Uncertainty, Variability, and Sensitivity Analysis in Physiological Pharmacokinetic Models

The Application of a Physiologically Based Toxicokinetic Model in Health Risk Assessment

Toxicokinetics plays a crucial role in the health risk assessments of xenobiotics. Classical compartmental models are limited in their ability to determine chemical concentrations in specific organs or tissues, particularly target organs or tissues, and their limited interspecific and exposure route extrapolation hinders satisfactory health risk assessment. In contrast, physiologically based toxicokinetic (PBTK) models quantitatively describe the absorption, distribution, metabolism, and excretion of chemicals across various exposure routes and doses in organisms, establishing correlations with toxic effects. Consequently, PBTK models serve as potent tools for extrapolation and provide a theoretical foundation for health risk assessment and management. This review outlines the construction and application of PBTK models in health risk assessment while analyzing their limitations and future perspectives.

Incorporation of stochastic variability in mechanistic population pharmacokinetic models: handling the physiological constraints using normal transformations

The utilisation of physiologically-based pharmacokinetic models for the analysis of population data is an approach with progressively increasing impact. However, as we move from empirical to complex mechanistic model structures, incorporation of stochastic variability in model parameters can be challenging due to the physiological constraints that may arise. Here, we investigated the most common types of constraints faced in mechanistic pharmacokinetic modelling and explored techniques for handling them during a population data analysis. An efficient way to impose stochastic variability on the parameters of interest without neglecting the underlying physiological constraints is through the assumption that they follow a distribution with support and properties matching the underlying physiology. It was found that two distributions that arise through transformations of the normal, the logit-normal generalisation and the logistic-normal, are excellent for such an application as not only they can satisfy the physiological constraints but also offer high flexibility during characterisation of the parameters' distribution. The statistical properties and practical advantages/disadvantages of these distributions for such an application were clearly displayed in the context of different modelling examples. Finally, a simulation study clearly illustrated the practical gains of the utilisation of the described techniques, as omission of population variability in physiological systems parameters leads to a biased/misplaced stochastic model with mechanistically incorrect variance structure. The current methodological work aims to facilitate the use of mechanistic/physiologically-based models for the analysis of population pharmacokinetic clinical data.

Quantifying uncertainty, variability and likelihood for ordinary differential equation models

Background

In many applications, ordinary differential equation (ODE) models are subject to uncertainty or variability in initial conditions and parameters. Both, uncertainty and variability can be quantified in terms of a probability density function on the state and parameter space.

Results

The partial differential equation that describes the evolution of this probability density function has a form that is particularly amenable to application of the well-known method of characteristics. The value of the density at some point in time is directly accessible by the solution of the original ODE extended by a single extra dimension (for the value of the density). This leads to simple methods for studying uncertainty, variability and likelihood, with significant advantages over more traditional Monte Carlo and related approaches especially when studying regions with low probability.

Conclusions

While such approaches based on the method of characteristics are common practice in other disciplines, their advantages for the study of biological systems have so far remained unrecognized. Several examples illustrate performance and accuracy of the approach and its limitations.

A comparison of approximation techniques for variance-based sensitivity analysis of biochemical reaction systems

Background

Sensitivity analysis is an indispensable tool for the analysis of complex systems. In a recent paper, we have introduced a thermodynamically consistent variance-based sensitivity analysis approach for studying the robustness and fragility properties of biochemical reaction systems under uncertainty in the standard chemical potentials of the activated complexes of the reactions and the standard chemical potentials of the molecular species. In that approach, key sensitivity indices were estimated by Monte Carlo sampling, which is computationally very demanding and impractical for large biochemical reaction systems. Computationally efficient algorithms are needed to make variance-based sensitivity analysis applicable to realistic cellular networks, modeled by biochemical reaction systems that consist of a large number of reactions and molecular species.

