Population Fisher Information Matrix Research Articles

Design optimisation for pharmacokinetic modeling of a cocktail of phenotyping drugs.

Our paper proposes a methodological strategy to select optimal sampling designs for phenotyping studies including a cocktail of drugs. A cocktail approach is of high interest to determine the simultaneous activity of enzymes responsible for drug metabolism and pharmacokinetics, therefore useful in anticipating drug-drug interactions and in personalized medicine. Phenotyping indexes, which are area under the concentration-time curves, can be derived from a few samples using nonlinear mixed effect models and maximum a posteriori estimation. Because of clinical constraints in phenotyping studies, the number of samples that can be collected in individuals is limited and the sampling times must be as flexible as possible. Therefore to optimize joint design for several drugs (i.e., to determine a compromise between informative times that best characterize each drug's kinetics), we proposed to use a compound optimality criterion based on the expected population Fisher information matrix in nonlinear mixed effect models. This criterion allows weighting different models, which might be useful to take into account the importance accorded to each target in a phenotyping test. We also computed windows around the optimal times based on recursive random sampling and Monte-Carlo simulation while maintaining a reasonable level of efficiency for parameter estimation. We illustrated this strategy for two drugs often included in phenotyping cocktails, midazolam (probe for CYP3A) and digoxin (P-glycoprotein), based on the data of a previous study, and were able to find a sparse and flexible design. The obtained design was evaluated by clinical trial simulations and shown to be efficient for the estimation of population and individual parameters.

Optimizing pharmacokinetic bridging studies in paediatric oncology using physiologically-based pharmacokinetic modelling: application to docetaxel.

Applying physiologically-based pharmacokinetic (PBPK) modelling in paediatric cancer drug development is still challenging. We aimed to demonstrate how PBPK modelling can be applied to optimize dose and sampling times for a paediatric pharmacokinetic (PK) bridging study in oncology and to compare with the allometric scaling population PK (AS-popPK) approach, using docetaxel as an example. A PBPK model for docetaxel was first developed for adult cancer patients using Simcyp® and subsequently used to predict its PK profiles in children by accounting for age-dependent physiological differences. Dose (mg m(-2) ) requirements for children aged 0-18 years were calculated to achieve targeted exposure in adults. Simulated data were then analyzed using population PK modelling with MONOLIX® in order to perform design optimization with the population Fisher information matrix (PFIM). In parallel, the AS-popPK approach was performed for the comparison. The PBPK model developed for docetaxel adequately predicted its PK profiles in both adult and paediatric cancer patients (predicted clearance and volume of distribution within 1.5 fold of observed data). The revised dose of docetaxel for a child over 1.5 years old was higher than the adult dose. Considering clinical constraints, the optimal design contained two groups of 15 patients, having three or four sampling times and had good predicted relative standard errors (RSE<30%) for almost all parameters. The AS-popPK approach performed reasonably well but could not predict for very young children. This research shows the clinical utility of PBPK modelling in combination with population PK modelling and optimal design to support paediatric oncology development.

Adaptive Optimal Design for Bridging Studies with an Application to Population Pharmacokinetic Studies

To develop and evaluate methods for conducting adaptive population pharmacokinetic bridging studies. An adaptive D-optimal design based on optimization of the population Fisher information matrix was used to determine the best sampling schedule for a target-population. Recruitment of the target-population was divided into batches and patients are assumed to enroll by batch. A prior-population model was used to determine the optimal sampling schedule for the first batch and to stabilise the data analysis in the interim iteration. Simulation studies were performed under two scenarios (1) the prior- and target-populations have similar pharmacokinetic profiles and (2) the pharmacokinetic profiles diverge significantly. A design criterion to determine early full enrollment was also proposed. The target-population estimates obtained using the proposed method were compared to estimates obtained if the target-population was studied with a design optimized based on the prior-population model. The proposed method is shown to be not inferior in scenario (1) and superior in scenario (2). The criterion to determine early full enrollment was proven to be effective. An adaptive optimal design method together with an early full enrollment criterion were evaluated and resulted in more accurate estimates for the target-population in bridging studies.

