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

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. A general equation is derived for the sum of all separated and superposed widths of SCPs in a chromatogram. The equation is a function of the saturation α, a metric of chromatographic crowdedness, and is expressed in dimensionless form by dividing by the duration of the chromatogram. This dimensionless function, f(α), is also the cumulative distribution function of the probability of separating adjacent SCPs. Simulations based on the clustering of line segments representing SCPs verify expressions for f(α) calculated from five functions for the distribution of intervals between adjacent SCPs. Synthetic chromatograms are computed with different saturations, distributions of intervals, and distribution of SCP amplitudes. The chromatograms are analyzed by calculating the sum of the widths of peaks at different relative responses, dividing the sum by the duration of the chromatograms, and graphing the reduced sum against relative response. For small values of relative response, the reduced sum approaches the fraction of baseline that is occupied by chromatographic peaks. This fraction can be identified with f(α), if the saturation α is defined with the average minimum resolution equaling 1.5. The identification is general and is independent of the saturation, the interval distribution, or the amplitude distribution. This constant value of resolution corresponds to baseline resolution, which simplifies the interpretation of SOT.

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. The Weibull and exponential pdfs, which have a higher probability of small-concentration components than the log-normal pdf, are also investigated. Results show that assessments of the loss of low-intensity signals by curve fitting are ambiguous. Next, we simulate synthetic chromatograms to compare the loss of peaks from superposition (overlap) with neighboring peaks to the loss arising from lying below the limit of detection (LOD) imposed by a finite signal-to-noise ratio (SNR). The simulations are made using amplitude pdfs based on the Enke-Nagels data as functions of relative column efficiency, i.e., saturation, and SNR. Results show that at the highest efficiencies, the lowest-amplitude peaks are lost below the LOD. However, at small and medium efficiencies, peak overlap is the dominant loss mechanism, suggesting that low-level components will not be found easily in liquid chromatography with single channel detectors regardless of SNR. A simple treatment shows that a multichannel detector, e.g., a mass spectrometer, is necessary to expose more low-level components.

Effect of response factor variations on the response distribution of complex mixtures

The goal of "comprehensive" or "untargeted" analysis is to generate a list of all the compounds of a particular type in a complex mixture. To this end, modern analytical techniques, including high-resolution mass spectrometry, enable the individual detection of hundreds to tens of thousands of compounds, which can have response magnitudes that span several orders of magnitude. Moreover, we and others have found that many comprehensive analyses of complex mixtures result in a log-normal distribution of measured responses. The response distribution is not the concentration distribution but is related to it through the distribution of component response factors (RFs). Here, we present a theoretical investigation of how the shape of the RF distribution can impact the distribution of measured responses. We show that several distributions of RFs combined with a log-normal distribution of concentrations result in log-normal distributions of responses. Additionally, we find that deviations from log-normal responses depend on the shape of the RF distribution, and that, when they occur, these deviations occur in the low-response regions of the response distribution, which would often be below the detection limits for real measurements. In all cases, applying a distribution of RFs to a concentration distribution results in a response distribution that is both broader than the original concentration distribution and has a lower mean value than it would if the RFs for all the components were unity. Further, the distorting effect of the RF distribution can impair the determination of the original concentration distribution and the prediction of the number and concentrations of the undetected components.