A program for the optimum design of pharmacokinetic, pharmacodynamic, drug metabolism and drug-drug interaction models

Planning any experiment includes issues such as how many samples are to be taken and their location given some predictor variable. Often a model is used to explain these data; hence including this formally in the design will be beneficial for any subsequent parameter estimation and modelling. A number of criteria for model oriented experiments, which maximise the information content of the collected data are available. In this paper we present a program, Optdes, to investigate the optimal design of pharmacokinetic, pharmacodynamic, drug metabolism and drug-drug interaction models. Using the developed software the location of either a predetermined number of design points (exact designs) or together with the proportion of samples at each point (continuous designs) can be determined. Local as well as Bayesian designs can be optimised by either D- or A-optimality criteria. Although often the optimal design cannot be applied for practical reasons, alternative designs can be readily evaluated.     

MODDIS: a microcomputer program for model discrimination

A computer program for a microcomputer (HP 86) is presented to discriminate between different models and to design new experiments for model discrimination. By a non-linear fitting algorithm a set of experimental data is fitted with different models suggested by the user. The parameters characterizing each model are estimated by minimizing the sum of squared residuals; different criteria are used to test the choice between two or more models at different levels of probability and the smallest number of additional experiments required for discrimination is computed. If discrimination is not achieved a direct search method is used to find the local maxima of the divergence (or information for discrimination) over a user-chosen domain of independent variables (x R'). The x value corresponding to the absolute maximum of the divergence is the best choice to run a new experiment for discrimination.     

Nonparametric identification of population models via Gaussian processes

Population models are used to describe the dynamics of different subjects belonging to a population and play an important role in drug pharmacokinetics. A nonparametric identification scheme is proposed in which both the average impulse response of the population and the individual ones are modelled as Gaussian stochastic processes. Assuming that the average curve is an integrated Wiener process, it is shown that its estimate is a cubic spline. An empirical Bayes algorithm for estimating both the average and the individual curves is worked out. The model is tested on simulated data sets as well as on xenobiotics pharmacokinetic data.     

Simulation-based fully Bayesian experimental design for mixed effects models

Bayesian inference has commonly been performed on nonlinear mixed effects models. However, there is a lack of research into performing Bayesian optimal design for nonlinear mixed effects models, especially those that require searches to be performed over several design variables. This is likely due to the fact that it is much more computationally intensive to perform optimal experimental design for nonlinear mixed effects models than it is to perform inference in the Bayesian framework. Fully Bayesian experimental designs for nonlinear mixed effects models are presented, which involve the use of simulation-based optimal design methods to search over both continuous and discrete design spaces. The design problem is to determine the optimal number of subjects and samples per subject, as well as the (near) optimal urine sampling times for a population pharmacokinetic study in horses, so that the population pharmacokinetic parameters can be precisely estimated, subject to cost constraints. The optimal sampling strategies, in terms of the number of subjects and the number of samples per subject, were found to be substantially different between the examples considered in this work, which highlights the fact that the designs are rather problem-dependent and can be addressed using the methods presented.     

Experimental design for regression analysis when the responses are subject to censoring

Experimental design issues are investigated for regression models with possibly censored responses, arising typically from pharmacokinetic or virus dynamic experiments. Examples are provided for both locally and Bayesian optimal designs. A case study in pharmacokinetics is provided.     

A new generic approach to model reduction for complex physiologically based drug models

This paper explores the identification problem when dealing with physiological models relating to anaesthetic drugs such as fentanyl. The Mapleson model for drug concentration, which will be the focus of this study, is described by algebraic equations, which are derived from the laws of physics and chemistry, and there are some limitations in its system's analysis, i.e. in the study of its relevant dynamics, and its exploitation from a control design viewpoint. Hence, we propose to represent this model via dynamic differential equations with a reduced number of variables using MATLAB–SIMULINK. Using Mapleson's approach for modelling, the input–output data for each organ can be obtained under a particular drug regimen which in turn can be used to obtain a continuous time-transfer function fit for each of these organs.     