Design evaluation and optimisation in crossover pharmacokinetic studies analysed by nonlinear mixed effects models

Bioequivalence or interaction trials are commonly studied in crossover design and can be analysed by nonlinear mixed effects models as an alternative to noncompartmental approach. We propose an extension of the population Fisher information matrix in nonlinear mixed effects models to design crossover pharmacokinetic trials, using a linearisation of the model around the random effect expectation, including within-subject variability and discrete covariates fixed or changing between periods. We use the expected standard errors of treatment effect to compute the power for the Wald test of comparison or equivalence and the number of subjects needed for a given power. We perform various simulations mimicking crossover two-period trials to show the relevance of these developments. We then apply these developments to design a crossover pharmacokinetic study of amoxicillin in piglets and implement them in the new version 3.2 of the r function PFIM.

Population Fisher information matrix and optimal design of discrete data responses in population pharmacodynamic experiments

In the recent years, interest in the application of experimental design theory to population pharmacokinetic (PK) and pharmacodynamic (PD) experiments has increased. The aim is to improve the efficiency and the precision with which parameters are estimated during data analysis and sometimes to increase the power and reduce the sample size required for hypothesis testing. The population Fisher information matrix (PFIM) has been described for uniresponse and multiresponse population PK experiments for design evaluation and optimisation. Despite these developments and availability of tools for optimal design of population PK and PD experiments much of the effort has been focused on repeated continuous variable measurements with less work being done on repeated discrete type measurements. Discrete data arise mainly in PDs e.g. ordinal, nominal, dichotomous or count measurements. This paper implements expressions for the PFIM for repeated ordinal, dichotomous and count measurements based on analysis by a mixed-effects modelling technique. Three simulation studies were used to investigate the performance of the expressions. Example 1 is based on repeated dichotomous measurements, Example 2 is based on repeated count measurements and Example 3 is based on repeated ordinal measurements. Data simulated in MATLAB were analysed using NONMEM (Laplace method) and the glmmML package in R (Laplace and adaptive Gauss-Hermite quadrature methods). The results obtained for Examples 1 and 2 showed good agreement between the relative standard errors obtained using the PFIM and simulations. The results obtained for Example 3 showed the importance of sampling at the most informative time points. Implementation of these expressions will provide the opportunity for efficient design of population PD experiments that involve discrete type data through design evaluation and optimisation.

High-dimensional Ising model selection using ℓ1-regularized logistic regression

We consider the problem of estimating the graph associated with a binary Ising Markov random field. We describe a method based on $\ell_1$-regularized logistic regression, in which the neighborhood of any given node is estimated by performing logistic regression subject to an $\ell_1$-constraint. The method is analyzed under high-dimensional scaling in which both the number of nodes $p$ and maximum neighborhood size $d$ are allowed to grow as a function of the number of observations $n$. Our main results provide sufficient conditions on the triple $(n,p,d)$ and the model parameters for the method to succeed in consistently estimating the neighborhood of every node in the graph simultaneously. With coherence conditions imposed on the population Fisher information matrix, we prove that consistent neighborhood selection can be obtained for sample sizes $n=\Omega(d^3\log p)$ with exponentially decaying error. When these same conditions are imposed directly on the sample matrices, we show that a reduced sample size of $n=\Omega(d^2\log p)$ suffices for the method to estimate neighborhoods consistently. Although this paper focuses on the binary graphical models, we indicate how a generalization of the method of the paper would apply to general discrete Markov random fields.

