Endodontics: principles and practice, 4th edition

Abstract

A family of models is proposed for the description of skewed chromatographic peaks, based on the modification of the standard deviation of a pure Gaussian peak, by the use of a polynomial function, h(t)) He-(1/2)([t-t R ]/[s 0 +s 1 (t-t R)+s 2 (t-t R) 2 +...]) 2 , where H and t R are the height and time at the peak maximum, respectively. The model has demonstrated a high flexibility with peaks of a wide range of asymmetry and can be used to accurately predict the profile of asymmetrical peaks, using the values of efficiency and asymmetry factor measured on experimental chromatograms. This possibility permits the simulation of chromatograms and the optimization of the separation of mixtures of compounds producing skewed peaks, where both the position and peak shape are considered. A first-degree polynomial was adequate for peaks of moderate asymmetry, but higher degree poly-nomials were preferable for peaks showing a high asymmetry , including those with negative skewness. The proposed models can be employed in the resolution of overlapped peaks in binary and ternary mixtures of compounds, or to improve the accuracy in the evaluation of peak shape parameters. The results obtained with the proposed models were comparable or even superior to those achieved with the exponentially modified Gaussian model. The fitting and resolution of peaks is of great importance in the field of analytical chemistry. In the literature, there are a number of reports where numerical methods are used to describe individual peaks and to achieve the deconvolution of overlapped peaks in a chromatogram. 1-5 The most simple chromatographic models predict Gaussian elution profiles. However, in practice, skewed peaks with low efficiencies may be obtained, and, in such cases, the assumption of a Gaussian model yields large errors. 6,7 The reasons for the deviation from the ideal behavior are diverse, but the main explanation is the slow mass transfer of the solutes between stationary and mobile phases and, to a lesser extent, extracolumn effects. The complexity of the chromatographic process does not facilitate the proposal of a simple function to accurately describe the peak profile. Many of the proposed models lack a physical meaning, such as the bi-Gaussian approach, which describes the peaks using two Gaussian equations with different standard deviation for the leading and tailing halves, the Gaussian-Lorentzian model, which substitutes a Lorentzian function by the Gaussian equation describing the tailing half of the peak, 8 and a model that combines the bi-Gaussian approach with an exponential decay. 9 Other attemps to explain the shape of chromatographic profiles have employed frequency distributions based on hypothetical equilibrium steps. These models have been applied to slightly distorted peaks, which can approximately be described by binomial and Poisson distributions. 10 One of the most popular models used in the literature for chromatographic peaks is the exponentially modified Gaussian model (EMG), 11-15 which has a physical justification. The model arises from the fact that the ideal Gaussian peak is distorted by first-order decays, caused by several intra-and extracolumn factors. The equation describing a pure Gaussian peak is where t is the time, H and t R are the height and time at the peak maximum, respectively, and σ G is the standard deviation. The EMG model considers that, in an asymmetrical peak, the Gaussian model is modified by an exponential function: where τ is a constant that quantifies the decay time of the system. In principle, several decay processes can exist in a real chromato-graphic system, which implies several time constants and a more complex decay function. However, one of the processes is usually dominant, and only one constant can be used to characterize the whole behavior. If t is negative, f(t) is nullified to avoid an infinite increase in the exponential contribution. When τ) 0, the exponential function is also null, and the EMG model describes a symmetrical peak. Convolution of both functions, the pure (1) Foley, J. P.