Jimmy Garnier

François Hamel

Lionel Roques

Transition fronts and stretching phenomena for a general class of reaction-dispersion equations

We consider a general form of reaction-dispersion equations with non-local dispersal and local reaction. Under some general conditions, we prove the non-existence of transition fronts, as well as some stretching properties at large time for the solutions of the Cauchy problem. These conditions are satisfied in particular when the reaction is monostable and when the dispersal operator is either the fractional Laplacian, a convolution operator with a fat-tailed kernel or a nonlinear fast diffusion operator.

Introduction

This note is concerned with fast propagation phenomena and large time qualitative properties of dispersion-reaction equations of the type u t (t, x) = Du(t, x) + f (u(t, x)), t ∈ R, x ∈ R.

(1.1)

The given reaction function f : [0, 1] → R is of class C 1 and such that f (0) = f (1) = 0. Throughout the paper, we assume that, for any u 0 ∈ L ∞ (R; [0, 1]), the Cauchy problem u t (t, x) = Du(t, x) + f (u(t, x)), t > 0, x ∈ R, u(0, •) = u 0 (1.2) admits a unique mild solution, such that u(t, •) ∈ L ∞ (R; [0, 1]) for every t > 0 and u is uniformly continuous with respect to t in [δ, +∞) × R for every δ > 0. The equation (1.1) is assumed to be autonomous in the sense that, for any solution u of the Cauchy problem (1.2) and for any T > 0, the function (t, x) → u(T +t, x) solves (1.2) with initial condition u(T, •).

We also assume that a comparison principle holds for (1.2), that is, for any two solutions u and v with respective initial conditions u 0 and v 0 ,

0 ≤ u 0 ≤ v 0 ≤ 1 a.e. in R =⇒ ∀ t > 0, 0 ≤ u(t, •) ≤ v(t, •) ≤ 1 a.e. in R . (1.3)
Lastly, the solutions u of (1.2) are assumed to have infinite spreading speeds (to the right) in the sense that

u 0 ∈ L ∞ (R; [0, 1])\{0} =⇒ ∀ c > 0, ess inf (0,ct)
u(t, •) → 1 as t → +∞ .

(1.4)

The first typical example of equations (1.1) for which these conditions are fulfilled is the monostable reaction-diffusion equation with fractional diffusion u t = -(-∆ x ) α u + f (u), (1.5) where the reaction function f is monostable, that is f (0) = f (1) = 0, f (u) > 0 for all u ∈ (0, 1), f (0) > 0, (1.6) and the dispersal operator D is the fractional Laplacian Du = -(-∆ x ) α u with 0 < α < 1, see [START_REF] Cabré | The influence of fractional diffusion in Fisher-KPP equations[END_REF] (see also [START_REF] Cabré | Propagation in Fisher-KPP type equations with fractional diffusion in periodic media[END_REF] for similar equations in periodic media). In [START_REF] Cabré | The influence of fractional diffusion in Fisher-KPP equations[END_REF], the function f is also assumed to be concave in [0, 1], but the condition (1.4) actually holds under the condition (1.6) by the comparison principle and putting below f a concave function g : [0, 1 -ε] → R on the interval [0, 1 -ε], with ε ∈ (0, 1) as small as wanted. The set of functions f satisfying (1.6) includes the class of Fisher-KPP [START_REF] Fisher | The advance of advantageous genes[END_REF][START_REF] Kolmogorov | Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique[END_REF] nonlinearities f for which, in addition to these assumptions, f satisfies f (u) ≤ f (0)u in [0, 1]. It also contains the set of C 2 concave functions f with f (0) = f (1) = 0 and f > 0 on (0, 1), whose archetype is the logistic nonlinearity f (u) = u(1 -u).

A second important class consists of integro-differential equations with fat-tailed dispersal kernels

u t = J * u -u + f (u), (1.7)
where f is of the type (1.6) and the dispersal operator is a convolution operator

Du(t, x) = J * u(t, x) -u(t, x) := R J(x -y)(u(t, y) -u(t, x)) dy,
with a fat-tailed positive even kernel

J ∈ L 1 (R) such that    J L 1 (R) = 1, R |x|J(x)dx < +∞,
J is of class C 1 for large |x| and J (x) = o(J(x)) as |x| → +∞, see [START_REF] Garnier | Fast propagation in integro-differential equations[END_REF]. Notice that such kernels J are called "fat tailed" since they decay slowly as |x| → +∞ in the sense that J(x)e η|x| → +∞ as |x| → +∞ for every η > 0. Archetypes are J(x) = c α,β e -β|x| α with 0 < α < 1, β > 0 and some normalization constant c α,β > 0, or J(x) = c α /(1 + |x| α ) with α > 2 and some c α > 0. Such operators and equations arise in many physical or ecological models, see e.g. [START_REF] Clark | Why trees migrate so fast: Confronting theory with dispersal biology and the paleorecord[END_REF][START_REF] Del-Castillo-Negrete | Truncation effects in superdiffusive front propagation with Lévy flights[END_REF][START_REF] Del-Castillo-Negrete | Front propagation and segregation in a reactiondiffusion model with cross-diffusion[END_REF][START_REF] Fife | Mathematical Aspects of Reacting and Diffusing Systems[END_REF][START_REF] Kot | Dispersal data and the spread of invading organisms[END_REF][START_REF] Mancinelli | Front propagation in reactive systems with anomalous diffusion[END_REF][START_REF] Medlock | Spreading disease: Integro-differential equations old and new[END_REF][START_REF] Mollison | Spatial contact models for ecological and epidemic spread[END_REF]. Lastly, the conditions (1.3) and (1.4) are also fulfilled when the dispersal operator D corresponds to fast nonlinear diffusion. Using a detailed formal analysis of the equation

u t = (u γ ) xx + f (u), (1.8)
with 0 < γ < 1 and where f satisfies (1.6), it was shown in [START_REF] King | On the Fisher-KPP equation with fast nonlinear diffusion[END_REF] that the assumption (1.4) was fulfilled. In [START_REF] Stan | The Fisher-KPP equation with nonlinear fractional diffusion[END_REF], this result was extended to the case of nonlinear fractional diffusion equations:

u t = -(-∆ x ) α (u γ ) + f (u), (1.9)
with 0 < α < 1 and max(1 -2 α, 0) < γ and where f satisfies (1.6) and is a concave function. Due to the comparison principle (1.3), which is valid for both (1.8) and (1.9), it can be shown that the condition (1.4) holds without the concavity assumption, as for the standard fractional Laplacian.

Given the few general assumptions (1.3) and (1.4), the goal of the paper is twofold. Firstly, we will prove the non-existence of front-like entire solutions. Secondly, further stretching properties of the solutions of the Cauchy problem (1.2) at large time will be shown. As a corollary of the second main theorem, we will show that the solutions of (1.2) are in some sense flat at some large times in left and right neighborhoods of any level set. We again insist on the fact that these results will hold for the four main examples (1.5), (1.7), (1.8) and (1.9).

Non-existence of transition fronts

When propagating solutions are mentioned, one immediately has in mind standard traveling fronts u(t, x) = ϕ(±x -ct), with velocity c ∈ R and front profile ϕ : R → [0, 1] such that ϕ(-∞) = 1 and ϕ(+∞) = 0. On the one hand, without the assumption (1.4), standard traveling fronts ϕ c (±x -ct) are known to exist when we consider equations (1.8) with the standard (local) Laplacian Du = ∆ x u = u xx or a nonlocal convolution operator Du = J * uu with thin-tailed dispersal kernel J ∈ L 1 (R), which is nonnegative even and exponentially bounded in the sense that J L 1 (R) = 1 and R J(x) e λx dx < +∞ for some λ > 0. More precisely, when f is of the type (1.6), these fronts exist for every speed c ≥ c * , with a positive minimal speed c * . Furthermore, uniqueness of the profile ϕ c and stability results for a given c ≥ c * have been shown, with possibly heterogeneous nonlinearities f (x, u), see [START_REF] Aronson | Multidimensional nonlinear diffusions arising in population genetics[END_REF][START_REF] Carr | Uniqueness of travelling waves for nonlocal monostable equations[END_REF][START_REF] Coville | On uniqueness and monotonicity of solutions of non-local reaction diffusion equation[END_REF][START_REF] Coville | Non-local anisotropic dispersal with monostable nonlinearity[END_REF][START_REF] Coville | Propagation speed of travelling fronts in non local reaction-diffusion equations[END_REF][START_REF] Coville | On a nonlocal reaction diffusion equation arising in population dynamics[END_REF][START_REF] Kolmogorov | Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique[END_REF][START_REF] Sattinger | Weighted norms for the stability of traveling waves[END_REF][START_REF] Schumacher | Travelling-front solutions for integro-differential equations[END_REF][START_REF] Shen | Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats[END_REF][START_REF] Shen | Traveling wave solutions of spatially periodic nonlocal monostable equations[END_REF][START_REF] Uchiyama | The behavior of solutions of some semilinear diffusion equation for large time[END_REF]. However, even for the reaction-diffusion equation u t = u xx + f (u) with local diffusion and f of the type (1.6), fast propagation phenomena with infinite spreading speed as in (1.4) is known to occur when u 0 decays to 0 as x → ±∞ more slowly than any exponentially decaying function, see [START_REF] Hamel | Fast propagation for KPP equations with slowly decaying initial conditions[END_REF]. On the other hand, some existence, uniqueness and stability results of standard traveling fronts have also been shown for some nonlocal equations of the type (1.1) with fractional Laplacians or fattailed dispersal convolution kernels J, but for other types of nonlinearities f (u) or f (x, u) than (1.6), see e.g. [START_REF] Bates | Traveling waves in a convolution model for phase transitions[END_REF][START_REF] Chen | Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations[END_REF][START_REF] Chmaj | Existence of traveling waves in the fractional bistable equation[END_REF][START_REF] Coville | On uniqueness and monotonicity of solutions of non-local reaction diffusion equation[END_REF][START_REF] Coville | Pulsating fronts for nonlocal dispersion and KPP nonlinearity[END_REF][START_REF] Coville | Propagation speed of travelling fronts in non local reaction-diffusion equations[END_REF][START_REF] Coville | On a nonlocal reaction diffusion equation arising in population dynamics[END_REF][START_REF] Gui | Traveling wave solutions to some reaction diffusion equations with fractional Laplacians[END_REF][START_REF] Gui | Traveling wave solutions of Allen-Cahn equation with a fractional Laplacian[END_REF][START_REF] Mellet | Existence and asymptotics of fronts in non local combustion models[END_REF].

