Identifiability of PBPK models with applications to dimethylarsinic acid exposure

Parameter grouping and co-estimation in physiologically based kinetic models using genetic algorithms

Physiologically based kinetic (PBK) models are widely used in pharmacology and toxicology for predicting the internal disposition of substances upon exposure, voluntarily or not. Due to their complexity, a large number of model parameters need to be estimated, either through in silico tools, in vitro experiments, or by fitting the model to in vivo data. In the latter case, fitting complex structural models on in vivo data can result in overparameterization and produce unrealistic parameter estimates. To address these issues, we propose a novel parameter grouping approach, which reduces the parametric space by co-estimating groups of parameters across compartments. Grouping of parameters is performed using genetic algorithms and is fully automated, based on a novel goodness-of-fit metric. To illustrate the practical application of the proposed methodology, two case studies were conducted. The first case study demonstrates the development of a new PBK model, while the second focuses on model refinement. In the first case study, a PBK model was developed to elucidate the biodistribution of titanium dioxide (TiO2) nanoparticles in rats following intravenous injection. A variety of parameter estimation schemes were employed. Comparative analysis based on goodness-of-fit metrics demonstrated that the proposed methodology yields models that outperform standard estimation approaches, while utilizing a reduced number of parameters. In the second case study, an existing PBK model for perfluorooctanoic acid (PFOA) in rats was extended to incorporate additional tissues, providing a more comprehensive portrayal of PFOA biodistribution. Both models were validated through independent in vivo studies to ensure their reliability.

Mechanistic Modeling of In Vitro Skin Permeation and Extrapolation to In Vivo for Topically Applied Metronidazole Drug Products Using a Physiologically Based Pharmacokinetic Model

Physiologically based pharmacokinetic (PBPK) modeling has increasingly been employed in dermal drug development and regulatory assessment, providing a framework to integrate relevant information including drug and drug product attributes, skin physiology parameters, and population variability. The current study aimed to develop a stepwise modeling workflow with knowledge gained from modeling in vitro skin permeation testing (IVPT) to describe in vivo exposure of metronidazole locally in the stratum corneum following topical application of complex semisolid drug products. The initial PBPK model of metronidazole in vitro skin permeation was developed using infinite and finite dose aqueous metronidazole solution. Parameters such as stratum corneum lipid-water partition coefficient (K_sclip/water) and stratum corneum lipid diffusion coefficient (D_sclip) of metronidazole were optimized using IVPT data from simple aqueous solutions (infinite) and MetroGel (10 mg/cm² dose application), respectively. The optimized model, when parameterized with physical and structural characteristics of the drug products, was able to accurately predict the mean cumulative amount permeated (cm²/h) and flux (μg/cm²/h) profiles of metronidazole following application of different doses of MetroGel and MetroCream. Thus, the model was able to capture the impact of differences in drug product microstructure and metamorphosis of the dosage form on in vitro metronidazole permeation. The PBPK model informed by IVPT study data was able to predict the metronidazole amount in the stratum corneum as reported in clinical studies. In summary, the proposed model provides an enhanced understanding of the potential impact of drug product attributes in influencing in vitro skin permeation of metronidazole. Key kinetic parameters derived from modeling the metronidazole IVPT data improved the predictions of the developed PBPK model of in vivo local metronidazole concentrations in the stratum corneum. Overall, this work improves our confidence in the proposed workflow that accounts for drug product attributes and utilizes IVPT data toward improving predictions from advanced modeling and simulation tools.

Quantitative Characterization of Population-Wide Tissue- and Metabolite-Specific Variability in Perchloroethylene Toxicokinetics in Male Mice

Quantification of interindividual variability is a continuing challenge in risk assessment, particularly for compounds with complex metabolism and multi-organ toxicity. Toxicokinetic variability for perchloroethylene (perc) was previously characterized across 3 mouse strains and in 1 mouse strain with various degrees of liver steatosis. To further characterize the role of genetic variability in toxicokinetics of perc, we applied Bayesian population physiologically based pharmacokinetic (PBPK) modeling to the data on perc and metabolites in blood/plasma and tissues of male mice from 45 inbred strains from the Collaborative Cross (CC) mouse population. After identifying the most influential PBPK parameters based on global sensitivity analysis, we fit the model with a hierarchical Bayesian population analysis using Markov chain Monte Carlo simulation. We found that the data from 3 commonly used strains were not representative of the full range of variability in perc and metabolite blood/plasma and tissue concentrations across the CC population. Using interstrain variability as a surrogate for human interindividual variability, we calculated dose-dependent, chemical-, and tissue-specific toxicokinetic variability factors (TKVFs) as candidate science-based replacements for the default uncertainty factor for human toxicokinetic variability of 100.5. We found that toxicokinetic variability factors for glutathione conjugation metabolites of perc showed the greatest variability, often exceeding the default, whereas those for oxidative metabolites and perc itself were generally less than the default. Overall, we demonstrate how a combination of a population-based mouse model such as the CC with Bayesian population PBPK modeling can reduce uncertainty in human toxicokinetic variability and increase accuracy and precision in quantitative risk assessment.

