Optimal Sampling Times for Bayesian Estimation of the Pharmacokinetic Parameters of Nortriptyline During Therapeutic Drug Monitoring

Sampling times for Bayesian estimation of the pharmacokinetic parameters of an antidepressant drug, nortriptyline, during its therapeutic drug monitoring were optimized. Our attention was focused on designs including a limited number of measurements: one, two, and three sample designs in which sampling times had to be chosen between 0 and 24 hr after the last intake of a test-dose study. The optimization was conducted in four groups of patients defined by their gender and the administration or not of concomitant drugs inhibiting the metabolism of nortriptyline. The Bayesian design criterion was defined as the expected information provided by an experiment. A stochastic approximation algorithm, the Kiefer–Wolfowitz algorithm, was used for the criterion maximization under experimental constraints. Results showed that optimal Bayesian sampling times differ between patients in monotherapy and polytherapy. For one-sample designs the measurements have to be performed either at the lower (0 hr) or at the upper (24 hr) bound of the admissible interval. Replications were often found for 2- and 3-point designs. Other sampling designs can lead to criterion close to the optimum and can therefore be performed without great loss of information. In contrast, we found that several designs lead to low values of the information criterion, which justifies the approach.     

Application of optimal sampling theory to the determination of metacycline pharmacokinetic parameters: Effect of model misspecification

Use of optimal sampling theory (OST) in pharmacokinetic studies allows the number of sampling times to be greatly reduced without loss in parameter estimation precision. OST has been applied to the determination of the bioavailability parameters (area under the curve (AUC), maximal concentration (C_max), time to reach maximal concentration (T_max), elimination half-life (T_1/2), of metacycline in 16 healthy volunteers. Five different models were used to fit the data and to define the optimal sampling times: one-compartment first-order, two-compartment first-order, twocompartment zero-order, two-compartment with Michaelis-Menten absorption kinetics, and a stochastic model. The adequacy of these models was first evaluated in a 6-subject pilot study. Only the stochastic model with zero-order absorption kinetics was adequate. Then, bioavailability parameters were estimated in a group of 16 subjects by means of noncompartmental analysis (with 19 samples per subject) using each optimal sampling schedule based procedure (with 6 to 9 samples depending on the model). Bias (PE) and precision (RMSE) of each bioavailability parameter estimation were calculated by reference to noncompartmental analysis, and were satisfactory for the 3 adequate models. The most relevant criteria for discrimination of the best model were the coefficient of determination, the standard deviation, and the mean residual error vs. time plot. Additional criteria were the number of required sampling times and the coefficient of variation of the estimates. In this context, the stochastic model was superior and yielded very good estimates of the bioavailability parameters with only 8 samples per subject.     

Comparison of Robust Criteria for D-Optimal Designs

This study compared the performance of a local and three robust optimality criteria in terms of the standard error for a one-parameter and a two-parameter nonlinear model with uncertainty in the parameter values. The designs were also compared in conditions where there was misspecification in the prior parameter distribution. The impact of different correlation between parameters on the optimal design was examined in the two-parameter model. The designs and standard errors were solved analytically whenever possible and numerically otherwise.     

Robust Optimal Design for the Estimation of Hyperparameters in Population Pharmacokinetics

The expectation of the determinant of the inverse of the population Fisher information matrix is proposed as a criterion to evaluate and optimize designs for the estimation of population pharmacokinetic (PK) parameters. Given a PK model, a measurement error model, a parametric distribution of the parameters and a prior distribution representing the belief about the hyperparameters to be estimated, the EID criterion is minimized in order to find the optimal population design. In this approach, a group is defined as a number of subjects to whom the same sampling schedule (i.e., the number of samples and their timing) is applied. The constraints, which are defined a priori, are the number of groups, the size of each group and the number of samples per subject in each group. The goal of the optimization is to determine the optimal sampling times in each group. This criterion is applied to a one-compartment open model with first-order absorption. The error model is either homoscedastic or heteroscedastic with constant coefficient of variation. Individual parameters are assumed to arise from a lognormal distribution with mean vector M and covariance matrix C. Uncertainties about the M and C are accounted for by a prior distribution which is normal for M and Wishart for C. Sampling times are optimized by using a stochastic gradient algorithm. Influence of the number of different sampling schemes, the number of subjects per sampling schedule, the number of samples per subject in each sampling scheme, the uncertainties on M and C and the assumption about the error model and the dose have been investigated.     

