Cyrielle Dumont
email: cyrielle.dumont@inserm.fr

Marylore Chenel

France Mentré

INFLUENCE OF COVARIANCE BETWEEN RANDOM EFFECTS IN DE-SIGN FOR NONLINEAR MIXED EFFECT MODELS WITH AN ILLUSTRA-TION IN PAEDIATRIC PHARMACOKINETICS

Keywords: D-optimality, Fisher information matrix, Nonlinear mixed effect models, Optimal design, Paediatric studies, PFIM, Population pharmacokinetics

come

Influence of covariance between random effects in design

for nonlinear mixed-effect models with an illustration in pediatric pharmacokinetics.

INTRODUCTION

Nonlinear mixed-effect models (NLMEM) are used increasingly during drug development for the analysis of dose-concentration-response data. They allow for a sparse sampling design with few data points per individual in a set of individuals and estimation of mean parameters, their inter-individual variability and covariate effects. This can be particularly useful in clinical trials with specific populations, such as children (Tod et al., 2008), where classical studies with a large number of samples are often limited for ethical or physiological reasons.

An appropriate choice of experimental design for estimating parameters in NLMEM is required. NLMEM are also now commonly used for the joint modelling of several biological responses, such as the joint analysis of the pharmacokinetics (PK) and pharmacodynamics (PD) of a drug, or the PK of parent drugs and their active metabolites.

Designs in NLMEM are called population designs. Population designs consist of a set of elementary designs to be carried out in a set of individuals [START_REF] Mentré | Optimal design in random-effects regression models[END_REF]. Elementary designs are composed of the design variables to be performed for each individual. Design variables are for example the number of samples by individual and the location of sampling times but can be more general. The choice of the design variables and the balance between the number of individuals and the amount of information by individual have a large impact on the precision of population parameter estimates. In this work, the sampling times were considered as the only design variables. Approaches to evaluating and optimizing the designs are required. To avoid simulations, which are time consuming, designs can be evaluated using the Fisher information matrix (M F ) and the optimization of its determinant. The expression of M F for the single-response model was proposed by [START_REF] Mentré | Optimal design in random-effects regression models[END_REF] and [START_REF] Retout | Development and implementation of the population Fisher information matrix for the evaluation of population pharmacokinetic designs[END_REF], using a first-order Taylor expansion of the model around the expectation of random effects. The expression of M F for multiple responses was first extended by [START_REF] Hooker | Simultaneous population optimal design for pharmacokineticpharmacodynamic experiments[END_REF], using the same development as for a single-response model. The expression of M F for nonlinear mixed-effect multiple-response models was also implemented in PFIM, an R function dedicated to design evaluation and optimization, and was evaluated by simulations [START_REF] Bazzoli | Fisher information matrix for nonlinear mixed effects multiple response models: evaluation of the appropriateness of the first order linearization using a pharmacokinetic/pharmacodynamic model[END_REF]. Several software packages, which implement an evaluation of the population M F and optimization of the experimental designs in the context of NLMEM, are available, notably PFIM in R [START_REF] Bazzoli | Design evaluation and optimisation in multiple response non linear mixed effects models: PFIM 3.0[END_REF].

For M F evaluation, it is often assumed that the variance-covariance matrix of random effects is diagonal, which means that individual random effects are assumed to be independent. The consequence is that covariance terms between parameters are ignored at the design stage. [START_REF] Ogungbenro | Incorporating correlation in interindividual variability for the optimal design of multiresponse pharmacokinetic experiments[END_REF] developed an expression of M F that accounts for covariance between random effects for multiple-response population PK and for population PK/pharmacodynamics (PD). The objectives of Ogungbenro et al. were to investigate the implications of including and excluding correlation between random effects and to investigate the effect of correlation between responses on the optimal sampling times. They performed a simulation and presented the empirical and expected coefficients of variation of the parameter estimates for a PK/PD example as well as the mean relative error and root mean square error for the same example, with and without correlation, to assess the level of bias and precision of the estimates. The aim of the present work, is to investigate the impact of the size of covariance between random effects on the optimal sampling-time design for a pharmacokinetic example and in addition to study its effect on predicted standard errors (SE), predicted relative standard errors (RSE) and the amount of information, taking into consideration covariance between random effects. Moreover and conversely to the work of Ogungbenro et al., in the present work, design optimization allowed several groups of elementary designs. In addition, it is shown how the SEs in the framework of rich individual data can be predicted analytically, without using the model. Assuming rich individual data, it can be assumed that individual parameters of each subject are known without error. Estimation of fixed effects and variance for these parameters is standard and predicted SEs can be derived analytically.

The results are illustrated by applying these developments to the design of a clinical PK study in children of a molecule in development. To illustrate the results, a molecule and its main metabolite were chosen. Indeed, as the parent drug is metabolised into an active metabolite, concentration-time data of both parent drug and its metabolite are of interest. Developments are applied to the single-response model (parent-drug model) and to the multiple-response model (joint parent-and metabolite-model). To optimize sampling times, it is necessary to have a priori information, and in this case, a priori information was model and parameters obtained from data generated by the SIMCYP software using the adults-tochildren extrapolation tool, the knowledge of the drug in adults, its physico-chemical properties and in vitro experimental data. SIMCYP is a software program of Physiologically-Based PharmacoKinetic (PBPK) modelling [START_REF] Jamei | The simcyp populationbased ADME simulator[END_REF]. PBPK models make it possible to describe the biodistribution of a substance (absorption, distribution, metabolism, excretion) and are based on a physiological reality. SIMCYP can predict the evolution of concentration versus time for a given molecule and in a target population (healthy volunteers, Caucasians, children, etc.). More specifically for the present study, the paediatric module of SIMCYP took into account all the changes occurring during childhood [START_REF] Rostami-Hodjegan | Prediction of the clearance of eleven drugs and associated variability in neonates, infants and children[END_REF]. Concentration-time profiles for parent and metabolite in children from SIMCYP simulations have already been analysed by NLMEM with the NONMEM software [START_REF] Brendel | Using Modelling & Simulation techniques to optimise the design of a paediatric PK/PD study. Population Approach Group in Europe[END_REF].

The present article is divided into four sections including the introduction. Section 2 presents the methodological development with notation definitions in 2.1. The calculation of the SEs of correlation and covariance terms is presented in subsection 2.2, methodology to approximate M F and the SEs in the framework of individual-rich data is presented in subsection 2.3. In the fourth subsection (2.4), the focus is on the implementation and optimization of M F considering covariance between random effects in PFIM. Section 3 first presents the PK example in paediatrics and the models (3.1). Then, the methods and results concerning the evaluation of the extension of M F in PFIM are shown (3.2). The two last subsections present the methods and the results of the approximation of the M F calculation (3.3) and the impact of covariance on SEs, the amount of information and optimal designs (3.4).

METHODOLOGICAL DEVELOPMENT

Model and notations

A nonlinear mixed-effect multiple-response model, or a multiple response population model, is defined as follows. The vector of observations Y i for the i th individual is defined as the vector of the K different responses:

Y i = [y T i1 , y T i2 , ..., y T iK ] T , where y ik , k = 1, ..., K,
is the vector of n ik observations for the k th response. Each of these responses is associated with a known function f k , which defines the nonlinear structural model. The K functions f k can be grouped in a vector of multiple-response model F, such as

F (θ i , ξ i ) = [f 1 (θ i , ξ i1 ) T , f 2 (θ i , ξ i2 ) T , ..., f K (θ i , ξ iK ) T ] T
, where ξ i is an elementary design for one individual i, composed of K sub-designs such that ξ i = (ξ i1 , ξ i2 , ..., ξ iK ). ξ i is defined by n i sampling times and its sub-design ξ ik is defined by (t ik1 , t ik2 , ..., t ikn ik ), with n ik sampling times for the observations of the k th response, so that

n i = K k=1 n ik .
θ i is the vector of all the individual parameters for all the response models in individual i. The vector of individual parameters θ i depends on β, the p-vector of the fixed effects parameters and on b i , the vector of the p random effects for individual i. The relation between

θ i and (β,b i ) is modelled by a function g, that is θ i = g(β, b i ), which is usually additive, so that θ i = β + b i , or exponential so that θ i = exp(β * + b i ), where exp(β * ) = β. It is assumed that b i ∼ N (0, Ω)
with Ω defined as a p × p-positive definite matrix, for which, each diagonal element ω 2 rr , r = 1, ..., p, represents the variance of the r th component of the vector b i and ω rs , r = 1, ..., p and s = 1, ...p, with r = s, represents the covariance between the r th and the s th components of the vector b i .

