Muriel Boulakia

Erik Burman

Miguel A Fernández

Colette Voisembert

Data assimilation finite element method for the linearized Navier-Stokes equations in the low Reynolds regime

In this paper we are interested in designing and analyzing a finite element data assimilation method for laminar steady flow described by the linearized incompressible Navier-Stokes equation. We propose a weakly consistent stabilized finite element method which reconstructs the whole fluid flow from velocity measurements in a subset of the computational domain. Using the stability of the continuous problem in the form of a three balls inequality, we derive quantitative local error estimates for the velocity. Numerical simulations illustrate these convergences properties and we finally apply our method to the flow reconstruction in a blood vessel.

Introduction

The question of how to assimilate measured data into large scale computations of flow problems is receiving increasing attention from the computational community. There are several different situations where such data assimilation problems arise. One situation is when the data necessary to make the flow problem well posed is lacking, that is, boundary data can not be obtained on parts of the boundary, but some other measured data on the boundary or in the bulk is available to make up for this shortfall, or alternatively the position of the boundary itself is unknown. Another situation is when the initial data is unknown, but measurements are available in the (space-time) bulk domain. In such situations, the problem is ill-posed and numerical simulations are much more delicate to handle than for the well-posed flow equations. The traditional approach is to regularize the continuous problem to obtain a well-posed continuous problem, often using a variational framework, that can then be discretized using standard techniques. The regularization parameter then has to be tuned to have an optimal value with respect to noise in the data. The granularity of the computational mesh is chosen afterwards to resolve all scales of the regularized problem. An example of this strategy is the quasi-reversibility method (see the references [START_REF] Bourgeois | A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace's equation[END_REF][START_REF] Bourgeois | A quasi-reversibility approach to solve the inverse obstacle problem[END_REF][START_REF] Bourgeois | The "exterior approach" to solve the inverse obstacle problem for the Stokes system[END_REF][START_REF] Bourgeois | A mixed formulation of the Tikhonov regularization and its application to inverse PDE problems[END_REF]). For examples directly related to flow problems, we refer to [START_REF] Bourgeois | The "exterior approach" to solve the inverse obstacle problem for the Stokes system[END_REF] for an application to inverse identification of boundaries subject to Stokes problem. Other examples can be found in [START_REF] Elia | Uncertainty quantification for data assimilation in a steady incompressible Navier-Stokes problem[END_REF], where additional measured data is used to compensate for a lack of knowledge of the boundary conditions in hemodynamics or [START_REF] Prathish | Combining existing numerical models with data assimilation using weighted least squares finite element methods[END_REF] where a least squares method is proposed for combining and enhancing the results from an existing computational fluid dynamics model with experimental data.

Regardless of the application, success hinges on the existence of some stability property of the ill-posed problem. Fortunately, it is known that a relatively large class of ill-posed problems has some conditional stability property. Stability estimates give a precise information on the effect of perturbations on the system. In particular, they imply that, if the data are compatible with the PDE, in the sense that there exists a solution, in some suitable Sobolev space, satisfying both the PDE and the data, then this solution is unique. For the Stokes equation, this unique continuation property was originally proven by Fabre and Lebeau [START_REF] Fabre | Prolongement unique des solutions de l'equation de Stokes[END_REF]. The analysis of the stability properties of ill-posed problems based on the Navier-Stokes equations is a very active field of research and we refer to the works [START_REF] Lin | Optimal three-ball inequalities and quantitative uniqueness for the Stokes system[END_REF][START_REF] Ballerini | Stable determination of an immersed body in a stationary Stokes fluid[END_REF][START_REF] Boulakia | Stability estimates for the unique continuation property of the Stokes system and for an inverse boundary coefficient problem[END_REF][START_REF] Yu | Remark on boundary data for inverse boundary value problems for the Navier-Stokes equations [Addendum to MR3319370[END_REF][START_REF] Yu | Global uniqueness in inverse boundary value problems for the Navier-Stokes equations and Lamé system in two dimensions[END_REF][START_REF] Bellassoued | Carleman estimate for the Navier-Stokes equations and an application to a lateral Cauchy problem[END_REF][START_REF] Badra | Stability estimates for Navier-Stokes equations and application to inverse problems[END_REF] for recent results. Stability estimates for inverse problems classically rely on Carleman inequalities or three-balls inequalities, two tools which are strongly related. The idea of applying Carleman estimates for the stability analysis of inverse problems is introduced in the seminal paper [START_REF] Bukhgeim | Uniqueness in the large of a class of multidimensional inverse problems[END_REF] by Bukhgeim and Klibanov.

There appears to be relatively few results in the literature discussing the combined error due to regularization, discretization and perturbations for inverse problems subject to the equations of fluid mechanics. To the best of our knowledge such a combined analysis has only been performed in the recent paper [START_REF] Burman | Stabilized nonconforming finite element methods for data assimilation in incompressible flows[END_REF], where a nonconforming finite element method was used, together with regularization techniques developed in the context of discontinuous Galerkin methods, to analyze a data assimilation problem for Stokes problem. One of the reasons for this is that there is in general a gap between the stability estimates that can be proven analytically and the stability required to perform a numerical analysis. An approach allowing to bridge this gap was proposed in [START_REF] Burman | Stabilized finite element methods for nonsymmetric, noncoercive, and ill-posed problems. Part I: Elliptic equations[END_REF][START_REF] Burman | Error estimates for stabilized finite element methods applied to ill-posed problems[END_REF][START_REF] Burman | Building bridges: connections and challenges in modern approaches to numerical partial differential equations[END_REF][START_REF] Burman | Solving ill-posed control problems by stabilized finite element methods: an alternative to Tikhonov regularization[END_REF], drawing on earlier ideas for well-posed problems in [START_REF] Bramble | A least-squares approach based on a discrete minus one inner product for first order systems[END_REF][START_REF] Bramble | Leastsquares for second-order elliptic problems[END_REF]. This framework combines stabilized finite element methods designed for well-posed problems with variational formulations for data assimilation and sharp stability estimates for the continuous problems based on three balls inequalities or Carleman estimates. Recent developments include finite element data assimilation methods with optimal error estimates for the heat equation [START_REF] Burman | Data assimilation for the heat equation using stabilized finite element methods[END_REF][START_REF] Burman | Fully discrete finite element data assimilation method for the heat equation[END_REF] and design of methods for indefinite or nonsymmetric scalar elliptic problems analyzed using Carleman estimates with explicit dependence on the physical parameters [START_REF] Burman | Unique continuation for the Helmholtz equation using stabilized finite element methods[END_REF][START_REF] Burman | A stabilized finite element method for inverse problems subject to the convection-diffusion equation. Part 1: Diffusion dominated refime[END_REF].

In this paper, our aim is to build on these results and use known techniques for the approximation of the (well-posed) Navier-Stokes equation in an optimisation framework in order to assimilate data with computation. Contrary to the previous work [START_REF] Burman | Stabilized nonconforming finite element methods for data assimilation in incompressible flows[END_REF], we consider the linearized Navier-Stokes equations and use standard H 1 -conforming, piecewise affine, finite element spaces. The key idea is that the ill-posed continuous problem is not regularized. Instead, we discretize the equation and set up a constrained optimization problem where we minimize the distance between the discrete solution and the measured data. To counter the instabilities in the discrete system, we introduce weakly consistent regularization terms in such a way that the error with respect to the approximation and the underlying conditional stability is optimized. In our framework, the only regularization parameter, up to a constant scaling factor, is the mesh parameter.

The analysis is applicable to incompressible laminar steady flow in the low Reynolds regime. We are interested in a situation where a known laminar base flow U is available. A perturbation of the velocities of the base flow has been measured in some subset ω M of the computational domain Ω. Assuming that the perturbation is small, we then consider a linearized equation and design a stabilized finite element method for the approximation of the global perturbation. Using a three balls inequality derived by Lin, Uhlmann and Wang [START_REF] Lin | Optimal three-ball inequalities and quantitative uniqueness for the Stokes system[END_REF] we derive quantitative local error estimates for the perturbations in some target subdomain ω T . Herein, we only consider error estimates for local L 2 -norms, but errors in local H 1 -norms of velocity together with L 2norms of pressure are also possible to analyse using three balls inequalities derived in [START_REF] Boulakia | Stability estimates for the unique continuation property of the Stokes system and for an inverse boundary coefficient problem[END_REF], provided measurements of both velocities and pressures are available in ω M .

The order of the estimate depends on the Hölder coefficient of the continuous stability estimate which depends on the size of the measure domain and the distance between the target domain and the boundary of the computational domain. For simplicity, we restrict the discussion to piecewise affine continuous approximation spaces, but the arguments can be extended to higher order finite element spaces, with the expected improvement of convergence order, following the ideas of [START_REF] Burman | Building bridges: connections and challenges in modern approaches to numerical partial differential equations[END_REF]. It should however be noted that the system matrix becomes increasingly ill-posed as the polynomial order increases and the computation becomes more sensitive to noise in the measured data, so the practical interest in using high order approximation spaces remains to be proven.

The rest of the paper is organized as follows. In Section 2, we introduce the considered inverse problem and some related stability estimates. Section 3 presents the proposed stabilized finite element approximation of the data assimilation problem. The numerical analysis of the method is carried out in Section 4. Finally, Section 5 presents a series of numerical examples which illustrate the performance of the proposed method. In particular, in Section 5.3, we explore how the present approach can be applied to the estimation of relative pressure in blood flow from MRI velocity measurements.

Presentation of the inverse problem and stability results for the continuous problem

Let Ω be a bounded open polyhedral domain in R d with d = 2, 3. We denote by (U, P ) a solution of the stationary incompressible Navier-Stokes equations and we consider some perturbation (u, p) of this base flow. It is then known that, if the quadratic term is neglected, the linearized Navier-Stokes equations for (u, p) may be written

(U • ∇)u + (u • ∇)U + ∇p -ν∆u = f in Ω ∇ • u = 0 in Ω. ( 1
)
The above system is linear, but its approximation is nevertheless nontrivial, since without further assumptions on the size of ∇U , one can not prove the coercivity of the system. We assume that U belongs to [W 1,∞ (Ω)] d and that the flow perturbation satisfies the regularity

(u, p) ∈ [H 2 (Ω)] d × H 1 (Ω).
In what follows, we consider that measurements on u are available in some subdomain ω M ⊂ Ω having a nonempty interior and that these measure-ments are polluted by a small noise δu

∈ [L 2 (ω M )] d .
Let us now introduce some useful notations. We will consider the following spaces:

V := [H 1 (Ω)] d , V 0 := [H 1 0 (Ω)] d , L 0 := L 2 0 (Ω), and
L := L 2 (Ω).
and the following norms, for k = 1 or d,

• L := • [L 2 (Ω)] k , • V := • [H 1 (Ω)] k , • V 0 := • [H -1 (Ω)] d .
Let us notice that, in the first two definitions, with some abuse of notation, we use the same notation for k = 1 and k = d. For any subdomain ω ⊂ Ω, we set

|v| ω := L 2 (ω) |v| 2 1 2 , ∀ v ∈ L 2 (ω).
Besides, we introduce the bilinear forms

a(u, v) := Ω ((U • ∇)u + (u • ∇)U ) • v + ν Ω ∇u : ∇v, (2)
where

H : G := d i,j=1 H ij G ij and b(p, v) := Ω p∇ • v. (3)
We will assume that the non-homogeneous linearized Navier-Stokes problem completed by homogeneous Dirichlet boundary data is well-posed:

Assumption A For all f ∈ V 0 and g ∈ L 0 , we consider the problem:

find (u, p) ∈ V 0 × L 0 such that a(u, v) -b(p, v) + b(q, u) = f, v V 0 ,V 0 + (g, v) L 2 (Ω) (4)
for all (v, q) ∈ V 0 × L 0 . We assume that this problem admits a unique solution (u, p) ∈ V 0 × L 0 . Moreover, we assume that there exists C S > 0 depending only on U and Ω such that, for all f ∈ V 0 and g ∈ L 0

u V + p L ≤ C S ( f V 0 + g L ). (5)
In particular, if ∇U [L ∞ (Ω)] d×d is small enough, it is straightforward to verify that Assumption A holds according to Lax-Milgram lemma. This assumption of smallness on ∇U however is a sufficient condition and there are reasons to believe that the well-posedness of system (4) holds in more general cases.

