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.     

Statistical Moments in Pharmacokinetics: Models and Assumptions

Abstract— The modelling basis of statistical moments in pharmacokinetics is considered and the associated assumptions and restrictions highlighted. Both deterministic and statistical models are described and they are seen to give identical results on the basis of equivalent assumptions. It is shown that it is not necessary to assume that all kinetic processes are first order and that some of the pharmacokinetic parameters may be dependent on the dose of administered drug and on the route of administration.     

脑组织生理药代动力学模型研究进展

在新药研发过程中,了解药物在脑内的转运与分布情况可以预测药物效应和不良反应。然而,人脑内的药物浓度难以通过现有分析技术直接测定。生理药代动力学（PBPK）模型通过数学建模的方式模拟药物在脑内的转运与分布情况,为预测脑内药物浓度提供帮助。综述影响药物在脑内转运分布的生理因素,以及近年来文献中报道的研究药物脑组织分布的PBPK模型及其在药物研发中的应用。     

Lumping of Whole-Body Physiologically Based Pharmacokinetic Models

Lumping is a common pragmatic approach aimed at the reduction of whole-body physiologically based pharmacokinetic (PBPK) model dimensionality and complexity. Incorrect lumping is equivalent to model misspecification with all the negative consequences to the subsequent model implementation. Proper lumping should guarantee that no useful information about the kinetics of the underlying processes is lost. To enforce this guarantee, formal standard lumping procedures and techniques need to be defined and implemented. This study examines the lumping process from a system theory point of view, which provides a formal basis for the derivation of principles and standard procedures of lumping. The lumping principle in PBPK modeling is defined as follows: Only tissues with identical model specification, and occupying identical positions in the system structure should be lumped together at each lumping iteration. In order to lump together parallel tissues, they should have similar or close time constants. In order to lump together serial tissues, they should equilibrate very rapidly with one another. The lumping procedure should include the following stages: (i) tissue specification conversion (when tissues with different model specifications are to be lumped together); (ii) classification of the tissues into classes with significantly different kinetics, according to the basic principle of lumping above; (iii) calculation of the parameters of the lumped compartments; (iv) simulation of the lumped system; (v) lumping of the experimental data; and (vi) verification of the lumped model. The use of the lumping principles and procedures to be adopted is illustrated with an example of a commonly implemented whole-body physiologically based pharmacokinetic model structure to characterize the pharmacokinetics of a homologous series of barbiturates in the rat.     

Structural Identifiability of Physiologically Based Pharmacokinetic Models

When starting a project in drug kinetics it is necessary to test a priori whether there is sufficient information in the experimental input-output design to estimate unique values of internal rate constants. This is an important test if the pharmacokinetics of a drug are to be characterised in some way by the parameter values estimated from the observed plasma or blood concentration profile. Various modifications of the well-perfused Physiologically Based Pharmacokinetic model (PBPK) are considered here. More complex PBPK models can be considered to consist of subsystems, representing groups of tissues, which are connected in parallel to the central compartment. A novel method of structural identifiability analysis is presented here that considers these subsystems individually. This makes analysis of subsequently modified models much simpler. It is found in a number of cases that these more complex systems remain globally identifiable and at worst reduce to locally identifiable for the additional parameters. A caveat is added about having more than one eliminating peripheral tissue.     

Optimal Design for Multivariate Response Pharmacokinetic Models

We address the problem of designing pharmacokinetic experiments in multivariate response situations. Criteria, based on the Fisher information matrix, whose inverse according to the Rao–Cramer inequality is the lower bound of the variance–covariance matrix of any unbiased estimator of the parameters, have previously been developed for univariate response for an individual and a population. We extend these criteria to design individual and population studies where more than one response is measured, for example, when both parent drug and metabolites are measured in plasma, multi-compartment models, where measurements are taken at more than one site, or when drug concentration and pharmacodynamic data are collected simultaneously. We assume that measurements made at distinct times are independent, but measurements made of each concentration are correlated with a response variance–covariance matrix. We investigated a number of optimisation algorithms, namely simplex, exchange, adaptive random search, simulated annealing and a hybrid, to maximise the determinant of the Fisher information matrix as required by the D-optimality criterion. The multiresponse optimal design methodology developed was applied in two case studies, where the aim was to suggest optimal sampling times. The first was a restrospective iv infusion experiment aimed to characterise the disposition kinetics of tolcapone and its two metabolites in healthy volunteers. The second was a prospective iv bolus experiment designed to estimate the tissue disposition kinetics of eight beta-blockers in rat.     

