample preconcentration prior to injection. GRAM is also demonstrated as a means to resolve overlapped signals, while enhancing the S/N. Four alkyl benzene signals of low S/N which were not resolved by GC×GC are mathematically resolved and quantified.     

Simultaneous determination of metronidazole and tinidazole in plasma by using HPLC–DAD coupled with second-order calibration

<正>A method using HPLC-DAD coupled with second-order calibration was developed to simultaneously determine metronidazole and tinidazole in plasma samples in this paper.The second-order calibration method based on APTLD(alternating penalty trilinear decomposition) algorithm was proposed to analyze the three-way HPLC-DAD data from both standard and prediction samples, which makes it possible that calibration can be performed even in the presence of unknown interferences with a simple and green chromatographic condition and short analysis time.The results showed that good recoveries were obtained although the chromatographic and spectral profiles of the analytes of interest as well as background were partially overlapped with each other in plasma samples.     

Parallel column liquid chromatography with a single multi-wavelength absorbance detector for enhanced selectivity using chemometric analysis

The selectivity of high performance liquid chromatography (HPLC) separations is increased using a parallel column configuration. In this system, an injected sample is first split between two HPLC columns that provide complementary separations. The effluent from the two columns is recombined prior to detection with a single multiwavelength absorbance detector. Complementary stationary phases are used so that each chemical component produces a detected concentration profile consisting of two peaks. A parallel column configuration, when coupled with multivariate detection, provides increased chemical selectivity relative to a single column configuration with the same multivariate detection. This enhanced selectivity is achieved by doubling the number of peaks in the chromatographic dimension while keeping the run time constant. Unlike traditional single column separation methodology, the parallel column system sacrifices chromatographic resolution while actually increasing the chemical selectivity, thus allowing chemometric data analysis methods to mathematically resolve the multivariate chromatographic data. The parallel column system can be used to reduce analysis times for partially resolved peaks and simplify initial method development as well as provide a more robust methodology if and when subsequent changes in the sample matrix occur (such as when new interferences show up in subsequent samples). Here, a mixture of common aromatic compounds were separated with this system and analyzed using the generalized rank annihilation method (GRAM). Analytes that were significantly overlapped on both stationary phases applied, ZirChrom PBD and CARB phases, when used in traditional single column format, were successfully quantified with a R.S.D.% of typically 2% when the same stationary phases were used in the parallel column format. These results indicate that a parallel column system should substantially improve the chemical selectivity and quantitative precision of the analysis relative to a single-column instrument.     

Guide to Semi-Quantitative Non-Targeted Screening Using LC/ESI/HRMS

Non-targeted screening (NTS) with reversed phase liquid chromatography electrospray ionization high resolution mass spectrometry (LC/ESI/HRMS) is increasingly employed as an alternative to targeted analysis; however, it is not possible to quantify all compounds found in a sample with analytical standards. As an alternative, semi-quantification strategies are, or at least should be, used to estimate the concentrations of the unknown compounds before final decision making. All steps in the analytical chain, from sample preparation to ionization conditions and data processing can influence the signals obtained, and thus the estimated concentrations. Therefore, each step needs to be considered carefully. Generally, less is more when it comes to choosing sample preparation as well as chromatographic and ionization conditions in NTS. By combining the positive and negative ionization mode, the performance of NTS can be improved, since different compounds ionize better in one or the other mode. Furthermore, NTS gives opportunities for retrospective analysis. In this tutorial, strategies for semi-quantification are described, sources potentially decreasing the signals are identified and possibilities to improve NTS are discussed. Additionally, examples of retrospective analysis are presented. Finally, we present a checklist for carrying out semi-quantitative NTS.     