Drug–drug interaction predictions with PBPK models and optimal multiresponse sampling time designs: application to midazolam and a phase I compound. Part 1: comparison of uniresponse and multiresponse designs using PopDes

To determine the optimal sampling time design of a drug-drug interaction (DDI) study for the estimation of apparent clearances (CL/F) of two co-administered drugs (SX, a phase I compound, potentially a CYP3A4 inhibitor, and MDZ, a reference CYP3A4 substrate) without any in vivo data using physiologically based pharmacokinetic (PBPK) predictions, population PK modelling and multiresponse optimal design. PBPK models were developed with AcslXtreme using only in vitro data to simulate PK profiles of both drugs when they were co-administered. Then, using simulated data, population PK models were developed with NONMEM and optimal sampling times were determined by optimizing the determinant of the population Fisher information matrix with PopDes using either two uniresponse designs (UD) or a multiresponse design (MD) with joint sampling times for both drugs. Finally, the D-optimal sampling time designs were evaluated by simulation and re-estimation with NONMEM by computing the relative root mean squared error (RMSE) and empirical relative standard errors (RSE) of CL/F. There were four and five optimal sampling times (=nine different sampling times) in the UDs for SX and MDZ, respectively, whereas there were only five sampling times in the MD. Whatever design and compound, CL/F was well estimated (RSE < 20% for MDZ and <25% for SX) and expected RSEs from PopDes were in the same range as empirical RSEs. Moreover, there was no bias in CL/F estimation. Since MD required only five sampling times compared to the two UDs, D-optimal sampling times of the MD were included into a full empirical design for the proposed clinical trial. A joint paper compares the designs with real data. This global approach including PBPK simulations, population PK modelling and multiresponse optimal design allowed, without any in vivo data, the design of a clinical trial, using sparse sampling, capable of estimating CL/F of the CYP3A4 substrate and potential inhibitor when co-administered together.

Incorporating Correlation in Interindividual Variability for the Optimal Design of Multiresponse Pharmacokinetic Experiments

This paper presents a method for optimal design of multiresponse population pharmacokinetic experiments taking into account correlations between interindividual variances. Expressions for the population Fisher information matrix have been defined for uniresponse and multiresponse pharmacokinetic experiments. A major assumption often made is that the variance–covariance matrix of the interindividual variance components has only diagonal elements so that whenever intersubject covariance elements are present, they are ignored during the design of the experiment. Recently expressions that accounted for these off diagonal elements were developed for uniresponse population pharmacokinetic experiments. The work presented here extends these expressions to multiresponse population pharmacokinetic experiments. These were applied to a population pharmacokinetic model, a population pharmacokinetic–pharmacodynamic model, and a parent–metabolite pharmacokinetic model example. The results obtained showed that optimal designs are different with diagonal omega matrix and full omega matrix and ignoring the off diagonal elements can lead to a design that produces more biased and less precise parameter estimates compared to a design that includes the off diagonal elements. The results also showed correlation between residual components of the responses has an effect on the optimal design.

Design in nonlinear mixed effects models: Optimization using the Fedorov–Wynn algorithm and power of the Wald test for binary covariates

We extend the methodology for designs evaluation and optimization in nonlinear mixed effects models with an illustration of the decrease of human immunodeficiency virus viral load after antiretroviral treatment initiation described by a bi-exponential model. We first show the relevance of the predicted standard errors (SEs) given by the computation of the population Fisher information matrix using the R function PFIM, in comparison to those computed with the stochastic approximation expectation-maximization algorithm, implemented in the Monolix software. We then highlight the usefulness of the Fedorov-Wynn (FW) algorithm for designs optimization compared to the Simplex algorithm. From the predicted SE of PFIM, we compute the predicted power of the Wald test to detect a treatment effect as well as the number of subjects needed to achieve a given power. Using the FW algorithm, we investigate the influence of the design on the power and show that, for optimized designs with the same total number of samples, the power increases when the number of subjects increases and the number of samples per subject decreases. A simulation study is also performed with the nlme function of R to confirm this result and show the relevance of the predicted powers compared to those observed by simulation.

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

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

The use of a modified Fedorov exchange algorithm to optimise sampling times for population pharmacokinetic experiments

We propose a new algorithm for optimising sampling times for population pharmacokinetic experiments using D-optimality. The algorithm was used in conjunction with the population Fisher information matrix as implemented in MATLAB (PFIM 1.1 and 1.2) to evaluate population pharmacokinetic designs. The new algorithm based on the classical Fedorov exchange algorithm optimises the determinant of the population Fisher information matrix. The performance of the new algorithm has been compared with other existing algorithms including simplex, simulated annealing and adaptive random search. The new algorithm performed better especially when dealing with complex designs at the expense of longer computing times.