The statistical model is then given by Y i = F (g(β, b i ), ξ i ) + ε i , where ε i is the vector composed of the K vectors of residual errors ε ik , k = 1, ..., K, associated with the K re-

sponses. It is also supposed that ε ik ∼ N (0, Σ ik ) with Σ ik a n ik × n ik -diagonal matrix such that: Σ ik (β, b i , σ inter k , σ slope k , ξ ik ) = diag(σ inter k + σ slope k × f k (g(β, b i ), ξ ik )) 2
where σ inter k and σ slope k are two parameters of the model for the variance of the residual error of the k th response. The case σ slope k = 0 returns a homoscedastic error model, whereas the case σ inter k = 0 returns a constant coefficient of variation error model. The case where the two parameters differ from zero is called a combined error model. The variance of ε i over the K responses is then noted Σ i (β, b i , σ inter , σ slope , ξ i ), where Σ i is a n i × n i -diagonal matrix composed of each diagonal element of Σ ik with k = 1, ..., K and σ inter and σ slope are two vectors of the K components σ inter k and σ slope k , k = 1, ..., K, respectively. Finally, conditionnally on the value of b i , it is assumed that the ε i errors are independently distributed.

Let λ be the vector of the unknown variance-covariance terms, corresponding to the vector of the lower triangular of Ω, which contains p(p+1)/2 variance-covariance terms and all the parameters of the error model.

Then, λ is expressed as λ T = (ω 2 11 , ..., ω 2 pp , ω 21 , ω 31 , ω 32 , ..., ω p,p-1 , σ T inter , σ T slope ). Let Ψ be the vector of the population parameters to be estimated such as Ψ T = (β T , λ T ).

For N individuals, a population design is composed of the N elementary designs ξ i , i = 1, ..., N . A population design is therefore described by the N elementary designs for a total number n tot of observations such as n tot = N i=1 n i and Ξ = {ξ 1 , ..., ξ N }. Usually, population designs are composed of a limited number Q of groups of individuals with identical designs and identical covariates within each group. Each of these groups is defined by an elementary design ξ q , q = 1, ..., Q, which is composed, for the k th response, especially of n qk sampling times (t qk1 , t qk2 , ..., t qkn qk ) to be performed in a number N q of individuals. The population design can then be written as follows:

Ξ = {[ξ 1 , N 1 ]; [ξ 2 , N 2 ]; ...; [ξ Q , N Q ]}.

Calculation of the standard errors of correlation and covariance

The population Fisher information matrix for a population design Ξ is computed as the sum of the N elementary Fisher information matrices M F (Ψ, ξ i ) for each individual i:

M F (Ψ, Ξ) = N i=1 M F (Ψ, ξ i ).
In the case of a limited number Q of groups, it is expressed by [START_REF] Ogungbenro | Incorporating correlation in interindividual variability for the optimal design of multiresponse pharmacokinetic experiments[END_REF] have developed the expression of M F in NLMEM that accounts for covariance between random effects for multiple responses, i.e. with a non-diagonal Ω matrix. This expression is implemented in a working version of PFIM and the corresponding development is detailed in the appendix.

M F (Ψ, Ξ) = Q q=1 N q M F (Ψ, ξ q ).
From the square roots of the diagonal elements of M -1 F , the predicted SE for estimated parameters can be calculated. The aim is to predict the SE for estimated correlation, the correlation being the ratio of the covariance and the product of the standard deviations. It should be noted that, if ϕ(θ) is a function of θ, then M F (θ) = JM F (ϕ(θ))J T , with J = ∂ϕ(θ) T ∂θ . In the following, B 11 , B 22 , B 33 , B 12 , B 13 and B 23 are terms of the 3 × 3 sub-matrix of B, formed of lines and columns corresponding to ω mm , ω ll and ω ml , respectively. Thus:

SE 2 ( ρ ml ) = ρ 2 ml ( 1 4ω 4 mm B -1 11 + 1 4ω 4 ll B -1 22 + 1 ω 2 mm ω 2 ll ρ 2 ml B -1 33 + 1 2ω 2 mm ω 2 ll B -1 12 - 1 ω 3 mm ω ll ρ ml B -1 13 - 1 ωmmω 3 ll ρ ml B -1 23 )
And vice versa, if B is now computed as a function of ω mm , ω ll and ρ ml , it becomes

SE 2 ( ω ml ) = ω 2 ml ( 1 4ω 4 mm B -1 11 + 1 4ω 4 ll B -1 22 + ω 2 mm ω 2 ll ω 2 ml B -1 33 + 1 2ω 2 mm ω 2 ll B -1 12 + ω ll ωmmω ml B -1 13 + ωmm ω ll ω ml B -1 23 )
where estimators are designated with .

Approximation of the Fisher information matrix for rich design

In NLMEM, the parameters themselves are not estimated, but their mean and variance are. Let's consider that the vector of individual parameters θ i of each subject is known without error, and estimation of β and Ω is performed from this set of N individual parameters.

The SEs for β and Ω can be predicted analytically without using the model and will depend on the design only through N, the number of subjects, therefore Ξ = N . To derive the expression of the M F in that case, we assume that we have N vectors of observation θ i with θ ∼ N (β, Ω). In that case, Ψ * T = (β T , λ * T ), where λ * T = vec(Ω). The general term of the m th line and l th column of the elementary Fisher information matrix for β and Ω is given by expression:

M * F (Ψ * , N ) m,l = ∂β T ∂Ψ * m Ω -1 ∂β ∂Ψ * l + 1 2 tr(Ω -1 ∂Ω ∂Ψ * m Ω -1 ∂Ω ∂Ψ * l )
where tr denotes the trace of a matrix.

The derivative of β with respect to fixed effects is equal to 1, and with respect to variance terms is equal to 0. The derivative of Ω with respect to fixed effects is 0 while the derivative of Ω with respect to variance terms is equal to a matrix composed of 1 for the corresponding term and 0 elsewhere. Therefore, the information matrix is block diagonal.

The block of M * F for the fixed effects, called block A, is given by M * F (β, N ) = Ω -1 . The block for λ * , called block B, has the following components:

M * F (λ * , N ) m,l = 1 2 tr(Ω -1 ∂Ω ∂λ * m Ω -1 ∂Ω ∂λ * l ), m, l = 1, ..., dim(λ * ).
The population Fisher information matrix for N individuals is obtained with the above formula multiplied by N. It is then inverted to obtain SEs.

From these expressions, it can be shown that for β, the diagonal terms of block A -1 are then equal to 1 N × ω 2 mm in the case of additive random effect and to 1 N × ω 2 mm × β 2 m in the case of exponential random effect, m = 1, ..., p. The standard errors for the fixed effects are therefore given by

SE( β m ) = 1 √ N × ω mm , for additive random effect, SE( β m ) = 1 √ N × ω mm × β m , for exponential random effect, respectively.
For Ω, the diagonal terms of the block B -1 are equal to 1 N × 2 × ω 4 mm when the index corresponds to a variance term and to 1 N ×(ω 2 mm ω 2 ll +ω 2 ml ) when it corresponds to a covariance term. The SE of a variance parameter is equal to

SE( ω 2 mm ) = 1 √ N × √ 2 × ω 2 mm
and that of a covariance parameter is

SE( ω ml ) = 1 √ N × ω 2 mm × ω 2 ll + ω 2 ml .
Similarly, the results for the correlation can be obtained:

SE( ρ ml ) = 1 √ N × (1 -ρ 2 ml ).
For variance components, the results are the same for additive and exponential random effect models.

Using this approach, the SEs can be viewed as lower limits of the expected SEs that we would get in the case of a rich design, where individual parameters are estimated rather precisely. They are the lower bound of any design Ξ with N subjects.

Implementation in PFIM and optimization

Having carried out the extension of the M F for NLMEM with covariance between random effects, the calculation was implemented into a working version of PFIM, making it possible to evaluate and optimize designs with covariance between random effects for single-and multiple-response models. The impact of covariance size on SE and amount of information was examined. The total information was evaluated through the criterion:

criterion = det(M F ) 1/P
with P being the total number of parameters in Ψ.

Optimization was performed using the Federov-Wynn algorithm implemented in PFIM. The Federov-Wynn algorithm maximises the determinant of M F using statistical design within a finite set of sampling times. It can be used to optimize both the structure (number of groups, number of subjects per group, number of samples per group) and the sampling times from a finite set of times [START_REF] Retout | Design in nonlinear mixed effects models: Optimization using the Federov-Wynn algorithm and power of the Wald test for binary covariates[END_REF]. Only one elementary design in all patients (i.e. one group, with a fixed number of samples) can be assumed.

EVALUATION VIA A PHARMACOKINETIC EX-AMPLE IN PAEDIATRICS

Presentation of the example

In this section, the PK examples, to which the new developments will be applied, are presented. Data used to derive the models were "simulated" plasma concentration versus time in patients between 0 and 25 years old, obtained via the SIMCYP software using the adultsto-children extrapolation tool, the knowledge of the drug in adults, its physico-chemical properties and in vitro experimental data [START_REF] Perdaems | Predictions of metabolic drug-drug interactions using physiologically based modelling[END_REF]. As the parent drug is metabolised into an active metabolite, concentration-time profiles of both the parent drug and its active metabolite were predicted in children. The plasma concentration-time profiles resulted from an intravenous bolus dose of 0.1 mg/kg in 400 children.

