Simultaneous vs. Sequential Analysis for Population PK/PD Data II: Robustness of Methods

A model can be fit to joint PK/PD data (concentration and effect) either simultaneously or sequentially. The results of a companion paper suggested that when the data-analytic and true models agree, a particular sequential approach is computationally faster than the simultaneous one, yet produces hardly less precise PD parameter estimates, and for suitable designs, about as accurate PD standard error estimates. In this paper, we compare the performance of various methods for the case that the data-analytic model is misspecified. We illustrate these methods by applying them to a set of real data. Using NONMEM, population PK/PD observations were simulated under various study designs according to a one- or two-compartment PK model and direct Emax or sigmoid Emax model. A one-compartment PK model and Emax PD model were fit to the simulated observations by simultaneous and sequential methods. Predictive performance (interpolation and extrapolation) of PD and the type-I error rate of a likelihood ratio test are compared. The real data set consists of PK and (more frequent) PD observations after administration of the muscle relaxant vecuronium. When only the PK data-analytic model is misspecified, the simultaneous method has greater precision than the sequential methods. However a sequential method that uses a non-parametric PK model performs better than both other methods when PK model misspecification is severe. When the PD data-analytic model is misspecified, sequential and simultaneous methods perform similarly. The analysis of the real data shows that the PK fitted with the simultaneous method can be quite sensitive to PD model misspecification, yielding a possible diagnostic for this type of misspecification.     

抗真菌药PK/PD研究进展

药代动力学/药效学(PK/PD)结合了PK与PD的研究方法,同步研究药代动力学特征及与药效和疗效的关系,定量描述药物的效应-时间过程。PK/PD研究有利于更有效地利用现有的抗真菌药物和提高其在许多严重真菌感染时的抗菌效果。本文就抗真菌药的PK/PD研究进展进行综述。     

Performance of an Iterative Two-Stage Bayesian Technique for Population Pharmacokinetic Analysis of Rich Data Sets

Purpose To test the suitability of an Iterative Two-Stage Bayesian (ITSB) technique for population pharmacokinetic analysis of rich data sets, and to compare ITSB with Standard Two-Stage (STS) analysis and nonlinear Mixed Effect Modeling (MEM). Materials and Methods Data from a clinical study with rapacuronium and data generated by Monte Carlo simulation were analyzed by an ITSB technique described in literature, with some modifications, by STS, and by MEM (using NONMEM). The results were evaluated by comparing the mean error (accuracy) and root mean squared error (precision) of the estimated parameter values, their interindividual standard deviation, correlation coefficients, and residual standard deviation. In addition, the influence of initial estimates, number of subjects, number of measurements, and level of residual error on the performance of ITSB were investigated. Results ITSB yielded best results, and provided precise and virtually unbiased estimates of the population parameter means, interindividual variability, and residual standard deviation. The accuracy and precision of STS was poor, whereas ITSB performed better than MEM. Conclusions ITSB is a suitable technique for population pharmacokinetic analysis of rich data sets, and in the presented data set it is superior to STS and MEM. An erratum to this article can be found at     

Comparison of some control strategies for three-compartment PK/PD models

In drug therapy, effective dosage strategies are needed to maintain target drug effects. The relationship between drug dose and drug effect is often described by pharmacokinetic/pharmacodynamic (PK/PD) models where typically the PK model has a multicompartment form and the PD model is the sigmoidal Emax model. The parameters in the PK/PD model are generally unknown in the individual patient, although prior knowledge may be available and can be updated after measurements of drug effect are taken during the therapy. This fact, together with the complexity of the PK/PD model, makes the control problem complex. This paper investigates several control strategies in the framework of a three-compartment PK model plus an effect site with a PD model. Using computer simulations under different assumptions, we show that a MAP (maximum a posteriori) Bayesian type of strategy is effective, nevertheless in high-risk situations a stochastic control strategy hedging against estimation errors provides better performance at computational cost. Partially funded by Palo Alto Institute for Research and Education Inc., and the Veterans Administration Merit Review Program.     