The problem that we are interested in approximating on the other hand is ill-posed. It can be expressed in the following form:

f ∈ V 0 being given, find (u, p) ∈ V × L 0 such that u = u M in ω M (6) and a(u, v) -b(p, v) + b(q, u) = f, v V 0 ,V 0 , ∀ (v, q) ∈ V 0 × L (7)
Here, u M is the measurement of the velocity made on ω M and we assume that u M is given by u M := u| ω M +δu where δu corresponds to a noise term in the velocity measurements. We observe that, compared to problem (4), the test function space has been enlarged since there are no boundary conditions on u. As a consequence, the spaces of the solution and of the test functions do not match.

In the homogeneous case, the solution (u, p) of (1) satisfies a threeballs inequality which only involves u. This result (or more precisely its non-homogeneous version given in Corollary 2.1) will be capital in the convergence study of the numerical method presented in the sequel. It is stated in [START_REF] Lin | Optimal three-ball inequalities and quantitative uniqueness for the Stokes system[END_REF] (with their notations, A corresponds to U and B to ∇U ): Theorem 2.1 (Conditional stability for the linearized Navier-Stokes problem) There exists R ∈ (0, 1) such that for all 0

< R 1 < R 2 < R 3 ≤ R 0 and x 0 ∈ Ω satisfying R 1 /R 3 < R 2 /R 3 < R and B R 0 (x 0 ) ⊂ Ω, we have B R 2 (x 0 ) |u| 2 ≤ C B R 3 (x 0 ) |u| 2 1-τ B R 1 (x 0 ) |u| 2 τ (8) for (u, p) ∈ [H 1 (B R 0 (x 0 ))] d × H 1 (B R 0 (x 0 )), solution of (1) with f = 0 in B R 0 (x 0 ). In this inequality, C depends on R 2 /R 3 and 0 < τ < 1 depends on R 1 /R 3 , R 2 /R 3 and d.
To avoid the proliferation of multiplicative constants independent of the mesh size, we will frequently use the notation a b for a ≤ Cb for some C > 0.

Theorem 2.1 with the help of Assumption A implies the following local stability inequality in the non-homogeneous case: Corollary 2.1 Let f ∈ V 0 and g ∈ L 0 be given. For all ω T ⊂⊂ Ω, there exist C > 0 and 0 < τ < 1 such that

|u| ω T ≤ C( f V 0 + g L + u L ) 1-τ ( f V 0 + g L + |u| ω M ) τ (9) for all (u, p) ∈ [H 1 (Ω)] d × H 1 (Ω) solution of (U • ∇)u + (u • ∇)U -ν∆u + ∇p = f in Ω ∇ • u = g in Ω. ( 10
)
Proof: Let us first assume that there exist

x 0 ∈ ω M and 0 < R 1 < R 2 < R 3 ≤ R 0 such that B R 1 (x 0 ) ⊂ ω M , ω T ⊂ B R 2 (x 0 ), B R 0 (x 0 ) ⊂⊂ Ω and R 1 /R 3 < R 2 /R 3 < R where R is defined in Theorem 2.1.
According to Assumption A, problem (4) admits a unique solution that we denote (ũ, p) ∈ V 0 × L 0 . Then ( w, ỹ) := (u -ũ, p -q) satisfies ( 1) with f = 0. Using the interior regularity of solution of Stokes problem (see for instance [START_REF] Seregin | Lecture notes on regularity theory for the Navier-Stokes equations[END_REF]), ( w, ỹ)

∈ [H 1 (B R 0 (x 0 ))] d × H 1 (B R 0 (x 0 )
). We may then write

|u| ω T ≤ |ũ| ω T + | w| ω T .
For the first term in the right hand side, we use that (ũ, p) satisfies the stability inequality [START_REF] Bertoglio | Relative pressure estimation from velocity measurements in blood flows: state-of-the-art and new approaches[END_REF]:

|ũ| ω T ( f V 0 + g L ).
For the second term, according to Theorem 2.1, ( w, ỹ) satisfies ( 8) and thus:

| w| ω T ≤ w [L 2 (B R 2 (x 0 ))] d w 1-τ [L 2 (B R 3 (x 0 ))] d w τ [L 2 (B R 1 (x 0 ))] d w 1-τ L | w| τ ω M .
We now revert back to u in the right hand side:

| w| ω T ( ũ L + u L ) 1-τ (|ũ| ω M + |u| ω M ) τ ( f V 0 + g L + u L ) 1-τ ( f V 0 + g L + |u| ω M ) τ .
We conclude the proof by collecting the estimates for ũ and w.

If ω M and ω T do not satisfy the assumptions for the construction of the balls B R 1 (x 0 ) and B R 0 (x 0 ), we introduce a finite sequence of intermediate balls in order to link ω T to ω M (as it is done for instance in [START_REF] Robbiano | Théorème d'unicité adapté au contrôle des solutions des problèmes hyperboliques[END_REF]) and we get again the estimate.

In a classical way for ill-posed problems [START_REF] Alessandrini | The stability for the Cauchy problem for elliptic equations[END_REF], Corollary 2.1 gives a conditional stability result in the sense that, to be useful, this estimate has to be accompanied with an a priori bound on the solution on the global domain (due to the presence of u L in the right hand side). Let us notice that Corollary 2.1 implies in particular the uniqueness of a solution (u, p) in [H 1 (Ω)] d × H 1 (Ω) for problem ( 6)-( 7), up to a constant for p.

In what follows, we assume that f ∈ L 2 (Ω) and we introduce the operator

A defined on (V × L 0 ) × (V 0 × L) by A[(u, p), (v, q)] := a(u, v) -b(p, v) + b(q, u) (11)
where a and b are respectively defined by ( 2) and ( 3). Thus, we look for (u, p)

∈ V × L 0 such that A[(u, p), (v, q)] = (f, v) L 2 (Ω) , ∀ (v, q) ∈ V 0 × L (12)
and ( 6) holds.

Finite element formulation

On the domain Ω, we consider a family {T h } h of shape regular, conforming, quasi-uniform meshes consisting of shape regular simplices K. This family is indexed by h defined as the maximum over the diameters h K of elements in the mesh. For a fixed h > 0, we denote by F i the set of interior faces of the mesh T h . We define the jump over a face F shared by the elements K and K (which means that F = K ∩ K ) as follows: if ζ is a scalar,

ζ F = ζ| K -ζ| K and, if ζ is a vector, ζ F = ζ| K • n K + ζ| K • n K
where n K denotes the outward pointing normal of the element K.

We denote by X h the standard H 1 -conforming finite element space of piecewise affine functions defined on T h and we introduce V

h := [X h ] d , W h := V h ∩ V 0 , Q h := X h and Q 0 h := X h ∩ L 0 .
We may then write the finite element approximation of (12): find

(u h , p h ) ∈ V h × Q 0 h such that A[(u h , p h ), (v h , q h )] = (f, v h ) L 2 (Ω)
for all (v h , q h ) ∈ W h × Q h . This linear system is not invertible and to regularize it, we introduce the following operators s u :

V h × V h → R, s * u : W h × W h → R, and for the pressure s p : Q 0 h × Q 0 h → R and s * p : Q h × Q h → R defined by s u (u h , v h ) := γ u F ∈F i F h F ∇u h ∇v h + γ div Ω (∇ • u h )(∇ • v h ), s p (p h , q h ) := γ p Ω h 2 ∇p h • ∇q h and s * u (z h , w h ) := γ * u Ω ∇z h : ∇w h , s * p (y h , x h ) := γ * p Ω y h x h ,
where γ u , γ p , γ * u and γ * p are positive user-defined parameters. Let us notice that the stabilization term s p for the pressure of the direct problem comes from the Brezzi-Pitkäranta method [START_REF] Brezzi | On the stabilization of finite element approximations of the Stokes equations[END_REF].

For compactness, we introduce the primal and dual stabilizers: for all

(u h , p h ), (v h , q h ) ∈ V h × Q 0 h S[(u h , p h ), (v h , q h )] = s u (u h , v h ) + s p (p h , q h ) and, for all (z h , y h ), (w h , x h ) ∈ W h × Q h S * [(z h , y h ), (w h , x h )] = s * u (z h , w h ) + s * p (y h , x h ).
Finally, we introduce the measurement bi-linear form

m(u, v) = γ M ω M uv,
where γ M > 0 is a free parameter representing the relative confidence in the measurements. We may then write the discrete Lagrangian L :

(V h × Q 0 h ) × (W h × Q h )
→ R that will form the basis of our method as: for all

(u h , p h ) ∈ (V h × Q 0 h ) and (z h , y h ) ∈ (W h × Q h ) L[(u h , p h ), (z h , y h )] := 1 2 m(u M -u h , u M -u h ) + A[(u h , p h ), (z h , y h )] -(f, z h ) L 2 (Ω) + 1 2 S[(u h , p h ), (u h , p h )] - 1 2 S * [(z h , y h ), (z h , y h )]. ( 13
)
If we differentiate with respect to (u h , p h ) and (z h , y h ), we get the following optimality system: find

(u h , p h ) ∈ V h × Q 0 h and (z h , y h ) ∈ W h × Q h such that A[(u h , p h ), (w h , x h )] -S * [(z h , y h ), (w h , x h )] =(f, w h ) L 2 (Ω) A[(v h , q h ), (z h , y h )] + S[(u h , p h ), (v h , q h )] + m(u h , v h ) =m(u M , v h ) ( 14
)
for all (v h , q h ) ∈ V h × Q 0 h and all (w h , x h ) ∈ W h × Q h . We can already notice that formulation ( 14) is weakly consistent in the sense that we have a modified Galerkin orthogonality relation with respect to the scalar product associated to A: Lemma 3.1 (Consistency) Let (u, p) satisfy (1) and (u h , p h ) be a solution of [START_REF] Ch | Effective downstream boundary conditions for incompressible navierstokes equations[END_REF]. Then there holds

A[(u -u h , p -p h ), (w h , x h )] = -S * [(z h , y h ), (w h , x h )] ( 15
)
for all (w h , x h ) ∈ W h × Q h .
Proof: The result is immediate by taking the difference between ( 12) and the first equation of [START_REF] Ch | Effective downstream boundary conditions for incompressible navierstokes equations[END_REF].