After intravenous injection in children, concentration-time data of the parent drug were described by a 3-compartment model (single-response model), which is written in ordinary differential equations as follows:

             dA 1 (t) dt = k 21 A 2 (t) + k 31 A 3 (t) -(k 12 + k 13 + k 10 )A 1 (t) dA 2 (t) dt = k 12 A 1 (t) -k 21 A 2 (t) dA 3 (t) dt = k 13 A 1 (t) -k 31 A 3 (t)
A 1 , A 2 and A 3 being the amount of parent drug in the first, second, third compartment respectively, k ij being the distribution rate constant from compartment i to compartment j. For each compartment, C i (t) = A i (t) V i and C 1 is the concentration of the parent drug in plasma. According to these notations, initial conditions are:

A 1 (0) = Dose, A 2 (0) = 0, A 3 (0) = 0.
The model is composed of 6 parameters: CLP = k 10 ×V 1 , which represents the elimination clearance,

Q 2 = k 12 × V 1 = k 21 × V 2 , the inter-compartmental clearance from compartment 1 to compartment 2, Q 3 = k 13 × V 1 = k 31 × V 3
, the inter-compartmental clearance from compartment 1 to compartment 3, and V 1 , V 2 , V 3 the volumes of the central compartment and of the two peripheral compartments, respectively.

The distribution for the parameters was exponential. Inter-individual variabilities were found on the CLP clearance and on the volume of the central compartment V 1 and were significantly correlated. The residual error model was a combined error model. σ inter P corresponds to the additive error for the parent molecule and σ slope P corresponds to the proportional part. Table 1 shows the parameter values: as the dose was given per kilogram, all the parameters and dose are expressed by kilogram.

Regarding the multiple-response model, both parent and metabolite concentration-time profiles were described by a 4-compartment model. Figure 1 represents the model and the ordinary differential equations describing the model are presented below:

                   dA 1 (t) dt = k 21 A 2 (t) + k 31 A 3 (t) -(k 12 + k 13 + k 14 + k 10 )A 1 (t) dA 2 (t) dt = k 12 A 1 (t) -k 21 A 2 (t) dA 3 (t) dt = k 13 A 1 (t) -k 31 A 3 (t) dA 4 (t) dt = k 14 A 1 (t) -k 40 A 4 (t)
A 4 is the amount of metabolite and C 4 (t) = A 4 (t) V 4 is the metabolite concentration in plasma. The initial conditions are the same as before, with A 4 (0) = 0.

For reasons of model identifiability, metabolite volume (V 4 ) was fixed as being the sum of the three volumes of the parent, that is,

V 4 = V 1 + V 2 + V 3 .
As before, CLP represents the elimination clearance for the parent, CLPM, defined as k 14 × V 1 , represents the clearance from the parent to the metabolite, and CLM, defined as k 40 × V 4 , represents the elimination clearance for the metabolite.

The parameter distribution was also exponential. Inter-individual variabilities were found on the CLP, CLPM and CLM clearances, and on the volume of the central compartment of the parent V 1 . As some of the corresponding random effects were significantly correlated, a full covariance matrix for those four parameters was retained. The residual error model was a combined error model. σ inter P (σ inter M ) corresponds to the additive error for the parent molecule (metabolite, respectively) and σ slope P (σ slope M ) its proportional part. The parameter values are presented in Table 1 and Figure 2 shows the mean PK profiles of the subjects; the parent on the left, and metabolite on the right.

Evaluation of the extension of the Fisher information matrix

including covariance between random effects in PFIM

Methods

To evaluate the extension of M F including covariance between random effects in PFIM for single-and multiple-responses, the results, with and without covariance, predicted by PFIM were compared with those obtained by both NONMEM and MONOLIX obtained from simulated data, using the models presented in the previous section. The parameter values used for the simulation are described in Table 1.

Data was simulated, using a rich design with 22 sampling times at 0.1, 0.2, 0.4, 0.6, 0.8, 1, 1.3, 1.6, 1.8, 2, 2.5, 3, 4, 5, 6, 7, 8, 9, 10, 12, 16 and 24 hours after dose injection, with 82 subjects and a dose equal to 0.1 mg/kg. The simulations with and without covariance were performed for both the single-response model (parent drug only) and for the multipleresponse model (joint parent-and metabolite-model). In each case, only one simulation was performed because we put ourself in the framework of rich design. The model without covariance was the same as that with covariance, except for the covariance terms, which were equal to zero. Two estimation methods were used to estimate parameters: First Order Conditional Estimate with Interaction (FOCEI) method in NONMEM version 6 (Sheiner and Beal, 1998) and the Stochastic Approximation Expectation-Maximization (SAEM) algorithm in MONOLIX version 3.2 (Samson et al., 2006). The estimation methods are different and so too were the calculations of M F . NONMEM estimates the covariance and its SE, whereas MONOLIX estimates the correlation and its SE. Estimating parameters with two different software programs provides a more accurate evaluation of the prediction using PFIM to observed SE. Parameter estimates obtained were called N and M for NONMEM and MONOLIX, respectively, whereas the parameters of the model are called the reference parameters (P) (Table 1).

The multiple-response model was implemented in PFIM, whereas the single-response model was available in the PFIM library (Dubois et al., 2011). The SE of parameters predicted by PFIM for the 2 models using the reference parameter values (Table 1) were then compared to SEs obtained by NONMEM and MONOLIX. Given that MONOLIX gives the SE of correlation and not of covariance, the SE of covariance was then derived by the approach detailed in section 2.3. Moreover, a comparison between results obtained by PopDes, the software for optimal design developed by Ogungbenro et al. [START_REF] Gueorguieva | A program for individual and population optimal design for univariate and multivariate response pharmacokinetic-pharmacodynamic models[END_REF], and results obtained by PFIM, was performed for the multiple-response model with and without covariance.

Results

The objective was to compare the SEs predicted by PFIM to those obtained with both NONMEM and MONOLIX, in both cases, i.e., with and without covariance. Table 2 presents the reference parameters with and without covariance of the single-response model (parent molecule model), as well as the parameter estimates obtained with NONMEM (N) and MONOLIX (M). The SEs predicted by PFIM, and those obtained by NONMEM and MONO-LIX with and without covariance for the PK model of the parent molecule, are also shown in Table 2. First of all, it can be seen that all parameters were accurately estimated and close to the reference parameter values. The SE values obtained in the case of the model without covariance with NONMEM and MONOLIX were very close to those predicted by PFIM for the single-response model. The largest difference between the SE of parameters was for Q 3 between MONOLIX and both PFIM and NONMEM, and was equal to 0.030. It corresponds to a relative standard error (RSE) equal to 12% for both PFIM and NONMEM and to 9% for MONOLIX. Similar differences were observed for the RSE of variance parameters (an absolute difference respectively equal to 3% and 4% between the RSEs for the variances of CLP and V 1 obtained from PFIM and those obtained from MONOLIX). It can be observed that the differences between the SE of the other parameters were small (lower than 0.010).

In the case of model with covariance, the SEs given by NONMEM and MONOLIX were also very close to the SE predicted by PFIM (the largest difference is still for Q 3 and is equal to 0.020 between NONMEM and MONOLIX and to 0.010 between PFIM and NONMEM). In terms of RSE, the differences between PFIM, NONMEM and MONOLIX for Q 3 were very small since the RSEs were equal to 12%, 12% and 11%, respectively. For the covariance term, the difference between the SE predicted by PFIM and the SEs obtained by both NONMEM and MONOLIX was equal to 0.0060, which is small. The main differences regarding the RSE of parameters were for the variance and covariance terms: the largest difference was equal to 10% between PFIM and NONMEM for the variance of V 1 . The SEs predicted by PFIM were similar to those obtained with NONMEM and MONOLIX, in both cases, with and without covariance, for the single-response model. Moreover, it was observed that the SEs of parameters were almost identical with and without covariance for PFIM, whether for fixed effects or variance components. The same results can be observed from the SEs obtained for both NONMEM and MONOLIX.

Table 3 displays the parameters and corresponding SE predicted by PFIM and obtained with NONMEM and MONOLIX, with and without covariance, for the PK model of parent and metabolite. The same conclusions can be drawn with the multiple-response model. As for the single-response model, it can be noted that PFIM accurately predicted the SE with and without covariance when compared to both NONMEM and MONOLIX. For some covariance terms corresponding to very small correlations, it was observed that the parameter values estimated by NONMEM and MONOLIX were different from the simulated values.

