Chunhui Dang
email: chunhui.dang@cea.fr

Mathieu Peybernes

Romain Le Tellier

Laurent Saas

Numerical Simulations of the Rayleigh-Bénard-Marangoni Convections in a Thin Metallic Layer

Keywords: Rayleigh-Bénard convection, Bénard-Marangoni convection, In-Vessel Retention, severe accident, TrioCFD

Within the context of nuclear power safety, the In-Vessel Retention (IVR) strategy consists in sustaining the integrity of the reactor pressure vessel by External Reactor Vessel Cooling (ERVC). However, due to the ablation of the vessel wall and the thermochemical effects, a thin metallic layer may be formed on top of the corium pool and lead to the vessel failure. In this article, numerical simulations of the thermal-hydraulic behavior of the thin metallic layer are carried out to reduce the uncertainties in the heat flux evaluation. Specifically, the Rayleigh-Bénard and Bénard-Marangoni convections are considered in the heat transfer models of the metallic layer. The TrioCFD code is validated for the Bénard-Marangoni convection, then applied for the numerical simulations of the metallic layer with different top boundary conditions: pure radiative heat transfer, and radiative heat transfer plus positive or negative Marangoni effects. The mean temperature and focusing effect of the metallic layer are evaluated and compared for various cases, as well as the temperature profile inside the layer. Results show that the Marangoni effects have to be considered for small metallic layer thicknesses. Furthermore, the layer can be divided into two regions, of which the heat flux correlations can be evaluated independently, giving a better estimation of the heat flux profile.

Introduction

In nuclear power safety, various strategies can be applied to mitigate a reactor severe accident. The IVR strategy, consists in sustaining the integrity of the reactor pressure vessel by ERVC, hence keeping the radioactive materials inside the vessel. The assessment of IVR strategy relies on the evaluation of the heat flux near the vessel wall, which should be lower than the specific 5 vessel Critical Heat Flux (CHF). An in-depth understanding of the melt pool thermal behavior is required to give a reliable evaluation.

Intensive researches have been carried out on this topic, several experimental studies indicated that the melt pool divides into several layers: the heavy metal layer, the oxide layer, and the light metal layer (metallic layer) [START_REF] Tellier | Transient stratification modelling of a corium pool in a LWR vessel lower head[END_REF]. The thin metallic layer is heated from below by the corium pool 10 (oxide layer) and cooled from its lateral and top surfaces, it has an evolving thickness during the accident transients, ranging from centimetric to decimetric [START_REF] Asmolov | Challenges left in the area of in-vessel melt retention[END_REF]. The heat and mass transfer model of the metallic layer is shown in Figure 1, where e is the thickness, L is the diameter of the vessel, ϕ lat and ϕ up correspond to the heat flux through the lateral wall and the top surface, separately, ϕ down is the incoming heat flux from the corium pool, T is the mean temperature of the metallic layer, T up and T lat are the temperatures at the top and lateral surfaces, separately. The mass transfer dm fusion dt is neglected in current study. Due to the high thermal conductivity and small thickness of the metallic layer, the heat flux would be concentrated on its lateral interface if the top cooling is insufficient. The lateral heat flux increases sharply as the thickness decreases, inducing the so called 'focusing effect', when it exceeds the CHF of the RPV external wall, the RPV would be melt down, which is one of the major risks for Reactor Pressure Vessel (RPV) failure during the IVR. The concentration factor Ω FE is used to quantify the focusing effect, and taken as a major concern in the BALI experiment [START_REF] Bonnet | In-Vessel Corium Pool Thermalhydraulics for Bounding Cases[END_REF], it is defined as

Ω FE = ϕ lat /ϕ down . (1)
A number of simplified analytical models [START_REF] Esmaili | Analysis of likelihood of lower head failure and ex-vessel fuel coolant interaction energetics for AP1000[END_REF] and severe accident scenario codes [START_REF] Henry | MAAP4-Modular Accident Analysis Program for LWR Power Plants[END_REF][START_REF] Van Dorsselaere | The ASTEC integral code for severe accident simulation[END_REF]7] based on steady state heat exchange correlations [START_REF] Globe | Natural convection heat transfer in liquids confined by two horizontal plates and heated from below[END_REF][START_REF] Chawla | Heat transfer from vertical/inclined boundaries of heat generating boiling pools[END_REF] are commonly used to evaluate the thermal behavior of the corium pool and the metallic layer. In a preliminary study of Saas et al. [START_REF] Saas | Rayleigh-Bénard and Bénard-Marangoni convection in a thin metallic layer on top of corium pool[END_REF], a simplified analytical model was proposed introducing the so-called Bénard-Marangoni convection into the metallic layer heat transfer model. Marangoni effect is induced by the surface tension on a free surface, for example, on the top surface of a metallic layer. The temperature gradients introduced by the Rayleigh-Bénard natural convections activate surface flows, which would further initiate convection cells due to the mass conservation. This article focuses on the numerical simulations of Rayleigh-Bénard-Marangoni convections of a thin metallic layer with physical properties of steel. The main goal is to understand the influences of various boundary conditions, i.e. radiative heat transfer, radiative heat transfer plus positive or negative Marangoni effects, on the natural convections and on the concentration factors of the metallic layer. Moreover, different layer thicknesses, emissivities, and ambient temperatures are investigated by numerical simulations. The non-dimensionalized heat transfer equations are introduced in Section 2, taking into consideration the surface tension driving Marangoni effect, then used in the numerical simulations by the TrioCFD code. In Section 3, the TrioCFD code is validated for the thin metallic layer configuration with high aspect ratios and various boundary conditions, simulating the Rayleigh-Bénard and Bénard-Marangoni convections separately. Finally, numerical simulations of the thin metallic layer with various top boundary conditions are carried out in Section 4, the mean temperatures and concentration factors of various cases are compared, as well as the temperature profiles at the top surface. Conclusions are given in Section 5.

Mathematical model of the thin metallic layer

The considered thin metallic layer geometry is a 2D slab, similar to that used in the BALImetal experiments [START_REF] Bonnet | Thermal hydraulic phenomena in corium pools: the Bali experiment[END_REF] . It is considered as incompressible fluid with Boussinesq approximation, following the system of equations:

∇ • v = 0 ( 2
)
ρ 0 ∂ v ∂t + v • ∇ v = -∇p + µ∆ v -ρ 0 β(T -T 0 ) g (3)
ρ 0 C p ∂T ∂t + v • ∇T = λ ∆T (4)
with v the velocity, p the pressure, µ the dynamic viscosity, β the thermal expansion coefficient,

T the temperature, g the gravity acceleration, ν the kinematic viscosity, C p the heat capacity and λ the thermal conductivity. The ρ 0 and T 0 stand for the reference density and temperature, respectively.

The thermal boundary conditions of the model are as following:

• Upper boundary condition: -λ∇T • n = ϕ up , the top heat flux is obtained considering a radiative heat transfer to the upper structures assimilated to an infinite planar surface (a constant ambient temperature T ∞ is assumed), that is,

ϕ up = σ s (T 4 up -T 4 ∞ )
with the emissivity, σ s the Stefan-Boltzmann constant;

• Lateral boundary condition: T lat = T fu , T fu is the melting temperature of the RPV steel;

• Bottom boundary condition:

-λ∇T • n = ϕ down .
As to the mechanical boundary conditions, the lateral and bottom boundaries are considered as no-slip wall. While for the top boundary, two different conditions are studied: i) a rigid surface assumption; ii) a free surface considering the surface tension, defining two types of convections: the Rayleigh-Bénard convection and Bénard-Marangoni convection.

Rayleigh-Bénard convection

Considering the rigid surface upper boundary condition, the nondimensionalization analysis is performed in 2D by applying the following scalings:

• t = t 0 t * , t 0 = e 2
α is the thermal diffusion time scale, with α = λ ρCp the thermal diffusivity.

• z = ez * and x = ex * Γ , with the length scale as the thickness e of the layer.

• v x = v 0 v * x and v z = v 0 v * z , v 0 = e t 0 is the velocity scale. • T = T ref + δT T * , δT = T fu -T 0 is the temperature scale.
• p = p ref + p 0 p * , p 0 = ρ 0 ge is the pressure scale, p ref is taken as the gas pressure above the metallic layer.

Introducing the aspect ratio Γ = e/L, we have the non-dimensionalized system equations:

Γ ∂v * x ∂x * + ∂v * z ∂z * = 0 (5) ∂v * x ∂t * + Γv * x ∂v * x ∂x * + v * z ∂v * x ∂z * = -Ga P r 2 Γ ∂p * ∂x * + P r Γ 2 ∂ 2 v * x ∂(x * ) 2 (6) ∂v * z ∂t * + Γv * z ∂v * x ∂x * + v * z ∂v * z ∂z * = -Ga P r 2 ∂p * ∂z * + P r Γ 2 ∂ 2 v * z ∂(z * ) 2 -Ra P r T * (7)
∂T * ∂t * + Γv * x ∂T * ∂x * + v * z ∂T * ∂z * = Γ 2 ∂ 2 T * ∂(x * ) 2 + ∂ 2 T * ∂(z * ) 2 (8)
with the dimensionless numbers:

• P e = v 0 e α , Peclet number

• Ga = ge 3 ν 2 , Galileo number • Gr = gβ∆T e 3 ν 2
, Grashof number Classically, the Grashof number Gr corresponds to the ratio of buoyancy force to viscous force characterizing the flow regime in natural convection, while the Prandtl number P r corresponds to the ratio of momentum diffusivity and thermal diffusivity. With the previous choice of scales, we fix the velocity according to the thermal diffusivity by setting P e = 1 and consequently the Reynolds number is given by Re = 1/P r. Indeed, the relation P e = P r • Re leads us to fix the Peclet number out of these three parameters, as the range of flows investigated in the light metallic layer study (from laminar to quasi-turbulent flows) and the varying physical properties (from water in the BALI experiment to prototypical fluid simulation with metal) lead to a large variation of Reynolds and Prandtl numbers. Also, the Ra number verifies Ra = Gr P r, characterizing the Rayleigh-Bénard convection driven by the buoyancy force, see Figure 2.

