A New Approach to Modeling Covariate Effects and Individualization in Population Pharmacokinetics-Pharmacodynamics

By combining Laplace’s approximation and Monte Carlo methods to evaluate multiple integrals, this paper develops a new approach to estimation in nonlinear mixed effects models that are widely used in population pharmacokinetics and pharmacodynamics. Estimation here involves not only estimating the model parameters from Phase I and II studies but also using the fitted model to estimate the concentration versus time curve or the drug effects of a subject who has covariate information but sparse measurements. Because of its computational tractability, the proposed approach can model the covariate effects nonparametrically by using (i) regression splines or neural networks as basis functions and (ii) AIC or BIC for model selection. Its computational and statistical advantages are illustrated in simulation studies and in Phase I trials.     

广义线性混合效应模型在临床疗效评价中的应用

目的:探讨临床疗效评价中分类重复测量资料的广义线性混合效应模型(GLMMs)及SAS8.0的GLIMMIX宏实现。方法:利用GLIMMIX宏ERROR和LINK语句来指示疗效指标的分布及连接函数,通过REPEATED和RANDOM语句的TYPE选项选择合适方差-协方差结构矩阵来模拟不同时间疗效指标的相关性,采用基于线性的伪似然函数进行模型参数估计。结果:广义线性混合效应模型允许临床疗效评价指标是指数家族中任意分布(如:连续分布包括正态分布、beta分布、卡方分布等;离散分布包括二项分布、泊松分布、负二项分布等),可以通过连接函数将疗效指标的均数向量与模型参数建立线性关系,简化运算过程。结论:广义线性混合效应模型建模灵活,可为临床疗效评价提供更丰富的信息。     

Further Developments of the Fisher Information Matrix in Nonlinear Mixed Effects Models with Evaluation in Population Pharmacokinetics

We extend the development of the expression of the Fisher information matrix in nonlinear mixed effects models for designs evaluation. We consider the dependence of the marginal variance of the observations with the mean parameters and assume an heteroscedastic variance error model. Complex models with interoccasions variability and parameters quantifying the influence of covariates are introduced. Two methods using a Taylor expansion of the model around the expectation of the random effects or a simulated value, using then Monte Carlo integration, are proposed and compared. Relevance of the resulting standard errors is investigated in a simulation study with NONMEM.     

Population Pharmacokinetics of Terfenadine

Purpose. After oral administration of terfenadine, plasma concentrations of the parent drug are usually below the limits of quantitation of conventional analytical methods because of extensive first-pass metabolism. Data are usually reported on the carboxylic acid metabolite (Ml) but there are no published reports of pharmacokinetic parameters for terfenadine itself. The present study was undertaken to evaluate the population pharmacokinetics of terfenadine. Methods. Data from 132 healthy male subjects who participated in several different studies were included in this analysis. After an overnight fast, each subject received a single 120 mg oral dose of terfenadine; blood samples were collected for 72 hours. Terfenadine plasma concentrations were measured using HPLC with mass spectrometry detection and Ml plasma concentrations were measured using HPLC with fluorescence detection. A 2-compartment model was fitted to the terfenadine data using NONMEM; terfenadine and Ml data were also analyzed by noncompartmental methods. Results. Population mean Ka was 2.80 hr^–1, T_lag was 0.33 hr, Cl/F was 4.42 × 10³ 1/hr, V_C/F was 89.8 ×10³1, Q/F was 1.85 ×10³ 1/hr and V_p/F was 29.1 × 10³1. Intersubject CV ranged from 66 to 244% and the residual intrasubject CV was 21%. Based on noncompartmental methods, mean terfenadine C_max was 1.54 ng/ml, T_max was 1.3 hr, t_{1/2 Z} was 15.1 hr, Cl/F was 5.48 × 10³ 1/hr and V_z/F was 119.2 × 10³1. Ml concentrations exceeded terfenadine concentrations by more than 100 fold and showed less intersubject variability. Conclusions. Terfenadine disposition was characterized by a 2-compartment model with large intersubject variability, consistent with its significant first-pass effect.     

