Shijie Zhu

Céline Bouby

Abel Cherouat

Ben Zineb

3D reconstitution and numerical analysis of superelastic behavior of porous shape memory alloy

Keywords: Shape Memory Alloy, Foams, 3D reconstitution, Superelasticity

In this work, a CAD 3D foam structure of Ni-Ti Shape Memory Alloy (SMA) is rebuilt using ellipsoid unit cells in order to study the superelastic behavior of porous SMAs. A taking and placing algorithm based on a uniform and a normal distributions is adopted for the 3D reconstitution process. In the design sample, dimensions, positions and orientations of the ellipsoid unit cells are randomly selected. A SMA constitutive model including phase transformation, martensitic reorientation and twins accommodation is selected to simulate the superelastic behavior of the SMA foam. As an application, the compression of SMA porous sample with random ellipsoid unit cell distribution is simulated by finite element in order to show the efficiency of the proposed methodology to predict its superelastic response and to analyze the porosity effect. The obtained results show that the force-displacement or stress-strain responses of the porous material are highly depend on the foam porosity. The effects of porosity, size, orientation, ratio of long axis and short axis, and distribution of unit cell on the superelastic behavior of SMA porous material are also discussed in both closed cells and open cells cases.

Introduction

As a new class of materials, metallic foams have attracted increasing interest in different fields of engineering in the last past decades [START_REF] Gibson | Cellular solids: structure and properties[END_REF][START_REF] Ashby | Metal foams: a design guide[END_REF][START_REF] Deshpande | Isotropic constitutive models for metallic foams[END_REF][START_REF] Forest | Continuum modeling of strain localization phenomena in metallic foams[END_REF]. These kind of materials are cellular structures containing solid and gaseous phase, where the cavities occupy a certain percentage of the overall volume. Based on the morphology of their internal cells, the pores can be sealed (closed cell foam) or interconnected (open cell foam). They are particularly versatile because of their interesting mechanical and physical properties: for example, the relatively low density make it possible to obtain a high stiffness/weight ratio, the existence of cavities results in the ability to energy absorption and damping, and also gives them thermal and acoustic insulation properties [START_REF] Gibson | Cellular solids: structure and properties[END_REF][START_REF] Ashby | Metal foams: a design guide[END_REF].

As a well-known material for reversible inelastic deformation, Shape Memory Alloys (SMAs) have been paid attention on over last few years [START_REF] Otsuka | Shape memory materials[END_REF][START_REF] Lagoudas | Shape memory alloys: modeling and engineering applications[END_REF][START_REF] Lexcellent | Shape-memory alloys handbook[END_REF]. Thanks to the thermoelastic martensitic transformation, SMA materials exhibit two main important properties: superelasticity and shape memory effect. Due to the possibility of these behaviors, it could become interesting if this kind of material can be combined with an attractive structure like porous SMAs. Cellular structures in SMAs are particularly interesting for their potential to provide superelasticity and shape memory effect in a light-weight material. Lot of processes have been investigated to fabricate the SMA foams [START_REF] Bansiddhi | Shape-memory NiTi foams produced by solidstate replication with NaF[END_REF][START_REF] Bansiddhi | Shape-memory NiTi foams produced by replication of NaCl space-holders[END_REF][START_REF] Castrodeza | Processing of shape memory CuZnAl open-cell foam by molten metal infiltration[END_REF][START_REF] Bansiddhi | Processing of NiTi foams by transient liquid phase sintering[END_REF]. Besides, many experiments and characterizations have been done to understand the properties of SMA foams [START_REF] Nemat-Nasser | Experimental characterization and micromechanical modeling of superelastic response of a porous NiTi shape-memory alloy[END_REF][START_REF] Wu | Pore formation mechanism and characterization of porous NiTi shape memory alloys synthesized by capsule-free hot isostatic pressing[END_REF][START_REF] Scalzo | Mechanical and microstructural characterization of porous NiTi shape memory alloys[END_REF]. To complete these experimental data on SMA foams, numerical modeling is needed to predict accurately the properties of this kind of materials. In particular, it might be very useful to study the effective macroscopic behaviors of SMA foams depending on their microstructure.

