Christoffer Wenzel Tornøe

Non-Linear Mixed-Effects Models with Stochastic Differential Equations - Implementation of an Estimation Algorithm

Pharmacokinetic/pharmacodynamic modelling is most often performed using non-linear mixed-effects models based on ordinary differential equations with uncorrelated intra-individual residuals. More sophisticated residual error models as e.g. stochastic differential equations (SDEs) with measurement noise can in many cases provide a better description of the variations, which could be useful in various aspects of modelling. This general approach enables a decomposition of the intra-individual residual variation ε into system noise w and measurement noise e. The present work describes implementation of SDEs in a non-linear mixed-effects model, where parameter estimation was performed by a novel approximation of the likelihood function. This approximation is constructed by combining the First-Order Conditional Estimation (FOCE) method used in non-linear mixed-effects modelling with the Extended Kalman Filter used in models with SDEs. Fundamental issues concerning the proposed model and estimation algorithm are addressed by simulation studies, concluding that system noise can successfully be separated from measurement noise and inter-individual variability.

Stochastic differential equations in NONMEM: implementation, application, and comparison with ordinary differential equations.

PurposeThe objective of the present analysis was to explore the use of stochastic differential equations (SDEs) in population pharmacokinetic/pharmacodynamic (PK/PD) modeling.MethodsThe intra-individual variability in nonlinear mixed-effects models based on SDEs is decomposed into two types of noise: a measurement and a system noise term. The measurement noise represents uncorrelated error due to, for example, assay error while the system noise accounts for structural misspecifications, approximations of the dynamical model, and true random physiological fluctuations. Since the system noise accounts for model misspecifications, the SDEs provide a diagnostic tool for model appropriateness. The focus of the article is on the implementation of the Extended Kalman Filter (EKF) in NONMEM® for parameter estimation in SDE models.ResultsVarious applications of SDEs in population PK/PD modeling are illustrated through a systematic model development example using clinical PK data of the gonadotropin releasing hormone (GnRH) antagonist degarelix. The dynamic noise estimates were used to track variations in model parameters and systematically build an absorption model for subcutaneously administered degarelix.ConclusionsThe EKF-based algorithm was successfully implemented in NONMEM for parameter estimation in population PK/PD models described by systems of SDEs. The example indicated that it was possible to pinpoint structural model deficiencies, and that valuable information may be obtained by tracking unexplained variations in parameters.

Creation of a Knowledge Management System for QT Analyses

An increasing number of thorough QT (TQT) reports are being submitted to the US Food and Drug Administrations interdisciplinary review team for QT (IRT‐QT), requiring time‐intensive quantitative analyses by a multidisciplinary review team within 45 days. This calls for systematic learning to guide future trials and policies by standardizing and automating the QT analyses to improve review efficiency, provide consistent advice, and enable pooled data analyses to answer key regulatory questions. The QT interval represents the time from initiation of ventricular depolarization to completion of ventricular repolarization recorded by electrocardiograph (ECG) and is used in the proarrhythmic risk assessment. The developed QT knowledge management system is implemented in the R package “QT.” Data from 11 crossover TQT studies including time‐matched ECGs and pharmacokinetic measurements following single doses of 400 to 1200 mg moxifloxacin were used for the QT analysis example. The automated workflow was divided into 3 components (data management, analysis, and archival). The generated data sets, scripts, tables, and graphs are automatically stored in a queryable repository and summarized in an analysis report. More than 100 TQT studies have been analyzed using the system since 2007. This has dramatically reduced the time needed to review TQT studies and has made the IRT‐QT reviews consistent across reviewers. Furthermore, the system enables leveraging prior knowledge through pooled data analyses to answer policy‐related questions and to understand the various effects that influence study results.

