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.     

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.     

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.     

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.     

迭代法求取一室模型血管外给药药物动力学的参数

提出给一种新的求取一室模型血管外给药动力学参数的方法,该法首先用残差法求取参数的初值,再使用迭代法求取参数的准确值。本法原理和计算较非线性最小二乘法更为简单方便,且例子的计算结果优于参数法和非线性最小二乘拟合法。     

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.     

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.     

A comparison of the accuracy of a least squares regression, a Bayesian, Chiou's and the steady-state clearance method of individualising theophylline dosage

We compared a least squares regression method, used prospectively to individualise the intravenous aminophylline and oral theophylline dosage of 48 patients, with 3 other pharmacokinetic methods - Chiou's, the steady-state clearance and the Bayesian - used retrospectively to analyse the same patient data. Methods were compared on the basis of the similarity of their parameter estimates and the accuracy with which serum concentrations during subsequent intravenous and oral therapy could be forecast, assuming each method's parameter estimates. The least squares and Bayesian programs were able to fit data from all but 4 and 2 patients, respectively. Mean absolute prediction errors were of the order of 20% for serum concentrations during intravenous therapy, and of the order of 40% for serum concentrations during oral therapy. The accuracy of the least squares, Bayesian and steady-state clearance methods were similar, but the accuracy of Chiou's method was comparable only when the 2 serum concentrations needed for the method were measured between 11 and 17 hours apart; an interval which corresponds to the 1.0 to 1.5 half-lives previously suggested as desirable for implementation of the Chiou method.     

The influence of weight with assay error on gentamicin pharmacokinetics using the Bayesian and nonlinear least square regression analysis in appendicitis patients

The purpose of this study was to determine the influence of weight with gentamicin assay error on the Bayesian and nonlinear least squares regression analysis in 12 Korean appendicitis patients. Gentamicin was administered intravenously over 0.5 h every 8 h. Three specimens were collected 48 h after the first dose from all patients at the following times, just before the regularly scheduled infusion, at 0.5 h and 2 h after the end of the 0.5 h infusion. Serum gentamicin levels were analysed by fluorescence polarization immunoassay technique with TDxFLx. The standard deviation (SD) of the assay over its working range had been determined at the serum gentamicin concentrations of 0, 2, 4, 8, 12 and 16 microg/ml in quadruplicate. The polynominal equation of gentamicin assay error was found to be SD (microg/ml) = 0.0246-(0.0495C) + (0.00203C(2)). There were differences in the influence of weight with gentamicin assay error on pharmacokinetic parameters of gentamicin using the nonlinear least squares regression analysis but there were no differences on the Bayesian analysis. This polynominal equation can be used to improve the precision of fitting of pharmacokinetic models to optimize the process of model simulation both for population and for individualized pharmacokinetic models. The result would be improved dosage regimens and the better, safer care of patients receiving gentamicin.     

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.     

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.     

Quantitative Structure-Pharmacokinetic Relationships for drug distribution properties by using general regression neural network

Quantitative Structure-Pharmacokinetic Relationships (QSPkR) have increasingly been used for developing models for the prediction of the pharmacokinetic properties of drug leads. QSPkR models are primarily developed by means of statistical methods such as multiple linear regression (MLR). These methods often explore a linear relationship between the pharmacokinetic property of interest and the structural and physicochemical properties of the studied compounds, which are not applicable to those agents with nonlinear relationships. Hence, statistical methods capable of modeling nonlinear relationships need to be developed. In this work, a relatively new kind of nonlinear method, general regression neural network (GRNN), was explored for modeling three drug distribution properties based on diverse sets of drugs. The three properties are blood-brain barrier penetration, binding to human serum albumin, and milk-plasma distribution. The prediction capability of GRNN-developed models was compared to those developed using MLR and a nonlinear multilayer feedforward neural network (MLFN) method. For blood-brain barrier penetration, the computed r(2) and MSE values of the GRNN-, MLR-, and MLFN-developed models are 0.701 and 0.130, 0.649 and 0.154, and 0.662 and 0.147, respectively, by using an independent validation set. The corresponding values for human serum albumin binding are 0.851 and 0.041, 0.770 and 0.079, and 0.749 and 0.089, respectively, and that for milk-plasma distribution are 0.677 and 0.206, 0.224 and 0.647, and 0.201 and 0.587, respectively. These suggest that GRNN is potentially useful for predicting QSPkR properties of chemical agents.     

Measuring the predictive performance of computer-controlled infusion pumps.