药动学模型的辨识问题研究

目的:研究药动学房室模型的辨识问题。方法:采用Laplace变换方法对一次性给药的经典房室模型的辨识问题进行系统讨论。结果:一次性给药的经典房室模型一般不具有唯一性,存在房室模型的辨识问题。结论:在两种经典房室模型不可辨识的情况下,血药浓度与分布器官中药物浓度成正比,此时通过测定血药浓度来预测分布器官或靶器官中药物浓度是有理论根据和有意义的;反之,在两种经典房室模型可辨识的情况下,血药浓度与分布器官或靶器官中药物浓度不成比例关系,此时欲通过血药浓度监测来预测分布器官或靶器官中药物浓度,就需首先确定血药浓度与分布器官或靶器官中药物浓度之间的关系,否则是没有理论根据和无意义的。     

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.     

Approaches for Applications of Physiologically Based Pharmacokinetic Models in Risk Assessment

Physiologically based pharmacokinetic (PBPK) models are particularly useful for simulating exposures to environmental toxicants for which, unlike pharmaceuticals, there is often little or no human data available to estimate the internal dose of a putative toxic moiety in a target tissue or an appropriate surrogate. This article reviews the current state of knowledge and approaches for application of PBPK models in the process of deriving reference dose, reference concentration, and cancer risk estimates. Examples drawn from previous U.S. Environmental Protection Agency (EPA) risk assessments and human health risk assessments in peer-reviewed literature illustrate the ways and means of using PBPK models to quantify the pharmacokinetic component of the interspecies and intraspecies uncertainty factors as well as to conduct route to route, high dose to low dose and duration extrapolations. The choice of the appropriate dose metric is key to the use of the PBPK models for the various applications in risk assessment. Issues related to whether uncertainty factors are most appropriately applied before or after derivation of human equivalent dose (or concentration) continue to be explored. Scientific progress in the understanding of life stage and genetic differences in dosimetry and their impacts on variability in susceptibility, as well as ongoing development of analytical methods to characterize uncertainty in PBPK models, will make their use in risk assessment increasingly likely. As such, it is anticipated that when PBPK models are used to express adverse tissue responses in terms of the internal target tissue dose of the toxic moiety rather than the external concentration, the scientific basis of, and confidence in, risk assessments will be enhanced.     

药代动力学/药效动力学模型在降糖药物评价中的应用

本文通过检索国外相关文献,对采用PK/PD方法对降糖药物及其制剂进行评价的研究进行归纳整理。以介绍药代动力学/药效动力学（pharmacokinetic/pharmacodynamic,PK/PD）模型在降糖药物及其制剂评价中的应用。检索结果表明血糖和胰岛素的相关性是PK/PD模型中的重点部分,PK/PD模型能够评价不同制剂或给药方式下的降糖效果。PK/PD模型是降糖药物及其制剂研究中一种较好的评价方法。     

General Pharmacokinetic Model for Drugs Exhibiting Target-Mediated Drug Disposition

Drugs that bind with high affinity and to a significant extent (relative to dose) to a pharmacologic target such as an enzyme, receptor, or transporter may exhibit nonlinear pharmacokinetic (PK) behavior. Processes such as receptor-mediated endocytosis may result in drug elimination. A general PK model for characterizing such behavior is described and explored through computer simulations and applications to several therapeutic agents. Simulations show that model predicted plasma concentration vs. time profiles are expected to be polyexponential with steeper distribution phases for lower doses and similar terminal disposition phases. Noncompartmental parameters always show apparent V_ss and CL_D decreasing with dose, but apparent clearance decreases only when the binding process produces drug elimination. The proposed model well captured the time-course of drug concentrations for the aldose reductase inhibitor imirestat, the endothelin receptor antagonist bosentan, and recombinant human interferon- 1a. This type of model has a mechanistic basis and considerable utility for fully describing the kinetics for various doses of relevant drugs.     