Here, the assumptions (1.3) and (1.4) forbid standard traveling fronts to exist, since all non-trivial solutions of the Cauchy problem accelerate with infinite speed as t → +∞ in the sense of (1.4). This non-existence result has already been known in the case of the fractional Laplacian Du = -(-∆ x ) α u [START_REF] Cabré | The influence of fractional diffusion in Fisher-KPP equations[END_REF][START_REF] Gui | Traveling wave solutions to some reaction diffusion equations with fractional Laplacians[END_REF] as well as in the convolution case Du = J * u -u with non-exponentially-bounded kernels J [START_REF] Garnier | Fast propagation in integro-differential equations[END_REF][START_REF] Yagisita | Existence and nonexistence of traveling waves for a nonlocal monostable equation[END_REF]. Non-existence of standard traveling fronts has also been derived for the fast nonlinear diffusion equations (1.8) and (1.9) [START_REF] King | On the Fisher-KPP equation with fast nonlinear diffusion[END_REF][START_REF] Stan | The Fisher-KPP equation with nonlinear fractional diffusion[END_REF]. However, other propagating solutions connecting 0 and 1, more general than the standard traveling fronts, can be investigated. These solutions, called transition fronts, have been defined in [START_REF] Berestycki | Generalized travelling waves for reaction-diffusion equations[END_REF][START_REF] Berestycki | Generalized transition waves and their properties[END_REF] in more general situations. In the one-dimensional situation considered here, as a wave-like solution defined in [START_REF] Shen | Traveling waves in diffusive random media[END_REF], the following definition holds.

Definition 2.1 (Transition front)

A transition front connecting 0 (say, on the right) and 1 (on the left) for (1.1) is a time-global solution u : R × R → [0, 1] (that is, the unique mild solution of the Cauchy problem (1.2) starting at any time t 0 ∈ R with "initial" condition u(t 0 , •)) such that there exists a family (ξ t ) t∈R of real numbers with

     ess inf (-∞,x) u(t, ξ t + •) → 1 as x → -∞, ess sup (x,+∞) u(t, ξ t + •) → 0 as x → +∞, uniformly in t ∈ R. (2.1)
The real numbers ξ t therefore reflect the positions of a transition front u as a function of time. However, the ξ t 's are not uniquely defined since any family ( ξ t ) t∈R satisfies (2.1) as soon as (ξ t ) t∈R does and ( ξ t -ξ t ) t∈R remains bounded. Roughly speaking, condition (2.1) means that the diameter of the transition zone between the sets where u 1 and u 0 is uniformly bounded in time in the sense that, for every 0 < a ≤ b < 1, there is a nonnegative real number C such that, for all t ∈ R,

x ∈ R; a ≤ u(t, x) ≤ b ⊂ X(t) -C, X(t) + C]
up to a negligible set. Transition fronts connecting 0 on the left and 1 on the right can be defined similarly by permuting the limits as x → ±∞ in (2.1). We will therefore only consider here transition fronts connecting 0 on the right and 1 on the left. Obviously, any standard traveling front u(t, x) = ϕ(x -ct) with ϕ(-∞) = 1 and ϕ(+∞) = 0 would be a transition front connecting 0 and 1 with ξ t = ct, but, as already emphasized, the assumption (1.4) excludes the existence of standard traveling fronts for (1.1). Nevertheless, transition fronts can a priori be much more general than standard traveling fronts, since no assumption is made on the family of front positions (ξ t ) t∈R . In particular, transition fronts different from the standard traveling fronts have been constructed recently for some homogeneous or heterogeneous local reaction-diffusion equations [START_REF] Hamel | Transition fronts for the Fisher-KPP equation[END_REF][START_REF] Hamel | Admissible speeds of transition fronts for non-autonomous monostable equations[END_REF][START_REF] Mellet | Stability of generalized transition fronts[END_REF][START_REF] Mellet | Generalized fronts for one-dimensional reaction-diffusion equations[END_REF][START_REF] Nadin | Propagation phenomena for time heterogeneous KPP reaction-diffusion equations[END_REF][START_REF] Nolen | Existence and non-existence of Fisher-KPP transition fronts[END_REF][START_REF] Nolen | Traveling waves in a one-dimensional heterogeneous medium[END_REF][START_REF] Zlatoš | Transition fronts in inhomogeneous Fisher-KPP reaction-diffusion equations[END_REF][START_REF] Zlatoš | Generalized traveling waves in disordered media: existence, uniqueness, and stability[END_REF], and fronts with global speed ξ t /t having different limits as t → ±∞ are also known to exist even for the homogeneous local Fisher-KPP reaction-diffusion equation u t = u xx + f (u) [START_REF] Hamel | Travelling waves and entire solutions of the Fisher-KPP equation in R N[END_REF][START_REF] Hamel | Transition fronts for the Fisher-KPP equation[END_REF][START_REF] Zlatoš | Transition fronts in inhomogeneous Fisher-KPP reaction-diffusion equations[END_REF]. For our equation (1.1), transition fronts with global speed ξ t /t growing arbitrarily as t → ±∞ are a priori not excluded.

Our first result shows actually that this is anyway impossible, since transition fronts cannot exist whatever the family of positions (ξ t ) t∈R may be. Theorem 2.2 generalizes the known non-existence result of standard traveling fronts for the equations of the type (1.5) with the fractional Laplacian, (1.7) with a fat-tailed dispersal kernel J,and (1.8) and (1.9) with fast nonlinear diffusion since the standard fronts are particular classes of transition fronts.

Roughly speaking, this non-existence result can be heuristically explained as follows: for any transition transition front connecting 0 and 1, the uniformity (with respect to time) of the limits (2.1) prevents the front from traveling too fast (see also [START_REF] Hamel | Transition fronts for the Fisher-KPP equation[END_REF] for a similar phenomenon for local reaction-diffusion equations), and this last property is finally in contradiction with the spreading properties (1.4) combined with the comparison principle (1.3).

Proof of Theorem 2.2. We argue by contradiction. So assume that, under the assumptions (1.3) and (1.4), equation (1.1) admits a transition front u connecting 0 and 1. Let (ξ t ) t∈R be a family of real numbers such that (2.1) holds.

For every t ∈ R, denote

ξ - t = sup x ∈ R; ess inf (-∞,x) u(t, •) ≥ 2 3 and ξ + t = inf x ∈ R; ess sup (x,+∞) u(t, •) ≤ 1 3 .
It follows from (2.1) that ξ ± t are real numbers. The definitions of ξ ± t imply that ess inf

(-∞,ξ - t ) u(t, •) ≥ 2 3 , ess inf (ξ - t ,ξ - t +1) u(t, •) < 2 3 (2.2)
and ess sup

(ξ + t ,+∞) u(t, •) ≤ 1 3
, ess sup

(ξ + t -1,ξ + t ) u(t, •) > 1 3 (2.3)
together with ξ - t ≤ ξ + t . Property (2.1) also yields

sup t∈R ξ t -ξ - t < +∞.
Indeed, otherwise, there would exist a sequence (t n ) n∈N of real numbers such that

ξ tn -ξ - tn → +∞ as n → +∞, whence ess inf (-∞,ξ - tn +1) u(t n , •) → 1 as n → +∞, contradicting the property ess inf (ξ - tn ,ξ - tn +1) u(t n , •) < 2 3 .
Similarly, there holds inf

t∈R ξ t -ξ + t > -∞.
Eventually, using again ξ - t ≤ ξ + t , one infers that the families (ξ t -ξ ± t ) t∈R and (ξ + t -ξ - t ) t∈R are all bounded.

From the general regularity assumptions made in the paper, the function u is actually uniformly continuous with respect to t in [T, +∞) × R, for every T ∈ R. By considering T = 0, it follows in particular that there exists δ > 0 such that, for all t ≥ 0, s ≥ 0 and x ∈ R,

|t -s| ≤ δ =⇒ |u(t, x) -u(s, x)| ≤ 1 12 .
As a consequence, for every t ≥ 0, denoting A ± t the non-negligible measurable sets defined as

A - t = x ∈ (ξ - t , ξ - t + 1); u(t, x) < 2 3 and A + t = x ∈ (ξ + t -1, ξ + t ); u(t, x) > 1

Stretching at large time for the Cauchy problem (1.2)

The second main result is concerned with flattening and stretching properties at large time for the solutions of the Cauchy problem (1.2). We assume in this section that the equation (1.1) is homogeneous, in the sense that, for any u 0 ∈ L ∞ (R; [0, 1]) and for any h ∈ R, there holds u(t,

• + h) = v(t,
that if u 0 ∈ L ∞ (R; [0, 1]) is nonincreasing, then 0 ≤ u 0 (• + h) ≤ u 0 ≤ 1 a.e. in R for every h > 0, whence 0 ≤ u(t, • + h) ≤ u(t,
•) ≤ 1 a.e. in R for every t > 0. In other words, u(t, •) is nonincreasing for every t > 0. Under these conditions, for every t ≥ 0, denote

m(t) = ess inf R u(t, •) = ess inf (x,+∞) u(t, •) for every x ∈ R and M (t) = ess sup R u(t, •) = ess sup (-∞,x) u(t, •) for every x ∈ R,
and observe that 0 ≤ m(t) ≤ M (t) ≤ 1. Lastly, for every λ ∈ (0, 1), set Theorem 3.1 means that the function u(t, •) becomes regularly as stretched as wanted as t → +∞, that is, for any given 0 < ε < 1/2, the "reaction" zone where u(t, •) ranges between ε and 1 -ε cannot stay bounded at large time. However, the following reasonable conjecture remains open:

x λ (t) = inf x ∈ R; ess sup (x,+∞) u(t, •) < λ . (3.2) Notice that x λ (t) = +∞ if λ ≤ m(t), while x λ (t) = -∞ if λ > M (t). Moreover, -∞ ≤ x b (t) ≤ x a (t) ≤ +∞ if 0 < a ≤ b < 1.
Conjecture 3.2 Under the assumptions of Theorem 3.1, x a (t) -x b (t) → +∞ as t → +∞ for any 0 < a < b < 1.
The assumptions m(t) → 0 and M (t) → 1 are satisfied for instance if Du = -(-∆ x ) α u with 0 < α < 1, Du = J * u -u, Du = (u γ ) xx with 0 < γ < 1, and Du = -(-∆ x ) α (u γ ) with 0 < α < 1 and max(1 -2 α, 0) < γ, and if m(0) = 0 and M (0) = 1 (remember that the C 1 ([0, 1]) function f is such that f (0) = f (1) = 0). These conditions are also satisfied for the same type of dispersal Du in the case m(0) = 0, 0 < M (0) ≤ 1 with f > 0 on (0, 1).

Theorem 3.1 is in spirit similar to a result obtained in [START_REF] Hamel | Fast propagation for KPP equations with slowly decaying initial conditions[END_REF] for the solutions of the Cauchy problem of the equation u t = u xx + f (u): if f satisfies (1.6) and f (u)/u is non-increasing over (0, 1] and if the initial condition u 0 : R → (0, 1] satisfies lim inf x→-∞ u 0 (x) > 0, u 0 (+∞) = 0, together with lim x→+∞ u 0 (x)e ηx = +∞ for every η > 0 and u 0 /u 0 ∈ L p (R) ∩ C 2,θ (R) for some p ∈ (1, +∞) and θ ∈ (0, 1), then u x (t, •) L ∞ (R) → 0 as t → +∞. This last conclusion is clearly stronger than (3.3). But, for our equation (1.1), we make weaker regularity assumptions on u 0 . In particular, u 0 may well be discontinuous and, since there is in general no spatial regularizing effect for (1.1), u(t, •) may not be differentiable or even continuous. But the property (3.3) still gives some information about the profile of u(t, •) at large time. In particular, even if u(t, •) may be discontinuous at some points, the discontinuity jumps cannot stay large as time goes to +∞.