However, the SEs of the parameters that were predicted by PFIM were in a similar range to those obtained with NONMEM and MONOLIX, with and without covariance, for the multiple-response model. Same conclusions are observed for the RSE. Moreover, the predictions were identical for the fixed effects and variance parameters, with and without covariance.

It has therefore been demonstrated that the development of M F for covariance terms and its implementation in a working version of PFIM in R were satisfactory because the SEs of parameters predicted by PFIM were similar to those obtained with NONMEM and MONO-LIX, and that was also true for the SE of covariance terms. Table 4 also shows a comparison of RSEs obtained by PopDes and RSEs obtained by PFIM, for the multiple-response model with and without covariance. This evaluation shows the appropriateness of the extension of the population Fisher information matrix with covariance in PFIM. Moreover, it was observed that the results with covariance were similar to those without covariance for the SE and RSE of fixed effects and variance components.

Evaluation of approximation of the Fisher information matrix for rich design

To evaluate the results of the approximation of M F in the framework of individual-rich data, the SEs obtained by analytical predictions, using the formula of section 2.3 assuming N = 82, were compared with those predicted by PFIM for both single-and multiple-response models for the parameters with inter-individual variability (Table 5). This approach can only be applied to parameters with non-null variance of random effects.

The analytical SEs were very close to the SE predicted by PFIM for both single-and multiple-response models. The results obtained by analytical predictions assuming rich data were similar to those given by PFIM using the model on a rich design and the same number of patients. They represent the lower limits of what could be observed with a sparse design.

Influence of covariance on designs

Methods

First, the impact of the size of covariance on the SEs, the RSEs and the amount of information for the single-response model (parent-drug model) was considered. Using PFIM, the SE and the RSE on fixed effects and variance components assuming different values of covariance between the two random effects, i.e. CLP and V 1 , were predicted. The influence of the covariance on SE, RSE and amount of information was tested on a range of covariance from -0.12 to 0.12 (correlation from -0.9 to 0.9).

Second, the sampling time optimization for both the single-and multiple-response models with and without covariance is taken into consideration. The impact of covariance was studied on an optimal design with 6 sampling times out of 12 for the single-response model and with 4 out of 12 for the multiple-response model. For the latter, both response measurements (i.e. parent and metabolite concentrations) were carried out at each sampling time, i.e. a total of 8 measurements. The number of measurements was chosen depending on the number of fixed effects, 6 for the parent model and 8 for the joint parent and metabolite model. The design domain took place over a 24-hour period with allowed sampling times at 0.1, 0.2, 0.4, 0.8, 1, 2, 4, 6, 8, 12, 16 and 24 hours after dose injection.

Sampling time optimization was performed using the Federov-Wynn algorithm implemented in PFIM, which makes it possible to test for design with only one group (one) or with several groups (several ). Optimal designs for the single-response model (single) and the multiple-response model (multiple) without covariance (nocov ) were studied, as well as the covariance values of Table 1 (cov ). For the single-response model, which had a covariance equal to 0.10, design was optimized for a covariance of 0.12 (highcov ), corresponding to a correlation of 0.9. Optimizing design with the highest covariance value makes it possible to be in the furthest case and thus to have an idea of the impact of the covariance value when the latter is high. The corresponding designs were called Ξ single nocov one , Ξ single cov one , Ξ single highcov one , Ξ single nocov several , Ξ single cov several , Ξ single highcov several for the single-response model and Ξ multi nocov one , Ξ multi cov one , Ξ multi nocov several and Ξ multi cov several for the multiple-response model. Then, the optimized designs obtained with and without covariance were evaluated considering a model with a covariance equal to 0.12 for the single-response model and with covariance terms of Table 1 for the multiple-response model. Therefore, the optimized designs obtained with and without covariance were then evaluated taking into consideration a model with covariance, i.e. the same model was used with these two different designs, concerning the parent in a first time and concerning both parent and metabolite in a second time. The focus was then on the relative standard error (RSE) of parameters and the criterion.

Results

As expected, regardless of the value of covariance between -0.12 and 0.12, the SEs and the RSEs for fixed effects, variances and error terms were similar (results not shown). Figure 3 shows how the covariance had an impact on the SE and the RSE of covariance and the amount of information. The SEs of covariance increased for large values of covariance, but were similar for covariance between -0.03 and 0.03. On the other hand, the RSE decreased and the amount of information increased for large value of covariance.

Focus on the influence of covariance on optimal designs: for the single-response model and one group, the optimized designs with 6 times were identical for a covariance equal to 0 and equal to 0.10 (covariance value of Table 1) and the optimal sampling times were 0.1, 0.4, 0.8, 2, 6 and 16 hours after dose injection (Ξ single nocov one and Ξ single cov one ). For the Ξ single highcov one design, where cov = 0.12, the optimal sampling times were the same as those of the Ξ single nocov one and Ξ single cov one designs, except for the second sampling time which was 0.2 instead of 0.4. These designs were then evaluated taking into consideration a model with covariance equal to 0.12. Figure 4 shows the RSE of parameters obtained for each design. It can be seen that the RSEs were very similar between these two evaluations. The criterion, was equal to 1955 for evaluation of the Ξ single nocov one and Ξ single cov one designs, and 1971 for evaluation of the Ξ single highcov one design. So, the efficiency loss was minor as it was below 1%.

Concerning the comparison of the optimal Ξ single nocov several and Ξ single highcov several designs: Table 6 shows the elementary design and the number of subjects for the two different designs. As previously demonstrated, several optimal sampling times obtained were not the same, whether taking into consideration the covariance or not. However, similar to the optimizations with one group, when the criteria were compared between the Ξ single nocov several and Ξ single highcov several designs, respectively, equal to 1996 and 2025, the efficiency loss was very low (1.5%). Therefore, even if the optimal designs were not exactly the same with or without covariance, the design efficiency was barely affected.

Focus on the influence of the covariance on optimal designs for the multiple-response model: one optimal sampling time differed between the Ξ multi nocov one and the Ξ multi cov one designs. The Ξ multi nocov one design was composed of optimal sampling times at 0.2, 2, 8 and 24 hours after dose injection and in the Ξ multi cov one design, the second sampling time was not at 0.2 but rather at 0.1. Then, these two optimal designs were evaluated with a covariance term model. It was observed that RSEs were very similar between the Ξ multi nocov one and Ξ multi cov one designs, be it for fixed effect parameters, variance parameters or covariance terms. Moreover, when the two criteria obtained for these two designs were compared, they were equal to 1612 (Ξ multi nocov one design) and 1622 (Ξ multi cov one design), respectively.

Therefore, the efficiency loss here was still below 1%.

Considering several groups of subjects with 4 sampling times, Table 7 shows the different elementary designs with the corresponding number of subjects. The number of groups was different and some groups were present in the Ξ multi nocov several design and not in the Ξ multi cov several design and vice versa. Evaluating these designs with a covariance model, the RSEs of Ξ multi nocov several design and the RSEs of Ξ multi cov several design were very similar. Similarly, the criteria for these two designs were equal to 1747 and 1758, respectively. Therefore, the efficiency loss here is still minor (lower than 1%).

DISCUSSION

The first objective was to extend the Fisher information matrix in PFIM software for covariance between random effects and to evaluate it. For the evaluation, a comparison of the SEs of parameters predicted by PFIM and those given by two nonlinear mixed effect estimation software, NONMEM and MONOLIX, was carried out. To that end, data were simulated, using a rich design with 22 sampling times, a dose equal to 0.1 mg/kg in 82 individuals and with the single-response model and a more complex joint parent and metabolite model. Only one simulation was performed, it should not affect the results because it is a rich design with many subjects, so only very little changes are expected across replication. Also, the objective was the comparison between observed SE and predicted SE and not an evaluation of empirical SE. This comparison showed agreement between results predicted by PFIM and those obtained by both the NONMEM and MONOLIX software.

To carry out an optimal design, it is necessary to have a priori information, and in this paediatric example, the a priori information was model and parameters obtained from data generated by the SIMCYP software. The population PK models were fitted to the predicted SIMCYP data, which were "simulated" plasma concentration versus time for a parent drug and its active metabolite.

Then, the impact of the size of covariance was studied. In their work, [START_REF] Ogungbenro | Incorporating correlation in interindividual variability for the optimal design of multiresponse pharmacokinetic experiments[END_REF] showed that RSE were very similar for fixed effect parameters, while for variance parameters, they are slightly higher for design without covariance compared to design with covariance, especially for pharmacodynamic parameters. Regarding the SEs for the present work, they did not differ for the fixed parameters or the variance parameters, no matter the size of the covariance. Similarly, regardless of the value of covariance, the RSEs did not differ for the fixed parameters or the variance parameters. Therefore, it is perhaps not necessarily useful to incorporate covariance in design stage for those parameters. But further comparisons would be needed. On the other hand, it was observed that the SE of covariance increased when covariance was large whereas the RSE of covariance decreased. Concerning the amount of information, it increased for very large values of covariance. The amount of information was lower without covariance, which means that the corresponding M F was a lower limit of the information computed with covariance terms.

