Mirko Fiacchini
email: fiacchini@laas.fr.

Sophie Tarbouriech
email: sophie.tarbouriech@laas.fr.

Christophe Prieur
email: christophe.prieur@gipsa-lab.grenoble-inp.fr.

Polytopic control invariant sets for differential inclusion systems: a viability theory approach

This paper presents a criterion to characterize control invariant polytopes for differential inclusion systems. The practice-oriented method, based on viability theory and convex analysis, can be applied to determine computational procedures to obtain families of control invariant polytopes. The criterion is based on a necessary and sufficient condition for viability to hold at any point on the boundary of a polytope.

I. INTRODUCTION

The importance of invariance in control and systems analysis has been increasing since the first results on the topic, see the pioneering work [START_REF] Bertsekas | Infinite-time reachability of state-space regions by using feedback control[END_REF]. Currently, well established results are available, mainly for linear systems, [START_REF] Gilbert | Linear systems with state and control constraints: The theory and application of maximal output admissible sets[END_REF], [START_REF] Kolmanovsky | Theory and computation of disturbance invariant sets for discrete-time linear systems[END_REF], but also for nonlinear systems [START_REF] Alamo | Convex invariant sets for discrete-time Lur'e systems[END_REF], [START_REF] Cannon | Nonlinear model predictive control with polytopic invariant sets[END_REF], [START_REF] Fiacchini | On the computation of convex robust control invariant sets for nonlinear systems[END_REF], see the monograph on invariance [START_REF] Blanchini | Set-Theoretic Methods in Control[END_REF]. The problem of characterization and computation of invariant sets for particular classes of continuoustime nonlinear systems has been tackled in [START_REF] Chen | Optimisation of attraction domains of nonlinear MPC via LMI methods[END_REF], [START_REF] Gomes Da | Antiwindup design with guaranteed regions of stability: an LMI-based approach[END_REF], [START_REF] Hu | Exact characterization of invariant ellipsoids for single input linear systems subject to actuator saturation[END_REF], mostly using LMI related approaches. The attention devoted to invariance is strongly due to its tight relation with many basic topics in control and systems analysis, such as stability, domain of attraction estimation and hard constraints satisfaction, among others. On the other hand, few computationoriented results are available for generic nonlinear systems.

Viability theory, considered from the control point of view, concerns systems whose dynamics are given by a differential inclusion. Roughly speaking, they are systems for which the variation (or the successor, for discrete-time systems) of a state is a set rather than a point in the state space. Viability theory provides mathematical tools to characterize conditions for a set to be robust or control invariant for the system. Viability has been mainly developed by Aubin and co-authors, see [START_REF] Aubin | A survey of viability theory[END_REF]- [START_REF] Aubin | Set-valued analysis[END_REF], and applies to very general families of sets. See also [START_REF] Broucke | Viability kernels for nonlinear control systems using bang controls[END_REF], which proposes methods to construct viability kernels for multi-input single-ouput nonlinear systems affine in the control.

The main objective of this paper is to provide a characterization of control invariance and contractiveness of polytopes for differential inclusion systems. As common modeling frameworks, like uncertain systems or constrained controlled ones, can be cast in terms of differential inclusion systems, the proposed approach applies to a wide class of systems. Differential inclusion can also be used to approximate nonlinear systems. In this paper, we focus on linear systems with state-dependent bounds on the input, which are intrinsically nonlinear. It has to be stressed that, while ellipsoidal invariant sets are more common in the context of nonlinear continuous-time systems, results in literature involving polytopes concern mainly linear systems, see [START_REF] Castelan | On invariant polyhedra of continuoustime linear systems[END_REF], [START_REF] Farina | Invariant polytopes of linear systems[END_REF]. The aim is to apply analytical tools proper of viability theory and convex analysis, see [START_REF] Bertsekas | Convex analysis and optimization[END_REF], [START_REF] Boyd | Convex Optimization[END_REF], [START_REF] Rockafellar | Convex Analysis[END_REF], [START_REF] Schneider | Convex bodies: The Brunn-Minkowski theory[END_REF], to determine a computation-oriented criterion for characterizing polytopic control invariant sets for constrained continuoustime linear systems. Restricting our attention to polytopes, rather than to the generic sets dealt with in viability theory, allows us to use properties of convex analysis, which lead to tractable problems and more practical solutions. The price to pay is the introduction of a certain conservativeness.

The paper is organized as follows: Section II presents the problem statement, Section III recalls some definitions and results on viability theory. In Section IV the main results on control invariance of a polytope are stated. In Section V the presented method is applied to numerical examples. The paper ends with a section of conclusions.

Notation

Given n ∈ N, define

N n = {x ∈ N : 1 ≤ x ≤ n}. Given A ∈ R n×m , A i with i ∈ N n denotes its i-
R n → S (R m ), its domain is dom(F) = {x ∈ R n : F(x) = / 0} and its graph is graph(F) = {(x, y) ∈ R n × R m : y ∈ F(x)}.

II. PROBLEM STATEMENT

Consider the continuous-time system given by:

ẋ(t) ∈ F(x(t)), (1)
where x(t) ∈ R n is the state at time t, and with F : R n → S (R n ), set valued map. Notice that this modeling framework, which is referred to as differential inclusion, encloses common systems such as uncertain systems and controlled ones. In fact, the solutions of the uncertain system ẋ(t) = f (x(t), w(t)), where w(t) ∈ W (x(t)) is the uncertainty (or the parameter) with W (x) ⊆ R n , are those of (1) with

F(x) = f (x,W (x)) = {y ∈ R n : y = f (x, w), w ∈ W (x)}.
Analogously, the trajectories of the controlled system ẋ(t) = f (x(t), u(t)), with bounds on the input u(t) ∈ U(x(t)) ⊆ R m , are those of system (1) with

F(x) = f (x,U(x)) = {y ∈ R n : y = f (x, u), u ∈ U(x)}.
Moreover, differential inclusion can be used to approximate the evolutions of a nonlinear system ẋ(t) = f (x(t)), provided that f (x) ∈ F(x) for all x ∈ R n . The objective of this work is to design a computationoriented method for obtaining polytopic control invariant sets for a controlled system with state-dependent bounds on the input. This means, as formalized below, that our aim is the characterization and computation of a (family of) set K ⊆ R n such that, for all x(0) = x 0 ∈ K, there exists an admissible control signal u(t) ∈ U(x(t)) which permits to maintain the state x(t), solution of (1), in K for t ≥ 0.

III. VIABILITY THEORY

We recall here some general definitions and results on viability theory, which is strongly associated to the research of Aubin and co-authors, see [START_REF] Aubin | A survey of viability theory[END_REF]- [START_REF] Aubin | Set-valued analysis[END_REF]. Many of those results are developed in the cited works, and references therein, under assumptions which are more general than those required in this paper. Since we are interested in characterizing and computing polytopic control invariant sets, we give the definitions and the properties for the case under analysis.

Definition 1 (Viability properties [START_REF] Aubin | Viability theory[END_REF]):

Consider the set K ⊆ dom(F). A function x(•) from [0, T ] to R n , solution of (1), is called viable if x(t) ∈ K for all [0, T ].
We say that K enjoys the local viability property or control invariance (for the set valued map F) if, for any initial state x 0 in K, there exist T > 0 and a viable solution on [0, T ] to differential inclusion [START_REF] Alamo | Convex invariant sets for discrete-time Lur'e systems[END_REF]. It enjoys the global viability property (or, simply, the viability property) if we can take T = +∞.

Many of the results provided in the context of viability theory analysis are valid for generic nonempty sets K in the state space. An important tool on which those results are based is the contingent cone of set K at x ∈ K, denoted as T K (x), see [START_REF] Aubin | Viability theory[END_REF]. When K is closed and convex, the contingent cone is equal to the closure of the tangent cone.

Theorem 1 (Tangent cones of closed, convex subsets [START_REF] Aubin | Viability theory[END_REF]): For K ⊆ R n , closed and convex, its contingent cone T K (x) coincides with the closure of the tangent cone, given by

C k (x) = h>0 K -x h ,
that is the closed cone spanned by Kx.

Then, for convex, closed set K, we have that T K (x) = C K (x), which are closed and convex cones. Notice that the tangent cone of a closed, convex set K is R n at any point x ∈ int(K). Hence, the definition of viability domain, involving the contingent cone in the general case, can be given directly in terms of the tangent cone for closed convex sets.

Definition 2 (Viability domain [START_REF] Aubin | Viability theory[END_REF]):

Let F : R n → S (R n ) be a nontrivial set valued map. A closed, convex set K ⊆ dom(F) is a viability domain of F if and only if ∀x ∈ K, F(x) ∩C K (x) = / 0. ( 2
)
Also some assumptions on the set valued map F(•) have to be imposed in order to apply the Viability Theorem. A set valued map F : R n → S (R n ) fulfilling such assumptions is defined as Marchaud map, which amounts to say that its graph and its domain are closed, the values of F(•) are convex, and the growth of F(•) is linear, see [START_REF] Aubin | Viability theory[END_REF]. Such preliminaries are useful since it has been proved that any closed, convex viability domain K for system [START_REF] Alamo | Convex invariant sets for discrete-time Lur'e systems[END_REF] with F(•) Marchaud map enjoys the viability property.

Theorem 2 (Viability theorem [START_REF] Aubin | Viability theory[END_REF]): Consider a Marchaud map F : R n → S (R n ) and a closed, convex K ⊆ dom(F). If K is a viability domain, then for any state x 0 ∈ K, there exists a viable solution on [0, +∞) to the differential inclusion [START_REF] Alamo | Convex invariant sets for discrete-time Lur'e systems[END_REF].