Moreover, we can describe the profile of the solution u around the position of each level sets. More precisely, for any level λ ∈ (0, 1), u(t, • + x λ (t)) becomes regularly as flat as wanted at large times and becomes as close as wanted to λ locally on the left and on the right of the position x λ (t).

Corollary 3.3 Under the same assumptions as in Theorem 3.1, for every 0 < λ < 1 and R > 0, there holds

       lim inf t→+∞ ess sup x∈[-R,0] |u(t, x + x λ (t)) -λ| = 0, lim inf t→+∞ ess sup x∈[0,R] |u(t, x + x λ (t)) -λ| = 0. (3.4)
Proof of Theorem 3.1. Let u be a solution of the Cauchy problem (1.2) with a nonincreasing initial condition u 0 ∈ L ∞ (R; [0, 1])\{0}. As already emphasized, u(t, •) is nonincreasing in R for every t > 0. In particular, if m(t) < λ < M (t), then x λ (t) defined in (3.2) is a real number and ess sup

(x λ (t),+∞) u(t, •) ≤ λ and ess inf (-∞,x λ (t)) u(t, •) ≥ λ. (3.5)
Let now a and b be two given real numbers such that 0 < a < b < 1 and assume by contradiction that lim sup t→+∞ x a (t) -x b (t) < +∞, that is, there exist T > 0 and

M > 0 such that 0 ≤ x a (t) -x b (t) ≤ M for all t ≥ T. (3.6)
Since m(t) → 0 and M (t) → 1 as t → +∞, one can assume without loss of generality that m(t) < a < b < M (t) for all t ≥ T , whence x a (t) and x b (t) are real numbers for t ≥ T . Notice that

x b (t) ≤ x (a+b)/2 (t) ≤ x a (t) for all t ≥ T. (3.7)
From the uniform continuity of u with respect to t in [T, +∞) × R, there is δ > 0 such that, for every (t, s, x)

∈ [T, +∞) × [T, +∞) × R, |t -s| ≤ δ =⇒ |u(t, x) -u(s, x)| ≤ b -a 4 .
Now, for every t ≥ T , one has u(t, •) ≥ b a.e. in (-∞, x b (t)), whence

u(t + δ, •) ≥ b - b -a 4 = 3b + a 4 a.e. in (-∞, x b (t)),
whereas u(t, •) ≤ a a.e. in (x a (t), +∞) and

u(t + δ, •) ≤ a + b -a 4 = 3a + b 4 a.e. in (x a (t), +∞). Since (3a + b)/4 < (a + b)/2 < (3b + a)/4, one infers from the definition of x (a+b)/2 (t + δ) that x b (t) ≤ x (a+b)/2 (t + δ) ≤ x a (t), whence x (a+b)/2 (t + δ) -x (a+b)/2 (t) ≤ M for all t ≥ T. (3.8)
from (3.6) and (3.7).

On the other hand, since u(T, •) is nonincreasing and ranges in [0, 1], it follows again from the definition of x (a+b)/2 (T ) that

1 ≥ u(T, •) ≥ a + b 2 1 (-∞,x (a+b)/2 (T )) =: v 0 a.e. in R.
Therefore, the comparison principle (1.3) implies that, for every t > 0, 1 ≥ u(t + T, •) ≥ v(t, •) a.e. in R, where v denotes the solution of the Cauchy problem (1.2) with initial condition v 0 ∈ L ∞ (R; [0, 1])\{0}. The assumption (1.4) applied to v therefore yields ess inf (0,ct) v(t, •) → 1 as t → +∞ for every c > 0, whence ess inf

(0,ct) u(t + T, •) → 1 as t → +∞, for every c > 0.
Finally, for any arbitrary positive real number c, there holds ess inf ∆ x ) α u with 0 < α < 1. For this equation, there is a regularizing effect in the spatial variable and, under the assumptions of Theorem 3.1, it follows that the solution u is of class C 1 in [δ, +∞) × R for every δ > 0, with bounded derivatives with respect to t and x.

For any given λ ∈ (0, 1), there is a real number T λ > 0 such that, for every t ≥ T λ , there is a unique x λ (t) ∈ R such that u(t, x λ (t)) = λ. We now claim that lim inf

t→+∞ |u x (t, x λ (t))| = 0.
Indeed, otherwise, there are ε > 0 and T ≥ T λ such that u x (t, x λ (t)) ≤ -ε for all t ≥ T . It follows then from the implicit function theorem that the function x λ is of class C 1 ([T , +∞)) and that

x λ (t) = - u t (t, x λ (t)) u x (t, x λ (t))
for all t ≥ T . Therefore, there is M ≥ 0 such that |x λ (t)| ≤ M for all t ≥ T and one can get a contradiction as in the end of the proof of Theorem 3.1.

Again for (1.5) with the fractional Laplacian Du = -(-∆ x ) α u, and with a Fisher-KPP function f , Roquejoffre and Tarfulea [START_REF] Roquejoffre | Gradient estimates and symmetrization for Fisher-KPP front propagation with fractional diffusion[END_REF] recently showed that the partial derivatives of u with respect to x are all bounded by u times an exponentially decaying function, that is, for every k ∈ N, there are positive constants C k and δ k such that ∂ k x u(t, •) ∞ ≤ C k u(t, x)e -δ k t for large t. This result has been proved under the additional condition that u 0 is exponentially decaying as |x| → +∞, and it actually holds in any spatial dimension. It implies in particular that u x (t, •) L ∞ (R) → 0 as t → +∞. Moreover, it is proved in [START_REF] Roquejoffre | Gradient estimates and symmetrization for Fisher-KPP front propagation with fractional diffusion[END_REF] that the large time dynamics of the KPP problem (1.5) is in some sense the same as that of the ODE θ (t) = f (θ).

More generally speaking, under the general conditions (1.3), (1.4) and (3.1), we conjecture that fast propagation should lead to a flattening of the solution in the sense that, at least if f > 0 on (0, 1), then the only x-monotone solutions u : R × R → (0, 1) of (1.5) are actually x-independent, that is, they can be written as u(t, x) = θ(t), where θ (t) = f (θ(t)) for all t ∈ R with θ(-∞) = 0 and θ(+∞) = 1.

An intermediate result that we prove here under the sole conditions (1.3), (1.4) and (3.1) is the flattening result stated in Corollary 3.3. We point out that this result holds for all equations (1.5), (1.7), (1.8) and (1.9) under the same conditions as in Section 1.

Proof of Corollary 3.3. Let u be a solution of the Cauchy problem (1.2) with a nonincreasing initial condition u 0 ∈ L ∞ (R; [0, 1])\{0}. As already emphasized, u(t, •) is nonincreasing in R for every t > 0. In particular, if m(t) < λ < M (t), then x λ (t) defined in (3.2) is a real number and (3.5) holds.

Let then λ be any real number such that 0 < λ < 1, R be any positive real number and ε > 0 be any positive real number such that 0 < λ -ε < λ < λ + ε < 1. From Theorem 3.1, there exist two increasing sequences (t n ) n∈N and (t n ) n∈N of positive real numbers such that

x λ (t n ) -x λ+ε (t n ) → +∞ and x λ-ε (t n ) -x λ (t n ) → +∞ as n → +∞.
As a consequence, there exists N ∈ N such that, for all n ≥ N ,

x λ+ε (t n ) ≤ x + x λ (t n ) for all x ∈ [-R, 0], x + x λ (t n ) ≤ x λ-ε (t n ) for all x ∈ [0, R].
Thus, by (3.5) applied with λ -ε, λ and λ + ε, one infers that, for all n ≥ N ,

λ ≤ u(t n , x + x λ (t n )) ≤ λ + ε for a.e. x ∈ [-R, 0], λ -ε ≤ u(t n , x + x λ (t n )) ≤ λ for a.e. x ∈ [0, R], whence ess sup x∈[-R,0] |u(t n , x + x λ (t n )) -λ| ≤ ε and ess sup x∈[0,R] |u(t n , x + x λ (t n )) -λ| ≤ ε
for all n ≥ N . Since ε > 0 can be arbitrarily small and lim n→+∞ t n = lim n→+∞ t n = +∞, the proof of Corollary 3.3 is thereby complete. Remark 3.5 It easily follows from the proof of Corollary 3.3 that, if Conjecture 3.2 is true, then for any level λ ∈ (0, 1), the solution u of the Cauchy problem (1.2) converges to λ locally around each position of the level set as t → +∞, that is,

ess sup x∈[-R,R] |u(t, x + x λ (t)) -λ| → 0 as t → +∞
for every R > 0. This last convergence result would then hold automatically for the whole family t → +∞ and on both the left and right of x λ (t) simultaneously.

Lastly, in order to illustrate the theoretical results on the flattening and stretching properties of the solutions of the Cauchy problem (1.2), we have performed some numerical simulations that are depicted in figures 1 and 2. They have been done with the KPP nonlinearity f (u) = u (1 -u), for the three examples Du = -(-∆ x ) α u with α = 0.9, Du = J * u -u with J(x) = exp(-|x|)/4, and Du = (u γ ) xx with γ = 1/2 (Fig. 1, a, b andc, respectively). These properties, and the acceleration of the solutions in each of these three cases are to be compared with the well-known convergence to a standard travelling front which can be observed with the standard diffusion operator Du = u xx (Fig. 1,d).

To compute these solutions, we used (a and b): the Strang splitting, which consists in splitting the equation (1.2) into two simpler evolution problems [START_REF] Coulon | Propagation in reaction-diffusion equations with fractional diffusion[END_REF]:

v t = -(-∆ x

Theorem 2 . 2

 22 Under the assumptions (1.3) and (1.4), equation (1.1) does not admit any transition front connecting 0 and 1.

  •) a.e. in R for every t > 0, (3.1) where u and v denote the solutions of the Cauchy problem (1.2) with initial conditions u 0 and v 0 := u 0 (• + h). The property (3.1) is satisfied by the previous examples of dispersal operators Du: fractional Laplacian Du = -(-∆ x ) α u, the convolution operator Du = J * u -u, the nonlinear fast diffusion Du = (u γ ) xx and the nonlinear fractional diffusion Du = -(-∆ x ) α (u γ ). If follows from assumptions (1.3) and (3.1)

Theorem 3 . 1

 31 Assume that (1.1) satisfies (1.3), (1.4) and (3.1) and let u be the solution of the Cauchy problem (1.2) with a nonincreasing initial condition u 0 ∈ L ∞ (R; [0, 1])\{0} such that m(t) → 0 and M (t) → 1 as t → +∞. Then, for every 0 < a < b < 1, lim sup t→+∞ x a (t) -x b (t)) = +∞. (3.3)

Remark 3 . 4

 34 (0,cnδ) u(T + nδ, •) > a + b 2 for n ∈ N large enough, whence x (a+b)/2 (T + nδ) ≥ cnδ for n large enough, since u(T + nδ, •) is nonincreasing. As a consequence, lim inf n→+∞ x (a+b)/2 (T + nδ) nδ ≥ c and, since c > 0 is arbitrary, it follows that x (a+b)/2 (T + nδ)/(nδ) → +∞ as n → +∞. This contradicts (3.8) and the proof of Theorem 3.1 is thereby complete. Consider here the case of equation (1.5) with the fractional Laplacian Du = -(-

  ) α v and v t = J * v -v which are treated with fast Fourier transform techniques and w t = w (1 -w) which solution can be computed explicitly; (c and d): the time-dependent finite element solver of Comsol Multiphysics c .