A crucial inequality to get the stability of the method is the following Poincaré inequality that we recall from [START_REF] Burman | Solving ill-posed control problems by stabilized finite element methods: an alternative to Tikhonov regularization[END_REF] (Lemma 2). For all

v h ∈ V h , we have h v h V (s u (v h , v h ) + γ M |v h | 2 ω M ) 1 2 . ( 16
)
If we take in the variational formulation ( 14) the test functions

w h = -z h , x h = -y h and v h = u h , q h = p h we see that, for any solution (u h , p h ) ∈ V h × Q 0 h , (z h , y h ) ∈ W h × Q h , there holds S[(u h , p h ), (u h , p h )]+S * [(z h , y h ), (z h , y h )]+γ M |u h | 2 ω M = -(f, z h ) L +m(u M , u h ). (17)
Then, according to [START_REF] Burman | A stabilized finite element method for inverse problems subject to the convection-diffusion equation. Part 1: Diffusion dominated refime[END_REF], the left hand side in equation ( 17) is the square of a norm in (V h × Q 0 h ) × (W h × Q h ) and, according to Babuska-Necas-Brezzi theorem (see [START_REF] Ern | Theory and practice of finite elements[END_REF]), we conclude that the square linear system defined by ( 14) admits a unique solution for all h > 0.

In the analysis below, we will use the following classical inverse and trace inequalities:

• Inverse inequality (see [26, Section 1.4.3]), |v| H 1 (K) h -1 K v L 2 (K) ∀v ∈ P 1 (K). ( 18
)
Here P 1 (K) denotes the set of polynomials of degree less than or equal to 1 on the simplex K.

• Trace inequalities (see [26, Section 1.4.3]), v L 2 (∂K) ≤ C h -1 2 K v L 2 (K) + h 1 2 K v H 1 (K) ∀v ∈ H 1 (K). ( 19
)
At last, by combining ( 19) and ( 18)

v L 2 (∂K) ≤ Ch -1 2 K v L 2 (K) ∀v ∈ P 1 (K). ( 20
)

Stability and error analysis

Let us first define the semi-norms associated to the stabilization operators defined on ([

H 2 (Ω)] d + V h ) × (H 1 (Ω) + Q h ) |||(v, q)||| = h v V + S[(v, q), (v, q)] 1 2 , |||(v, q)||| * = S * [(v, q), (v, q)] 1 2 .
We also introduce the norm

(v, q) # := v V + q L .
Let i h : L 2 (Ω) → X h be the Scott-Zhang interpolant. We refer to [START_REF] Ern | Theory and practice of finite elements[END_REF] for the following results: for t = 1, 2, we have, for all u ∈ H t (Ω)

u-i h u Ω +h ∇(u-i h u) Ω +h 1 2 K u-i h u 2 ∂K 1/2 ≤ Ch t |u| H t (Ω) (21)
and for all u ∈ H 2 (Ω)

F ∈F i ∇(u -i h u) • n 2 F 1/2 ≤ h 1 2 |u| H 2 (Ω) .
Using the componentwise extension of i h to vectorial functions, we deduce the approximation bounds:

∀(v, q) ∈ [H 2 (Ω)] d × H 1 (Ω) |||(v -i h v, q -i h q)||| + (v -i h v, q -i h q) # h( v H 2 (Ω) + q H 1 (Ω) ). ( 22
)
The following continuity results for the bilinear form motivates the definition of the triple norms.

Lemma 4.1 (Continuity) For all ς

∈ V h + [H 2 (Ω)] d and ∈ Q h + H 1 (Ω) there holds A[(ς, ), (v h , q h )] (ς, ) # |||(v h , q h )||| * , ∀(v h , q h ) ∈ W h × Q h (23)
and, for all

(v h , q h ) ∈ V h × Q 0 h , for all w ∈ [H 1 0 (Ω)] d and y ∈ L 2 (Ω) A[(v h , q h ), (w -w h , y -y h )] ≤ |||(v h , q h )|||( w V + y L ) (24)
where (w h , y h ) = (i h w, i h y).

Proof: The proof of ( 23) directly comes from the Cauchy-Schwarz inequality applied termwise in the definition (11) of A. For the second inequality [START_REF] Burman | Data assimilation for the heat equation using stabilized finite element methods[END_REF], we set w = w -w h and ỹ = y -y h and notice that

A[(v h , q h ), ( w, ỹ)] := a(v h , w) -b(q h , w) + b(ỹ, v h ). ( 25
)
For the first term in the right hand side, an integration by parts in the viscous term gives, observing that w| ∂Ω = 0,

a(v h , w) U [W 1,∞ (Ω)] d hv h V h -1 w L + 1 2 F ∈F i F ∇v h • n • w ds.
Using Cauchy-Schwarz inequality with the right scaling in h, we get

a(v h , w) U [W 1,∞ (Ω)] d hv h V h -1 w L + F ∈F i h 1 2 ∇v h F h -1 2 w F .
Applying the trace inequality [START_REF] Burman | Error estimates for stabilized finite element methods applied to ill-posed problems[END_REF], we notice that

h -1 w K + h -1 2 w ∂K h -1 w K + ∇ w K .
Then, if we take the square, sum over K and use the H 1 -stability inequality of i h (21) for t = 1, this inequality becomes

h -1 w L + F ∈F i h -1 2 w 2 F 1/2 w V . As a consequence a(v h , w) |||(v h , 0)||| w V .
Similarly, for the second term in [START_REF] Elia | Uncertainty quantification for data assimilation in a steady incompressible Navier-Stokes problem[END_REF], an integration by parts gives

|b(q h , w)| ≤ h∇q h L h -1 w L |||(0, q h )||| w V .
Finally, the bound for the third term in ( 25) is immediate by the Cauchy-Schwarz inequality and the

L 2 -stability of i h |b(ỹ, v h )| ≤ ∇ • v h L ỹ L |||(v h , 0)||| y L .
Gathering these results, we get [START_REF] Burman | Data assimilation for the heat equation using stabilized finite element methods[END_REF].

Remark 4.1 The augmented Lagrangian stabilization on the divergence in the operator s u is used in the proof of Lemma 4.1 to bound in a direct way term III but it is not strictly necessary. Indeed, if y h is chosen as the L 2projection of y we see that for all

x h ∈ Q h , b(ỹ, v h ) = (ỹ, ∇ • v h -x h ) L
and recalling that inf

x h ∈Q h ∇ • v h -x h L   F ∈F i h F ∇ • v h 2 F   1 2 we conclude that b(ỹ, v h ) y L   F ∈F i h F ∇v h 2 F   1 2
.

Hence, the stabilization of the gradient jump is sufficient to bound this term.

In practice however, it can be useful to add the stabilization term on the divergence since it allows to get a stronger coercivity estimate.

Lemma 4.2

We assume that the solution (u, p) of (12) belongs to [H 2 (Ω)] d × H 1 (Ω) and we consider [START_REF] Ch | Effective downstream boundary conditions for incompressible navierstokes equations[END_REF]. Then there holds

(u h , p h ) ∈ V h × Q 0 h , (z h , y h ) ∈ W h × Q h the discrete solution of
|||(u -u h , p -p h )||| + |||(z h , y h )||| * + γ 1/2 M |u -u h | ω M ≤ Ch( u [H 2 (Ω)] d + p H 1 (Ω) ) + γ 1/2 M |δu| ω M . Proof: We introduce the discrete errors ξ h = i h u -u h , η h = i h p -p h . By this way, (u -u h , p -p h ) = (u -i h u, p -i h p) + (ξ h , η h ). First we observe that |||(u -u h , p -p h )||| + γ 1/2 M |u -u h | ω M ≤ |||(u -i h u, p -i h p)||| + γ 1/2 M |u -i h u| ω M + |||(ξ h , η h )||| + γ 1/2 M |ξ h | ω M .
Using inequalities [START_REF] Burman | Solving ill-posed control problems by stabilized finite element methods: an alternative to Tikhonov regularization[END_REF] and ( 21) for t = 1, we can directly bound the first two terms in the right hand side. For the last two terms, according to inequality [START_REF] Burman | A stabilized finite element method for inverse problems subject to the convection-diffusion equation. Part 1: Diffusion dominated refime[END_REF], we have

|||(ξ h , η h )||| 2 + γ M |ξ h | 2 ω M S[(ξ h , η h ), (ξ h , η h )] + γ M |ξ h | 2 ω M .
To estimate the right hand side, we notice that, using the second equation of ( 14) with (v h , q h ) = (ξ h , η h )

S[(ξ h , η h ), (ξ h , η h )] + γ M |ξ h | 2 ω M -A[(ξ h , η h ), (z h , y h )] = S[(i h u, i h p), (ξ h , η h )] + m(i h u -u, ξ h ) -m(δu, ξ h ).
Next, according to [START_REF] Bukhgeim | Uniqueness in the large of a class of multidimensional inverse problems[END_REF] with (w h , x h ) = (z h , y h ), we have

|||(z h , y h )||| 2 * + A[(ξ h , η h ), (z h , y h )] = A[(i h u -u, i h p -p), (z h , y h )].
Thus, adding these two equalities, we get

S[(ξ h , η h ), (ξ h , η h )] + |ξ h | 2 ω M + |||(z h , y h )||| 2 * = A[(i h u -u, i h p -p), (z h , y h )] I + S[(i h u, i h p), (ξ h , η h )] II + m(i h u -u, ξ h ) -m(δu, ξ h ) III .
We bound the terms I-III term by term. By Lemma 4.1 and the approximation bound [START_REF] Burman | Solving ill-posed control problems by stabilized finite element methods: an alternative to Tikhonov regularization[END_REF], we have for term I

I (i h u-u, i h p-p) # |||(z h , y h )||| * h( u [H 2 (Ω)] d + p H 1 (Ω) )|||(z h , y h )||| * .
For term II, we have

S[(i h u, i h p), (ξ h , η h )] = S[(i h u -u, 0), (ξ h , η h )] + γ p Ω h 2 ∇i h p • ∇η h .
Thus, using [START_REF] Burman | Solving ill-posed control problems by stabilized finite element methods: an alternative to Tikhonov regularization[END_REF] with (v, q) = (u, p) for the first term and the H1 -stability of i h for the second term, we get

II h( u [H 2 (Ω)] d + p H 1 (Ω) )|||(ξ h , η h )|||.
For term III, according to [START_REF] Burman | Stabilized nonconforming finite element methods for data assimilation in incompressible flows[END_REF] with t = 2, we have

III ≤ γ M (|i h u -u| ω M + |δu| ω M )|ξ h | ω M ≤ (Ch 2 u H 2 (Ω) + γ 1/2 M |δu| ω M )γ 1/2 M |ξ h | ω M .
Thus, collecting the above bounds, we get

|||(ξ h , η h )||| 2 + |||(z h , y h )||| 2 * + γ M |ξ h | 2 ω M (h( u [H 2 (Ω)] d + p H 1 (Ω) )+γ 1/2 M |δu| ω M )(|||(ξ h , η h )||| 2 +γ M |ξ h | 2 ω M +|||(z h , y h )||| 2 * )
and we conclude by dividing by

(|||(ξ h , η h )||| 2 + γ 1/2 M |ξ h | 2 ω M + |||(z h , y h )||| 2 * ) 1 2 .
The above lemma does not give the convergence of the error, it only asserts that some residual must converge with optimal order if the exact solution is smooth enough. Nevertheless, this lemma implies the following a priori bounds on the finite element solution.