• P r = ν k ,

Bénard-Marangoni convection

Considering the surface tension and the temperature gradient on the top surface, surface flow may establish, which would stimulate a flow circulation cell inside the fluid bulk due to the mass conservation, namely, the Bénard-Marangoni convection. The top free surface is given by z(x, t) = e + η(x, t), where z is the coordinate of the vertical axis (at the bottom z = 0), η corresponds to the surface deformation at horizontal location x and time t. The norm of the free surface normal at (x, z) is given by:

N = 1 + ∂z ∂x 2 = 1 + ∂η(t, x) ∂x 2 . ( 9
)
The top mechanical boundary condition is determined by the surface deformation equation and the balance equations of pressure, viscous, and surface tension forces:

∂η ∂t + v x ∂η ∂x = v z ( 10
) (p -p gas ) -2 µ N 2 ∂v z ∂z + ∂v z ∂x ∂η ∂x 2 - ∂η ∂x ∂v x ∂z + ∂v z ∂x = - σ N 3 ∂ 2 η ∂x 2 (11)
µ 1 -( ∂η ∂x ) 2 ∂v x ∂z + ∂v z ∂x + 2 ∂η ∂x ∂v z ∂z - ∂v x ∂x = N ∂σ ∂x + ∂η ∂x ∂σ ∂z . (12)
! It is assumed that the surface tension varies linearly with temperature:

σ = σ 0 -γ • (T -T 0 ), 90
with σ the surface tension at temperature T , σ 0 the surface tension at reference temperature T 0 , γ a temperature coefficient.

Applying N * = 1 + Γ 2 ( ∂z * ∂x * ) 2 as the non-dimensionalized norm of the top surface normal, and the dimensionless scales in Section 2.1, the non-dimensionalized top boundary conditions are:

∂z * ∂t * + Γv * x ∂z * ∂x * = v * z (13) Ga P r p * - 2 (N * ) 2 ∂v * z ∂z * + Γ 3 ∂v * z ∂x * ∂z * ∂x * 2 -Γ ∂z * ∂x * ∂v * x ∂z * + Γ ∂v * z ∂x * = - Γ 2 (N * ) 3 1 Cr ∂ 2 z * ∂(x * ) 2 (14) 1 -Γ 2 ∂z * ∂x * 2 ∂v * x ∂z * + Γ ∂v * z ∂x * + 2Γ ∂z * ∂x * ∂v * z ∂z * -Γ ∂v * x ∂x * = ΓN * M a ∂T * ∂x * + ∂z * ∂x * ∂T * ∂z * , (15)
where M a = γ∆T e ραν is the Marangoni number, Cr = ρνα σe is the Crispation number. Notice that, when M a << 1, the mechanical boundary condition corresponds to a no-slip boundary condition:

v * z = 0, p * = 0 and v * x = 0.
surface tension decreases with temperature, i.e. σ + up > σ - up . In this case, the surface flow is from high temperature to low temperature. Considering the surface temperature gradient induced by Rayleigh-Bénard convection (refer Figure 2), the Rayleigh-Bénard and Bénard-Marangoni convection cells are in the same direction. Therefore, in a macroscopic point of view, it could be assumed that the Marangoni effects enhance the heat transfer on the top surface and diminish the lateral focusing effect of the thin metallic layer. However, when γ is negative, the Bénard-Marangoni convection cells flow in the opposite direction of Rayleigh-Bénard convection cells. In this context, the flow structures inside the fluid have to be investigated further to evaluate the effects on heat transfer. S. Ozawa et al. [START_REF] Ozawa | Influence of Oxygen Partial Pressure on Surface Tension of Molten Type 304 and 316 Stainless Steels Measured by Oscillating Droplet Method Using Electromagnetic Levitation[END_REF] measured the surface tensions of molten SS304 and 316 stainless steels. Results showed that the pure molten metals usually have a negative γ, whereas under a high oxygen partial pressure in the ambient γ would turn to positive. Moreover, γ changes with the potential impurities in the steel, e.g. sulphur and oxygen contents. Thus, for the thin metallic layer, its surface tension temperature coefficient γ can be either positive or negative, or even switch between positive and negative at different temperatures, depending on its compositions, ambient compositions and temperature [START_REF] Chikhi | Influence of steel properties on the in-vessel retention strategy: measurement of 304L and 16MND5 steel density and surface tension[END_REF].

Additionally, for a Crispation number Cr < 10 -4 and a Galileo number Ga >> 1, which is applicable for the thin metallic layer [START_REF] Sarma | INTERACTION OF THE BUOYANCY AND SURFACE TENSION MECHANISMS IN A ROTATING LIQUID LAYER SUBJECTED TO A TRANSVERSE MAGNETIC FIELD[END_REF], a classical hypothesis is applied that the deformation of the free surface is neglected, i.e. η << 1, then we have:

∂z * ∂t * , ∂z * ∂x * = 0, and N * = 1. ( 16
)
The mechanical boundary conditions implemented in current study become:

v * z = 0, p * = 0 and ∂v * x ∂z * = Γ M a ∂T * ∂x * . ( 17
)

TrioCFD code

In present study, the Finite Volume Differences (FVD) method with cartesian meshing is employed for numerical simulations. This method combines the finite difference and the finite volume methods for incompressible Naiver-Stokes problems solving. It is realized in the open-source code TrioCFD. TrioCFD is dedicated to CFD simulations of laminar and turbulent flows, fluid/solid coupling, multiphase flows or flows in porous media [START_REF] Angeli | CFD analysis of the turbulent flow in a PWR 5x5 rod bundle with staggered structural grids[END_REF][START_REF] Angeli | Overview of the TrioCFD code: main features, V and V procedures and typical applications to nuclear engineering[END_REF]. It can be used to investigate a wide range of nuclear industrial flows at different scales, combining various complex physical phenomena such as flows in pressurized water reactors, sodium fast reactors and sub-assembly bundles [START_REF] Bieder | CFD analysis of the flow in the near wake of a generic PWR mixing grid[END_REF]. Moreover, it is designed for massively parallel computers thanks to a MPI parallelization with a quite ideal scalability up to 10,000 processor cores. In the FVD discretization, the QUICK scheme is used for the convective terms in this study, and a 2nd order centered scheme for the diffusion terms [START_REF] Angeli | Overview of the TrioCFD code: main features, V and V procedures and typical applications to nuclear engineering[END_REF].

In a previous work [START_REF] Guennic | Contribution of CFD studies to IVR assessment[END_REF], TrioCFD has been validated based on the BALI experiment with water in a Parallelepiped geometry configuration. Especially, TrioCFD allowed to recover the heat flux splitting ratio in 3D for the BALI-metal experiment. However, the Marangoni effect is not considered in the experiments, so that we present here a complementary validation using academical cases from literatures with high domain aspect ratio and various boundary conditions (including the Marangoni effect). Finally, in the present work, the simulation of the metallic layer is carried out in a configuration similar to the BALI-metal experiment.

Numerical validations of the TrioCFD

The TrioCFD code has been modified to perform simulations with Bénard-Marangoni type boundary conditions, thus it should be validated by academic cases with simple configurations before applying into the thin metallic layer simulations. The simulations of Rayleigh-Bénard convection and Bénard-Marangoni convection are carried out separately for validation purpose. Moreover, this kind of simulations would give us a better understanding about the mechanisms of Rayleigh-Bénard and Bénard-Marangoni convections, separately. The simulation geometry is a 2D rectangular, with a Γ ranging from 0.66 to 8, and two different thermal boundary conditions exerted at the lateral walls, see Figure 4,'A' denotes the adiabatic wall, 'C' denotes the cooled wall, while 'H' denotes the hot wall. As we can see, we have the bottom (floor) as the hot wall (H), the top (ceiling) as the cold wall (C), while the left and right lateral walls have two different conditions: CC and CA. All the walls are no-slipping walls, the Ra number ranges from 10 3 to 10 6 , the P r number is 0.71, corresponding to that of air. The dimensionless scales used in [START_REF] Corcione | Effects of the thermal boundary conditions at the sidewalls upon natural convection in rectangular enclosures heated from below and cooled from above[END_REF] are: the velocity scale U * = u/(ν/H), pressure scale P * = (p + ρ 0 gy)/ρ 0 (ν/H) 2 , the length and temperature scales are the same as in Section 2.1, the dimensionless temperature is -0.5 for cold wall and 0.5 for hot wall.

The averaging Nusselt number N u along the four walls at different Ra are computed and compared with the values from [START_REF] Corcione | Effects of the thermal boundary conditions at the sidewalls upon natural convection in rectangular enclosures heated from below and cooled from above[END_REF]. Figure 5 shows the variation of N u with Γ along the horizontal and vertical walls at Ra = 10 5 . As we can see, the N u are comparable between the simulation results and the literature results for both CA and CC cases, a similar trend is observed, and the differences decrease with Γ. To explain this undesirable deviation for low aspect ratios, several comparative cases were simulated with different meshes, schemes, and initial states. Results show that all these parameters have influences on the N u numbers, especially the schemes. Noting that the difference is more pronounced for low Γ cases where the lateral boundary condition is expected to have more influences, while a high Γ is relevant to the metallic layer case we focus on.