It is obvious that microstructural properties have an influence on macroscopic behaviors, essential for designers, but it is still difficult to derive macroscopic behaviors from microstructure. In this case, computational analyses based on microstructure of porous SMA are undoubtedly necessary. Some approaches adopted to build the microstructure of porous SMA are based on the assumption of a periodic distribution of pores. Qidwai et al. [START_REF] Qidwai | Modeling of the thermomechanical behavior of porous shape memory alloys[END_REF] used the unit cell finite element method (UCFEM) to display periodic distribution of pores in SMA matrix by cutting an in-plane hexagonal arrangement of pores. Sepe et al. [START_REF] Sepe | Micromechanical analysis of porous SMA[END_REF] have also done some computational micromechanical analyses of porous SMA to investigate their properties. They used a porous 3D unit cell with transition from spherical to ellipsoidal, then to cylindrical void (at the same time, from closed to open void) and in particular, in terms of the parametric studies, they simplified the model into a 2D one instead. Moreover, Karamooz Ravari et al. [START_REF] Ravari | A unit cell model for simulating the stress-strain response of porous shape memory alloys[END_REF] used a unit cell model with repetitive pores to simulate the stress-strain response of porous SMA by assuming the porous microstructure as an interpenetration of spheres. However, the assumption of a periodic distribution of pores seems to be far away from the irregularity of real microstructures, which will cause the overestimation of material response. In addition, although the material characteristics can be captured, the approach of Sepe et al. [START_REF] Sepe | Micromechanical analysis of porous SMA[END_REF] is limited by the fact that only two-dimension model is considered.

In order to create a random model instead of a periodic one, De Giorgi and Qidwai [START_REF] De Giorgi | A computational mesoscale evaluation of material characteristics of porous shape memory alloys[END_REF] began some researches on mesoscale analysis techniques based on a two-dimensional finite element method, in which pores are allowed to vary in size and shape without limit on maximum pore size. Panico and Brinson [START_REF] Panico | Computational modeling of porous shape memory alloys[END_REF] used a Representative Volume Element (RVE), by assigning randomly to some elements elastic material properties with negligible stiffness. They also studied another kind of porous SMA in Bewerse et al. [START_REF] Bewerse | Porous shape-memory niti-nb with microchannel arrays[END_REF] with a structured pores obtained starting from scaffold in magnesium. This leads to pores with vertical and horizontal ribbon shape. In addition, Ashrafi et al. [START_REF] Ashrafi | A 3-D constitutive model for pressure-dependent phase transformation of porous shape memory alloys[END_REF][START_REF] Ashrafi | Theoretical and numerical modeling of dense and porous shape memory 33 alloys accounting for coupling effects of plasticity and transformation[END_REF] and Maîtrejean et al. [START_REF] Maîtrejean | Density dependence of the superelastic behavior of porous shape memory alloys: representative volume element and scaling relation approaches[END_REF] used a similar RVE approach where the brick element is also randomly picked. These approaches allow to create the irregularity of foam microstructure, however, the irregularity seems not to satisfy the nature smooth and coherent of inner surfaces encountered in the real microstructure.

In this paper, the design of porous structure (open or closed ellipsoid cell), by using a random "taking" and "placing" algorithm and uniform or normal distribution [START_REF] De Giorgi | Aluminium foams structural modelling[END_REF], is proposed to build the internal microstructure of porous SMAs.

The macroscopic compressive superelastic behavior of NiTi alloy foams is evaluated. In addition of considering an SMA sample with superelastic behavior, the other novelties come from: (i) generating randomly open and closed porous samples and perform an exhaustive analysis, of the porosity control and optimization (arrangement, size, orientation and volume fraction), (ii) applying the Monte Carlo method in order to predict and optimize the discrete porosity obtained by meshing of the CAD structure, and choosing a normal distribution with a specific limit, (iii) modeling with superelastic thermo-mechanical behavior under loading/unloading for different periodic and random pore distribution considering the effect of their shape, size, and orientation.