Results

We present four techniques, derivative approximation (DA), polynomial approximation (PA), Gauss-Hermite integration (GHI), and orthonormal Hermite approximation (OHA), for analytically approximating the variance-based sensitivity indices associated with a biochemical reaction system. By using a well-known model of the mitogen-activated protein kinase signaling cascade as a case study, we numerically compare the approximation quality of these techniques against traditional Monte Carlo sampling. Our results indicate that, although DA is computationally the most attractive technique, special care should be exercised when using it for sensitivity analysis, since it may only be accurate at low levels of uncertainty. On the other hand, PA, GHI, and OHA are computationally more demanding than DA but can work well at high levels of uncertainty. GHI results in a slightly better accuracy than PA, but it is more difficult to implement. OHA produces the most accurate approximation results and can be implemented in a straightforward manner. It turns out that the computational cost of the four approximation techniques considered in this paper is orders of magnitude smaller than traditional Monte Carlo estimation. Software, coded in MATLAB^®, which implements all sensitivity analysis techniques discussed in this paper, is available free of charge.

Conclusions

Estimating variance-based sensitivity indices of a large biochemical reaction system is a computationally challenging task that can only be addressed via approximations. Among the methods presented in this paper, a technique based on orthonormal Hermite polynomials seems to be an acceptable candidate for the job, producing very good approximation results for a wide range of uncertainty levels in a fraction of the time required by traditional Monte Carlo sampling.

Probabilistic sensitivity analysis of biochemical reaction systems

Sensitivity analysis is an indispensable tool for studying the robustness and fragility properties of biochemical reaction systems as well as for designing optimal approaches for selective perturbation and intervention. Deterministic sensitivity analysis techniques, using derivatives of the system response, have been extensively used in the literature. However, these techniques suffer from several drawbacks, which must be carefully considered before using them in problems of systems biology. We develop here a probabilistic approach to sensitivity analysis of biochemical reaction systems. The proposed technique employs a biophysically derived model for parameter fluctuations and, by using a recently suggested variance-based approach to sensitivity analysis [Saltelli et al., Chem. Rev. (Washington, D.C.) 105, 2811 (2005)], it leads to a powerful sensitivity analysis methodology for biochemical reaction systems. The approach presented in this paper addresses many problems associated with derivative-based sensitivity analysis techniques. Most importantly, it produces thermodynamically consistent sensitivity analysis results, can easily accommodate appreciable parameter variations, and allows for systematic investigation of high-order interaction effects. By employing a computational model of the mitogen-activated protein kinase signaling cascade, we demonstrate that our approach is well suited for sensitivity analysis of biochemical reaction systems and can produce a wealth of information about the sensitivity properties of such systems. The price to be paid, however, is a substantial increase in computational complexity over derivative-based techniques, which must be effectively addressed in order to make the proposed approach to sensitivity analysis more practical.

Pharmacokinetic predictions in children by using the physiologically based pharmacokinetic modelling

Nowadays, 50-90% of drugs used in children have never been actually studied in this population. Consequently, either our children are often exposed to the risk of adverse drug events or to lack of efficacy, or they are unable to benefit from a number of therapeutic advances offered to adults, as no clinical study has been properly performed in children. Actually the main methods used to calculate the dose for a child are based on allometric methods taking into account different categories of age, the body weight and/or the body surface area. Unfortunately, these calculation methods consider the children as small adults, which is not the case. Physiologically based pharmacokinetics is one way to integrate the physiological changes occurring in the childhood and to anticipate their impact on the pharmacokinetic processes: absorption, distribution, metabolism and excretion/elimination. From different examples, the application of this modelling approach is discussed as a possible and valuable method to minimize the ethical and technical difficulties of conducting research in children.