Moreover, the temperature profiles close to the walls (δx = δy = 0.01) are plotted to study the flow structures inside the domain. Considering the case with Γ = 4.0, Ra = 10 5 and CA configuration, Figure 6(b) shows the temperature profiles along the top and bottom walls, from which the influences of the cooling wall (at x = 0) can be clearly seen: the temperature at the top wall increases sharply from x = 0 to x = 0.1 then it fluctuates in a small temperature range at x > 0.1, which is dominated by the convection cells. This behavior is also illustrated by the temperature profile along the left wall in Figure 6(a). It is noted that the influence of the lateral cooling is restricted only within a small region close to the wall.

Validation for Bénard-Marangoni effect

In this section, both 2D and 3D simulations are carried out by TrioCFD with pure Bénard-Marangoni convection, neglecting the buoyancy force. The researches of Boeck et al. [START_REF] Boeck | Turbulent Bénard-Marangoni convection: Results of two-dimensional simulations[END_REF][START_REF] Boeck | Bénard-Marangoni convection at large Marangoni numbers: results of numerical simulations[END_REF] are used as reference cases, with a low P r = 0.1 and various M a.

The studied domain is heated from bottom with a constant dimensionless temperature 1.0, and cooled from the top with a constant dimensionless heat flux, the lateral walls are periodic. As to the mechanical boundary conditions, the bottom and lateral walls are rigid, while the top wall is set as rigid only in z direction (vertical velocity v z = 0), the velocities v x and v y in horizontal directions are determined by Marangoni effect (v y is not considered in 2D). That is,

v z = 0, v x = v y = 0, T = 1, at z = 0, v z = 0, ∂ z T = -1, ∂ z v i = -M a • ∂ i T, i = x, y, at z = 1. ( 18
)
Since the heat flux at the top surface is fixed, the convection in the 2D domain reduces the temperature difference across the layer, thus, the Nusselt number can be obtained from N u = 1/ ∆T , with ∆T the mean temperature difference between the bottom and top surfaces.

2D cases

For 2D cases, similar geometry is used as in Figure 4 with a fixed aspect ratio Γ = 2, the meshing size is 512 × 128. The dimensionless time and temperature scales are taken as e 2 /α and ed/λ, respectively.

A total of six different M a from 4 × 10 4 to 12 × 10 4 are studied with two different spatial discretization schemes (upwind and Quick schemes) to evaluate the influence of the spatial discretization order, of which the N u numbers are calculated and compared with the reference values, as presented in Figure 7. As we can see, the N u obtained from TrioCFD agree well with the reference values at low M a, the maximum error is around 20% at very high M a. According to Boeck et al. [START_REF] Boeck | Turbulent Bénard-Marangoni convection: Results of two-dimensional simulations[END_REF], at high M a the flow fields become spatially irregular, the time signals display irregularity on the scale of a roll turnover time, so that the differences between the FVD method used in our simulations and the pseudo-spectral numerical method based on a Fourier?Chebyshev expansion used in Boeck et al. would be large.

Besides, the evolution of N u along time at M a = 5 000 and 60 000 are plotted in Figure 8. Comparing with the literature [START_REF] Boeck | Turbulent Bénard-Marangoni convection: Results of two-dimensional simulations[END_REF], the tendency and periodic behavior are comparable for both M a. Moreover, two quasi-periodic dynamics on different time scales can be identified: i) a short time oscillation due to the evolving sizes of the convection cells; ii) a long time oscillation related to the amplitude of the mean flow.

3D cases

The geometry considered in 3D validation is a slab, with the layer thickness e (z direction) as the dimensionless length scale, the horizontal lengths in x and y directions are set to be 2π, which is much larger than in 2D cases to avoid a strong mean velocity in the domain. The meshing matrix in Boeck et al. [START_REF] Boeck | Bénard-Marangoni convection at large Marangoni numbers: results of numerical simulations[END_REF] is 256 × 256 × 32 for M a ≤ 1500, and 512 × 512 × 32 for M a ≥ 2000, to capture the small vortex at large M a. The same meshing parameters are used in TrioCFD. Seven different M a are studied from 500 to 4000 with the upwind and Quick schemes, the N u at different M a are plotted in comparison with the literature values [START_REF] Boeck | Bénard-Marangoni convection at large Marangoni numbers: results of numerical simulations[END_REF], as shown in Figure 9.

The N u values are considerably close to each other, the error is within 8% for various M a.

Numerical simulations of the thin metallic layer 205

In this section, the thermal-hydraulic behavior of the thin metallic layer is simulated with physical properties of steel, its geometry is assumed to be a sliced slab, with a length (along x axis) of L = 2 m and a width (along y axis) of 0.13 m. Three different thickness (along z axis) e = 1 cm, 2.5 cm, 4 cm are simulated, considering the Marangoni effect can be neglected for e > 3 cm [START_REF] Saas | Rayleigh-Bénard and Bénard-Marangoni convection in a thin metallic layer on top of corium pool[END_REF].

For the thermal boundary conditions, the bottom surface (z = 0) is subjected to a constant 210 input heat flux of 10 6 W/m 2 . The front and back walls are assumed adiabatic, as well as one of the lateral wall at x = 2 m. The other lateral wall at x = 0 is subjected to a fixed temperature T lat =1658 K, corresponding to the steel melting temperature. At the top surface, three different conditions are studied: radiative heat transfer, and radiative heat transfer plus positive or negative Marangoni effects. For the radiative heat transfer, the ambient temperature could be between 400 K and 1658 K, the emissivity could be from 0.2 to 0.8, corresponding to the minimum and maximum values usually taken. No-slipping boundary conditions are considered for all the boundaries except for the top surface where the Marangoni effect is applied. The physical properties of the steel are listed in Table 1, surface tension coefficients of steel (γ steel ) and iron (γ iron ) are considered for positive and negative M a, respectively. Unit kg/m 3 J/(kg

• K) K -1 m 2 /s W/(m • K) N/(m • K) N/(m • K)
Table 1: Physical properties of the steel with surface tension coefficients of steel γ steel and iron γiron.

The magnitudes of the dimensionless numbers of the metallic layer considered in present study are listed in Table 2. They are calculated with a thickness e =0.1 m, a temperature difference ∆T = 400 K, and the physical properties in Table 1 ((using γ iron )), which corresponds to one of the simulation configurations. Table 2: Order of magnitudes of the dimensionless numbers for steel (e = 0.1 m, ∆T = 400 K).

A mesh convergence study has been carried out on the 1 cm thickness case with fixed temper-ature. Firstly, the concentration factors for different mesh sizes are plotted in Figure 10(a). We observe a convergence for a grid cell size close to 1 mm (difference lower than 0.5% between the 1 mm grid cell size case and the 0.5 mm one). Then, to supplement this analysis, the Kolmogorov length scale defined by η

= ν 3 ε 1/4
is computed for the case with 1 mm cell size (Prandtl number verifying P r << 1, the mesh size is constrained by the viscous scale rather than the thermal diffusivity scale). Figure 10 plots the number of cells versus the ratio of the Kolmogorov length scale to the cell size for the 2000 × 130 × 10 mesh case (corresponding to 1 mm cell size). It is observed that most of grid cells have a size comparable to the Kolmogorov length scale. Thus, a mesh corresponding to 1 mm cell size is chosen for the following simulations.

Results analysis

Table 3 presents the T mean and Ω FE for the four different top boundary conditions, with the emissivity = 0.2 and the ambient temperature T env = 1658 K. It can be seen that the Marangoni effect has a particular influence on Ω FE for e = 1 cm case: the Ω FE at a negative M a is almost 2 times larger than the pure Ra case. While for e ≥ 2.5 cm cases, the Ω FE are comparable for all cases, corroborating with the 0D analysis in [START_REF] Saas | Rayleigh-Bénard and Bénard-Marangoni convection in a thin metallic layer on top of corium pool[END_REF]. A possible reason is that the Marangoni effect decreases as e increases, such that the numerical deviation induced by the mesh and numerical scheme is more noticeable.

As explained in Section 3.2, the negative Marangoni convection cells are in the opposite direction of the Rayleigh-Bénard convection cells, which reduces the top cooling and enhances the focusing effect. However, the effects of positive M a on Ω FE seem unclear: the positive M a cases give higher Ω FE than the pure Ra cases. That is, both positive and negative M a enhance the focusing effect of the metallic layer. Regarding the T mean , a positive M a leads to a lower T mean whereas a negative M a increases the T mean .

The results are further validated through simulation cases with = 0.8 and T env = 400 K. The results are shown in

Flow structures analysis

To understand the non-obvious effects of positive M a on the Ω FE shown in the former section, the temperature profiles and fields in the domain under the stationary state are analyzed for the simulation cases with = 0.2, T env = 1658 K, and e = 1 cm, 2.5 cm. Figure 11 shows the temperature profiles along the x axis located in the middle of the top surface.