Current measures of the performance of computer-controlled infusion pumps (CCIPs) are poorly defined, of little use to the clinician using the CCIP, and pharmacostatistically incorrect. We propose four measures be used to quantitate the performance of CCIPs: median absolute performance error (MDAPE), median performance error (MDPE), divergence, and wobble. These measures offer several significant advantages over previous measures. First, their definitions are based on the performance error as a fraction of the predicted (rather than measured) drug concentration, making the measures much more useful to the clinician. Second, the measures are defined in a way that addresses the pharmacostatistical issue of appropriate estimation of population parameters. Finally, the measure of inaccuracy, MDAPE, is defined in a way that is consistent with iteratively reweighted least squares nonlinear regression, a commonly used method of estimating pharmacokinetic parameters. These measures make it possible to quantitate the overall performance of a CCIP or to compare the predictive performance of CCIPs which differ in either general approach (e.g., compartmental model driven vs. plasma efflux approach), pump mechanics, software algorithms, or pharmacokinetic parameter sets.     

Measuring the predictive performance of computer-controlled infusion pumps

Current measures of the performance of computer-controlled infusion pumps (CCIPs) are poorly defined, of little use to the clinician using the CCIP, and pharmacostatistically incorrect. We propose four measures be used to quantitate the performance of CCIPs: median absolute performance error (MDAPE), median performance error (MDPE), divergence, and wobble. These measures offer several significant advantages over previous measures. First, their definitions are based on the performance error as a fraction of the predicted (rather than measured) drug concentration, making the measures much more useful to the clinician. Second, the measures are defined in a way that addresses the pharmacostatistical issue of appropriate estimation of population parameters. Finally, the measure of inaccuracy, MDAPE, is defined in a way that is consistent with iteratively reweighted least squares nonlinear regression, a commonly used method of estimating pharmacokinetic parameters. These measures make it possible to quantitate the overall performance of a CCIP or to compare the predictive performance of CCIPs which differ in either general approach (e.g., compartmental model driven vs. plasma efflux approach), pump mechanics, software algorithms, or pharmacokinetic parameter sets.     

Numerical deconvolution using system identification methods

A deconvolution method is presented for use in pharmacokinetic applications involving continuous models and small samples of discrete observations. The method is based on the continuous-time counterpart of discrete-time least squares system identification, well established in control engineering. The same technique, requiring only the solution of a linear regression problem, is used both in system identification and input identification steps. The deconvolution requires no a priori information, since the proposed procedure performs system identification (including optimal selection of model order), selects the form of the input function and calculates its parametric representation and its values at specified time points.     

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.     

Individualising gentamicin dosage regimens. A comparative review of selected models, data fitting methods and monitoring strategies.

The various components required for individualising clinical drug dosage regimens are reviewed, including a study of 3 types of fitting procedures, 2 types of gentamicin pharmacokinetic model and the utility of D-optimal times for obtaining serum gentamicin concentrations. The combination of the current Bayesian fitting procedure, the kslope pharmacokinetic model [in which the elimination rate constant (kel) can change from dose to dose with changing creatinine clearance] and the explicit measurement of the assay error pattern yielded predictions of future serum gentamicin concentrations which were (a) slightly better than those found using weighted nonlinear least squares; (b) somewhat better than those found with Bayesian fitting and a fixed-kel model; (c) better than those found using the traditional linear regression fitting procedure and a fixed kel model. D-Optimally timed pairs of concentrations also predicted future concentrations at least as well, and more cost effectively.     

Statistical methodology to determine kinetically derived maximum tolerated dose in repeat dose toxicity studies

Several statistical approaches were evaluated to identify an optimum method for determining a point of nonlinearity (PONL) in toxicokinetic data. (1) A second-order least squares regression model was fit iteratively starting with data from all doses. If the second order term was significant (α < 0.05), the dataset was reevaluated with successive removal of the highest dose until the second-order term became non-significant. This dose, whose removal made the second order term non-significant, is an estimate of the PONL. (2) A least squares linear model was fit iteratively starting with data from all doses except the highest. The mean response for the omitted dose was compared to the 95% prediction interval. If the omitted dose falls outside the confidence interval it is an estimate of the PONL. (3) Slopes of least squares linear regression lines for sections of contiguous doses were compared. Nonlinearity was suggested when slopes of compared sections differed. A total of 33 dose–response datasets were evaluated. For these toxicokinetic data, the best statistical approach was the least squares regression analysis with a second-order term. Changing the α level for the second-order term and weighting the second-order analysis by the inverse of feed consumption were also considered. This technique has been shown to give reproducible identification of nonlinearities in TK datasets.     

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.