This means that, for every initial condition in K, closed and convex, there exists a trajectory of system (1) which remains in K at any time t ∈ [0, +∞), if there exists a "direction" belonging to the map F(x(t)) and to the tangent cone of K at x(t), see Figure 1. In this case, in fact, considering such direction at any time, the trajectory would always head towards the interior of set K (or on the boundary).

C K (x) F(x) K Fig. 1. Viability condition.

IV. POLYTOPIC CONTROL INVARIANT SETS

The results presented in this section, representing the main contributions of that paper, provide a computation-oriented characterization of control invariance for polytopes.

Consider a polytope in the state space containing the origin in its interior, Ω = {x ∈ R n : Hx ≤ 1}, with H ∈ R n h ×n , and the linear controlled system

ẋ(t) = Ax(t) + Bu(t), (3)
with u(t) ∈ U(x(t)) control input. The input bounding set U(x(t)) ⊆ R m is assumed to be the state-dependent polytope

U(x) = {u ∈ R m : Lu ≤ P(x)}, (4)
with L ∈ R n u ×m and P : R n → R n u , and such that

F(x) = Ax ⊕ BU (x) is Marchaud. Notice that if U(x) is Marchaud then F(x) = Ax ⊕ BU (x) is Marchaud too, see [3]. Remark 1:
No loss of generality is induced by considering system (3)-( 4) in spite of (1). In fact, given F(x) determining (1), Marchaud and with F(x) polytopic, for any A ∈ R n×n and defining B = I n and U(x) = (-Ax) ⊕ F(x), the differential inclusion (1) can be written in terms of (3).

The Minkowski function is introduced here, see [START_REF] Blanchini | Set-Theoretic Methods in Control[END_REF] and references therein for some properties.

Definition 3: Given a compact, convex set

K ∈ R n with 0 ∈ int(K), the Minkowski function of K at x ⊆ R n is Ψ K (x) = min α≥0 {α ∈ R : x ∈ αK}.
In the case of a polytopic set Ω containing the origin in its interior, the Minkowski function is given by

Ψ Ω (x) = min α≥0 {α ∈ R : H j x ≤ α, ∀ j ∈ N n h } = max j∈N n h {H j x}.
The objective is to determine a condition for the level sets of the Minkowski function to be control invariant sets, within a region Γ ⊆ R n . To achieve the purpose, it is sufficient to prove that there exists u(x) ∈ U(x) such that ΨΩ (x) ≤ 0, for all x ∈ Γ, since it implies that Ψ Ω (x) is a nonincreasing function and since αΩ ⊆ β Ω if and only if α ≤ β . In what follows we provide conditions, based on the concept of viability, to ensure ΨΩ (x) ≤ 0, which implies that the level sets of Ψ Ω (•) are control invariant sets for system (3) with constrained input. Furthermore, we aim at determining the greatest region in the state space where such conditions are satisfied and then stability can be ensured by a proper selection of the control input.

Condition ΨΩ (x) ≤ 0 is equivalent to prove that, for every x ∈ Γ, there exists a u(x) ∈ U(x) such that (Ax + Bu(x)) lies within the tangent cone of the level set of function

Ψ Ω (•). Then, denoted Ω(x) = Ψ Ω (x)Ω, the condition (2) is ∀x ∈ Γ, (Ax ⊕ BU (x)) ∩C Ω(x) (x) = / 0. ( 5
)
Notice that Ω(x) is the smallest level set of Ψ Ω (•) containing x. The tangent cone of Ω(x) at x (which lies on the boundary of Ω(x), by construction) is given by

C Ω(x) (x) = {v ∈ R n : H k v ≤ 0, ∀k ∈ arg max j∈N n h {H j x}}.
Hence, suppose that the state x is known and denote with k

= k(x) ∈ N n h an index such that H j x ≤ H k x for all j ∈ N n h , that is, such that H k x = Ψ Ω (x). This is equivalent to k ∈ k Ω (x),
where k Ω (x) ⊆ N n h is defined as

k Ω (x) = arg max j∈N n h {H j x}. ( 6
) Given k ∈ N n h define R k = {x ∈ R n : H i x ≤ H k x, ∀i ∈ N n h }, (7)
that is, the region of points x ∈ R n such that k ∈ k Ω (x).

Notice that regions R k , with k ∈ N n h , can have nonempty intersections. The viability condition ( 5) is satisfied at x if there exists u = u(x) ∈ U(x) such that

H k Ax + H k Bu ≤ 0, (8)
for all k ∈ k Ω (x). For every k ∈ N n h and any x ∈ R k , as in [START_REF] Blanchini | Set-Theoretic Methods in Control[END_REF], we define an optimization problem as follows.

Definition 4 (Primal problem): Consider the system (3) with input bounds (4). Given k ∈ N n h and x ∈ R k , consider the following optimization problem:

α * k (x) = min α, u α, s.t. H i x ≤ α, ∀i ∈ N n h , τ(H k Ax + H k Bu) + (H k x -α) ≤ 0, L j u ≤ P j (x), ∀ j ∈ N n u , α ≥ 0, (9)
with τ > 0.

Remark 2: There is no direct connection between the second inequality in [START_REF] Broucke | Viability kernels for nonlinear control systems using bang controls[END_REF] and the Euler Approximating System (EAS), which is a discrete-time system often used in spite of the continuous-time one for computational purposes, see [START_REF] Blanchini | Set-Theoretic Methods in Control[END_REF]. The inequality has a geometrical meaning, it is valid for every positive τ and, most importantly, it does not introduce any approximation error. In fact, the condition for viability based on ( 9) is necessary and sufficient, as illustrated below.

A computation-oriented condition for the viability to be satisfied at a point x ∈ R n stems from the following property.

Proposition 1: Given k ∈ N n h and x ∈ R k , the optimal solution α * k (x) of the primal problem ( 9), is such that

Ψ Ω (x) = α * k (x)
if and only if there exists u ∈ U(x) such that condition (8) holds at x. Furthermore, Ψ Ω (x) < α * k (x) if and only if condition [START_REF] Boyd | Convex Optimization[END_REF] is not satisfied at x for any u ∈ U(x).

Proof: First notice that k ∈ k Ω (x) since x ∈ R k . Suppose that condition (8) holds for a proper u ∈ U(x). Then the second constraint in ( 9) is satisfied by the value α = Ψ Ω (x) = H k x and the first set of constraints are fulfilled, by definition of Minkowski function. On the other hand if the optimal value of problem ( 9) is given by the Minkowski function at x, i.e. α * k (x) = Ψ Ω (x), then the first set of constraints are satisfied by definition, in fact H i x ≤ max j∈N n h {H j x} = Ψ Ω (x) = α, for all i ∈ N n h . Moreover, from [START_REF] Bertsekas | Convex analysis and optimization[END_REF], it follows that H k x = α and then the second constraint in (9) becomes the condition [START_REF] Boyd | Convex Optimization[END_REF]. Hence, we can conclude that the solution α * k (x) is equal to the Minkowski function at x if and only if the condition ( 8) is satisfied at x.

Furthermore it is easy to see that Ψ Ω (x) ≤ α * k (x). In fact, the value of the Minkowski function at x would be attained by removing the second constraint (if the set U(x) is nonempty, clearly), that means, it would be the optimal over a greater feasibility region, and then a smaller or equal value should be obtained. Hence it can be concluded that the optimal solution of the optimization problem is equal to the Minkowski function at x if and only if condition ( 8) is satisfied at x. This implies that the optimal value is greater than Ψ Ω (x) if and only if [START_REF] Boyd | Convex Optimization[END_REF] is not fulfilled at x.

In what follows, we are going to use the Lagrange multipliers and the dual optimization problem to pose geometric conditions for a region of the state space to be a control invariant set. First, it is worth recalling that for the case under analysis strong duality holds if the primal is feasible, being (9) a linear problem in the optimization variables α and u, see [START_REF] Bertsekas | Convex analysis and optimization[END_REF], [START_REF] Boyd | Convex Optimization[END_REF].

Applying classical results from duality for convex opti-mization problems and defining the function

L k (β , δ , σ ; x) = n h ∑ i=1 β i H i x + δ τH k Ax + δ H k x - n u ∑ j=1 σ j P j (x),
we obtain the dual problem:

L * k (x) = max β ,δ ,σ L k (β , δ , σ ; x), s.t. n h ∑ i=1 β i + δ ≤ 1, δ τH k B + n u ∑ j=1 σ j L j = 0, β ≥ 0, δ ≥ 0, σ ≥ 0, (10)
whose optimal value is such that

L k (β , δ , σ ; x) ≤ L * k (x) for all feasible (β , δ , σ ), clearly. Hence L * k (x) is the maximal lower bound of α * k (x) and, from strong duality, L * k (x) = α * k (x). Then, L k (β , δ , σ ; x) ≤ α * k (x)
for any feasible solution of (10). Proposition 2: A necessary and sufficient condition for inequality [START_REF] Boyd | Convex Optimization[END_REF] to be satisfied at x ∈ R k is

L * k (x) ≤ H k x, ( 11
)
or, equivalently L k (β , δ , σ ; x) ≤ H k x for every feasible solution (β , δ , σ ) of ( 10). Furthermore, condition ( 8) is violated at x ∈ R k if and only if

L * k (x) > H k x, (12)
or, equivalently, if there exists a feasible solution (β , δ , σ ) of ( 10)

such that L k (β , δ , σ ; x) > H k x. Proof: Recall that Ψ Ω (x) ≤ α * k (x)
and that Ψ Ω (x) < α * k (x) if and only if [START_REF] Boyd | Convex Optimization[END_REF] does not hold at x ∈ R k . Then (8) holds at x if and only if

H k x = Ψ Ω (x) = L * k (x) = α * k (x)
, which is implied by [START_REF] Castelan | On invariant polyhedra of continuoustime linear systems[END_REF]. Analogously, it can be proved that [START_REF] Chen | Optimisation of attraction domains of nonlinear MPC via LMI methods[END_REF] entails that α * k (x) > Ψ Ω (x) and then, as previously shown, violation of condition [START_REF] Boyd | Convex Optimization[END_REF].