药代动力学-药效动力学结合模型在中药研究中的应用

药代动力学-药效动力学(Pharmacokinetic-pharmacodynamic,PK/PD)结合模型是研究中药体内代谢过程、药物效应及二者联系的有效工具,对于中药作用机制研究、临床用药优化有重要的参考价值。建立能体现中医药特色的PK/PD结合模型十分必要。该文针对目前PK/PD结合模型在中药研究领域的应用现状作了系统的阐述,并就中药效应物质基础的确定、效应指标的选择等关键问题进行探讨并提出建议,以期为今后的相关研究提供参考。     

Propagation of Population PK and PD Information Using a Bayesian Approach: Dealing with Non-Exchangeability

The aim of this work is to implement a conservative prior that safeguards against population non-exchangeability of prior and data likelihood, in the framework of population pharmacokinetic/pharmacodynamic analysis, incorporating multi-level hierarchical modelling. Three different exercises were performed: (i) we investigated the use of parametric priors in the multilevel hierarchical modelling framework; (ii) we assessed the average performance of the multilevel hierarchical model compared to the standard mixed effect model, considering also some interesting extreme cases and (iii) we implemented an application with a small Proof of Principle (PoP) study, which demonstrates the propagation of information across PD studies using multilevel modelling. Fitting with the 4-level model and informative parametric priors performed similar to a meta-analysis of the test datasets combined with the datasets that the priors came from, demonstrating that parametric priors can be used alternatively to meta-analysis. Further, the 4-level model gave posterior distributions which had larger uncertainty but at the same time were unbiased, compared to the 3-level model, and therefore implements a more conservative prior in a formal way, which is appropriate when the prior and the test populations are not exchangeable. For the application with the PoP study, the statistical power of detecting the difference in potency between two drugs, when inter-study variability was present, was greater when an extra level in the hierarchical model to account for it, was used. In conclusion, by applying the prior one hierarchical level above the level of the parameters of interest, we implemented a more conservative prior, compared to applying the prior directly on the parameters of interest. The approach is equivalent to Bayesian individualization, offers a safeguard against bias from the prior and also avoids the danger of the data being overwhelmed by a strong prior.     

Sample Size Computations for PK/PD Population Models

We describe an accurate, yet simple and fast sample size computation method for hypothesis testing in population PK/PD studies. We use a first order approximation to the nonlinear mixed effects model and chi-square distributed Wald statistic to compute the minimum sample size to achieve given degree of power in rejecting a null hypothesis in population PK/PD studies. The method is an extension of Rochon’s sample size computation method for repeated measurement experiments. We compute sample sizes for PK and PK/PD models with different conditions, and use Monte Carlo simulation to show that the computed sample size retrieves the required power. We also show the effect of different sampling strategies, such as minimal, i.e., as many observations per individual as parameters in the model, and intensive on sample size. The proposed sample size computation method can produce estimates of minimum sample size to achieve the desired power in hypothesis testing in a greatly reduced time than currently available simulation-based methods. The method is rapid and efficient for sample size computation in population PK/PD study using nonlinear mixed effect models. The method is general and can accommodate any type of hierarchical models. Simulation results suggest that intensive sampling allows the reduction of the number of patients enrolled in a clinical study.     

利用药代动力学/药效动力学优化中国艾滋病患者的药物治疗方案

艾滋病患者需要终生进行药物治疗。我国常用的抗人类免疫缺陷病毒（HIV）药物主要有依非韦伦、替诺福韦、奈韦拉平、拉米夫定、齐多夫定、洛匹那韦/利托那韦等。为了明确国产抗反转录病毒药物在中国人群的药代动力学/药效动力学（PK/PD）特性,进一步明确和优化治疗方案,开展中国成人HIV感染者服用抗反转录病毒药物的PK/PD系列研究,结果表明这些药物在中国HIV感染患者的PK参数和安全性与外国人群不尽相同,主要表现为峰浓度（Cmax）和药时曲线下面积（AUC）较高,表观清除率（Cl/F）较低,肝毒性、皮疹、骨骼和肾脏毒性等不良反应发生率较高,其原因可能与体质量和不同种族人的药物遗传学差异有关,提示应根据中国人的PK/PD特点调整给药方案。综述5种常用抗反转录病毒药物的PK/PD,以期为优化中国HIV感染者的临床治疗提供参考。     

Population Pharmacokinetic/Pharmacodynamic Modeling of Systemic Corticosteroid Inhibition of Whole Blood Lymphocytes: Modeling Interoccasion Pharmacodynamic Variability