As discussed in Section 3.1, only the region close to the lateral cooling wall is affected by the lateral cooling, the rest part is dominated by the convection cells. Thus, the temperature rises up quickly in the region close to the lateral cooling wall, then fluctuates due to the convection cells. In Figure 11(a), the temperature rises up quickly at x < 0.3 m (the lateral cooling region), then fluctuates at 0.3 < x < 2 m (the convection dominate region), which agrees well with the temperature profiles of the CA configuration in Section 3.1. Similar trend is observed in Figure 11(b), except that the lateral cooling region is 0 < x < 0.6 m, which is longer than that in e = 1 cm case. Moreover, it is observed in Figure 11(b) that in the convection dominate region, the mean temperatures of the pure Ra and positive M a cases are higher than that of the negative M a case. Considering the radiative heat flux is a power of 4 to the temperature, higher radiative heat flux are emitted to the ambient in the pure Ra and positive M a cases than in the negative M a case. On the opposite, the radiative heat flux in the lateral cooling region are larger for the negative M a case. This compensation effect explains the similar Ω FE for e = 2.5 cm case shown in Table 3. Similar mechanisms hold for e = 1 cm case (refer Figure 11(a)), while the lateral cooling region is shorter, which explains the Ω FE for e = 1 cm case in Table 3.

Furthermore, the temperature fields at the top and the lateral cooling surface for pure Ra case with e = 2.5 cm are shown in Figure 12. From Figure 12(a) we can see that the temperature rises before x ≈ 0.5 m, after that the convection cells are observed. While in Figure 12(b) and 12(c) the cold stripe induced by the lateral cooling can be clearly seen close to the cooling wall.

Figure 13 shows the temperature fields at the top surface for positive and negative M a cases. As we can see, the temperature fields of both cases are in the same pattern with pure Ra case, while the length of the lateral cooling region is different. While in the convection dominate region, the positive M a case gives stronger convection cells, the negative M a case gives a more homogeneous temperature field, which confirms that the convection cells induced by a negative M a are in the opposite direction to that induced by Rayleigh-Bénard convection.

Additionally, the temperature fields at the top surface for e = 1 cm and 4 cm cases with positive M a are shown in Figure 14. They are in the same patterns with e = 2.5 cm cases, while it is noticed that the length of the lateral cooling region is increasing with e.

To sum up, the temperature field of the metallic layer can be divided into two regions: i) the region close to the lateral cooling wall is affected by the lateral cooling, no convection cell is initiated, the temperature is rising continuously with x; ii) the other region is dominated by convection, it has fully-developed convection cells induced by Rayleigh-Bénard and Bénard-Marangoni convections. The length of each part is dependent on the top boundary conditions and the thickness of the layer e.

Summary and conclusion

In this article, numerical simulations are carried out to investigate the thermal behavior of a metallic layer appearing above the corium pool under the IVR context during a severe accident. Specifically, the Rayleigh-Bénard and Bénard-Marangoni convections are considered in the heat transfer models, their influences on the heat transfer behavior of the metallic layer are studied with various layer thickness. The TrioCFD code is validated with academic cases with separate Rayleigh-Bénard and Bénard-Marangoni effects, then applied in the numerical simulations of the metallic layer with a set of top boundary conditions and layer thickness.

Results show that the Marangoni effects have to be considered for low metallic layer thicknesses: the numerical simulations give an indication that the Marangoni effect actually can enhance the focusing effect of the metallic layer only for very thicknesses, no matter it is positive or negative. The flow structures inside the metallic layer are analyzed to explain this unexpected result. The temperature fields at the top surface and the cold stripes near the lateral cooling wall for various configurations show that the thermal behavior of the metallic layer can be divided into two regions: i) one region is dominated by the lateral cooling, in which no convection cell is initiated; ii) the other region has fully-developed convection cells induced by Rayleigh-Bénard-Marangoni convections. Therefore, it seems necessary to take into account this complex interaction between the lateral and upper heat flux in the computation models widely used in severe accident scenario codes.

Furthermore, new correlations based on the Rayleigh-Bénard-Marangoni convections may allow a better evaluation of the focusing effect. For future improvements of the present work, more comparison and material internal heat source analysis of the boundary conditions are appealing. analysis was granted access to the HPC resources of Très Grand Centre de Calcul under the allocation A0012A07691 made by GENCI.

Figure 1 :

 1 Figure 1: Heat and mass transfer model of the metallic layer (the mass transfer is neglected in current study).

  Prandtl number • Ra = Gr P r = gβ∆T e 3 αν , Rayleigh number with ν the kinematic viscosity.

Figure 2 :

 2 Figure 2: Rayleigh-Bénard convection cells discarding the coupling with the lateral cooling. The convection cells drive the hot plumes up to the surface and cold plumes down to the bottom, as the body is heated from the bottom and cooled by top radiative heat transfer.

Figure 3 :

 3 Figure 3: Bénard-Marangoni convection cells.

3. 1 .

 1 Validation for Rayleigh-Bénard convection Corcione et al. [19] studied a number of cases with various geometrical aspect ratios Γ and boundary conditions in a wide range of Ra (10 3 to 10 6 ) close to those of the thin metallic layer in the IVR context. The results in their paper are used for the validation of TrioCFD for pure Rayleigh-Bénard convection.

Figure 4 :

 4 Figure4: Thermal configurations of the simulation geometry, adapted from[START_REF] Corcione | Effects of the thermal boundary conditions at the sidewalls upon natural convection in rectangular enclosures heated from below and cooled from above[END_REF]

  Figure 5: Surface N u number varies with the aspect ratio Γ, Ra = 10 5 .

Figure 6 :

 6 Figure 6: Temperature profile along four walls in CA configuration, with Γ = 4.0, Ra = 10 6 .

Figure 7 :Figure 8 :

 78 Figure 7: Nusselt number varies with Marangoni number M a in 2D cases.

Figure 9 :

 9 Figure 9: Nusselt number N u versus Marangoni number M a in 3D cases.

3 ×

 3 10 -5 4.5696 × 10 -3 20 -9.6 × 10 -4 3.97 × 10 -4

Value 1 . 5 ×

 15 10 -1 2.2 × 10 10 3.93 × 10 7 7.81 × 10 5 1.04 × 10 -7

Figure 10 :

 10 Figure 10: Concentration factors for different mesh sizes (left); and the number of cells versus the ratio between the Kolmogorov length scale and the grid cell size for the 2000 × 130 × 10 mesh case (right).

Figure 11 :

 11 Figure 11: Temperature profiles in the middle of top surface at y = 0.065 m, for layer thickness e = 1 cm and 2.5 cm.

Figure 12 :

 12 Figure 12: Temperature fields at the top and lateral surfaces in pure Ra case (e = 2.5 cm).

Figure 13 :

 13 Figure 13: Temperature fields at the top surface with positive and negative M a (e = 2.5 cm).

Figure 14 :

 14 Figure 14: Temperature fields at the top surface for positive M a, for e = 1 cm and 4 cm.

Table 4 ,

 4 the conclusions are the same with the previous cases about the Cases 1 cm 2.5 cm 4 cm

		pure Ra	8.1	10.2	10.8
	Concentration factor	positive M a 16.1	11.2	11.3
	Ω FE	negative M a 19.2	10.4	9.9
		pure Ra	3152 3090 3004
	Mean Temperature (K)	positive M a 3065 3010 2915
	T mean	negative M a 3182 3168 3094

Table 3 :

 3 Results of TrioCFD with different top boundary conditions and = 0.2, Tenv = 1658 KMarangoni effects. Noticing that the T mean and Ω FE with = 0.8 and T env = 400 K are both lower than the previous values due to the higher radiation heat transfer at the top surface.

		Cases	1 cm 2.5 cm 4 cm
		Pure Ra	1.90	2.43	2.65
	Concentration factor	Positive M a	4.1	2.72	2.88
	Ω FE	Negative M a 6.46	3.59	2.95
		Pure Ra	2228 2237 2230
	Mean Temperature (K)	Positive M a 2176 2172 2156
	T mean	Negative M a 2289 2307 2312

Table 4 :

 4 Results of TrioCFD with different top boundary conditions and = 0.8, Tenv = 400 K

The dimensionless number M a characterizes the Bénard-Marangoni convection. Figure3shows the convection cells induced by Bénard-Marangoni convection with a positive γ > 0, where the

ACKNOWLEDGMENTS

This work is funded by the IVMR grant agreement number 662157 (H2020). The TrioCFD


Alexandre J Vivanti

Isabelle Monier

Eleonora Salakos

Caroline Elie

Vassilis Tsatsaris

Marie-Victoire Senat

Jacques Jani

Jean-Marie Jouannic

Norbert Winer

Jennifer Zeitlin

Charlotte Mougneaud

Jean-Claude Souberbielle

Marie Courbebaisse

Alexandra Benachi

Vitamin D and pregnancy outcomes: Overall results of the FEPED study

Keywords: Vitamin D, 25(OH)-vitamin D3, preterm birth, preeclampsia, gestational diabetes, FEPED

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.

Vitamin D plays a role in the homeostasis of many organs. It is particularly involved in calcium absorption, leading to skeletal calcium balance and bone maintenance. It has also been associated with immune system interaction: inhibition of B or T cell proliferation [START_REF] Bhalla | 1,25-Dihydroxyvitamin D3 inhibits antigen-induced T cell activation[END_REF], induction of T cell regulation [START_REF] Boonstra | 1alpha,25-Dihydroxyvitamin d3 has a direct effect on naive CD4(+) T cells to enhance the development of Th2 cells[END_REF], effects on dendritic cells and monocytes. In addition to its protective effect against bone demineralization, vitamin D sufficiency is associated with a reduced risk of many chronic diseases including type 2 diabetes mellitus, cardiovascular diseases, cancers, auto-immune and infectious diseases [START_REF] Wacker | Vitamin D -effects on skeletal and extraskeletal health and the need for supplementation[END_REF]. Vitamin D status can be assessed by serum measurement of the serum concentration of 25(OH)-vitamin D (25(OH)D).