Posing the condition for viability as in [START_REF] Castelan | On invariant polyhedra of continuoustime linear systems[END_REF], rather than by means of an equality constraint, permits to obtain convex optimization problems under adequate assumptions on U(x).

Consider now the dual problem [START_REF] Cannon | Nonlinear model predictive control with polytopic invariant sets[END_REF]. Given k ∈ N n h and x ∈ R k , the problem is the maximization of a linear function over a polyhedral set in the space of variables β , δ and σ . In general case, the maximum is attained at some extreme point or the problem is unbounded. Since the primal optimum exists and is bounded, the analysis can be reduced to the extreme points of the feasibility region of the dual problem.

Property 1: The optimal value of the dual problem ( 10) is attained at an extreme point of the feasibility region.

Proof: Since the origin is an extreme point of the feasibility region of the dual problem [START_REF] Cannon | Nonlinear model predictive control with polytopic invariant sets[END_REF], which is bounded above by the primal optimal value, the result is implied by Fundamental Theorem of Linear Programming, see [START_REF] Bertsekas | Convex analysis and optimization[END_REF].

It is important to stress the fact that the feasibility region of the dual problem does not depend on the value of x but only on H k (and on the structure of the system, clearly). Then the dual problem feasibility set is valid for every x ∈ R k and its extreme points can be precomputed knowing H k only.

Proposition 3: Given k ∈ N n h , denote with (β p , δ p , σ p ) the p-th extreme point of the feasibility region of the dual problem [START_REF] Cannon | Nonlinear model predictive control with polytopic invariant sets[END_REF], with p ∈ N n v . The subset of R k , defined in [START_REF] Blanchini | Set-Theoretic Methods in Control[END_REF], such that the condition ( 8) is satisfied at x ∈ R k by a u(x) ∈ U(x) is given by

V k = p∈N nv {x ∈ R n : L k (β p , δ p , σ p ; x) ≤ H k x}. ( 13
)
Furthermore, the region of points x ∈ R k for which the condition ( 8) is violated for every

u ∈ U(x) is Vk = p∈N nv {x ∈ R n : L k (β p , δ p , σ p ; x) > H k x}. ( 14
)
Proof: From Property 1 we have that, for every x ∈ R k , there exists a p * = p * (x) ∈ N n v such that

L k (β , δ , σ ; x) ≤ L k (β p * , δ p * , σ p * ; x) = L * k (x)
, for any feasible (β , δ , σ ). From this and Proposition 2 the first claim follows. Analogous considerations prove the second claim.

It is worth stressing that V k , given by the intersection of subsets of the state space, is the exact region of all x ∈ R k where condition ( 8) is satisfied for an adequate u(x) ∈ U(x). The only optimization problem to solve for characterizing V k concerns the computation of the extremes of the dual problem, neither the computation of u(x) is required.

Remark 3: Notice that V k is the set of points in R k for which condition ( 8) is satisfied by a u = u(x) ∈ U(x), for a particular k ∈ k Ω (x). This is equivalent to the condition of viability for all x ∈ int(R k ), that is if k Ω (x) = {k}. Viability, in fact, should concern a condition on u ∈ U(x) involving every k ∈ k Ω (x). Intriguing phenomena (as Zeno solutions, for instance) could occur at x ∈ ∂ R k , with k ∈ N n h , and deserve more accurate considerations. The analysis of such boundary phenomena is one the objective of our future research.

From the computational point of view, it is important to notice that if P j (x) is concave in x for all j ∈ N n u , the functions L k (β p , δ p , σ p ; x) are convex and then V k is a convex set. Analogously, if P j (x) is convex in x for all j ∈ N n u , then the L k (β p , δ p , σ p ; x) is a concave function and Vk is a union of convex sets. The analysis of the different n h regions R k , one for any H k with k ∈ N n h , permits to obtain a polytopic viable domain.

V. ILLUSTRATIVE EXAMPLES

Example 1: Consider the linear system (3) with matrices

A = 0 -1 1 0 , B = 1 0 0 1 , (15)
and constraints on the input is

U(x) = U = {u ∈ R 2 : u ∞ ≤ 1}.
The trajectories in absence of control are given by the circumferences of the circles centered in the origin. It is, then, immediate to check that any circle in the state space is a viable set for the dynamic system. On the other hand, our aim here is to use this simple explanatory example to illustrate how the proposed results can be used for computing a family of control invariant sets and a region where viability condition holds for Ω(x).

Consider the set Ω = {x ∈ R 2 : x 1 ≤ 1}. The objective is to compute the maximum γ ≥ 0 such that µΩ are a control invariant polytopes for every µ such that 0 ≤ µ ≤ γ.

Considering H k = [1 1], the region under analysis is the first quadrant, i.e. R k = {x ∈ R 2 : x ≥ 0}. By geometric inspection it can be noticed that the "critical" point in R k for viability of γΩ is x = [γ 0] T , see Figure 2. Actually, moving x along the facet of γΩ from [γ 0] T to [0 γ] T , the direction of the autonomous system, i.e. Ax, is such that H k Ax decreases, becoming negative from x = [0.5γ 0.5γ] T . Notice that if H k Ax ≤ 0 then viability condition holds at x simply posing u = [0 0] T . By geometric inspection it can also be concluded that the maximal γ k for which γ k Ω satisfies the viability condition in the region R k is γ k = 2, see Figure 3. We expect to recover the same value applying the presented results. The dual problem feasibility region for the case under analysis is given by the following constraints:

           δ τ + σ 1 -σ 3 = 0, δ τ + σ 2 -σ 4 = 0, n h ∑ i=1 β i + δ ≤ 1, β ≥ 0, δ ≥ 0, σ ≥ 0.
Notice that, for any possible value of δ , there exist admissible values of σ such that the linear equality constraints hold. The first constraint, for instance, is satisfied by every pair of values σ 3 ≥ 0 and σ 1 ≥ 0 such that their difference is equal to δ τ. The extreme values of β and δ are given by the extreme points of the region, in their subspace, contained between the simplex and the origin. Then the extreme values of δ are 0 or 1. For δ = 0 we have that σ is such that σ 1 = σ 3 and σ 2 = σ 4 , and then the extremes are σ 1 = σ 3 = 0 or σ 1 = σ 3 = +∞ and σ 2 = σ 4 = 0 or σ 2 = σ 4 = +∞. The infinite values can be discarded, since the related function L k (β , δ , σ ; x) would be equal to -∞, then leading to trivial inequalities in the definition of V k and Vk , (see ( 13) and ( 14)). Then the only interesting extremes are given by σ i = 0, for i ∈ N 4 . The other possibility is δ = 1 and then the finite extreme value of σ is σ = [0 0 τ τ] T . Thus the finite extreme values of the dual problem feasibility region are (β 1 , δ 1 , σ 1 ) = 1 0 0 0 0 0 0 0 0 T , (β 2 , δ 2 , σ 2 ) = 0 1 0 0 0 0 0 0 0 T , (β 3 , δ 3 , σ 3 ) = 0 0 1 0 0 0 0 0 0 T , (β 4 , δ 4 , σ 4 ) = 0 0 0 1 0 0 0 0 0 T , (β 5 , δ 5 , σ 5 ) = 0 0 0 0 1 0 0 τ τ T , (β 6 , δ 6 , σ 6 ) = 0 0 0 0 0 0 0 0 0 T . The resulting set of (nontrivial) constraints determining x ∈ R k such that the viability condition holds for Ω(x) are

               L k (β 1 , δ 1 , σ 1 ; x) = [1 1]x ≤ [1 1]x, L k (β 2 , δ 2 , σ 2 ; x) = [1 -1]x ≤ [1 1]x, L k (β 3 , δ 3 , σ 3 ; x) = [-1 1]x ≤ [1 1]x, L k (β 4 , δ 4 , σ 4 ; x) = [-1 -1]x ≤ [1 1]x, L k (β 5 , δ 5 , σ 5 ; x) = [1 1]x + τ[1 -1]x -2τ ≤ [1 1]x, L k (β 6 , δ 6 , σ 6 ; x) = 0 ≤ [1 1]x, and then        x 2 ≥ 0, x 1 ≥ 0, [-1 -1]x ≤ 0, [1 -1]x ≤ 2. ( 16
)
The region V k , and the half-spaces that determine it as in [START_REF] Gomes Da | Antiwindup design with guaranteed regions of stability: an LMI-based approach[END_REF], are depicted in Figure 4. The shadowed regions are those points in the state space that violate constraints ( 16), and then the white portion of the space represents V k . Thus, the maximal γ k such that γ k Ω satisfies the viability condition in R k is 2, as expected. From symmetry, analogous results are obtained for every R i , with i ∈ N 4 , and the resulting γ, obtained as the minimal γ i , is 2. It is important to stress that no extreme point computation for Ω is required, the halfspace representation of the polytope is sufficient. Example 2: We consider now the same continuous-time dynamic system [START_REF] Gilbert | Linear systems with state and control constraints: The theory and application of maximal output admissible sets[END_REF], with state-dependent bounds on the control input, that is U(x) = {u ∈ R 2 : Lu ≤ P(x)} with