Figure 1 :

 1 Figure 1: Propagation to the right of the solution of the Cauchy problem (1.2), at successive times t = 0, 1, . . . , 20, with (a) Du = -(-∆ x ) α u with α = 0.9; (b) Du = J * u -u with J(x) = exp(-|x|)/4; (c) Du = (u γ ) xx with γ = 1/2; and (d) Du = u xx . In all cases, the initial condition was u 0 (x) = exp(-x 2 /100) and the function f was of the KPP type f (u) = u (1 -u).

Figure 2 :

 2 Figure 2: Distance x 0.4 (t) -x 0.6 (t) between two level sets of the solution of the Cauchy problem (1.2), for t ∈ (0, 20), with: (solid line) Du = -(-∆ x ) α u with α = 0.9; (dashed line) Du = J * u -u with J(x) = exp(-|x|)/4; (dash-dot line) Du = (u γ ) xx with γ = 1/2; and (circles) Du = u xx . The assumptions on f and u 0 are the same as in Fig. 1.

  Finally, notice that 1 ≥ u(0, •) ≥ (2/3)1 (-∞,ξ - 0 ) a.e. in R by (2.2), where 1 E denotes the characteristic function of a measurable set E. The comparison principle (1.3) implies that, for every t > 0, 1 ≥ u(t, •) ≥ v(t, •) a.e. in R, +∞, which contradicts (2.5). The proof of Theorem 2.2 is thereby complete.

	3 t have a positive measure, one infers from (2.4) that for all x ∈ A -t and u(t + δ, x) > 1 4 for all x ∈ A + t . u(t, x) ≥ 3 4 3 4 for a.Since, for every t ≥ 0, the sets A ± it follows that u(t + δ, x) <      ξ -t + 1 > ξ t+δ -M and ξ + t -1 < ξ t+δ + M. It follows then from the spreading property (1.4) that ess inf (0,ct) v(t, •) → 1 as t → +∞ for , (2.4) 0 ) . every c > 0, whence ess inf (0,ct) u(t, •) → 1 as t → +∞ for every c > 0. For any given c > 0, property (2.3) then yields ξ + t ≥ ct for all t > 0 large enough, whence lim inf t→+∞ ξ + t /t ≥ c. Therefore, lim inf t→+∞ ξ t t ≥ c for all c > 0, Remembering that the quantities (ξ where v denotes the solution of the Cauchy problem (1.2) with initial condition (2/3)1 (-∞,ξ -since (ξ

On the other hand, from (2.1), there is M ≥ 0 such that, for every t ∈ R, e. x ∈ (-∞, ξ t -M ),

u(t,

x) ≤ 1 4 for a.e. x ∈ (ξ t + M, +∞). t -ξ ± t ) t∈R are bounded, one gets that sup t≥0 |ξ t+δ -ξ t | < +∞. (2.5) t -ξ + t ) t∈R is bounded. As a conclusion, ξ t /t → +∞ as t →

* This work has been carried out in the framework of Archimède LabEx (ANR-11-LABX-0033) and of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the "Investissements d'Avenir" French Government program managed by the French National Research Agency (ANR). The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n.321186 -ReaDi -Reaction-Diffusion Equations, Propagation and Modelling, and from the ANR project NONLOCAL (ANR-14-CE25-0013).


Maria Melchior
email: maria.melchior@inserm.fr

Robleh Hersi

Judith Van Der Waerden

Béatrice Larroque

Marie-Josèphe J Saurel-Cubizolles

Aude Chollet

Cédric Galéra

Eden Mother

I Annesi-Maesano

J Botton

M A Charles

P Dargent- Molina

B De Lauzon-Guillain

P Ducimetière

M De Agostini

B Foliguet

A Forhan

X Fritel

A Germa

V Goua

R Hankard

B Heude

M Kaminski

N Lelong

J Lepeule

G Magnin

L Marchand

C Nabet

R Slama

M Schweitzer

O Thiebaugeorge

Maternal tobacco smoking in pregnancy and children's socio-emotional development at age 5: the EDEN mother-child birth cohort study

Keywords: tobacco use, pregnancy, ADHD, cohort study cigarette, ADHD symptoms, addiction, mental health

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

Introduction

Maternal tobacco use in pregnancy can have negative effects on children's birth characteristics and perinatal health (ex. increased risk of low birth weight and prematurity) [START_REF] Andres | Perinatal complications associated with maternal tobacco use[END_REF][START_REF] Iniguez | Maternal smoking during pregnancy and fetal biometry: the INMA Mother and Child Cohort Study[END_REF]. Moreover, children whose mothers smoke appear to have elevated levels of behavioral and substance-related problems [START_REF] Ernst | Behavioral and neural consequences of prenatal exposure to nicotine[END_REF][START_REF] Roza | Maternal smoking during pregnancy and child behaviour problems: the Generation R Study[END_REF][START_REF] Gaysina | Maternal Smoking During Pregnancy and Offspring Conduct Problems: Evidence From 3 Independent Genetically Sensitive Research Designs[END_REF][START_REF] Hellstrom-Lindahl | Smoking during pregnancy: a way to transfer the addiction to the next generation?[END_REF]. In particular, research has consistently shown high rates of symptoms of hyperactivity/inattention and Attention Deficit and Hyperactivity Disorder (ADHD) [START_REF] Linnet | Smoking during pregnancy and the risk for hyperkinetic disorder in offspring[END_REF][START_REF] Thapar | Do prenatal risk factors cause psychiatric disorder? Be wary of causal claims[END_REF][START_REF] Thapar | Prenatal smoking might not cause attention-deficit/hyperactivity disorder: evidence from a novel design[END_REF][START_REF] Obel | Is maternal smoking during pregnancy a risk factor for hyperkinetic disorder?--Findings from a sibling design[END_REF][START_REF] Langley | Maternal and paternal smoking during pregnancy and risk of ADHD symptoms in offspring: testing for intrauterine effects[END_REF][START_REF] Keyes | Associations of prenatal maternal smoking with offspring hyperactivity: causal or confounded?[END_REF]. The question of whether this association is causal has been widely debated [START_REF] Thapar | Do prenatal risk factors cause psychiatric disorder? Be wary of causal claims[END_REF] and findings from genetically informative studies suggest that this is probably not the case: rather, high levels of hyperactivity/inattention in children of smoking mothers could reflect the intergenerational transmission of a genetic vulnerability to behavioral difficulties or incomplete control for confounding factors such as socioeconomic position [START_REF] Thapar | Prenatal smoking might not cause attention-deficit/hyperactivity disorder: evidence from a novel design[END_REF][START_REF] Langley | Maternal and paternal smoking during pregnancy and risk of ADHD symptoms in offspring: testing for intrauterine effects[END_REF][START_REF] Skoglund | Familial confounding of the association between maternal smoking during pregnancy and ADHD in offspring[END_REF].

Nonetheless, past research on this topic had several limitations. First, the extent of genetic confounding is yet uncertain. Studies conducted among siblings discordant with regard to maternal tobacco use show no effect of this exposure on children's behavior [START_REF] Obel | Is maternal smoking during pregnancy a risk factor for hyperkinetic disorder?--Findings from a sibling design[END_REF][START_REF] Skoglund | Familial confounding of the association between maternal smoking during pregnancy and ADHD in offspring[END_REF], however these results could be biased due to sample selection and an uneven distribution of confounders across siblings [START_REF] Frisell | Sibling comparison designs: bias from nonshared confounders and measurement error[END_REF]. In twin studies, estimates of heritability could be inflated because of gene x environment interactions that are not entirely taken into account. Moreover, if genetic vulnerability plays a key role, one expects paternal tobacco smoking to be associated with children's behavior; however this is not always the case [START_REF] Roza | Maternal smoking during pregnancy and child behaviour problems: the Generation R Study[END_REF][START_REF] Thapar | Prenatal smoking might not cause attention-deficit/hyperactivity disorder: evidence from a novel design[END_REF][START_REF] Langley | Maternal and paternal smoking during pregnancy and risk of ADHD symptoms in offspring: testing for intrauterine effects[END_REF][START_REF] Keyes | Associations of prenatal maternal smoking with offspring hyperactivity: causal or confounded?[END_REF][START_REF] Brion | Maternal smoking and child psychological problems: disentangling causal and noncausal effects[END_REF]. Additionally, the likelihood of symptoms of hyperactivity/inattention in the offspring should only moderately vary with the timing of maternal smoking (pre, during or post-pregnancy). Yet studies on this topic have reported elevated levels of behavioral problems only in children exposed to tobacco smoking in utero, including in historical times when maternal smoking in pregnancy was widespread [START_REF] Keyes | Associations of prenatal maternal smoking with offspring hyperactivity: causal or confounded?[END_REF]. Second, due to reliance on small samples [START_REF] Linnet | Smoking during pregnancy and the risk for hyperkinetic disorder in offspring[END_REF][START_REF] Thapar | Prenatal smoking might not cause attention-deficit/hyperactivity disorder: evidence from a novel design[END_REF], merged datasets [START_REF] Gaysina | Maternal Smoking During Pregnancy and Offspring Conduct Problems: Evidence From 3 Independent Genetically Sensitive Research Designs[END_REF][START_REF] Obel | Is maternal smoking during pregnancy a risk factor for hyperkinetic disorder?--Findings from a sibling design[END_REF] or data from birth registers [START_REF] Obel | Is maternal smoking during pregnancy a risk factor for hyperkinetic disorder?--Findings from a sibling design[END_REF], prior studies have not always accounted for all relevant covariates, particularly social and economic factors. Third, with rare exceptions [START_REF] Brion | Maternal smoking and child psychological problems: disentangling causal and noncausal effects[END_REF], the specificity of the association between maternal smoking in pregnancy and symptoms of hyperactivity/ inattention in the offspring has not been thoroughly studied [START_REF] Obel | Is maternal smoking during pregnancy a risk factor for hyperkinetic disorder?--Findings from a sibling design[END_REF].

Research primarily examined children's conduct problems or overall behavior [START_REF] Roza | Maternal smoking during pregnancy and child behaviour problems: the Generation R Study[END_REF] while a systematic comparison between different types of difficulties that children may experience could help make inferences about underlying mechanisms.

In the present study, using data from the French EDEN mother-child birth cohort study, we examine the hypothesis that maternal tobacco smoking in pregnancy is associated with children's behavior at age 5 years. To account for limitations of past research, we analyze the data: 1) using negative controls (paternal smoking), 2) comparing different time periods of exposure to maternal smoking (pre-, during and post-pregnancy), 3) comprehensively assessing children's social and family environment, 4) using inverse probability weights (IPWs) to account for covariates, and 5) studying multiple aspects of children's behavior.