For the organization of this paper, section 2 presents the SMA model developed by Duval and Chemisky et al. [START_REF] Duval | Modelling of localization and propagation of phase transformation in superelastic SMA by a gradient nonlocal approach[END_REF][START_REF] Chemisky | Constitutive model for shape memory alloys including phase transformation, martensitic reorientation and twins accommodation[END_REF] which is used to predict the thermo-mechanical response of a sample of both closed-cells and open-cells NiTi foams. Section 3.1 shows the procedure of 3D design of foam structure using Monte Carlo and CAD method. In Section 3.2, a numerical simulation of the sample superelastic behavior is presented for a compressive loading-unloading.

In Section 4, the effects of porosity, size, orientation, distribution and ratio of long axis and short axis of unit cell on the superelastic behavior of SMA foams are also numerically studied.

Constitutive equations of Shape Memory Alloy

In this paper, a constitutive model for SMA including phase transformation, and martensitic reorientation developed by Duval and Chemisky (see [START_REF] Duval | Modelling of localization and propagation of phase transformation in superelastic SMA by a gradient nonlocal approach[END_REF][START_REF] Chemisky | Constitutive model for shape memory alloys including phase transformation, martensitic reorientation and twins accommodation[END_REF]) is used to simulate the superelastic behavior of the SMA foam. Thus, two internal variables are defined: the martensite volume fraction f (0 ≤ f ≤ 1), the mean transformation strain tensor in martensite volume ε T .

The total macroscopic strain tensor can be decomposed as:

ε = ε e + ε th + ε T (1)
where (ε e , ε th , ε T ) are respectively the elastic strain, the thermal expansion strain, the inelastic strain due to martensitic transformation.

Further, the equation ( 1) can be expanded as:

ε = S : σ + α(T -T ref ) + f ε T (2)
where S is the isotropic fourth order compliance tensor and σ is the mean stress tensor, α is the isotropic thermal expansion tensor, T ref is the reference temperature and T is the current temperature.

The derivation of the phenomenological constitutive equations for SMAs begins with the choice of a thermodynamic free energy potential. The Clausius-Duhem inequality can be written in terms of the Gibbs free energy:

- ∂G ∂σ : σ - ∂G ∂T Ṫ - ∂G ∂V k • Vk -σ : ε -q • --→ gradT T ≥ 0 (3)
where V k are internal variables, q is the heat flux vector and the Gibbs free energy potential G (σ, T, V k ) is defined as follows:

G =U A -T S A + B(T -T 0 )f - 1 2 σ : S : σ -σ : α(T -T ref ) -σ : ε T f + 1 2 f H ε ε T : ε T + 1 2 H f f 2 + C v (T -T 0 ) -T log T T 0 (4)
where B = -∆S = -(S M -S A ), the equilibrium temperature of transformation T 0 = ∆U ∆S = U M -U A S M -S A (U A and U M are the internal energies of the austenitic and martensitic phases, S A and S M are the entropy of the austenitic and martensitic phases), (H ε and H f ) are constant factors describing interactions between grains and variants, C v is the heat flux for the RVE.

The force variables linked to internal variables V k (f, ε T ) are identified as:

-Transformation force linked to f :

F f = - ∂G ∂f = σ : ε T -B(T -T 0 ) - 1 2 H T ε ε T : ε T -H f f (5)
-Orientation force linked to ε T : The evolution laws of the internal variables can be obtained from the definition of a dissipation potential (see Table . 1):

F ε T = - ∂G ∂ε T = f σ d -f H T ε ε T (6)
-For the transformation force:

-

If |F f | < F crit f or |F f | = F crit f and | d|F f |
dt | < 0, the transformation force is not activated. In this case, the linked variable is constant, therefore ḟ = 0.