Purpose To develop a population pharmacokinetic/pharmacodynamic (PK/PD) model that characterizes the effects of major systemic corticosteroids on lymphocyte trafficking and responsiveness. Materials and Methods Single, presumably equivalent, doses of intravenous hydrocortisone (HC), dexamethasone (DEX), methylprednisolone (MPL), and oral prednisolone (PNL) were administered to five healthy male subjects in a five - way crossover, placebo - controlled study. Measurements included plasma drug and cortisol concentrations, total lymphocyte counts, and whole blood lymphocyte proliferation (WBLP). Population data analysis was performed using a Monte Carlo-Parametric Expectation Maximization algorithm. Results The final indirect, multi-component, mechanism-based model well captured the circadian rhythm exhibited in cortisol production and suppression, lymphocyte trafficking, and WBLP temporal profiles. In contrast to PK parameters, variability of drug concentrations producing 50% maximal immunosuppression (IC₅₀) were larger between subjects (73–118%). The individual log-transformed reciprocal posterior Bayesian estimates of IC₅₀ for ex vivo WBLP were highly correlated with those determined in vitro for the four drugs (r ²  = 0.928). Conclusions The immunosuppressive dynamics of the four corticosteroids was well described by the population PK/PD model with the incorporation of inter-occasion variability for several model components. This study provides improvements in modeling systemic corticosteroid effects and demonstrates greater variability of system and dynamic parameters compared to pharmacokinetics. Electronic Supplementary Material The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

基于PK/PD效应相关的目标获得概率论述美罗培南延长输注给药的适宜性

目的： 探讨美罗培南延长输注给药的适宜性。方法： 以美罗培南0.5 g q 8 h、1 g q 8 h和2 g q 8 h为模型方案，以40%、60%和80%fT>MIC为PK/PD靶标，以目标获得概率（PTA）为PK/PD效应指标，通过蒙特卡洛模拟计算各方案对MIC分别为0.125，0.25，0.5，1，2，4，8，16，32，64 mg·L^-1分别通过0.5，1，2，4，6 h输注给药达到各靶标的PTA。同一方案对同一MIC在2种不同的输注时间下可得到2个PTA值，结果组成一个对子，对所有MIC可组成10个对子。对所有对子进行配对资料检验，考察输注时间对PTA的影响。结果： 无论PK/PD靶标和给药方案如何，均存在这样一个MIC临界值，使得对于那些小于该值的MIC，即使美罗培南0.5 h输注，PTA也远远大于90%；而对于那些大于该值的MIC，即使美罗培南6 h输注，PTA也远远小于90%；对于这些非临界值上的MIC，延长输注给药并不能有效增加PTA。但对于部分处在临界值上的MIC，美罗培南延长输注可大大增加PTA。但任何2个不同时间的输液组的PTA进行比较，统计学分析结果显示差异均无统计学意义（P>0.05）。结论： 美罗培南延长输注给药获得的PK/PD增效对分离菌株或MIC具有一定的选择性，并非所有菌株均适宜采用延长输注给药。对处于MIC临界值上的分离菌株，更适合延长输注给药；对处于非MIC临界值上的分离菌株，无需采用延长输注给药。     

Using spline-enhanced ordinary differential equations for PK/PD model development

A spline-enhanced ordinary differential equation (ODE) method is proposed for developing a proper parametric kinetic ODE model and is shown to be a useful approach to PK/PD model development. The new method differs substantially from a previously proposed model development approach using a stochastic differential equation (SDE)-based method. In the SDE-based method, a Gaussian diffusion term is introduced into an ODE to quantify the system noise. In our proposed method, we assume an ODE system with form dx/dt = A(t)x + B(t) where B(t) is a nonparametric function vector that is estimated using penalized splines. B(t) is used to construct a quantitative measure of model uncertainty useful for finding the proper model structure for a given data set. By means of two examples with simulated data, we demonstrate that the spline-enhanced ODE method can provide model diagnostics and serve as a basis for systematic model development similar to the SDE-based method. We compare and highlight the differences between the SDE-based and the spline-enhanced ODE methods of model development. We conclude that the spline-enhanced ODE method can be useful for PK/PD modeling since it is based on a relatively uncomplicated estimation algorithm which can be implemented with readily available software, provides numerically stable, robust estimation for many models, is distribution-free and allows for identification and accommodation of model deficiencies due to model misspecification.     