Methods

Study population

The EDEN mother-child cohort study ('Etude sur les Déterminants pré et postnatals précoces du développement psychomoteur et de la santé de l'ENfant') aims to examine pre-and postnatal determinants of child growth, development and health. Pregnant women were recruited in two French university hospitals (Nancy and Poitiers, 2003-2005) before 24 weeks of gestation. Exclusion criteria were: multiple pregnancies, pre-pregnancy diabetes, illiteracy, and planned move outside the region in the following 3 years. 2,002 women were initially included during pregnancy and 1,893 agreed to have their newborns take part in the study. Average age, proportion of unmarried couples, and of children born with low birth weight in the EDEN study were comparable to national estimates, but maternal educational level was somewhat higher [START_REF] Blondel | Trends in perinatal health in France between 1995 and 2010: Results from the National Perinatal Surveys[END_REF]. Over the follow-up period, attrition rates were highest in families in which the mother was young, had low educational level and low income, did not live with the child's father, smoked tobacco or had psychological difficulties in pregnancy, and whose child did not have low birth weight.

Since study inception, there have been eight waves of data collection (pregnancy, the child's birth, 4, 8, 12, 24, 36 months and 5 years). Data were collected by trained interviewers (pregnancy, birth, 36 months), by mothers' and fathers' self-reports at all study waves, and from medical records (pregnancy, birth).

The EDEN cohort received approval from the Bicêtre Hospital ethics committee (CCPPRB) and the Commission Nationale Informatique et Libertés (CNIL) overseeing ethical data collection in France.

Measures

Maternal tobacco smoking

Mothers reported on their tobacco smoking before and during pregnancy on the initial EDEN study questionnaire at 24-28 weeks of gestation ('Did you smoke in the three months prior to pregnancy?', 'Did you smoke in the first trimester of pregnancy?', 'Do you smoke now, and if yes how many cigarettes per day?') and once the child was born ('Did you smoke prior to pregnancy?', 'Did you smoke in the third trimester of pregnancy, and if yes how many cigarettes per day?', 'If you quit smoking during pregnancy, when did you quit?'). Based on this information, we created a fourcategory variable assessing maternal tobacco use: non-smoker (reference category), quit smoking prior to pregnancy, smoked in the first trimester of pregnancy (>=1 cigarette/day), smoked throughout pregnancy (>=1 cigarette/day). In additional analyses, we divided the group of mothers who quit smoking prior to pregnancy into two subgroups depending on whether they smoked during follow-up.

Children's socio-emotional development

Children's socio-emotional development was ascertained using the Strengths and Difficulties Questionnaire (SDQ) completed by the mothers when children were 5 years of age [START_REF] Goodman | The Strengths and Difficulties Questionnaire: a research note[END_REF][START_REF] Marzocchi | The use of the Strengths and Difficulties Questionnaire (SDQ) in Southern European countries[END_REF]. This validated instrument includes 25 items on 5 dimensions of children's behaviour: emotional symptoms, conduct problems, hyperactivity/inattention, peer relationship problems and prosocial behaviour. Each item is scored 0 (never), 1 (sometimes true) or 2 (certainly true) and items composing each subscale are summed to generate a score ranging from 0 to 10. Following prior research, children's scores on all SDQ subscales were dichotomized at the 85 th percentile, to study a high, potentially clinically significant, level of symptoms [START_REF] Amone-P'olak | Socioeconomic position and mental health problems in pre-and early-adolescents : The TRAILS study[END_REF].

Covariates

Covariates included were all potentially associated with children's socio-emotional development: characteristics of parental smoking, characteristics of children, mothers and families measured at study baseline or longitudinally, that is combining all measures obtained across the eight waves of EDEN study follow-up.

Parental smoking, other than maternal tobacco use before and during pregnancy, was ascertained through maternal smoking post-pregnancy (yes vs. no), paternal smoking in pregnancy (yes vs. no), and paternal smoking post-pregnancy (yes vs. no).

Children's characteristics included the child's sex (male vs. female), premature (<37 weeks gestation) birth (yes vs. no), birth weight (<=2500 vs. >2500 g) ascertained from the child's medical records, and duration of breastfeeding (< 3 vs. >=3 months) ascertained from maternal reports up to the child's 1st birthday [START_REF] Betoko | Infant feeding patterns over the first year of life: influence of family characteristics[END_REF].

Maternal characteristics included age at the child's birth (<30 vs. >=30 years), psychological difficulties in pregnancy ascertained via high symptoms of depression measured with the CES-D (score >=16) [START_REF] Radloff | The CES-D scale : a self report depression scale for research in the general population[END_REF] and/or high anxiety measured with the STAI [START_REF] Spielberger | State-Trait Anxiety Inventory (Form Y)[END_REF] (yes vs. no), maternal depression post-pregnancy measured with the EPDS at 4, 8, and 12 months post-partum [START_REF] Cox | Validation of the Edinburgh Postnatal Depression Scale (EPDS) in non-postnatal women[END_REF] and with the CES-D at 36 months and 5 years [START_REF] Radloff | The CES-D scale : a self report depression scale for research in the general population[END_REF] (yes vs. no), maternal alcohol use in pregnancy (>=1 vs. <1 glass of alcohol/week).

Family characteristics included study center (Poitiers vs. Nancy), parents' educational attainment ascertained as highest of mother's and father's (< vs. >= any higher education), family income (<1500 vs. >=1500 euros/month, which corresponds to the lowest quartile of the distribution in our study population [START_REF] Melchior | Maternal depression, socioeconomic position, and temperament in early childhood: the EDEN motherchild cohort[END_REF]), parental separation between pregnancy and the age 5 assessment (yes vs. no), number of children in the family (>=1 vs. 0), and negative life events from pregnancy to the age 5 assessment : death of a close person, death of a child, home eviction, fire or other major damage to the home, serious parental health problems, parental unemployment, legal problems, maternal exposure to domestic or sexual violence (yes vs. no).

Statistical analysis

Our aim was to examine the association between maternal tobacco smoking in pregnancy and children's behavioural difficulties. Analyses were based on the complete study sample from pregnancy to age 5 years (n=1,113). First, we conducted logistic regression analyses adjusted for children's sex and study centre. Second, to control for factors which covary with maternal smoking in pregnancy and can bias the association with children's behaviour, we calculated Inverse Probability Weights (IPW) of exposure [START_REF] Hernan | IP weighting and marginal structural models[END_REF] based on all covariates potentially associated with our study outcome [START_REF] Austin | An introduction to propensity score methods for reducing the effects of confounding in observational studies[END_REF]. This method renders study groups comparable on all measured covariates, thereby mimicking the random allocation of exposure. Third, we conducted logistic regression analyses testing the association between maternal tobacco smoking in pregnancy and children's behaviour weighting the analysis on IPWs. Fourth, we tested the association between the average number of cigarettes smoked daily by the mother and children's behaviour. Fifth, we tested for interactions with the child's sex, parental educational attainment and income level. Sixth, we calculated the fraction of cases of high symptoms of hyperactivity/inattention potentially attributable to maternal smoking in pregnancy, using the following formula: Attributable Fraction=[OR-1]/[n° exposed cases/n°cases] [START_REF] Hanley | A heuristic approach to the formulas for population attributable fraction[END_REF].

All analyses were conducted using SAS 9.2 (Carey, NC).

Results

Characteristics of EDEN cohort children and families according to maternal tobacco use in pregnancy

are shown in Table 1. Overall, 7.8% of study mothers smoked only during the first trimester of pregnancy and 12.9% throughout pregnancy. 30.5% of mothers smoked post-pregnancy, while 36.0% of fathers smoked in pregnancy and 43.3% post-pregnancy. Factors associated with maternal tobacco use in pregnancy include maternal smoking post-pregnancy (p<0.0001), paternal smoking in pregnancy (p<0.0001), paternal smoking post-pregnancy (p<0.0001), duration of breastfeeding (p<0.0001), maternal age (p<0.0001), maternal psychological difficulties (p=0.0006), maternal depression post-pregnancy (p=0.04), maternal alcohol use (p<0.0001), study center (p=0.0006), parents' educational attainment (p<0.0001), family income (p<0.0001), parental separation (p=0.0003) and negative life events (p=0.0007).

Table 2 shows associations between maternal smoking in pregnancy and children's behaviour at age 5, controlling for children's sex and study centre. Maternal smoking in pregnancy was only associated with children's symptoms of hyperactivity/inattention (tobacco smoking during the 1st trimester of pregnancy: OR=1.95, 95% CI 1.13-3.38; tobacco smoking throughout pregnancy: OR=2.11, 95% CI 1.36-3.27).

Table 3 shows associations between all potential covariates and children's symptoms of hyperactivity/inattention. Table 4 shows IPW-controlled associations between maternal tobacco smoking in pregnancy and children's symptoms of hyperactivity/inattention. The OR associated with maternal smoking during the 1 st trimester of pregnancy was reduced compared to the sex and study center-adjusted model and became statistically non-significant (OR=1.25, 95% CI 0.58-2.68). However, the OR associated with maternal tobacco smoking throughout pregnancy was marginally higher and remained statistically significant (OR=2.20, 95% CI 1.21-4.00).

In additional analyses, we observed a dose-response association between maternal tobacco smoking and children's levels of hyperactivity/inattention, however, due to a small sample size, the ORs associated with different levels of maternal smoking were not statistically significant (1-9 cigarettes/ day: IPW-adjusted OR=1.49, 95% CI 0.52-4.21; >=10 cigarettes/ day: IPW-adjusted OR=1.64, 0.53-4.64). The association between maternal smoking and children's symptoms of hyperactivity/ inattention did not vary with the child's sex, parental educational attainment or family income. Overall, approximately 13% of cases of high symptoms of hyperactivity/inattention in children were potentially attributable to maternal smoking in pregnancy.

Discussion

Our findings suggest that maternal smoking during pregnancy may specifically contribute to symptoms of hyperactivity/inattention in children, which differs from the results of studies conducted among siblings [START_REF] Obel | Is maternal smoking during pregnancy a risk factor for hyperkinetic disorder?--Findings from a sibling design[END_REF][START_REF] Skoglund | Familial confounding of the association between maternal smoking during pregnancy and ADHD in offspring[END_REF] or among children born using assisted reproduction technologies [START_REF] Thapar | Prenatal smoking might not cause attention-deficit/hyperactivity disorder: evidence from a novel design[END_REF].

However, these genetically-informative designs may suffer from bias due to participants' selection and the uneven distribution of confounders among the exposed and the non-exposed [START_REF] Frisell | Sibling comparison designs: bias from nonshared confounders and measurement error[END_REF], thus the jury is still out on the nature of the association between maternal tobacco use in pregnancy and children's behavior [START_REF] Keyes | Commentary: Smoking in pregnancy and offspring health: early insights into family-based and 'negative control' studies?[END_REF]. In the present study, we found a unique association between maternal smoking in pregnancy and children's behaviour, even after accounting for maternal smoking postpregnancy, paternal smoking and multiple characteristics of children and their families using the propensity score technique which is more robust than traditional methods to control for covariates.

