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(3)

2.3 PKPD simulation example

In this paper, we use a simple and typical PKPD model as an example to evaluate MF by simulation. It is derived from the one used by Hooker et al. [25] to illustrate the development of the Fisher information matrix for a multiple response model. The PK model for drug concentration is a one compartment with bolus input and first order elimination given as follows for the sampling time tPK:

fPK(θPK,tPK)=

where θPK=(Cl,VC)is the vector of the PK parameters with Cl and VC, the clearance and the volume in the central compartment, respectively.

T

doseCl

exp( tPK) VCVC

(12)

The PD model for drug effect is a simple Emax model with baseline, expressed as a function of the predicted concentrationsfPK, and given as follows for the sampling times tPD :

fPD(θPK,θPD,tPD)=E0+

where θPD=(E0,Emax,C50) is the vector of the PD parameters with E0, Emax andC50, the effect at baseline, the maximum effect and the concentration needed to observe half of the maximum effect, respectively.

We assumed an exponential model of the random effects for both the PK and the PD parameters. We associated a proportional error model with the PK model characterized by the parameter σslopePK and a homoscedastic error model with the PD model characterized by the parameter σinterPD. Thus, the vector of population parameters Ψ is described by the vector of the fixed effects βT=βCl,βV,βE,βE,βC and by λT the vector composed by the

C0max50variance of the random effects and by the parameters for the error models such that

22222

λT=(ωCl,ωV,ωE,ωE,ωC,σslopePK,σinterPD). The dose was fixed to 1 and the parameter

C

max

50

EmaxfPK(θPK,tPD)C50+fPK(θPK,tPD)

(13)

T

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()

values used in this paper are given in Table 1.

We determined a population design associated with this PKPD example. This determination was empirical, without any optimization. The population design was composed of one group of N=100 individuals. They all had 3 sampling times at 0.166, 6 and 12 for PK and 4 sampling times for PD at 0.166, 6, 12 and 20 hours. Therefore, we had one elementary design

(ξPK,ξPD) with ξPK=(0.166,6,12) and ξPD=(0.166,6,12,20). The population design was

thus defined byΞ= ξPK,ξPD,N . The curve profiles of the PK and the PD model for the

fixed effects are displayed in Figure 1; the sampling times for each response are overlaid.

2.4 Evaluation of MF for multiple responses

2.4.1 Comparison of MF with and without linearization

In this section, we propose to compare the predicted SE obtained from the approximate MF for multiple responses computed by PFIM 3.0 to the SE obtained from more “exact” approaches using the SAEM estimation algorithm. This latter algorithm was used by Retout et al. [20]

{(

)

}

and Samson et al. [29] to show the appropriateness of this approximation in a single response model. This SAEM algorithm allows the observed population Fisher information matrix to be computed according to two approaches. The first approach was developed by Samson et al. [29] and has been used to evaluate an “exact” population Fisher information matrix using the Louis’s principle [30]. It does not require any linearization and can thus be considered as the “true” population Fisher information matrix. The second approach evaluates the Fisher information matrix using a linearization of the model around the conditional expectation of the individual parameters previously estimated by SAEM without any linearization. To perform this comparison, we first computed the predicted MF for the population design associated with the PKPD example using PFIM 3.0, based on the linearization. We then simulated a dataset of PK and PD observations for 10 000 individuals in order to achieve asymptotic properties using the software R 2.4.1. To do that, we used the parameter values given in Table 1 and the sampling times shown in Figure 1, defining the PKPD example (section 2.3). For each individual i, we simulated a vector of random effects bi in N(0, ), where the diagonal elements of are the variance of the random effects, and we calculated the individual parameters usingθi=βexp(bi). We then calculated the individual PK concentrations fPK(θPK,tPK) predicted by the model at each time tPKof ξPK. We also computed the individual PK concentrations at each time tPD of ξPD to derive the concentration fPK(θPK,tPD) for the PD response using Equation (13). PD observations

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fPD(θPK,θPD,tPD) were then generated. Finally, for each response, we simulated the random

errors εPKand εPD from a normal distribution with zero mean and variance derived from Equation (6) using the parameters σslopePK andσinterPD, respectively. Those errors were added to the previously generated PK and PD data to form the simulated observations for the PK and the PD response respectively.

Using MONOLIX (Version 2.1) with SAEM as the estimation algorithm, we estimated the parameters using this simulated dataset and we then derived the observed population Fisher information matrix with the Louis’s principle procedure and the linearization method of SAEM. For these two Fisher information matrices, we then transformed the observed SE for each component of the population vector Ψ obtained with a simulation of Nsim=10000 individuals into predicted SE of a population of N=100 individuals to be adapted to the

design of the example using SEN(Ψi)=SENsim(Ψi, for the ithcomponent of Ψ.

For estimation with the SAEM algorithm, we used an initial set of parameters with the values

(0.2,0.05,1.2,5,1.5) for the fixed effects, (1,1,1,1,0.5) for the variance of the random effects

and (0.5,0.5) for the …… 此处隐藏：6040字，全部文档内容请下载后查看。喜欢就下载吧 ……