Concerning the approximation of the calculation of the SEs without using the model, which mimics the case of individual-rich data, it was possible to obtain an idea of the precision of estimates quickly, and to see if the number of subjects was large enough. Indeed, these analytical predictions were the lower limits of the SEs that could be obtained by the population approach, and could not be improved by increasing the number of samples per patient.

Focusing then on the influence of covariance on optimal designs, as Ogungbenro et al.

(2008) demonstrated, it was shown that optimal design depends on the covariance between parameters. Indeed, the values of covariance terms can affect optimal sampling times, however it was rather slight in the case of a one-group design. When considering several groups of subjects, optimal sampling times and groups can be different with and without covariance, but the efficiency loss was very low for both one group and several groups. Further comparisons should be performed. Also, the design variables chosen in this work were the location of sampling times but it would be possible to extend to other design variables, as for example the dose [START_REF] Nyberg | Simultaneous optimal experimental design on dose and sample times[END_REF].

In the present work, the calculation of M F was proposed using a first-order linearization of the model by the Taylor expansion. The derivative of the variance of observations according to β was assumed to be zero. Therefore, M F was expressed as a block diagonal matrix.

An extension of M F considering the dependence of the observations on the parameters of the model, that is, a complete M F with an additional off-diagonal block, was also proposed [START_REF] Retout | Further developments of the Fisher information matrix in nonlinear mixed effects models with evaluation in population pharmacokinetics[END_REF]. However, [START_REF] Mentré | Comparison of results of the different software for design evaluation in population pharmacokinetics and pharmacodynamics. Population Approach Group in Europe[END_REF] showed on two examples that if a linearization of the model was carried out, it is better to have a block diagonal expression for M F . [START_REF] Mielke | Some considerations of the Fisher information in nonlinear mixed effects models[END_REF] also showed that an approach with block diagonal M F was more reliable than one with the full M F . But it is not the case for all situations, as shown by [START_REF] Nyberg | Different approximations and methods for calculating the FIM and their consequences[END_REF]. In the MONOLIX software, the observed M F calculated by linearization of the model also had a block diagonal expression. It is known that linearization can lead to problems when a very nonlinear model is used. An alternative to linearization, for example the Gaussian quadrature [START_REF] Guedj | Practical identifiability of HIV dynamics models[END_REF][START_REF] Nguyen | Design evaluation and optimisation in crossover pharmacokinetic studies analysed by nonlinear mixed effects models[END_REF], could also be developed for models with covariance.

In conclusion, the extension of M F including covariance between random effects in PFIM provides a useful computing tool for design evaluation and optimization, and this extension will be available in the next version of PFIM. The size of covariance had no impact on the SE of fixed effects or variance parameters. Omitting the covariance at the design stage influenced the optimal sampling times, but barely had an influence on efficiency, and predicted SEs were only slightly changed.

APPENDIX

The Fisher information matrix M F (Ψ, ξ i ) for multiple-response models for the individual i with design ξ i is given by M

F (Ψ, ξ i ) = E(-∂ 2 L i (Ψ;Y i ) ∂Ψ∂Ψ T )
, where L i (Ψ; Y i ) is the log-likelihood of the vector of observations Y i of that individual for the population parameters Ψ. Because F is nonlinear, there is no analytical expression for the log-likelihood and a first-order Taylor expansion of the model F (θ i , ξ i ) = F (g(β, b i ), ξ i ), around the expectation of b i , that is to say around 0, is used:

F (g(β, b i ), ξ i ) ∼ = F (g(β, 0), ξ i ) + ( ∂F T (g(β,b i ),ξ i ) ∂b i ) b i=0 b i . Therefore, Y i ∼ = F (g(β, 0), ξ i ) + ( ∂F T (g(β,b i ),ξ i ) ∂b i ) b i =0 b i + ε i .
Using this linearization, the approximated marginal expectation E i and variance V i of Y i are given by:

E(Y i ) ∼ = E i = F (g(β, 0), ξ i ) V ar(Y i ) ∼ = V i = ( ∂F T (g(β, b i ), ξ i ) ∂b i ) b i =0 Ω( ∂F (g(β, b i ), ξ i ) ∂b T i ) b i =0 + Σ(β, 0, σ inter , σ slope , ξ i )
The log-likelihood L i is approximated by:

-2L i (Ψ; Y i ) ∼ = n i ln(2π) + ln(|V i |) + (Y i -E i ) T V -1 i (Y i -E i )
The elementary Fisher information matrix for a multiple-response model with non-diagonal variance matrix is derived from this expression of the log-likelihood. In the following, the i index for the individual is omitted for the sake of simplicity.

M F (Ψ, ξ) = E Y (- ∂ 2 (L(Ψ, Y ) ∂Ψ∂Ψ T )
The elementary M F depends on the approximated marginal expectation E and variance V of the observations. Assuming that the derivative of V does or does not depend on the fixed effects, the elementary M F is a full matrix or a block diagonal matrix. It was suggested [START_REF] Mentré | Comparison of results of the different software for design evaluation in population pharmacokinetics and pharmacodynamics. Population Approach Group in Europe[END_REF][START_REF] Mielke | Some considerations of the Fisher information in nonlinear mixed effects models[END_REF]) that for derivation of M F using first-order linearization, the block diagonal expression might be better. This is not always the case, as presented by [START_REF] Nyberg | Different approximations and methods for calculating the FIM and their consequences[END_REF]. Here, the block diagonal expression for M F was chosen and is expressed as:

M F (Ψ, ξ) ∼ = 1 2   A(E, V ) 0 0 B(E, V )   where (A(E, V )) ml = 2 ∂F T ∂β m V -1 ∂F ∂β l
with m and l = 1, ..., p and

(B(E, V )) ml = tr( ∂V ∂λ m V -1 ∂V ∂λ l V -1 )
with m and l = 1, ..., dim(λ).

The calculations of the Fisher information matrix with all covariance terms in Ω were developed. For block A, it is the same calculation with and without covariance, except that the Ω matrix is full when covariance is considered. On the other hand, the calculations are different for block B. Indeed

∂V ∂λ m = ( ∂F T (g(β, b), ξ) ∂b ) b=0 ∂Ω ∂λ m ( ∂F (g(β, b), ξ) ∂b T ) b=0 + ( ∂Σ(β, b, σ inter , σ slope , ξ) ∂λ m ) b=0
with m = 1, ..., dim(λ).

And depending on whether m corresponds to a variance or covariance term, ∂Ω ∂λ m contains one or two terms different from zero, respectively. For a variance term, that is for λ 1 to λ p (corresponding to ω 2 11 to ω 2 pp ), the general term for the derivation of the matrix V according to λ m , m varying from 1 to p, for one response, is expressed as:

∂f k (g(β, 0), t j ) ∂b m ∂f k (g(β, 0), t j ′ ) ∂b m
with j = 1, ..., n k the line index, j ′ = 1, ..., n k the column index and k corresponding to the k th response.

For a covariance term from λ p+1 to λ p(p+1)/2 (corresponding to ω 21 to ω p,p-1 ), the derivation of the matrix V, for one response, according to λ m is expressed as:

∂f k (g(β, 0), t j ) ∂b r ∂f k (g(β, 0), t j ′ ) ∂b s + ∂f k (g(β, 0), t j ) ∂b s ∂f k (g(β, 0), t j ′ ) ∂b r
with r, s = 1, ..., p, r = s, j = 1, ..., n k the line index, j ′ = 1, ..., n k the column index, and k corresponding to the k th response.

Note that ( ∂Σ(β,b,σ inter ,σ slope ,t j ) ∂λ m

) b=0 is equal to zero when m does not correspond to an item of an error parameter. Otherwise, the derivation of the matrix V according to λ m ,

λ m = σ inter k , is equal to 2 × diag(σ inter + σ slope × f k (g(β, 0), t j )
) and the derivation of the matrix V according to = 0 and with a covariance = 0.10) and the Ξ single highcov one design in white (optimized with covariance = 0.12) for the single-response model with covariance = 0.12.

λ m = σ slope k is 2 × diag(f k (g(β, 0), t j )(σ inter + σ slope × f k (g(β, 0), t j ))), with j = 1, ...,
β CLP M (L.h -1 .kg -1 ) - 0.11 β V 1 (L.kg -1 ) 0.88 0.97 β Q 2 (L.h -1 .kg -1 ) 0.58 0.12 β V 2 (L.kg -1 ) 1.0 0.41 β Q 3 (L.h -1 .kg -1 ) 1.3 0.69 β V 3 (L.kg -1 ) 0.39 0.87 β CLM (L.h -1 .kg -1 ) - 0.91 ω 2 CLP 0.23 0.25 ω 2 CLP M - 1.4 ω 2 V 1 0.075 0.070 ω 2 CLM - 0.27 ω CLP,V 1 0.10 0.045 ω CLP,CLP M - 0.036 ω V 1 ,CLP M - 0.024 ω CLP,CLM - 0.091 ω V 1 ,CLM - 0.083 ω CLP M,CLM - - 0

Figure 1 :

 1 Figure 1: Structural PK model with 4 compartments: 3 compartments for the parent drug and one additional compartment for the metabolite.