Corollary 4.1 Under the same assumptions as for Lemma 4.2 there holds

|||(u h , p h )||| h( u [H 2 (Ω)] d + p H 1 (Ω) ) + γ 1/2 M |δu| ω M ( 26
)
and

u h V + p h H 1 (Ω) u [H 2 (Ω)] d + p H 1 (Ω) + h -1 γ 1/2 M |δu| ω M . ( 27
)
Proof: In an evident way, we have

|||(u h , p h )||| ≤ |||(u h -u, p h -p)||| + |||(u, p)|||.
Thus, according to Lemma 4.2 and the fact that

|||(u, p)||| = h u V + s p (p, p) 1 2
, we get [START_REF] Antonio | Mathematical aspects of discontinuous Galerkin methods[END_REF]. Moreover, by definition of ||| • |||, we have

u h V + p h H 1 (Ω) h -1 |||(u h , p h )|||.
Thus inequality [START_REF] Antonio | Mathematical aspects of discontinuous Galerkin methods[END_REF] directly implies [START_REF] Ern | Theory and practice of finite elements[END_REF].

In the following theorem, we state an error estimate. The proof of this result relies on the conditional stability estimate for the continuous problem given by Corollary 2.1 and the convergence of the residual quantities given by Lemma 4.2.

Theorem 4.1 We assume that the solution (u, p) of (12 [START_REF] Ch | Effective downstream boundary conditions for incompressible navierstokes equations[END_REF]. Then, for all ω T ⊂⊂ Ω, there exists τ ∈ (0, 1) such that

) belongs to [H 2 (Ω)] d × H 1 (Ω) and we consider (u h , p h ) ∈ V h × Q 0 h , (z h , y h ) ∈ W h × Q h the discrete solution of
|u -u h | ω T ≤ Ch τ ( u [H 2 (Ω)] d + p H 1 (Ω) + h -1 |δu| ω M ) + h f L . (28)
The constant C depends on the mesh geometry, the geometry of ω M and ω T and on

U [W 1,∞ (Ω)] d×d .
Proof: Let us first introduce the weak formulation of the problem satisfied by (ξ, η) := (u -u h , p -p h ). By equation ( 12), we have for all w ∈ V 0 and q ∈ L, A[(ξ, η), (w, q)] = (f, w)

L 2 (Ω) -A[(u h , p h ), (w, q)]
We introduce, u h and p h being fixed, the linear forms r f and r g on V 0 and L respectively defined by: for all w ∈ V 0 and q ∈ L,

r f , w V 0 ,V 0 + (r g , q) L := (f, w) L 2 (Ω) -A[(u h , p h ), (w, q)].
It follows that (ξ, η) is solution of [START_REF] Bourgeois | A mixed formulation of the Tikhonov regularization and its application to inverse PDE problems[END_REF] with the functions f and g in the right hand sides replaced respectively by r f and r g . Applying now Corollary 2.1, we directly get

|ξ| ω T ≤ C( r f V 0 + r g L + ξ L ) 1-τ ( r f V 0 + r g L + |ξ| ω M ) τ . (29)
Using the first equation of ( 14), we can write the residuals: for all (w h , q h ) ∈

W h × Q h r f , w V 0 ,V 0 + (r g , q) L = (f, w -w h ) L 2 (Ω) -A[(u h , p h ), (w -w h , q -q h )] -S * [(z h , y h ), (w h , q h )].
We take w h = i h w and q h = i h q in this equality. For the first term, according to (21) for t = 1, we have

|(f, w -w h ) L 2 (Ω) | ≤ h f L w V .
The second term can be bounded by using the relations ( 24) and [START_REF] Antonio | Mathematical aspects of discontinuous Galerkin methods[END_REF]. For the last term, we have, according to Lemma 4.2

|S * [(z h , y h ), (w h , q h )]| ≤ |||(z h , y h )||| * (w h , q h ) # (hC(u, p) + |δu| ω M ) (w, q) # where C(u, p) := u [H 2 (Ω)] d + p H 1 (Ω) .
We thus get

r f , w V 0 ,V 0 + (r g , q) L hC(u, p) + |δu| ω M + h f L + |||(ξ, η)||| (w, q) # .
Since this bound holds for all w ∈ V 0 and q ∈ L, we conclude that

r f V 0 + r g L hC(u, p) + |δu| ω M + h f L + |||(ξ, η)|||.
Thus, we can bound the terms in the right hand side of (29) in the following way:

r f V 0 + r g L + ξ L C(u, p) + h f L + h -1 |δu| ω M
according to inequalities ( 26) and ( 27) and

r f V 0 + r g L + |ξ| ω M hC(u, p) + h f L + |δu| ω M according to Lemma 4.2.
Using these two bounds in [START_REF] Hecht | New development in FreeFem++[END_REF], we conclude that

|ξ| ω T (C(u, p) + h -1 |δu| ω M ) 1-τ (hC(u, p) + |δu| ω M ) τ + h f L h τ (C(u, p) + h -1 |δu| ω M ) + h f L ,
which completes the proof. Thus, if the measurement noise δu is equal to 0, this theorem gives the convergence of the error u -u h when h tends to 0 on any subset ω T ⊂⊂ Ω. In the case of perturbed data, the accuracy of the error is limited and inequality [START_REF] Fabre | Prolongement unique des solutions de l'equation de Stokes[END_REF] shows that the mesh size has to balance the error due to the discretization and the error due to the noise.

Numerical simulations

In this section, we apply the method introduced in Section 3 in different two-dimensional different numerical examples. The free parameters in [START_REF] Ch | Effective downstream boundary conditions for incompressible navierstokes equations[END_REF] are set to

γ u = γ div = γ p = γ * u = γ * p = 10 -1 , γ M = 1000,
in all the numerical examples. The numerical computations have been performed with FreeFEM++ (see [START_REF] Hecht | New development in FreeFem++[END_REF]).

Convergence study: Stokes example

In order to illustrate the convergence behavior of the method introduced in Section 3, we take up the test case presented in [START_REF] Burman | Stabilized nonconforming finite element methods for data assimilation in incompressible flows[END_REF] for the Stokes system. In the unit square Ω = (0, 1) 2 , we consider the velocity and pressure fields given by u(x, y) = (20xy 3 , 5x 4 -5y 4 ), p(x, y) = 60x 2 y -20y 3 -5.

It is straightforward to verify that (u, p) is a solution to the homogeneous Stokes problem with ν = 1, which corresponds to system (1) with U = 0 First, we perform the computation with unperturbed data. In Figure 1 (left), we report the velocity and pressure errors both in the global L 2norm and the local L 2 -norm in the subdomain ω T . We also provide the convergence history of the residual quantity for the velocity stabilization:

  F ∈F i γ u F h 1 2 ∇u h 2 F   1 2
.

The observed global asymptotic behaviors of the local velocity error (filled squares) and residual (filled circles) are in agreement with the convergence rates obtained in Theorem 4.1 and Corollary 4.1 with |δu| ω M = 0. It should be noted that, for the finest grids, the local velocity error tends to stagnate or increase, which can be related either to the impact of the rounding-off errors or to ill-conditioning issues of the system matrix, so that |δu| ω M > 0. The other error quantities, global velocity error (empty squares) and local and global pressure errors (filled and empty triangles, respectively) show a convergent behavior which also tends to stagnate for the smallest values of h. Figure 1 (right) presents the convergence history of the same quantities with a 10% Gaussian noise. The impact of the noise is clearly visible. In particular, it is worth noting that the convergence history of the local and global velocity and pressure errors is not monotone anymore and the residual loses first-order convergence rate. This is also in agreement with Theorem 4.1 and Corollary 4.1 with |δu| ω M > 0.

Convergence study: linearized Navier-Stokes example

In this subsection, we will use Taylor-Green vortices to construct exact solutions of [START_REF] Alessandrini | The stability for the Cauchy problem for elliptic equations[END_REF]. Let us first introduce a flow of size Rπ described by a Taylor-Green vortex:

u R (x, y) := -sin(x/R) cos(y/R) exp(-2νt/R 2 ) cos(x/R) sin(y/R) exp(-2νt/R 2 ) , p R (x, y) := 1 4 (cos(2x/R) + cos(2y/R)) exp(-4νt/R 2 ).
For any R > 0, we can check that (u R , p R ) is solution of the unsteady Navier-Stokes equations. In our numerical example, we take Ω = (0, 2π) 2 and consider the following system which admits (u

1 2 , p 1 2
) as solution:

(u 1 • ∇)u + (u • ∇)u 1 -ν∆u + ∇p = f in Ω ∇ • u = 0 in Ω
where f is given by

f = -(u 1 2 • ∇)u 1 2 + (u 1 • ∇)u 1 2 + (u 1 2 • ∇)u 1 -∂ t u 1 2 .
Thus, f and u 1 being given, we can use the method presented in Section 3 to reconstruct u and p from measurements on ω M . The measurement and target subdomains are defined by

ω M :=(0, π/2) × (π/2, 3π/2) ∪ (3π/2, 2π) × (π/2, 3π/2), ω T :=(π/2, 2π) × (π/2, 3π/2).
As in the previous case, we have performed numerical tests for unperturbed data and for data perturbed with a 10% noise and we have studied the convergence of the method. The obtained results are illustrated in Figure 2. The convergence curves present similarities with the ones obtained in Figure 1. We can all the same notice that, for unperturbed data, the evolution of the local velocity error is more satisfactory: it is close to the linear behavior and the error reaches much smaller values.

