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.     

Numerical deconvolution by least squares: Use of prescribed input functions

A new method for numerical deconvolution is described, for use in calculating drug input rates. The method is based on the least-squares criterion and is applicable when the input function can be assumed to take a prescribed form. In particular, an exponential input function and an input function derived from the cube-root dissolution law are considered. The stability of the method to data noise is shown by means of examples, using simulated data.     

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.     

Estimation of active pharmaceutical ingredients content using locally weighted partial least squares and statistical wavelength selection

Development of quality estimation models using near infrared spectroscopy (NIRS) and multivariate analysis has been accelerated as a process analytical technology (PAT) tool in the pharmaceutical industry. Although linear regression methods such as partial least squares (PLS) are widely used, they cannot always achieve high estimation accuracy because physical and chemical properties of a measuring object have a complex effect on NIR spectra. In this research, locally weighted PLS (LW-PLS) which utilizes a newly defined similarity between samples is proposed to estimate active pharmaceutical ingredient (API) content in granules for tableting. In addition, a statistical wavelength selection method which quantifies the effect of API content and other factors on NIR spectra is proposed. LW-PLS and the proposed wavelength selection method were applied to real process data provided by Daiichi Sankyo Co., Ltd., and the estimation accuracy was improved by 38.6% in root mean square error of prediction (RMSEP) compared to the conventional PLS using wavelengths selected on the basis of variable importance on the projection (VIP). The results clearly show that the proposed calibration modeling technique is useful for API content estimation and is superior to the conventional one.     

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 curve fitting: a comparison of SIPHAR and MKMODEL.

SIPHAR and MKMODEL in their extended least squares modes have been compared when fitting a triexponential declining function to simulated data. The data were simulated on SAS incorporating normally distributed random error, having coefficients of variation (CV) of 5, 10, 15, and 25 per cent. At each error level 100 data sets, consisting of 21 data pairs, were simulated. Non-parametric tests were used to compare the accuracy and precision of the estimates produced by the packages. The comparison was repeated with two different sets of exponent values incorporating error at the 15 per cent level. MKMODEL was also compared to ELSFIT and ELSMOS at the same error level. SIPHAR was consistently less accurate and less precise than MKMODEL in estimating the structural model parameters. SIPHAR was also sensitive to the concentration units used for the input data. Estimates of the variance model power produced by SIPHAR were very variable while those from MKMODEL covered a much tighter range. For both packages there was generally a linear increase in the CV on each mean parameter estimate with increase in the CV on the error model. Good agreement was observed between MKMODEL, ELSFIT, and ELSMOS. The presence of an additive constant in the variance model used in SIPHAR was shown to be responsible for its poorer accuracy and precision.     

Error estimates on normal least squares linear regression with replicate injection of calibration standards

Background: In bioanalysis, including replicate injection of calibration standards, a normal least squares linear regression model can be generated from, A: the first injection dataset; B: the second injection dataset; C: averaged data from both injection datasets; and D: all data from both injections. Sample results, and their estimated confidence intervals, are expected to be different among these four models. Results: Models C and D yield same slopes and intercepts, which are the mathematical means of respective values in models A and B. Relatively narrower confidence intervals on sample results are estimated in both models C and D as the former reduces the overall standard error of the curve and the latter increases the total number of calibration points and leads to a lower Student's t-value. Conclusion: Replicate injection of calibration standards provides added benefits for an analytical measurement on instrument sensitivity compensation and relatively improved precision of results.     

Representation theorem for convex nonparametric least squares

Summary We examine a nonparametric least‐squares regression model that endogenously selects the functional form of the regression function from the family of continuous, monotonic increasing and globally concave functions that can be nondifferentiable. We show that this family of functions can be characterized without a loss of generality by a subset of continuous, piece‐wise linear functions whose intercept and slope coefficients are constrained to satisfy the required monotonicity and concavity conditions. This representation theorem is useful at least in three respects. First, it enables us to derive an explicit representation for the regression function, which can be used for assessing marginal properties and for the purposes of forecasting and ex post economic modelling. Second, it enables us to transform the infinite dimensional regression problem into a tractable quadratic programming (QP) form, which can be solved by standard QP algorithms and solver software. Importantly, the QP formulation applies to the general multiple regression setting. Third, an operational computational procedure enables us to apply bootstrap techniques to draw statistical inference.     

Pharmacokinetic analysis of drug concentration data obtained during repetitive drug administration

A digital computer method that uses all drug concentration-time data collected during repetitive dosing studies was developed to simultaneously estimate pharmacokinetic parameters and fit the data. The method utilizes the nonlinear least-squares regression program NONLIN. It can accommodate changes in dose and dosing interval during the regimen and it permits curve fitting on the basis of one of a number of linear pharmacokinetic models. Theoretical data containing 10% uniformly distributed pseudorandom error and experimental data from three clinical studies were used to validate the method. Reasonably precise and accurate parameter estimates and good curve fits were obtained.Supported in part by Grant GM-20852 from the National Institute of General Medical Sciences, National Institutes of Health.     