L =     1 0 0 1 -1 0 0 -1     , P =     -x 2 1 + 16 -x 2 2 + 9 -x 2 1 + 9 -x 2 2 + 4     .
Thus, the bounds on the input are boxes whose extreme values dependent on the state. Notice that the bounds are tighter as the state is further from the origin. We have to expect, then, that viability condition is satisfied in a region around the origin. Moreover, since P j (•) are concave in x, then functions L k (β , δ , σ ; •) are convex and then the viable region V k is also convex, being the intersection of convex sets. Functions L k (β p , δ p , σ p ; x) are the same as in the previous example for all p = 5. For p = 5 the convex constraint results in:

[1 1]x + τ[1 -1]x -τ(-x 2 1 -x 2 2 + 13) ≤ [1 1]x, ⇔ x 2 1 + x 2 2 + x 1 -x 2 -13 ≤ 0. ( 17
)
Constraints and the region V k are depicted in Figure 5. The points where the circumference of ( 17) intersects the axis are [3.14 0] T , [-4.14 0] T , [0 3.14] T and [0 -4.14] T . The result can be checked from a geometric point of view. In fact, consider the point x = [3.14 0] T and the set valued map at this point. The input bounding set at x is given by U( x) = {u ∈ R 2 : 0.86 ≤ u 1 ≤ 25.86, -4 ≤ u 2 ≤ 9} and A x = [0 3.14] T . Then the vector v( x) = [0.86 -4] T is admissible, i.e. v( x) ∈ U( x), and A x + v( x) lies in the tangent cone of Ω( x) = 3.14Ω at x, as depicted in Figure 6.

VI. CONCLUSIONS

The paper presented a method to characterize control invariance of polytopes for differential inclusion systems. Properties related to viability theory and to convex sets and functions have been used to propose a practice-oriented method for analysis and computation of control invariant polytopes. Several problems and possible directions of further research are open, such as the analysis of border phenomena like Zeno solutions, the characterization of polyhedral Lyapunov functions and the problem of control design.

  th row. Operators ≤, ≥,< and > are intended to apply element-wise to vectors a, b ∈ R n . Given two sets D and E and a nonnegative scalar α ≥ 0, denote the set αD = {αx : x ∈ D}, and the Minkowski set addition is D ⊕ E = {x = d + e : d ∈ D, e ∈ E}. The interior of D is denoted as int(D), its boundary is ∂ D. With S (D) we denote the set of subsets of D. Given a set valued map F :

Fig. 2 .

 2 Fig. 2. Autonomous directions on the boundary of γΩ in R 1 .

Fig. 3 .

 3 Fig. 3. Sets γΩ and system dynamics.

Fig. 4 .

 4 Fig. 4. Viability region in R k .

Fig. 5 .

 5 Fig. 5. Viability region in R k .

Fig. 6 .

 6 Fig. 6. Critical point of viability region in R 1 .

This work was partially supported by the ANR project ArHyCo, ARPEGE, contract number ANR-2008 SEGI 004 01-30011459.

M. Fiacchini and S.


Barbara Heude

Olivier Thiébaugeorges

Valérie Goua

Anne Forhan

Monique Kaminski

Bernard Foliguet

Michel Schweitzer

Guillaume Magnin

Marie-Aline Charles

Pre-pregnancy body mass index and weight

Keywords: obesity, pregnancy, gestational diabetes, hypertension, birth weight

Introduction

The prevalence of obesity has been increasing worldwide, especially among young women susceptible to become pregnant. In France, although the prevalence is lower than in the United States, the frequency of obesity has nearly doubled between 1997 and 2006 [START_REF] Charles | Monitoring the obesity epidemic in France: The Obepi Surveys 1997-2006[END_REF]. In 20-39 year old women, it increased from 5.2% to 11.0% over this period. Obesity and associated cardio-metabolic complications deserve particular attention in the context of pregnancy. Indeed, it is now well recognized that maternal obesity at conception increases the risk of complications in pregnancy, labor, and birth for both the mother and the neonate [START_REF] Castro | Maternal obesity and pregnancy outcomes[END_REF][START_REF] Yu | Obesity in pregnancy[END_REF][START_REF] Raatikainen | Transition from overweight to obesity worsens pregnancy outcome in a BMI-dependent manner[END_REF][START_REF] Heslehurst | The impact of maternal BMI status on pregnancy outcomes with immediate short-term obstetric resource implications: a meta-analysis[END_REF].

In particular, gestational diabetes mellitus and hypertensive disorders are more prevalent in obese pregnant women (6; 7). These complications themselves confer higher risks of adverse fetal and neonatal outcome, such as large-for-gestational-age (LGA) neonates in mothers with gestational diabetes, or intra-uterine growth restriction in mothers with gestational hypertension [START_REF] Ehrenberg | The influence of obesity and diabetes on the prevalence of macrosomia[END_REF][START_REF] Rosenberg | Maternal obesity and diabetes as risk factors for adverse pregnancy outcomes: differences among 4 racial/ethnic groups[END_REF][START_REF] Grisaru-Granovsky | Hypertensive disorders of pregnancy and the small for gestational age neonate: not a simple relationship[END_REF][START_REF] Steer | Maternal blood pressure in pregnancy, birth weight, and perinatal mortality in first births: prospective study[END_REF].

Another important factor influencing pregnancy and birth outcomes is weight gain during pregnancy (12; 13). Most studies show that the higher the pre-pregnancy BMI, the lower the pregnancy weight gain (14-20). It has been shown that the risks of gestational diabetes and gestational hypertension could be different according to the amount of weight gain, independent of pre-pregnancy BMI [START_REF] Nohr | Combined associations of prepregnancy body mass index and gestational weight gain with the outcome of pregnancy[END_REF]. Most of the studies interested in weight gain during pregnancy analyzed total weight gain thus including the weight of the fetus and the annexes; this obviously overestimates the link between weight gain and birth weight, but also between weight gain and length of gestation. This is why some studies, but very few, analyzed "net" weight gain, defined by subtracting birth weight from total weight gained during pregnancy (21; 22).

In the present study, we aimed to investigate the role of pre-pregnancy BMI and net gestational weight gain on risks of gestational diabetes and hypertension, preterm delivery, small for gestational age and large for gestational age neonates.

Population and Methods

Population studied

The EDEN study (Pre-and post-natal determinants of children's growth and development) is an on-going mother-child cohort, with a follow-up of the child until their fifth birthday. The primary aim of the EDEN cohort is the study of pre-natal and early post-natal nutritional, environmental, and social determinants of children's development and health. The study was proposed to all women presenting before the 24 th week of gestation (week of amenorrhea) at the prenatal clinics of the University hospitals in Nancy and Poitiers, two middle-sized cities of semi rural areas in France. Enrolment started in February 2003 in Poitiers and September 2003 in Nancy, lasted 27 months in each center and 2002 women were included. Criteria of exclusion were twin pregnancies, known diabetes before pregnancy, moving outside the region planned in the next three years and not being able to speak and read French. Among all eligible women, 55% agreed to participate. Written consent was obtained from the mother for herself at inclusion and for the newborn after delivery. The study was approved by the ethics committee (CCPPRB) of Kremlin Bicêtre and by the Data Protection Authority "Comission Nationale de l'Informatique et des Libertés" (CNIL).

Among the 2002 eligible women who accepted to participate, 1884 were included in the present analysis as they had information on BMI before pregnancy and at least one pregnancy, delivery or birth outcome available. Of the other women, 18 had moved away or been lost to follow-up, 19 had a miscarriage or a termination of pregnancy, and 44 withdrew for personal reasons. For the remaining 37 women, information on weight before pregnancy was not available.

Measurements

At 24-28 weeks, the mother had a clinical examination performed by midwife research assistants and she completed a self-administered questionnaire. Maternal height was measured with a wall Seca 206 stadiometer (Hamburg, Germany), to the nearest 0.2 cm. A second research examination was performed by the same research assistants, on average, 2.3 days (± sd: 1.5) after delivery. The mother's weight was measured using electronic Terraillon SL 351 scales (Hanson Ltd, UK) to the nearest 0.1 kg. At birth, neonates were weighed using an electronic Seca scales (Hamburg, Germany: Seca 737 in Nancy and Seca 335 in Poitiers).

At 24-28 weeks, maternal plasma glucose concentrations were measured 1 h after a 50-g glucose challenge. Women with 1-h glucose concentrations over 130 mg/dL in Nancy and 140 mg/dL in Poitiers were scheduled for a 3-h 100-g oral-glucose tolerance test. Gestational diabetes was diagnosed by using the oral-glucose-tolerance test according to the Carpenter and Coustan criteria [START_REF] Steer | Maternal blood pressure in pregnancy, birth weight, and perinatal mortality in first births: prospective study[END_REF] when there were >2 plasma glucose concentrations greater than the following cutpoints: fasting = 95 mg/dL, at 1 h = 180 mg/dL, at 2 h = 155 mg/ dL, and at 3 h = 140 mg/dL. Some additional cases of gestational diabetes were diagnosed by clinicians either before or after the 24-28-week examination, and extracted from medical records of the maternity.