A pragmatic approach to the design of population pharmacokinetic studies

The publication of a seminal article on nonlinear mixed-effect modeling led to a revolution in pharmacokinetics (PKs) with the introduction of the population approach. Since then, interest in obtaining accurate and precise estimates of population PK parameters has led to work on population PK study design that extended previous work on optimal sampling designs for individual PK parameter estimation. The issues and developments in the design of population PK studies are reviewed as a prelude to investigating, via simulation, the performance of 2 approaches (population Fisher information matrix D-optimal design and informative block [profile] randomized [IBR] design) for designing population PK studies. The results of our simulation study indicate that the designs based on the 2 approaches yielded efficient parameter estimates. The designs based on the 2 approaches performed similarly, and in some cases designs based on the IBR approach were slightly better. The ease with which the IBR designs can be generated makes them preferable in drug development, where pragmatism and time are of great consideration. We, therefore, refer to the IBR designs as pragmatic designs. Pragmatic designs that achieve high efficiency in the estimation parameters should be used in the design of population PK studies, and simulation should be used to determine the efficiency of the designs.

The use of simulated annealing for finding optimal population designs

The development of functions for MATLAB and S-PLUS that can be used for the evaluation of specific population pharmacokinetic designs has been described recently. These functions are based on the evaluation of an approximation of the population Fisher information matrix. Optimisation of the design of the population experiment can be made on the basis of D-optimal design techniques, where the determinant of the population Fisher information matrix is maximised. This maximisation is complex due to the convoluted nature of the surface of the determinant. Four optimisation algorithms (simplex, non-adaptive random search, non-adaptive random search followed by simplex and simulated annealing) are compared in their ability to optimise the sampling times for various design structures for three examples of population pharmacokinetic models. In all cases, despite more computing time, simulated annealing was superior to the other methods for finding optimal designs with greater benefits being seen over the other algorithms for the more complex designs.

Development and implementation of the population Fisher information matrix for the evaluation of population pharmacokinetic designs

In population pharmacokinetic studies, the precision of parameter estimates is dependent on the population design. Methods based on the Fisher information matrix have been developed and extended to population studies to evaluate and optimize designs. In this paper we propose simple programming tools to evaluate population pharmacokinetic designs. This involved the development of an expression for the Fisher information matrix for nonlinear mixed-effects models, including estimation of the variance of the residual error. We implemented this expression as a generic function for two software applications: S-PLUS and MATLAB. The evaluation of population designs based on two pharmacokinetic examples from the literature is shown to illustrate the efficiency and the simplicity of this theoretic approach. Although no optimization method of the design is provided, these functions can be used to select and compare population designs among a large set of possible designs, avoiding a lot of simulations.

Optimal design of a population pharmacodynamic experiment for ivabradine.

To design a parsimonious population pharmacodynamic experiment that has the same or greater efficiency than that provided by two phase I studies. The design was based on optimization of the population Fisher information matrix. Options for optimization were (1) determination of the optimal sampling times for each group ("group" represents a group of subjects that have identical design characteristics), (2) determination of the optimal doses for each group, and (3) determination of the optimal group structure. (1) Optimizing the sampling times, while retaining only four unique times per group, provided a more parsimonious experiment with the same efficiency as the original "study" that involved on average 10 samples per subject. Splitting sampling times between the first dose and a steady-state dose gave the most informative design. (2) The optimal dose was the same in all groups and was the upper bound of the dose range. (3) The optimal population design consisted of only one group with four unique sampling times that are the same for all subjects. A population pharmacodynamic trial design is presented that is more parsimonious than the original study and would be appropriate for inclusion in a premarketing clinical study.

Robust optimal design for the estimation of hyperparameters in population pharmacokinetics.

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