Calibration by the gradient ratio-standard addition method in flow injection flame atomic absorption spectrometry

A calibration method has been developed which is realised in the flow injection analysis (FIA) by the gradient technique. According to this method two transient peaks, one for a sample and the other for a sample with standard addition, are recorded and compared point by point in the entire defined time range. The analytical result is estimated on the basis of information gained about the local analyte concentrations in the sample zone. The method allows the results to be reliable when both the non-linear calibration dependence and the interference effect occur. As an example, calcium in synthetic samples containing silicon, phosphate, aluminium, vanadium and titanium, and also in iron ore sample, were determined by flame atomic absorption spectrometry (FAAS). It has been proved, that the method can be effective in overcoming even extremely strong interferences, providing analytical results with average accuracy equal to ca. 5% and with precision 2–3 times inferior to that obtained by conventional FI calibration.     

Fast chromatographic separation techniques as competitors to chemical and biochemical sensor systems

Fast chromatographic methods, e.g. chromatography with supercritical fluids, ion chromatography and especially capillary electrophoresis, are important competitors of chemical sensors in environmental monitoring and process control. These methods show high selectivity and low interferences and it is possible to determine several parameters within a single chromatographic run. In order to show the potential and limitation of sensor systems and chromatographic separation methods in environmental analysis this review is limited to applications in this field.With regard to practical problems a comparison of ion chromatography (IC), capillary electrophoresis (CE) and electrochemical sensors is given for a special example, the ion analysis in waste water. Authentic water samples with a high surfactant content from a car-wash are examined, proving the suitability of IC, CE and electrochemical sensors in terms of specifity, sensitivity, reproducibility, analysis time and calibration linearity. The results show that the chromatographic methods are useful techniques in water analysis, yielding good sensitivity, high resolution and short analysis times. In comparison the chemical sensor also shows short analysis times, good sensitivity and a simple instrumental set-up. The disadvantage is the lack of selectivity and the instability of the sensor signal, when the sensor is exposed to the extremely complex matrix. Good results could only be achieved by pretreatment of the sample solution.     

Improving the performance of hollow waveguide-based infrared gas sensors via tailored chemometrics

The use of chemometrics in order to improve the molecular selectivity of infrared (IR) spectra has been evaluated using classic least squares (CLS), partial least squares (PLS), science-based calibration (SBC), and multivariate curve resolution-alternate least squares (MCR-ALS) techniques for improving the discriminatory and quantitative performance of infrared hollow waveguide gas sensors. Spectra of mixtures of isobutylene, methane, carbon dioxide, butane, and cyclopropane were recorded, analyzed, and validated for optimizing the prediction of associated concentrations. PLS, CLS, and SBC provided equivalent results in the absence of interferences. After addition of the spectral characteristics of water by humidifying the sample mixtures, CLS and SBC results were similar to those obtained by PLS only if the water spectrum was included in the calibration model. In the presence of an unknown interferant, CLS revealed errors up to six times higher than those obtained by PLS. However, SBC provided similar results compared to PLS by adding a measured noise matrix to the model. Using MCR-ALS provided an excellent estimation of the spectra of the unknown interference. Furthermore, this method also provided a qualitative and quantitative estimation of the components of an unknown set of samples. In summary, using the most suitable chemometrics approach could improve the selectivity and quality of the calibration model derived for a sensor system, and may avoid the need to analyze expensive calibration data sets. The results obtained in the present study demonstrated that (1) if all sample components of the system are known, CLS provides a sufficiently accurate solution; (2) the selection between PLS and SBC methods depends on whether it is easier to measure a calibration data set or a noise matrix; and (3) MCR-ALS appears to be the most suitable method for detecting interferences within a sample. However, the latter approach requires the most extensive calculations and may thus result in limited temporal resolution, if the concentration of a component should be continuously monitored.     

Comparison of GC-ECD,GC-MS and GC-AED for the determination of polychlorinated biphenyls in highly contaminated marine sediments

Summary Electron capture detection (ECD), low- and highresolution mass spectrometry (LR- and HRMS), and atomic emission detection (AED) were compared for the gas chromatographic (GC) detection of polychlorinated biphenyls (PCBs) present in highly contaminated marine sediments. With ECD, LRMS, and even HRMS, detection was seriously disturbed by the complex matrix of the sediments, whereas AED in the chlorine-selective mode provided excellent PCB profiles without interferences. In addition, GC-AED provided congener independent responses, which enabled accurate quantitation of all PCBs based on a single calibration curve. However, because GC-AED was less sensitive than the other techniques studied, preparation of relatively large amounts of sample (10–20 g dry sediment) was required for most analyses.     