Weight before pregnancy, educational level and smoking habits were obtained by interview at inclusion. Some additional data were extracted from clinical records: weight at first maternity visit, clinical diagnosis of gestational hypertension, treatment for hypertension, gestational age at delivery and number of previous pregnancies. In this study, only women treated for hypertension were considered as hypertensive. Existing hypertension before pregnancy was also collected from clinical records, and permitted to confirm cases of hypertension appearing during pregnancy only (pregnancy-induced hypertension).

Generated variables

Large-and small-for-gestational-age (LGA and SGA) births were defined when birth weights were over the 90 th and below the 10 th percentiles, respectively, using French gestational ageand gender-specific reference curves [START_REF] Mamelle | Fetal growth from the AUDIPOG study. I. Establishment of reference curves[END_REF]. Pre-term delivery was defined as gestational age <37 weeks.

Pre-pregnancy body mass index (BMI) was computed as reported weight (kg) divided by square of measured height (m) and categorized into four groups as underweight (<18.5 kg/m 2 ), normal (≥18.5 and <25 kg/m 2 ), overweight (≥25 and <30 kg/m 2 ) and obese (≥30 kg/m 2 ). Net gestational weight gain per week (netGWG) was computed as the difference between weight measured after delivery and reported weight before pregnancy, divided by the number of weeks of gestation and expressed in g/week. This variable was also categorized into four groups. Since one objective of this paper was to compare the strength of the association of BMI with pregnancy and birth outcome, to the strength of the association of gestational weight gain with the same outcomes, the categorization was performed so that the distribution of the number of women in each group was similar to that of the corresponding groups of the pre-pregnancy BMI categorization. Thus, pregnancy weight gain was considered as low under 74g per week, normal between 74 and 282g per week, medium high between 282 and 389g per week and high over 389g per week. The equivalence in net weight gain during a full-term pregnancy (41 weeks) was <3, 3 to 12, 12 to 16 and over 16kg respectively for low, normal, medium high and high net gestational weight gain.

Statistical analysis

We described the characteristics of the mothers and of the neonates during pregnancy and birth outcomes using mean + standard deviation or percentages (number). As weight gain is strongly and inversely related with pre-pregnancy BMI, the associations between maternal weight gain and pregnancy outcomes were investigated in multivariable models including both variables together. We modeled the risks of gestational diabetes, treated hypertension, pre-term delivery, SGA and LGA, according to categories of pre-pregnancy BMI and netGWG, using logistic regression models adjusted for center, maternal age and height, parity and average number of cigarettes smoked per day during pregnancy. Reference categories for ORs were normal pre-pregnancy BMI and normal pregnancy weight gain as defined above.

We considered the interactions between the categorical variables for pre-pregnancy BMI and netGWG, but we also tested for interactions between the two variables considered as continuous traits, introducing a multiplicative term in the model, as this is a more powerful method.

For each of the three birth outcomes (pre-term delivery, SGA and LGA) we analyzed the impact of excluding women with gestational diabetes or/and hypertension. As sensitivity analyses, we performed the same analyses using weight measured at booking instead of recalled weight for the calculation of pre-pregnancy BMI, in the 1351 mothers who booked before 15 weeks.

RESULTS

Description of study population (Table 1)

Among the 2002 women enrolled in the cohort, the 118 women excluded from the present analysis (see Methods) were on average 2 years younger at inclusion (27 vs. 29 years). Birth weight was available for 1855 neonates. Gestational diabetes and pregnancy-induced hypertension were diagnosed in about respectively 6% and 4% of the mothers. Six percent of the births were pre-term. Small-for-gestational age (SGA) and large-for-gestational age (LGA) each accounted for 8% of the newborns.

Maternal weight gain during pregnancy decreased with increasing pre-pregnancy BMI, and obese women differed from their peers with a mean of 4 kg gained vs. about 9 kg for the other women (results not shown).

Pre-pregnancy BMI, gestational weight gain and associations with gestational hypertension and gestational diabetes (Table 2)

Gestational diabetes and gestational hypertension were diagnosed in respectively 19% and 11% of obese women vs. only 4% and 2% in women with a normal pre-pregnancy BMI (P<0.0001). In the multivariate analysis, a J-shaped relationship was observed for the association between the risk of gestational hypertension and categories of increasing netGWG, but only the highest weight gain was significantly associated with more hypertension (OR [95% CI] =3.02 [1.39;6.58] compared to normal category).

The risk of gestational diabetes increased steadily with pre-pregnancy BMI even after adjustment for weight gain (P-trend<0.001). The risk was significantly increased in both overweight and obese mothers. There was a trend, at the limit of significance (P=0.06), for a decreasing risk of gestational diabetes with increasing categories of netGWG. The interaction between pre-pregnancy BMI and maternal gestational weight gain considered as continuous traits was positive and significant (P=0.011, results not shown): the association between pre-pregnancy BMI and gestational diabetes increased with higher netGWG, or reciprocally, the association between netGWG and gestational diabetes, which was marginally negative, tends to be positive for higher pre-pregnancy BMI.

Pre-pregnancy BMI, gestational weight gain and associations with birth outcomes (Table 3) The rates of LGA increased with categories of pre-pregnancy BMI. Even if not significant, pre-term delivery tended to be more frequent in obese women. In contrast, the rate of SGA was highest in underweight women (13% vs. 5% in obese women). Before adjustment none of the birth outcome was significantly associated with maternal net gestational weight gain.

There was no significant interaction between the effects of pre-pregnancy BMI and netGWG for the different outcomes considered. The risk of pre-term delivery was not significantly associated with pre-pregnancy BMI (P-trend=0.49), although the lowest risk was observed in underweight women. In contrast, an inverse J-shaped relationship was observed with increasing gestational weight gain. Indeed, the risk was highest in low weight gain mothers (OR=2.18 [1.20;3.99]), and lowest in the medium-high weight gain group (OR=0.46 [0.23;0.91]). When mothers with gestational diabetes or hypertension were excluded, the association of low weight gain with pre-term delivery was reinforced (OR=2.70 [1.37;5.34], Table 4). For SGA, a significant decreasing risk was found with increasing pre-pregnancy BMI (P=0.005) and increasing maternal weight gain (P=0.008), and this was not strongly modified after selection of mothers without gestational diabetes or hypertension. Lastly, the risk of LGA increased steadily with pre-pregnancy BMI (P-trend<0.0001) and more weakly with categories of netGWG (P-trend=0.027). Obese mothers were at increased risk for LGA, and this risk was already present in overweight women (OR=1.66 [1.07;2.57] compared to normal category). Excluding women with gestational diabetes or pregnancy hypertension from the analysis had the strongest impact on the results for LGA: the association between categories of pre-pregnancy BMI and LGA was weakened, whereas the relationship with netGWG was strengthened (Table 4). Even medium-high netGWG was significantly associated with higher risk of LGA in this subsample.

DISCUSSION

Our results showed that weight gain in pregnancy was lower when pre-pregnancy BMI was higher. In particular, women who were overweight or obese before pregnancy generally gained less weight during pregnancy and very few had an excessive weight gain. Conversely, only 2 of the 161 lean women had a low weight gain. This relationship has been well established (14-20) but the reason for this is not obvious. It can be speculated that medical management and/or personal efforts for not gaining too much weight could be responsible for a part of this association. However, during pregnancy, fat is stored to secure energy supply during fetal growth and lactation. In obese women, no additional storage is necessary, which suggests that pregnancy weight gain could be restricted because of physiological mechanisms.

Our study was based on self-reported measures of weight before pregnancy whereas many studies have found that weight is more likely to be under-reported by women with a high prepregnancy BMI [START_REF] Rowland | Self-reported weight and height[END_REF]. Such a bias would therefore impact on the calculation of gestational weight gain and lead to an over-estimation of weight gain in overweight and obese women.

We performed a sensitivity analysis and used weight measured at booking instead of recalled weight for the calculation of gestational weight gain and BMI. Gestational weight gain was divided by the number of remaining weeks of gestation, and then multiplicated by 41 to obtain a full term pregnancy. The results obtained on the 1351 mothers who booked before 15 weeks were very similar, gestational weight gain was respectively of 10.9, 10.2, 8.4 and 4.3 kg in lean, normal, overweight obese mothers respectively (results not shown). We also performed all the association analyses presented in the paper with those new two variables and the results were unchanged, even if less significant because of the loss of power (results not shown).

As expected [START_REF] Guelinckx | Maternal obesity: pregnancy complications, gestational weight gain and nutrition[END_REF], gestational hypertension and gestational diabetes were more frequent in women with higher pre-pregnancy BMI. High weight gain was also associated with an increased risk of gestational hypertension, but in contrast, a tendency for an inverse relation was observed between weight gain and risk of gestational diabetes. This last result was also obtained in previous studies [START_REF] Catalano | Weight gain in women with gestational diabetes[END_REF], and in particular in a recent Danish study on about 60 000 term-pregnancies [START_REF] Nohr | Combined associations of prepregnancy body mass index and gestational weight gain with the outcome of pregnancy[END_REF]. This could be attributable to reverse causation, since diagnosis of gestational diabetes may be accompanied with diet recommendations resulting in a decreased weight gain in late pregnancy, as Catalano et al. previously suggested [START_REF] Catalano | Weight gain in women with gestational diabetes[END_REF]. On the other hand, it is also possible that common factors such as weak insulin resistance present from the beginning of pregnancy favor both gestational diabetes and low weight gain during pregnancy.

Because of different habits in the two centres, the criteria for inviting mothers to a second glucose load test were different. Indeed, the number of women performing this test was higher in Nancy (290) than in Poitiers (179), but the final number of cases was not significantly higher in Nancy (6.6%) than in Poitiers (6%, p=0.50). Therefore, those methodological differences between the centres may not have impacted on our results.