Physiologically-Based Pharmacokinetic Modeling to Support Determination of Bioequivalence for Dermatological Drug Products: Scientific and Regulatory Considerations

Physiologically-based pharmacokinetic (PBPK) modeling and simulation provides mechanism-based predictions of the pharmacokinetics of an active ingredient following its administration in humans. Dermal PBPK models describe the skin permeation and disposition of the active ingredient following the application of a dermatological product on the skin of virtual healthy and diseased human subjects. These models take into account information on product quality attributes, physicochemical properties of the active ingredient and skin (patho)physiology, and their interplay with each other. Regulatory and product development decision makers can leverage these quantitative tools to identify factors impacting local and systemic exposure. In the realm of generic drug products, the number of US Food and Drug Administratioin (FDA) interactions that use dermal PBPK modeling to support alternative bioequivalence (BE) approaches is increasing. In this report, we share scientific considerations on the development, verification and validation (V&V), and application of PBPK models within the context of a virtual BE assessment for dermatological drug products. We discuss the challenges associated with model V&V for these drug products stemming from the fact that target-site active ingredient concentrations are typically not measurable. Additionally, there are no established relationships between local and systemic PK profiles, when the latter are quantifiable. To that end, we detail a multilevel model V&V approach involving validation for the model of the drug product of interest coupled with the overall assessment of the modeling platform in use while leveraging in vitro and in vivo data related to local and systemic bioavailability.

pksensi: An R package to apply global sensitivity analysis in physiologically based kinetic modeling

Sensitivity analysis (SA) is an essential tool for modelers to understand the influence of model parameters on model outputs. It is also increasingly used in developing and assessing physiologically based kinetic (PBK) models. For instance, several studies have applied global SA to reduce the computational burden in the Bayesian Markov chain Monte Carlo-based calibration process PBK models. Although several SA algorithms and software packages are available, no comprehensive software package exists that allows users to seamlessly solve differential equations in a PBK model, conduct and visualize SA results, and discriminate between the non-influential model parameters that can be fixed and those that need calibration. Therefore, we developed an R package, named pksensi, to make global SA more accessible in PBK modeling. This package can investigate both uncertainty and sensitivity in PBK models, including those with multivariate model outputs. It also includes functions to check the convergence of the global SA results. Overall, pksensi improves the user experience of performing global SA and can create robust and reproducible results for decision making in PBK model calibration.

PBPK model reporting template for chemical risk assessment applications

Physiologically-based pharmacokinetic (PBPK) modeling analysis does not stand on its own for regulatory purposes but is a robust tool to support drug/chemical safety assessment. While the development of PBPK models have grown steadily since their emergence, only a handful of models have been accepted to support regulatory purposes due to obstacles such as the lack of a standardized template for reporting PBPK analysis. Here, we expand the existing guidances designed for pharmaceutical applications by recommending additional elements that are relevant to environmental chemicals. This harmonized reporting template can be adopted and customized by public health agencies receiving PBPK model submission, and it can also serve as general guidance for submitting PBPK-related studies for publication in journals or other modeling sharing purposes. The current effort represents one of several ongoing collaborations among the PBPK modeling and risk assessment communities to promote, when appropriate, incorporating PBPK modeling to characterize the influence of pharmacokinetics on safety decisions made by regulatory agencies.