According to the Endocrine Society Clinical Practice Guideline, vitamin D deficiency is defined as 25(OH)D below 20 ng/mL (50 nmol/L) and vitamin D insufficiency as 25(OH)D of 21-29 ng/mL (525-725 nmol/L) [START_REF] Holick | Evaluation, Treatment, and Prevention of Vitamin D Deficiency: an Endocrine Society Clinical Practice Guideline[END_REF]. During pregnancy, the definition of vitamin D deficiency remains unclear [START_REF] Nassar | Systematic review of first-trimester vitamin D normative levels and outcomes of pregnancy[END_REF]. It is admitted that vitamin D insufficiency among pregnant women is highly prevalent in most countries [START_REF] Kimball | Vitamin D: a growing perspective[END_REF]: 18% percent of pregnant women in the United Kingdom, 25% in the United Arab Emirates and nearly 80% in the Netherlands have vitamin D levels less than 25 nmol/L [START_REF] Datta | Vitamin D deficiency in pregnant women from a non-European ethnic minority population--an interventional study[END_REF].

The FEPED study was designed to assess the impact of maternal vitamin D status in the first trimester on the risks of preeclampsia (primary outcome). This observational prospective cohort included 3129 women with a singleton pregnancy between April 2012 and July 2014 in five maternity units in France (Béclère, Bicêtre, Cochin, Trousseau and Nantes university hospitals) and one in Belgium (University Hospital Brugmann). Inclusion criteria were women aged over 18 years, singleton pregnancies, gestational age ranged from 10 to <15WG at enrollment and having a healthcare coverage. The exclusion criteria were hypercalcemia (>2.65 mmol/L), high blood pression (>140/90 mmHg), renal failure (creatinine > 120 µmol/L), bone disease, lithium therapy, bowel malabsorption and kidney stones disease. Maternal vitamin D levels were assessed at three points of pregnancy: in the first trimester (between 10 and 14 weeks), in the third trimester of pregnancy and at birth (at cord blood). The aim of this review article is to summarize the results of the FEPED study regarding the potential links between vitamin D levels and previously described major obstetrical issues.

Vitamin D levels throughout pregnancy among women of the FEPED cohort

Several international guidelines recommend vitamin D supplementation during pregnancy [START_REF]to Review Dietary Reference Intakes for Vitamin D and Calcium. Dietary Reference Intakes for Calcium and Vitamin D[END_REF][START_REF] Berkane | The use of supplements in pregnancy[END_REF][START_REF] Rcog | Vitamin D in Pregnancy[END_REF]. However, most of these studies focused on first trimester vitamin D levels.

Vitamin D levels throughout pregnancy (first and third trimester) and at delivery (cord blood) were studied in the FEPED cohort [START_REF] Courbebaisse | Vitamin D status during pregnancy and in cord blood in a large prospective French cohort[END_REF]. This study assessed the vitamin D status of the French pregnant women and their newborns and the impact of the French guidelines regarding recommended vitamin D supplementation during pregnancy (cholecalciferol 100 000 IU administered in the 6th month of pregnancy) [START_REF] Cngof | Recommandations pour la pratique clinique : SUPPLÉMENTATIONS au cours de la GROSSESSE[END_REF]. A total of 2803 women were included in this analysis. Vitamin D status was below 20 ng/mL in 46.5% of pregnant women in the first trimester of pregnancy. In multivariate analysis, overweight before pregnancy, dark phototype, sampling during fall, winter, or spring, and absence of vitamin D supplementation at the very beginning of pregnancy were independently associated with vitamin D insufficiency in the first trimester.

The recommended supplementation, which was received by 88.6% of participants, had a significant impact on serum vitamin D levels during the third trimester (32.5 ± 11.4 ng/mL in supplemented women versus 25.8 ± 11.4 ng/mL in non-supplemented women, p<0.001). However, 25(OH)D levels in cord blood, reflecting the vitamin D status of newborns [START_REF] Wierzejska | Maternal and Cord Blood Vitamin D Status and Anthropometric Measurements in Term Newborns at Birth[END_REF], was only marginally higher in case of supplementation (17.2 ± 7.2 ng/mL versus 15.5 ± 7.1 ng/mL, p<0.01). Importantly, the FEPED study demonstrates that this high dose of cholecalciferol given once, 100,000 IU theoretically corresponding to 1100 IU daily during 3 months, is clearly insufficient to obtain serum 25(OH)D levels above 20 ng/mL in most newborns and is also insufficient to completely prevent severe vitamin D deficiency (defined as serum 25(OH)D concentration <10 ng/mL), which was present in 12.4% of newborns whose mothers had received the supplementation. In conclusion, vitamin D insufficiency is highly prevalent in the middle-north of France at the beginning of pregnancy and in cord blood, despite the recommended supplementation.

Preeclampsia

In France, prevalence of preeclampsia is around 0.5 to 7% of all pregnancies depending on maternal risk factors. [START_REF] Abalos | Global and regional estimates of preeclampsia and eclampsia: a systematic review[END_REF] , [START_REF] Blondel | Trends in perinatal health in metropolitan France from 1995 to 2016: Results from the French National Perinatal Surveys[END_REF] Preeclampsia was defined as a blood pressure >140 mmHg or a diastolic blood pressure >90 mmHg and proteinuria >0.3 g/24h. This definition is the one currently used in France and it was also used internationally at the time of the FEPED study. Prevention of preeclampsia remains a cornerstone of therapeutic management because of its high morbidity and mortality and because there is currently no curative treatment. Several drugs and vitamin supplementations have been reported to reduce the incidence of preeclampsia. Vitamin D insufficiency in early pregnancy has been identified as an independent risk factor of preeclampsia. [START_REF] Bodnar | Maternal Vitamin D Deficiency Increases the Risk of Preeclampsia[END_REF] In the study of Bodnar and colleagues in 274 women, serum 25(OH)D levels in early pregnancy were 15% lower in women who developed preeclampsia compared with healthy controls. Furthermore, vitamin D insufficiency at less than 22 weeks of gestation was more prevalent among women who developed preeclampsia than among women who did not. Some studies suggest that sufficient vitamin D levels in early pregnancy may be protective against preeclampsia [START_REF] Achkar | Vitamin D status in early pregnancy and risk of preeclampsia[END_REF][START_REF] Baker | A Nested Case-Control Study of Midgestation Vitamin D Deficiency and Risk of Severe Preeclampsia[END_REF][START_REF] Kiely | Vitamin D status is associated with uteroplacental dysfunction indicated by pre-eclampsia and small-forgestational-age birth in a large prospective pregnancy cohort in Ireland with low vitamin D status[END_REF]. Others imply that normal levels in the second or mid-third trimester maintain the protective effect against preeclampsia [START_REF] Bärebring | Preeclampsia and Blood Pressure Trajectory during Pregnancy in Relation to Vitamin D Status[END_REF][START_REF] Robinson | Maternal vitamin D and fetal growth in early-onset severe preeclampsia[END_REF]. The FEPED study explored the relationship between vitamin D insufficiency (defined as a serum 25(OH)D level below 30 ng/mL) in the first trimester (main outcome) and the occurrence of preeclampsia later in pregnancy [START_REF] Benachi | Relationship between vitamin D status in pregnancy and the risk for preeclampsia: A nested case-control study[END_REF]. A secondary objective was to evaluate the relationship between preeclampsia and vitamin D levels during the third trimester of pregnancy. Vitamin D insufficiency was defined as a serum 25(OH)D level below 30 ng/mL. The overall prevalence of preeclampsia was 3% (mean term at the onset of preeclampsia was 36.2 ± 3.8 weeks of gestation and nine cases before 32 weeks of gestation). Eighty-three cases of preeclampsia were matched to 319 controls randomly selected in a 1:4 ratio (table 1). In the first trimester, the risk for preeclampsia decreased in women with vitamin D sufficiency, but did not achieve statistical significance (OR=0.57 95% CI, 0.30-1.01; p=0.09). The overall prevalence of preeclampsia in this cohort was 3.0%. Nine cases of preeclampsia occurred before 32 weeks of gestation. The probability of preeclampsia decreased with the increase of 25(OH)D levels in the first trimester of pregnancy. For each 10 ng/mL increase, the risk of preeclampsia decreased by 19%, without however achieving statistical significance. In the third trimester, the risk for preeclampsia decreased in women with vitamin D sufficiency (OR = 0.43 95% CI, 0.23-0.80; p=0.008). In the first trimester, the risk for preeclampsia was decreased in these patients, but did not achieve statistical significance (OR = 0.57 95% CI, 0.30-1.01; p=0.09). The mean 25(OH)D level was significantly higher in controls compared with women with preeclampsia (30.8 ± 11.0 vs. 27.7 ± 12.2 ng/mL; p < 0.05). Finally, vitamin D sufficiency at the first and third trimesters had a protective effect against preeclampsia compared with women with 25(OH)D < 30 ng/mL in both the first and third trimesters (OR = 0.34; 95% CI, 0.13-0.86; p=0.023).

In order to assess the interactions between vitamin D status and the pathogenesis of preeclampsia, global gene expression profiling was proposed [START_REF] Mirzakhani | Early pregnancy vitamin D status and risk of preeclampsia[END_REF]. The aim of this study was to identify vitamin D gene signatures related to preeclampsia during early pregnancy (transcriptomic analysis). A gene expression analysis was performed among pregnant women with 25(OH)D sufficiency and insufficiency. A set of vitamin D-associated genes related to preeclampsia was identified. Most of the dysregulated genes were associated with immune and inflammatory pathways (eg., downregulation of IL-10, which is already known to be decreased in the serum and placenta of women with preeclampsia).

