Nonlinear mixed effects modelling for estimation of steady state attainment

Assessment of the time needed to attain steady state is a key pharmacokinetic objective during drug development. Traditional approaches for assessing steady state include ANOVA‐based methods for comparing mean plasma concentration values from each sampling day, with either a difference or equivalence test. However, hypothesis‐testing approaches are ill suited for assessment of steady state. This paper presents a nonlinear mixed effects modelling approach for estimation of steady state attainment, based on fitting a simple nonlinear mixed model to observed trough plasma concentrations. The simple nonlinear mixed model is developed and proposed for use under certain pharmacokinetic assumptions. The nonlinear mixed modelling estimation approach is described and illustrated by application to trough data from a multiple dose trial in healthy subjects. The performance of the nonlinear mixed modelling approach is compared to ANOVA‐based approaches by means of simulation techniques. Copyright © 2005 John Wiley & Sons, Ltd.     

On Level Crossing Analysis of Queues

In this note we introduce a new level crossing analysis and using it derive an integral equation for the steady state waiting time in the GI/G/1 Queue. For the GI/M/1 queue we derive the rates of up- and down-crossings of the virtual delay process and two integral equations, one for the steady state time spent in the system and the other for the steady state waiting time in the queue. Also, the steady state probability distributions of the time spent in the system and the waiting time in the queue are obtained by solving these integral equations.     

Nonparametric confidence intervals for Tmax in sequence-stratified crossover studies

Tmax is the time associated with the maximum serum or plasma drug concentration achieved following a dose. While Tmax is continuous in theory, it is usually discrete in practice because it is equated to a nominal sampling time in the noncompartmental pharmacokinetics approach. For a 2-treatment crossover design, a Hodges-Lehmann method exists for a confidence interval on treatment differences. For appropriately designed crossover studies with more than two treatments, a new median-scaling method is proposed to obtain estimates and confidence intervals for treatment effects. A simulation study was done comparing this new method with two previously described rank-based nonparametric methods, a stratified ranks method and a signed ranks method due to Ohrvik. The Normal theory, a nonparametric confidence interval approach without adjustment for periods, and a nonparametric bootstrap method were also compared. Results show that less dense sampling and period effects cause increases in confidence interval length. The Normal theory method can be liberal (i.e. less than nominal coverage) if there is a true treatment effect. The nonparametric methods tend to be conservative with regard to coverage probability and among them the median-scaling method is least conservative and has shortest confidence intervals. The stratified ranks method was the most conservative and had very long confidence intervals. The bootstrap method was generally less conservative than the median-scaling method, but it tended to have longer confidence intervals. Overall, the median-scaling method had the best combination of coverage and confidence interval length. All methods performed adequately with respect to bias.     

Markov Chain Monte Carlo Techniques for Studying Interoccasion and Intersubject Variability: Application to Pharmacokinetic Data

Values of pharmacokinetic parameters may seem to vary randomly between dosing occasions. An accurate explanation of the pharmacokinetic behaviour of a particular drug within a population therefore requires two major sources of variability to be accounted for, namely interoccasion variability and intersubject variability. A hierarchical model that recognizes these two sources of variation has been developed. Standard Bayesian techniques were applied to this statistical model, and a mathematical algorithm based on a Gibbs sampling strategy was derived. The accuracy of this algorithm's determination of the interoccasion and intersubject variation in pharmacokinetic parameters was evaluated from various population analyses of several sets of simulated data. A comparison of results from these analyses with those obtained from parallel maximum likelihood analyses (NONMEM) showed that, for simple problems, the outputs from the two algorithms agreed well, whereas for more complex situations the NONMEM approach may be less accurate. Statistical analyses of a multioccasion data set of pharmacokinetic measurements on the drug metoprolol (the measurements being of concentrations of drug in blood plasma from human subjects) revealed substantial interoccasion variability for all structural model parameters. For some parameters, interoccasion variability appears to be the primary source of pharmacokinetic variation.     