Assessing Toxicokinetic Uncertainty and Variability in Risk Prioritization

High(er) throughput toxicokinetics (HTTK) encompasses in vitro measures of key determinants of chemical toxicokinetics and reverse dosimetry approaches for in vitro-in vivo extrapolation (IVIVE). With HTTK, the bioactivity identified by any in vitro assay can be converted to human equivalent doses and compared with chemical intake estimates. Biological variability in HTTK has been previously considered, but the relative impact of measurement uncertainty has not. Bayesian methods were developed to provide chemical-specific uncertainty estimates for 2 in vitro toxicokinetic parameters: unbound fraction in plasma (fup) and intrinsic hepatic clearance (Clint). New experimental measurements of fup and Clint are reported for 418 and 467 chemicals, respectively. These data raise the HTTK chemical coverage of the ToxCast Phase I and II libraries to 57%. Although the standard protocol for Clint was followed, a revised protocol for fup measured unbound chemical at 10%, 30%, and 100% of physiologic plasma protein concentrations, allowing estimation of protein binding affinity. This protocol reduced the occurrence of chemicals with fup too low to measure from 44% to 9.1%. Uncertainty in fup was also reduced, with the median coefficient of variation dropping from 0.4 to 0.1. Monte Carlo simulation was used to propagate both measurement uncertainty and biological variability into IVIVE. The uncertainty propagation techniques used here also allow incorporation of other sources of uncertainty such as in silico predictors of HTTK parameters. These methods have the potential to inform risk-based prioritization based on the relationship between in vitro bioactivities and exposures.

Better prediction of the local concentration-effect relationship: the role of physiologically based pharmacokinetics and quantitative systems pharmacology and toxicology in the evolution of model-informed drug discovery and development

Model-informed drug discovery and development (MID3) is an umbrella term under which sit several computational approaches: quantitative systems pharmacology (QSP), quantitative systems toxicology (QST) and physiologically based pharmacokinetics (PBPK). QSP models are built using mechanistic knowledge of the pharmacological pathway focusing on the putative mechanism of drug efficacy; whereas QST models focus on safety and toxicity issues and the molecular pathways and networks that drive these adverse effects. These can be mediated through exaggerated on-target or off-target pharmacology, immunogenicity or the physiochemical nature of the compound. PBPK models provide a mechanistic description of individual organs and tissues to allow the prediction of the intra- and extra-cellular concentration of the parent drug and metabolites under different conditions. Information on biophase concentration enables the prediction of a drug effect in different organs and assessment of the potential for drug-drug interactions. Together, these modelling approaches can inform the exposure-response relationship and hence support hypothesis generation and testing, compound selection, hazard identification and risk assessment through to clinical proof of concept (POC) and beyond to the market.

Population pharmacokinetic reanalysis of a Diazepam PBPK model: a comparison of Stan and GNU MCSim

The aim of this study is to benchmark two Bayesian software tools, namely Stan and GNU MCSim, that use different Markov chain Monte Carlo (MCMC) methods for the estimation of physiologically based pharmacokinetic (PBPK) model parameters. The software tools were applied and compared on the problem of updating the parameters of a Diazepam PBPK model, using time-concentration human data. Both tools produced very good fits at the individual and population levels, despite the fact that GNU MCSim is not able to consider multivariate distributions. Stan outperformed GNU MCSim in sampling efficiency, due to its almost uncorrelated sampling. However, GNU MCSim exhibited much faster convergence and performed better in terms of effective samples produced per unit of time.

Applying a Global Sensitivity Analysis Workflow to Improve the Computational Efficiencies in Physiologically-Based Pharmacokinetic Modeling

Traditionally, the solution to reduce parameter dimensionality in a physiologically-based pharmacokinetic (PBPK) model is through expert judgment. However, this approach may lead to bias in parameter estimates and model predictions if important parameters are fixed at uncertain or inappropriate values. The purpose of this study was to explore the application of global sensitivity analysis (GSA) to ascertain which parameters in the PBPK model are non-influential, and therefore can be assigned fixed values in Bayesian parameter estimation with minimal bias. We compared the elementary effect-based Morris method and three variance-based Sobol indices in their ability to distinguish “influential” parameters to be estimated and “non-influential” parameters to be fixed. We illustrated this approach using a published human PBPK model for acetaminophen (APAP) and its two primary metabolites APAP-glucuronide and APAP-sulfate. We first applied GSA to the original published model, comparing Bayesian model calibration results using all the 21 originally calibrated model parameters (OMP, determined by “expert judgment”-based approach) vs. the subset of original influential parameters (OIP, determined by GSA from the OMP). We then applied GSA to all the PBPK parameters, including those fixed in the published model, comparing the model calibration results using this full set of 58 model parameters (FMP) vs. the full set influential parameters (FIP, determined by GSA from FMP). We also examined the impact of different cut-off points to distinguish the influential and non-influential parameters. We found that Sobol indices calculated by eFAST provided the best combination of reliability (consistency with other variance-based methods) and efficiency (lowest computational cost to achieve convergence) in identifying influential parameters. We identified several originally calibrated parameters that were not influential, and could be fixed to improve computational efficiency without discernable changes in prediction accuracy or precision. We further found six previously fixed parameters that were actually influential to the model predictions. Adding these additional influential parameters improved the model performance beyond that of the original publication while maintaining similar computational efficiency. We conclude that GSA provides an objective, transparent, and reproducible approach to improve the performance and computational efficiency of PBPK models.