A note on confidence intervals with extended least squares parameter estimates

It has previously been shown that the extended least squares (ELS) method for fitting pharmacokinetic models behaves better than other methods when there is possible heteroscedasticity (unequal error variance) in the data. Confidence intervals for pharmacokinetic parameters, at the target confidence level of 95%, computed in simulations with several pharmacokinetic and error variance models, using a theoretically reasonable approximation to the asymptotic covariance matrix of the ELS parameter estimator, are found to include the true parameter values considerably less than 95% of the time. Intervals with the ordinary least squares method perform better. Two adjustments to the ELS confidence intervals, taken together, result in better performance. These are: (i) apply a bias correction to the ELS estimate of variance, which results in wider confidence intervals, and (ii) use confidence intervals with a target level of 99% to obtain confidence intervals with actual level closer to 95%. Kineticists wishing to use the ELS method may wish to use these adjustments.     

A note on confidence intervals with extended least squares parameter estimates

It has previously been shown that the extended least squares (ELS) method for fitting pharmacokinetic models behaves better than other methods when there is possible heteroscedasticity (unequal error variance) in the data. Confidence intervals for pharmacokinetic parameters, at the target confidence level of 95%, computed in simulations with several pharmacokinetic and error variance models, using a theoretically reasonable approximation to the asymptotic covariance matrix of the ELS parameter estimator, are found to include the true parameter values considerably less than 95% of the time. Intervals with the ordinary least squares method perform better. Two adjustments to the ELS confidence intervals, taken together, result in better performance. These are: (i) apply a bias correction to the ELS estimate of variance, which results in wider confidence intervals, and (ii) use confidence intervals with a target level of 99% to obtain confidence intervals with actual level closer to 95%. Kineticists wishing to use the ELS method may wish to use these adjustments.     

An Experimental Design Strategy for Quantitating Complex Pharmacokinetic Models: Enterohepatic Circulation with Time-Varying Gallbladder Emptying as an Example

A four-step strategy is proposed for determining appropriate experimental designs for investigating the pharmacokinetics of drugs characterized by complex compartmental models and this strategy has been applied to the pharmacokinetics of enterohepatic circulation (EHC). The four steps are (1) to establish an appropriate pharmacokinetic model, (2) to complete an identifiability analysis for the model to determine the route(s) of administration and sampling compartment(s) that are theoretically adequate for the quantitation of model parameters, (3) to carry out nonlinear least-squares fitting for the proposed number and timing of simulated error-free data points, and (4) to complete nonlinear least-squares fits of the model to data obtained by adding random error to the simulated data in step 3. The four-compartment model chosen for EHC of unchanged drug contained central, peripheral, gallbladder, and intestinal compartments and an intermittent gallbladder emptying rate constant. Identifiability analysis demonstrated that three alternative experimental designs for route(s) of administration and sampling compartment(s) are adequate for quantitating all model parameters, when the gallbladder emptying rate constant as a function of time is known (using controlled emptying from an engineered gallbladder in an animal model or quantitation in humans or animals using imaging techniques). Parameter estimates from fitting error-free data matched closely with the known values for all three experimental designs, indicating an adequate number and appropriate timing of data points. Results from fitting simulated data containing ±10% random error indicated unacceptable coefficients of variation and a nonrandom pattern in residual plots for one of the experimental designs. Of the two remaining designs, one was less resilient relative to poor initial estimates and relative to timing of gallbladder emptying simultaneously with the distribution process. It is clear that application of this new strategy permits the elimination of experimental designs that are inadequate (from either a theoretical or an experimental standpoint) prior to initiating in vivo experiments. As such, it represents a major advance in reliability over methods used previously for complex models.     

Comparison of ordinary, weighted, and generalized least-squares straight-line calibrations for LC-MS-MS, GC-MS, HPLC, GC, and enzymatic assay

The impact of experimental errors in one or both variables on the use of linear least-squares was investigated for method calibrations (response = intercept plus slope times concentration, or equivalently, Y = a(1) + a(2)X ) frequently used in analytical toxicology. In principle, the most reliable calibrations should consider errors from all sources, but consideration of concentration (X) uncertainties has not been common due to complex fitting algorithm requirements. Data were obtained for liquid chromatography-tandem mass spectrometry, gas chromatography-mass spectrometry, high-performance liquid chromatography, gas chromatography, and enzymatic assay. The required experimental uncertainties in response were obtained from replicate measurements. The required experimental uncertainties in concentration were determined from manufacturers' furnished uncertainties in stock solutions coupled with uncertainties imparted by dilution techniques. The mathematical fitting techniques used in the investigation were ordinary least-squares, weighted least-squares (WOLS), and generalized least-squares (GLS). GLS best-fit results, obtained with an efficient iteration algorithm implemented in a spreadsheet format, are used with a modified WOLS-based formula to derive reliable uncertainties in calculated concentrations. It was found that while the values of the intercepts and slopes were not markedly different for the different techniques, the derived uncertainties in parameters were different. Such differences can significantly affect the predicted uncertainties in concentrations derived from the use of the different linear least-squares equations.     