Consistent with previous studies, we found that low weight gain was significantly associated with a higher risk of pre-term delivery (13; 17; 22; 27-29). We show that this is true even if only maternal weight gain is considered. Moreover, the association was strengthened after excluding women with gestational diabetes and gestational hypertension. The lowest risk of prematurity was observed in medium-high weight gain (12 to 16 kg net maternal weight gain for a term pregnancy, 16 to 20 kg equivalent total weight gain during pregnancy). Some studies found that the magnitude of the association decreases as pre-pregnancy BMI increases (22; 27), whereas in our study, the interaction with pre-pregnancy BMI was not significant, probably because of the small number of cases in the extreme groups. Recently, Nhor et al.

found a statistically significant interaction between the effects of pre-pregnancy BMI group and weight gain on preterm birth (13; 27): the risk was potentiated at the extremes, namely among underweight women with a low weight gain and obese women with a high weight gain. However, this interaction was observed only for induced preterm deliveries, not for spontaneous preterm births, suggesting the predominant role of obstetric care in obesity related diseases.

Low maternal weight gain may indicate deficiencies in nutrients, a lack of expansion of plasma volume, infection, or other unidentified problems. Further understanding of these associations is needed, because it remains unclear whether they are causal and therefore amenable to nutritional interventions.

We acknowledge that the relationship between maternal health conditions and birth outcome is complex since especially maternal anthropometry is the consequence of many other genetic or environmental factors such as eating habits, exercise and social economic conditions [START_REF] Trotter | Testing for racial/ethnic differences in the association between childhood socioeconomic position and adult adiposity[END_REF] that themselves can impact on pregnancy outcome. Additional adjustment for educational level did not change the result, but this is a very crude estimation of socioeconomic condition and the presence of residual confounding cannot be excluded.

Another limitation of this study is that the EDEN cohort was conducted in two regional maternity centers and is not representative of the general French population. We adjusted for center in order to take into account regional specificities of way of life and obstetrical practices. Moreover, a comparison with a representative survey carried out on about 15 000 pregnant French women in 2003 [START_REF] Blondel | Trends in perinatal health in metropolitan France between 1995 and 2003: results from the National Perinatal Surveys[END_REF] indicates that women in the EDEN study were more educated and more often employed, showed healthier life-styles and were more likely to exclusively breast-feed when in the maternity hospital, than the general French population.

We acknowledge that this limits the possibility to generalize our results to the whole French pregnant women.

Our results show clearly that both prepregnancy BMI and gestational weight gain are independently and negatively associated with SGA at birth, and the lowest risk of SGA will concern obese women who gain more weight, as shown by Ay et al. in the Generation R Study in the Netherlands [START_REF] Ay | Maternal anthropometrics are associated with fetal size in different periods of pregnancy and at birth. The Generation R Study[END_REF]. For many years, clinical recommendations have focused mainly on the limitation of low birth weight [START_REF] Mamelle | Fetal growth from the AUDIPOG study. I. Establishment of reference curves[END_REF]. However, the nutritional and clinical context has changed since then: more and more mothers start their pregnancy being overweight or obese, and the risk of LGA has to also be put in the balance.

It has been explicitly recognized since at least 1990, that both pre-pregnancy BMI and gestational weight gain are positively associated with birth weight (14; 15; 33-35). Few studies have considered the impact of taking into account both blood pressure and glucose tolerance disorders, on the quantification of the association between prepregnancy BMI and gestational weight gain on extreme birth weight. Indeed, as seen earlier, gestational diabetes and gestational hypertension are seen more frequently in obese women and can be important confounders for the association with birth weight. Some studies have excluded women with overt gestational diabetes, and showed that even milder degrees of gestational glucose intolerance increases the risk of LGA (36; 37). In our study, we examined the relationship after excluding women with gestational diabetes, and could observe that the association weakened but the risk of LGA remained doubled in obese women. We also performed further adjustments for blood glucose and blood pressure measured during pregnancy, and the conclusions remained unchanged (results not shown in order to simplify the presentation). This suggests that mechanisms other than maternal glucose concentration may be involved in the risk of macrosomia. This also confirms the major role of BMI before the start of pregnancy and that peculiar attention should be driven to preconception counseling for patients with obesity. Moreover, after excluding cases of gestational diabetes, a stronger association between maternal weight gain and LGA appeared. This is the first time this result has been reported, and it suggests that the known relationship between maternal glucose and birth weight obscures, in part, the association between maternal weight gain and birth weight, and this must be taken into account. It is possible that the large weight gain in women without glucose or blood pressure disorders may receive less attention from clinicians and it is therefore associated with a stronger risk of LGA than in other women.

In conclusion, our analysis using net maternal weight gain and taking into account glucose and blood pressure disorders reinforces the association between low maternal weight gain with premature birth and between high weight gain with LGA compared to that with prepregnancy BMI. Regarding the risk of LGA, our results suggest that monitoring weight gain in obese women may reduce but not alleviate the increased risk of LGA amongst these women. Moreover, high weight gain in women without gestational diabetes and/or gestational hypertension increases the risk of LGA, suggesting that more attention should be paid to mothers without apparent risk factors. However, weight gain has to be monitored tightly as low weight gain is also associated with the risk of pre-term delivery.

Details of ethics approval

The study was approved by the ethics committee (CCPPRB) of Kremlin Bicêtre and by the Data Protection Authority "Comission Nationale de l'Informatique et des Libertés" (CNIL). Written consents were obtained from the mother for herself at inclusion and for her newborn child after delivery.

Authors have no conflict of interest to declare.

List of tables


Peggy Drouillet-Pinard

Guy Huel

Rémy Slama

Anne Forhan

Josiane Sahuquillo

Valérie Goua

Olivier Thiébaugeorges

Bernard Foliguet

Guillaume Magnin

Monique Kaminski

Sylvaine Cordier

Marie-Aline Charles
email: marie-aline.charles@inserm.fr

Prenatal mercury contamination: relationship with maternal seafood consumption during pregnancy and fetal growth in the 'EDEN

Keywords: Mercury, seafood consumption, prenatal exposure, fetal growth

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Human exposure to Methylmercury (MeHg) occurs mainly via consumption of fish [START_REF] Bjornberg | Methyl mercury and inorganic mercury in Swedish pregnant women and in cord blood: influence of fish consumption[END_REF][START_REF] Mahaffey | Fish and shellfish as dietary sources of methylmercury and the omega-3 fatty acids, eicosahexaenoic acid and docosahexaenoic acid: risks and benefits[END_REF][START_REF] Morrissette | Temporal variation of blood and hair mercury levels in pregnancy in relation to fish consumption history in a population living along the St. Lawrence River[END_REF][START_REF] Johnsson | Impact of consumption of freshwater fish on mercury levels in hair, blood, urine, and alveolar air[END_REF][START_REF] Díez | Prenatal and early childhood exposure to mercury and methylmercury in Spain, a high-fish-consumer country[END_REF] . As MeHg is transferred to the children through placenta, maternal exposition represents a risk for the offspring [START_REF] Kajiwara | Methylmercury transport across the placenta via neutral amino acid carrier[END_REF][START_REF] Bjornberg | Transport of methylmercury and inorganic mercury to the fetus and breast-fed infant[END_REF] . Adverse health effects following prenatal exposure to MeHg have been described from several prospective cohort studies conducted in fish-eating population: low mean birthweight in Greenland [START_REF] Foldspang | Dietary intake of methylmercury as a correlate of gestational length and birth weight among newborns in Greenland[END_REF] , adverse neuropsychological and behavioural effects in Faroe Islands [START_REF] Grandjean | Cognitive performance of children prenatally exposed to "safe" levels of methylmercury[END_REF][START_REF] Murata | Delayed brainstem auditory evoked potential latencies in 14-year-old children exposed to methylmercury[END_REF] , risk of preterm delivery in Michigan [START_REF] Xue | Maternal fish consumption, mercury levels, and risk of preterm delivery[END_REF] .

Nevertheless, in the Seychelles Child development Study [START_REF] Davidson | Effects of prenatal and postnatal methylmercury exposure from fish consumption on neurodevelopment: outcomes at 66 months of age in the Seychelles Child Development Study[END_REF][START_REF] Palumbo | Association between prenatal exposure to methylmercury and cognitive functioning in Seychellois children: a reanalysis of the McCarthy Scales of Children's Ability from the main cohort study[END_REF][START_REF] Myers | Prenatal methylmercury exposure from ocean fish consumption in the Seychelles child development study[END_REF] , no adverse effects of MeHg exposure had been found. The first hypothesis to explain these controversial results was the different level of contamination of the study populations, the second was a difference in the kind of fish consumed which resulting in differences in nutrients intake (n-3 Fatty Acids (FA)) [START_REF] Davidson | Effects of prenatal and postnatal methylmercury exposure from fish consumption on neurodevelopment: outcomes at 66 months of age in the Seychelles Child Development Study[END_REF] , selenium and other contaminants exposure (PCBs).