Optimal Blood Sampling Time Windows for Parameter Estimation Using a Population Approach: Design of a Phase II Clinical Trial

The objective of this paper is to determine optimal blood sampling time windows for the estimation of pharmacokinetic (PK) parameters by a population approach within the clinical constraints. A population PK model was developed to describe a reference phase II PK dataset. Using this model and the parameter estimates, D-optimal sampling times were determined by optimising the determinant of the population Fisher information matrix (PFIM) using PFIM_ _M 1.2 and the modified Fedorov exchange algorithm. Optimal sampling time windows were then determined by allowing the D-optimal windows design to result in a specified level of efficiency when compared to the fixed-times D-optimal design. The best results were obtained when K_a and IIV on K_a were fixed. Windows were determined using this approach assuming 90% level of efficiency and uniform sample distribution. Four optimal sampling time windows were determined as follow: at trough between 22 h and new drug administration; between 2 and 4 h after dose for all patients; and for 1/3 of the patients only 2 sampling time windows between 4 and 10 h after dose, equal to [4 h–5 h 05] and [9 h 10–10 h]. This work permitted the determination of an optimal design, with suitable sampling time windows which was then evaluated by simulations. The sampling time windows will be used to define the sampling schedule in a prospective phase II study     

Optimal sampling times for pharmacokinetic experiments

A sequential estimation procedure is presented which uses optimal sampling times to estimate the parameters of a model from data obtained from a group of subjects. This optimal sampling sequential estimation procedure utilizes parameter estimates from previous subjects in the group to determine the optimal sampling times for the next subject. Parameter estimates obtained from the optimal sampling procedure are compared to those obtained from a conventional sampling scheme by using Monte Carlo simulations which include noise terms for both assay error and intersubject variability. The results of these numerical experiments, for the two examples considered here, show that the parameter estimates obtained from data collected at optimal sampling times have significantly less variability than those generated using the conventional sampling procedure. We conclude that optimal sampling and preexperiment simulation may be useful tools for designing informative pharmacokinetic experiments.Presented at the First Annual Conference of the American College of Clinical Pharmacy, Boston, July 1980.     

Construction of an Optimal Destructive Sampling Design for Noncompartmental AUC Estimation

Based on toxicokinetic studies of a destructive sampling design, this work was aimed at selecting the number of time points, their locations, and the number of replicates per time point in order to obtain the most accurate and precise noncompartmental estimate of the area under the concentration-time curve (AUC). From a prior population pharmacokinetic model, the design is selected to minimize the scaled mean squared error of AUC. Designs are found for various sample sizes, number of time points, and a distribution of animals across time points from being very unbalanced to balanced. Their efficiencies are compared both theoretically and based on simulations. An algorithm has been implemented for this purpose using the symbolic resolution and numerical minimization capabilities of Mathematica ^TM and an example of its use is provided. This method provides efficient tools for constructing, validating, and comparing optimal sampling designs for destructive sampled toxicokinetic studies.     

Experimental design and efficient parameter estimation in population pharmacokinetics

A computer simulation technique used to evaluate the influence of several aspects of sampling designs on the efficiency of population pharmacokinetic parameter estimation is described. Although the simulations are restricted to the one-compartment one-exponential model, they provide the basis for a discussion of the structural aspects involved in designing a population study. These aspects include number of subjects required, number of samples per subject, and timing of these samples. Parameter estimates obtained from different sampling schedules based on two- and three-point designs are evaluated in terms of accuracy and precision. These simulated data sets include noise terms for both inter- and intraindividual variability. The results show that the population fixed-effect parameters (mean clearance and mean volume of distribution) for this simple pharmacokinetic model are efficiently estimated for most of the sampling schedules when two or three points are used, but the random-effect parameters (describing inter- and intraindividual variability) are inaccurate and imprecise for most of the sampling schedules when only two points are used. This drawback was remedied by increasing the number of data points per individual to three.Supported by the Scottish Home and Health Department (Biomedical Research Committee).     