A recent Cochrane meta-analysis including four randomized trials for a total of 499 women concluded that vitamin D supplementation may reduce the risk of preeclampsia compared to no intervention or placebo (RR 0.48, 95% CI 0.30 to 0.79) [START_REF] Palacios | Vitamin D supplementation for women during pregnancy[END_REF] (table 2).

Although there is an association between 25(OH)D levels and the risk for preeclampsia, timing of vitamin D supplementation remains controversial. Vitamin D supplementation (4400 IU/day or 400 IU/day) initiated in weeks 10-18 of pregnancy failed to reduce the incidence of preeclampsia. [START_REF] Mirzakhani | Early pregnancy vitamin D status and risk of preeclampsia[END_REF] However, women with sufficient first-trimester levels of 25(OH)D (≥30 ng/mL) had lower rates of preeclampsia. In France, vitamin D supplementation is recommended with 100,000 IU of cholecalciferol administered at month 7 of gestation. This strategy of vitamin D supplementation at the middle of the second trimester should be questioned because i) sufficient levels of 25(OH)D seem to be needed in the first trimester to improve placentation and reduce the risk of preeclampsia and ii) although sufficient levels of 25(OH)D seem also to be needed at the third trimester, the results from the FEPED study have shown that this supplementation is insufficient to achieve serum 25(OH)D levels ≥30 ng/mL in nearly half of the women at the third trimester (vitamin D sufficiency was reported in only 58.2% of the supplemented women) [START_REF] Courbebaisse | Vitamin D status during pregnancy and in cord blood in a large prospective French cohort[END_REF]. For instance, the WHO does not yet support a policy of systematic vitamin D supplementation during pregnancy to improve maternal and perinatal outcomes [START_REF]WHO | WHO recommendations on antenatal care for a positive pregnancy experience[END_REF]. Studies aimed at evaluating vitamin D supplementation from early pregnancy and even before pregnancy in at-risk women should be undertaken.

Gestational diabetes mellitus

Attempts to assess a potential link between 25(OH)D insufficiency and gestational diabetes mellitus (GDM) have been made for many years. Studies of an Iranian cohort and of an Australian cohort suggested an inverse correlation between maternal 25(OH)D levels during the mid-second trimester and the incidence of GDM [START_REF] Maghbooli | Correlation between vitamin D3 deficiency and insulin resistance in pregnancy[END_REF][START_REF] Mcelduff | Maternal vitamin D deficiency, ethnicity and gestational diabetes[END_REF]. Lu and colleagues published a large meta-analysis including a total of 20 observational studies and 16,515 pregnant women (cohort, cross-sectional, case control or nested case control) [START_REF] Lu | Association between vitamin D status and the risk of gestational diabetes mellitus: a meta-analysis[END_REF]. The authors showed that 25(OH)D insufficiency during pregnancy was associated with increased GDM risk (RR 1.45; 95% CI 1.15-1.83; p<0.001). Most of their findings were consistent with the findings of previously reported observational studies, however substantial heterogeneity was detected across the studies included (I 2 = 67%; p<0.001). Most studies included were cross-sectional, with small sample, different cut-offs and lack of adjustments for confounding factors. Because of such heterogeneity, a subgroup analysis was performed. It seems that the link between 25(OH)D insufficiency and GDM was still present if the studies were performed in developed countries. Taken together, their results should be viewed with caution because of the low quality of several of the studies included. Similar results were also found in two previous meta-analyses in which low concentrations of 25(OH)D increased the risk of GDM in nearly 50% of cases [START_REF] Aghajafari | Association between maternal serum 25-hydroxyvitamin D level and pregnancy and neonatal outcomes: systematic review and meta-analysis of observational studies[END_REF][START_REF] Wei | Maternal vitamin D status and adverse pregnancy outcomes: a systematic review and meta-analysis[END_REF]. Data extracted from the FEPED study were analyzed to assess the risk of GDM according to 25(OH)D concentration during the first trimester of pregnancy [START_REF] Salakos | Relationship between vitamin D status in the first trimester of pregnancy and gestational diabetes mellitus -A nested case-control study[END_REF] (table 1). Cases of GDM were matched to eligible controls randomly selected in a 1:4 ratio. The overall rate of GDM was 8.3% in the FEPED cohort: 250 cases were matched with 941 controls. The risk of GDM was higher among women with 25(OH)D insufficiency defficiency below 20 ng/mL at the first trimester (OR=1.42; 95% CI 1.06-1.91; p=0.02). However, this association was no longer significant when modifying the cut-off to consider 25(OH)D insufficiency.

The latest Cochrane meta-analysis also evaluated the impact of systematic vitamin D supplementation on the risk of GDM [START_REF] Palacios | Vitamin D supplementation for women during pregnancy[END_REF] (table 2). Four RCT's were included (446 pregnant women). Vitamin D supplementation was considered as a protecting event (RR 0.51 95% CI 0.27-0.97) with a moderate certainty of evidence according to GRADE methodology. Taken together, the evidence in the literature suggests that low 25(OH)D status in the first trimester of pregnancy may increase GDM risk. However, the use of different thresholds in literature reports suggests that the strength of the link between 25(OH)D concentration and the risk of GDM should be questioned.

Preterm birth

Vitamin D may play a role in the prevention of infections. Low levels of 25(OH)D seem to be associated with bacterial vaginosis during pregnancy [START_REF] Hensel | Pregnancy-specific association of vitamin D deficiency and bacterial vaginosis[END_REF], which is known to be a risk factor for preterm rupture of membranes. Several studies have therefore suggested that vitamin D supplementation during pregnancy could reduce the risk of preterm birth, but the results differ between A large observational cohort supports the idea of a protective effect of vitamin D sufficiency on preterm birth [START_REF] Bodnar | Early-pregnancy vitamin D deficiency and risk of preterm birth subtypes[END_REF], but those results are counterbalanced by other observational studies [START_REF] Baker | A nested case-control study of first-trimester maternal vitamin D status and risk for spontaneous preterm birth[END_REF][START_REF] Rodriguez | Associations of maternal circulating 25-hydroxyvitamin D3 concentration with pregnancy and birth outcomes[END_REF][START_REF] Fernández-Alonso | First-trimester maternal serum 25-hydroxyvitamin D₃ status and pregnancy outcome[END_REF]. The FEPED study evaluated the association between maternal 25(OH)D levels in the first trimester of pregnancy and the risk of preterm birth [START_REF] Monier | First Trimester Maternal Vitamin D Status and Risks of Preterm Birth and Small-For-Gestational Age[END_REF] (table 1). A total of 2813 women were included. Forty-five percent of pregnant women had a serum 25(OH)D concentration below 20 ng/mL and 27% lower than 15 ng/mL in the first trimester (1 st quartile). Low 25(OH)D concentrations were associated with obesity, ethnicity, skin color and season at blood sampling. Nearly 7% of the women included experienced a preterm birth. Logistic regression was used to estimate adjusted odds ratios (aOR) for preterm birth. OR were adjusted on maternal age, parity, BMI before pregnancy, medical or obstetrical history, smoking, ethnicity, skin color and season at blood draw. For the risk of preterm birth, the adjusted odds ratio associated with 25(OH)D concentrations below 15 ng/mL (first quartile of 25(OH)D levels) was 1.53 (95% CI 0.97-2. [START_REF] Roth | Vitamin D supplementation during pregnancy: state of the evidence from a systematic review of randomised trials[END_REF] in the whole population compared to the fourth quartile (30+ ng/mL). Analyses stratified by skin color found that dark-skinned women with low 25(OH)D levels in early pregnancy had a higher risk of preterm birth (aOR=2.89, 95% CI 1.02-8.18) compared to those with higher 25(OH)D levels. No association was found in women with lighter skin. These results are concordant with a previous study in the USA stratified by ethnicity which showed protective effects of vitamin D for spontaneous preterm births among black women, but not white women [START_REF] Bodnar | Maternal vitamin D status and spontaneous preterm birth by placental histology in the US Collaborative Perinatal Project[END_REF].

A recent Cochrane meta-analysis of seven randomized trials including a total of 1640 women (table 2) found that vitamin D supplementation may not reduce the risk of preterm birth compared to no intervention or placebo (RR 0.66, 95% CI 0.34 to 1.30). It is important to note that the numbers of women included differed greatly from one trial to another (from 19 to 486) and the baseline characteristics could have been different from one trial to another (in particular, baseline concentration of 25(OH)D).

Small-for-gestational age

Vitamin D could play a role in the regulation of genes involved in trophoblast invasion and in angiogenesis for placenta implantation, which could impact fetal growth. Several observational studies have suggested an association between low 25(OH)D levels in early pregnancy [START_REF] Bodnar | Maternal serum 25-hydroxyvitamin D concentrations are associated with small-for-gestational age births in white women[END_REF][START_REF] Ertl | Maternal serum vitamin D at 11-13 weeks in pregnancies delivering small for gestational age neonates[END_REF] and the second trimester [START_REF] Burris | Plasma 25-hydroxyvitamin D during pregnancy and small-for-gestational age in black and white infants[END_REF] and a higher risk of SGA at birth for Caucasian women. However, this difference no longer remained significant for African women [START_REF] Burris | Plasma 25-hydroxyvitamin D during pregnancy and small-for-gestational age in black and white infants[END_REF]. In the FEPED cohort, 336 women (11.9%) had an SGA neonate (birthweight <10th percentile for gestational age). No association was found between low maternal 25(OH)D levels in the first trimester of pregnancy and the risk of SGA at birth in the overall sample using clinical cut-offs and quartiles (respectively: aOR=1.10, 95% CI 0.80-1.52 for 25(OH)D levels <20 ng/mL and aOR=1.07, 95% CI 0.75-1.54 for 25(OH)D levels <15 ng/mL (first quartile) compared to the fourth quartile). Similar results were found for analysis stratified by skin color.