Likelihood-Based Diagnostics for Influential Individuals in Non-Linear Mixed Effects Model Selection

Purpose. Data from single individuals, or a small group of subjects may influence non-linear mixed effects model selection. Diagnostics routinely applied in model building may identify such individuals, but these methods are not specifically designed for that purpose and are, therefore, not optimal. We describe two likelihood-based diagnostics for identifying individuals that can influence the choice between two competing models. Methods. One method is based on a jackknife of the raw data on the individual level and refitting the model to each new data set. The second method is a calculation which utilises the contribution each individual make to the objective function values under each of the two models. The two methods were applied to model selection during analysis of a real data set. Results. The agreement between the methods was high. Individuals for whom there was a discrepancy between the methods tended to be those for which neither of the contending models described the data appropriately. Both methods identified individuals that influenced the model selection. Conclusions. Two objective, specific and quantitative methods for identifying influential individuals in nonlinear mixed effects model selection have been presented. One of the methods doesn't require additional model fitting and is therefore particularly attractive.     

Application of Semiparametric Mixed Models and Simultaneous Confidence Bands in a Cardiovascular Safety Experiment with Longitudinal Data

Several pharmacological studies involve experiments aimed at testing for a difference between experimental groups wherein the data are longitudinal in nature, frequently with long sequences per subject. Oftentimes, treatment effect, if present, is not constant over time. In such situations, imposing a parametric mean structure can be too complicated and/or restrictive. A more flexible approach is to model the mean using a semiparametric smooth function, estimated using, for example, penalized smoothing splines. We formulate a series of models exhibiting how the group-specific mean profiles could possibly differ. Once an appropriate model is chosen, interest lies in identifying specific time points where the groups differ. For this purpose, we propose the use of simultaneous confidence bands around the fitted models wherein the bands take into account within and between-subject variability, as well as variability arising from smoothing.     

Analyzing Rich Data Using Different Methods Provided by NONMEM: Pharmacokinetics of Telmisartan Following Intravenous Infusion to Healthy Volunteers

Pharmaceutical Research -     

Interaction Effects of Client and Treatment Program Characteristics on Retention: An Exploratory Analysis Using Hierarchical Linear Models

This paper applied a hierarchical linear modeling approach to explore the interaction effects of treatment program and client characteristics on client retention in treatment for drug users. Program characteristics included services provision, funding sources, and staff-client gender congruence, and client characteristics included gender, age at admission, and drug use level prior to admission. The same model was applied separately to three modalities: residential, methadone maintenance, and outpatient drug-free programs. Data were obtained from 59 treatment programs and 3,764 of their clients who had discharge records. The most noteworthy significant interaction effect detected was program's funding source and client's gender on treatment retention in the outpatient drug-free modality. for example, female clients remained less time in the programs that accepted only public funding than in the programs that accepted both public and private funding. Male clients remained in the treatment an average of 25.3 fewer days than female clients in drug-free programs that only accepted public fund, but stayed about the same time as females if the programs received mixed funding.     

混合效应线性模型与单因素方差分析在重复测量数据中的应用比较

目的:通过混合效应线性模型与单因素方差分析在重复测量资料中的应用比较,旨在说明两方法在处理重复测量资料时的应用特点。方法:用混合效应线性模型和单因素方差分析处理重复测量资料并比较。结果:混合效应线性模型和单因素方差分析都是处理重复测量资料的重要统计方法,前者在选择协方差结构下可对重复测量资料的固定效应和随机效应参数及协方差矩阵进行参数估计和统计检验,后者可对重复测量资料的固定效应做出统计推断。结论:混合效应线性模型是处理重复测量资料的有力方法,它对资料的协方差结构要求宽松,且结论可靠;单因素方差分析对资料的协方差结构有严格的限定。     

Prediction of an Outcome Using Trajectories Estimated from a Linear Mixed Model

In longitudinal data, interest is usually focused on the repeatedly measured variable itself. In some situations, however, the pattern of variation of the variable over time may contain information about a separate outcome variable. In such situations, longitudinal data provide an opportunity to develop predictive models for future observations of the separate outcome variable given the current data for an individual. In particular, longitudinally changing patterns of repeated measurements of a variable measured up to time t , or trajectories, can be used to predict an outcome measure or event that occurs after time t .