Optimal Experimental Design for Precise Estimation of the Parameters of the Axial Dispersion Model of Hepatic Elimination

The axial dispersion model of hepatic drug elimination is characterized by two dimensionless parameters, the dispersion number, D_N , and the efficiency number, R_N , corresponding to the relative dispersion of material on transit through the organ and the relative efficiency of elimination of drug by the organ, respectively. Optimal design theory was applied to the estimation of these two parameters based on changes in availability (F) of drug at steady state for the closed boundary condition model, with particular attention to variations in the fraction of drug unbound in the perfusate (fu_B ). Sensitivity analysis indicates that precision in parameter estimation is greatest when F is low and that correlation between R_N and D_N is high, which is desirable for parameter estimation, when D_N lies between 0.1 and 100. Optimal design points were obtained using D-optimization, taking into account the error variance model. If the error variance model is unknown, it is shown that choosing Poisson error model is reasonable. Furthermore, although not optimal, geometric spacing of fu_B values is often reasonable and definitively superior to a uniform spacing strategy. In practice, the range of fu_B available for selection may be limited by such practical considerations as assay sensitivity and acceptable concentration range of binding protein. Notwithstanding, optimal design theory provides a rational approach to precise parameter estimation.     

药品质量抽查检验的组织、抽样和收检要求新旧对比研究

目的:加深相关人员对《药品质量抽查检验管理办法》的理解,以有效地落实抽检规定,加强药品监管。方法:对比研究《药品质量抽查检验管理办法》和《药品质量抽查检验管理规定》中组织、抽样和收检要求,分析新增和修订内容的必要性与意义,提出实施建议。结果与结论:《药品质量抽查检验管理办法》基于药品监管需要,相对于《药品质量抽查检验管理规定》新增和修订的组织、抽样和收检相关内容,对加强抽检管理,提高抽样、收检工作效能,进而打击假冒伪劣药品,防控潜在质量安全隐患具有重要意义。     

Evaluation of the Pre-Posterior Distribution of Optimized Sampling Times for the Design of Pharmacokinetic Studies

Information theoretic methods are often used to design studies that aim to learn about pharmacokinetic and linked pharmacokinetic–pharmacodynamic systems. These design techniques, such as D-optimality, provide the optimum experimental conditions. The performance of the optimum design will depend on the ability of the investigator to comply with the proposed study conditions. However, in clinical settings it is not possible to comply exactly with the optimum design and hence some degree of unplanned suboptimality occurs due to error in the execution of the study. In addition, due to the nonlinear relationship of the parameters of these models to the data, the designs are also locally dependent on an arbitrary choice of a nominal set of parameter values. A design that is robust to both study conditions and uncertainty in the nominal set of parameter values is likely to be of use clinically. We propose an adaptive design strategy to account for both execution error and uncertainty in the parameter values. In this study we investigate designs for a one-compartment first-order pharmacokinetic model. We do this in a Bayesian framework using Markov-chain Monte Carlo (MCMC) methods. We consider log-normal prior distributions on the parameters and investigate several prior distributions on the sampling times. An adaptive design was used to find the sampling window for the current sampling time conditional on the actual times of all previous samples.     

Population Pharmacokinetic Measures,Their Estimation and Selection of Sampling Times

In pharmacokinetic (PK) studies, including bioavailability assessment, various population PK measures, such as area under the curve (AUC), maximal concentration (C _max ) and time to maximal concentration (T _max ) are estimated. In this paper we compare a model-based approach, where parameters of a compartmental model are estimated and the explicit formulae for PK measures are used, and a model-independent approach, where numerical integration algorithms are used for AUC and sample estimates for C _max and T _max . Since regulatory agencies usually require the model-independent estimation of PK measures, we focus on the empirical approach while using the model-based approach and corresponding measures as a benchmark. We show how to “split” a single sampling grid into two or more subsets, which substantially reduces the number of samples taken for each patient, but often has little effect on the precision of estimation of PK measures in terms of mean squared error (MSE). We give explicit formulae for the MSE of the empirical estimator of AUC for a simple example and discuss how costs may be taken into account.     

Optimal Design Criteria for Discrimination and Estimation in Nonlinear Models

Nonlinear models are common in pharmacokinetics and pharmacodynamics. To date, most work in design in this area has concentrated on parameter estimation. Here, we introduce the idea of optimization of both estimation and model selection. However, experimental designs that provide powerful discrimination between a pair of competing model structures are rarely efficient in terms of estimating the parameters under each model. Conversely, designs which are efficient for parameter estimation may not provide suitable power to discriminate between the models. Several different methods of addressing both of these objectives simultaneously are introduced in this paper and are compared to an existing optimality criterion.     