- In this case, ḟ = 0.

If |F f | = F crit
-For the orientation force:

-If |F ε T | < F crit ε T or |F ε T | = F crit ε T and | d|F ε T |
dt | < 0, there is no orientation. In this case, if the domain does not change, εT = 0. Otherwise, if the domain evolve, εT evolves to respect the equilibrium.

-

If |F ε T | = F crit ε T and | d|F ε T |
dt | = 0, the orientation force is activated. In this case, εT = 0.

Table 2 summarizes the constitutive equations which leads to a non linear differential system of first order, solved by the Newton-Raphson scheme as in [START_REF] Chemisky | Numerical tool for SMA material simulation: application to composite structure design[END_REF]. The activation of the different specific behaviors of the SMA are detected after an elastic prediction. A corresponding correction is carried out in order to lead to the final behavior.

ε = ε e + ε th + ε T
Thermodynamical potential:

G = U A -T S A + B(T -T 0 )f -1 2 σ : S : σ -σ : α(T -T ref ) -σ : ε T f + 1 2 f H ε ε T : ε T + 1 2 H f f 2 + C v (T -T 0 ) -T log T T0
Clausius-Duhem inequality:

-∂G ∂σ : σ -∂G ∂T Ṫ -∂G ∂V k • Vk -σ : ε -q • --→ gradT T ≥ 0
Thermo-elastic balance conditions:

S : σ + α(T -T ref ) + f ε T -ε = 0 S A -f B + σ : α -C v log T T 0 = 0
Thermodynamics forces:

F f = -∂G ∂f = σ : ε T -B(T -T 0 ) -1 2 H T ε ε T : ε T -H f f F ε T = -∂G ∂ε T = f σ d -f H T ε ε T Criterion functions: |F f | ≤ F yield f |F ε T | ≤ f F yield ε T
It is worth mentioning that the SMA model has been implemented in ABAQUS via user's material UMAT, which allows us to carry out the simulations by FE on the sample.

Numerical analysis of shape memory NiTi foams

In this section, an iterative 3D reconstitution of foam structure using CAD method is detailed, which will be used for numerical analysis of superelastic behavior of porous SMA sample under compression load.

3D reconstitution of foam structure

The sample contains a series of ellipsoids with random variable ranges: dimension, position and orientation. Each ellipsoid is located randomly. In order to not repeat again the existing position, a specific checking method of intersection is developed. Two conditions should be taken into consideration: firstly, the ellipsoid to be checked must be intersected with at least one of the existing valid ellipsoids; secondly, the intersecting part of the two ellipsoids should not occupy a large part of the existing valid ellipsoids because of the foam structure authenticity. The real porosity was estimated by the analysis of digital images of flat sections of a given microstructure. By the fact, pores dimensions and distribution in the flat section is assumed to be close to the real tridimensional one for a random pore distributions and sizes. Based on the obtained random sizes and distribution of pores, the realistic microstructure statistical distribution is performed. In order to build porous structure with random cell morphology, an ellipsoid shape where two of the three axes are the same is used to define the foam pore 150 using the following parameters (see Fig. 1):

✞ ✁ ✂ ✄☎ ✆ ✝✟ ✠ ✡ ☛ ☞ ✌ ✍ P c (✎ ✏ | ✑ ✒ , ✓ ✔ , ✕ ✖ ) C 0, c ✗ ✘✙ ✚ ✛✜ ✢ ✣✤ ✥ ✦ ✧ ★ ✩ ✪ C 0, e ✫ ✬ ✭ ✮ ✯ ✰ [M c ] [M e ]
Position: C 0 (x 0 , y 0 , z 0 ) are the coordinates of the ellipsoid center; Dimension: (a, b) are the lengths of the semi-major and the semi-minor axis; Orientation: (α, β, γ) are the orientation angles of the ellipsoid (Euler angles) (see Fig.

2).