In this article, we propose a method for predicting an outcome variable based on a generalized linear model, specifically, a logistic regression model, the covariates of which are variables that characterize the trajectory of an individual. Since the trajectory of an individual contains estimation error, the proposed logistic regression model constitutes a measurement error model. The model is fitted in two steps. First, a linear mixed model is fitted to the longitudinal data to estimate the random effect that characterizes the trajectory for each individual while adjusting for other covariates. In the second step, a conditional likelihood approach is applied to account for the estimation error in the trajectory. Prediction of an outcome variable is based on the logistic regression model in the second step. The receiver operating characteristic curve is used to compare the discrimination ability of a model with trajectories to one without trajectories as covariates. A simulation study is used to assess the performance of the proposed method, and the method is applied to clinical trial data.     

The Effect of Collinearity on Parameter Estimates in Nonlinear Mixed Effect Models

Purpose. To demonstrate how correlations among predictor variables in a population pharmacokinetic model affect the ability to discern which covariates should enter into the structural pharmacokinetic model. Methods. Monte Carlo simulation was used to generate multiple-dose concentration-time data similar to that seen in a Phase III clinical trial. The drugs' pharmacokinetics were dependent on two covariates. Five data sets were simulated with increasing correlation between the two covariates. All data sets were analyzed using NONMEM both with and without inclusion of the covariates in the structural pharmacokinetic model. Summary measures for ill-conditioning and sensitivity analysis were used to examine how increasing correlation among covariates affects the accuracy and precision of the parameter estimates. Results. When covariates were included in the structural pharmacokinetic model and the correlation between covariates increased, the standard error of the parameter estimates increased and the value of parameter estimates themselves became increasingly biased. When the correlation between predictor variables was 0.75, the standard errors of the parameter estimates were too large to declare statistical significance. Conclusions. Correlations among predictor variables greater than 0.5 when entered into the model simultaneously should be a warning to researchers because the (1) the accuracy of the parameter estimates themselves may be biased and (2) the precision of the estimates may be inflated due to ill-conditioning.     

非线性混合效应模型法在卡马西平治疗儿童癫痫中的群体药动学应用

目的探讨非线性混合效应模型法在卡马西平治疗儿童癫痫中的群体药动学应用。方法选取2010年1月至2011年6月我院收治的癫痫患儿共180例,采用非线性混合效应模型法估算儿童癫痫卡马西平的群体药动学参数,并建立群体药动学模型。结果年龄、体质量及每日服药剂量均为卡马西平清除率的影响因素。经自举法验证,本模型可靠、稳定。结论用NONMEM软件成功建立我院癫痫患者服用卡马西平的PPK模型。根据我院癫痫患者的PPK模型,结合患者年龄、体质重和合并用药可估算其清除率,优化临床个体化用药方案。     

Robust Small-Sample Inference for Fixed Effects in General Gaussian Linear Models

Although asymptotically, the empirical covariance estimator is consistent and robust with respect to the selection of the working correlation matrix, when the sample size is small, its bias may not be negligible. This article proposes a small sample correction for the empirical covariance estimator in general Gaussian linear models. Inference for the fixed effects based on the corrected covariance matrix is also derived. A two-way analysis of variance (ANOVA) model with repeated measures, which evaluates the effectiveness of a CB1 receptor antagonist, and a four-period crossover design, which assesses the treatment effect in subjects with intermittent claudication, serve as examples to illustrate the proposed and other investigated methods. Simulation studies show that the proposed method generally performs better than other bias-correction methods, including Mancl and DeRouen (2001 Mancl , L. A. , DeRouen , T. A. ( 2001 ). A covariance estimator for GEE with improved small-sample properties . Biometrics 57 : 126 – 134 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), Kauermann and Carroll (2001 Kauermann , G. , Carroll , R. J. ( 2001 ). A note on the efficiency of sandwich covariance matrix estimation . Journal of the American Statistical Association 96 : 1387 – 1396 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), and Fay and Graubard (2001 Fay , M. P. , Graubard , B. I. ( 2001 ). Small-sample adjustments for Wald-type tests using sandwich estimators . Biometrics 57 : 1198 – 1206 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), in the investigated balanced designs.     