Analyzing Multi-Response Data Using Forcing Functions

Introduction. Two analytic strategies can be taken to the analysis of multi-response data: a multivariate output model can be fit to all the response components simultaneously (SIM), or each response component can be fit separately to a univariate output model, conditioning in some way on the non-modeled components, the so-called forcing function approach (FFA). Focusing on a special case of multi-response model corresponding to a (pharmacokinetic) physiological f low model (PFM), the aims of this study are to (i) provide an algorithm for applying FFA to multi-response data from a PFM; (ii) examine the performance of FFA vs. SIM under optimal conditions for both, and in the presence of model misspecification; (iii) make recommendations regarding the use of FFA for multi-response data analysis. Methods. The basic PFM we use (variants of the basic model are used for simulation) has four homogenous compartments among which drug distributes. All are sampled arterial blood (A), non-eliminating tissue (N), eliminating tissue (E), and venous blood (V), which is also the drug dosing site. Parameters are blood f low rates to E and N, volumes of distribution of A, E, N, and V, elimination rate constant from E, and observation error variances. Observations from a generic individual under various study designs and parameter values are simulated. Using data-analytic models (DAM) both the same as, and different than the data simulation model (DSM), SIM fits the PFM to all data simultaneously; FFA first fits each type of response (one per tissue) separately, approximating the tissue’s input by linearly interpolating the observed concentrations from the donor tissue(s), estimates the identifiable parameter combinations for the response type, and then solves the simultaneous equations linking these across tissues, to obtain the primary model parameters of interest. This simulation and analysis steps are repeated to generate reliable performance statistics. Performances are compared with respect to parameter estimation error (when DAM and DSM are identical), and interpolated prediction error (when DAM and DSM are/are-not identical).The ability of SIM and FFA to identify the correct analytic model is also examined by comparing their failure rates in rejecting the wrong DAM. Results. The parameter estimation errors with FFA are generally about two times greater than those with SIM when the DAM is identical to the DSM. The prediction errors of FFA are about ten times greater than those of SIM when the DAM is identical to the DSM, and are about three times greater when the two are different. However, SIM fails to identify the correct model twice as often as FFA. Conclusions. Despite its greater convenience for model building, and its clear advantages for model identification, FFA’s final parameter estimates cannot be trusted when the multi-response system being modeled involves feedback. The size of the ratio of the two FFA residuals (obtained from the response-specific fits and from predictions made with the final FFA parameters) can, however, be used to indicate when FFA’s final estimates may be trustworthy.     

Analysis of Pharmacokinetics,Pharmacodynamics, and Pharmacogenomics Data Sets Using VizStruct,a Novel Multidimensional Visualization Technique

PURPOSE: Data visualization techniques for the pharmaceutical sciences have not been extensively investigated. The purpose of this study was to evaluate the usefulness of VizStruct, a multidimensional visualization tool, for applications in pharmacokinetics, pharmacodynamics, and pharmacogenomics. METHODS: The VizStruct tool uses the first harmonic of the discrete Fourier transform to map multidimensional data to two dimensions for visualization. The mapping was used to visualize several published pharmacokinetic, pharmacodynamic, and pharmacogenomic data sets. The VizStruct approach was evaluated using simulated population pharmacokinetics data sets, the data from Dalen and colleagues (Clin. PharmacoL Ther. 63:444-452, 1998) on the kinetics of nortriptyline and its 10-hydroxynortriptyline metabolite in subjects with differing number of copies of the CYP2D6, and the gene expression profiling data of Bohen and colleagues (Proc. Natl. Acad. Sci. USA 100:1926-1930, 2003) on follicular lymphoma patients responsive and nonresponsive to rituximab. RESULTS: The VizStruct mapping preserves the key characteristics of multidimensional data in two dimensions in a manner that facilitates visualization. The mapping is computationally efficient and can be used for cluster detection and class prediction in pharmaceutical data sets. The VizStruct visualization succinctly summarized the salient similarities and differences in the nortriptyline and 10-hydroxynortriptyline pharmacokinetic profiles in subjects with increasing number of CYP2D6 gene copies. In the simulated population pharmacokinetic data sets, it was capable of discriminating the subtle differences between pharmacokinetic profiles derived from 1- and 2-compartment models with the same area under the curve. The two-dimensional VizStruct mapping computed from a subset of 102 informative genes from the Bohen and colleagues data set effectively separated the rituximab responder, rituximab nonresponder, and control subject groups. CONCLUSIONS: The VizStruct approach is a computationally efficient and effective approach for visualizing complex, multidimensional data sets. It could have many useful applications in the pharmaceutical sciences.     