Our study contributes to research suggesting that maternal smoking during pregnancy may directly influence the likelihood of symptoms of hyperactivity/inattention in the offspring.

Strengths and limitations

Key strengths of our study are: 1) a large size community based sample with prospective follow-up; 2) repeated assessments of multiple characteristics of children and their families from pregnancy to age 5; 3) the use of propensity scores -a statistical method designed to account for confounding as well as selection effects. Nevertheless, we also need to acknowledge limitations. First, families who did not take part in the follow-up were more likely to experience socioeconomic difficulties, therefore the relationship between maternal smoking in pregnancy and children's behaviour may be underestimated. Second, symptoms of hyperactivity/inattention were reported by children's mothers and could suffer from information bias [START_REF] Weissman | Assessing psychiatric disorders in children. Discrepancies between mothers' and children's reports[END_REF]. Maternal reports of high levels of hyperactivity/inattention are probably valid; nonetheless, future studies should aim to use multiple sources of information (ex. mother, father or teacher, and child him/herself) [START_REF] Weiler | Mother and teacher reports of ADHD symptoms: DSM-IV Questionnaire data[END_REF] or study clinical diagnoses of ADHD. Third, tobacco use in pregnancy was ascertained by the mothers and could be underreported, which again would bias our results towards the null. Future studies should consider ascertaining maternal tobacco use through questionnaires as well as biological measures. Fourth, we did not assess potential confounders such as paternal psychopathology, use of substances other than tobacco and genetic/biological factors; the inclusion of these elements in future investigations can help disentangle the complex processes implicated in the etiology of hyperactivity/inattention.

Putative mechanisms

The association between maternal tobacco smoking in pregnancy and children's symptoms of hyperactivity/inattention may reflect several mechanisms. First, mothers who do not quit smoking in pregnancy probably have especially high levels of nicotine dependence and can transmit this heightened vulnerability to addiction and behavioral difficulties to their children [START_REF] Mcclernon | ADHD and smoking: from genes to brain to behavior[END_REF]. We found no association between paternal smoking and children's behavior, which goes against the hypothesis of genetic confounding, yet fathers are much less likely than mothers to reduce tobacco use before a child's birth and genetic factors may play a lesser role with regard to paternal than maternal smoking.

Second, maternal smoking in pregnancy may be a marker of environmental confounding by factors such as maternal psychological difficulties related both to a decreased ability to quit smoking (despite warnings and known fetus toxicity) and less positive parenting after the child's birth. For example, high levels of maternal smoking in pregnancy have been associated with mothers' reduced feelings of attachment towards the fetus [START_REF] Magee | The relationship between maternal-fetal attachment and cigarette smoking over pregnancy[END_REF] and worse mother-child attachment in infancy [START_REF] Chittleborough | Prenatal prediction of poor maternal and offspring outcomes: implications for selection into intensive parent support programs[END_REF]. This lower quality of mother-child attachment could in turn contribute to later ADHD symptoms [START_REF] Storebo | Association between insecure attachment and ADHD: environmental mediating factors[END_REF]. A recent review [START_REF] Massey | Psychological differences between smokers who spontaneously quit during pregnancy and those who do not: a review of observational studies and directions for future research[END_REF] reports that compared to persistent smokers, women who spontaneously quit during pregnancy show a more secure pattern of attachment to their infant and higher levels of parenting competence, implying that the ability to quit smoking may in part reflect the mother's investment in her pregnancy and child. Failure in quitting appears to be associated with maternal conduct disorder but not ADHD, again going against the hypothesis that the association between maternal tobacco smoking and children's symptoms of hyperactivity/inattention is solely due to genetic factors. Third, maternal tobacco smoking may have direct teratological effects on fetal brain development in utero and immediately after birth [START_REF] Cornelius | Developmental consequences of prenatal tobacco exposure[END_REF][START_REF] Dwyer | The dynamic effects of nicotine on the developing brain[END_REF]. These effects could occur through the impact of tobacco on cerebral structures and connectivity, neurotransmission, as well as neuronal differentiation and migration [START_REF] Dwyer | The dynamic effects of nicotine on the developing brain[END_REF][START_REF] Bublitz | Maternal smoking during pregnancy and offspring brain structure and function: review and agenda for future research[END_REF][START_REF] Heath | Cortico-thalamic connectivity is vulnerable to nicotine exposure during early postnatal development through alpha4/beta2/alpha5 nicotinic acetylcholine receptors[END_REF]. In addition, tobacco products and nicotine could impair placental function, resulting in chronic fetal hypoxia and subsequent fetal growth retardation [START_REF] Herrmann | Prenatal tobacco smoke and postnatal secondhand smoke exposure and child neurodevelopment[END_REF][START_REF] Jauniaux | Morphological and biological effects of maternal exposure to tobacco smoke on the feto-placental unit[END_REF]. Fourth, growing evidence suggests that prenatal tobacco smoking may have epigenetic effects both on fetal and placental tissues [START_REF] Suter | Genetic and epigenetic influences associated with intrauterine growth restriction due to in utero tobacco exposure[END_REF][START_REF] Suter | Maternal tobacco use modestly alters correlated epigenome-wide placental DNA methylation and gene expression[END_REF][START_REF] Maccani | Maternal cigarette smoking during pregnancy is associated with downregulation of miR-16, miR-21, and miR-146a in the placenta[END_REF].

Implications

Tobacco smoking during pregnancy is one of the most common and preventable risk factors of poor pregnancy and infant outcomes. Quitting smoking has many health benefits for the mother and her child. Importantly, our findings suggest that smoking in the first trimester only is not associated with children's behavioral difficulties, underlining the potential benefits of smoking cessation during the course of pregnancy. However, it is still possible that smoking during pregnancy is a marker of biological or psychological risk which is transmitted intergenerationally. This implies that health professionals should screen for these factors in women who do not quit smoking during pregnancy in order to provide adequate support, potentially targeting characteristics such as insecure attachment, parenting difficulties, maternal conduct disorder, or parental substance use disorders. Early action on these characteristics could positively influence a range of outcomes in the offspring among which symptoms of ADHD. As suggested by Massey et al. [START_REF] Massey | Psychological differences between smokers who spontaneously quit during pregnancy and those who do not: a review of observational studies and directions for future research[END_REF], among pregnant women who smoke tobacco, efforts could be dedicated not only to smoking cessation but also to the development of psychoeducational and parenting interventions. We are not able to entirely reject the possibility that genetic factors play a role in this association, either directly or via epigenetic processes, nonetheless, maternal smoking in pregnancy may also have effects on children's behavior, through physiological mechanisms which are not yet fully uncovered. Future etiological research on the causes of ADHD, should integrate measures of early environmental, biological, psychological and social processes provided to the child by the mother and her surroundings.

Conclusions

Maternal tobacco smoking in pregnancy may contribute directly or through epigenetic mechanisms to children's symptoms of hyperactivity/inattention.

Table 1 . Characteristics of mothers and children participating in the EDEN mother-child birth cohort study according to maternal smoking in pregnancy (n=1113, %) Maternal smoking in pregnancy

 1

			Quit smoking	Smoked during	Smoked
	Parental tobacco smoking	Non-smoker (n=777)	prior to pregnancy	the 1 st trimester of pregnancy	throughout pregnancy
			(n=105)	(n=87)	(n=144)
	Maternal smoking post pregnancy: No	91.7	31.4	29.1	3.5
	Yes	8.3	68.6	70.9	96.5
	Missing data n=11		p<0.0001
	Paternal smoking in pregnancy: No	74.8	50.0	41.0	27.9
	Yes	25.2	50.0	59.0	72.1
	Missing data n=74		p<0.0001
	Paternal smoking post pregnancy: No	68.4	35.0	37.2	20.0
	Yes	31.6	65.1	62.8	80.0
	Missing data n=14		p<0.0001
	Children's characteristics
	Sex : Female	46.6	48.6	48.3	47.9
	Male	53.4	51.4	51.7	52.1
	Missing data n=0		p=0.97
	Born premature : No	94.0	93.3	97.7	95.8
	Yes	6.1	6.7	2.3	4.2
	Missing data n=0		p=0.41
	Birthweight : >2500 g	95.4	95.2	95.4	94.4
	<=2500 g	4.6	4.8	4.6	5.6
	Missing data n=0		p=0.97
	Duration of breastfeeding : >=3 months	60.4	57.1	50.6	40.3
	< 3 months	39.6	42.9	49.4	59.7
	Missing data n=0		p<0.0001
	Maternal characteristics
	Maternal age at child's birth :<30 years	41.6	61.9	55.2	53.5
	>=30 years	58.4	38.1	44.8	46.5
	Missing data n=0		p<0.0001
	Maternal psychological difficulties in pregnancy: No	81.4	69.5	68.2	71.8
	Yes	18.7	30.5	31.8	28.2
	Missing data n=9		p=0.0006
	Maternal depression post pregnancy: No	80.1	77.1	71.3	70.8
	Yes	20.0	22.9	28.7	29.2
	Missing data n=0		p=0.04
	Maternal alcohol use in pregnancy: <1 glass/week	83.1	75.0	60.9	69.7
	>=1 glass/week	17.0	25.0	39.1	30.3
	Missing data n=274		p<0.0001
	Family characteristics
	Study centre: Nancy	48.8	59.1	42.5	34.0
	Poitiers	51.2	41.0	57.5	66.0
	Missing data n=0		p=0.0006

Table 2 . Maternal tobacco smoking in pregnancy and children's emotional and behavioral symptoms at age 5 years (EDEN mother-child birth cohort study, n=1113, logistic regression models adjusted for children's sex and study center, OR, 95% CI)

 2

	High emotional	High conduct problems	High symptoms of	High peer relationship	Low prosocial
	symptoms	(174/1112)	hyperactivity/	problems	behavior
	(238/1113)		Inattention	(160/1112)	(365/1110)
			(176/1113)

Table 3 . Characteristics of mothers and children participating in the EDEN mother-child birth cohort study in relation to high symptoms of hyperactivity/inattention (n=1113, OR, 95% CI)

 3

	N	ORs, 95% CI , p-value

66 (1.19-2.32)

	Parents' educational attainment : >=Higher education	1105	1
	< Higher education		2.06 (1.48-2.86)
	Family income : >1500 euros/month	1100	1
	<=1500 euros/month		2.29 (1.64-3.20)
	Parental separation: No	1094	1
	Yes		1.59 (1.03-2.44)
	Number of siblings: None	1113	1
	>=1		0.65 (0.47-0.90)
	Negative life events: No	1111	1
	Yes		1.62 (1.15-2.28)

Table 4 . Maternal tobacco smoking in pregnancy and children's high symptoms of hyperactivity/inattention at age 5 years (EDEN mother-child birth cohort study, n=997, Inverse probability weights-controlled ORs, 95% CI) a a

 4 Multivariate analyses are controlled for inverse probability weights (IPW) based on children's sex, premature birth, birth weight, duration of breastfeeding, maternal age at birth, maternal psychological difficulties in pregnancy, maternal depression post-pregnancy, maternal alcohol use in pregnancy, maternal smoking post pregnancy, paternal tobacco smoking during and post pregnancy, study center, parents' educational attainment, family income, parental separation, number of siblings, negative life events.