Bayesian Population Pharmacokinetic and Pharmacodynamic Analyses Using Mixture Models

Population studies of the pharmacokinetics or pharmacodynamics or drugs help us learn about the variability in drug disposition and effects, information that can be used to treat future patients at safe and effective doses. We present a new approach to population modeling based on a weighted mixture of normal distributions having random weights and means. This method allows estimation of underlying continuous population distributions without prespecifying the parametric form or shape of these probability distributions. Additionally, this method can carry out nonparametric regression of pharmacokinetic or dynamic parameters on patient covariates while estimating the underlying distributions. Two examples illustrate the method and its flexibility.     

Prospective Use of Population Pharmacokinetics/ Pharmacodynamics in the Development of Cisatracurium

Purpose. The population PK/PD approach was prospectively used to determine the PK/PD of cisatracurium in various subgroups of healthy surgical patients. Methods. Plasma concentration (Cp) and neuromuscular block data from 241 patients in 8 prospectively-designed Phase I–III trials were pooled and analyzed using NONMEM. The analyses included limited Cp-time data randomly collected from 186 patients in efficacy/safety studies and full Cp-time data from 55 patients in pharmacokinetic studies. The effects of covariates on the PK/PD parameters of cisatracurium were evaluated. The time course of neuromuscular block was predicted for various patient subgroups. Results. The population PK/PD model for cisatracurium revealed that anesthesia type, gender, age, creatinine clearance, and presence of obesity were associated with statistically significant (p < 0.01) effects on the PK/PD parameters of cisatracurium. These covariates were not associated with any clinically significant changes in the predicted recovery profile of cisatracurium. Slight differences in onset were predicted in patients with renal impairment and patients receiving inhalation anesthesia. Based on the validation procedure, the model appears to be accurate and precise. Conclusions. The prospective incorporation of a population PK/PD strategy into the clinical development of cisatracurium generated information which influenced product labeling and reduced the number of studies needed during development.     

Application of Resampling Techniques to Estimate Exact Significance Levels for Covariate Selection During Nonlinear Mixed Effects Model Building: Some Inferences

Purpose. One of the main objectives of the nonlinear mixed effects modeling is to provide rational individualized dosing strategies by explaining the interindividual variability using intrinsic and/or extrinsic factors (covariates). The aim of the current study was to evaluate, using computer simulations and real data, methods for estimating the exact significance level for including or excluding a covariate during model building. Methods. Original data were simulated using a simple one-compartment pharmacokinetic model with (full model) or without (null model) covariates (one or two). The covariate values in the original data were resampled (using either permutations or parametric bootstrap methods) to generate data under the null hypothesis that there is no covariate effect. The original and permuted data were fitted to null and full models, using first-order and first-order condition estimation (with or without interaction) methods in NONMEM, to compare the asymptotic and conditional p-value. Target log-likelihood ratio cutoffs for assessing covariate effects were derived. Results. The simulations showed that for sparse as well as dense data, the first-order condition estimation methods yielded the best results while the first-order method performs somewhat better for sparse data. Depending on the modeling objective, the appropriate asymptotic p-value can be substituted for the conditional significance level. Target log-likelihood ratio cutoffs should be determined separately for each covariate when exact p-values are important. Conclusions. Resampling methods can be employed to estimate the exact significance level for including a covariate during nonlinear mixed effects model building. Some reasonable inferences can be drawn for potential application to design future population analyses.     

