PkStaMp Library for Constructing Optimal Population Designs for PK/PD Studies

We discuss a Matlab-based library for constructing optimal sampling schemes for pharmacokinetic (PK) and pharmacodynamic (PD) studies. The software relies on optimal design theory for nonlinear mixed effects models and, in particular, on the first-order optimization algorithm. The library includes a number of popular compartmental PK and combined PK/PD models and can be extended to include more models. An outline of inputs/outputs is provided, some algorithmic details and examples are presented, and future work is discussed.     

A semi‐parametric approach to fitting a nonlinear mixed PK/PD model with an effect compartment using SAS

Modelling of the relationship between concentration (PK) and response (PD) plays an important role in drug development. The modelling becomes complicated when the drug concentration and response measurements are not taken simultaneously and/or hysteresis occurs between the response and the concentration. A model‐based approach fits a joint pharmacokinetic (PK) and concentration–response (PK/PD) model, including an effect compartment if necessary, to concentration and response data. However, this approach relies on the PK data being well described by a common PK model. We propose an algorithm for a semi‐parametric approach to fitting nonlinear mixed PK/PD models including an effect compartment using linear interpolation and extrapolation for concentration data. This approach is independent of the PK model, and the algorithm can easily be implemented using SAS PROC NLMIXED. Practical issues in programming and computing are also discussed. The properties of this approach are examined using simulations. This approach is used to analyse data from a study of the PK/PD relationship between insulin and glucose levels. Copyright © 2005 John Wiley & Sons, Ltd.     

Constructing optimal designs for fitting pharmacokinetic models

We consider some computational issues that arise when searching for optimal designs for pharmacokinetic (PK) studies. Special factors that distinguish these are (i) repeated observations are taken from each subject and the observations are usually described by a nonlinear mixed model (NLMM), (ii) design criteria depend on the model fitting procedure, (iii) in addition to providing efficient parameter estimates, the design must also permit model checking, (iv) in practice there are several design constraints, (v) the design criteria are computationally expensive to evaluate and often numerical integration is needed and finally (vi) local optimisation procedures may fail to converge or get trapped at local optima.We review current optimal design algorithms and explore the possibility of using global optimisation procedures. We use these latter procedures to find some optimal designs.For multi-purpose designs we suggest two surrogate design criteria for model checking and illustrate their use.     

Minimax A-, c-, and I-optimal regression designs for models with heteroscedastic errors

It is well known that it is difficult to construct minimax optimal designs. Furthermore, since in practice we never know the true error variance, it is important to allow small deviations and construct robust optimal designs. We investigate a class of minimax optimal regression designs for models with heteroscedastic errors that are robust against possible misspecification of the error variance. Commonly used A-, c-, and I-optimality criteria are included in this class of minimax optimal designs. Several theoretical results are obtained, including a necessary condition and a reflection symmetry for these minimax optimal designs. In this article, we focus mainly on linear models and assume that an approximate error variance function is available. However, we also briefly discuss how the methodology works for nonlinear models. We then propose an effective algorithm to solve challenging nonconvex optimization problems to find minimax designs on discrete design spaces. Examples are given to illustrate minimax optimal designs and their properties.     

Sample reuse simulation in optimal design for Tmax in pharmacokinetic experiments

T _max and C _max are important pharmacokinetic parameters in drug development processes. Often a nonparametric procedure is needed to estimate them when model independence is required. This paper proposes a simulation-based optimal design procedure for finding optimal sampling times for nonparametric estimates of T _max and C _max for each subject, assuming that the drug concentration follows a non-linear mixed model. The main difficulty of using standard optimal design procedures is that the property of the nonparametric estimate is very complicated. This procedure uses a sample reuse simulation to calculate the design criterion, which is an integral of multiple dimension, so that effective optimization procedures such as Newton-type procedures can be used directly to find optimal designs. This procedure is used to construct optimal designs for an open one-compartment model. An approximation based on the Taylor expansion is also derived and showed results that were consistent with those based on the sample reuse simulation.     

Applications and Implementations of Continuous Robust Designs

In the literature concerning the construction of robust optimal designs, many resulting designs turn out to have densities. In practice, an exact design should tell the experimenter what the support points are and how many subjects should be allocated to each of these points. In particular, we consider a practical situation in which the number of support points allowed is constrained. We discuss an intuitive approach, which motivates a new implementation scheme that minimizes the loss function based on the Kolmogorov and Smirnov distance between an exact design and the optimal design having a density. We present three examples to illustrate the application and implementation of a robust design constructed: one for a nonlinear dose-response experiment and the other two for general linear regression. Additionally, we perform some simulation studies to compare the efficiencies of the exact designs obtained by our optimal implementation with those by other commonly used implementation methods.     