Figure 2 :

 2 Figure 2: Mean concentration-time profiles for the parent (left) and for the metabolite (right) for 22 sampling times per subject.

Figure 3 :

 3 Figure 3: Influence of covariance on SE (left, ) and RSE (right, •) of covariance for the single-response model (top), and influence of covariance on criterion for the single-response model (bottom).

Figure 4 :

 4 Figure 4: Predicted RSE (%) of fixed effect parameters (top) and variance components (bottom) for Ξ single nocov one and Ξ single cov one designs in black (optimized with a covariance

Table 1 :

 1 n k and k corresponding to the k th response. Population pharmacokinetic parameter values for the parent drug and the joint model of the parent drug and its metabolite

	linear mixed-effects model: Application to HIV dynamics model. Computational Statistics
	and Data Analysis, 2006; 51(3): 1562-1574.
	Sheiner L, Beal S. NONMEM Version 5.1. University of California, NONMEM Project Group; San
	Francisco, 1998.
	Tod M, Jullien V, Pons G. Facilitation of drug evaluation in children by population methods and
	modelling. Clinical Pharmacokinetics, 2008; 47(4): 231-243.

Table 2 :

 2 SE of the parameters with and without covariance predicted by PFIM (P), and obtained by NONMEM (N) and MONOLIX (M) on a simulated data set for the parent molecule and the rich design in 82 patients (22 observations per patient)

	.045

Table 3 :

 3 SE of the parameters with and without covariance predicted by PFIM (P), and obtained by NONMEM (N) and

	MONOLIX (M) on a simulated data set for parent and metabolite and the rich design in 82 patients (22 observations per
	patient)
			Model without covariance			Model with covariance
		Reference	Estimated	Predicted	Obtained	Reference	Estimated	Predicted	Obtained
		parameters	parameters	SE	SE		parameters	parameters	SE	SE
	Parameters	P	N	M	P	N	M	P	N	M	P	N	M
	β CLP	0.61	0.67	0.67	0.034	0.041	0.040	0.61	0.68	0.67	0.034	0.041	0.041
	β V 1	0.97	0.97	0.95	0.030	0.028	0.027	0.97	0.98	0.98	0.030	0.026	0.026
	β Q 2	0.12	0.098	0.13	0.044	0.038	0.045	0.12	0.11	0.14	0.044	0.041	0.047
	β V 2	0.41	0.37	0.44	0.093	0.079	0.089	0.41	0.39	0.47	0.093	0.083	0.094
	β Q 3	0.69	0.69	0.65	0.037	0.031	0.038	0.69	0.68	0.65	0.037	0.034	0.041
	β V 3	0.87	0.89	0.80	0.097	0.085	0.094	0.87	0.86	0.78	0.097	0.089	0.099
	β CLP M	0.11	0.11	0.11	0.015	0.015	0.015	0.11	0.11	0.11	0.015	0.015	0.016
	β CLM	0.91	0.93	0.91	0.054	0.048	0.050	0.91	0.95	0.95	0.054	0.048	0.050
	ω 2 CLP	0.25	0.30	0.29	0.040	0.039	0.046	0.25	0.30	0.30	0.040	0.037	0.046
	ω 2 V 1	0.070	0.063	0.056	0.012	0.012	0.010	0.070	0.053	0.048	0.012	0.011	0.0090
	ω 2 CLP M	1.4	1.6	1.6	0.21	0.22	0.25	1.4	1.6	1.6	0.21	0.22	0.25
	ω 2 CLM	0.27	0.23	0.23	0.043	0.036	0.036	0.27	0.21	0.21	0.043	0.039	0.034
	ω (CLP,V 1 ) (ρ CLP,V 1 )	-	-	-	-	-	-	0.045(0.34)	0.044(0.35)	0.043(0.36)	0.016	0.012	0.014
	ω CLP,CLP M (ρ CLP,CLP M )	-	-	-	-	-	-	0.036(0.061)	0.13(0.19)	0.12(0.18)	0.065	0.073	0.077
	ω V 1 ,CLP M (ρ V 1 ,CLP M )	-	-	-	-	-	-	0.024(0.078)	-0.015(-0.052)	-0.0050(-0.018)	0.036	0.040	0.031
	ω CLP,CLM (ρ CLP,CLM )	-	-	-	-	-	-	0.091(0.35)	0.082(0.33)	0.081(0.32)	0.031	0.028	0.029
	ω V 1 ,CLM (ρ V 1 ,CLM )	-	-	-	-	-	-	0.083(0.61)	0.052(0.49)	0.049(0.49)	0.019	0.017	0.012
	ω CLP M,CLM (ρ CLP M,CLM )	-	-	-	-	-	-	-0.045(-0.075)	-0.14(-0.23)	-0.13(-0.22)	0.068	0.073	0.066
	σ inter P	0.072	0.095	0.076	0.0061	0.0045	0.0052	0.072	0.094	0.076	0.0061	0.0047	0.0052
	σ slope P	0.16	0.19	0.17	0.0032	0.0031	0.0036	0.16	0.19	0.17	0.0032	0.0030	0.0036
	σ inter M	0.011	0.015	0.011	0.00082	0.00073	0.00062	0.011	0.014	0.012	0.00082	0.00064	0.00062
	σ slope M	0.053	0.057	0.049	0.0014	0.0012	0.0013	0.053	0.057	0.049	0.0014	0.0011	0.0013

Table 4 :

 4 RSE (%) of the parameters with and without covariance predicted by PopDes and PFIM for the parent and metabolite and the rich design in 82 patients (22 observations per patient)

			Model without covariance		Model with covariance
	Parameters	Parameter values	PopDes	PFIM	Parameter values	PopDes	PFIM
	β CLP	0.61	5.6	5.6	0.61	5.6	5.6
	β V 1	0.97	3.0	3.1	0.97	3.0	3.1
	β Q 2	0.12	32	38	0.12	32	38
	β V 2	0.41	20	23	0.41	20	23
	β Q 3	0.69	4.7	5.4	0.69	4.7	5.4
	β V 3	0.87	9.7	11	0.87	9.7	11
	β CLP M	0.11	13	13	0.11	13	13
	β CLM	0.91	5.8	5.8	0.91	5.8	5.8
	ω 2 CLP	0.25	16	16	0.25	16	16
	ω 2 V 1	0.070	17	17	0.070	17	17
	ω 2 CLP M	1.4	16	16	1.4	16	16
	ω 2 CLM	0.27	16	16	0.27	16	16
	ω (CLP,V 1 )	-	-	-	0.045	35	36
	ω CLP,CLP M	-	-	-	0.036	183	183
	ω V 1 ,CLP M	-	-	-	0.024	147	148
	ω CLP,CLM	-	-	-	0.091	34	34
	ω V 1 ,CLM	-	-	-	0.083	22	22
	ω CLP M,CLM	-	-	-	-0.045	151	151
	σ inter P	0.072	7.7	8.3	0.072	7.7	8.3
	σ slope P	0.16	1.8	2.0	0.16	1.8	2.0
	σ inter M	0.011	5.1	5.3	0.011	5.1	5.3
	σ slope M	0.053	2.0	2.4	0.053	2.0	2.4

Table 6 :

 6 Optimal designs with 6 points with several groups for the single-response model: Ξ single nocov several design is for a model with covariance = 0 and Ξ single highcov several design

	for a model with covariance = 0.12
	Time (in hour)	Number of subjects
		Ξ single nocov several Ξ single highcov several
	0.1, 0.2, 0.4, 2, 6, 16	0	16
	0.1, 0.2, 0.8, 2, 6, 16	0	28
	0.1, 0.2, 0.8, 4, 12, 16	5	22
	0.1, 0.4, 0.8, 2, 4, 12	5	14
	0.1, 0.4, 0.8, 2, 6, 16	52	0
	0.1, 0.4, 1, 4, 12, 16	18	0
	0.1, 0.4, 2, 6, 16, 24	2	2

Table 7 :

 7 Optimal designs with 4 points with several groups for the multiple-response model: Ξ multi nocov several design is for a model with every covariance = 0 and Ξ multi cov several design with at least one covariance = 0 (see Table1)

	Time (in hour)	Number of subjects
		Ξ multi nocov several Ξ multi cov several
	0.1, 0.8, 4, 12	22	21
	0.1, 0.8, 12, 24	3	2
	0.1, 2, 6, 16	0	5
	0.1, 2, 8, 24	37	42
	0.1, 2, 12, 24	0	2
	0.1, 6, 16, 24	14	10
	0.2, 2, 8, 24	1	0
	0.8, 4, 12, 24	4	0
	2, 6, 16, 24	1	0

ACKNOWLEDGMENTS

C. Dumont was supported by a grant from Institut de Recherches Internationales Servier, France.

The authors would like to thank Kay Ogungbenro (University of Manchester) for running PopDes on the example and for providing the results presented in Table 4.


