Fisher信息矩阵用于非线性混合效应的多重效应模型：用于的药代动(5)

approximation around the expectation of the random effects thus differ for design evaluation where derivatives of log-likelihood are computed and for parameter estimation. Several studies have considered the single response, stressing the limitations of this linearization for design evaluation. Using a simple model with few random effects, Han [32] found that MF was quite different when computed by linearization than by an adaptative Gaussian quadrature method. Merlé et al. [33] compared the Fisher information matrix computed by linearization to one computed by stochastic simulation and showed that the linearization seems to have no impact on the population D-optimal designs obtained but only on the true efficiency of the designs. Whether via adapative Gaussian quadrature or via stochastic simulation, the evaluation of the Fisher information matrix without linearization is computationally intensive and is also limited to a matrix of small dimensions. Finally, in the context of Bayesian design where prior distributions are used, Han et al. [34] proposed an attractive solution to compute MF; however, it is also time consuming.

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Another alternative to the linearization method is to compute the expected Fisher information matrix using the SAEM estimation algorithm, as proposed here. A large dataset is simulated to be close to the asymptotic properties. The observed MF is then estimated for the estimation performed in this dataset using one of the two methods of deriving MF proposed for SAEM. This simulation study shows a broader distribution of the observed RSE when MF is computed by the Louis’s principle [30] compared to the RSE observed with the linearization procedure. Nevertheless, correct results are obtained from both methods. Although the approach using the SAEM algorithm of MONOLIX version 2.1 can be applied for problems with a large number of random effects, it is time consuming (hours) compared to PFIM (seconds). It may be required in specific cases such as when design evaluation is used to predict the power of a test to detect covariates [20]. The appropriateness of the RSEs of PFIM combined with its fast execution emphasizes its advantage in cases of design optimization where a large number of MF often have to be computed. Moreover, in design evaluation and optimization for nonlinear models, some a priori values of the parameters are required. Often, they are not precisely known, and therefore the need to use exact methods to predict MF is questionable.

In this study, we empirically determined a population design associated with a simple PKPD example for which the dose was equal to 1. No change in the dose was envisaged because the main purpose of this work was to evaluate MF for a PKPD model associated with one population design. However, it would be interesting to study the influence of dose on the population design and thus to plan dose optimization in order to have a better understanding of the relationship between the PK and the PD model.

In the present study, we considered only the case of a diagonal matrix with no correlation between the random effects of the PK and the PD parameters. This could be extended by exploring the appropriateness of MF when the PK parameters are directly correlated with the PD parameters, and thus the development of MF for a full matrix. This development was performed by Mentré et al. [17] for the correlation of the random effects parameters of single response models and recently by Ogungbenro et al. [26] for multiple response models. Furthermore, the population Fisher information matrix implemented in PFIM 3.0 is approximated by a block diagonal matrix assuming that the variance of the observations with respect to the mean parameters is constant (see Appendix). It would therefore be interesting to investigate the influence of this assumption on the computation of the Fisher information matrix.

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In conclusion, the comparison of RSE predicted by PFIM to RSE computed without any linearization by SAEM as well as to RSE obtained from the simulation study, supports the appropriateness of the approximated Fisher information matrix for multiple response models. Its implementation in PFIM 3.0 provides a useful computing tool for design evaluation and optimization in the development of PKPD or PK studies.

ACKNOWLEDGEMENTS

Part of this work was supported by a grant from F. Hoffmann La Roche Ltd, Basel, Switzerland. The authors would like to acknowledge Pr Marc Lavielle for his help regarding the use of the MONOLIX software as well as the IFR02 of INSERM and Hervé Le Nagard for the use of the “centre de biomodélisation”.

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