Optimal designs for discriminating between some extensions of the Michaelis–Menten model

In this paper some results on the computation of optimal designs for discriminating between nonlinear models are provided. In particular, some typical deviations of the Michaelis–Menten model are considered. A common deviation of this pharmacokinetic model consists on adding a linear term. If two linear models differ in one parameter the T-optimal design for discriminating between them is c-optimal for estimating the added linear term. This is not the case for nonlinear models.     

D-optimal minimax design criterion for two-level fractional factorial designs

A D-optimal minimax design criterion is proposed to construct two-level fractional factorial designs, which can be used to estimate a linear model with main effects and some specified interactions. D-optimal minimax designs are robust against model misspecification and have small biases if the linear model contains more interaction terms. When the D-optimal minimax criterion is compared with the D-optimal design criterion, we find that the D-optimal design criterion is quite robust against model misspecification. Lower and upper bounds derived for the loss functions of optimal designs can be used to estimate the efficiencies of any design and evaluate the effectiveness of a search algorithm. Four algorithms to search for optimal designs for any run size are discussed and compared through several examples. An annealing algorithm and a sequential algorithm are particularly effective to search for optimal designs.     

A general method to determine sampling windows for nonlinear mixed effects models with an application to population pharmacokinetic studies

Optimal design methods have been proposed to determine the best sampling times when sparse blood sampling is required in clinical pharmacokinetic studies. However, the optimal blood sampling time points may not be feasible in clinical practice. Sampling windows, a time interval for blood sample collection, have been proposed to provide flexibility in blood sampling times while preserving efficient parameter estimation. Because of the complexity of the population pharmacokinetic models, which are generally nonlinear mixed effects models, there is no analytical solution available to determine sampling windows. We propose a method for determination of sampling windows based on MCMC sampling techniques. The proposed method attains a stationary distribution rapidly and provides time-sensitive windows around the optimal design points. The proposed method is applicable to determine sampling windows for any nonlinear mixed effects model although our work focuses on an application to population pharmacokinetic models.     

Joint modelling of longitudinal and repeated time-to-event data using nonlinear mixed-effects models and the stochastic approximation expectation–maximization algorithm

We propose a nonlinear mixed-effects framework to jointly model longitudinal and repeated time-to-event data. A parametric nonlinear mixed-effects model is used for the longitudinal observations and a parametric mixed-effects hazard model for repeated event times. We show the importance for parameter estimation of properly calculating the conditional density of the observations (given the individual parameters) in the presence of interval and/or right censoring. Parameters are estimated by maximizing the exact joint likelihood with the stochastic approximation expectation–maximization algorithm. This workflow for joint models is now implemented in the Monolix software, and illustrated here on five simulated and two real datasets.     

Robust designs for experiments with blocks

ABSTRACT

For experiments running in field plots or over time, the observations are often correlated due to spatial or serial correlation, which leads to correlated errors in a linear model analyzing the treatment means. Without knowing the exact correlation matrix of the errors, it is not possible to compute the generalized least-squares estimator for the treatment means and use it to construct optimal designs for the experiments. In this paper, we propose to use neighborhoods to model the covariance matrix of the errors, and apply a modified generalized least-squares estimator to construct robust designs for experiments with blocks. A minimax design criterion is investigated, and a simulated annealing algorithm is developed to find robust designs. We have derived several theoretical results, and representative examples are presented.     

Optimal designs for copula models

Copula modelling has in the past decade become a standard tool in many areas of applied statistics. However, a largely neglected aspect concerns the design of related experiments. Particularly the issue of whether the estimation of copula parameters can be enhanced by optimizing experimental conditions and how robust all the parameter estimates for the model are with respect to the type of copula employed. In this paper an equivalence theorem for (bivariate) copula models is provided that allows formulation of efficient design algorithms and quick checks of whether designs are optimal or at least efficient. Some examples illustrate that in practical situations considerable gains in design efficiency can be achieved. A natural comparison between different copula models with respect to design efficiency is provided as well.     

A generalisation of T‐optimality for discriminating between competing models with an application to pharmacokinetic studies

The T‐optimality criterion is used in optimal design to derive designs for model selection. To set up the method, it is required that one of the models is considered to be true. We term this local T‐optimality. In this work, we propose a generalisation of T‐optimality (termed robust T‐optimality) that relaxes the requirement that one of the candidate models is set as true. We then show an application to a nonlinear mixed effects model with two candidate non‐nested models and combine robust T‐optimality with robust D‐optimality. Optimal design under local T‐optimality was found to provide adequate power when the a priori assumed true model was the true model but poor power if the a priori assumed true model was not the true model. The robust T‐optimality method provided adequate power irrespective of which model was true. The robust T‐optimality method appears to have useful properties for nonlinear models, where both the parameter values and model structure are required to be known a priori, and the most likely model that would be applied to any new experiment is not known with certainty. Copyright © 2012 John Wiley & Sons, Ltd.     