Application: relative blood pressure estimation from velocity measurements

To evaluate the risks related to a constriction (also called stenosis) in a blood vessel, the relative pressure difference (RPD) is a standard clinical bio-marker. Direct blood pressure measurements can however only be obtained through invasive procedures like catheterization. Non-invasive measurements are limited to the blood velocity. In particular, 4D-MRI provides a measurement of the velocity field in the whole vessel. A natural question is hence to reconstruct the RPD from these velocity measurements. We refer to [START_REF] Bertoglio | Relative pressure estimation from velocity measurements in blood flows: state-of-the-art and new approaches[END_REF] for a review on direct based estimation methods for this problem. The purpose of this example is to illustrate how the method introduced in Section 3 can be used to estimate the RPD from full velocity measurements. We assume that blood flow is described by the Navier-Stokes equations and that we have velocity measurements in the whole domain Ω (see Figure 3) at a given set of time instants. We denote by (0, T ) the time interval, by N the number of measurements instants and set the time-step length to ∆t := T N -1 . For all 0 ≤ n ≤ N -1, the symbol u n M stands for the measured velocity at time t n = n∆t. Then, for all 0 ≤ n ≤ N -2, we consider the following Oseen type equation in terms of (u n , p n ): Note that no boundary data is prescribed in [START_REF] Yu | Global uniqueness in inverse boundary value problems for the Navier-Stokes equations and Lamé system in two dimensions[END_REF], which can be cumbersome in practice since the measured velocity u n M is not necessarily divergence free (and hence incompatible with Dirichlet data on the whole boundary ∂Ω). We hence propose to estimate u n and p n (up to a constant) from the data assimilation problem [START_REF] Bramble | Leastsquares for second-order elliptic problems[END_REF], with f and a (in the definition (11) of A) given respectively by

   (u n M • ∇)u n -2ν∇ • ε(u n ) + ∇p n = - u n+1 M -u n Γ i Γ o Γ w Γ w
f = - u n+1 M -u n M ∆t , a(u, v) = Ω ((u n M • ∇)u) • v + ν Ω ε(u) : ε(v).
Note that, here, the measurement and target sets coincide, ω M = ω T = Ω, so that the estimated velocity field u n has to be seen as a physically driven regularization of the full velocity measure u n M . Yet, the main target is to estimate the RPD, defined by the following quantity:

δp = 1 |Γ i | Γ i p - 1 |Γ o | Γo p.
In oder to investigate this new approach, we consider a two-dimensional version of the test case reported in [START_REF] Bertoglio | Relative pressure estimation from velocity measurements in blood flows: state-of-the-art and new approaches[END_REF]Section 6]. The stenotic blood vessel represented in Figure 3 corresponds to a contraction of 60%. The radii of inlet and outlet are 1 cm, the length of the vessel is 6 cm and the dynamic viscosity is given by ν = 0.035 Poise. Synthetic measurements are first generated by numerically solving the incompressible Navier-Stokes system ∂ t u + (u • ∇)u -2ν∇ • ε(u) + ∇p = 0 in Ω × (0, T ), ∇ • u = 0 in Ω × (0, T ), [START_REF] Yu | Remark on boundary data for inverse boundary value problems for the Navier-Stokes equations [Addendum to MR3319370[END_REF] with the following boundary and initial conditions: This direct problem (31)-( 32) is discretized in space by continuous piecewise affine finite element approximations based on the SUPG/PSPG stabilization method. The time discretization consists in a backward Euler scheme with a semi-implicit treatment of the convective term. A standard backflow stabilization term is also applied on the outlet boundary Γ o in order to guarantee the overall stability of the numerical scheme (see, e.g., [START_REF] Ch | Effective downstream boundary conditions for incompressible navierstokes equations[END_REF]). The discretization parameters are set to ∆t = 0.002 s and h = 0.01 cm. This space-time grid generates a set of synthetic velocity measurements which can be perturbed either by noise or by space-time subsampling.

               u =
Figure 4 represents the estimate RPD with the same discretization parameters as for the direct problem (no subsampling). When the data are unperturbed, we see that the reconstructed curve is perfectly superimposed with the exact curve (Figure 4, left). With a 10% Gaussian noise, we observe that the reconstructed curve (which corresponds to the mean curve obtained from 30 tests with variable noises) succeeds in following accurately the variations of the reference data (Figure 4, right).

In Figure 5, the measurements are perturbed by a subsampling both in time and in space (the time step is 10 times larger and the mesh size is 8 times larger). We then solve the data assimilation problem with the time step or the mesh size corresponding to this subsampling. Figure 5 shows that the proposed approach is able to provide a reasonable estimation of the RPD with and without noise (10% Gaussian noise). In particular, we can clearly observe that the RPD peak is well captured in both cases.

Figure 1 :

 1 Figure 1: Relative errors against mesh size for the Stokes problem. Left: without noise. Right: with 10% noise.

Figure 3 :

 3 Figure 3: Geometric description of the domain Ω representing a stenotic blood vessel

  0 on Γ w × (0, T ),u = (-60(y 2 -1) sin 5π 2 t , 0 on Γ i × (0, T ), 2νε(u)n -pn = 0 on Γ o × (0,T ), pressure (dynes/cm2) h=0.01, dt=0.002, 10% noise (mean over 30 samples) reference

Figure 4 :

 4 Figure 4: Left: Noise free data, right: 10% Gaussian noise on the data with exact sampling. The exact RPD is represented in full line whereas the reconstructed RPD is represented in dotted line.

Figure 5 :

 5 Figure 5: Left: Time subsampling of 0.02 s, right: Space subsampling of 0.08 cm. The exact RPD is represented in full line whereas the reconstructed RPD is represented in dotted line.

Figure 6 :

 6 Figure 6: Velocity magnitude at t = 0.082, 0.162, 0.242 (from top to bottom). Right: reference. Left: reconstruction with space-time subsampling and 10% of Gaussian noise.

M∆tin Ω,∇ • u n = 0 in Ω. (30)


Sylvaine Cordier
email: sylvaine.cordier@inserm.fr.

Nadine Forget-Dubois

Mireille Desrochers-Couture

Florence Rouget

Leah Michineau

Christine Monfort

Jean Pierre Thome

Philippe Kadhel

Luc Multigner

Gina Muckle

Prenatal and childhood exposure to chlordecone and sex-typed toy preference of 7-year-old Guadeloupean children AUTHORS

Keywords: child development, sex-typed play behavior, chlordecone, endocrine-disrupting chemicals, French West Indies, insecticides, organochlorine

Chlordecone was used intensively as an insecticide in the French West Indies. Because of its high persistence, the resulting contamination of food and water has led to chronic exposure of the general population as evidenced by its presence in the blood of people of Guadeloupe, in particular in pregnant women and newborns, and in maternal breast milk. Chlordecone is recognized as a reproductive and developmental toxicant, is neurotoxic and carcinogenic in rodents, and considered as an endocrine-disrupting compound with well-established estrogenic and progestogenic properties both in vitro and in vivo. The question arises of its potential consequences on child neurodevelopment following prenatal and childhood exposure, in particular on behavioral sexual dimorphism in childhood.

We followed 116 children from the TIMOUN mother-child cohort study in Guadeloupe, who were examined at age 7. These children were invited to participate in a seven-minute structured play session in which they could choose between different toys considered as feminine, masculine or neutral. The play session was video recorded, and the percentage of the time spent playing with feminine or masculine toys was calculated. We estimated associations between playtime and prenatal exposure to chlordecone (assessed by concentration in cord blood) or childhood exposure (determined from concentrations in child blood obtained at the 7-year followup), taking into account confounders and co-exposures to other environmental chemicals. We used a two-group regression model to take into account sex differences in play behavior. Our results do not indicate any modification in sex-typed toy preference among 7-year old children in relation with either prenatal or childhood exposure to chlordecone.

Introduction

Sexual differentiation of brain starts during fetal development under the parallel influence of sex hormones and sex chromosomes [START_REF] Mccarthy | Reframing sexual differentiation of the brain[END_REF][START_REF] Lombardo | Fetal Testosterone Influences Sexually Dimorphic Gray Matter in the Human Brain[END_REF]. Expression of this differentiation includes both reproductive and non-reproductive behaviors that show sex differences. One of its first manifestation is observed early during childhood as toy preference (females choose dolls, males prefer toy-cars and balls), and this is observed among both human [START_REF] Alexander | Sex Differences in Infants' Visual Interest in Toys[END_REF]) and monkey offspring (Hasset et al. 2008). Compared to other behavioral sex differences such as aggression level, childhood play behavior gives the strongest evidence for an influence of steroid hormones during early fetal life [START_REF] Collaer | Human behavioral sex differences: A role for gonadal hormones during early development?[END_REF]). An argument against the role of social influence in this sex-typed behavior can be drawn from studies of individuals who have experienced abnormal gonadal exposure but normal socialisation. [START_REF] Muthén | Sex-typed toy play behavior correlates with the degree of prenatal androgen exposure assessed by CYP21 genotype in girls with congenital adrenal hyperplasia[END_REF], for instance, reported that girls with congenital adrenal hyperplasia, a syndrome resulting in overproduction of adrenal androgens from early fetal life, showed a preference for masculine toys in a structured play situation, and a dose-response relationship between the degree of preference and disease severity.

Endocrine-disrupting chemicals (EDCs), environmental chemicals that perturb hormonal systems, may interfere with normal brain foetal development when exposure occurs during pregnancy, including its sexual differentiation [START_REF] Frye | Endocrine disrupters: A review of some sources, effects, and mechanisms of actions on behaviour and neuroendocrine systems[END_REF]. A number of studies have been conducted on the association between perinatal exposure to some EDCs and sexual dimorphic neurobehaviors. Some individual congeners of polychlorinated biphenyls (PCBs), a large class of persistent contaminants, are known for their neurotoxic effects in animals and humans [START_REF] Faroon | Effects of polychlorinated biphenyls on the nervous system[END_REF], and for their ability to disrupt steroid hormone signalling via estrogen or aryl hydrocarbon receptors [START_REF] Bell | Endocrine-disrupting actions of PCBs on brain development and social and reproductive behaviors[END_REF]. Sexually dimorphic neurobehaviors following perinatal exposure to selected PCBs were reported in animals [START_REF] Schantz | Spatial learning deficits in adult rats exposed to ortho-substituted PCB congeners during gestation and lactation[END_REF] and humans in the Yu-Cheng cohort of children born to mothers accidentally exposed to high levels of PCBs and dioxins in rice oil in Japan and Taiwan [START_REF] Guo | Gender-related decrease in Raven's progressive matrices scores in children prenatally exposed to polychlorinated biphenyls and related contaminants[END_REF]. Using the Pre-School Activities Inventory (PSAI) to rate sexually dimorphic patterns of play behavior in school-age children, Vreugdenhill et al. (2002) and Winneke et al. (2014) reported similar associations: a masculinization of typical sex play behavior in girls as well as a feminization of typical sex play behavior in boys with increasing prenatal exposures to PCBs and dioxins. A similar alteration of sex-typed behaviors in boys using the PSAI was reported in relation with prenatal exposure to phthalates, a category of EDCs with antiandrogenic properties (Swan et al. 2010).