155

The flowchart of the geometric modeling of foams with random distributions of ellipsoidal cells using MATLAB and Abaqus/CAE is shown on Fig. 3. The basic steps of the taking and placing algorithm are described thereafter.

Step 1 Definition of porous sample ellipsoid characteristics

To obtain a foam structure, we define a series of ellipsoids with random To this end, a method of intersection checking is developed. Two conditions should be taken into consideration: firstly, the ellipsoid to be checked must be intersected with at least one of the existing valid ellipsoids; secondly, the intersecting part of the two ellipsoids should not occupy a large part of the existing valid ellipsoid because of the foam structure authenticity. On the basis of these two conditions, the points on the surface of the ellipsoid to be checked are used to verify whether and how much the ellipsoid to be checked intersects with the existing valid ellipsoid. The coordinates of each point P, using spherical coordinate system of the ellipsoid to be checked (e xc , e yc and e zc ), are defined as (see Fig. 1): and the orientation angles (α, β, γ) are:

         x c = a c cos θ c
205 [M c ] or [M e ] =      M 11 M 12 M 13 M 21 M 22 M 23 M 31 M 32 M 33                                                   M 11 = cos α cos γ -sin α cos β sin γ M 12 = cos α sin γ -sin α cos β cos γ M 13 = sin α sin β M 21 = -sin α cos γ -cos α cos β sin γ M 22 = -sin α sin γ + cos α cos β cos γ M 23 = cos α sin β M 31 = sin β sin γ M 32 = -sin β cos γ M 33 = cos β (8)
Therefore, the coordinates of the ellipsoid to be checked after transformation are:

     x e y e z e      = [M e ] -1      [M c ]      x c y c z c      +      x 0,c -x 0,e y 0,c -y 0,e z 0,c -z 0,e           (9)
where C 0,c (x 0,c , y 0,c , z 0,c ) and C 0,e (x 0,e , y 0,e , z 0,e ) are respectively the coordinates of the ellipsoid centers in the sample coordinate system (see Fig. 1).

210

The verification is done using the equation: -π ≤ ϕ < π, and the spacing as 2π 64 , 1984 points to be tested are created for one ellipsoid to be checked. Here, only 200 points are considered as the upper limit for the intersecting of the ellipsoid to be checked and the existing valid ellipsoid. This is a compromise between computational time of porous foam reconstruction and validation of intersection of the ellipsoid to be checked.

Step 3 Control of the porosity

It is difficult to calculate exactly the porosity of open cells foam even if

all the parameters of the ellipsoid unit cells are known. It is due to the complexity of the intersections between two ellipsoids.

To solve this problem, a statistical method is used to approximate the porosity. First, a point P p (x p , y p , z p ) is generated randomly inside the sample cube by a uniform distribution. Then using the coordinate transformation, the coordinates of P p in the coordinate system of the existing valid ellipsoid (e xe , e ye and e ze ) is:

     x pe y pe z pe      = [M e ] -1      x p -x 0,e y p -y 0,e z p -z 0,e      (11)
and the equation for verification is:

ξ p = x 2 pe a 2 e + y 2 pe b 2 e + z 2 pe b 2 e ( 12
)
If ξ p < 1, then the point P p is inside of the existing valid ellipsoid. In this way, we note the successful index S as 1, otherwise we note S as 0.

Furthermore, the Monte Carlo method is applied to predict the porosity [START_REF] Reichenberger | Monte Carlo simulation of random, porous (foam) structures for neutron detection[END_REF]. Precisely, a large number of the times to generate P p is executed, here we define it as 10000. By calculating how many times T in the situation where S equals 1, and by calculating T 10000 , the porosity of foam structure could be well approximated.

Step 4 Numerical models

On the basis of the secondary development of Abaqus language script Python, by repeating the process of generating ellipsoid pores, the foam structure is built on the following steps:

For the detailed steps:

Iteration 1 A cube of sample is built (see Fig. 3).

Iteration 2 Three points inside the cube are chosen randomly to constitute a plane of sketch. The coordinates of these points are used to calculate the orientation angles (α, β, γ) in order to satisfy the conditions discussed above.