Statistical model and inference for pharmacokinetic data

In this paper, we consider a statistical model for the drug concentration–time profiles that are obtained in a pharmacokinetic (PK) study when the drug is orally administered. In the proposed statistical PK model, the subject-specific concentration–time curve is described by the one-compartment PK model with first-order absorption and elimination. Moreover, a multivariate generalized gamma distribution is developed for the joint distribution of the drug concentrations that are repeatedly measured from the same subject. We then construct confidence intervals for the subject–exposure parameters which provide a further insight into the individual exposure of the drug under study. The proposed statistical PK model and the associated inference are then applied to illustrate a real data set. A simulation study is also implemented to investigate the performances of the coverage probability and expected length of the proposed confidence intervals. Finally, we give conclusions and discussions on the application of the proposed procedures.     

Approximated Transient Queue Length and Waiting Time Distributions via Steady State Analysis

Abstract

We propose a method to approximate the transient performance measures of a discrete time queueing system via a steady state analysis. The main idea is to approximate the system state at time slot t or on the n-th arrival–-depending on whether we are studying the transient queue length or waiting time distribution–-by the system state after a negative binomially distributed number of slots or arrivals. By increasing the number of phases k of the negative binomial distribution, an accurate approximation of the transient distribution of interest can be obtained.

In order to efficiently obtain the system state after a negative binomially distributed number of slots or arrivals, we introduce so-called reset Markov chains, by inserting reset events into the evolution of the queueing system under consideration. When computing the steady state vector of such a reset Markov chain, we exploit the block triangular block Toeplitz structure of the transition matrices involved and we directly obtain the approximation from its steady state vector. The concept of the reset Markov chains can be applied to a broad class of queueing systems and is demonstrated in full detail on a discrete-time queue with Markovian arrivals and phase-type services (i.e., the D-MAP/PH/1 queue). We focus on the queue length distribution at time t and the waiting time distribution of the n-th customer. Other distributions, e.g., the amount of work left behind by the n-th customer, that can be acquired in a similar way, are briefly touched upon.

Using various numerical examples, it is shown that the method provides good to excellent approximations at low computational costs–-as opposed to a recursive algorithm or a numerical inversion of the Laplace transform or generating function involved–-offering new perspectives to the transient analysis of practical queueing systems.     

Comparison of sample size formulae for 2 × 2 cross‐over designs applied to bioequivalence studies

We consider the comparison of two formulations in terms of average bioequivalence using the 2 × 2 cross‐over design. In a bioequivalence study, the primary outcome is a pharmacokinetic measure, such as the area under the plasma concentration by time curve, which is usually assumed to have a lognormal distribution. The criterion typically used for claiming bioequivalence is that the 90% confidence interval for the ratio of the means should lie within the interval (0.80, 1.25), or equivalently the 90% confidence interval for the differences in the means on the natural log scale should be within the interval (?0.2231, 0.2231). We compare the gold standard method for calculation of the sample size based on the non‐central t distribution with those based on the central t and normal distributions. In practice, the differences between the various approaches are likely to be small. Further approximations to the power function are sometimes used to simplify the calculations. These approximations should be used with caution, because the sample size required for a desirable level of power might be under‐ or overestimated compared to the gold standard method. However, in some situations the approximate methods produce very similar sample sizes to the gold standard method. Copyright © 2005 John Wiley & Sons, Ltd.     

Pharmacokinetic parameters estimation using adaptive Bayesian P-splines models

In preclinical and clinical experiments, pharmacokinetic (PK) studies are designed to analyse the evolution of drug concentration in plasma over time i.e. the PK profile. Some PK parameters are estimated in order to summarize the complete drug's kinetic profile: area under the curve (AUC), maximal concentration (C(max)), time at which the maximal concentration occurs (t(max)) and half-life time (t(1/2)).Several methods have been proposed to estimate these PK parameters. A first method relies on interpolating between observed concentrations. The interpolation method is often chosen linear. This method is simple and fast. Another method relies on compartmental modelling. In this case, nonlinear methods are used to estimate parameters of a chosen compartmental model. This method provides generally good results. However, if the data are sparse and noisy, two difficulties can arise with this method. The first one is related to the choice of the suitable compartmental model given the small number of data available in preclinical experiment for instance. Second, nonlinear methods can fail to converge. Much work has been done recently to circumvent these problems (J. Pharmacokinet. Pharmacodyn. 2007; 34:229-249, Stat. Comput., to appear, Biometrical J., to appear, ESAIM P&S 2004; 8:115-131).In this paper, we propose a Bayesian nonparametric model based on P-splines. This method provides good PK parameters estimation, whatever be the number of available observations and the level of noise in the data. Simulations show that the proposed method provides better PK parameters estimations than the interpolation method, both in terms of bias and precision. The Bayesian nonparametric method provides also better AUC and t(1/2) estimations than a correctly specified compartmental model, whereas this last method performs better in t(max) and C(max) estimations.We extend the basic model to a hierarchical one that treats the case where we have concentrations from different subjects. We are then able to get individual PK parameter estimations. Finally, with Bayesian methods, we can get easily some uncertainty measures by obtaining credibility sets for each PK parameter.     