Chlordecone (Kepone) is a synthetic organochlorine compound, a recognized reproductive and A c c e p t e d m a n u s c r i p t developmental toxicant, neurotoxic and carcinogenic in rodents [START_REF] Faroon | Atsdr Evaluation of Health Effects of Chemicals[END_REF]. Chlordecone is an EDC with well-established estrogenic and progestogenic properties both in vitro and in vivo [START_REF] Hammond | Estrogenic activity of the insecticide chlordecone (Kepone) and interaction with uterine estrogen receptors[END_REF][START_REF] Lemaire | Activation of α-and β-estrogen receptors by persistent pesticides in reporter cell lines[END_REF][START_REF] Scippo | Recombinant human estrogen, androgen and progesterone receptors for detection of potential endocrine disruptors[END_REF]Vonier et al. 1996). The toxicity of chlordecone to humans was revealed following intoxication of workers in a production facility in Hopewell, Virginia, USA in the 1970s. Intoxicated workers showed reduced fertility and signs of neurotoxicity such as tremors and memory loss [START_REF] Cannon | Epidemic kepone poisoning in chemical workers[END_REF]. Experimental studies with rodents have shown that gestational and perinatal chlordecone exposure impairs neurobehavior (acquisition and retention of active and passive avoidance response, acoustic startle reflex, motor coordination) during pre-weaning and post-weaning development [START_REF] Mactutus | Evaluation of long-term consequences in behavioral and/or neural function following neonatal chlordecone exposure[END_REF][START_REF] Mactutus | Neonatal chlordecone exposure impairs early learning and retention of active avoidance in the rat[END_REF][START_REF] Mactutus | Neonatal chlordecone exposure impairs early learning and memory in the rat on a multiple passive avoidance task[END_REF][START_REF] Squibb | Effects of gestational and perinatal exposure to chlordecone (Kepone) on the neurobehavioral development of Fischer-344 rats[END_REF]. Such alterations appear generally more pronounced in males, suggesting sex-dependent neurotoxic effects. Chlordecone exposure during the sensitive period for neurodevelopment can affect sexual differentiation of the brain, resulting in significant permanent sexually differentiated behaviors in adult rats [START_REF] Laessig | Effects of prenatal chlordecone on sexually differentiated behavior in adult rats[END_REF].

Chlordecone was intensively used as an insecticide in the French West Indies from 1973 to 1993 to control banana root borers. This pesticide undergoes no significant biotic or abiotic degradation in the environment and, as a consequence, permanently polluted soils and waters led to contamination of foodstuffs. It was only from 1999 that French authorities became aware of the extent of this pollution [START_REF] Dubuisson | Impact of subsistence production on the management options to reduce the food exposure of the Martinican population to Chlordecone[END_REF][START_REF] Guldner | Pesticide exposure of pregnant women in Guadeloupe: Ability of a food frequency questionnaire to estimate blood concentration of chlordecone[END_REF][START_REF] Seurin | Dietary exposure of 18-month-old Guadeloupian toddlers to chlordecone[END_REF]. The first population contamination data are from the early 2000s and were in the ppb range (µg/L wet weight) in blood and milk [START_REF] Multigner | Chlordecone exposure and adverse effects in French West Indies populations[END_REF]. There is evidence that chlordecone is still present in the blood of the general population of Guadeloupe, in particular in pregnant women and newborns, and in maternal breast milk [START_REF] Guldner | Pesticide exposure of pregnant women in Guadeloupe: Ability of a food frequency questionnaire to estimate blood concentration of chlordecone[END_REF][START_REF] Dallaire | Cognitive, visual, and motor development of 7-month-old Guadeloupean infants exposed to chlordecone[END_REF]. To evaluate the possible health effects resulting from exposure to chlordecone during pregnancy and childhood, we implemented a mother-child cohort in Guadeloupe (TIMOUN cohort) from 2004. We reported in follow-up conducted during infancy significant associations between cord chlordecone concentrations and poorer neurodevelopmental outcomes, more precisely with reduced novelty preference at 7 months (above 0.31µg/L in cord blood) [START_REF] Dallaire | Cognitive, visual, and motor development of 7-month-old Guadeloupean infants exposed to chlordecone[END_REF]) and poorer fine motor development at 7 (above 0.06µg/L in cord blood) and 18 months (above 0.24µg/L in cord blood), age at which only males were affected [START_REF] Boucher | Exposure to an organochlorine pesticide (chlordecone) and development of 18-month-old infants[END_REF]).

Here we report the evaluation of sex-typed play behavior of 7-year-old children in the TIMOUN cohort, in relation with prenatal and childhood exposure to chlordecone.

Methods

Population and data collection

The TIMOUN study is a prospective mother-child cohort currently underway in the Guadeloupe archipelago.

Between November 2004 and December 2007, 1068 pregnant women were enrolled during their late prenatal visit at the public hospitals of Pointe-à-Pitre or Basse-Terre or at a local antenatal care dispensary. A prenatal maternal face-to-face interview was conducted at enrollment by trained midwives to assess obstetrical, medical, personal, and socioeconomic characteristics. Maternal diseases, adverse delivery incidents, as well as newborn anthropometric parameters and health information, were collected from child and maternal medical records.

For assessment of child neurodevelopment at 3, 7 or 18 months, a subsample was selected. Exclusion criteria associated with maternal diseases were history of diabetes, gestational diabetes mellitus, hypertension, epilepsy, human immunodeficiency virus infection, and long-term corticotherapy (n = 254). Exclusion criteria for newborns (n = 225) were multiple birth, stillbirth, major birth defects, gestational age < 37 wks, APGAR < 7 at 5 min, intrauterine growth restriction < 10 th percentile of birth weight for gestational age, severe respiratory distress, severe icterus, hypoglycemia and maternofoetal infection. For the remaining 589 participants, 287 were not contacted because of incorrect address or refusal to participate. Altogether, 302 children participated in at least one follow-up at 3, 7 or 18 months of age [START_REF] Dallaire | Cognitive, visual, and motor development of 7-month-old Guadeloupean infants exposed to chlordecone[END_REF][START_REF] Boucher | Exposure to an organochlorine pesticide (chlordecone) and development of 18-month-old infants[END_REF]. Of the 302 children, 70 did not have a cord blood sample. At the 7-year follow-up, 62 were not contacted because of incorrect address or for exceeding the age window, and 15 refused; 155 came at the Pointe-à-Pitre hospital for neuropsychological and medical evaluation and collection of a blood sample, 125 of them completed a play session as part of the neuropsychological evaluation. Lack of access to the playroom because of other activities was the main reason for the missing play session for 30 children.

A maternal interview was conducted to provide information on familial environment of the child at the 7-year follow-up, and to document the non-verbal intellectual functioning of the mother assessed with the Raven's Progressive Matrices continuous score [START_REF] Raven | Manual for Raven's Progressive Matrices and Vocabulary Scales[END_REF]. Additionally, at the 7 and 18 months followed-ups, the Home Observation for Measurement of the Environment (HOME, [START_REF] Caldwell | HOME Inventory and Administration Manual[END_REF]) was used to document the quality of stimulation received by the child.

Written informed consent was obtained from the principal caregiver, usually the mother, of each participating child. The research procedures were approved by the Guadeloupean Ethics Committee for biomedical studies involving human subjects.

A c c e p t e d m a n u s c r i p t

Assessment of sex-typed play behavior was organized in a room dedicated to this task, as a structured toy session [START_REF] Muthén | Sex-typed toy play behavior correlates with the degree of prenatal androgen exposure assessed by CYP21 genotype in girls with congenital adrenal hyperplasia[END_REF]. Seven different toys chosen as typically masculine, feminine or neutral for children of that age, were placed in a defined order in a semi-circle on a carpet, every other toy being masculine and feminine and the neutral toys in between. Feminine toys included a doll with a blanket and a bottle, a female doll's head with a comb, a brush and colored hairpins and elastics, and a set comprised of a pink pony camper with 3 small colored ponies and a bucket. Masculine toys were a set of 5 miniature cars, a set composed of a fire truck and an helicopter with two figurines, and a set of two superheroes figurines with their accessories. Neutral toys were a set of 3 children's books. The child tester instructed the child to sit in the middle of the carpet and play with the toys until her return. The playing time was videotaped for a total of 7 min. The classification of either feminine, masculine or neutral play was done according to which toy was touched by the child. Instances where children played with toys of two categories at the time were counted separately from masculine, feminine or neutral play. Two examiners unaware of the child exposure did the coding. A between-rater agreement was established first, and then calculated every 10 videos to maintain it. The raters were unaware of which videos would be used for this purpose. Agreement on a play sequence duration was considered acceptable if the durations observed by the two raters differed by 5% or less. Feminine and masculine play are reported as a proportion of the complete playing time. We realized during training of video coding that even though the pony camper was considered a feminine toy, the boys played with it as with a big truck, considerably more than with any other feminine toy. We therefore counted the time allotted to playing with the pony camper separately and removed it from feminine playtime for all participants. We kept in the analyses only the participants whose total playtime was at least 3 minutes after removal of the pony camper sequences (N = 116).

Assessment of exposure to contaminants and nutrients

A cord blood sample, and a child blood sample at the 7 years visit, were collected in EDTA tubes to document respectively prenatal and childhood exposure to chlordecone and to other environmental contaminants. Plasma Trace 2000). Preparation of samples and quantification method were previously described [START_REF] Debier | Quantitative dynamics of PCB transfer from mother to pup during lactation in UK grey seals Halichoerus grypus[END_REF][START_REF] Multigner | Chlordecone Exposure and Risk of Prostate Cancer[END_REF]. The limit of detection (LOD) was 0.05 µg/L for the cord chlordecone blood measures and 0.02 µg/L for the childhood chlordecone measures, and 0.05 µg/L for all measures of PCB-153 and DDE. Values under the LOD were replaced by random values <LOD with the R package RobCompositions (Templ et al. 2011) for the cord and child chlordecone, cord and child PCB-153 and cord DDE measures; child DDE had only one value under LOD, which was kept as measured. Total cholesterol and triglyceride in plasma were determined by standard enzymatic procedures (DiaSys Diagnostic Systems GmbH; Holzheim, Germany), and total lipid concentrations were calculated as described by [START_REF] Bernert | Calculation of serum "total lipid" concentrations for the adjustment of persistent organohalogen toxicant measurements in human samples[END_REF]. Blood Hg and Pb concentrations were measured by inductively coupled plasma mass spectrometry (ICP-MS) at the laboratory of the Centre de toxicologie du Québec. For Hg determination, blood samples were diluted 20 fold in a solution containing ammonium hydroxide before analysis. The LOD for Hg and Pb were 0.4 µg/L and 2 µg/L respectively; child Hg had four values under LOD, which were kept as measured.