	High symptoms of
	hyperactivity/
	inattention

Funding and other support

The authors thank the EDEN study team for help in implementing the study. This work was supported by: Fondation pour la Recherche Médicale (FRM); French Ministry of Research: IFR Program; INSERM Human Nutrition National Research Program; Diabetes National Research Program (through a collaboration with the French Association of Diabetic Patients (AFD)); French Ministry of Health; French Agency for Environment Security (AFSSET); French National Institute for Population Health Surveillance (InVS); Paris-Sud University; French National Institute for Health Education (INPES); Nestlé; Mutuelle Générale de l'Education Nationale (MGEN); French speaking association for the study of diabetes and metabolism (ALFEDIAM); National Agency for Research (ANR nonthematic programme); and National Institute for Research in Public Health (IRESP: TGIR Cohorte Santé 2008 programme). Maria Melchior is the recipient of a Young Researcher Award from the French National Research Agency (ANR). The founders had no further role in study design; in the collection, analysis and interpretation of data; in the writing of the report; or in the decision to submit the paper for publication.

Conflict of interest

The authors declare no conflicts of interest.


MD Amedeo Anselmi
email: amedeo.anselmi@alice.it

MD Vito Giovanni Ruggieri

MD PhD Sophie Mascle +

MD Vincent Auffret +

MD Hervé Le Breton +

MD Xavier Beneux

MD Jean-Philippe Verhoye

Is the EuroSCORE II best suited for Reoperative Risk Estimation in Patients with Structural Deterioraion of Aortic Bioprostheses?

Keywords: Valve implantation, Operative risk, Structural valve deterioration

Introduction

The transcatheter aortic valve implantation (TAVI) is a feasible option for patients with severe aortic valve disease and deemed to have excessive surgical risk on the basis of preoperative comorbidities. Generally, the population of patients referred for TAVI reflects a logistic EuroSCORE I >20%, a STS score >10% and an EuroSCORE II >7% for perioperative mortality [START_REF] Durand | Performance analysis of EuroSCORE II compared to the original logistic EuroSCORE and STS Scores for predicting 30-day mortality after transcatheter aortic valve replacement[END_REF].The use of risk prediction systems has gained growing importance for the systematic decision for either strategy (surgery or TAVI) in the individual patients, although several other factors are taken into consideration [START_REF] Zahn | Transcatheter aortic valve implantation : first results from a multi-center real-world registry[END_REF].

Nonetheless, the reliability of scoring systems in the prediction of operative mortality for aortic valve replacement (AVR) has been largely questioned. The EuroSCORE I models tend to overestimate the risk of primary AVR and to be inappropriate for the definition of candidates to TAVI [START_REF] Brown | Is the European System for Cardiac Operative Risk Evaluation model valid for estimating the operative risk of patients considered for percutaneous aortic valve replacement?[END_REF]. Such overestimation is particularly evident in the 'high risk' cases, i.e. the subgroup in which we need more frequently to make a decision among alternative strategies [START_REF] Osswald | Overestimation of aortic valve replacement risk by EuroSCORE: implications for percutaneous valve replacement[END_REF]. The EuroSCORE II is reportedly more reliable than previous versions in the prediction of mortality in primary aortic valve replacement.

In such a scenario, the feasibility of the Valve-in-Valve procedure [START_REF] Piazza | Transcatheter aortic valve implantation for failing surgical aortic bioprosthetic valve. From concept to clinical application and evaluation (Part 1)[END_REF][START_REF] Piazza | Transcatheter aortic valve implantation for failing surgical aortic bioprosthetic valve. From concept to clinical application and evaluation (Part 2)[END_REF] in patients with structural degeneration (SVD) of aortic valve bioprosthesis imposes an even more complex decision-making process. The Valve-in-Valve technique consists in delivering a transcatheter valve within a failing bioprosthesis (Figure 1). In fact, still limited data are available over the immediate results of this approach and population-specific variables exist (i.e. different mechanisms of bioprosthetic failure, characteristics of the primary surgery). Additionally, the risk of reoperation on the aortic valve has considerably decreased during the recent years [START_REF] Langanay | Current hospital mortality of aortic valve replacement in octogenarians[END_REF][START_REF] Ruggieri | Logistic EuroSCORE I risk analysis in aortic valve reoperations after bioprosthetic replacement[END_REF][START_REF] Jamieson | Reoperation for bioprosthetic aortic structural failure -risk assessment[END_REF], and this may further challenge the validity of risk assessment.

Hypothesis

We postulate that the EuroSCORE II performs better than other risk algorithms in patients undergoing reoperation for SVD of an aortic bioprosthesis. The limits of the EuroSCORE I in these individuals have been recently evidenced [START_REF] Ruggieri | Logistic EuroSCORE I risk analysis in aortic valve reoperations after bioprosthetic replacement[END_REF].

Evaluation of the Hypothesis

Preliminary testing of the hypothesis was based on a comparison of the performance of three risk prediction systems (logistic EuroSCORE I, STS Score and EuroSCORE II) within a population of patients reoperated for SVD in a single center. A prospective database is conducted in our Institution for all patients undergoing cardiac surgery since 1975, and includes both pre-and intraoperative characteristics, as far as the early postoperative results. In May 2012, the database was queried in order to identify the records of patients who received reoperation on the aortic valve due to a diagnosis of structural valve deterioration (SVD) of a previously implanted bioprosthesis (index reoperation). The performance of any associated procedure to AVR at the time of primary operation did not represent an exclusion criterion. Conversely, we excluded all individuals whose primary indication to redo surgery was not a dysfunction of the aortic bioprosthesis and the patients undergoing multiple-valve procedures at the time of the index reoperation. Other mechanisms of bioprosthetic dysfunction than SVD (namely nonstructural valve dysfunction, NSVD, and infective endocarditis, IE) were excluded as they do not represent an indication to the Valve-in-Valve procedure. We adopted the definitions accepted by the current recommendations for reporting morbidity and mortality after cardiac valve interventions [START_REF] Akins | Guidelines for reporting morbidity and mortality after cardiac valve interventions[END_REF]. In particular, SVD was defined as any dysfunction or deterioration of the bioprosthesis which is intrinsic to the valve components, such as wear, fracture, calcification, tear, and suture line disruption, and which determines clinically significant stenosis and/or regurgitation. NSVD was any abnormality not intrinsic to the bioprosthesis itself, such as entrapment by pannus, tissue or suture, paravalvular leak or inappropriate positioning, which result in clinically significant stenosis and/or regurgitation. Finally, IE was defined as any infection of the bioprosthesis as confirmed by either findings at reoperation/autopsy or by the Duke criteria. Operative mortality was defined as death within 30 days after surgery or before hospital discharge.

The terms 'scoring system', 'risk prediction system' and 'risk calculation system' were interchangeably used to indicate the Logistic EuroSCORE I, the EuroSCORE II and the Society of Thoracic Surgeons (STS) score.

Full clinical records for each subject were retrieved. The clinical data contained in the full records were retrospectively used to calculate the predicted risk at the index reoperation for each enrolled patient. The logistic EuroSCORE I and the EuroSCORE II were determined using the calculator available online at www.euroscore.org. The STS score was obtained by the calculator publicly available at www.sts.org. The additive EuroSCORE I was not considered, given its known unreliability to the purposes of risk assessment in candidates to TAVI [START_REF] Zingone | Logistic versus additive EuroSCORE. A comparative assessment of the two models in an independent population sample[END_REF]. .

Statistical analysis.

Data are presented as either mean ± standard deviation (continuous variables) or as percentages (categorical variables). The calibration of scoring systems in predicting the mortality at index reoperation was assessed through the Hosmer-Lemeshow goodness-offit test. For this test, a p value higher than 0.05 indicates an adequate calibration. A Receiver Operating Characteristic (ROC) curve was built for each scoring system and the area under the ROC curve (AUC) was quantified as a measure of discrimination (c statistic: discriminatory ability of the system). Opposed ROC curves (and the corresponding AUCs) were compared according to the Hanley and McNeil methodology.

The alpha level was 0.05. The analyses were conducted using the SAS ver. 9.3 software for Windows.

Empirical Data

The study population consisted of 81 patients whose preoperative risk profile is reported in Table 1. The mean age was 65.8 years, entailing a wide age range (from 32.6 to 86.9 years). Fifty-one patients (63%) came to index reoperation while in NYHA class III or IV, while the priority was non-elective in 16% of cases. The prevalent bioprosthetic dysfunction observed at preoperative echocardiography and/or left heart catheterization was regurgitation in 33.4% of cases and stenosis in 18.5% of cases, while a mixed stenoregurgitant lesion was observed in the majority of patients (48.1%). Concomitant coronary bypass surgery had been realized at the time of primary surgery in 6.2% of patients, while 4.9% had received concomitant surgery on any other valve than the aortic. The failing bioprosthesis was replaced with a novel biological valve in 51.9 of patients, while the remainders received a mechanical prosthesis. Associated procedures at index reoperation were surgery of the ascending aorta in 7.4% of patients and septal myectomy in 1.2%.

The operative mortality for index reoperation was 4.9% (4 patients). Death was cardiacrelated in 3 out of 4 cases (postoperative myocardial failure) and non-cardiac in one case (multiorgan failure). Operative morbidity included a 2.5% rate of surgical revision for bleeding, a 6.2% rate of prolonged (>24 hrs) mechanical ventilation, and a 7.4% rate of acute renal insufficiency.

Compared with a 4.9% operative mortality rate, the average logistic EuroSCORE I was 15.8% ± 13.4, the average EuroSCORE II was 7.3% ± 7.4 and the average STS score for operative mortality was 15% ± 9.8. Hence, all the available scoring systems overestimated the operative mortality. Concerning calibration, the results of the Hosmer-Lemeshow test indicated that all of the three scoring systems taken into analysis are adequate (p=0.847, p=0.999 and p=0.9948 for the logistic EuroSCORE, the EuroSCORE II and the STS Score for mortality, respectively). The calculation of AUC of the ROC curves indicated that the three scoring systems were good predictors of mortality in the entire population.

Nonetheless, the EuroSCORE II yielded an excellent 0.9903 AUC, which was significantly higher than the AUC obtained from the logistic EuroSCORE I (AUC=0.8994, p=0.044), but not statistically different that the AUC obtained from the STS Score (AUC=0.9643, p=0.293 vs. the EuroSCORE II). The STS Score did not display a statistically higher AUC than the EuroSCORE I (p=0.118) (Figure 2). The odds ratios obtained from logistic regression for each score were 1.079 (confidence interval: 1.021 to 1.141) for the logistic EuroSCORE I; 1.474 (confidence interval: 1.079 to 2.013) for the EuroSCORE II; 1.223 (range: 1.056 to 1.416) for the STS Score. Thus, a one unit increment in the calculated EuroSCORE II for each patient is associated with the greater increase in the actual operative mortality, compared with the other systems.