Population pharmacokinetics of clomethiazole and its effect on the natural course of sedation in acute stroke patients

AIMS: This analysis was performed to investigate the population pharmacokinetics of clomethiazole and its effect on the natural course of sedation in acute stroke patients using a nonlinear mixed effects modelling approach. METHODS: One thousand five hundred and forty-six acute stroke patients (774 on active treatment) from 166 centres were included in three randomized, double-blind, placebo-controlled phase III efficacy and safety studies. A total dose of 68 mg kg(-1) clomethiazole edisilate was given as a three-phase i.v.-infusion over 24 h. Three blood samples were drawn from all patients to characterize the pharmacokinetics. Sedation was monitored throughout the entire treatment period and the degree of sedation was measured on a discrete ordinal scale with six levels. Models were fitted to the data using the software NONMEM. RESULTS: Clomethiazole was characterized by a two-compartment pharmacokinetic model with interindividual variability in all structural parameters. For a patient weighing 75 kg, the average CL, V1, Q, and V2 was estimated to be 52.7 l h(-1), 82.5 l, 167 l h(-1) and 335 l, respectively. The interindividual variability in CL, V1, Q and V2 was estimated to be 48%, 53%, 42% and 54%, respectively. Increasing body weight and concomitant administration of liver enzyme inducing drugs were found to increase clearance (by 0.5 l h(-1) kg(-1) and 40%, respectively). Increasing weight also increased the volume of distribution (1.1 l kg(-1) for V1 and 4.7 l kg(-1) for V2). A six-category proportional odds model with a component including the natural course of sedation following placebo administration, a drug component (present or absent) and an interindividual variability component described the degree of sedation. Stroke severity as measured on the NIH-stroke scale on admission and drug treatment were the most important predictors of sedation, but a nonlinear increase in sedation with increasing age was also found. Increasing body weight increased the sedative drug effect. CONCLUSIONS: The pharmacokinetics of clomethiazole were characterized in acute stroke patients and the analysis excluded several possible covariates of interest in drug development. The time course of sedation could be quantitatively described during the first 24 h following an acute stroke in the presence or absence of clomethiazole treatment.     

Modelling a Spontaneously Reported Side Effect by Use of a Markov Mixed-Effects Model

Aims: To present a method for analyzing side-effect data where change in severity is spontaneously reported during the experiment. Methods: A clinical study in 12 healthy volunteers aimed to investigate the concentration-response characteristics of a CNS-specific side-effect was conducted. After an open session where the subjects experienced the side-effect and where the individual pharmacokinetic parameters were evaluated they were randomized to a sequence of three different infusion rates of the drug in a double-blinded crossover way. The infusion rates were individualized to achieve the same target concentration in all subjects and different drug input rates were selected to mimic absorption profiles from different formulations. The occurrence of the specific side-effect and any subsequent change in severity was self-reported by the subjects. Severity was recorded as 0 = no side-effect, 1 = mild side-effect and 2 = moderate or severe side-effect. Results: The side-effect data were analyzed using a mixed-effects model for ordered categorical data with and without Markov elements. The former model estimated the probability of having a certain side-effect score conditioned on the preceding observation and drug exposure. The observed numbers of transitions between scores were from 0 −> 1: 24, from 0− > 2: 11, from 1 − >, 2: 23, from 2− > 1: 1, from 2− > 0: 32 and from 1 − >0: 2. The side-effect model consisted of an effect-compartment model with a tolerance compartment. The predictive performance of the Markov model was investigated by a posterior predictive check (PPC), where 100 datasets were simulated from the final model. Average number of the different transitions from the PPC was from 0 − > 1: 26, from 0 − > 2: 11, from 1 − > 2: 25, from 2 − >1: 1, from 2 − >0: 35 and from 1 − > 0: 1. A similar PPC for the model without Markov elements was at considerable disparity with the data. Conclusion: This approach of incorporating Markov elements in an analysis of spontaneously reported categorical side-effect data could adequately predict the observed side-effect time course and could be considered in analyses of categorical data where dependence between observations is an issue.     