Non-linear mixed-effects pharmacokinetic/pharmacodynamic modelling in NLME using differential equations

The standard software for non-linear mixed-effect analysis of pharmacokinetic/pharmacodynamic (PK/PD) data is NONMEM while the non-linear mixed-effects package NLME is an alternative as long as the models are fairly simple. We present the nlmeODE package which combines the ordinary differential equation (ODE) solver package odesolve and the non-linear mixed effects package NLME thereby enabling the analysis of complicated systems of ODEs by non-linear mixed-effects modelling. The pharmacokinetics of the anti-asthmatic drug theophylline is used to illustrate the applicability of the nlmeODE package for population PK/PD analysis using the available data analysis tools in R for model inspection and validation. The nlmeODE package is numerically stable and provides accurate parameter estimates which are consistent with NONMEM estimates.

Grey-box Modelling of Pharmacokinetic /Pharmacodynamic Systems

Grey-box pharmacokinetic/pharmacodynamic (PK/PD) modelling is presented as a promising way of modelling PK/PD systems. The concept behind grey-box modelling is based on combining physiological knowledge along with information from data in the estimation of model parameters. Grey-box modelling consists of using stochastic differential equations (SDEs) where the stochastic term in the differential equations represents unknown or incorrectly modelled dynamics of the system. The methodology behind the grey-box PK/PD modelling framework for systematic model improvement is illustrated using simulated data and furthermore applied to Bergman’s minimal model of glucose kinetics using clinical data from an intravenous glucose tolerance test (IVGTT). The grey-box estimates of the stochastic system noise parameters indicate that the glucose minimal model is too simple and should preferably be revised in order to describe the complicated in vivo system of insulin and glucose following an IVGTT.

Pharmacokinetic/Pharmacodynamic Modelling of GnRH Antagonist Degarelix: A Comparison of the Non-linear Mixed-Effects Programs NONMEM and NLME

In this paper, the two non-linear mixed-effects programs NONMEM and NLME were compared for their use in population pharmacokinetic/pharmacodynamic (PK/PD) modelling. We have described the first-order conditional estimation (FOCE) method as implemented in NONMEM and the alternating algorithm in NLME proposed by Lindstrom and Bates. The two programs were tested using clinical PK/PD data of a new gonadotropin-releasing hormone (GnRH) antagonist degarelix currently being developed for prostate cancer treatment. The pharmacokinetics of intravenous administered degarelix was analysed using a three compartment model while the pharmacodynamics was analysed using a turnover model with a pool compartment. The results indicated that the two algorithms produce consistent parameter estimates. The bias and precision of the two algorithms were further investigated using a parametric bootstrap procedure which showed that NONMEM produced more accurate results than NLME together with the nlmeODE package for this specific study.

Esomeprazole FDA Approval in Children With GERD: Exposure-Matching and Exposure-Response.

Objectives: Food and Drug Administration approval of proton-pump inhibitors for infantile gastroesophageal reflux disease has been limited by intrapatient variability in the clinical assessment of gastroesophageal reflux disease. For children 1 to 17 years old, extrapolating efficacy from adults for IV esomeprazole was accepted. The oral formulation was previously approved in children. Exposure-response and exposure matching analyses were sought to identify approvable pediatric doses. Methods: Intragastric pH biomarker comparisons between children and adults were conducted. Pediatric doses were selected to match exposures in adults and were based on population pharmacokinetic (PK) modeling and simulations with pediatric esomeprazole data. Observed IV or oral esomeprazole PK data were available from 50 and 117 children, between birth and 17 years, respectively, and from 65 adults, between 20 and 48 years. A population PK model developed using these data was used to simulate steady-state esomeprazole exposures for children at different doses to match the observed exposures in adults. Results: Exposure-response relationships of intragastric pH measures were similar between children and adults. The PK simulations identified a dosing regimen for children that results in comparable steady-state area under the curve to that observed after 20 mg in adults. For IV esomeprazole, increasing the infusion duration to 10 to 30 minutes in children achieves matching Cmax values with adults. Conclusions: The exposure-matching analysis permitted approval of an esomeprazole regimen not studied directly in clinical trials. Exposure-response for intragastric pH-permitted approval for the treatment of gastroesophageal reflux disease in children in whom it was not possible to evaluate the adult primary endpoint, mucosal healing assessed by endoscopy.