药物动力学模型的Cp选择法

目的:为药物动力学(PK)模型的选择提供一个新定量指标。方法:将多元回归分析中选择“最优”回归模型的统计量Cp的计算公式赋予新含义使之适用于PK房室模型选择。结果:经实例计算,与AIC准则判定的PK模型相同,但Cp选择法更为灵敏。     

Bayesian Optimal Designs for Pharmacokinetic Models: Sensitivity to Uncertainty

We studied the sensitivity of the number of unique design points and their placement, in Bayesian optimal designs for pharmacokinetic models, with respect to the magnitude of prior uncertainty. We used two and three-parameter pharmacokinetic models with fixed and mixed effects and two Bayesian optimal design criteria, namely ED and API, using different error weighting schemes. We found that by increasing the magnitude of the uncertainty, in most cases, additional design points appear, compared to the corresponding local design, and this happens gradually, forming bifurcation patterns. These bifurcation patterns were interpreted as high sensitivity of the design from the magnitude of the uncertainty.     

Simplification of Complex Physiologically Based Pharmacokinetic Models of Monoclonal Antibodies

Monoclonal antibodies (mAbs) exhibit biexponential profiles in plasma that are commonly described with a standard two-compartment model with elimination from the central compartment. These models adequately describe mAb plasma PK. However, these models ignore elimination from the peripheral compartment. This may lead to underestimation of the volume of distribution of the peripheral compartment and thus over-predicts concentration in the peripheral compartment. We developed a simple and physiologically relevant model that incorporates information on binding and dissociation rates between mAb and FcRn receptor, mAb uptake, reflection, and catabolic degradation. We employed a previously published PBPK model and, with assumptions regarding rates of processes controlling mAb disposition, reduced the complex PBPK model to a simpler circular model with central, peripheral, and lymph compartments specifying elimination from both central and peripheral. We successfully applied the model to describe the PK of an investigational mAb. Our model presents an improvement over standard two-compartmental models in predicting whole-body average tissue concentrations while adequately describing plasma PK with minimal complexity and physiologically more meaningful parameters.     

Evaluating Pharmacokinetic/Pharmacodynamic Models Using the Posterior Predictive Check

The posterior predictive check (PPC) is a model evaluation tool. It assigns a value (p ^PPC ) to the probability that the value of a given statistic computed from data arising under an analysis model is as or more extreme than the value computed from the real data themselves. If this probability is too small, the analysis model is regarded as invalid for the given statistic. Properties of the PPC for pharmacokinetic (PK) and pharmacodynamic (PD) model evaluation are examined herein for a particularly simple simulation setting: extensive sampling of a single individual's data arising from simple PK/PD and error models. To test the performance characteristics of the PPC, repeatedly, real data are simulated and for a variety of statistics, the PPC is applied to an analysis model, which may (null hypothesis) or may not (alternative hypothesis) be identical to the simulation model. Five models are used here: (PK1) mono-exponential with proportional error, (PK2) biexponential with proportional error, (PK2) biexponential with additive error, (PD1) E _max model with additive error under the logit transform, and (PD2) sigmoid E _max model with additive error under the logit transform. Six simulation/analysis settings are studied. The first three, (PK1/PK1), (PK2/PK2), and (PD1/PD1) evaluate whether the PPC has appropriate type-I error level, whereas the second three (PK2/PK1), (PK2/PK2), and (PD2/PD1) evaluate whether the PPC has adequate power. For a set of 100 data sets simulated/analyzed under each model pair according to a stipulated extensive sampling design, the p ^PPC is computed for a number of statistics in three different ways (each way uses a different approximation to the posterior distribution on the model parameters). We find that in general; (i) The PPC is conservative under the null in the sense that for many statistics, prob(p ^PPC )< for small . With respect to such statistics, this means that useful models will rarely be regarded incorrectly as invalid. A high correlation of a statistic with the parameter estimates obtained from the same data used to compute the statistic (a measure of statistical sufficiency) tends to identify the most conservative statistics. (ii) Power is not very great, at least for the alternative models we tested, and it is especially poor with statistics that are in part a function of parameters as well as data. Although there is a tendency for nonsufficient statistics (as we have measured this) to have greater power, this is by no means an infallible diagnostic. (iii) No clear advantage for one or another method of approximating the posterior distribution on model parameters is found.     

