Weighting functions and data truncation in the fitting of multi-exponential functions

Error structure for the HPLC analysis for atenolol, metoprolol and propranolol: a useful weighting method in parameter estimation

Three reversed-phase high performance liquid chromatography (HPLC) methods with UV detection were developed and fully validated for the quantification of three beta-blockers: atenolol, metoprolol and propranolol. After validation, error structures for the HPLC analysis were established using a convenient and practical procedure. The mean percentage of relative standard deviation (RSD) of the experimental concentrations (C), were less than 4.29% for proportionality and less than 3.68% for precision for any of the drugs, which allowed the quantitation of beta-blockers assayed at concentrations in the range 25-0.78 micrograms.ml-1. The error structures for the HPLC analysis were: SD (micrograms.ml-1) = 5.02 x 10(-2) C for atenolol, SD (micrograms.ml-1) = 4.55 x 10(-2) + 0.63 x 10(-2) C - 7.58 x 10(-6) C3 for metoprolol and SD (micrograms.ml-1) = 2.73 x 10(-2) C - 3.49 x 10(-4) C2 for propranolol. The reciprocal of the square of the SD of the drug concentrations measured within the calibration curve could be used as weighting methods in parameter estimation by non-linear regression.

Determination of analytical error function for beta-blockers as a possible weighting method for the estimation of the regression parameters

Three analytical methods have been developed and validated for the quantification of beta-blockers (celiprolol, bisoprolol and oxprenolol) using high performance liquid chromatography (HPLC) with UV detection. The methods were determined to be linear, precise and accurate (RSDs were lower than 5%), which allowed the quantitation of beta-blockers assayed at concentrations in the range 25-0.78 micrograms ml-1. After validation of reversed-phase HPLC methods, their analytical error functions were established by a rapid, simple and economical procedure. The discrimination of the best function for each active principle was performed by an appropriate polynomial statistical analysis, yielding SD (microgram ml-1) = 0.0295 + 0.0124C - 3.88 x 10(-4)C2 for celiprolol, 0.0199 + 0.011C - 1.27 x 10(-5)C3 for bisoprolol; and 0.0183 + 0.0089C - 9.68 x 10(-6)C3 for oxprenolol. These analytical error functions are an alternative to the weighting methods used in parameter estimation of beta-blockers.

Pharmacokinetic parameter estimations by minimum relative entropy method

For estimating pharmacokinetic parameters, we introduce the minimum relative entropy (MRE) method and compare its performance with least squares methods. There are several variants of least squares, such as ordinary least squares (OLS), weighted least squares, and iteratively reweighted least squares. In addition to these traditional methods, even extended least squares (ELS), a relatively new approach to nonlinear regression analysis, can be regarded as a variant of least squares. These methods are different from each other in their manner of handling weights. It has been recognized that least squares methods with an inadequate weighting scheme may cause misleading results (the "choice of weights" problem). Although least squares with uniform weights, i.e., OLS, is rarely used in pharmacokinetic analysis, it offers the principle of least squares. The objective function of OLS can be regarded as a distance between observed and theoretical pharmacokinetic values on the Euclidean space RN, where N is the number of observations. Thus OLS produces its estimates by minimizing the Euclidean distance. On the other hand, MRE works by minimizing the relative entropy which expresses discrepancy between two probability densities. Because pharmacokinetic functions are not density function in general, we use a particular form of the relative entropy whose domain is extended to the space of all positive functions. MRE never assumes any distribution of errors involved in observations. Thus, it can be a possible solution to the choice of weights problem. Moreover, since the mathematical form of the relative entropy, i.e., an expectation of the log-ratio of two probability density functions, is different from that of a usual Euclidean distance, the behavior of MRE may be different from those of least squares methods. To clarify the behavior of MRE, we have compared the performance of MRE with those of ELS and OLS by carrying out an intensive simulation study, where four pharmaco-kinetic models (mono- or biexponential, Bateman, Michaelis-Menten) and several variance models for distribution of observation errors are employed. The relative precision of each method was investigated by examining the absolute deviation of each individual parameter estimate from the known value. OLS is the best method and MRE is not a good one when the actual observation error magnitude conforms to the assumption of OLS, that is, error variance is constant, but OLS always behaves poorly with the other variance models. On the other hand, MRE performs better than ELS and OLS when the variance of observation is proportional to its mean. In contrast, ELS is superior to MRE and OLS when the standard deviation of observation is proportional to its mean. In either case the difference between MRE and ELS is relatively small. Generally, the performance of MRE is comparable to that of ELS. Thus MRE provides as reliable a method as ELS for estimating pharmacokinetic parameters.

Pharmacokinetic parameter estimates from several least squares procedures: superiority of extended least squares

The precision of pharmacokinetic parameter estimates from several least squares parameter estimation methods are compared. The methods can be thought of as differing with respect to the way they weight data. Three standard methods, Ordinary Least Squares (OLS-equal weighting), Weighted Least Squares with reciprocal squared observation weighting [WLS(y-2)], and log transform OLS (OLS(ln))--the log of the pharmacokinetic model is fit to the log of the observations--are compared along with two newer methods, Iteratively Reweighted Least Squares with reciprocal squared prediction weighting (IRLS,(f-2)), and Extended Least Squares with power function "weighting" (ELS(f-xi)--here xi is regarded as an unknown parameter). The values of the weights are more influenced by the data with the ELS(f-xi) method than they are with the other methods. The methods are compared using simulated data from several pharmacokinetic models (monoexponential, Bateman, Michaelis-Menten) and several models for the observation error magnitude. For all methods, the true structural model form is assumed known. Each of the standard methods performs best when the actual observation error magnitude conforms to the assumption of the method, but OLS is generally least perturbed by wrong error models. In contrast, WLS(y-2) is the worst of all methods for all error models violating its assumption (and even for the one that does not, it is out performed by OLS(ln)). Regarding the newer methods, IRLS(f-2) improves on OLS(ln), but is still often inferior to OLS. ELS(f-xi), however, is nearly as good as OLS (OLS is only 1-2% better) when the OLS assumption obtains, and in all other cases ELS(f-xi) does better than OLS. Thus, ELS(f-xi) provides a flexible and robust method for estimating pharmacokinetic parameters.

Extended least squares nonlinear regression: a possible solution to the "choice of weights" problem in analysis of individual pharmacokinetic data

It is often difficult to specify weights for weighted least squares nonlinear regression analysis of pharmacokinetic data. Improper choice of weights may lead to inaccurate and/or imprecise estimates of pharmacokinetic parameters. Extended least squares nonlinear regression provides a possible solution to this problem by allowing the incorporation of a general parametric variance model. Weighted least squares and extended least squares analyses of data from a simulated pharmacokinetic experiment were compared. Weighted least squares analysis of the simulated data, using commonly used weighting schemes, yielded estimates of pharmacokinetic parameters that were significantly biased, whereas extended least squares estimates were unbiased. Extended least squares estimates were often significantly more precise than were weighted least squares estimates. It is suggested that extended least squares regression should be further investigated for individual pharmacokinetic data analysis.

Fitting biological equations to data using non-parametric methods

Simulated experimental data were generated for these equations: a straight line, the integrated Michaelis--Menten equation, plus a linear term, the Hill equation, a two-exponential function and a double Michaelis--Menten equation. The equations were fitted to the data using (i) least-squares and (ii) non-parametric methods. The precision and accuracy of the parameter estimates obtained by each method were compared and the methods assessed. For several of the equations, non-parametric methods provided robust techniques for parameter estimation. For the remainder, the results were poor. The reasons for this are discussed.