Multiple-objective response-adaptive repeated measurement designs for clinical trials

In a response-adaptive design, we review and update the trial on the basis of outcomes in order to achieve a specific goal. Response-adaptive designs for clinical trials are usually constructed to achieve a single objective. In this paper, we develop a new adaptive allocation rule to improve current strategies for building response-adaptive designs to construct multiple-objective repeated measurement designs. This new rule is designed to increase estimation precision and treatment benefit by assigning more patients to a better treatment sequence. We demonstrate that designs constructed under the new proposed allocation rule can be nearly as efficient as fixed optimal designs in terms of the mean squared error, while leading to improved patient care.     

General orthogonal designs for parameter estimation of ANOVA models under weighted sum-to-zero constraints

ABSTRACT

Traditional studies on optimal designs for ANOVA parameter estimation are based on the framework of equal probabilities of appearance for each factor's levels. However, this premise does not hold in a variety of experimental problems, and it is of theoretical and practical interest to investigate optimal designs for parameters with unequal appearing odds. In this paper, we propose a general orthogonal design via matrix image, in which all columns’ matrix images are orthogonal with each other. Our main results show that such designs have A- and E-optimalities on the estimation of ANOVA parameters which have unequal appearing odds. In addition, we develop two simple methods to construct the proposed designs. The optimality of the design is also validated by a simulation study.     

Pharmacokinetic parameters estimation using adaptive Bayesian P-splines models

In preclinical and clinical experiments, pharmacokinetic (PK) studies are designed to analyse the evolution of drug concentration in plasma over time i.e. the PK profile. Some PK parameters are estimated in order to summarize the complete drug's kinetic profile: area under the curve (AUC), maximal concentration (C(max)), time at which the maximal concentration occurs (t(max)) and half-life time (t(1/2)).Several methods have been proposed to estimate these PK parameters. A first method relies on interpolating between observed concentrations. The interpolation method is often chosen linear. This method is simple and fast. Another method relies on compartmental modelling. In this case, nonlinear methods are used to estimate parameters of a chosen compartmental model. This method provides generally good results. However, if the data are sparse and noisy, two difficulties can arise with this method. The first one is related to the choice of the suitable compartmental model given the small number of data available in preclinical experiment for instance. Second, nonlinear methods can fail to converge. Much work has been done recently to circumvent these problems (J. Pharmacokinet. Pharmacodyn. 2007; 34:229-249, Stat. Comput., to appear, Biometrical J., to appear, ESAIM P&S 2004; 8:115-131).In this paper, we propose a Bayesian nonparametric model based on P-splines. This method provides good PK parameters estimation, whatever be the number of available observations and the level of noise in the data. Simulations show that the proposed method provides better PK parameters estimations than the interpolation method, both in terms of bias and precision. The Bayesian nonparametric method provides also better AUC and t(1/2) estimations than a correctly specified compartmental model, whereas this last method performs better in t(max) and C(max) estimations.We extend the basic model to a hierarchical one that treats the case where we have concentrations from different subjects. We are then able to get individual PK parameter estimations. Finally, with Bayesian methods, we can get easily some uncertainty measures by obtaining credibility sets for each PK parameter.     

Design of robust parameter experiments in a continuous space using an evolutionary optimization algorithm

Two common experimental designs used in robust parameter design (RPD) are crossed array and mixed resolution designs. However, the prohibited number of runs, constraints in the design space or special model requirements render some of these designs inadequate. This paper presents the application of an evolutionary strategy to produce nearly optimal design matrices for RPD. The designs are derived by solving a nonlinear optimization problem involving both 𝒟- and 𝒢-efficiency simultaneously. The methodology presented allows the user to obtain new exact designs for a specific number of runs, and a particular experimental region. The combination of 𝒟- and 𝒢-efficiency results in experimental designs that outperform the corresponding benchmarks.     

Optimal cross-over designs for nonlinear mixed models using a first-order expansion

We consider the construction of optimal cross-over designs for nonlinear mixed effect models based on the first-order expansion. We show that for AB/BA designs a balanced subject allocation is optimal when the parameters depend on treatments only. For multiple period, multiple sequence designs, uniform designs are optimal among dual balanced designs under the same conditions. As a by-product, the same results hold for multivariate linear mixed models with variances depending on treatments.     

CVX‐based algorithms for constructing various optimal regression designs

CVX‐based numerical algorithms are widely and freely available for solving convex optimization problems but their applications to solve optimal design problems are limited. Using the CVX programs in MATLAB, we demonstrate their utility and flexibility over traditional algorithms in statistics for finding different types of optimal approximate designs under a convex criterion for nonlinear models. They are generally fast and easy to implement for any model and any convex optimality criterion. We derive theoretical properties of the algorithms and use them to generate new A‐, c‐, D‐ and E‐optimal designs for various nonlinear models, including multi‐stage and multi‐objective optimal designs. We report properties of the optimal designs and provide sample CVX program codes for some of our examples that users can amend to find tailored optimal designs for their problems. The Canadian Journal of Statistics 47: 374–391; 2019 © 2019 Statistical Society of Canada