The impact of vitamin D supplementation has been questioned by several authors. A meta-analysis published in 2015, which included three RCT's (between 1980 and 2014) and 232 patients, did not suggest any association between vitamin D supplementation and risk for low birthweight (under 2500g) [START_REF] Pérez-López | Effect of vitamin D supplementation during pregnancy on maternal and neonatal outcomes: a systematic review and meta-analysis of randomized controlled trials[END_REF]. This meta-analysis was counterbalanced by another meta-analysis [START_REF] Roth | Vitamin D supplementation during pregnancy: state of the evidence from a systematic review of randomised trials[END_REF] which pooled 741 women of five RCT's and concluded that vitamin D supplementation could help prevent SGA birth (RR=0.60, 95% CI 0.40-0.90, I 2 =0%). The latest Cochrane meta-analysis evaluated the risk of low birthweight (under 2500g) and included 697 women from five trials [START_REF] Palacios | Vitamin D supplementation for women during pregnancy[END_REF] (table 2). The authors concluded that systematic vitamin D supplementation could help reduce the risk of low birthweight compared to no intervention or placebo (RR=0.55, 95% CI 0.35-0.87, I 2 =36%).

Conclusion

The FEPED study has highlighted that vitamin D insufficiency is highly prevalent at the beginning of pregnancy and in cord blood in the middle-north of France. The supplementation with cholecalciferol 100,000 IU during the seventh month of pregnancy is insufficient to prevent vitamin D insufficiency and deficiency in newborns and should therefore be reevaluated. In this study, a significant decrease risk of preeclampsia was associated with maternal serum 25(OH)D concentrations ≥ 30 ng/mL in the first and third trimesters (OR=0.34; 95% CI: 0.13-0.86. p=0.023). No linear relationship was found between GDM and 25(OH)D levels in the first trimester of pregnancy since GDM risk does not continuously decrease as 25(OH)D concentrations increase. Finally, no association was found between low maternal 25(OH)D levels in the first trimester and the risks of preterm birth (aOR=1.53; 95% CI: 0.97-2.43) or SGA (aOR=1.07; 95% CI: 0.75-1.54). However, subgroup analyses showed that women with darker skin color and lower 25(OH)D levels had an increased risk of preterm birth (aOR=2.89; 95% CI: 1.02-8.18 in the first quartile (<15 ng/mL) compared to the fourth quartile (30+ ng/mL).

FEPED is the first European study who explicitly assessed the effects of maternal 25(OH)D status throughout pregnancy on birth outcomes. Maternity units were at similar latitudes limiting variability in sun exposure and 25(OH)D blood was measured at three cut points of pregnancy: in the first trimester of pregnancy (10-14 weeks), during the third trimester and at birth (cord blood sampling). Additionally, strengths of the study were the large sample of women, data collected on skin color using the Fitzpatrick scale and the few missing data.

However, the FEPED study has several limitations. The first include its observational design, which means it is not possible to demonstrate a causal relationship. For the preeclampsia and GDM case-control study, which was nested in a large cohort, each case was paired with controls according to known risk factors of these diseases (parity (primipara or more), age (<35 or ≥35 years), body mass index before pregnancy (<25 kg/m2 or ≥25 kg/m2), season of conception (autumn/winter or spring/summer) and phototype (<V or ≥V Fitzpatrick scale). Although all precautions were taken in matching cases with controls, it cannot be excluded that some unknown confounding factors could have introduced a bias, even after matching for known risk factors. Moreover, the potential for residual confounding remains and, in particular, the mother's socioeconomic characteristics were not considered, but are known to be linked to preterm birth and SGA at birth.

The literature regarding potential links between 25(OH)D concentrations and/or

Vitamin D supplementation during pregnancy is controversial because of a very high heterogeneity in observational studies, trials and meta-analyses. Aggregated evidence suggests that some differences can depend on the populations studied, including skin phenotype, nutrition, physical activity and place of residence (latitude). Moreover, several studies have been compared and pooled but differ on many points: baseline levels of 25(OH)D, timing of supplementation, cut-offs set to define vitamin D insufficiency, technical measurement of 25(OH)D concentrations, and so forth. Taken together, the conclusions of most studies should be considered in light of the low grade of evidence. However, according to the biological functions of vitamin D and the literature analysis, 25(OH)D levels and/or vitamin D supplementation may interact with the risks of preeclampsia and GDM. Rigorous and high-quality randomized prospective trials should be started to solve all the mysteries of vitamin D and its interaction with major obstetrical issues.

Table 1 :

 1 Summary of the obstetrical outcomes according to the FEPED observational cohort

	Risk	Prevalence	25(OH)D levels	25(OH)D levels	p	25(OH)D insufficiency	25(OH)D insufficiency	OR [95% CI]	p
			(ng/mL)	(ng/mL)		<30 ng/mL	<30 ng/mL
			1 st trimester	1 st trimester		1 st trimester -cases	1 st trimester -controls
			cases	controls
	Preeclampsia	3.0% (84/2797)	20.1 ± 9.3	22.3 ± 11.1	0.09	83.1% (69/83)	74.0% (236/319)	0.57 [0.30-1.01] 0.09
	Risk	Prevalence	25(OH)D levels	25(OH)D levels	p	25(OH)D insufficiency	25(OH)D insufficiency	OR [95% CI]	p
			(ng/mL)	(ng/mL)		defficiency	defficiency
			1 st trimester	1 st trimester		<20 ng/mL	<20 ng/mL
			cases	controls		1 st trimester -cases	1 st trimester -controls
	Gestational	8.3% (258/2690)	21.1 ± 10.0	22.7 ± 10.0	0.03	50% (125/250)	41.9% (394/941)	1.42 [1.06-1.91] 0.02
	diabetes
	Risk	Prevalence				25(OH)D insufficiency	25(OH)D insufficiency	aOR [95% CI]
						defficiency	defficiency
						<15 ng/mL	<15 ng/mL
						1 st trimester	1 st trimester
	Gestational age	6.7% (189/2813)				<37 weeks	≥37 weeks
	at birth					34.4% (65/189)	26.6% (698/2624)	1.53 [0.97-2.43]
	Birthweight	11.9% (336/2813)				<10 th percentile	≥10 th percentile
	(<10 th percentile)					27.4% (92/336)	7.0% (669/2477)	1.07 [0.75-1.54]

Table 2 :

 2 Obstetrical outcomes regarding vitamin D supplementation for women during pregnancy (adapted from Palacios et al, Cochrane Database of Systematic Reviews 2019)

	Risk	Total events	Total events	Relative risk (95% CI) Heterogeneity	Quality of the	Studies included
		Group vitamin D	Placebo group/no		I -	evidence
			supplementation			(GRADE)
	Preeclampsia	7.7% (21/273)	16.8% (38/226)	0.48 (0.3-0.79)	I 2 =0% -p=0.91	Moderate	[44,45,56,57]
	Gestational
	diabetes

Acknowledgment

We would like to thank A. Baptiste, J. Taieb, J. Guibourdenche, H. Haidar, M.C. Haguet, D.

Masson and T. Rabeony for their consistent support and valuable feedback. We would also like to thank S. Albert, V. Buth, V. da Costa, S. Larrède, E. Etienne, L. Peaudecerf and I. Rieger for monitoring the data, and A. Bellino and M. Delattre for coordinating the study. We are grateful to the participating doctors, midwives and nurses for patient management. We would also like to thank the "Centre de Ressources Biologiques", Centre Hospitalier Intercommunal de Créteil, for the management and storage of biospecimens. We sincerely thank D. Marsh for his advice on language editing.

Fundings

The study was sponsored by the Assistance Publique-Hôpitaux de Paris (AP-HP) and was funded by a grant from the Programme Hospitalier de Recherche Publique -PHRC national 2010 (Ministry of Health -AOM10113). It is registered with the ClinicalTrials.gov identifier NCT01648842.

Declaration of interests

JC. Souberbielle reports lecture fees and/or travel/hotel expenses from DiaSorin, Roche Diagnosis, Abbott, Amgen, Shire, MSD, Lilly, and Rottapharm/Meda.

The other authors declare no conflicts of interest.


Boudissa Mehdi
email: mehdiboudissamboudissa@chu-grenoble.frnepasutiliser

Résumé:

Le système de navigation Surgivisio est un dispositif de navigation par imagerie tridimensionnelle nouvelle generation que nous utilisons depuis janvier 2018 dans notre institution pour les insertions pédiculaires lors de la chirurgie rachidienne mini invasive. L'objectif de cette note de technique est de décrire le fonctionnement du système de navigation Surgivisio, les avantages et les limites de son utilisation en chirurgie rachidienne percutanée. Les résultats de la première année d'utilisation, soit une série de 29 patients, ont été analysés pour évaluer la précision du système en terme de visée pédiculaire, mais L'objectif de cette note de technique est de décrire le fonctionnement du système de navigation Surgivisio, les avantages et les limites de son utilisation en chirurgie rachidienne percutanée. Les résultats de la première année d'utilisation ont été analysés pour évaluer la précision du système en termes de visée pédiculaire, mais aussi la durée opératoire totale et le taux d'irradiation.