For cord docosahexaenoic acid (DHA, an omega-3 fatty acid) quantification, total lipids were first extracted from a 200 µl of total whole blood with 6 ml of chloroformethanol 2:1 (v/v) containing butylated hydroxytoluene to minimize oxidation of polyunsaturated fatty acids. This method was previously validated [START_REF] Chajes | Association between serum trans-monounsaturated fatty acids and breast cancer risk in the E3N-EPIC Study[END_REF]. Fatty acids were methylated with Methyl Prep II (Alltech, Templemars, France) at room temperature for 10 minutes. Fatty acids methyl esters were extracted with 4 ml of hexane. Analyses of DHA were performed by gas chromatography (6890N Agilent, Massy France) with a capillary column (SGE, Courtaboeuf, France) used for separation of fatty acid methyl esters. DHA levels were expressed as percentage of total fatty acids by integrating the area under the peak and dividing the result by total area. Identification of DHA was made by a known mixture of fatty acid methyl esters (SUPELCO, St Quentin Fallavier, France).

Confounding variables

Sex, child age, and maternal Raven's score were treated as obligatory covariates. The following variables were considered as potential maternal and family characteristics confounders: age and parity of the mother at delivery, marital status (single or not), maternal education, family income. Child characteristics considered were gestational age and number of weeks breastfed. The presence of siblings of opposite sex was included among participant characteristics. The following contaminants and nutrients were also considered as potential confounders: cord DHA, Hg, Pb, PCB-153, and DDE.

A c c e p t e d m a n u s c r i p t

Statistical analysis

We investigated the means, variances, and distribution of each variable. All exposure variables and maternal breastfeeding duration were log 2 -transformed to reduce skewness. An examination of the descriptive statistics and the residuals for the proportions of masculine and feminine play showed bimodal distributions and nonnormal residuals, because girls played a lot more with feminine toys and boys played a lot more with masculine toys (see Table 2). To take potential sex differences into account without splitting the sample altogether, we chose to test the association between chlordecone and sex-typed play with two-group models. In the first step, we let the parameters in each regression models vary freely between the boy group and the girl group (saturated model with 0 degrees of freedom). In a second step, we constrained to equality all corresponding parameters (beta, means, variances, covariances) between the two groups. A deterioration of the overall fit of the two-group model at this step indicates that at least one of the parameters differs significantly between boys and girls. In a third step, we allowed the means and variances to vary between boys and girls. If the overall fit was good at this step, the model was considered the most parsimonious; in cases where the fit would still be deteriorated (i.e. significantly different from the saturated model, p < 0.05), then we would allow the beta representing the association between the chlordecone measure and the sex-typed play outcome to vary between boys and girls to formally test the hypothesis of a moderation by sex of the contaminant-outcome association. Because of the small sample and non-normal data, we used a bootstrapping method (n = 1000) to correct the standard errors and confidence intervals of the parameters. The proportion of missing data ranged from 0% to 25% (Child DDE; see Table 1). Little's Missing Completely at Random test [START_REF] Little | A test of missing completely at random for multivariate data with missing values[END_REF]) performed in SPSS on all contaminants measures, outcomes, and confounders, was not significant (Little's Chi-Square = 453.42, DF = 453, p = 0.49), therefore the missing data were considered unbiased. Descriptive statistics, bivariate correlations and patterns of missing data analyses were computed with IBM SPSS 25.0 for Windows (IBM, Chicago, IL, USA). All multivariate models were tested with Mplus 8.2 [START_REF] Muthén | Sex-typed toy play behavior correlates with the degree of prenatal androgen exposure assessed by CYP21 genotype in girls with congenital adrenal hyperplasia[END_REF][START_REF] Muthén | Sex-typed toy play behavior correlates with the degree of prenatal androgen exposure assessed by CYP21 genotype in girls with congenital adrenal hyperplasia[END_REF].

Results

Descriptive statistics and bivariate associations

Table 1 shows that less than half of the children had a sibling of the opposite sex at home, which could affect their familiarity with sex-typed toys. Further analysis showed that girls who had brothers (n = 27) actually played less with masculine toys (mean masculine play for girls without brothers = 0.29; mean masculine play for girls with brothers = 0.13; T(67) = 3.22, p = 0.002). Contaminant exposure generally decreased between birth A c c e p t e d m a n u s c r i p t (cord blood concentrations) and childhood (7-year blood), except for Pb, which increased, and DDE, which stayed relatively stable. Boys and girls did not differ in exposure for any of the contaminants (non-significant Ttests not shown). Table 2 shows the proportion of time spend playing with feminine and masculine toys. As expected, boys spent largely more time playing with masculine toys (77% of the playing time) while girls were spending more time playing with feminine toys (68% of the playing time).

Results from Table 3 shows that cord and child chlordecone concentrations are not significantly correlated, as well as they are not related with the other contaminants documented. Significant correlations are found between persistent organochlorine compounds PCB-153 and DDE. Cord and child Pb concentrations are significantly correlated while only child Pb concentrations are correlated with cord PCB-153 and cord Hg.

Two-group models predicting sex-typed play

The unconstrained models showing the association of chlordecone exposure with masculine and feminine play for boys and girls are reported in Table 4, and the subsequent tests of model fit after constraining the parameters are shown in Table 5. We found no significant association of prenatal and childhood chlordecone exposure with masculine or feminine play, and no moderation of these associations by sex. However, the standardized beta larger than 0.20 between childhood chlordecone and feminine play in both boys and girls suggest that the lack of significance could be a power issue. In all models, constraining all parameters between boys and girls resulted in a significant deterioration of model fit (Chi-square difference between the saturated model and the constrained model at p < 0.000). In all cases, freeing the means and variances of the measures of sex-typed play between boys and girls resulted in models with good fit; this means that models with the same parameters for boys and girls except for the means and variance of masculine and feminine play were the most parsimonious.

Discussion

This is the first study of its kind conducted in a population environmentally exposed to chlordecone, an EDC with well-established estrogenic and progestogenic properties. Our results show no association between prenatal chlordecone exposure and the time spent playing with toys boys and girls are typically attracted to. An increase in playtime with feminine toys (feminization) was observed in association with childhood chlordecone exposure for children of both sexes, but did not reach statistical significance (p=0.17).

Chlordecone blood levels observed in our population are thousand times lower than those measured among the intoxicated Hopewell workers, but similar to those experienced by a community population around A c c e p t e d m a n u s c r i p t the Hopewell factory in the 1970's, in which 12% reported tremor symptoms compared to 46% of the workers [START_REF] Cannon | Epidemic kepone poisoning in chemical workers[END_REF]. There is currently no data on exposure to chlordecone in other populations.

Early experimental studies in rodents have suggested sex-dependent, persistent alterations in neurobehavioral function as responses to neonatal exposure to chlordecone [START_REF] Squibb | Effects of gestational and perinatal exposure to chlordecone (Kepone) on the neurobehavioral development of Fischer-344 rats[END_REF][START_REF] Mactutus | Neonatal chlordecone exposure impairs early learning and retention of active avoidance in the rat[END_REF][START_REF] Mactutus | Evaluation of long-term consequences in behavioral and/or neural function following neonatal chlordecone exposure[END_REF][START_REF] Cooper | Effects of perinatal Kepone exposure on sexual differentiation of the rat brain[END_REF]. A more recent study reported both masculinized and feminized reproductive (mounting, lordosis) and non-reproductive behaviors (exploratory behavior, anxiety) in adult rats following gestational exposure to chlordecone, consistent with both estrogenic and anti-estrogenic effects [START_REF] Laessig | Effects of prenatal chlordecone on sexually differentiated behavior in adult rats[END_REF]), depending on the estrogen receptor subtypes that are being activated or antagonized [START_REF] Lemaire | Activation of α-and β-estrogen receptors by persistent pesticides in reporter cell lines[END_REF]. In previous analyses of the same TIMOUN birth cohort, reduced novelty preference was observed at 7 months in children of both sexes [START_REF] Dallaire | Cognitive, visual, and motor development of 7-month-old Guadeloupean infants exposed to chlordecone[END_REF], and poorer fine motor function among boys at 18 months [START_REF] Boucher | Exposure to an organochlorine pesticide (chlordecone) and development of 18-month-old infants[END_REF] in association with increased prenatal exposure to chlordecone. These previous reports suggest a sex-dependent impact of prenatal exposure to chlordecone on child neurobehavioral development at 18 months, which is not apparent in our present report.

Several reasons may explain these negative findings. First, the small sample implies a loss of statistical power. However, effect size of prenatal exposure, in both sexes, is small (standardized beta <0.123 for feminine play, <0.096 for masculine play). Compared to the whole cohort, families engaged in the neurodevelopmental follow-up (no maternal diseases, no adverse birth outcomes) were more educated and slightly less exposed to chlordecone (as expected due to the existing association between chlordecone exposure and risk of preterm birth - [START_REF] Kadhel | Chlordecone exposure, length of gestation, and risk of preterm birth[END_REF]. Children who participated in the play session were not different from other children in the follow-up. The choice of a "healthy sample" for the investigation of possible neurodevelopmental consequences of chlordecone was justified by the high prevalence of maternal diseases during pregnancy in Guadeloupe [START_REF] Rouget | Medical and sociodemographic risk factors for preterm birth in a French Caribbean population of African descent[END_REF], that may have blurred the association with environmental exposure. This choice, however, limits the generalization of our results. Another potential limitation relates to the instrument used. The structured play session was adapted from a similar protocol used in the general population to study the evolution of toy preference with age (from 1 to 5 years) [START_REF] Servin | Sex differences in 1-, 3-, and 5-year-olds' toy-choice in a structured playsession[END_REF] and in clinical setting among girls with congenital adrenal hyperplasia, therefore submitted to overproduction of androgens in utero [START_REF] Muthén | Sex-typed toy play behavior correlates with the degree of prenatal androgen exposure assessed by CYP21 genotype in girls with congenital adrenal hyperplasia[END_REF]. In both situations, this protocol proved to be a very sensitive tool. Previous studies investigating the influence of prenatal exposure to PCBs/PCDDs (Vreugdenhill 2002;Winneke 2014) or of phthalates (Swan 2010) on sextyped play behavior, used the Pre-School Activities Inventory (PSAI). The PSAI is based on parental, mainly maternal, reports of a combination of aspects of sexual behavioral dimorphism along three categories, namely A c c e p t e d m a n u s c r i p t toys preference (e.g. cars vs dolls), preferred activities (e.g. rough-and-tumble vs homework play) and other characteristics (e.g. risk taking vs risk avoidance) [START_REF] Golombok | The Pre-School Activities Inventory: A standardized assessment of gender role in children[END_REF]. Our instrument, unlike the PSAI, is based on only one of these aspects, namely direct standardized observations of toys preference. Nevertheless, we believe that our standardized play session, although representing a more narrow behavioral spectrum than the PSAI, is more objective, more specific and more adapted to the hypothesis tested. The choice of toys in the structured play session is important however. The greater proportion of time playing with toys typically designed for the child's sex somewhat validate our selection of the toys for the play session, but we had to exclude a posteriori a toy that was not clearly feminine, masculine or neutral, reducing the overall time of observation. This may have reduced our ability to fully capture the outcome measured.