Simultaneous versus sequential pharmacokinetic-pharmacodynamic population analysis using an iterative two-stage Bayesian technique

A method for simultaneous pharmacokinetic-pharmacodynamic (PK-PD) population analysis using an Iterative Two-Stage Bayesian (ITSB) algorithm was developed. The method was evaluated using clinical data and Monte Carlo simulations.Data from a clinical study with rocuronium in nine anesthetized patients and data generated by Monte Carlo simulation using a similar study design were analysed by sequential PK-PD analysis, PD analysis with nonparametric PK data and simultaneous PK-PD analysis. Both PK and PD data sets were 'rich' with respect to the number of measurements per individual. The accuracy and precision of the estimated population parameters were evaluated by comparing their mean error (ME) and root mean squared error (RMSE), respectively. The influence of PD model misspecification on the results was also investigated.The simultaneous PK-PD analysis resulted in slightly more precise population parameter estimates than the sequential PK-PD analysis and the nonparametric PK method. In the presence of PD model misspecification, however, simultaneous analysis resulted in poor PK parameter estimates, while sequential PK-PD analysis performed well.In conclusion, ITSB is a valuable technique for PK-PD population analysis of rich data sets. The sequential PK-PD method is better suited for the analysis of rich data than the simultaneous analysis.     

Sample Size/Power Calculations for Population Pharmacodynamic Experiments Involving Repeated-Count Measurements

Repeated discrete outcome variables such as count measurements often arise in pharmacodynamic experiments. Count measurements can only take nonnegative integer values; this and correlation between repeated measurements from an individual make the design and analysis of repeated-count data special. Sample size/power calculation is an important part of clinical trial design to ensure adequate power for detecting significant effect, and it is often based on the procedure for analysis. This paper describes an approach for calculating sample size/power for population pharmacokinetic/pharmacodynamic experiments involving repeated-count measurements modeled as a Poisson process based on mixed-effects modeling technique. The noncentral version of the Wald χ² test is used for testing parameter/treatment significance. The approach was applied to two examples and the results were compared to results obtained from simulations in NONMEM. The first example involves calculating the power of a design to detect parameter significance between two groups: placebo and treatment group. The second example involves characterization of the dose-efficacy relationship of oxybutynin using a mixed-effects modeling approach. Weekly urge urinary incontinence episodes (a discrete count variable) is the primary efficacy variable and is modeled as a Poisson variable. A prospective study based on two different formulations of oxybutynin was designed using published population pharmacokinetic/pharmacodynamic model. The results of simulation studies showed good agreement between the proposed method and NONMEM simulations.     

Bioavailability Assessment from Pharmacologic Data: Method and Clinical Evaluation

A novel method is described for assessing drug bioavailability from pharmacologic data. The method is based upon a generalized model for the relationship between the observed effect (E) and the input rate (f): E = (c_e * f), where * denotes convolution, c_e is effect site unit impulse response (amount of drug at the effect site resulting from the instantaneous input of a unit amount of drug) and is transduction function (relates amount of drug at the effect site to E). The functions and c_e are expressed as cubic splines for maximum versatility. Pharmacologic data collected after the administration of two different doses by iv infusion are analyzed simultaneously to estimate the function parameters. This experimental design addresses the fact that and c_e cannot be uniquely estimated from the results of a single dose experiment. The unknown f from a test treatment is then estimated by applying an implicit deconvolution method to the pharmacologic data collected during that treatment. The method was tested with simulated data. The method and the model were further evaluated by application to a clinical study of verapamil (V) pharmacodynamics in 6 healthy volunteers. Simulations showed that the method is accurate and precise in the presence of a high degree of measurement error, but large intrasubject variability in the model functions can result in biased estimates of the amount absorbed. The method produced reasonably accurate estimates of the V input rate and systemic availability (F) in the 6 human volunteers though there was a trend towards underestimation (estimated total F%=93.6±14 vs. the true F% of 100).