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.This work was supported in part by USUHS Grant RO-7516 and NIH Grants GM26676 and GM26691.     

General approach to handling nonuniform variance in assay calibration

A general method for handling nonuniform variance in data from assay calibrations is discussed. Calibration data from the analysis of ibuprofen and aspirin by high-performance liquid chromatography was analysed by the traditional least squares method; nonuniform error variance was found to be significant. Weighted least squares analysis overcomes the problem of nonuniform variance but relies on good estimates of the error variance. The method of extended least squares, a maximum likelihood method, is described which incorporates handling of the weighting in the regression analysis. The extended least squares method produces accurate and precise estimates of the parameters of the calibration and allows precise estimates of the variance of future predictions, provided that a sufficient number of calibrators are used.     

Analysis of pharmacokinetic data using parametric models. II. Point estimates of an individual's parameters

This is the second in a series of tutorial articles discussing the analysis of pharmacokinetic data using parametric models. In this article the basic issue is how to estimate the parameters of such models. Primary emphasis is placed on point estimates of the parameters of the structural (pharmacokinetic) model. All the estimation methods discussed are least squares (LS) methods: ordinary least squares, weighted least squares, iteratively reweighted least squares, and extended least squares. The choice of LS method depends on the variance model. Some discussion is also provided of computer methods used to find the LS estimates, identifiability, and robust LS-based estimation methods.     

Fitting heteroscedastic regression models to individual pharmacokinetic data using standard statistical software

In the analysis of individual pharmacokinetic data by nonlinear regression it is important to allow for possible heterogeneity of variance in the response. Two common methods of doing this are weighted least squares with appropriate weights or data transformation using a suitable transform. With either approach it is appealing to let the data determine the appropriate choice of weighting scheme or transformation. This article describes two methods of doing this which are easy to compute using standard statistical software. The first method is a generalized least squares scheme for the case where the variance is assumed proportional to an unknown power of the mean. The second involves applying a power transformation to both sides of the regression equation. It is shown that both techniques may be implemented using only nonlinear regression routines. Sample code is provided for their implementation using the SAS software package. However, the proposed methods are feasible using any software package that incorporates a nonlinear least squares routine, and are thus well suited to routine use.     

Analysis of pharmacokinetic data using parametric models. II. Point estimates of an individual's parameters

This is the second in a series of tutorial articles discussing the analysis of pharmacokinetic data using parametric models. In this article the basic issue is how to estimate the parameters of such models. Primary emphasis is placed on point estimates of the parameters of the structural (pharmacokinetic) model. All the estimation methods discussed are least squares (LS) methods: ordinary least squares, weighted least squares, iteratively reweighted least squares, and extended least squares. The choice of LS method depends on the variance model. Some discussion is also provided of computer methods used to find the LS estimates, identifiability, and robust LS-based estimation methods.Work supported in part by NIH grants GM26676 and GM 26691.     

Fitting heteroscedastic regression models to individual pharmacokinetic data using standard statistical software

In the analysis of individual pharmacokinetic data by nonlinear regression it is important to allow for possible heterogeneity of variance in the response. Two common methods of doing this are weighted least squares with appropriate weights or data transformation using a suitable transform. With either approach it is appealing to let the data determine the appropriate choice of weighting scheme or transformation. This article describes two methods of doing this which are easy to compute using standard statistical software. The first method is a generalized least squares scheme for the case where the variance is assumed proportional to an unknown power of the mean. The second involves applying a power transformation to both sides of the regression equation. It is shown that both techniques may be implemented using only nonlinear regression routines. Sample code is provided for their implementation using the SAS software package. However, the proposed methods are feasible using any software package that incorporates a nonlinear least squares routine, and are thus well suited to routine use.     

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 ℝ^N, whereN 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 pharmacokinetic 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 (ELS) for pharmacokinetic models

An important part of pharmacokinetic research is fitting models to observed data and estimating the parameters in the model. In general, parameter estimation in pharmacokinetics is a subset of the general problem of nonlinear regression or parameter estimation in nonlinear regression models. The same criteria, algorithms, and software used in other areas of science have been used in pharmacokinetics. Nonlinear modeling is a difficult mathematical and statistical task, often presenting problems. Any proposed new tool is of interest, and extended least squares (ELS) has been suggested as being better than the methods usually used. This suggestion and the evidence supporting it are examined; additional simulations are reported. With the evidence presently available, ELS does not seem to be superior to traditional least squares methods.     

Extended quasi-likelihood is useful for analyzing intra-individual variability in pharmacokinetic regression models

The method of maximum extended quasi-likelihood (MEQL) can be viewed as an estimation method in the framework of generalized linear models. The method was applied to a pharmacokinetic problem in which the pharmacokinetic model was a nonlinear function of its parameters. The behavior of the method toward the estimation of a variance function was numerically compared with those of the generalized least squares (GLS) and extended least squares methods. In general, the MEQL and GLS methods were equally better. However, the MEQL estimator often showed smaller mean squared errors for the scaling parameter than the other two estimators. Such a generally comparable but partially distinct property of the MEQL method, as compared with the GLS method, is useful to pharmacokinetic analysis.     