Evaluation of physiologically based pharmacokinetic models for use in risk assessment

Physiologically based pharmacokinetic (PBPK) models are sophisticated dosimetry models that offer great flexibility in modeling exposure scenarios for which there are limited data. This is particularly of relevance to assessing human exposure to environmental toxicants, which often requires a number of extrapolations across species, route, or dose levels. The continued development of PBPK models ensures that regulatory agencies will increasingly experience the need to evaluate available models for their application in risk assessment. To date, there are few published criteria or well-defined standards for evaluating these models. Herein, important considerations for evaluating such models are described. The evaluation of PBPK models intended for risk assessment applications should include a consideration of: model purpose, model structure, mathematical representation, parameter estimation, computer implementation, predictive capacity and statistical analyses. Model purpose and structure require qualitative checks on the biological plausibility of a model. Mathematical representation, parameter estimation, computer implementation involve an assessment of the coding of the model, as well as the selection and justification of the physical, physicochemical and biochemical parameters chosen to represent a biological organism. Finally, the predictive capacity and sensitivity, variability and uncertainty of the model are analysed so that the applicability of a model for risk assessment can be determined. Published in 2007 by John Wiley & Sons, Ltd.

A human PBPK model for ethanol describing inhibition of gastric motility

A physiologically based pharmacokinetic model for investigating inter-individual and inter-racial variability in ethanol pharmacokinetics is presented. The model is a substantial modification of an existing model which described some genetic polymorphisms in the hepatic alcohol dehydrogenase enzymes. The model was modified to incorporate a description of ethanol absorption from the stomach and gastro-intestinal tract and the retardation of gastric emptying due to a concentration-dependent inhibition of gastric peristalsis. In addition, intra-venous and intra-arterial routes of administration were added to investigate whether the biological structure of the model provided a core which may be easily adapted for any route of exposure. The model is proposed as suitable for the investigation of the effects of both acute and chronic ethanol exposure.

Whole body pharmacokinetic models

The aim of the current review is to summarise the present status of physiologically based pharmacokinetic (PBPK) modelling and its applications in drug research, and thus serve as a reference point to people interested in the methodology. The review is structured into three major sections. The first discusses the existing methodologies and techniques of PBPK model development. The second describes some of the most interesting PBPK model implementations published. The final section is devoted to a discussion of the current limitations and the possible future developments of the PBPK modelling approach. The current review is focused on papers dealing with the pharmacokinetics and/or toxicokinetics of medicinal compounds; references discussing PBPK models of environmental compounds are mentioned only if they represent considerable methodological developments or reveal interesting interpretations and/or applications.The major conclusion of the review is that, despite its significant potential, PBPK modelling has not seen the development and implementation it deserves, especially in the drug discovery, research and development processes. The main reason for this is that the successful development and implementation of a PBPK model is seen to require the investment of significant experience, effort, time and resources. Yet, a substantial body of PBPK-related research has been accumulated that can facilitate the PBPK modelling and implementation process. What is probably lagging behind is the expertise component, where the demand for appropriately qualified staff far outreaches availability.

Multidisciplinary modelling of biomedical systems

This paper describes general principles and example results of a new software tool being developed for physiologically-based modelling of biomedical systems within a multidisciplinary framework. The aim is to overcome some limitations of currently available software designed either for general purpose or for highly specialised modelling applications. In fact, general purpose tools usually impose explicit coding of mathematical model equations or non-intuitive system representations, whereas specialised software use domain-specific notations that allow efficient and convenient model building only for special classes of systems. The aim of the present study is to pursue intuitive representation of various, possibly interacting, types of biological systems described as interconnected physical components, such as mass and energy storage elements, active and passive transport or biochemical transformations. The presented software generates automatically the mathematical model equations that can be coded in different formats. This allows interoperability with other existing software, e.g. for numerical simulation, symbolic analysis or text processing. A multi-domain structural language has been defined for an intuitive, hierarchical and self-explanatory specification of physiological models. The proposed strategies may become useful for dissemination and integration of multidisciplinary modelling knowledge.