Empirical models for fitting of oral concentration time curves with and without an intravenous reference

Appropriate model selection is important in fitting oral concentration–time data due to the complex character of the absorption process. When IV reference data are available, the problem is the selection of an empirical input function (absorption model). In the present examples a weighted sum of inverse Gaussian density functions (IG) was found most useful. It is shown that alternative models (gamma and Weibull density) are only valid if the input function is log-concave. Furthermore, it is demonstrated for the first time that the sum of IGs model can be also applied to fit oral data directly (without IV data). In the present examples, a weighted sum of two or three IGs was sufficient. From the parameters of this function, the model-independent measures AUC and mean residence time can be calculated. It turned out that a good fit of the data in the terminal phase is essential to avoid parameter biased estimates. The time course of fractional elimination rate and the concept of log-concavity have proved as useful tools in model selection.     

Estimation problem due to multiple solutions in pharmacokinetic curve fitting to two-compartment model and its avoidance

A size of error in observed data for fitting curves and an estimation problem due to multiple solutions in a two-compartment model were studied by using two different non-linear least-squares regression programs, SALS and NONLIN. It was found that bolus intravenous data have generally 5-10 per cent errors and oral data contain 10-25 per cent errors against the fitted data with respect to total 151 data sets of 11 different drugs. Parameters of five drugs reported in references were used to obtain simulated concentrations at the sampling times, and five different data sets containing 25 per cent normally distributed random errors as a coefficient of variation were generated using each data set of these simulated concentration. In the two-compartment model with tri-exponential equations, unreasonable estimates were occasionally observed, resulting in reversed relative values to the theoretical ones of L/M, L/N, M/N or Ka/alpha, which are analogous to the well-known flip-flop phenomenon in the one-compartment model, when number of parameters to be estimated is not less than five or errors of data exceed about 10 per cent. In an attempt to avoid such unreasonable values, initial estimates for curve fitting was successfully obtained by using a microcomputer program SIMPLEX based on a simplex method. On the basis of these results, some problems in curve fitting of plasma drug concentration data are discussed.     

A model selection method for S‐estimation

Summary Cleaning data or removing some data periods in least squares (LS) regression analysis is not unusual. This practice indicates that a researcher sometimes desires to estimate the parameter value, with which the regression function fits a large fraction of individuals or events in the population (behind the original data set), possibly exhibiting poor fits to some atypical individuals or events. The S‐estimators are a class of estimators that are consistent with the researcher's desire in such situations. In this paper, we propose a method of model selection suitable in the S‐estimation. The proposed method chooses a model that minimizes a criterion named the penalised S‐scale criterion (PSC), which is decreasing in the sample S‐scale of fitted residuals and increasing in the number of parameters. We study the large sample behavior of the PSC in nonlinear regression with dependent, heterogeneous data, to establish sets of conditions sufficient for the PSC to consistently select the best‐fitting, most parsimonious model. Our analysis allows for partial unidentifiability, which is an important possibility when selecting one among non‐linear regression models. We conduct Monte Carlo simulations to verify that a particular PSC called the PSC‐S is at least as trustworthy as the Schwarz information criterion, often used in the LS regression.     

Numerical deconvolution by least squares: Use of polynomials to represent the input function

A new method for numerical deconvolution is described, for use in calculating drug input rates. The method is based on the least-squares criterion and approximates the input rate by a polynomial function. Ill-conditioning of the normal equations is avoided by using orthogonal functions. The use of the method is illustrated by means of examples, using simulated data.     

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^–)-here is regarded as an unknown parameter). Tne values of the weights are more influenced by the data with the ELS(f^–) 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^–), 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^–) does better than OLS. Thus, ELS(f^–.This work supported by NIH Grants GM 26676 and GM 26691.     

模糊数据分析的可能性线性系统

H.Tanaka等提出的可能性线性系统是模糊数据分析的重要分支,本文把求解模糊参数的均值和展作先后处理,提出4种可能性线性系统模型,这些模型对模糊效据的拟合,均较Tanaka模型为优,本文论证新模型解的存在性、解法以及数据分析的性质,并与Tanaka的3种模型作比较。