Fitting nonlinear regression models with correlated errors to individual pharmacodynamic data using SAS software

Nonlinear regression is widely used in pharmacokinetic and pharmacodynamic modeling by applying nonlinear ordinary least squares. Although the assumption of independent errors is frequently not fulfilled, this has received scant attention in the pharmacokinetic literature. As in linear regression, leaving correlation of errors out of account leads to an underestimation of the standard deviations of parameter estimates. On the other hand, the use of models that accommodate correlated errors requires more care and more computation. This paper describes a method to fit log-normal functions to individual response curves containing correlated errors by means of statistical software for time series. A sample computer program is given in which the SAS/ETS procedure MODEL is used. In particular, the problem of finding appropriate starting values for nonlinear iterative algorithms is considered. A linear weighted least squares approach for initial parameter estimation is developed. The adequacy of the method is investigated by means of Monte Carlo simulations. Furthermore, the statistical properties of nonlinear least squares with and without accommodating correlated errors are compared. Time action profiles of a long-acting insulin preparation injected subcutaneously in humans are analyzed to illustrate the usefulness of the method proposed.     

OMNE: a new robust membership-set estimator for the parameters of nonlinear models

A new method for estimating parameters and their uncertainty is presented. Data are assumed to be corrupted by a noise whose statistical properties are unknown but for which bounds are available at each sampling time. The method estimates the set of all parameter vectors consistent with this hypothesis. Its results are compared with those of the weighted least squares, extended least squares, and biweight robust regression approaches on two data sets, one of which includes 33% outliers. On the basis of these preliminary results, the new method appears to have attractive properties of reliability and robustness.     

OMNE: A new robust membership-set estimator for the parameters of nonlinear models

A new method for estimating parameters and their uncertainty is presented. Data are assumed to be corrupted by a noise whose statistical properties are unknown but for which bounds are available at each sampling time. The method estimates the set of all parameter vectors consistent with this hypothesis. Its results are compared with those of the weighted least squares, extended least squares, and biweight robust regression approaches on two data sets, one of which includes 33% outliers. On the basis of these preliminary results, the new method appears to have attractive properties of reliability and robustness.     

MicroPharm-K,a Microcomputer Interactive Program for the Analysis and Simulation of Pharmacokinetic Processes

Purpose. The microcomputer program, MicroPharm-K (MP-K) was developed for pharmacokinetic modeling, including analysis of experimental data and estimation of relevant parameters, and simulation. The intention was to provide a user-friendly, interactive, event-driven program for PC computers. Methods. The data are ascribed to a predefined model from a library including various routes of administration, oral or intra-venous, bolus or infusion, and various compartmental interpretations, 1 to 3. Single and multiple administrations are supported. The program provides initial estimates of the parameters in most cases, and the parameters are then fitted to the model by non linear model fitting using either the Simplex, Evol, Gauss-Newton, Levenberg-Marquardt or Fletcher-Powell algorithms. The non linear model fitting is based on the maximum likelihood method, and the criterion to minimize is either the weighted least squares (Chi² criterion) or the extended least squares. Graphical representations of non-fitted or curve-fitted data are immediately available (including log-scale representation), as well as pharmacokinetic typical parameters such as area under the curve, clearance, volumes, time-rate constants, transfer rate constants, etc. Results. Simulated and experimental data were analysed and the results were similar to those obtained by other programs. Conclusions. This non linear fitting program has been proved in our laboratory to be a very effective package for pharmacokinetic studies, including estimation and simulation. Because it is easy-to-use and runs on basic computers, the program could also be used for educational purposes.     

Validation of assay methodology used in pharmacokinetic studies

Statistical methods for validating assays used in pharmacokinetic studies are discussed. Methods for assessing linearity, variability, and sensitivity are derived. It is proposed that a calibration experiment, in which samples are replicated at three different concentrations, will provide the necessary information to validate an assay prior to commencement of a pharmacokinetic study and during its routine use. A two-stage weighted least squares regression is used to analyze the calibration data. High performance liquid chromatography calibration data are used to illustrate the principles and techniques involved.     