The application of physiologically based pharmacokinetic modelling in the analysis of occupational exposure to perchloroethylene

A physiologically based pharmacokinetic model for perchloroethylene was parameterized, calibrated and validated using anatomic, physiologic, biochemical and physicochemical data obtained from the literature. The model was used to analyse human exposure data obtained under controlled conditions and from dry cleaning establishments in the Padua area of northern Italy. Whilst the model satisfactorily simulated the urinary excretion of trichloroacetic acid, following experimental inhalation exposure to 10, 20 and 40 ppm perchloroethylene under controlled conditions the opposite was true for the occupational exposure data. However, further model refinement to incorporate inter-individual variability of anatomical, physiological and biochemical parameters which have an impact on model output, would further improve the predictive capabilities of the model. The possibility of perchloroethylene and trichloroethylene co-exposure in the occupational setting was indicated by the model.

The use of toxicodynamics in risk assessment

Risk assessment of xenobiotics is a qualitative and quantitative assessment of toxic properties conventionally based on data resulting from tests in animals exposed to the substance. The assessment of dose-effect relationship includes evaluation of exposure at the site of action. More recently, emphasis is put on understanding the relationship between exposure at the site of action and the resulting effect, i.e. toxicodynamic. In this respect, results from genotoxicity studies may be a measure for exposure and at the same time of an effect. Results of toxicodynamic endpoints such as binding to receptors or release of hormones have been used when replacing default values for interspecies extrapolation. It may also be envisaged to use toxicodynamic endpoints in order to get an estimate of intraspecies variability. It was demonstrated that this approach may be helpful only if the relationship between the toxicodynamic endpoint and the definite endpoint is known by using the example of bisphenol A. Whereas there are clear effects of bisphenol A in in vitro and ex vivo studies, the classical two generation study has not been able to detect an effect on reproduction and/or fertility. Looking in the future development of toxicodynamic endpoints, gene profiling and the analysis of proteins ('proteomics') may be helpful tools employed in screening and being related to the mode of action are explored for their suitability in terms of toxicodynamic endpoints.

Modelling and simulation of variability and uncertainty in toxicokinetics and pharmacokinetics

Two important methodological issues within the framework of the variability and uncertainty analysis of toxicokinetic and pharmacokinetic systems are discussed: (i) modelling and simulation of the existing physiologic variability in a population; and (ii) modelling and simulation of variability and uncertainty when there is insufficient or not well defined (e.g. small sample, semiquantitative, qualitative and vague) information available. Physiologically based pharmacokinetic models are especially suited for separating and characterising the physiologic variability from the overall variability and uncertainty in the system. Monte Carlo sampling should draw from multivariate distributions, which reflect all levels of existing dependencies in the intact organism. The population characteristics should be taken into account. A fuzzy simulation approach is proposed to model variability and uncertainty when there is semiquantitative, qualitative and vague information about the model parameters and their statistical distributions cannot be defined reliably.

Characterization of uncertainty and variability in residential radon cancer risks

Radon, a naturally occurring gas found at some level in most homes, is an established risk factor for human lung cancer. The U.S. National Research Council has recently completed a comprehensive evaluation of the health risks of residential exposure to radon and developed models for projecting radon lung cancer risks to the general population. This analysis suggests that radon may play a role in the etiology of 10-15% of all lung cancer cases in the United States, although these estimates are subject to considerable uncertainty. In this article, we present a detailed analysis of uncertainty and variability in estimates of lung cancer risk due to residential exposure to radon. We use a general framework for the analysis of uncertainty and variability that we developed previously. Specifically, we focus on estimates of the age-specific excess relative risk (ERR) and lifetime relative risk (LRR), both of which vary substantially among individuals. We also consider estimates of the population attributable risk (PAR), which reflects the proportion of the lung cancer burden attributable to radon. Variability in the ERR and LRR is largely determined by variability in residential exposure levels and in the dosimetric K-factor used to extrapolate from occupational to environmental settings. Uncertainty in the ERR and LRR is due to uncertainty in the model parameters, notably those reflecting the carcinogenic potency of radon and the modifying effect of attained age. Uncertainty in the PAR is determined by uncertainty about the values of the parameters in the risk models used to estimate the PAR. Uncertainty in radon levels in homes and the dosimetric K-factor contribute comparatively little to uncertainty in the PAR. These results suggest that reduction in uncertainty about the PAR for radon induced lung cancer can only be achieved if more reliable risk projection models can be developed.