No human or experimental studies of the consequences of postnatal exposure to chlordecone on neurobehavior before puberty are available. In adult male rats, acute or chronic exposure to chlordecone induces neurobehavioral changes in a series of tests for sensorimotor functions [START_REF] Squibb | Effects of gestational and perinatal exposure to chlordecone (Kepone) on the neurobehavioral development of Fischer-344 rats[END_REF][START_REF] Pryor | Assessment of chemicals using a battery of neurobehavioral tests: a comparative study[END_REF]). In our study, although the increase in proportion of time playing with feminine toys (feminization) in children of both sexes in association with childhood chlordecone exposure did not reach statistical significance (p=0.17), the strength of the association (standardized regression coefficients >0.20) or the effect size is in the same range as those reported by [START_REF] Boucher | Exposure to an organochlorine pesticide (chlordecone) and development of 18-month-old infants[END_REF] when measuring associations between prenatal chlordecone exposure and fine motor scores at 18 months in the same TIMOUN cohort. Therefore, additional studies with larger sample size are needed to determine whether feminization of behavior, if any, could be attributed to the progestogenic action of chlordecone [START_REF] Birke | Progestin-induced changes in play behaviour of the prepubertal rat[END_REF]1984). In such studies a protocol involving the objective observation of toy preference, as used in the present study, could be combined with parental reporting of more comprehensive aspects of sexual dimorphism. Note: Constrained model = all parameters constrained to equality in the two groups (boys and girls). Significant p indicates a deterioration of the model fit caused by the constraint. Mean/variance parameters of play freed = the equality constraint between the groups was relaxed for the mean and variance of male play and female play. Non-significant p indicates that the relaxed model shows a good fit to the data.

A c c e p t e d m a n u s c r i p t

  samples were stored at -30 °C in Polypropylene Nunc® tubes following centrifugation. Chlordecone, p,p'dichlorodiphenylchloroethylene (DDE), PCB congener 153 (PCB-153) and lipids were measured in plasma. Total mercury (Hg) and lead (Pb) were quantified in whole blood. Determination of chlordecone, DDE and PCB-153 concentrations were done by the Center for Analytical and Research Technology at Liege University (Belgium). Contaminant concentrations were analyzed by high-resolution gas chromatography (Thermo Quest

  A c c e p t e d m a n u s c r i p t

Table 4 -

 4 Associations of contaminants with female and male play for boys and girls Two-group regression models tested with an FIML estimator. Standard errors and confidence intervals obtained by bootstrapping (n=1000). All models are adjusted for age at testing and mother's Raven score. Additional confounding factors: a Cord Hg, child PCB 153, child total lipid; b Child PCB 153, child total lipid; c Cord Hg, presence of siblings of opposite sex; d Mother's marital status (living with partner or not).

		Feminine Play
	Beta (s.e.)	95% C.I.	p	Stand. Beta

Table 5 -

 5 Changes in model fits in the sex X contaminants interaction tests

		Feminine Play
	Χ 2	DF	p

We thank Pierrich Plusquellec for his help in the design of the play situation. This work was supported by grants from the French General Health Directorate (DGS RMC11129NNA).

Androl 33: 259-267. doi: 10.1111/j.1365-2605[START_REF] Alexander | Sex Differences in Infants' Visual Interest in Toys[END_REF].01019.x Templ M, Hron K, Filzmoser P (2011). RobCompositions: an R-package for robust statistical analysis of compositional data. In V. editors,Compositional Data Analysis. Theory and Applications,John Wiley & Sons,Chichester (UK) Vonier PM, Crain DA, McLachlan JA, et al (1996)


Rubiela Carrillo
email: rubiela.carrillo-rozo@liris.cnrs.fr

Yannick Prié
email: yannick.prie@univ-nantes.fr

Élise Lavoué
email: elise.lavoue@univ-lyon3.fr

Observing learner engagement on mind mapping activities using Learning Analytics

Keywords: Engagement, Indicators, Mind map, Learning Analytics

Research on learner engagement has increased in recent years arguing that it favors academic success. Teachers want their learners to engage in meaningful learning activities, like mind mapping, but they lack clues for observing their engagement along the activity. In this paper, we propose indicators of behavioural and cognitive dimensions of learner engagement for mind mapping activities based on interaction traces. Our indicators have been defined from final mind maps as well as from the mind mapping processes. We discuss implications for the observation of learner engagement in learning activities similar to mind mapping.

Introduction

Mind maps are graphic representations composed of nodes and links, in which the nodes represent ideas and the links represent their connections. Their first educational uses were mainly to support instruction and reading, while latest uses support the mapping processes inspired by constructivist theories [START_REF] Novak | Learning, Creating, and Using Knowledge: Concept maps as facilitative tools in schools and corporations[END_REF] and the active learning benefits. Researchers in educational psychology have defined several dimensions of learner engagement (e.g. behavioural, cognitive) [START_REF] Fredricks | School Engagement: Potential of the Concept, State of the Evidence[END_REF] to facilitate its observation. Behavioural engagement refers to observable actions to carry out a learning task. Cognitive engagement relates to strategies for accomplishing a learning task. We consider that learners' actions to elaborate mind maps may describe their behaviour, and that mapping activity may expose their cognitive processes and strategies.

Indicators could highlight learner engagement in mind mapping activities and help teachers to adapt their pedagogical strategies. However, existing indicators mainly target the assessment of the final mind maps [START_REF] Cañas | Cmapanalysis: an extensible concept map analysis tool[END_REF], and those that are obtained from the mapping processes lack representations of such processes [START_REF] Yin | Comparison of two concept-mapping techniques: Implications for scoring, interpretation, and use[END_REF]. [START_REF] Miller | Use of the Cmaptools Recorder to explore acquisition of skill in concept mapping[END_REF] propose indicators from learners' traces and represent mind mapping processes but only for strictly hierarchical graphs. We think that mind mapping activities should allow learners to express freely their ideas, without imposing structures.

In this paper, we propose behavioural and cognitive indicators of learner engagement to assist teachers in monitoring and assessing mind mapping activities. Methods from Learning Analytics can be used to collect and analyse learners' interaction traces during mind mapping in order to obtain such indicators.

Theoretical model and data sources to define indicators

We propose a model of behavioural and cognitive engagement with four characteristics (i.e. participation, effort, meta-cognitive and cognitive strategies) that could be observed along mind mapping activities. Participation [START_REF] Newmann | Student Engagement and Achievement in American Secondary Schools[END_REF] is associated with the behavioural dimension. Effort is present in both dimensions, as behavioural effort (i.e. quantity) and cognitive effort (i.e. quality) [START_REF] Fredricks | The Measurement of Student Engagement: A Comparative Analysis of Various Methods and Student Self-report Instruments[END_REF]. Metacognitive and cognitive strategies [START_REF] Pintrich | The role of motivation in promoting and sustaining self-regulated learning[END_REF] are related to the cognitive dimension of engagement. We associate each of these characteristics with indicators obtained from the final mind maps and the sequences of actions carried out to construct them. Final maps refer to their elements (i.e. nodes and links), and to their properties (e.g. title, description, URL, position X, position Y). Sequences of actions are defined by the creation, deletion, modification and displacement actions, applied to the mind map elements. Each action is associated with its time stamp, which allows to reconstruct the activity.

Behavioural and cognitive engagement indicators

3.1 Indicators of participation -Number of elements: different trends can be found according to the type of elements created: definition of ideas (a greater number of nodes than links), or association of ideas (a greater number of links than nodes). -Duration of the construction: period of time between the first and the last action to build the mind map. -Time between two consecutive actions: a learner who interrupts his/her actions for a significant amount of time (determined by the teacher) may be in difficulty or poorly engaged behaviourally.

Indicators of behavioural effort

-Ratio between the number of elements and the minimum expected number: a mind map with a very poor proportion of expected elements may suggest a difficulty not overcome because of lack of knowledge or effort. -Number of elements with expected minimum properties: when learners do not define the properties of an element, considered crucial for interpreting the final map, they may have difficulties and lack effort. -Number of consecutive sequences of creation-deletion of a element:

the creation of the last element that is not removed may be interpreted as a possible solution to a difficulty, and to show the learner's effort.

-Number of displacements on the set of nodes: too high values may indicate difficulties with the subject of the map, or possible problems with the mapping tool interface. -Number of displacements of each node on the mind map: may identify concepts that require more structuring effort. -Number of modifications on the set of elements: may provide clues on the difficulties in defining the elements and the efforts made to solve them; or a lack of control over the mapping tool. -Number of modifications in the properties of each element: a very high value of the number of title modifications may indicate the difficulty in defining it.

Indicators of cognitive effort

-Pertinence of the element properties: inadequate values of the properties may be the result of a weak cognitive engagement to overcome a difficulty. -Pertinence of a created (and not deleted) element following consecutive sequences of creation-deletion: the last created element may be evaluated to judge if it brings quality to the map. It may reflect the learner's cognitive effort in face of a difficulty. -Pertinence of displacements of the same node: when the learner move repeatedly a node to a place that does not add meaning to the close and connected elements, the idea represented by the node may require more cognitive effort from the learner to be articulated. -Pertinence of modifications made to the set of elements: modifications improving the mind map quality may reflect cognitive effort. -Pertinence of modifications to the properties of a element: modifications not considered relevant may reflect a low cognitive effort of the learner.

Indicators of meta-cognitive strategies

-Construction approach: may be 1) based on a logical order (creating links between two nodes when these nodes are created, or creating links at the end once all the nodes have been created), or 2) based on a strong trial and error.

A trial and error approach does not show the follow-up of a meta-cognitive planning strategy because of difficulties to understand the subject and to use the mapping tool, or because of a low cognitive engagement in the activity. -Pertinence of the actions on the elements following a help request: learners self-regulate when they monitor their own activity, identify their needs, seek for help , and adapt their map according to the answers. -Pertinence of the actions on the elements following a search for information: learners self-regulate when they identify their information needs to build the map, search the information in a tool, and adapt their map according to the results obtained.

Indicators of cognitive strategies

-Ratio between the number of titled elements and the total number of elements: calculated from the final mind map, it focuses on the elaboration strategies used. -Length of the node titles: length of the node titles in the final map.

-Pertinence of the links: refers to the pertinence of the links between the nodes in the final map, and to the pertinence of the titles of these links. -Pertinence of the topological structure: pertinence of the spatial distribution of the nodes in the final graph.

-Pertinence of all actions on the elements over time: can be evaluated from the sequences of actions identified over time.

4 Discussion and future works

The next step will be to implement and evaluate our indicators on a dashboard. The visualization of indicators over time on a dashboard could favor the monitoring of learner engagement during mind mapping processes. Our main concern is to evaluate the relevance of the proposed indicators for teachers. Our future works will be directed toward the evaluation of our model of engagement for monitoring other learning activities that rely on the construction of resources such as text writing. We may consider the paragraphs as nodes, and the ideas that articulate them as links.