The postoperative follow-up yielded for analysis a total of 612 patient-years. The overall actuarial survival rate was 82.1% ± 4.7 at 5 years, 56.3% ± 6.8 at 10 years and 45.9% ± 7.3 at 15 years.

Consequences of the Hypothesis

The Valve-in-Valve technique to treat SVD is becoming an increasingly standardized procedure [START_REF] Piazza | Transcatheter aortic valve implantation for failing surgical aortic bioprosthetic valve. From concept to clinical application and evaluation (Part 1)[END_REF], and optimal candidate selection is pivotal for the achievement of adequate results. One most problematic issue in the decision-making between TAVI and conventional operation is the expected risk associated with open surgery. In fact, TAVI is currently reserved to patients whose operative risk is considered to be excessive due to comorbidities and other factors contraindicating surgery. At the current stage of evidence, conventional reoperation should not be denied to patients having reasonable reoperative risk. On the other hand, it is important to reliably identify those cases who are unlikely to survive surgery and who may benefit from a transcatheter procedure instead. Although the formal estimation of risk for conventional surgery is only one among several criteria, it plays a major role in these circumstances. Actually, there is growing need for formal and reproducible criteria to discriminate the indication to TAVI rather than to conventional surgery. At the same time, the inadequacy of some scoring systems has become evident in several investigations in the setting of primary aortic valve replacement. In the Mayo Clinic experience, a major overestimation of the risk of AVR was evident for both the additive and the logistic EuroSCORE [START_REF] Brown | Is the European System for Cardiac Operative Risk Evaluation model valid for estimating the operative risk of patients considered for percutaneous aortic valve replacement?[END_REF][START_REF] Osswald | Overestimation of aortic valve replacement risk by EuroSCORE: implications for percutaneous valve replacement[END_REF], and has prompted efforts to recalibrate the original logistic EuroSCORE algorithm by the introduction of correction factors [START_REF] Ranucci | An adusted EuroSCORE model for high-risk cardiac patients[END_REF].

The estimation of risk becomes even more challenging as the subject is moved to peculiar subgroups, such as the patients affected by SVD of a previously implanted bioprosthesis. TAVI is feasible for these patients under the form of the Valve-in-Valve procedure, and poses the same decision-making issues associated with reliability of quantitative scoring systems. The need for quantitative supports to prediction of reoperative risk is evident within all the published experiences on the Valve-in-Valve procedure [START_REF] Piazza | Transcatheter aortic valve implantation for failing surgical aortic bioprosthetic valve. From concept to clinical application and evaluation (Part 2)[END_REF][START_REF] Wilbring | Transcatheter valve-in-valve implantation for deteriorated aortic bioprosthesis: initial clinical results and follow-up in a series of high-risk patients[END_REF]. The hypothesis that EuroSCORE II could work better in this peculiar population was formulated by us after reports of its improved performance in primary AVR patients than the logistic EuroSCORE I. Additionally, the EuroSCORE II was developed on the basis of more recent patients cohorts, which better reflect the present status of surgical candidates and of perioperative care. The early mortality after Valve-in-Valve procedure in the first published registries ranges from 12% [START_REF] Bedogni | Transcatheter valve-in-valve implantation using CoreValve Revalving System for failed surgical aortic bioprostheses[END_REF] to 15% [START_REF] Piazza | Transcatheter aortic valve implantation for failing surgical aortic bioprosthetic valve. From concept to clinical application and evaluation (Part 2)[END_REF] and 17% [START_REF] Eggebrecht | Valve-in-Valve transcatheter aortic valve implantation for degenerated bioprosthetic heart valves[END_REF], faced to an average baseline EuroSCORE I equal to 31%, 27% and 35%, respectively, which indicates overestimation and raises questions about score calibration.. Optimal verification of such hypothesis would require large patients' cohorts and a multiinstitutional collaboration. Yet, preliminary testing on a more limited, single-Institution series was also needed as uncommon risk factors and their unusual combinations are more likely to occur in SVD patients [START_REF] Osswald | Overestimation of aortic valve replacement risk by EuroSCORE: implications for percutaneous valve replacement[END_REF]. The present preliminary testing of the hypothesis suggested that all of the tested models uniformly overestimated the mortality of index reoperation: predicted 15.8% for the EuroSCORE I, 7.3% for the EuroSCORE II and 15% for the STS Score, faced to a 4.9% observed operative mortality. SVD patients seem to behave like primary AVR candidates to this respect. This finding may be at least partly due to an excessive weight attributed to the factor 'Reoperation' in the constitution of the mathematical models. In the EuroSCORE I model, the 'Previous cardiac surgery' categorical variable is associated with a β coefficient of 1.00265 in the regression equation. Only a few major risk factors (severe left ventricular dysfunction, active endocarditis and postinfarct septal rupture) display a more important β coefficient.

Conversely, a trend towards decreasing operative mortality of redo cardiac surgery has been observed during the recent years, and the patients undergoing redo-AVR probably represent the lowest-risk category [START_REF] Luciani | Repeat valvular operations: bench optimization of conventional surgery[END_REF]. Secondly, our preliminary testing indicated that although all risk models showed adequate calibration, the EuroSCORE II presented the better discrimination (ROC curves), although limited by unsatisfactory calibration in the higher tertiles of risk in the overall cardiac surgical population [START_REF] Barili | Does EuroSCORE II perform better than its original versions? A multicentre validation study[END_REF]. This confirms the observations obtained in primary AVR candidates [START_REF] Nashef | EuroSCORE II[END_REF]. The SVD patients represent a distinctive population with different profile in terms of baseline features, early results and long-term prognosis compared to other indications to reoperative AVR (non-structural dysfunction and prosthetic endocarditis) [START_REF] Ruggieri | Logistic EuroSCORE I risk analysis in aortic valve reoperations after bioprosthetic replacement[END_REF]. The SVD patients who survive the reoperation show very good long-term survival: this features underline the need for optimal prediction in order to reserve TAVI to cases who pose actually a too elevated reoperative risk, since no data exist over the long-term durability of transcatheter prostheses. Therefore, this preliminary testing supports the worth of programming a multi-institutional collaboration in order to achieve greater statistical power and definitively verify the hypothesis that the EuroSCORE II would work better in SVD cases than the logistic EuroSCORE I and possibly the STS Score. Whether the EuroSCORE II will be found to display insufficient performance, dedicated algorithms may be developed de novo, as additional specific factors contribute to the risk of reoperative AVR but are not included in any previous scoring system (such as previous coronary surgery and the presence of a patent left internal thoracic artery graft) [START_REF] Park | What is the optimal myocardial preservation strategy at re-operation for aortic valve replacement in the presence of a patent internal thoracic artery?[END_REF][START_REF] He | Determinants of operative mortality in reoperative coronary artery bypass grafting[END_REF].

In conclusion, the hypothesis that the EuroSCORE II may be preferred to the logistic EuroSCORE I in the prediction of surgical risk in aortic valve reoperation for SVD deserves testing in large, multi-institutional cohorts. This represents a major element in the functioning of local Heart Teams. The place of the STS Score will need also to be defined.

On the basis of recent data concerning enhanced long-term durability of bioprostheses [START_REF] Ruggieri | Long-term results of the Carpentier-Edwards supraannular aortic valve prosthesis[END_REF][START_REF] Grunkemeier | Durability of pericardial versus porcine bioprosthetic heart valves[END_REF], we may expect a tendency towards implantation of biological prosthesis at an earlier age and an increase in the absolute number of patients presenting with SVD. In such scenario, we anticipate a growing importance of formal tools to assess the quality of care and to justify the clinical decisions even in a medical-legal environment. herewith declare that there is no commercial association which may pose a conflict of interest in connection with the presented material, that all authors have read and approved the manuscript, and that there has been no duplicate publication elsewhere. In case of acceptance, we shall transfer the copyright to the Publisher.

Rennes, March 7 th , 2014 Amedeo Anselmi, MD

Figure 1 .

 1 Figure 1.Fluoroscopic images of Valve-in-Valve implantation. Two models of

Figure

  Figure 2. Receiver Operating Characteristic (ROC) curves for the three risk prediction

Conclusions.

  The three investigated algorithms showed reasonable calibration in the prediction of mortality for reoperative aortic valve replacement, but they evenly overestimated the observed mortality. The hypothesis that the EuroSCORE II is better suited for the selection of candidates to Valve-in-Valve implantation is worth of further multi-Institutional investigations on the basis of our preliminary findings and due to the expanding role of transcatheter techniques. To the attention of: Mehar Manku, MD Editor-in-chief, Medical Hypotheses I undersigned Dr Amedeo Anselmi, on behalf of all co-authors of the manuscript entitled "Is the EuroSCORE II best suited for Reoperative Risk Estimation in Patients with Structural Deterioraion of Aortic Bioprostheses?", submitted for consideration in Medical Hypotheses,

Table 1 .

 1 Preoperative risk profile in the study population. *Based on the findings of preoperative echocardiography and left heart catheterization.

	Characteristic
	Male/Female

  2. Receiver Operating Characteristic (ROC) curves for the three risk prediction systems taken into analysis. The AUC value is 0.9903 for the EuroSCORE II, 0.9643 for the STS Score and 0.8994 for the logistic EuroSCORE I. The reported p-value is for the comparison between the EuroSCORE II and the logistic EuroSCORE I. EuroSCORE II was 7.3% ± 7.4 and the STS Score was 15% ± 9.8. The three systems provided sufficient adequacy (Hosmer-Lemeshow p=0.847, p=0.999 and p=0.9948, respectively). Yet, the area under the ROC curve was significantly higher for the EuroSCORE II (0.9903) vs. the EuroSCORE I (0.8994) (p=0.044). The STS Score yielded an intermediate figure (0.9643). The odds ratios (logistic regression) were 1.079 for EuroSCORE I, 1.223 for the STS Score and 1.474 for EuroSCORE II.

	Abstract
	Background. Operative risk prediction systems (logistic EuroSCORE I, EuroSCORE II
	and STS Score) are employed together with multidisciplinary discussion to contraindicate
	conventional surgery in patients with valvular heart disease and propose the employment
	of alternative transcatheter procedures. The EuroSCORE I has been reported to
	underperform in these circumstances; we hypothesize that the EuroSCORE II is best
	suited for the stratification of risk in patients with structural deterioration of valvular
	bioprostheses and potential candidates to the Valve-in-Valve procedure (deployment of a
	transcatheter valve within a failing valvular bioprosthesis).
	Methods and evaluation of the hypothesis. A multi-Institutional collaboration is required
	to fully address such hypothesis. Therefore, we performed a preliminary validation study
	by retrieval of the complete records of 81 patients undergoing reoperative aortic valve
	replacement for preoperative diagnosis of bioprosthetic SVD at our Institution. Logistic
	EuroSCORE I, EuroSCORE II and STS Score were calculated by preoperatively available
	data. Faced to an observed reoperative mortality of 4.9%, average EuroSCORE I was
	15.8% ± 13.4,

Acknowledgements

The authors are grateful to Mrs. Anne Ingels and to Mrs. Sylvie Marie for their invaluable support in data analysis and in conduction of the follow-up.