Sex Specificity in Methadone Analgesia in the Rat: A Population Pharmacokinetic and Pharmacodynamic Approach

Purpose. To quantify the extent to which a sex-specific dichotomy in the temporal evolution of the analgesic effect, after intravenous (i.v.) methadone injection in the rat, relates to the pharmacokinetics (PK) and pharmacodynamics (PD) that mediate the dose-to-effect pathway. Methods. Tail-flick analgesia was measured after i.v. methadone injection (0.35 mg/kg) in female (n = 16) and male (n = 16) Sprague-Dawley rats. The PK were evaluated in separate female (n = 56) and male (n = 56) rats after they had received the same dose of methadone i.v. (0.35 mg/kg). A bicompartmental model described the kinetics and a sigmoid E_max model-related drug effect vs. simulated concentrations (pharmacodynamics) at the times of effect measurement. All model parameters as well as interanimal and assay variabilities were estimated with a mixed-effects population method using the program NONMEM. Results. The area under the effect-time curve (AUCE_0-120) was (mean ± interanimal SD) 1859 ± 346 min in the females, which was significantly lower than the 4871 ± 393 min in the males (P < 0.0001). On the contrary, the profiles of concentration vs. time were higher in females and, therefore, corresponded inversely to the effect vs. time-relative magnitudes. The central volume of distribution, V1, was 1.94 ± 0.37 l/kg for female rats and 3.01 ± 0.33 l/kg for male rats. Also, the central clearance was 0.077 ± 0.006 l/min/kg and 0.102 ± 0.005 l/min/kg, respectively, for female and male rats. Both parameters differed significantly between sexes (P < 0.0001). The pharmacodynamic maximum observed effect parameter (E_max) was 37% ± 29% in female rats and 85% ± 16% in male rats, and these values were significantly different (P < 0.0001). The parameter for the concentration eliciting half of E_max (EC₅₀) was 24.1 ± 7.5 g/l in female rats and 20.3 ± 2.9 g/l in male rats, and the Hill-related exponent, , was 6.3 ± 3.9 in female rats and 5.5 ± 4.1 in male rats. These parameters did not differ significantly (at the P < 0.05 level). Conclusions. A sex-specific dichotomy in the methadone antinociceptive effect, in the rat, was not proportionally related to plasma concentrations. Each sex corresponded to a distinct subpopulation of the PK parameters and one of the pharmacodynamic parameters (E_max). When the course of a drug involves PK or PD subpopulations, PK/PD modeling can afford the safest prediction of the effect-time evolution for a particular dose.     

Optimization of Individual and Population Designs Using Splus

We address the problem of design optimization for individual and population pharmacokinetic studies. We develop Splus generic functions for pharmacokinetic design optimization: IFIM, a function for individual design optimization similar to the ADAPT II software, and PFIM_OPT, a function for population design optimization which is an extension of the Splus function PFIM for population design evaluation. Both evaluate and optimise designs using the Simplex algorithm. IFIM optimizes the sampling times in continuous intervals of times; PFIM_OPT optimizes either, for a given group structure of the population design, only the sampling times taken in some given continuous intervals or, both the sampling times and the group structure, performing then statistical optimization. A combined variance error model can be supplied with the possibility to include parameters of the error model as parameters to be estimated. The performance of the optimization with the Simplex algorithm is demonstrated with two pharmacokinetic examples: by comparison of the optimized designs to those of the ADAPT II software for IFIM, and to those obtained using a grid search or the Fedorov-Wynn algorithm for PFIM_OPT. The influence of the variance error model on design optimization was investigated. For a given total number of samples, different group structures of a population design are compared, showing their influence on the population design efficiency. The functions IFIM and PFIM_OPT offer new efficient solutions for the increasingly important task of optimization of individual or population pharmacokinetic designs.