S Nowakowski
email: samuel.nowakowski@loria.fr

N Issenmann
email: nathalie.issenmann@univ-lorraine.fr

I Cherqui-Houot

A Brun

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This paper asks how are being designed and expected modes of integration of students during and after their university studies in the specific context of development on private and public markets for applications such as e-digital portfolios. She also questioned the manner in which to deploy the strategies and institutional policies regarding the choice of digital interfaces for the enhancement of learning and using the integration of students and in particular the way is taken into account the research dimension a tool to select and deploy.

To do this, it relies on a study conducted as part of e-inclusion project supported by the Office for Students professional insertion (BAIP) and funded by the University of Lorraine and the Regional Council of Lorraine. This study is based on monitoring of a panel of about 250 students and fifteen teachers experimenting "Lorfolio" in their regular educational setting. Lorfolio is a portfolio of digital skills remotely accessible, for all the assets of a territory, and being developed in Lorraine at the initiative of the Regional Council. We propose in particular to highlight the returns through the use of specific questions that are generated when it comes to decide on their wide deployment of an institution, consortium or territory. What models to use is based does? What actual uses generates does? How to use these models and are they related to the digital strategies of institutions? After recalling the context of the study, the actors and the methodology adopted, we will build on the first qualitative and quantitative analyzes performed to establish the first profiles of the perception of the tool from the point of view of students as teachers to measure the impact of such a deployment at institutional level.

It's about understanding how and give teachers and students mobilize the ePortfolio as part of lessons usually taught strategies around employability. In parallel, an analysis of traces of use should generate typical profiles of students uses. Profiles that will adapt the best features found on the interface and to add, if any, resources customized news relevant to learning conditions and needs of specific integration of students.

This study is based on the following assumptions:

• formalization of the activity conducted by students of their formal and informal learning helps facilitate their positioning for their employability. • making available to students of an ePortfolio helps facilitate the formalization of their achievements and their visibility in the territory of Lorraine.

This multidisciplinary research contributions mobilizes science education that offer an analysis of the formalization of skills allied to the contributions of artificial intelligence in terms of modeling theory and uses of recommender systems. The results and prospects identified will be a basis for reflection by all departments of the University of Lorraine and components concerned with the employability of students.

To put the study in a broader context, we first redefine the concepts inherent in ePortfolio and we will review the implications and interactions and collaborations implemented between actors of this project.

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Currently, the ePortfolio or digital portfolio is growing both in the educational landscape that professional. Regions, departments, universities, schools, professionals will endow this tool. Nevertheless, the identification of objectives and interests of such a tool does not come naturally to the user. According to Thierry KARSENTI 3 , over 30 types of portfolio are identified (Developmental Portfolios, Reflective portfolios, Assessment Portfolios, Showcase Portfolios, Working Portfolios,...) with for each different purposes (show what has been learned, that one has reached the targeted skills, highlight reflexive work on their learning, show their skills in order to be evaluated, show that we continue to learn beyond the academic certification, showcases). To this, we can also add the difference made by the French Ministry of National Education between professional portfolio 4 (portfolio for teachers) and teaching portfolio 5 (students portfolio), which distinguish objectives centered on learning and objectives focused on Assessment i.e., process or product. Regarding Long Life Learning (LLL), ePortfolio is a powerful teaching tool to formalize the collection and valorization of competencies (see in particular, LAYEC, J. ( 2006)). Long Life Learning takes an important place in training institutions, particularly at the university, and brings new practices self-assessment of formal and informal learning. Thus, the ePortfolio, as a scalable set of materials, resources, … ie the "sèmes", identifies an individual in its singularity. We can then speak of individuation as a process leading from the undifferentiated to individuate. It is then placed in the footsteps of many authors such as Arthur Schopenhauer and especially Carl Jung and more recently Gilbert Simondon. Consequently, ePortfolios are at the crossroads of two processes of individuation: individual and technology. Thus, ePortfolio should not be seen as a process itself but as an outcome of a process of individuation linking people and technology [START_REF] Ravet | ePortfolios, Products of an Individuation Process?[END_REF].

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The Regional Council of Lorraine is in a regional framework for economical development: « To address the socio-economic challenges, the Region of Lorraine must address issues of vocational training based on three major principles :

• Enhance the human heritage of the Region,

• Give everyone the means to become subject and actor of his training course for its personal and professional development, • Mobilize training players to enhance its human heritage.

$ "Thierry Karsenti, Chair on information technology and communication (ICT) in Education, Faculty of Education, University of Montreal" % "http://www.educnet.education.fr/dossier/archives/portfolionumerique/usages-enseignement/portfolio-professionnel-delenseignant & "http://www.educnet.education.fr/dossier/archives/portfolionumerique/usages-enseignement/enseignement-superieur " One objective of the Region of Lorraine was to offer each people to built a custom tool to enable it to capitalize experiences and training skills». From these words approved by the Regional Council of Lorraine in 2006, the Lorraine ePortfolio project is born. Starting in 2007, the region of Lorraine entrusted at INFFOLOR 6 the realization of a opportunity and feasibility study for the development of a customized tool for capitalizing experiences and training. The choice was then focused on the development of an ePortfolio. First, to realize this study and to take the appropriate decisions in terms of technical and ergonomic specifications of the ePortfolio, INFFOLOR has worked with two experts 7 in normalization and digital systems interoperability. Second, INFFOLOR partnered with LISEC Lorraine in order to carry an experiment for characterizing the first uses and to imagine first deployment scenarios.

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The University of Lorraine has a Help Desk for students' professional integration (BAIP). It is a light structure designed to design the appropriate policies for a better professional integration of students. In this context, the research project "e-insertion" which aims to answer the question, «How digital environments can enforce professional integration strategies of the students?» aims, by its approach based on interdisciplinary research, to analyze the uses of populations of students et to offer tracks that will support the policy at the University of Lorraine level. The University of Lorraine eLearning department is also involved and works in the development of observation protocols and contributes to the analysis of results.

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The research teams involved will work on complementary approaches b a s e d o n uses modeling and qualitative analysis of pedagogical uses to have an analysis combining mathematical models of the real uses and qualitative analysis coming from questionnaires and interviews with both students and teachers. The diagram below shows the general process of interaction between the chosen approaches:

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The objective of the work carried out in the KIWI (Knowledge Information and Web intelligence) team in LORIA aims to improve the quality of service provided by a computer based system (such as a corporate intranet, an online digital library, an information portal, the Internet, an ePortfolio, ...) to a specified user or not. The general approach can be outlined as follows:

• Learning models of user behavior, from the observation of their interactions with the system, • From the observed behavior, planning actions that the system must perform in order to provide a customized service to user expectations. It is part of a holistic approach for uses analysis which can be outlined as follows: To provide the accurate content at the right time is a key factor for an efficient delivering and sharing information systems. Thus, recommender systems allow users to find resources that meet their needs by suggesting specific digital resources. Many approaches have been used to design recommender systems, including collaborative filtering, content based recommender, or users tracking by trajectories modeling in the recommender space. In this study, traces left by ePortfolio users will be the sets of data used to compute the models for the implementation of recommendation strategies. ATIP Team (Activity, Work, Professional identity) in LISEC 8 laboratory is working on many learning processes having in common the links between activity a n a l y s i s , identities and competencies in a professional environment. Whether the education market, social recognition of professional knowledge, procedures, guidance, work, the formalization of the activities of actors is at the heart of this research.

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The research is organized in three phases and includes three sets of qualitative and quantitative data:

• Set 1 consists of survey results, obtained before t o t h e first u s e o f LORFOLIO: survey by questionnaire carried out among 256 students who have been supported to elaborate their career plan (Personal and professional projects or specific course) and among 15 teachers. • Set 2 contains first feedbacks from students and teachers. It also includes the analysis of the traces.

• Set 3 contains interviews with subsets of students and teachers after the end of the experiment.

!"#$%&'2'3'4%+5&60'0"7&-".& All the interactions generated between the structures (research, teaching, administrative) will help to show the various processes involved in the use of LORFOLIO. Below are represented actors, actions and expected results.

