Mats O. Karlsson

Prediction-corrected visual predictive checks for diagnosing nonlinear mixed-effects models.

Informative diagnostic tools are vital to the development of useful mixed-effects models. The Visual Predictive Check (VPC) is a popular tool for evaluating the performance of population PK and PKPD models. Ideally, a VPC will diagnose both the fixed and random effects in a mixed-effects model. In many cases, this can be done by comparing different percentiles of the observed data to percentiles of simulated data, generally grouped together within bins of an independent variable. However, the diagnostic value of a VPC can be hampered by binning across a large variability in dose and/or influential covariates. VPCs can also be misleading if applied to data following adaptive designs such as dose adjustments. The prediction-corrected VPC (pcVPC) offers a solution to these problems while retaining the visual interpretation of the traditional VPC. In a pcVPC, the variability coming from binning across independent variables is removed by normalizing the observed and simulated dependent variable based on the typical population prediction for the median independent variable in the bin. The principal benefit with the pcVPC has been explored by application to both simulated and real examples of PK and PKPD models. The investigated examples demonstrate that pcVPCs have an enhanced ability to diagnose model misspecification especially with respect to random effects models in a range of situations. The pcVPC was in contrast to traditional VPCs shown to be readily applicable to data from studies with a priori and/or a posteriori dose adaptations.

Diagnosing Model Diagnostics

Conclusions from clinical trial results that are derived from model‐based analyses rely on the model adequately describing the underlying system. The traditionally used diagnostics intended to provide information about model adequacy have seldom discussed shortcomings. Without an understanding of the properties of these diagnostics, development and use of new diagnostics, and additional information pertaining to the diagnostics, there is risk that adequate models will be rejected and inadequate models accepted. Thus, a diagnosis of available diagnostics is desirable.

Importance of shrinkage in empirical bayes estimates for diagnostics: problems and solutions

Empirical Bayes (“post hoc”) estimates (EBEs) of ηs provide modelers with diagnostics: the EBEs themselves, individual prediction (IPRED), and residual errors (individual weighted residual (IWRES)). When data are uninformative at the individual level, the EBE distribution will shrink towards zero (η-shrinkage, quantified as 1-SD(ηEBE)/ω), IPREDs towards the corresponding observations, and IWRES towards zero (ε-shrinkage, quantified as 1-SD(IWRES)). These diagnostics are widely used in pharmacokinetic (PK) pharmacodynamic (PD) modeling; we investigate here their usefulness in the presence of shrinkage. Datasets were simulated from a range of PK PD models, EBEs estimated in non-linear mixed effects modeling based on the true or a misspecified model, and desired diagnostics evaluated both qualitatively and quantitatively. Identified consequences of η-shrinkage on EBE-based model diagnostics include non-normal and/or asymmetric distribution of EBEs with their mean values (“ETABAR”) significantly different from zero, even for a correctly specified model; EBE–EBE correlations and covariate relationships may be masked, falsely induced, or the shape of the true relationship distorted. Consequences of ε-shrinkage included low power of IPRED and IWRES to diagnose structural and residual error model misspecification, respectively. EBE-based diagnostics should be interpreted with caution whenever substantial η- or ε-shrinkage exists (usually greater than 20% to 30%). Reporting the magnitude of η- and ε-shrinkage will facilitate the informed use and interpretation of EBE-based diagnostics.

Implementation of a transit compartment model for describing drug absorption in pharmacokinetic studies

Purpose: To compare the performance of the standard lag time model (LAG model) with the performance of an analytical solution of the transit compartment model (TRANSIT model) in the evaluation of four pharmacokinetic studies with four different compounds. Methods: The population pharmacokinetic analyses were performed using NONMEM on concentration–time data of glibenclamide, furosemide, amiloride, and moxonidine. In the TRANSIT model, the optimal number of transit compartments was estimated from the data. This was based on an analytical solution for the change in drug concentration arising from a series of transit compartments with the same first-order transfer rate between each compartment. Goodness-of-fit was assessed by the decrease in objective function value (OFV) and by inspection of diagnostic graphs. Results: With the TRANSIT model, the OFV was significantly lower and the goodness-of-fit was markedly improved in the absorption phase compared with the LAG model for all drugs. The parameter estimates related to the absorption differed between the two models while the estimates of the pharmacokinetic disposition parameters were similar. Conclusion: Based on these results, the TRANSIT model is an attractive alternative for modeling drug absorption delay, especially when a LAG model poorly describes the drug absorption phase or is numerically unstable.