Fish is also known to have beneficial effects on fetal growth since it provides PUFA, especially n-3FA. In both epidemiologic [START_REF] Harper | Increased birth weight in northerly islands: is fish consumption a red herring?[END_REF][START_REF] Olsen | Does fish consumption during pregnancy increase fetal growth? A study of the size of the newborn, placental weight and gestational age in relation to fish consumption during pregnancy[END_REF][START_REF] Olsen | Frequency of seafood intake in pregnancy as a determinant of birth weight: evidence for a dose dependent relationship[END_REF][START_REF] Olsen | Duration of pregnancy in relation to seafood intake during early and mid pregnancy: prospective cohort[END_REF][START_REF] Muthayya | The effect of fish and omega-3 LCPUFA intake on low birth weight in Indian pregnant women[END_REF] and intervention studies [START_REF] Olsen | Randomised controlled trial of effect of fish-oil supplementation on pregnancy duration[END_REF][START_REF] Olsen | A possible preventive effect of low-dose fish oil on early delivery and pre-eclampsia: indications from a 50-year-old controlled trial[END_REF] mainly performed in women from Denmark and the Faroe Islands, intake of seafood or marine n-3FA by pregnant women was associated with an increased birthweight explained by both a prolonged duration of pregnancy and increased fetal growth rate. In the "EDEN mother-child" cohort, we have found a positive association between seafood consumption before pregnancy and fetal growth limited to overweight women [START_REF] Drouillet | Association between maternal seafood consumption before pregnancy and fetal growth: evidence for an association in overweight women. The "EDEN mother-child" cohort (study of pre and early postnatal determinants of the child's development and health)[END_REF] .

The public is faced with seemingly conflicting reports on the risks and benefits of seafood intake, resulting in controversery and confusion over the place of fish consumption in a healthy diet. Only recently, few studies in this field have focused on contaminant risks in the same time as nutrient benefits related to fish intake although both risk affect the same outcomes and are derived from the same foods [START_REF] Budtz-Jorgensen | Separation of risks and benefits of seafood intake[END_REF][START_REF] Mendez | Seafood consumption in pregnancy and infant size at birth: Results from a prospective Spanish cohort[END_REF][START_REF] Ramón | Fish consumption during pregnancy, prenatal mercury exposure, and anthropometric measures at birth in a prospective motherinfant cohort study in Spain[END_REF][START_REF] Lucas | Gestational age and birth weight in relation to n-3 fatty acids among Inuit (Canada)[END_REF] . Some recent studies hypothesized a confounding role of maternal nutrition in the assessment of Hg risk [START_REF] Strain | Associations of maternal long-chain polyunsaturated fatty acids, methyl mercury, and infant development in the Seychelles Child Development Nutrition Study[END_REF][START_REF] Davidson | Neurodevelopmental effects of maternal nutritional status and exposure to methylmercury from eating fish during pregnancy[END_REF] suggest opposite effect of maternal seafood intake and Hg exposure [START_REF] Oken | Maternal fish consumption, hair mercury, and infant cognition in a U.S. Cohort[END_REF][START_REF] Grandjean | Birthweight in a fishing community: significance of essential fatty acids and marine food contaminants[END_REF] .

Objectives

To further explore the relationship between seafood consumption prior to pregnancy and fetal growth reported in the "EDEN mother-child" cohort [START_REF] Drouillet | Association between maternal seafood consumption before pregnancy and fetal growth: evidence for an association in overweight women. The "EDEN mother-child" cohort (study of pre and early postnatal determinants of the child's development and health)[END_REF][START_REF] Drouillet | Maternal fatty acid intake and fetal growth: evidence for an association in overweight women. The 'EDEN mother-child' cohort (study of pre-and early postnatal determinants of the child's development and health)[END_REF] , the aim of the present analysis was, in the same French population, to study 1) the association between seafood consumption and Hg contamination, 2) potential risks of Hg exposure on fetal growth, 3) whether relationships between seafood consumption and fetal growth was modified after taking into account Hg and selenium exposure.

METHODS

Population and study design

Pregnant (n=2002) women were recruited in the University Hospitals of Nancy and Poitiers before 24 weeks of amenorrhea (WA). Standard ultrasound fetal measurements were recorded from routine examinations at 20-24 and 30-34 WA. Prepregnancy body mass index (pBMI) was computed as reported weight (kg) / measured height squared (m²). According to references of the International Obesity Task Force, overweight was defined as a BMI ≧25 kg/m² or more and obesity as a BMI ≧ 30kg/m². Birthweight and length were extracted from the hospital record. Head circumference (in duplicate) and tricipital and subscapular skinfolds (in triplicate) were performed on the newborn after delivery (1.8 days (range 0-16)), and averaged. Standard ultrasound fetal measurements were recorded from routine examinations performed between 20-24 and 30-34 WG. Measurements included biparietal diameter, head and abdominal circumferences and femur length.

The study was approved by the Ethic Committee of the Bicêtre Hospital. Written consents were obtained from the mother for herself at inclusion and for her newborn child after delivery.

Dietary assessment

Mothers completed two food frequency questionnaires: at inclusion, about diet in the year prior to pregnancy; after birth, about their diet in the last three months of pregnancy. We combined responses to the six questions that inquired about seafood consumption: 'At which frequency did you eat': (1) fresh or frozen fish (bought unprocessed), (2) oily fish, (3) smoked or salted fish, (4) breaded fish, (5) dishes containing fish, and (6) shellfish. We generated an average frequency of seafood servings per month for each woman, by weighing each frequency with the midpoint of the category (i.e. 2 for the category 1-3 servings/month).

Information about the type of fish was asked only in women who were regular eaters (more than 1 time/month). Regular fish eaters consumed both fatty and lean fish and we were not able to contrast women according to this characteristic. Determination of Hg exposure Determination of heavy metal exposure was planned for the first 700 women included in the cohort for cost reason. Hair samples were stored until analysis at room temperature. Chemical analyses were carried out at TOXILABO (Nantes, France), by cold-vapor atomic absorption spectrometry (Zeeman Perkin-Elmer AA600) for 691 mothers and only 87 newborns due to low hair mass. When hair mass was under 10mg for mothers and 7mg for newborns, measures were considered too inaccurate and were not taken into account. For samples of 82 women and 66 newborns with Hg levels too low to be detected, we arbitrarily attributed half of the limit level detectable with the hair mass. Hg concentration (expressed in micrograms per gram) was log transformed because of a skewed distribution.

Determination of Se concentration

Frozen samples at -80°C were thawed for Se measurements. Se concentrations in blood (expressed in micrograms per liter) were determined by fluorometric method which involves the reaction of 2,3-diaminonapthalene (DAN) with Se(IV) to form a fluorescent Se/DAN piazselenol.

Variable description and statistical analyses

Comparisons between groups were studied by Student's t test and correlations by Pearson's and Spearman's correlations. We studied relationships between seafood consumption before pregnancy as well as maternal Hg level and fetal growth, using multiple linear regressions adjusted for different sets of confounding variables. Most of included women were from Poitiers because this centre started recruitment earlier; therefore, comparisons were performed with adjustment for centre. Seafood consumption and Hg level were studied separately, then in the same model. We performed more analyses to evaluate impact of extreme values on the relationships; total hair-Hg level was studied in classes to separate the 15% lower levels (N1: Hg<0.23µg/g) and the 15% higher levels (N4: Hg>0.82µg/g) and two middle categories (N2: [0.23-0.52[; N3: [0.52-0.82[µg/g). As BMI modified relationships between seafood intake and fetal growth (p for interaction=0.0001 for birthweight), we studied separately non-overweight and overweight women (BMI<25 vs. >25 kg/m²). Adjustments for selenium concentration or educational level were also made.

All analyses were performed with SAS version 9.1 (Cary, N.C., USA).

RESULTS

Subjects characteristics

Among the 691 first women included in the study, 26 were excluded because of a hair's samples <10 mg, 15 because seafood consumption was unknown and 5 because delay between birth and newborn anthropometric measures were greater than one week.

Mean pBMI was 23 kg/m²; overweight women accounted for 26.7% of included women (n=645) and 28.1% of non-included women (n=1251). There was no differences in gestational age (39.2 weeks both) or parity (54% multipare both) between included and excluded women, except that included women were slightly younger (28.7 vs. 29.2y) and more often smokers (31% vs. 27.6%) than the others. Sex ratio (boys/girls) was similar: 1.1 and 1.2 in excluded and included women respectively. Mean consumption of seafood was on average 8.4 times per month (SD=7.75) before pregnancy in both groups. No difference for newborns' measures was observed between the two groups (mean birthweight=3280g).

The median hair-Hg level for mothers and newborns were 0.52 (Interquartile Range (IR):0.30-0.82) (SD=2.6) and 0.38 (IR:0.30-0.43) (SD=0.32) µg/g respectively. As correlation was strong between levels in mothers and their offspring (r=0.43 p<0.0001 (n=87)), and fewer measures were available for newborns, analyses were made with maternal Hg level only. The median blood Se level for mothers was 97.4 µg/l (IR:81.4-114.4) (SD=26.2).

Correlations between maternal total-Hg level and maternal characteristics

Total hair-Hg levels were higher with age and university level, in Poitiers, and in nonsmokers during pregnancy. BMI was not associated with Hg level. Spearman correlation between Hg and seafood consumption before pregnancy was 0.33 (p<0.0001) and 0.29 (p<0.0001) in the last three months of pregnancy. When the different items contributing to global seafood intake before pregnancy were considered separately, correlations with Hg contamination were stronger for "Fresh or frozen fish", "Smoked or salted fish", "Oily fish" and shellfish (r=0.39; 0.28; 0.20 and 0.17 respectively; p<0.0001).

Hg exposure, seafood consumption and fetal growth

In the whole sample of women, there was no association between maternal level of total hair-Hg and ultrasound measures as well as newborn anthropometric measures (data not shown).