A Theoretical Framework for Estimation of AUCs in Complete and Incomplete Sampling Designs

Nonclinical in vivo animal studies have to be completed before starting clinical studies of the pharmacokinetic behavior of a drug in humans. The drug exposure in animal studies is often measured by the area under the concentration versus time curve (AUC). The classic complete data design, where each animal is sampled for analysis once per time point, is usually only applicable for large animals. In the case of rats and mice, where blood sampling is restricted, the batch design or the serial sampling design needs to be considered. In batch designs samples are taken more than once from each animal, but not at all time points. In serial sampling designs only one sample is taken from each animal. In this article we present an estimator for the AUC from 0 to the last time point that is applicable to all three designs. The variance and asymptotic distribution of the estimator are derived and confidence intervals based upon the asymptotic results are discussed and evaluated in a simulation study. Further, we define an estimator for linear combinations of AUCs and investigate its asymptotic properties mathematically as well as in simulation.     

A computationally efficient approach for the design of population pharmacokinetic studies

A computationally efficient procedure was devised for designing experiments in which population pharmacokinetic parameters are estimated. The method, referred to as the large-sample approach, evaluates the variances of parameter estimates for a population pharmacostatistical model. The procedure utilizes the NONMEM program and requires a single simulation that assumes many, say 1000, subjects. The approach reduced CPU time by about a factor of 50 when compared with the evaluation of the same variances by the direct simulation of experiments. The large-sample and simulation approaches yielded generally similar values for the variances of parameter estimates. The variances calculated by the large-sample approach were, in the case of a simple model, close to the expected variances. The proposed method identified correctly the imprecise parameter estimates but somewhat underestimated their variances.This work was supported by the Medical Research Council of Canada.     

Errors in Clearance Estimation After Bolus Injection and Arterial Sampling: Nonexistence of a Central Compartment

In conventional pharmacokinetics monotonic decreasing drug disposition curves are usually assumed after bolus injection. For arterial sampling starting about 1 min after dosing the resulting neglect of the initial concentration peak leads to a relative overestimation of clearance which may be approximated by the percentage of systemic drug extraction. This error can be avoided by administering the drug as a short-term infusion.     

中国药品上市后抽验模式现状及问题

目的在综合分析中国药品上市后抽验模式现状及问题的基础上,提出药品上市后抽验的建议。方法通过文献研究的方法,发现中国药品上市后抽验模式中存在的问题,并提出相关建议。结果与结论在药品抽验中引入以风险为基础的抽验模式,将药品上市后抽验结果与药品监管相衔接。     

Optimal Design for Model Discrimination and Parameter Estimation for Itraconazole Population Pharmacokinetics in Cystic Fibrosis Patients

Optimal sampling times are found for a study in which one of the primary purposes is to develop a model of the pharmacokinetics of itraconazole in patients with cystic fibrosis for both capsule and solution doses. The optimal design is expected to produce reliable estimates of population parameters for two different structural PK models. Data collected at these sampling times are also expected to provide the researchers with sufficient information to reasonably discriminate between the two competing structural models     

Comparison of some practical sampling strategies for population pharmacokinetic studies

Using population analysis, sparsely sampled Phase 3 clinical data can be utilized to determine the pharmacokinetic characteristics of the target population. Data arising from such studies are likely to be constrained to certain sampling windows, i.e., the visiting hours at the study clinic. When the sampling window is narrow compared to the half-life of the drug, the advantage of taking more than one sample is not obvious. Study designs with one or two samples per visit have been compared with respect to (i) precision and bias of the population parameter estimates, (ii) the ability to identify the underlying pharmacokinetic model, and (iii) the estimation of individual parameter values. The first point was assessed using simulated data while the latter two were studied using a real data set. Results show: (i) Parameter estimates are more biased and imprecise when only one sample is taken compared to when two samples are obtained, this is true irrespective of the time span between the two samples. (ii) Ability to identify a more complex model is increased if two samples are taken. Specifically, the variability between occasions can be quantified. (iii) Two-sample designs are generally better with respect to prediction of individual parameter values. Even minor changes to commonly employed study designs, in this case the addition of one sample at each study occasion, can improve quality and quantity of the information obtained. During the course of this study E. Niclas Jonsson was paid by a grant from ASTRA AB.     

药品抽样中样本数确定的商榷

目的探讨国家《药品抽样指导原则》中,样本数确定可能存在的问题以及解决的办法。方法以概率论和数理统计原理诠释抽样样本数确定的理论基础。结果现行药物固体制剂抽样方法中,样本数的确定可能有悖抽样理论。结论应按照国家标准的相关抽样方案确定样本数。