Uncertainty in biomonitoring and kinetic modeling

Uncertainty in exposure assessment and uncertainty in kinetic models of early effects after exposure to a toxin are addressed in this paper. Sources of uncertainty in the determination of exposure of workers in chemical industry exposed to dioxins are exhibited and a simple kinetic model for biomonitor measurements of the concentrations from occupational exposure is derived. Model uncertainty, and uncertainty in the model parameters of physiologically-based pharmacokinetic models (PBPK models) are addressed when these models are used to estimate the effective dose in risk assessment. Uncertainty in the model parameters originating from the use of different statistical analysis methods is exhibited for Hill type nonlinear kinetics of enzyme induction mediated by a toxin.

Monograph: reassessment of human cancer risk of aldrin/dieldrin

In 1987, the US Environmental Protection Agency (EPA) classified aldrin and dieldrin as category B2 carcinogens, i.e. probable human carcinogens, based largely on the increase in liver tumors in mice fed either organochlorine insecticide. At that date, the relevant epidemiology was deemed inadequate to influence the cancer risk assessment. More time has now elapsed since early exposures of manufacturing workers to aldrin/dieldrin; therefore, updated epidemiological data possess more power to detect exposure-related differences in cancer risk and mortality. Also, recent experimental studies provide a plausible mode of action to explain the mouse specificity of dieldrin-induced hepatocarcinogenesis and call into question the relevance of this activity to human cancer risk. This monograph places this new information within the historic and current perspectives of human cancer risk assessment, including EPA's 1996 Proposed Guidelines for Carcinogen Risk Assessment. Updated epidemiological studies of manufacturing workers in which lifetime exposures to aldrin/dieldrin have been quantified do not indicate increased mortality or cancer risk. In fact, at the middle range of exposures, there is evidence of a decrease in both mortality from all causes and cancer. Recent experimental studies indicate that dieldrin-induced hepatocarcinogenesis in mice occurs through a nongenotoxic mode of action, in which the slow oxidative metabolism of dieldrin is accompanied by an increased production of reactive oxygen species, depletion of hepatic antioxidant defenses (particularly alpha-tocopherol), and peroxidation of liver lipids. Dieldrin-induced oxidative stress or its sequelae apparently result in modulation of gene expression that favors expansion of initiated mouse, but not rat, liver cells; thus, dieldrin acts as a nongenotoxic promoter/accelerator of background liver tumorigenesis in the mouse. Within the framework of EPA's Proposed Guidelines for Carcinogen Risk Assessment, it is proposed that the most appropriate cancer risk descriptor for aldrin/dieldrin, relating to the mouse liver tumor response, is 'not likely a human carcinogen', a descriptor consistent with the example of phenobarbital cited by EPA.

Uncertainty and variability analysis in multiplicative risk models

Currently, there is a trend away from the use of single (often conservative) estimates of risk to summarize the results of risk analyses in favor of stochastic methods which provide a more complete characterization of risk. The use of such stochastic methods leads to a distribution of possible values of risk, taking into account both uncertainty and variability in all of the factors affecting risk. In this article, we propose a general framework for the analysis of uncertainty and variability for use in the commonly encountered case of multireplicative risk models, in which risk may be expressed as a product of two or more risk factors. Our analytical methods facilitate the evaluation of overall uncertainty and variability in risk assessment, as well as the contributions of individual risk factors to both uncertainty and variability which is cumbersome using Monte Carlo methods. The use of these methods is illustrated in the analysis of potential cancer risks due to the ingestion of radon in drinking water.