Iteration 3 An approximate semi ellipse is drawn in the plane of sketch by using (a, b). By executing revolved cut in the cube, an ellipsoid pore can be generated.

By repeating the iterations 2 and 3 for every existing valid ellipsoid unit cell, the whole foam structure CAD is finally constructed.

Fig. 5 reports the level steps of a foam CAD reconstitution with 30 pores and the tetrahedral finite element mesh of porous sample using Optiform software [START_REF] Borouchaki | Enhancement of the accuracy of numerical eld computation using an adaptive three-dimensional remeshing scheme[END_REF][START_REF] Borouchaki | Improved 3d adaptive remeshing scheme applied in high electromagnetic field gradient computation[END_REF][START_REF] Zhang | 3d thermo-mechanical simulation coupled with adaptive remeshing for metal milling[END_REF][START_REF] Cherouat | Simulation of sheet metal forming processes using a fully rheological-damage constitutive model coupling and a specific 3d remeshing method[END_REF] .

FEM model and boundary conditions

Numerical simulation of super-elastic porous sample response under a compression loading-unloading and a constant temperature involves the use of the finite element analysis. It allows to compute, for each applied external load increment, the successive new body configurations verifying the equilibrium equations as well as the constitutive equations accounting for the above discussed coupled mechanical phenomena. For SMAs it consider phase transformation, martensitic reorientation. A contact with friction is considered between each rigid plane and one of the both horizontal sample edges (see Fig. 6). One of the rigid planes is fixed where the second is moved with control of the vertical

Numerical analysis of superelastic behavior

The predicted compression loading and unloading force versus displacement results obtained by the proposed methodology are presented in Fig. 7. We can notice that a typical form of the superelastic behavior response of a porous SMA is observed. Several key points, including the effective stiffness k slope (the slope of the line), some important forces in both loading and unloading procedure (the points A, B, C and D) and the dissipated energy (the area of the curve), are chosen here to reveal the superelastic behavior of porous SMA, which will be studied in the following sections. Fig. 8 presents the volume fraction of martensite for different tool displacements. It can be seen that at the largest compression level displacement, almost all of the sample are transformed.

SMA foam parameters study

Considering the fact that the performance of porous materials can be affected by the pore shape, size, distribution and orientation, different simulations are carried out to study these effects on the overall response of the sample. Here, the studies on both open cells and closed cells (see Fig. 9) are investigated and compared with other results published in [START_REF] Sepe | Micromechanical analysis of porous SMA[END_REF], [START_REF] Ravari | A unit cell model for simulating the stress-strain response of porous shape memory alloys[END_REF] and [START_REF] Dehaghani | 3D hierarchical multiscale analysis of heterogeneous SMA based materials[END_REF]. As the porosity volume fraction depends on the mesh of foam CAD, the mesh is controlled here according to the quality of the elements, the geometric curvatures of the surfaces of the pores and the gradation of the mesh inside the domain. The real porosity of the discrete CAD of the cell foam depends on the mesh size. If the mesh is fine then the porosity changes very little but the calculation time is very important. We adopt the meshes according to the desired porosity, the quality of the elements and the computation times. The table 3 shows the error between the real porosity and the discrete porosity for open and closed foams. We can note that for the proposed mesh in this paper, the average error is of the order of 7% for open cells and 11% for closed cells.