Evaluating In Vitro-In Vivo Extrapolation of Toxicokinetics

Prioritizing the risk posed by thousands of chemicals potentially present in the environment requires exposure, toxicity, and toxicokinetic (TK) data, which are often unavailable. Relatively high throughput, in vitro TK (HTTK) assays and in vitro-to-in vivo extrapolation (IVIVE) methods have been developed to predict TK, but most of the in vivo TK data available to benchmark these methods are from pharmaceuticals. Here we report on new, in vivo rat TK experiments for 26 non-pharmaceutical chemicals with environmental relevance. Both intravenous and oral dosing were used to calculate bioavailability. These chemicals, and an additional 19 chemicals (including some pharmaceuticals) from previously published in vivo rat studies, were systematically analyzed to estimate in vivo TK parameters (e.g., volume of distribution [Vd], elimination rate). For each of the chemicals, rat-specific HTTK data were available and key TK predictions were examined: oral bioavailability, clearance, Vd, and uncertainty. For the non-pharmaceutical chemicals, predictions for bioavailability were not effective. While no pharmaceutical was absorbed at less than 10%, the fraction bioavailable for non-pharmaceutical chemicals was as low as 0.3%. Total clearance was generally more under-estimated for nonpharmaceuticals and Vd methods calibrated to pharmaceuticals may not be appropriate for other chemicals. However, the steady-state, peak, and time-integrated plasma concentrations of nonpharmaceuticals were predicted with reasonable accuracy. The plasma concentration predictions improved when experimental measurements of bioavailability were incorporated. In summary, HTTK and IVIVE methods are adequately robust to be applied to high throughput in vitro toxicity screening data of environmentally relevant chemicals for prioritizing based on human health risks.

Physiologically Based Pharmacokinetic Model Qualification and Reporting Procedures for Regulatory Submissions: A Consortium Perspective

This work provides a perspective on the qualification and verification of physiologically based pharmacokinetic (PBPK) platforms/models intended for regulatory submission based on the collective experience of the Simcyp Consortium members. Examples of regulatory submission of PBPK analyses across various intended applications are presented and discussed. European Medicines Agency (EMA) and US Food and Drug Administration (FDA) recent draft guidelines regarding PBPK analyses and reporting are encouraging, and to advance the use and acceptability of PBPK analyses, more clarity and flexibility are warranted.

Evaluation and calibration of high-throughput predictions of chemical distribution to tissues

Toxicokinetics (TK) provides critical information for integrating chemical toxicity and exposure assessments in order to determine potential chemical risk (i.e., the margin between toxic doses and plausible exposures). For thousands of chemicals that are present in our environment, in vivo TK data are lacking. The publicly available R package "httk" (version 1.8, named for "high throughput TK") draws from a database of in vitro data and physico-chemical properties in order to run physiologically-based TK (PBTK) models for 553 compounds. The PBTK model parameters include tissue:plasma partition coefficients (K_p) which the httk software predicts using the model of Schmitt (Toxicol In Vitro 22 (2):457-467, 2008). In this paper we evaluated and modified httk predictions, and quantified confidence using in vivo literature data. We used 964 rat K_p measured by in vivo experiments for 143 compounds. Initially, predicted K_p were significantly larger than measured K_p for many lipophilic compounds (log₁₀ octanol:water partition coefficient > 3). Hence the approach for predicting K_p was revised to account for possible deficiencies in the in vitro protein binding assay, and the method for predicting membrane affinity was revised. These changes yielded improvements ranging from a factor of 10 to nearly a factor of 10,000 for 83 K_p across 23 compounds with only 3 K_p worsening by more than a factor of 10. The vast majority (92%) of K_p were predicted within a factor of 10 of the measured value (overall root mean squared error of 0.59 on log₁₀-transformed scale). After applying the adjustments, regressions were performed to calibrate and evaluate the predictions for 12 tissues. Predictions for some tissues (e.g., spleen, bone, gut, lung) were observed to be better than predictions for other tissues (e.g., skin, brain, fat), indicating that confidence in the application of in silico tools to predict chemical partitioning varies depending upon the tissues involved. Our calibrated model was then evaluated using a second data set of human in vivo measurements of volume of distribution (V_ss) for 498 compounds reviewed by Obach et al. (Drug Metab Dispos 36(7):1385-1405, 2008). We found that calibration of the model improved performance: a regression of the measured values as a function of the predictions has a slope of 1.03, intercept of - 0.04, and R² of 0.43. Through careful evaluation of predictive methods for chemical partitioning into tissues, we have improved and calibrated these methods and quantified confidence for TK predictions in humans and rats.

