Design evaluation and optimisation in multi-response nonlinear mixed effect models with cost functions: application to the pharmacokinetics of zidovudine and its active metabolite
Caroline Bazzoli, Sylvie Retout, France Mentré
UMR738, Inserm and, University Paris Diderot, Paris, France.
Introduction: Models with multiple responses within patients are increasingly used in population analyses. Main examples are joint pharmacokinetic-pharmacodynamic models, complex pharmacodynamic models and pharmacokinetic models of parent drug and metabolite(s). In this context, efficient tools for population designs evaluation and optimisation are necessary. For complex models it is indeed difficult to guess good empirical designs especially when limitations are imposed in the number of samples in each patient. The methodology for optimal population design based on the Fisher information matrix for nonlinear mixed effect models has been initially developed and evaluated [1, 2] for single response models. It has been implemented in several softwares including PFIM, an R function . Regarding design optimisation, algorithms are required either to optimise exact designs or statistical designs. The Fedorov-Wynn algorithm is particularly adapted to this last approach optimising both proportions of subject associated to each group (design structure) and the samples and their allocation in time.
Our objectives were: 1) to evaluate the expression of the Fisher information matrix for multiple response models, 2) to propose a new extension of the Fedorov-Wynn including cost functions, 3) to extend the R function PFIM for multiple response models with discrete covariates and intra-occasion variability, 4) to apply these new developments to the joint pharmacokinetic modeling of zidovudine and its active metabolite.
Design evaluation and optimisation for multiple response models
a) Expression of the Fisher information
We extended the expression of the Fisher information matrix for multiple response models [4, 5] using a linearisation of the model as proposed for a single response by Mentré et al. . Using a pharmacokinetic / pharmacodynamic model example , we evaluated the relevance of the predicted standard errors (SE) computed by linearisation. To do that, first, we compared those SE to those computed under asymptotic convergence assumption using the SAEM algorithm  through a simulation of 10000 subjects. We also compared those predicted SE to the empirical SE, defined as the standard deviation on the 1000 estimates, obtained with three algorithms: two algorithms based on a linearisation of the model (FO, FOCE) in the software NONMEM and the SAEM algorithm in MONOLIX. The SE computed by linearisation are equivalent to those predicted by SAEM and to the empirical ones obtained with FOCE and SAEM. Regarding FO, the empirical ones are much larger than the SE computed by linearisation and those obtained with FOCE or SAEM.
b) Design optimisation: extension of the Fedorov-Wynn algorithm
Usually, design optimisation is done for a fixed total number of samples without any consideration on the relative feasibility of the optimised sampling times or the group structure. Mentré et al.  proposed an approach allowing to take into account the cost of each sample in the context of single response model. From the extension of the Fisher information matrix for multiple responses, the Fedorov-Wynn algorithm was extended to the introduction of cost functions allowing design optimisation for several responses for a fixed total cost . The classical cost function defined the cost of an elementary design as the sum of the number of samples for each response. More complex cost functions can be implemented as for instance an additional cost for a new patient, different cost for the different responses, penalties for delay between samples.
Extensions of PFIM
a) PFIM 3.0
From the relevance of the expression of the Fisher information matrix for multiple responses and the interest of the use of the Fedorov-Wynn algorithm for design optimisation, we proposed extensions of the software tool PFIM. We first developed PFIM 3.0  to accommodate multiple response models. Other options were added in PFIM 3.0 for model specification or optimisation. Models can be specified either with their analytical form or by using a system of differential equations and library of analytical pharmacokinetic models was added. Design optimisation is performed using the D-optimal criterion optimization and the Fedorov-Wynn algorithm was implemented in PFIM 3.0 as an alternative to the Simplex algorithm.
b) PFIM 3.2
More recently, we proposed the version PFIM 3.2 based on an extension of the R function PFIM 3.0. This new version, released in January 2010, includes several new features in terms of model specification and expression of the Fisher information matrix. Regarding model specification, the library of standard pharmacokinetic models was completed and a library of pharmacodynamic models is now available. It is now also possible in PFIM 3.2 to use models including inter-occasion variability (IOV) with replicated designs at each occasion  and to compute the Fisher information matrix for models including fixed effects for the influence of discrete covariates on the parameters . It can be specified if covariates change or not accross occasions. The computation of the predicted power of the Wald test for comparison or equivalence tests, for a given distribution of the discrete covariate, as well as the number of subjects needed to achieve a given power can be computed.