Only a negative association was observed between total hair-Hg level and biparietal diameter measured at 20-24 WG (decrease of 0.24 mm by 1 SD of hair-Hg, p=0.06). Seafood intake was not associated with fetal growth in all women. When both seafood consumption and Hg level were included in the model, results did not change.

In overweight women total hair-Hg level and seafood intake before pregnancy were both associated with higher newborn anthropometric measures in separate regression models.

Seafood intake was also associated with increased placental weight and lower gestational length (Table 1). However, when adjusted on seafood intake, total hair-Hg level was no longer associated with newborn anthropometric measures, whereas relationship with seafood intake remained the same. Excluding shellfish intake from the computation of seafood intake did not change the results. The association between seafood intake and lower gestational length was reinforced when adjusted for total hair-Hg level but it became non significant after exclusion of the 8 preterm births.

In non-overweight women (data not shown), total hair-Hg level tended to be negatively associated with biparietal and head circumferences at 20-24 WG (decrease of 0.29 and -1.15 mm by 1SD of hair-Hg respectively, p<0.06) but not statistically significant for measures at 30-34 WG. Adjustment for seafood intake did not change these results. Seafood intake was associated with a lower birth length and head circumference, with an average decrease of 0.19 and 0.17 cm respectively for an increase of 1SD of seafood intake (p<0.02), even when adjusted for total hair-Hg level.

A similar trend was observed for associations with seafood intake in the last three months of pregnancy but associations were weaker.

We performed further analyses to evaluate impact of high values total hair-Hg maternal concentrations on the relationships. The relations reported above with total hair-Hg were consistent with linear relations with no thresholds effect for extreme values (data not shown).

In our study, correlation between selenium and seafood consumption were 0.10 (p=0.03) and 0.14 (p=0.001) before and in the last three months of pregnancy. Correlation between maternal hair-Hg after pregnancy and blood Se during pregnancy at 24-28WA was 0.10 (p=0.03). Adjustment for selenium level did not change the relation observed between total hair-Hg, seafood and fetal growth in the whole sample of women as well as in nonoverweight and overweight women (Data not shown).

DISCUSSION

In our study, total hair-Hg level was associated with seafood intake, but was not associated with fetal growth.

The lack of association between maternal total hair-Hg level and birthweight could be explained by low mean hair-Hg level in our population (0.52µg/g) compared to other studies: 12.7µg/g in French Guiana [START_REF] Cordier | Neurodevelopmental investigations among methylmercury-exposed children in French Guiana[END_REF] , 12.8µg/g in Amazonia [START_REF] Grandjean | Methylmercury neurotoxicity in Amazonian children downstream from gold mining[END_REF] . Studies where prenatal Hg exposition was associated with risk for fetal growth were cases of massive intoxication. The negative relationship with biparietal diameter early in pregnancy may be a chance finding because was not confirmed with measures at 30-34WA and at birth. Alternatively, it may disclose an early alteration in the neurological developmental process. Oken et al. [START_REF] Oken | Maternal fish consumption, hair mercury, and infant cognition in a U.S. Cohort[END_REF] found a negative effect on infant cognition when mean hair-Hg level was close to ours (0.55µg/g). .

As Lucas et al. [START_REF] Lucas | Gestational age and birth weight in relation to n-3 fatty acids among Inuit (Canada)[END_REF] , we did not find association between Hg exposition and length of gestation or risk of preterm delivery whereas Xue et al. [START_REF] Xue | Maternal fish consumption, mercury levels, and risk of preterm delivery[END_REF] or Ramirez et al. [START_REF] Ramirez | The Tagum study I: analysis and clinical correlates of mercury in maternal and cord blood, breast milk, meconium, and infants' hair[END_REF] found a decrease of length of gestation and risk of preterm delivery with increasing Hg contamination. However, low exposition induces low power and unstable results. Hg was measured in hair, however, measured in blood, Hg could be a better estimation of fetal exposition as hypothesized by Grandjean & Budtz-Jorgensen [START_REF] Grandjean | Total imprecision of exposure biomarkers: implications for calculating exposure limits[END_REF] . Problem with undetectable values could explain the differences with other studies; but similar results were found in our study when we took into account only detectable measures.

Health effects of Hg may partly be due to selenoprotein activation, which may be moderated by adequate intake of selenium [START_REF] Watanabe | Modification of mercury toxicity by selenium: practical importance?[END_REF][START_REF] Chen | The roles of serum selenium and selenoproteins on mercury toxicity in environmental and occupational exposure[END_REF] . Possible protective role of Se has been suggested in some studies to explain lack of negative effects of Hg [START_REF] Marsh | Fetal methylmercury study in a Peruvian 5 fish-eating population[END_REF] . In our study, relationships between total hair-Hg and fetal growth remained unchanged after adjustment for selenium level.

In conclusion, our data do not support a detrimental effect of low maternal Hg contamination on birthweight or other newborn anthropometric measurements. Taking into account maternal Hg contamination and blood selenium impregnation did not change the observed increase in birthweight associated with seafood consumption in overweight women [START_REF] Drouillet | Association between maternal seafood consumption before pregnancy and fetal growth: evidence for an association in overweight women. The "EDEN mother-child" cohort (study of pre and early postnatal determinants of the child's development and health)[END_REF] . The negative association between seafood intake and gestational length may be a chance finding as in most studies, opposite or no results were reported [START_REF] Olsen | Frequency of seafood intake in pregnancy as a determinant of birth weight: evidence for a dose dependent relationship[END_REF][START_REF] Olsen | Duration of pregnancy in relation to seafood intake during early and mid pregnancy: prospective cohort[END_REF][START_REF] Muthayya | The effect of fish and omega-3 LCPUFA intake on low birth weight in Indian pregnant women[END_REF] and was not found in our previous analysis, which was not restricted to women with Hg measurements [START_REF] Drouillet | Association between maternal seafood consumption before pregnancy and fetal growth: evidence for an association in overweight women. The "EDEN mother-child" cohort (study of pre and early postnatal determinants of the child's development and health)[END_REF] .

Most of the adverse effects of Hg exposure were negative effects on brain development [START_REF] Grandjean | Methylmercury neurotoxicity in Amazonian children downstream from gold mining[END_REF] and neuropsychological effects [START_REF] Cordier | Neurodevelopmental investigations among methylmercury-exposed children in French Guiana[END_REF] . Follow-up of the children included in our study on visual and neurodevelopmental outcomes will allow evaluating the consequences for the child of 5 maternal Hg exposure at levels currently found in France.

Table 1 .

 1 Fetal growth in relation to maternal total hair mercury level and average seafood intake before pregnancy (overweight women)

			Model 1 *	Model 2 *		Model 3 *
	n=159		Mercury	Seafood	Mercury	Seafood
		n	 	p †	 	p †	 	p †	 	p †
	ULTRASOUND MEASURES
	at 20-24 WG
	Biparietal diameter (mm)	156	-0.08	0.75	-0.07	0.75	-0.03	0.92	-0.06	0.79
	Head circumference (mm)	149	0.94	0.18	0.50	0.44	1.04	0.17	0.14	0.84
	Abdominal circumference (mm)	153	1.26	0.16	1.35	0.10	0.93	0.34	1.04	0.24
	Femoral length (mm)	154	0.18	0.32	0.24	0.16	0.12	0.53	0.20	0.28
	ULTRASOUND MEASURES
	at 30-34 WG
	Biparietal diameter (mm)	149	0.17	0.57	0.22	0.42	0.15	0.64	0.17	0.56
	Head circumference (mm)	145	0.64	0.71	-1.63	0.31	1.74	0.36	-2.19	0.20
	Abdominal circumference (mm)	147	0.54	0.68	1.51	0.21	0.34	0.81	1.40	0.28
	Femoral length (mm)	149	0.30	0.17	0.39	0.06	0.27	0.27	0.30	0.16
	ANTHROPOMETRIC
	MEASURES
	Birthweight (g)	151	79.80	0.04	115.09 0.001	41.91 0.30 100.52 0.008
	Birth length (cm)	147	0.34	0.05	0.39	0.01	0.23	0.21	0.32	0.07
	Head circumference (cm)	150	0.17	0.09	0.22	0.02	0.09	0.41	0.19	0.06
	Sum of skinfolds (mm)	150	0.24	0.15	0.39	0.01	0.10	0.57	0.35	0.04
	OTHER MEASURES
	Gestational length (WA)	156	0.13	0.41	-0.30	0.03	0.30	0.07	-0.40	0.008
	Placental weight (g)	109/123	4.05	0.75	31.58	0.005	0.43	0.97	31.48	0.007

  corresponds to variation of the outcome variable for 1 SD of MeHg level

(2.60)

or seafood intake (7.75) † Linear regression test 5 * Adjusted for centre, maternal age and height, smoking during pregnancy, parity (yes/no), gestational length (at ultrasound measures or at delivery), delay between birth and anthropometric measures (except for ultrasound measures and gestational length) and newborn's sex. Model 1: association with MeHg level / Model 2: association with seafood intake / Model 3: association with MeHg level and seafood intake, mutually adjusted on each other

ACKNOWLEDGMENTS

We are indebted to the participating families, to the midwife research assistants (L Douhaud, S Bedel, B Lortholary, S Gabriel, M Rogeon, M Malinbaum) for data collection and to P Lavoine for checking, coding and data entry.

We acknowledge all the funding sources for the EDEN study: Fondation pour la Recherche Médicale (FRM), French Ministry of Research: IFR program, INSERM Nutrition Research Program, French Ministry of Health Perinatality Program, French Agency for Environment Security (AFFSET), French National Institute for Population Health Surveillance (INVS),