Three novel approaches to structural identifiability analysis in mixed-effects models.

BACKGROUND AND OBJECTIVE: Structural identifiability is a concept that considers whether the structure of a model together with a set of input-output relations uniquely determines the model parameters. In the mathematical modelling of biological systems, structural identifiability is an important concept since biological interpretations are typically made from the parameter estimates. For a system defined by ordinary differential equations, several methods have been developed to analyse whether the model is structurally identifiable or otherwise. Another well-used modelling framework, which is particularly useful when the experimental data are sparsely sampled and the population variance is of interest, is mixed-effects modelling. However, established identifiability analysis techniques for ordinary differential equations are not directly applicable to such models. METHODS: In this paper, we present and apply three different methods that can be used to study structural identifiability in mixed-effects models. The first method, called the repeated measurement approach, is based on applying a set of previously established statistical theorems. The second method, called the augmented system approach, is based on augmenting the mixed-effects model to an extended state-space form. The third method, called the Laplace transform mixed-effects extension, is based on considering the moment invariants of the systems transfer function as functions of random variables. RESULTS: To illustrate, compare and contrast the application of the three methods, they are applied to a set of mixed-effects models. CONCLUSIONS: Three structural identifiability analysis methods applicable to mixed-effects models have been presented in this paper. As method development of structural identifiability techniques for mixed-effects models has been given very little attention, despite mixed-effects models being widely used, the methods presented in this paper provides a way of handling structural identifiability in mixed-effects models previously not possible.

Translational model to predict pulmonary pharmacokinetics and efficacy in man for inhaled bronchodilators.

Translational pharmacokinetic (PK) models are needed to describe and predict drug concentration-time profiles in lung tissue at the site of action to enable animal-to-man translation and prediction of efficacy in humans for inhaled medicines. Current pulmonary PK models are generally descriptive rather than predictive, drug/compound specific, and fail to show successful cross-species translation. The objective of this work was to develop a robust compartmental modeling approach that captures key features of lung and systemic PK after pulmonary administration of a set of 12 soluble drugs containing single basic, dibasic, or cationic functional groups. The model is shown to allow translation between animal species and predicts drug concentrations in human lungs that correlate with the forced expiratory volume for different classes of bronchodilators. Thus, the pulmonary modeling approach has potential to be a key component in the prediction of human PK, efficacy, and safety for future inhaled medicines.

Extending existing structural identifiability analysis methods to mixed-effects models.

The concept of structural identifiability for state-space models is expanded to cover mixed-effects state-space models. Two methods applicable for the analytical study of the structural identifiability of mixed-effects models are presented. The two methods are based on previously established techniques for non-mixed-effects models; namely the Taylor series expansion and the input-output form approach. By generating an exhaustive summary, and by assuming an infinite number of subjects, functions of random variables can be derived which in turn determine the distribution of the system's observation function(s). By considering the uniqueness of the analytical statistical moments of the derived functions of the random variables, the structural identifiability of the corresponding mixed-effects model can be determined. The two methods are applied to a set of examples of mixed-effects models to illustrate how they work in practice.

Input Estimation for Extended-Release Formulations Exemplified with Exenatide.

Estimating the in vivo absorption profile of a drug is essential when developing extended-release medications. Such estimates can be obtained by measuring plasma concentrations over time and inferring the absorption from a model of the drug's pharmacokinetics. Of particular interest is to predict the bioavailability-the fraction of the drug that is absorbed and enters the systemic circulation. This paper presents a framework for addressing this class of estimation problems and gives advice on the choice of method. In parametric methods, a model is constructed for the absorption process, which can be difficult when the absorption has a complicated profile. Here, we place emphasis on non-parametric methods that avoid making strong assumptions about the absorption. A modern estimation method that can address very general input-estimation problems has previously been presented. In this method, the absorption profile is modeled as a stochastic process, which is estimated using Markov chain Monte Carlo techniques. The applicability of this method for extended-release formulation development is evaluated by analyzing a dataset of Bydureon, an injectable extended-release suspension formulation of exenatide, a GLP-1 receptor agonist for treating diabetes. This drug is known to have non-linear pharmacokinetics. Its plasma concentration profile exhibits multiple peaks, something that can make parametric modeling challenging, but poses no major difficulties for non-parametric methods. The method is also validated on synthetic data, exploring the effects of sampling and noise on the accuracy of the estimates.

Parameter Identifiability of Fundamental Pharmacodynamic Models.

Issues of parameter identifiability of routinely used pharmacodynamics models are considered in this paper. The structural identifiability of 16 commonly applied pharmacodynamic model structures was analyzed analytically, using the input-output approach. Both fixed-effects versions (non-population, no between-subject variability) and mixed-effects versions (population, including between-subject variability) of each model structure were analyzed. All models were found to be structurally globally identifiable under conditions of fixing either one of two particular parameters. Furthermore, an example was constructed to illustrate the importance of sufficient data quality and show that structural identifiability is a prerequisite, but not a guarantee, for successful parameter estimation and practical parameter identifiability. This analysis was performed by generating artificial data of varying quality to a structurally identifiable model with known true parameter values, followed by re-estimation of the parameter values. In addition, to show the benefit of including structural identifiability as part of model development, a case study was performed applying an unidentifiable model to real experimental data. This case study shows how performing such an analysis prior to parameter estimation can improve the parameter estimation process and model performance. Finally, an unidentifiable model was fitted to simulated data using multiple initial parameter values, resulting in highly different estimated uncertainties. This example shows that although the standard errors of the parameter estimates often indicate a structural identifiability issue, reasonably "good" standard errors may sometimes mask unidentifiability issues.

Nonlinear mixed-effects modelling for single cell estimation: when, why, and how to use it.

BACKGROUND: Studies of cell-to-cell variation have in recent years grown in interest, due to improved bioanalytical techniques which facilitates determination of small changes with high uncertainty. Like much high-quality data, single-cell data is best analysed using a systems biology approach. The most common systems biology approach to single-cell data is the standard two-stage (STS) approach. In STS, data from each cell is analysed in a separate sub-problem, meaning that only data from the same cell is used to calculate the parameter values within that cell. Because only parts of the data are considered, problems with parameter unidentifiability are exaggerated in STS. In contrast, a related approach to data analysis has been developed for the studies of patient-to-patient variations. This approach, called nonlinear mixed-effects modelling (NLME), makes use of all data, when estimating the patient-specific parameters. NLME would therefore be advantageous compared to STS also for the study of cell-to-cell variation. However, no such systematic evaluation of the two approaches exists. RESULTS: Herein, such a systematic comparison between STS and NLME has been performed. Different examples, both linear and nonlinear, and both simulated and real experimental data, have been examined. With informative data, there is no significant difference in the results for either parameter or noise estimation. However, when data becomes uninformative, NLME is significantly superior to STS. These results hold independently of whether the loss of information is due to a low signal-to-noise ratio, too few data points, or a bad input signal. The improvement is shown to come from both the consideration of a joint likelihood (JLH) function, describing all parameters and data, and from an a priori postulated form of the population parameters. Finally, we provide a small tutorial that shows how to use NLME for single-cell analysis, using the free and user-friendly software Monolix. CONCLUSIONS: When considering uninformative single-cell data, NLME yields more accurate parameter and noise estimates, compared to more traditional approaches, such as STS and JLH.