Description de la technique

Tous les patients étaient opérés sous anesthésie générale en decubitus ventral sur une table carbone radio-transparante (Allen, Massachusetts, USA). Le système Surgivisio (Surgivisio Company, Gières, France) permet de réaliser de l'imagerie per-opératoire 2D et 3D ainsi qu'une navigation en temps réel grâce à une plateforme unique. La navigation se déroule en 3 étapes :

• Positionnement de la "référence patient" et centrage du "bras en C" (C-arm)

La référence est fixée sur les processus épineux à l'aide de broches (3 broches le plus souvent) connectées entre elles par un système de fixation rigide. Un fantôme de calibrage à usage unique en forme de papillon est positionné sur cette référence (figure 1).

Discussion

Ce nouveau système de navigation tout-en-un intégrant imagerie 2D et 3D avec navigation en temps réel est simple à utiliser et permet de sécuriser les visées pédiculaires en chirurgie rachidienne percutanée. La navigation peut servir en chirurgie ouverte ou en chirurgie de l'anneau pelvien, par exemple pour la réalisation de vissages ilio-sacrés. Avec 95.5% de vis pédiculaires acceptables, nos résultats semblent comparables à ceux des séries de la littérature évaluant les sytèmes de navigations concurrents. Dans une étude comparative récente entre système O-ARM et AIRO, Farah et collaborateurs retrouvaient respectivement 90.8% (O-ARM) et 92.2% (AIRO) de vis pédiculaires acceptables [START_REF] Farah | Prospective Comparative Study in Spine Surgery Between O-Arm and Airo Systems: Efficacy and Radiation Exposure[END_REF]. Selon les séries ce taux varie entre 89% et 100% [START_REF] Gelalis | Accuracy of pedicle screw placement: a systematic review of prospective in vivo studies comparing free hand, fluoroscopy guidance and navigation techniques[END_REF]. Nous pouvons supposer un effet "courbe d'apprentissage" pour les brêches médiales car survenues sur les premiers patients, au début de notre expérience [START_REF] Rivkin | Thoracolumbar instrumentation with CT-guided navigation (Oarm) in 270 consecutive patients: accuracy rates and lessons learned[END_REF][START_REF] Rahmathulla | Intraoperative image-guided spinal navigation: technical pitfalls and their avoidance[END_REF].

Avec une irradiation moyenne par vertèbre naviguée de 0.61 mSv, l'irradiation semble comparable, voir inférieure, aux systèmes de navigations concurrents évalués dans la

  aussi la durée opératoire totale et le taux d'irradiation. Conformément aux classifications de Heary et Gertzbein 95.5% des visées pédiculaires étaient satisfaisantes (107/112). La durée opératoire moyenne était de 29.3 min par vertèbre et l'irradiation moyenne par vertèbre naviguée était de 0.61 mSv. Le système Surgivisio est un nouvel outil de navigation efficace pour la visée pédiculaire en chirurgie rachidienne mini-invasive avec une irradiation et une durée opératoire moyenne acceptable pour chaque vertèbre naviguée. Niveau de preuve : II, étude de cohorte prospective Mots-clés : Chirurgie rachidienne; Imagerie 3D; Chirurgie assistée par ordinateur; Navigation; Radioprotection 1. Introduction Avec l'émergence de la chirurgie percutanée rachidienne, de nouvelles générations de système d'imagerie tri-dimensionnelle (3D) sont utilisées pour améliorer la précision des visées pédiculaires, permettant de réduire le taux de malposition entre 1 et 15% des visées [1-4]. Cette amélioration s'est faite au prix d'une irradiation supplémentaire dépendante de chaque système de navigation avec un taux moyen d'irradiation de 0.4 à 5.8 mSv dans la littérature [1-5]. Le système Surgivisio est un nouveau système d'imagerie deux et trois dimensions (2D/3D) avec navigation en temps réel de la visée pédiculaire. Simple d'utilisation, il ne requiert pas de bloquer la respiration du patient durant l'acquisition des images. Nous l'utilisons régulièrement depuis janvier 2018.

Figure 1 :

 1 Figure 1 : Le fantôme de calibrage à usage unique en "papillon" positionné sur la référence

Figure 2 :

 2 Figure 2 : Reconstruction tri-dimensionnelle, axiale, coronale et sagittale par le système

Figure 3 :

 3 Figure 3 : Matériel à usage unique de navigation. De gauche à droite : trocard de navigation,

Figure 4 :

 4 Figure 4 : Navigation du trocard et mise en correspondance du trajet du trocard sur les

Figure 5 :

 5 Figure 5 : Répartition des vertèbres naviguées et classification des fractures selon Magerl [6].

  Le taux d'irradiation en milliSievert (mSv) était calculé pour la procédure globale et pour chaque vertèbre naviguée, à l'aide du logiciel PCXMC 2.0 (Stuk, Helsinki, Finlande) ainsi que le produit dose surface (PDS) en Gy.cm 2 . La dose efficace pour le patient était évaluée par simulation de Monte Carlo[START_REF] Dedulle | Two-step validation of a Monte Carlo dosimetry framework for general radiology[END_REF]. Le repositionnement de vis pédiculaires en per-opératoire ainsi que les complications per et post-opératoires étaient recueillis. Un scanner en coupes fines millimètriques à haute résolution était réalisé en postopératoire et le positionnement des vis pédiculaires était évalué conformément aux classifications de Heary et Gertzbein par 2 observateurs indépendants[START_REF] Heary | Thoracic pedicle screws: postoperative computerized tomography scanning assessment[END_REF][START_REF] Gertzbein | Accuracy of pedicular screw placement in vivo[END_REF]. Le positionnement était jugé acceptable pour un grade de Heary ≤ 3 ou Gertzbein ≤ B[START_REF] Farah | Prospective Comparative Study in Spine Surgery Between O-Arm and Airo Systems: Efficacy and Radiation Exposure[END_REF]. En cas de discordance entre les 2 observateurs, le grade le moins bon était conservé.

	d'un scanner post-opératoire millimètrique haute résolution permettant d'évaluer 112 vis
	pédiculaires. Conformément aux classifications de Heary et Gertzbein, 95.5% des visées
	pédiculaires étaient satisfaisantes (107/112) [8,9]. Parmi les 25 effractions corticales, seules
	5 vis présentaient une brêche médiale de moins de 2 mm soit 4% des vis (5/112). Les
	résultats sont rapportés dans le tableau 2.
	Le produit dose surface moyen était de 8.07 Gy.cm² +/-5.73 (1.85-22.27). La dose efficace
	moyenne (E PCXMC ICRP 103) était de 1.97 mSv +/-1.52 (0.54-6.27). L'irradiation moyenne
	pour le chirurgien était de 0.88 mSv +/-0.78 (0.11-3.82) et l'irradiation moyenne par
	vertèbre était de 0.61 mSv +/-0.55 (0.16-2.43). Aucun repositionnement de vis n'était
	3. Résultats
	Un total de 29 patients, pour un total de 87 vertèbres naviguées et 173 vis pédiculaires,
	étaient opérés entre janvier et décembre 2018 pour une chirurgie percutanée à l'aide du
	système Surgivisio. Les données démographiques et cliniques, les indications chirurgicales, la
	distribution des fractures et leur classification sont présentées dans le tableau 1 et la figure
	5. La durée opératoire moyenne était de 95 min +/-37.1 (39-180) pour la procédure globale

• Acquisition d'images et reconstructions 3D Une trajectoire optimisée du C-arm est programmée automatiquement pour maximiser le volume reconstruit tout en évitant les collisions avec la table opératoire et en réduisant la dose d'irradiation grâce à l'acquisition de 90 ou 180 projections selon le choix de l'opérateur. L'équipe chirurgicale se protège des rayons X derrière un écran plombé durant cette étape. Le fantôme de calibrage est détecté automatiquement sur chaque projection et l'image 3D reconstruite en compensant les mouvements éventuels du patient dont sa respiration (Figure

2

). • Navigation Le fantôme de calibrage est remplacé par un traqueur optique et une mise en correspondance entre le traqueur et l'imagerie 3D est générée automatiquement. La navigation est réalisée à l'aide d'un trocard spécifique à usage unique équipé de traqueurs (figure 3). Le chirurgien peut alors naviguer en temps réel la position du trocard dans les images reconstruites, et réaliser une visée pédiculaire naviguée. La mesure de la longueur des vis se fait sur la graduation du trajet du trocard navigué au sein de l'image reconstruite (figure 4). La fixation pédiculaire était réalisée à l'aide du système ES2 (Stryker, Pusignan, France) ou du système Longitude2 (Medtronic, Boulogne-Billancourt, France) avec des vis cimentables quand cela était jugé necessaire. Entre janvier et décembre 2018, tous les patients opérés en chirurgie percutanée naviguée par le système Surgivisio étaient analysés. Les données démographiques, épidémiologiques et fracturaires selon la classification de Magerl, l'indication chirurgicale et la stratégie chirurgicale étaient recueillies [6]. La chirurgie était réalisée par trois chirurgiens expérimentés en chirurgie rachidienne. Les durées opératoires globales et par vertèbre naviguée étaient recueillies. L'analyse statistique était réalisée à l'aide du logiciel StatView 5.5 (SAS Institute, Cary, NC, USA). Les variables qualitatives étaient comparées par un test de Chi-2 et les variables quantitatives étaient comparées par un test de Student. Une valeur de p inférieure à 0.05 était considérée comme statistiquement significative. et de 29.3 min +/-18.7 (12-90) par vertèbre naviguée. Le nombre moyen de vertèbres instrumentées par procédure était de 3 +/-1 (2-5). Parmi les 29 patients, 20 ont bénéficié nécessaire en per-opératoire. Aucune complication et aucune ré-intervention n'étaient enregistrées durant le suivi.