The samples are meshed by home developed tool and is applied to each sample with a given volume fraction porosity leading to specific mesh for each sample. porosities for open cells. Also, it is observed that for porous SMA structure, the transition from elastic to phase transformation behavior is smoother than that 325 of bulk one. This phenomenon is due to the inhomogeneity of stress distribution in porous SMAs. The Fig. 11 shows that with the increasing of porosity, the effective stiffness k slope and the dissipated energy decrease. This is easy to understand because less material in total leads to less macroscopic stress. In addition, it can be seen that the hardening rate in phase transformation region increases with the increase of porosity. This is due to the reduction of the ratio of stresses for the beginning and the end of martensitic transformation as well as the elastic modulus. According to [START_REF] Ravari | A unit cell model for simulating the stress-strain response of porous shape memory alloys[END_REF], the increase of the porosity also leads to the increase of the stress concentration around the pores, which results in strain localization. Consequently, in order to obtain a specific strain value, smaller macroscopic stress for larger porosity is required, which demonstrates a smaller hardening rate. Another reason is that the increase of the porosity causes the reduction of martensite and austenite elastic modulus, resulting in the decrease of the total stiffness of the material. All the obtained results show the same tendency as those of the references [START_REF] Sepe | Micromechanical analysis of porous SMA[END_REF] and [START_REF] Ravari | A unit cell model for simulating the stress-strain response of porous shape memory alloys[END_REF]. It should be noticed that for the effects of these parameters, a periodic structure is more appropriate to exploit the results. Accordingly, the unit cell will be arranged periodically in the whole sample for every foam structure in the following to investigate the superelastic behavior.

Closed cell foam

4.2.

Pore size effect of periodic foam

Open cell foam

For a periodic structure porous SMA with constant porosity ρ = 80%, three unit cell sizes (a = 7.51, 3.76 and 2.50 mm) of the long axis (see Fig. 13) are studied numerically. Fig. 14 shows the predicted compression force-displacement curves. The results during both loading and unloading, show that when the long axis a decreases (which corresponds to a reduction of the size of unit cell), the effective stiffness and the forces in typical points (A, B, C and D) both increase. This could be explained by a decrease of the unit cell size leading to an increase of the stress localization around the pores of high porosity of foam (80%). In fact, our conclusion shows a greatly difference with that in [START_REF] Ravari | A unit cell model for simulating the stress-strain response of porous shape memory alloys[END_REF] in which the foam porosity is very small (13%).

Closed cell foam

In this section, the sample porosity is ρ = 30% and three unit cell sizes of the long axis (a = 4.95, 2.48 and 1.65 mm) of porous SMA (see Fig. 13) are investigated numerically. The obtained results of the predicted compression loading and unloading force-displacement allow to infer that in the closed cells, the long axis size of the periodic unit cell has no significant effect on the effective stiffness, the forces in typical points and the dissipated energy (see Fig. 15).

Pore axial ratio effect of periodic foam 4.3.1. Open cell foam

The effect of the ratio of long axis and short axis of periodic foam (r = a/b) is also studied for open cells (ρ = 80%). Fig. 17 shows that the obtained predicted force-displacement curves for different ratio values, r = 1 (sphere), r = 1.15 and r = 1.3 (ellipse) respectively (see Fig. 16). According to the results, the ratio r affects the effective stiffness and the forces in typical points. The results can be compared with those in [START_REF] Sepe | Micromechanical analysis of porous SMA[END_REF] and [START_REF] Dehaghani | 3D hierarchical multiscale analysis of heterogeneous SMA based materials[END_REF] even though they use only one unit cell resulting in the SMA foam with low porosity. Indeed, our results

show the same trend as those of the references [START_REF] Sepe | Micromechanical analysis of porous SMA[END_REF] and [START_REF] Dehaghani | 3D hierarchical multiscale analysis of heterogeneous SMA based materials[END_REF]. This could be explained by the increase of the ratio of long axis and short axis which leads to the increase of the stress concentration (estimated at about 5000 MPa, see Fig. 18). The difference should come from the fact that the direction of long axis of elliptical pores is orthogonal to that of the imposed displacement. Therefore, for the foam with higher ratio, the forward/reverse transformation occurs for a lower stress, resulting in a lower stress in the start and the finish of the forward/reverse transformation (points A, B, C and D). It is worth mentioning that this explanation is supported by the reference [START_REF] Dehaghani | 3D hierarchical multiscale analysis of heterogeneous SMA based materials[END_REF], where the distributions of the martensitic volume fraction show that the sample, with a higher ratio, presents the earliest starting martensitic transformation.

