Prediction by statistical overlap theory of fraction of baseline occupied by chromatographic peaks.

The average minimum resolution required for separating adjacent single-component peaks (SCPs) in one-dimensional chromatograms is an important metric in statistical overlap theory (SOT). However, its value changes with changing chromatographic conditions in non-intuitive ways, when SOT predicts the average number of peaks (maxima). A more stable and easily understood value of resolution is obtained on making a different prediction.

Estimation of low-level components lost through chromatographic separations with finite detection limits.

The search for biomarkers allowing the assessment of disease by early diagnosis is facilitated by liquid chromatography. However, it is not clear how many components are lost due to being present in concentrations below the detection limit and/or being obscured by chromatographic peak overlap. First, we extend the study of missing components undertaken by Enke and Nagels, who employed the log-normal probability density function (pdf) for the distribution of signal intensities (and concentrations) of three mixtures.

Theory of the probability of total resolution in chromatograms with systematic variation of average peak spacing and peak width.

An equation is proposed for the probability that all mixture constituents are separated, when the density (i.e., average number of eluting constituents per time) and width of single-component peaks (SCPs) vary systematically.

Likelihood of total resolution in selective comprehensive two-dimensional liquid chromatography with parallel processing: Simulation and theory.

The probability Pr(sLC×LC) that all peaks are separated by a resolution of 1.5 or more in selective comprehensive two-dimensional liquid chromatography (sLC × LC) is computed for simple model systems of 5 to 60 peaks and first-dimension (D) gradient times of 100 to 2000 s. The computations include mimics of a commercial instrument, whose fixed second-dimension (D) gradient time and use of one cycle time for initialization reduces Pr(sLC×LC) relative to an earlier report.

Orthogonality measurements for multidimensional chromatography in three and higher dimensional separations.

Orthogonality metrics (OMs) for three and higher dimensional separations are proposed as extensions of previously developed OMs, which were used to evaluate the zone utilization of two-dimensional (2D) separations. These OMs include correlation coefficients, dimensionality, information theory metrics and convex-hull metrics. In a number of these cases, lower dimensional subspace metrics exist and can be readily calculated.

Orthogonal separations: Comparison of orthogonality metrics by statistical analysis.

Twenty orthogonality metrics (OMs) derived from convex hull, information theory, fractal dimension, correlation coefficients, nearest neighbor distances and bin-density techniques were calculated from a diverse group of 47 experimental two-dimensional (2D) chromatograms. These chromatograms comprise two datasets; one dataset is a collection of 2D chromatograms from Peter Carr's laboratory at the University of Minnesota, and the other dataset is based on pairs of one-dimensional chromatograms previously published by Martin Gilar and coworkers (Waters Corp.).

Likelihood of total resolution in liquid chromatography: evaluation of one-dimensional, comprehensive two-dimensional, and selective comprehensive two-dimensional liquid chromatography.

Computer simulations of three methods of liquid chromatography (LC) are developed to understand better the conditions under which each method is superior to the others. The methods are one-dimensional LC (1D-LC), comprehensive two-dimensional LC (LC×LC), and selective comprehensive two-dimensional LC (sLC×LC). The criterion by which superiority is measured in this case is the probability that all peaks in a given sample are separated by a resolution equaling or exceeding unity.

New theory for distribution of minimum resolution in multi-component separations with noise/detection limits.

Equations were proposed recently for computing the distribution of minimum resolution (resolution distribution) of two Gaussian peaks with equal standard deviations, when peak heights in a multi-component separation follow a statistical distribution. The computation depended on the survival function of the peak-height ratio. Previously, an equation was derived for a first-order survival function that excluded peaks with heights less than a noise/detection limit.

Fractional coverage metrics based on ecological home range for calculation of the effective peak capacity in comprehensive two-dimensional separations.

Optimization of comprehensive two-dimensional separations frequently relies on the assessment of the peak capacity of the system. A correction is required for the fact that many pairs of separation systems are to some degree correlated, and consequently the entire separation space is not chemically accessible to solutes. This correction is essentially a measure of the fraction of separation space area where the solutes may elute.

The statistical overlap theory of chromatography using power law (fractal) statistics.

The chromatographic dimensionality was recently proposed as a measure of retention time spacing based on a power law (fractal) distribution. Using this model, a statistical overlap theory (SOT) for chromatographic peaks is developed that estimates the number of peak maxima as a function of the chromatographic dimension, saturation and scale. Power law models exhibit a threshold region whereby below a critical saturation value no loss of peak maxima due to peak fusion occurs as saturation increases.

Computation of distribution of minimum resolution for log-normal distribution of chromatographic peak heights.

General equations are derived for the distribution of minimum resolution between two chromatographic peaks, when peak heights in a multi-component chromatogram follow a continuous statistical distribution. The derivation draws on published theory by relating the area under the distribution of minimum resolution to the area under the distribution of the ratio of peak heights, which in turn is derived from the peak-height distribution. Two procedures are proposed for the equations' numerical solution.

Relationship between selectivity and average resolution in comprehensive two-dimensional separations with spectroscopic detection.

The average value of the multivariate selectivity (SEL) of randomly positioned peaks in a multi-component separation is shown to equal the average fraction of peaks that are singlets, as predicted by statistical-overlap theory (SOT). This equality is the basis for proposing a useful metric, specifically the average minimum resolution of nearest-neighbor peaks, for the performance of comprehensive two-dimensional (2D) separations. Furthermore this metric was computed both without ancillary spectroscopic information and with the assistance of such help, specifically multi-wavelength UV-vis spectra, acquired during the separation.

Dependence on effective saturation of numbers of singlet peaks in one- and two-dimensional separations.