The Importance of Villous Physiology and Morphology in Mechanistic Physiologically-Based Pharmacokinetic Models

Purpose

Existing PBPK models incorporating intestinal first-pass metabolism account for effect of drug permeability on accessible absorption surface area by use of “effective” permeability, P _eff , without adjusting number of enterocytes involved in absorption or proportion of intestinal CYP3A involved in metabolism. The current model expands on existing models by accounting for these factors.

Methods

The PBPK model was developed using SAAM II. Midazolam clinical data was generated at GlaxoSmithKline.

Results

The model simultaneously captures human midazolam blood concentration profile and previously reported intestinal availability, using values for CYP3A CLu _int , permeability and accessible surface area comparable to literature data. Simulations show: (1) failure to distinguish absorbing from non-absorbing enterocytes results in overestimation of intestinal metabolism of highly permeable drugs absorbed across the top portion of the villous surface only; (2) first-pass extraction of poorly permeable drugs occurs primarily in enterocytes, drugs with higher permeability are extracted by enterocytes and hepatocytes; (3) CYP3A distribution along crypt-villous axes does not significantly impact intestinal metabolism; (4) differences in permeability of perpetrator and victim drugs results in their spatial separation along the villous axis and intestinal length, diminishing drug-drug interaction magnitude.

Conclusions

The model provides a useful tool to interrogate intestinal absorption/metabolism of candidate drugs.     

General Pharmacokinetic Model for Topically Administered Ocular Drug Dosage Forms

Purpose

In ocular drug development, an early estimate of drug behavior before any in vivo experiments is important. The pharmacokinetics (PK) and bioavailability depend not only on active compound and excipients but also on physicochemical properties of the ocular drug formulation. We propose to utilize PK modelling to predict how drug and formulational properties affect drug bioavailability and pharmacokinetics.

Methods

A physiologically relevant PK model based on the rabbit eye was built to simulate the effect of formulation and physicochemical properties on PK of pilocarpine solutions and fluorometholone suspensions. The model consists of four compartments: solid and dissolved drug in tear fluid, drug in corneal epithelium and aqueous humor. Parameter values and in vivo PK data in rabbits were taken from published literature.

Results

The model predicted the pilocarpine and fluorometholone concentrations in the corneal epithelium and aqueous humor with a reasonable accuracy for many different formulations. The model includes a graphical user interface that enables the user to modify parameters easily and thus simulate various formulations.

Conclusions

The model is suitable for the development of ophthalmic formulations and the planning of bioequivalence studies.

     

Bootstrapping for Pharmacokinetic Models: Visualization of Predictive and Parameter Uncertainty

Purpose. We explore use of "bootstrapping methods to obtain a measure of reliability of predictions made in part from fits of individual drug level data with a pharmacokinetic (PK) model, and to help clarify parameter identifiability for such models. Methods. Simulation studies use four sets (A-D) of drug concentration data obtained following a single oral dose. Each set is fit with a two compartment PK model, and the "bootstrap is employed to examine the potential predictive variation in estimates of parameter sets. This yields an empirical distribution of plausible steady state (SS) drug concentration predictions that can be used to form a confidence interval for a prediction. Results. A distinct, narrow confidence region in parameter space is identified for subjects A and B. The bootstrapped sets have a relatively large coefficient of variation (CV) (35-90% for A), yet the corresponding SS drug levels are tightly clustered (CVs only 2-9%). The results for C and D are dramatically different. The CVs for both the parameters and predicted drug levels are larger by a factor of 5 and more. The results reveal that the original data for C and D, but not A and B, can be represented by at least two different PK model manifestations, yet only one provides reliable predictions. Conclusions. The insights gained can facilitate making decisions about parameter identifiability. In particular, the results for C and D have important implications for the degree of implicit overparameterization that may exist in the PK model. In cases where the data support only a single model manifestation, the "bootstrap method provides information needed to form a confidence interval for a prediction.