B "())*+,,---.3/?:CD4:C(:4C(:.:9," The analysis of the first set of data enables us to identify students' technological profile. Their privileged mean of communication is Internet with its different possibilities of exchange such as email, instant messaging and social networks as Facebook. The use of Internet is close to the use of mobile phones. Then, marginally, students use mail post and face-to-face exchanges. The preferred sources of information are of three categories:

• Internet: it is difficult to know what are the used online resources; Internet has been used as generic with no precision even if many cited examples are « wikipedia » and « google », • Traditional sources of information are Books, newspapers and sometimes libraries. We can also mention magazines, press and encyclopedia, • Television 98% of the students surveyed have a personal computer and 7.3% of them share it with their family. Only 2% of students have no computer. The large majority (96%) used the Internet daily. 48% say they spend 2 to 3 hours per day on their computer against 30% who spend less than 2 hours. They all have internet access. They say to connect from their mobile phone, WiFi hotspots in public places, streets, universities, among friends, at home and in restaurants … 30% of students consider themselves "regulars trying to get by" and over 60% identify themselves as "regulars who know their way". 99% of them sent their first email before the age of 19. Their Internet use can be classified into five categories listed below in order of importance: Relating to job search, the best way they think to find a job is to use relationship. The words most frequently cited are:

!"#$%&'@'3'A&/0'?1B/'0+'>".,'1'5+:' Less than 9% use an ePortfolio. Most students cannot say what is an ePortfolio. For others, the functionalities of an ePortfolio are collecting competencies (regrouping, collecting, sorting, summarizing…) and presenting competencies (showing, highlighting, presenting) then marginally reflexivity (analyzing, self-knowing).

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The goal of the users on this portfolio website is the construction of their cv and portfolio. To reach this goal, they have to perform some tasks, such as filling some personal information, competencies, etc. Each task is may be made up of several actions. For example, filling personal information, can be divided into filling one page to provide with information about name, age, sex, and filling another page about address, phone number, etc.

The characteristic of this website is that the order of the actions that the users have to perform to build theit cv is not predefined, they can perform them in the order that fits them the best.

The question we ask is if they tend to perform these actions in the same order, or if there are some prototypes of behaviors. More formally, by exploiting the logs of this website, we aim at discovering frequent sequential patterns of usage, if exist. To perform this task, we propose to exploit data mining algorithms [START_REF] Han | Data Mining : Concepts and Techniques[END_REF].

Based on the assumption that the action a user performs at a given time on the website, depends on his/her past actions, we propose to exploit a K-order Markov Model [START_REF] Rabiner | A tutorial on hidden Markov models and selected applications in speech recognition[END_REF]. Such a model assumes that an action depends on exactly the K preceding actions. This model not only learns the sequences of actions of size K+1 that users perform, it is also a predictive model that may be used to predict the action a user should perform, given his/her last K actions.

This model may be viewed as too strict, as we can naturally imagine that an action may depend on less than the last K actions. Thus, we propose to exploit an all-kth-order Markov Model [START_REF] Pitkow | Mining Longest Repeating Subsequences to Predict World Wide Web Surfing[END_REF], which learns the sequences of actions of size from 1 to K and uses these sequences to predict the action a user should do.

Such models have been used to recommend actions in the frame of several domains, such as the Web [START_REF] Deshpande | Selective Markov Models for Predicting Web Page Accesses[END_REF]), e-commerce (Lu, 2009) On the observation period that runs from late October 2011 to January 3, 2012, visits of the website are mainly divided as follows:

• 234 connections, 229 only connected once and 21 connected 5 times Finally, the sequences of most frequently visited pages are: Over this period, we have obtained 21567 5grams 9 and 11094 are distinct. We chose 5grams because they represent the optimal length of navigation within LORFOLIO. The most significant results show that: 697 5grams lead users to generate a resume (PDF and/or Word files) 207 5grams show navigation through competencies topics 119 5grams show navigation through training topics Clearly, in the early days of the use of LORFOLIO, students have only export their resume to print it (PDF, rtf or sxf formats). Unfortunately, categories related to the formalization of competencies are not really used. This corresponds to the listed statements in the previous section.

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Teachers interviewed can be classified as "digital migrants". However, they are advanced users of the Internet, they are aware to leave traces on the networks and they are aware of the encountered. Unlike their students, they are not users of social networks. Thus, one of the teachers surveyed had already created personal web pages and only two teachers have profiles on social networks (facebook for one / facebook, viadeo and linkedin for the other). They all indicate that the digital environment is essential for their educational and administrative activities. Their objectives are then organized around three topics:

• Objectives rather focused on job search activity and the way to introduce it pedagogically, with the desire to standardize practices of students in job search (write a resume, prepare a job interview, provide evidences,…) • Objectives rather focused on supposed prerequisites to get a job, most often presented in terms of knowledge to be acquired (« Know company, the environment, … ») • Objectives rather focused on students to help them to achieve reflexive posture (« self-knowing, identify own means, … ») Teachers surveyed consider ePortfolio through three dimensions: self-learning, self-evaluation, selfactualization. For them, success regard to the p r o f e s s i o n a l i n s e r t i o n d e p e n d s o n t wo f a c t o r s: on the one hand the mobilization of "good" means, ( traditional and networks) and on the other hand self. Finally, according to their perception, added value of the ePortfolio is in its numerical characteristics (storage, ease of use, format) but especially in the educational opportunities it offers: "different way to motivate students, to be efficient, communicate more widely." However, they mention some risk: e risk of "formatting" the work done by the student, risk of standardization of documents produced, risk due to non respect of privacy.

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Students have a digital profile quite different from that of their teachers, they are more familiar with social networks. However, they include only marginally the question of their professional integration in their use of the web and they seem unaware of the impact of their current uses on their present and future social visibility. Most of them count on their relationship (family, friends and professional) to find a job. When they prefer networks, they ask mainly to private networks. Finally, the analysis shows that LORFOLIO has been only used as a technical tool (to write a resume and to export it to text files). From teachers' point of view, LORFOLIO is also seen as a tool (to produce documents related to a search for a job). But, the pedagogical considerations to support this production are mentioned in their speech. These two observations call for few remarks. These observations need to recall Jean Heutte when he says in "le Livre Blanc" concerning the reflective process underlying the use of an ePortfolio.

« Est considérée comme relevant d'une "démarche ePortfolio", toute démarche réflexive d'un étudiant sur son parcours, ses apprentissages, ses expériences, ses compétences ou encore ses réalisations et visant à capitaliser dans un environnement numérique un ensemble évolutif de documents et de ressources électroniques qui décrit et illustre toutes ces E "&54<F+"G:89:0C:"21"&"=/11:4:0)"H/:-:="*<5:?" dimensions biographiques » (cité par Heutte,J, livre blanc, DGSIP TICE, MEN).

We can see that their expectations reach this objective even if the related pedagogical processes remain to be invented. Two main aspects have to be considered:

-To support of the process other than in the form of a transmission of knowledge -To solve the problem they have to clearly identify what is related to professional integration strategy. In this perspective, the implementation of the ePortfolio may then appear as a lever because teachers see it as a way to interest and to mobilize their students more. The students whose responses show very clearly that they separate their personal and social activities of their academic activities, seem to develop a perception of their professional integration that does not predispose to a strong commitment to a reflective approach. It therefore remains to persuade them. The future results will show how teachers have set up their teaching methods:

-How a support system of this reflective work has implemented -With what observations? -What impacts on the students? There is indeed much to expect from these experimental moments implemented in institutions. Concerning the institution, the University of Lorraine, currently undergoing reconfiguration, intends to make the professional integration of its students one of its objectives and its major strengths. While it is too early to know what his policy on implementation of digital devices, the university has expressed strong expectations about the results of this experiment that ended in June 2012.

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Have a combined analysis of the uses and perceptions of students and teachers through the implementation of a numerical tool which can be assumed that it will impact the process of individuation is an extremely rich field for observing the development of teaching practice. The implementation of such a tool at the university is not trivial. Indeed, it appears attractive to teachers, due in part because they see it more fun, less austere, more readily used by students. However, the observed student practices show that the conditions for an intensive use of ePortfolios are not necessarily present. In particular, the pedagogical forms related to a reflexive approach remain to be invented. In this perspective, these experiments used to test and to formalize new forms of teaching. Some aspects have to be studied:

-Only one tool will not be able to support diversity of practices and teaching profiles -What added value of the ePortfolio in face of professional social networks as Linkedin?

•

  Social networks (140 occurrences) including « facebook » (135 occurrences) • Videos ( 7 1 o c c u r r e n c e s ) including « y o u t u b e » ( 6 3 occurrences) • Search engines (57 occurrences) • Email (28 occurrences) • Online music (18 occurrences) • Online Encyclopedia (14 occurrences) And marginally commercial, sport, news and entertainment websites.

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As part of the research project funded by the university of Lorraine and the Regional Council of Lorraine, LISEC Lorraine Lab 1 , KIWI team 2 of LORIA and the department of the digital uses of the University of Lorraine propose to study, analyze and support the use of ePortfolio Lorraine, also known Lorfolio in the university. 40<4="A3<0=/0"