The average numbers of singlet peaks in one-dimensional (1D) and two-dimensional (2D) separations of randomly distributed peaks are predicted by statistical-overlap theory and compared against the effective saturation. The effective saturation is a recently introduced metric of peak crowding that is more practitioner-friendly than the usual metric, the saturation. The effective saturation absorbs the average minimum resolution of statistical-overlap theory, facilitating the comparison of 1D and 2D separations by traditional metrics of resolution and peak capacity.

Estimation of migration-time and mobility distributions in organelle capillary electrophoresis with statistical-overlap theory.

The separation of organelles by capillary electrophoresis (CE) produces large numbers of narrow peaks, which commonly are assumed to originate from single particles. In this paper, we show this is not always true. Here, we use established methods to partition simulated and real organelle CEs into regions of constant peak density and then use statistical-overlap theory to calculate the number of peaks (single particles) in each region.

Evaluation of peak overlap in migration-time distributions determined by organelle capillary electrophoresis: Type-II error analogy based on statistical-overlap theory.

Organelles commonly are separated by capillary electrophoresis (CE) with laser-induced-fluorescence detection. Usually, it is assumed that peaks observed in the CE originate from single organelles, with negligible occurrence of peak overlap. Under this assumption, migration-time and mobility distributions are obtained by partitioning the CE into different regions and counting the number of observed peaks in each region.

Effective saturation: a more informative metric for comparing peak separation in one- and two-dimensional separations.

A theoretical comparison is made of the numbers of observed peaks in one-dimensional (1D) and two-dimensional (2D) separations having the same peak capacity, as calculated from the traditional metric of resolution. The shortcoming of the average minimum resolution of statistical overlap theory (SOT) for this comparison is described. A new metric called the "effective saturation" is introduced to ameliorate the shortcoming.

Dependence of effective peak capacity in comprehensive two-dimensional separations on the distribution of peak capacity between the two dimensions.

One of the basic tenets of comprehensive two-dimensional chromatography is that the total peak capacity is simply the product of the first- and second-dimension peak capacities. As formulated, the total peak capacity does not depend on the relative values of the individual dimensions but only on the product of the two. This concept is tested here for the experimentally realistic situation wherein the first-dimension separation is undersampled.

Effect of first-dimension undersampling on effective peak capacity in comprehensive two-dimensional separations.

The objective of this work is to establish a means of correcting the theoretical maximum peak capacity of comprehensive two-dimensional (2D) separations to account for the deleterious effect of undersampling first-dimension peaks. Simulations of comprehensive 2D separations of hundreds of randomly distributed sample constituents were carried out, and 2D statistical overlap theory was used to calculate an effective first-dimension peak width based on the number of observed peaks in the simulated separations. The distinguishing feature of this work is the determination of the effective first-dimension peak width using the number of observed peaks in the entire 2D separation as the defining metric of performance.

Observations on "orthogonality" in comprehensive two-dimensional separations.

The concept and definition of orthogonality in the context of comprehensive two-dimensional (2D) separations are interesting topics of active discussion. Over the years, several approaches have been taken to quantify the degree of orthogonality, primarily to serve as a metric to optimize (and compare) comprehensive 2D separations. Recently, a mathematical function was reported that is qualitatively instructive for the purpose of providing such a metric.

Comparison of micellar isotherms of benzene determined by headspace gas chromatography and micellar electrokinetic chromatography. Assessment on impact of buffer and solubilization-induced conductivity change.

The possibility is discussed that micellar isotherms determined by vacancy-micellar electrokinetic chromatography (vacancy-MEKC) differ from isotherms in electrolyte-free surfactants due to thermodynamic effects of buffer. Also discussed is the possibility that they are biased at high solute concentrations by solubilization-induced changes of electrical conductivity. Such bias could invalidate a theory on peak asymmetry of neutral solutes in MEKC that is based on thermodynamic interpretation of the isotherms.

Validation of theory of n-column separations with gas chromatograms predicted by commercial software.

A probability theory for the average number of compounds resolved by the partial separation of complex mixtures on n columns was tested using commercial-software predictions of gas chromatograms. Such n-column separations are traditional means for addressing peak overlap, in which one chooses additional columns of different selectivity to separate compounds that cannot be separated by a single column. Gas chromatograms of five types of complex mixtures containing from 99 to 283 compounds were predicted for eight stationary phases using both optimized and other temperature programs.

Dependence on saturation of average minimum resolution in two-dimensional statistical-overlap theory: peak overlap in saturated two-dimensional separations.

A theory is proposed for the dependence on saturation of the average minimum resolution R(*) in point-process statistical-overlap theory for two-dimensional separations. Peak maxima are modelled by clusters of overlapping circles in hexagonal arrangements similar to close-packed layers. Such clusters exist only for specific circle numbers, but equations are derived that facilitate prediction of equivalent cluster properties for any number of circles.

Origin of peak asymmetry and isotherm nonlinearity in micellar electrokinetic chromatography: variation of peak shape with buffer concentration.

The diffuse fronts and sharp rears of peaks of nitrobenzene (nbz) solubilized at high concentrations in 50 mM SDS and 2.5, 25, and 50 mM sodium tetraborate buffers were modeled in MEKC by measurements of, and fits to, concave upward isotherms, and by numerical solution of the continuity equation. The isotherms varied with buffer concentration, with the smallest limiting slope and largest curvature found for the 50 mM tetraborate buffer.

Probability theory for number of mixture components resolved by n independent columns.

A general theory is proposed for the probability of different outcomes of success and failure of component resolution, when complex mixtures are partially separated by n independent columns. Such a separation is called an n-column separation. An outcome of particular interest is component resolution by at least one column.