PKgraph: an R package for graphically diagnosing population pharmacokinetic models

Population pharmacokinetic (PopPK) modeling has become increasing important in drug development because it handles unbalanced design, sparse data and the study of individual variation. However, the increased complexity of the model makes it more of a challenge to diagnose the fit. Graphics can play an important and unique role in PopPK model diagnostics. The software described in this paper, PKgraph, provides a graphical user interface for PopPK model diagnosis. It also provides an integrated and comprehensive platform for the analysis of pharmacokinetic data including exploratory data analysis, goodness of model fit, model validation and model comparison. Results from a variety of modeling fitting software, including NONMEM, Monolix, SAS and R, can be used. PKgraph is programmed in R, and uses the R packages lattice, ggplot2 for static graphics, and rggobi for interactive graphics.     

Comparison of nonparametric methods in nonlinear mixed effects models

During the drug development, nonlinear mixed effects models are routinely used to study the drug’s pharmacokinetics and pharmacodynamics. The distribution of random effects is of special interest because it allows to describe the heterogeneity of the drug’s kinetics or dynamics in the population of individuals studied. Parametric models are widely used, but they rely on a normality assumption which may be too restrictive. In practice, this assumption is often checked using the empirical distribution of random effects’ empirical Bayes estimates. Unfortunately, when data are sparse (like in patients phase III clinical trials), this method is unreliable. In this context, nonparametric estimators of the random effects distribution are attractive. Several nonparametric methods (estimators and their associated computation algorithms) have been proposed but their use is limited. Indeed, their practical and theoretical properties are unclear and they have a reputation for being computationally expensive. Four nonparametric methods in comparison with the usual parametric method are evaluated. Statistical and computational features are reviewed and practical performances are compared in simulation studies mimicking real pharmacokinetic analyses. The nonparametric methods seemed very useful when data are sparse. On a simple pharmacokinetic model, all the nonparametric methods performed roughly equivalently. On a more challenging pharmacokinetic model, differences between the methods were clearer.     

Local identifiability and sensitivity analysis of neuromuscular blockade and depth of hypnosis models

This paper addresses the local identifiability and sensitivity properties of two classes of Wiener models for the neuromuscular blockade and depth of hypnosis, when drug dose profiles like the ones commonly administered in the clinical practice are used as model inputs. The local parameter identifiability was assessed based on the singular value decomposition of the normalized sensitivity matrix. For the given input signal excitation, the results show an over-parameterization of the standard pharmacokinetic/pharmacodynamic models. The same identifiability assessment was performed on recently proposed minimally parameterized parsimonious models for both the neuromuscular blockade and the depth of hypnosis. The results show that the majority of the model parameters are identifiable from the available input–output data. This indicates that any identification strategy based on the minimally parameterized parsimonious Wiener models for the neuromuscular blockade and for the depth of hypnosis is likely to be more successful than if standard models are used.     

Simulation of the USP drug delivery problem using CFD: experimental,numerical and mathematical aspects

The numerical simulation of the dissolution of drug-containing compacts in a stirred reactive medium is presented. This is of interest to the design of drug delivery systems in which the goal is to design compacts which release the drug according to certain desired release profiles. A moving boundary finite element approach is adopted to simulate dissolution of layered compacts made up of a number of layers of different acids which dissolve at different rates. The simulation results are compared to experimental measurements. Although a number of idealisations have been adopted in the numerical model, good agreement with experiment is achieved. A semi-analytic solution is also developed which leads to an expression for the mass flux from a dissolving cylinder. Results for this model are compared with the numerical and experimental data.     

Bifurcation analysis of PID-controlled neuromuscular blockade in closed-loop anesthesia

The problem of PID-controlled neuromuscular blockade (NMB) in closed-loop anesthesia is considered. Contrary to the usual practice of designing PID-controllers for nonlinear systems on the basis of a linearized model and online tests, bifurcation analysis is utilized in this paper for that purpose. Two nonlinear Wiener models for the NMB are considered: a conventional pharmacokinetic/pharmacodynamic (PK/PD) model and a parsimonious model suitable for online parameter estimation. The models under a PID feedback are analyzed in order to discern the safe intervals of the controller parameters that are not subject to complex dynamical phenomena. The parsimony of the mathematical model is instrumental in minimizing the number of bifurcation parameters. The analyses show that the closed-loop systems undergo Andronov–Hopf bifurcation at a point in the model parameter space giving rise to nonlinear oscillations. For steeper, but still feasible slopes of the nonlinear function parameterizing the static nonlinearity of the Wiener models, deterministic chaos can arise in the closed loop for lower concentrations of the anesthetic drug. A model-based PID-controller tuning procedure is suggested that guarantees a certain settling time and robustness margin of the resulting loop with respect to the bifurcation. The tuning procedure is illustrated on mathematical models identified from patient data and the corresponding PID controllers.     