Pharmacokinetics and Safety of Intravenous Voriconazole in Children after Single- or Multiple-Dose Administration

ABSTRACT We conducted a multicenter study of the safety, tolerability, and plasma pharmacokinetics of the parenteral formulation of voriconazole in immunocompromised pediatric patients (2 to 11 years old). Single doses of 3 or 4 mg/kg of body weight were administered to six and five children, respectively. In the multiple-dose study, 28 patients received loading doses of 6 mg/kg every 12 h on day 1, followed by 3 mg/kg every 12 h on day 2 to day 4 and 4 mg/kg every 12 h on day 4 to day 8. Standard population pharmacokinetic approaches and generalized additive modeling were used to construct the structural pharmacokinetic and covariate models used in this analysis. In contrast to that in adult healthy volunteers, elimination of voriconazole was linear in children following doses of 3 and 4 mg/kg every 12 h. Body weight was more influential than age in accounting for the observed variability in voriconazole pharmacokinetics. Elimination capacity correlated with the CYP2C19 genotype. Exposures were similar at 4 mg/kg every 12 h in children (median area under the concentration-time curve (AUC), 14,227 ng · h/ml) and 3 mg/kg in adults (median AUC, 13,855 ng · h/ml). Visual disturbances occurred in 5 (12.8%) of the 39 patients and were the only drug-related adverse events that occurred more than once. No withdrawals from the study were related to voriconazole. We conclude that pediatric patients have a higher capacity for elimination of voriconazole per kilogram of body weight than do adult healthy volunteers and that dosages of 4 mg/kg may be required in children to achieve exposures consistent with those in adults following dosages of 3 mg/kg.

Assessment of Actual Significance Levels for Covariate Effects in NONMEM

The objectives of this study were to assess the difference between actual and nominal significance levels, as judged by the likelihood ratio test, for hypothesis tests regarding covariate effects using NONMEM, and to study what factors influence these levels. Also, a strategy for obtaining closer agreement between nominal and actual significance levels was investigated. Pharmacokinetic (PK) data without covariate relationships were simulated from a one compartment iv bolus model for 50 individuals. Models with and without covariate relationships were then fitted to the data, and differences in the objective function values were calculated. Alterations were made to the simulation settings; the structural and error models, the number of individuals, the number of samples per individual and the covariate distribution. Different estimation methods in NONMEM were also tried. In addition, a strategy for estimating the actual significance levels for a specific data set, model and parameter was investigated using covariate randomization and a real data set. Under most conditions when the first-order (FO) method was used, the actual significance level for including a covariate relationship in a model was higher than the nominal significance level. Among factors with high impact were frequency of sampling and residual error magnitude. The use of the first-order conditional estimation method with interaction (FOCE-INTER) resulted in close agreement between actual and nominal significance levels. The results from the covariate randomization procedure of the real data set were in agreement with the results from the simulation study. With the FO method the actual significance levels were higher than the nominal, independent of the covariate type, but depending on the parameter influenced. When using FOCE-INTER the actual and nominal levels were similar. The most important factors influencing the actual significance levels for the FO method are the approximation of the influence of the random effects in a nonlinear model, a heteroscedastic error structure in which an existing interaction between interindividual and residual variability is not accounted for in the model, and a lognormal distribution of the residual error which is approximated by a symmetric distribution. Estimation with FOCE–INTER and the covariate randomization procedure provide means to achieve agreement between nominal and actual significance levels.