PFIM versions and extensive documentations [12, 13] are freely available on the PFIM website: http://www.pfim.biostat.fr/.
Application to the pharmacokinetic of zidovudine and its active metabolite
We applied these developments to the plasma and intracellular pharmacokinetics of zidovudine (ZDV), a nucleoside reverse transcriptase inhibitors (NRTI), in HIV patients. Indeed, all NRTI undergo a series of sequential phosphorylation reactions producing triphosphates (TP) in the cell. ZDV is thus metabolised intracellularly to its active metabolite (ZDV-TP), necessary for antiviral activity . We first determined the first joint population model of ZDV and its active metabolite ZDV-TP. Data are obtained from the COPHAR 2-ANRS 111 trial  in 75 naïve HIV patients receiving oral combination of ZDV, as part of their HAART treatment. Four blood samples per patient were taken after two weeks of treatment to measure the concentrations at steady state. Intracellular concentrations, costly to analyse, were measured in 62 patients. Using the SAEM algorithm implemented in the MONOLIX software, we estimated the pharmacokinetic parameters of ZDV and its active metabolite. We then aimed at designing new trial for this joint population analysis. Based upon the joint population pharmacokinetic model, we evaluated the empirical design used in COPHAR 2 assuming 50 subjects with 4 measurements of each response. We then explored D-optimal population designs with PFIM 3.0. First, the optimisation was done for a fixed total number of samples meaning that the cost of a design was proportional to the number of samples. We then optimised designs through the use of three different cost functions using a working version of PFIM. Optimisation was done for a same total cost defined by the total number of sampling times of the empirical design i.e. 400 for both responses.
A one compartment model with first order absorption and elimination best described plasma ZDV concentration, with an additional compartment describing the metabolism of the drug to intracellular ZDV -TP with a first order elimination . The optimal design with the classical cost function shows that a design with only three samples for ZDV and two samples for ZDV-TP with adequate allocation in time, allows to estimate parameters as precisely as the empirical design but with less samples per patient. In addition, optimal designs were different according to the cost functions used. They are different in terms of sampling times but also in terms of group structure, reflecting the imposed penalties. Indeed, the optimal design penalising for example the addition of a new patient involve more sampling times per patient and a smaller number of patients.
Conclusion: We evaluated the extension of the Fisher information matrix for nonlinear mixed effect models with multiple responses using the usual first order linearisation. We used simulation and showed its relevance. We then developed and illustrated the usefulness of the Fedorov-Wynn algorithm with cost functions for design optimisation especially when substantial constraints on the design are imposed. We implemented these developments in new versions of the R function PFIM and we applied them to plasma and intracellular pharmacokinetics of zidovudine, an antiretroviral drug. We performed the first joint population analysis of zidovudine and its active metabolite in patients. We showed that population design optimisation allows to derive efficient designs according to clinical and technical constraints for further joint population pharmacokinetic analysis of this drug.
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 Bazzoli C, Benech H, Rey E, Retout S, Tréluyer JMT, Salmon D, Duval X, Mentré F and the COPHAR2- ANRS 111 study group. Pharmacokinetics of zidovudine, lamivudine and their active metabolites in HIV patients using joint population models. 10th International Workshop on Clinical Pharmacology of HIV Therapy, April 15-17, 2009, Amsterdam, The Netherlands. (Poster)