Online model selection approach based on Unscented Kalman Filtering

Highly predictive mathematical models are of inestimable value in systems biology. Their application ranges from investigations of basic processes in living organisms up to model based drug design in the field of pharmacology. For the development of reliable models suitable model candidates and related model parameters have to be identified by minimising the difference between the model outcome and available measurement data. Due to the complexity of the analysed processes mathematical models capture only the essential features of interest. This approximate representation, which is usually combined with a vague knowledge of basic processes, leads in many cases to a variety of potential model candidates describing the real process almost equally well. To determine the most plausible model candidate is the objective of model selection or model discrimination methods. If under given operation conditions no sufficient discrimination can be achieved, Optimal Experimental Designs (OED) comes into play. OED searches for operation conditions which facilitate the overall selection process. In this work an online model selection framework is presented. Here, the Unscented Kalman Filter (UKF) provides statistical information which is used to assign probability values to every model candidate. These probability values are immediately updated as soon as new measurement data become available. In addition, during the experimental run the process is steered in a fashion which maximises the differences in these candidates. To overcome limitations caused by parameter uncertainties the most sensitive model parameters are simultaneously estimated in the course of the model selection framework. The combined application of the online framework and the joint estimation of sensitive model parameters provides a very efficient usage of measurement data reducing the overall number of experiments. The method is demonstrated for a well known motif in signalling pathways, the mitogen-activated protein (MAP) kinase.     

Pair-copula based mixture models and their application in clustering

Finite mixtures are often used to perform model based clustering of multivariate data sets. In real life applications, such data may exhibit complex nonlinear form of dependence among the variables. Also, the individual variables (margins) may follow different families of distributions. Most of the existing mixture models are unable to accommodate these two aspects of the data. This paper presents a finite mixture model that involves a pair-copula based construction of a multivariate distribution. Such a model de-couples the margins and the dependence structures. Hence, the margins can be modeled using different families. Again, many possible dependence structures can also be studied using different copulas. The resulting mixture model (called DVMM) is then capable of capturing a broad family of distributions including non-Gaussian models. Here we study DVMM in the context of clustering of multivariate data. We design an expectation maximization procedure for estimating the mixture parameters. We perform extensive experiments on the basis of a number of well-known data sets. A detailed evaluation of the clustering quality obtained by DVMM in comparison to other mixture models is presented. The experimental results show that the performance of DVMM is quite satisfactory.     

Fast estimation of Gaussian mixture models for image segmentation

The expectation maximization algorithm has been classically used to find the maximum likelihood estimates of parameters in probabilistic models with unobserved data, for instance, mixture models. A key issue in such problems is the choice of the model complexity. The higher the number of components in the mixture, the higher will be the data likelihood, but also the higher will be the computational burden and data overfitting. In this work, we propose a clustering method based on the expectation maximization algorithm that adapts online the number of components of a finite Gaussian mixture model from multivariate data or method estimates the number of components and their means and covariances sequentially, without requiring any careful initialization. Our methodology starts from a single mixture component covering the whole data set and sequentially splits it incrementally during expectation maximization steps. The coarse to fine nature of the algorithm reduce the overall number of computations to achieve a solution, which makes the method particularly suited to image segmentation applications whenever computational time is an issue. We show the effectiveness of the method in a series of experiments and compare it with a state-of-the-art alternative technique both with synthetic data and real images, including experiments with images acquired from the iCub humanoid robot.     

一种基于ICA空间高斯混合模型的新颖检测*

新颖检测中,可应用高斯混合模型建立已知数据模型,拟合数据分布,但当数据维数较高时,自由参数太多,训练需要巨大的数据采样,而ICA搜寻数据的最大统计独立表示,可以将数据从高维空间投影到低维空间。提出一种基于ICA空间高斯混合模型的新颖检测,可有效减少估测的自由参数,降低训练数据采样的苛刻要求,实验也验证了该方法的可行性。