Statistical research needs in mechanistic modelling for carcinogenic risk assessment

If the broad spectrum of mechanistic research conducted on an environmental carcinogen is to be used in quantifying cancer risks, statisticians must play a key role. Statistical methods are critically needed for a scientifically valid analysis of a complicated series of linked experimental findings. This will require a greater understanding of the underlying biology than is common in statistical consulting, aiding in the development of complicated mechanistically based mathematical descriptions of mean response and in the creation of statistical methods for the estimation of model parameters (e.g. likelihoods) able to use both the underlying model and much of the available data.

Health risk assessment of drinking water contaminants in Canada: the applicability of mixture risk assessment methods

The objectives of this article are: (i) to review the current approaches of Health Canada to the risk assessment of drinking water contaminants, and (ii) to examine the applicability of mixture risk assessment methods to drinking water contaminants. Health Canada's current approaches to drinking water risk assessment, like those of many regulatory agencies, focus almost solely on the effects of individual chemicals. As such, no formal method is currently used for developing mixtures guidelines or for modifying guidelines of individual chemicals to account for the possibility of the occurrence of interactions (supraadditive or infraadditive). Recent interest in the risk assessment of mixtures, at least in part, stems from concerns over the potential health risks of mixtures of very commonly occurring compounds in Canadian drinking water supplies, namely the disinfection by-products. Before any mixtures methods can be considered for incorporation into Health Canada's current approaches to the risk assessment of drinking water contaminants, it is essential to consider the limitations and data requirements of the various mixture risk assessment methods (i.e., whole mixture approach, similar mixture approach, components-based approaches, interactions-based assessment). Among the existing mixture risk assessment methods, the components-based and interactions-based approaches could be applicable to drinking water contaminants. Specifically, among the components-based approaches, dose-addition, response-addition, and the toxic equivalency factor approaches are the most applicable ones for drinking water contaminants. Until an interactions-based, mechanistic risk assessment approach (e.g., physiological model-based approach) becomes available for routine use, the components-based approaches remain the default methods for consideration. Progress in the development and validation of an interactions-based risk assessment methodology should facilitate a more realistic assessment of risk due to drinking water contaminants without increasing the levels of uncertainty in risk estimates above those associated with existing single-chemical methods.

A computer model and program for xenobiotic disposition during pregnancy

A physiologically based pharmacokinetic computer model and program have been developed that depict internal disposition of chemicals during pregnancy in the mother and embryo/fetus. The model is based on human physiology but has been extended to simulate laboratory animal data. The model represents the distribution, metabolism, and elimination of two chemicals in both the maternal and embryo/fetal systems; the program handles the two chemicals completely independently or interactively with the two chemicals sharing routes of metabolism and/or elimination. The FORTRAN program computes the concentration of the two chemicals in 26 organs/tissues in the pregnant mother and 15 organs/tissues in the embryo/fetus using a 486DX4 or Pentium PC. Adjustments for embryo/fetal organ and tissue volumes as a function of developmental age are made utilizing the Gompertz growth equation for the developing embryo/fetus and allometric relationships for the developing organs. Various changes in the maternal compartments which could affect the distribution of a xenobiotic during pregnancy are also included in the model. Input files require estimates of binding coefficients, first- and/or second-order metabolism constants, level of interaction between the two chemicals, and dosing information. Different possible routes of administration are included (e.g., i.v., infusion, oral, dermal, and inhalation, as well as repeated doses or exposures). Regression analysis can be conducted on any combination of these various parameters to fit actual data. Output concentration-time curves are available simultaneously from all 82 differential equations. An illustrative example compares observed data with simulations for imipramine and its demethylated metabolite, desipramine, in both the maternal rat and her fetuses. Methyl mercury data for the non-pregnant and pregnant rat also are compared with human data. Based on parameters determined from analysis of rat data, the model is readjusted for human physiology and predicts human maternal and fetal tissue concentrations as a function of time.