Evaluation and calibration of high-throughput predictions of chemical distribution to tissues

Toxicokinetics (TK) provides critical information for integrating chemical toxicity and exposure assessments in order to determine potential chemical risk (i.e., the margin between toxic doses and plausible exposures). For thousands of chemicals that are present in our environment, in vivo TK data are lacking. The publicly available R package "httk" (version 1.8, named for "high throughput TK") draws from a database of in vitro data and physico-chemical properties in order to run physiologically-based TK (PBTK) models for 553 compounds. The PBTK model parameters include tissue:plasma partition coefficients (Kp) which the httk software predicts using the model of Schmitt (Toxicol In Vitro 22 (2):457-467, 2008). In this paper we evaluated and modified httk predictions, and quantified confidence using in vivo literature data. We used 964 rat Kp measured by in vivo experiments for 143 compounds. Initially, predicted Kp were significantly larger than measured Kp for many lipophilic compounds (log10 octanol:water partition coefficient > 3). Hence the approach for predicting Kp was revised to account for possible deficiencies in the in vitro protein binding assay, and the method for predicting membrane affinity was revised. These changes yielded improvements ranging from a factor of 10 to nearly a factor of 10,000 for 83 Kp across 23 compounds with only 3 Kp worsening by more than a factor of 10. The vast majority (92%) of Kp were predicted within a factor of 10 of the measured value (overall root mean squared error of 0.59 on log10-transformed scale). After applying the adjustments, regressions were performed to calibrate and evaluate the predictions for 12 tissues. Predictions for some tissues (e.g., spleen, bone, gut, lung) were observed to be better than predictions for other tissues (e.g., skin, brain, fat), indicating that confidence in the application of in silico tools to predict chemical partitioning varies depending upon the tissues involved. Our calibrated model was then evaluated using a second data set of human in vivo measurements of volume of distribution (Vss) for 498 compounds reviewed by Obach et al. (Drug Metab Dispos 36(7):1385-1405, 2008). We found that calibration of the model improved performance: a regression of the measured values as a function of the predictions has a slope of 1.03, intercept of - 0.04, and R2 of 0.43. Through careful evaluation of predictive methods for chemical partitioning into tissues, we have improved and calibrated these methods and quantified confidence for TK predictions in humans and rats.

What do we mean by identifiability in mixed effects models?

We discuss the question of model identifiability within the context of nonlinear mixed effects models. Although there has been extensive research in the area of fixed effects models, much less attention has been paid to random effects models. In this context we distinguish between theoretical identifiability, in which different parameter values lead to non-identical probability distributions, structural identifiability which concerns the algebraic properties of the structural model, and practical identifiability, whereby the model may be theoretically identifiable but the design of the experiment may make parameter estimation difficult and imprecise. We explore a number of pharmacokinetic models which are known to be non-identifiable at an individual level but can become identifiable at the population level if a number of specific assumptions on the probabilistic model hold. Essentially if the probabilistic models are different, even though the structural models are non-identifiable, then they will lead to different likelihoods. The findings are supported through simulations.

Reduction of a Whole-Body Physiologically Based Pharmacokinetic Model to Stabilise the Bayesian Analysis of Clinical Data

Whole-body physiologically based pharmacokinetic (PBPK) models are increasingly used in drug development for their ability to predict drug concentrations in clinically relevant tissues and to extrapolate across species, experimental conditions and sub-populations. A whole-body PBPK model can be fitted to clinical data using a Bayesian population approach. However, the analysis might be time consuming and numerically unstable if prior information on the model parameters is too vague given the complexity of the system. We suggest an approach where (i) a whole-body PBPK model is formally reduced using a Bayesian proper lumping method to retain the mechanistic interpretation of the system and account for parameter uncertainty, (ii) the simplified model is fitted to clinical data using Markov Chain Monte Carlo techniques and (iii) the optimised reduced PBPK model is used for extrapolation. A previously developed 16-compartment whole-body PBPK model for mavoglurant was reduced to 7 compartments while preserving plasma concentration-time profiles (median and variance) and giving emphasis to the brain (target site) and the liver (elimination site). The reduced model was numerically more stable than the whole-body model for the Bayesian analysis of mavoglurant pharmacokinetic data in healthy adult volunteers. Finally, the reduced yet mechanistic model could easily be scaled from adults to children and predict mavoglurant pharmacokinetics in children aged from 3 to 11 years with similar performance compared with the whole-body model. This study is a first example of the practicality of formal reduction of complex mechanistic models for Bayesian inference in drug development.