Conditional Weighted Residuals (CWRES): A Model Diagnostic for the FOCE Method

PurposePopulation model analyses have shifted from using the first order (FO) to the first-order with conditional estimation (FOCE) approximation to the true model. However, the weighted residuals (WRES), a common diagnostic tool used to test for model misspecification, are calculated using the FO approximation. Utilizing WRES with the FOCE method may lead to misguided model development/evaluation. We present a new diagnostic tool, the conditional weighted residuals (CWRES), which are calculated based on the FOCE approximation.Materials and MethodsCWRES are calculated as the FOCE approximated difference between an individual’s data and the model prediction of that data divided by the root of the covariance of the data given the model.ResultsUsing real and simulated data the CWRES distributions behave as theoretically expected under the correct model. In contrast, in certain circumstances, the WRES have distributions that greatly deviate from the expected, falsely indicating model misspecification. CWRES/WRES comparisons can also indicate if the FOCE estimation method will improve the results of an FO model fit to data.ConclusionsUtilization of CWRES could improve model development and evaluation and give a more accurate picture of if and when a model is misspecified when using the FO or FOCE methods.

Modeling and Simulation Workbench for NONMEM: Tutorial on Pirana, PsN, and Xpose

Several software tools are available that facilitate the use of the NONMEM software and extend its functionality. This tutorial shows how three commonly used and freely available tools, Pirana, PsN, and Xpose, form a tightly integrated workbench for modeling and simulation with NONMEM. During the tutorial, we provide some guidance on what diagnostics we consider most useful in pharmacokinetic model development and how to construct them using these tools.

Likelihood based approaches to handling data below the quantification limit using NONMEM VI

Purpose To evaluate the likelihood-based methods for handling data below the quantification limit (BQL) using new features in NONMEM VI. Methods A two-compartment pharmacokinetic model with first-order absorption was chosen for investigation. Methods evaluated were: discarding BQL observations (M1), discarding BQL observations but adjusting the likelihood for the remaining data (M2), maximizing the likelihood for the data above the limit of quantification (LOQ) and treating BQL data as censored (M3), and like M3 but conditioning on the observation being greater than zero (M4). These four methods were compared using data simulated with a proportional error model. M2, M3, and M4 were also compared using data simulated from a positively truncated normal distribution. Successful terminations and bias and precision of parameter estimates were assessed. Results For the data simulated with a proportional error model, the overall performance was best for M3 followed by M2 and M1. M3 and M4 resulted in similar estimates in analyses without log transformation. For data simulated with the truncated normal distribution, M4 performed better than M3. Conclusions Analyses that maximized the likelihood of the data above the LOQ and treated BQL data as censored provided the most accurate and precise parameter estimates.

Automated covariate model building within NONMEM.

AbstractPurpose. One important task in population pharmacokinetic/pharmacodynamic model building is to identify the relationships between the parameters and demographic factors (covariates). The purpose of this study is to present an automated procedure that accomplishes this. The benefits of the proposed procedure over other commonly used methods are (i) the covariate model is built for all parameters simultaneously, (ii) the covariate model is built within the population modeling program (NONMEM) giving familiar meaning to the significance levels used, (iii) it can appropriately handle covariates that varies over time and (iv) it is not dependent on the quality of the posterior Bayes estimates of the individual parameter values. For situations in which the computer run-times are a limiting factor, a linearization of the non-linear mixed effects model is proposed and evaluated. Methods. The covariate model is built in a stepwise fashion in which both linear and non-linear relationships between the parameters and covariates are considered. The linearization is basically a linear mixed effects model in which the population predictions and their derivatives with respect to the parameters are fixed from a model without covariates. The stepwise procedure as well as the linearization was evaluated using simulations in which the covariates were taken from a real data set. Results. The covariate models identified agreed well with what could be expected based on the covariates that were actually supported in each of the simulated data sets. The predictive performance of the linearized model was close to that of the non-linearized model. Conclusions. The proposed procedure identifies covariate models that are close to the model supported by the data set as well as being useful in the prediction of new data. The linearized model performs nearly as well as the non-linearized model.