Use of iridium plus rhodium as permanent modifier to determine As, Cd and Pb in acids and ethanol by electrothermal atomic absorption spectrometry

The most severe interferences in atomic absorption spectrometry are caused by the presence of anions when they are in different concentrations in the samples and in the calibration solutions. The analyte addition technique or matrix matching calibration can be employed to minimize or compensate the non-spectral interferences, but they are time consuming or difficult to be carried out. The use of chemical modifiers usually allows higher pyrolysis temperatures and consequently the removal of components of the sample matrix, equalizing the analyte signal in the sample and in the calibration solution. In this work, a mixture of Ir and Rh is proposed as permanent modifier to determine As, Cd and Pb in diluted hydrochloric, sulfuric and phosphoric acids and in ethanol and methanol by electrothermal atomic absorption spectrometry (ET AAS) with calibration against 1% v/v nitric acid aqueous solutions. The performance of the proposed permanent modifier was compared to that of Pd plus Mg nitrates in solution. Better recoveries, low background levels and faster analysis were obtained with the permanent modifier. The permanent modifier was also successfully employed for the determination of As, Cd and Pb in different concentrations of sulfuric and hydrochloric acids. For the phosphoric acid, the proposed modifier was only efficient for acid concentrations up to 2% v/v for As and up to 5% v/v for Cd and Pb. The precision, expressed as the relative standard deviation (n=3), was lower than 10%, for all samples, including ethanol and methanol.     

Fast and simultaneous determination of 12 polyphenols in apple peel and pulp by using chemometrics‐assisted high‐performance liquid chromatography with diode array detection

In this work, a smart chemometrics‐enhanced strategy, high‐performance liquid chromatography, and diode array detection coupled with second‐order calibration method based on alternating trilinear decomposition algorithm was proposed to simultaneously quantify 12 polyphenols in different kinds of apple peel and pulp samples. The proposed strategy proved to be a powerful tool to solve the problems of coelution, unknown interferences, and chromatographic shifts in the process of high‐performance liquid chromatography analysis, making it possible for the determination of 12 polyphenols in complex apple matrices within 10 min under simple conditions of elution. The average recoveries with standard deviations, and figures of merit including sensitivity, selectivity, limit of detection, and limit of quantitation were calculated to validate the accuracy of the proposed method. Compared to the quantitative analysis results from the classic high‐performance liquid chromatography method, the statistical and graphical analysis showed that our proposed strategy obtained more reliable results. All results indicated that our proposed method used in the quantitative analysis of apple polyphenols was an accurate, fast, universal, simple, and green one, and it was expected to be developed as an attractive alternative method for simultaneous determination of multitargeted analytes in complex matrices.     

Quantifying multiple trace elements in uranium ore concentrates: an interlaboratory comparison

An intercomparison was organized, with six laboratories tasked to quantify sixty-nine impurities in two uranium materials. The main technique employed for analysis was inductively coupled plasma mass spectrometry in combination with matrix-matched external calibration. The results presented highlight the current state-of-the-practice; lessons learned include previously unaccounted polyatomic interferences, issues related to sample dissolution, blank correction and calibration, and the challenge of estimating measurement uncertainties. The exercise yielded consensus values for the two analysed materials, suitable for use as laboratory standards to partially fill a gap in the availability of uranium reference materials characterized for impurities.     

混合氯化石蜡中短链氯化石蜡定量分析方法的比较

分别采用总体响应因子校正法和多元线性回归法对混合氯化石蜡中短链氯化石蜡进行定量分析,结果表明:两种方法均能较好地排除中链氯化石蜡的干扰,从而提高了短链氯化石蜡的定量准确性;但是两种方法均对标准曲线的浓度范围存在一定的依赖性,在使用这两种方法进行定量分析时,应将样品稀释到标准曲线适用的浓度范围。另外,相对于总体响应因子校正法,多元线性回归法定量范围更广。