Pharmacokinetics of digoxin: Comparison of a two- and a three-compartment model in man

An experiment has been carried out in man designed to compare the fit of a two- and a three-compartment pharmacokinetic model to experimentally determined serum digoxin concentration-time data following rapid intravenous injection of 1.0 mg of the drug. Digoxin was administered to five healthy male volunteers, blood samples were withdrawn repetitively over a period of 72 hr, and samples were assayed using a ¹²⁵ I radioimmunoassay. Appropriate equations describing two- and three-compartment open models were fitted to the experimental data using weighted nonlinear least squares regression analysis. It was demonstrated that the three-compartment fit resulted in a statistically significant reduction in residual error, a marked improvement in the randomness of scatter of the experimental data about the serum digoxin-time curve, and better agreement of the predicted serum concentration-time curve with experimental serum digoxin concentrations. Thus the three-compartment open model is the simplest pharmacokinetic model consistent with the data observed in this experiment.This study was supported in part by Philips Roxane Laboratories, Inc., Columbus, Ohio.     

Predictive performance of a semiparametric method to estimate population pharmacokinetic parameters using NONMEM

Routine clinical pharmacokinetic (PK) data collected from patients receiving inulin were analyzed to estimate population PK parameters; 560 plasma concentration determinations for inulin were obtained from 90 patients. The data were analyzed using NONMEM. The population PK parameters were estimated using a Constrained Longitudinal Splines (CLS) semiparametric approach and a first-order conditional method (FOCE). The mean posterior individual clearance values were 7.73 L/hr using both parametric and semiparametric methods. This estimation was compared with clearances estimated using standard nonlinear weighted least squares approach (reference value, 7.64 L/hr). The bias was not statistically different from zero and the precision of the estimates was 0.415 L/hr using parametric method and 0.984 L/hr using semiparametric method. To evaluate the predictive performances of the population parameters, 17 new subjects were used. First, the individual inulin clearance values were estimated from drug concentration-time curve using a nonlinear weighted least-squares method then they were estimated using the NONMEM POSTHOC method obtained using parametric and CLS methods as well as an alternative method based on a Monte Carlo simulation approach. The population parameters combined with two individual inulin plasma concentrations (0.25 and 2 hr) led to an estimation of individual clearances without bias and with a good precision. This paper not only evaluates the relative performance of the parametric and the CLS methods for sparse data but also introduces a new method for individual estimation.     

The bayesian bias correction method of the first-order approximation of nonlinear mixed-effects models for population pharmacokinetics

Population pharmacokinetic analysis usually employs nonlinear mixed-effects models. To estimate the parameters, Beal and Sheiner (1982) proposed the first-order method that employs a first-order Taylor series expansion around the means of random individual parameters. Because of the small computational burden and the high convergence proportion of maximization of the log likelihood function, this method is often used in practice. However, it is known that the estimates are biased. This paper proposes a simple procedure to reduce the bias. The proposed method maximizes the nonapproximated log likelihood functions of each individual given estimates of the population parameters derived from the first-order method, and the derived Bayes estimates of the random individual parameters are utilized to improve the estimates of the population mean parameters. We confirmed that the proposed method reduced the bias using simulated data and actual erythropoietin concentration data.     

A new algorithm for computing the parameters of linear compartment models in pharmacokinetics

A new algorithm (FADHA) for computing pharmacokinetic parameter estimates has been developed. This technique is based on the simplex method which is used to minimize a nonlinear cost function. An important property of this program is that the convergence is ensured contrary to the well-known linear or nonlinear least-squares regression analysis which lead to a lack of convergence or to a false one. Two investigations of the comparative performances of FADHA program and other algorithms were undertaken (hexamethylmelamine and Piracetam pharmacokinetics). Least square analysis of data yielded biased estimates whereas FADHA estimates were unbiased and more precise. This new technique, takes into account all the possible observation errors and uses the concept of a weighting function rather than weights as such.     

Simulation of vancomycin peak and trough concentrations using five dosing methods in 37 patients

Five methods of dosing vancomycin (Matzke, Moellering, Nielsen, Lake-Peterson, and manufacturer's) were simulated in 37 patients. Ten serum samples were obtained after a 1-hour intravenous infusion of 6.2-20 mg/kg total body weight. A preinfusion serum sample was obtained from patients not studied on the first dose. Initial estimates of pharmacokinetic values were made using nonlinear iterative least squares regression and serum concentration-time data. These data were fitted to a two-compartment, open-infusion model. Simulations of the peak and trough serum concentrations at steady state for each patient were determined by multiple-dose simulated pharmacokinetics and each patient's pharmacokinetic values using the regimen suggested by each of the five methods. Steady-state serum concentrations, predicted systemic clearance by each method (except Lake-Peterson), and the daily dose for each patient recommended by each method were determined. All the methods underpredicted actual drug clearance, with the Nielsen method having the lowest prediction. The Matzke method recommended the largest dosage. Using each of the methods, only 3-16% of patients would have achieved recommended peak and trough serum concentrations. In the simulation model used, no method performed satisfactorily in attaining the desired vancomycin peak and trough concentrations. We suggest that the Lake-Peterson method could be used initially, provided that monitoring is also performed to adjust the dosage regimen further.