Repeated‐measures models in the analysis of QT interval

Because of the recent regulatory emphasis on issues related to drug‐induced cardiac repolarization that can potentially lead to sudden death, QT interval analysis has received much attention in the clinical trial literature. The analysis of QT data is complicated by the fact that the QT interval is correlated with heart rate and other prognostic factors. Several attempts have been made in the literature to derive an optimal method for correcting the QT interval for heart rate; however the QT correction formulae obtained are not universal because of substantial variability observed across different patient populations. It is demonstrated in this paper that the widely used fixed QT correction formulae do not provide an adequate fit to QT and RR data and bias estimates of treatment effect. It is also shown that QT correction formulae derived from baseline data in clinical trials are likely to lead to Type I error rate inflation. This paper develops a QT interval analysis framework based on repeated‐measures models accomodating the correlation between QT interval and heart rate and the correlation among QT measurements collected over time. The proposed method of QT analysis controls the Type I error rate and is at least as powerful as traditional QT correction methods with respect to detecting drug‐related QT interval prolongation. Copyright © 2003 John Wiley & Sons, Ltd.     

Workload Process,Waiting Times,and Sojourn Times in a Discrete Time MMAP[K]/SM[K]/1/FCFS Queue

Abstract

In this paper, we study the total workload process and waiting times in a queueing system with multiple types of customers and a first-come-first-served service discipline. An M/G/1 type Markov chain, which is closely related to the total workload in the queueing system, is constructed. A method is developed for computing the steady state distribution of that Markov chain. Using that steady state distribution, the distributions of total workload, batch waiting times, and waiting times of individual types of customers are obtained. Compared to the GI/M/1 and QBD approaches for waiting times and sojourn times in discrete time queues, the dimension of the matrix blocks involved in the M/G/1 approach can be significantly smaller.     

Meta‐analysis of QTc interval – pooling data from heterogeneous trials

A diverse range of non‐cardiovascular drugs are associated with QT interval prolongation, which may be associated with a potentially fatal ventricular arrhythmia known as torsade de pointes. QT interval has been assessed for two recent submissions at GlaxoSmithKline. Meta‐analyses of ECG data from several clinical pharmacology studies were conducted for the two submissions. A general fixed effects meta‐analysis approach using summaries of the individual studies was used to calculate a pooled estimate and 90% confidence interval for the difference between each active dose and placebo following both single and repeat dosing separately. The meta‐analysis approach described provided a pragmatic solution to pooling complex and varied studies, and is a good way of addressing regulatory questions on QTc prolongation. Copyright © 2002 John Wiley & Sons, Ltd.     

Modeling attainment of steady state of drug concentration in plasma by means of a Bayesian approach using MCMC methods

The approach of Bayesian mixed effects modeling is an appropriate method for estimating both population-specific as well as subject-specific times to steady state. In addition to pure estimation, the approach allows to determine the time until a certain fraction of individuals of a population has reached steady state with a pre-specified certainty. In this paper a mixed effects model for the parameters of a nonlinear pharmacokinetic model is used within a Bayesian framework. Model fitting by means of Markov Chain Monte Carlo methods as implemented in the Gibbs sampler as well as the extraction of estimates and probability statements of interest are described. Finally, the proposed approach is illustrated by application to trough data from a multiple dose clinical trial.     

The covariance of the backward and forward recurrence times in a renewal process: the stationary case and asymptotics for the ordinary case

In a Poisson process, it is well-known that the forward and backward recurrence times at a given time point t are independent random variables. In a renewal process, although the joint distribution of these quantities is known (asymptotically), it seems that very few results regarding their covariance function exist. In the present paper, we study this covariance and, in particular, we state both necessary and sufficient conditions for it to be positive, zero or negative in terms of reliability classifications and the coefficient of variation of the underlying inter-renewal and the associated equilibrium distribution. Our results apply either for an ordinary renewal process in the steady state or for a stationary process.     

