Statistics of Peak Overlap

With sufficiently complex samples, particularly biological and environmental samples, the frequency of overlap can be estimated by statistical means. In a statistical model developed by Davis and this author [33], far-reaching conclusions follow from a simple basic assumption the probability that any small interval dx along the separation path x is occupied by a component peak center is A dx, where A is a constant. This assumption defines a Poisson process and leads to well-known statistical conclusions. 

According to Poisson statistics the average spacing between adjacent component peaks is 1/A the number m of components expected to occupy an interval of length X along coordinate x is thus X divided by 1/A, or 

We note that m is a statistical number related to basic constant A it may differ slightly from the true component number m. However, because of peak overlap, the true m cannot be obtained by counting the number of peaks appearing in the chromatogram. Thus m, if obtainable, becomes our best approximation for m. 

Whether two adjacent component peaks are fused or separated depends on their resolution. We assume that some critical resolution / , not necessarily unity, is needed to successfully resolve the two the choice of R will depend on the type and accuracy of analytical data required. To achieve the designated R%s, the distance A A between two component peaks must be some minimum value jc0. We get x0 by using Eq. 5.52 

According to Poisson statistics, the probability that the gap A A between component peaks exceeds x0 is 

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Pietrogrande, M.C., Cavazzini, A., Dondi, F. (2000). Quantitative theory of the statistical degree of peak overlapping in chromatography. Rev. Anal. Chem. 19, 123. 

The statistical model of peak overlap clearly explains that the number of observed peaks is much smaller than the number of components present in the sample. The Fourier analysis of multicomponent chromatograms can not only identify the ordered or disordered retention pattern but also estimate the average spot size, the number of detectable components present in the sample, the spot capacity, and the saturation factor (Felinger et al., 1990). Fourier analysis has been applied to estimate the number of detectable components in several complex mixtures. 

Dondi, F., Bassi, A., Cavazzini, A., Pietrogrande, M.C. (1998). A quantitative theory of the statistical degree of peak overlapping in chromatography. Anal. Chem. 70, 766. Eckerskom, C., Strupat, K., Schleuder, D., Hochstrasser, D.F., Sanchez, J.-C., Lottspeich, F., Hillenkamp, F. (1997). Analysis of proteins by direct-scanning infrared-MALDI mass spectrometry after 2D-PAGE separation and electroblotting. Anal. Chem. 69, 2888. Expasy,http //www.expasy.ch. 

A more recent evaluation of peak overlap statistics has been reported by Delinger and Davis [41]. 

F. Dondi, A. Bassi, A. Cavazzini and M.C. Pietrogrande, A Quantitative Theory of the Statistical Degree of Peak Overlapping in Chromatography, Anal Chem. 70... 

Certainly two-dimensional techniques have far greater peak capacity than onedimensional techniques. However, the two-dimensional techniques don t utilize the separation space as efficiently as one-dimensional techniques do. These theories and simulations utilized circles as the basis function for a two-dimensional zone. This was later relaxed to an elliptical zone shape for a more realistic zone shape (Davis, 2005) with better understanding of the surrounding boundary effects. In addition, Oros and Davis (1992) showed how to use the two-dimensional statistical theory of spot overlap to estimate the number of component zones in a complex two-dimensional chromatogram. 

Liu, S., Davis, J.M. (2006). Dependence on saturation of average minimum resolution in two-dimensional statistical-overlap theory peak overlap in saturated two-dimensional separations. J. Chromatogr. A 1126, 244—256. 

It is important to realize that statistical-overlap theory is not constrained by the contour of area A, which does not have to be rectangular as in earlier studies (in addition to previous references, see Davis, 1991 Martin, 1991,1992). In other words, Equations 3.2 and 3.3 should apply to the spaces WEG, FAN, and PAR. In this chapter, the number of clusters of randomly distributed circles in such areas is compared to the predictions of Equations 3.2 and 3.3a to assess the relationship between nP and practical peak capacity. Similarly, the number of peak maxima formed by randomly distributed bi-Gaussians in such areas is compared to the predictions of Equations 3.2 and 3.3b, and to Fig. 3.2, to make another assessment. 

Single-column gas chromatographic analysis has become the standard approach for separation of volatile and semivolatile constituents in numerous applications however, this does not necessarily provide the best analytical result in terms of unique separation and identification. There is considerable opportunity for peak overlaps, both on a statistical basis and also on the basis of observed separations achieved for real samples [7-9]. 

With V— 102-5 x 104 and z = 1-10, as above, we see that from 10 to 800 distinct peaks can theoretically be resolved in electrophoresis. (In practice the number of resolvable components is much less than nc due to statistical peak overlap as explained in Section 6.7). This enormous resolving power is consistent with observation (see Figures 8.2 and 8.4). Theory thus provides an explanation of the unusual power of electrophoresis and an insight into the variables (z and V) that must be manipulated for increased performance. 

Origin and Characterization of Departures from the Statistical Model of Component-Peak Overlap in Chromatography, J. M. Davis and J. C. Giddings, J. Chromatogr., 289, 277 (1984). 

In terms of organization, the text has two main parts. The first six chapters constitute generic background material applicable to a wide range of separation methods. This part includes the theoretical foundations of separations, which are rooted in transport, flow, and equilibrium phenomena. It incorporates concepts that are broadly relevant to separations diffusion, capillary and packed bed flow, viscous phenomena, Gaussian zone formation, random walk processes, criteria of band broadening and resolution, steady-state zones, the statistics of overlapping peaks, two-dimensional separations, and so on. 

Figure 5 Examples of measured TOF spectra from liquid 6V, Hi, in a Nb can, at T 295 K. The error bars are due to counting statistics only. The full lines denote the fitted TOF spectra. For the scattering angle 0 = 65°, the C- and 7V6-recoil peaks overlap. For 0 = 132°, however, these two peaks are well resolved, thus facilitating a reliable determination of the C- and Nb-peak intensities.

The results of Tables I, II and III confirm the general applicability of the peak overlap model, developed from point statistics, to randomly generated chromatograms. Individual exceptions to the model will undoubtedly be found as experimental testing Is conducted, but, overall, we anticipate modestly good predictions of m from high resolution chromatographic separations when the components are distributed randomly-... 

As samples become more complex, the ability of a particular separation method to resolve all components decreases. A statistical study of component overlap has shown that a chromatogram must be approximately 95% vacant to provide a 90% probability that a given component of interest will appear as an isolated peak (9). This is shown graphically in Fig. 10, where the probability of separation is plotted as a function of the system peak capacity for cases where the number... 

Figure 11.5 Chromatographic resolution in various states. A resolution of 0.75 is quite usable in terms of determining the presence of multiple analyte components but makes quantitation problematic. A resolution of 1.0 is suitable for quantitation. A resolution of 1.5 (approximately a 0.3% statistical peak overlap) is considered to be baseline resolved and allows for essentially complete quantitation in terms of both peak height and area.

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