Estimation of Measures in M/M/1 Queue

Maximum likelihood and uniform minimum variance unbiased estimators of steady-state probability distribution of system size, probability of at least ? customers in the system in steady state, and certain steady-state measures of effectiveness in the M/M/1 queue are obtained/derived based on observations on X, the number of customer arrivals during a service time. The estimators are compared using Asympotic Expected Deficiency (AED) criterion leading to recommendation of uniform minimum variance unbiased estimators over maximum likelihood estimators for some measures.     

DISTRIBUTION OF THE NUMBER OF MARKOVIAN RENEWALS IN AN ARBITRARY INTERVAL

A Markov Renewal Process (M.R.P.) is one which records at each time t , the number of times a system visits each of m states in time t , if the transitions from state to state are according to a Markov chain and if the time required for each successive move is a random variable whose distribution function (d.f.) depends on the two states between which the move is made. In this paper, the distribution of the number of times each state is visited in an arbitrary interval (t₀, t₀+t) is derived. Asymptotic expressions for the mean and variance of this distribution are also obtained.     

Analysis of a longitudinal ordinal response clinical trial using dynamic models

Summary.  In many areas of pharmaceutical research, there has been increasing use of categorical data and more specifically ordinal responses. In many cases, complex models are required to account for different types of dependences among the responses. The clinical trial that is considered here involved patients who were required to remain in a particular state to enable the doctors to examine their heart. The aim of this trial was to study the relationship between the dose of the drug administered and the time that was spent by the patient in the state permitting examination. The patient's state was measured every second by a continuous Doppler signal which was categorized by the doctors into one of four ordered categories. Hence, the response consisted of repeated ordinal series. These series were of different lengths because the drug effect wore off faster (or slower) on certain patients depending on the drug dose administered and the infusion rate, and therefore the length of drug administration. A general method for generating new ordinal distributions is presented which is sufficiently flexible to handle unbalanced ordinal repeated measurements. It consists of obtaining a cumulative mixture distribution from a Laplace transform and introducing into it the integrated intensity of a binary logistic, continuation ratio or proportional odds model. Then, a multivariate distribution is constructed by a procedure that is similar to the updating process of the Kalman filter. Several types of history dependences are proposed.     

Likelihood approach for evaluating bioequivalence of highly variable drugs

Bioequivalence (BE) is required for approving a generic drug. The two one‐sided tests procedure (TOST, or the 90% confidence interval approach) has been used as the mainstream methodology to test average BE (ABE) on pharmacokinetic parameters such as the area under the blood concentration‐time curve and the peak concentration. However, for highly variable drugs (%CV > 30%), it is difficult to demonstrate ABE in a standard cross‐over study with the typical number of subjects using the TOST because of lack of power. Recently, the US Food and Drug Administration and the European Medicines Agency recommended similar but not identical reference‐scaled average BE (RSABE) approaches to address this issue. Although the power is improved, the new approaches may not guarantee a high level of confidence for the true difference between two drugs at the ABE boundaries. It is also difficult for these approaches to address the issues of population BE (PBE) and individual BE (IBE). We advocate the use of a likelihood approach for representing and interpreting BE data as evidence. Using example data from a full replicate 2 × 4 cross‐over study, we demonstrate how to present evidence using the profile likelihoods for the mean difference and standard deviation ratios of the two drugs for the pharmacokinetic parameters. With this approach, we present evidence for PBE and IBE as well as ABE within a unified framework. Our simulations show that the operating characteristics of the proposed likelihood approach are comparable with the RSABE approaches when the same criteria are applied. Copyright © 2014 John Wiley & Sons, Ltd.     

Costing Mixed Coxian Phase-type Systems with Poisson Arrivals

Previously, we developed a modeling framework which classifies individuals with respect to their length of stay (LOS) in the transient states of a continuous-time Markov model with a single absorbing state; phase-type models are used for each class of the Markov model. We here add costs and obtain results for moments of total costs in (0, t], for an individual, a cohort arriving at time zero and when arrivals are Poisson. Based on stroke patient data from the Belfast City Hospital we use the overall modelling framework to obtain results for total cost in a given time interval.