Closed cell foam

In this section, the effect of the ratio is investigated for closed periodic cell foam. Fig. 19 displays the obtained predicted force-displacement curves for different ratio values, r = 1 (sphere), r = 1.15 and r = 1.3 (ellipse) respectively.

The results reflect that the ratio of long axis and short axis does not have an apparent effect on the overall performance of SMA foams in the case of closed cells. In order to better investigate the effect on the internal structure, the distributions of the martensite volume fraction at the end of loading (0.8 mm) for the three ratios are shown in Fig. 20. It can be seen that for the three samples with different ratios, most parts of them have completed the martensite transformation, while for some areas between the pores in the horizontal direction and the areas in the bottom, the martensite transformation just starts. If we consider the distribution of the martensite volume fraction in the main view (in the X-Y plan) for the three samples, it can be observed that there is not a big difference among them, which might proves that, the effect of the ratio is not apparent for closed cells. In order to study the effect of the orientation of unit cell pore, the angle between the directions of long axis and the horizontal directions is defined by an angle θ. The periodic model with the same porosity (ρ = 80%), same size (a = 2.50 mm) and same ratio (r = 1.3) is used (see Fig. 21). Fig. 22 displays the numerical compressive force-displacement curves for different orientations θ = 0 o , 45 o and 90 o respectively. Based on these results, one can note that, the increase of the orientation leads to an increase of the effective stiffness and the forces in typical points. It can be observed that, with the increase of the orientation, the length of pores in the longitudinal direction (here defined as a ) decreases, while that in the transverse direction (here defined as b ) increases (see Fig. 21). From this point of view, the study of the orientation effect can be regarded as that of the effect of long axis' and short axis' varieties. Thus, our results can be compared with those in [START_REF] Sepe | Micromechanical analysis of porous SMA[END_REF] and [START_REF] Dehaghani | 3D hierarchical multiscale analysis of heterogeneous SMA based materials[END_REF], which provide the cases a > b and a < b corresponding to the orientations 0 o and 90 o . The result presented in the references [START_REF] Sepe | Micromechanical analysis of porous SMA[END_REF] and [START_REF] Dehaghani | 3D hierarchical multiscale analysis of heterogeneous SMA based materials[END_REF] show the similar tendencies.

Closed cell foam

In this section, the porosity of the periodic model is set at ρ =30%. The results of the numerical compressive response (see Fig. Fig. 26 shows the obtained predicted force-displacement curves for the three different types of distribution, from case 1 to case 3. According to the results, in both loading and unloading steps, the effective stiffness and the forces in typical points in case 1 are higher than those in cases 2 and 3. The reason could be that the inhomogeneity of pore distribution in cases 2 and 3 causes an increasing of the stress concentration, which makes their effective stiffness and forces lower than those of homogeneous pore distribution case 1. Here, it can be said that case 3 (bigger pore size in the longitudinal direction) is less influenced by the transverse loading direction than case 2 (bigger pore size in the transverse direction). that the superelastic behavior of porous SMA also depends on whether the unit cells intersect with each other or not. It exhibits that the connectivity of unit cell in the foam structure is also an important parameter for the performance of this kind of material, which in addition provides the idea for foam structure design.

Conclusion

A study of 3D foam microstructure reconstitution and behavior has been

  Activated transformation Activated orientation Evolution of f and ε T σ d denotes the deviatoric part of the stress tensor σ.

  dt | = 0, the transformation force is activated.

  130

Figure 1 :Figure 2 :

 12 Figure 1: The ellipsoids with geometrical parameters.

160Figure 3 :

 3 Figure 3: Flowchart of foam modeling with random distributions of ellipsoid pores.

Figure 4 :Step 2

 42 Figure 4: Normal distribution for a specific limit for a and b.

  y c = b c sin θ c cos ϕ c z c = b c sin θ c sin ϕ c (7) where θ c is the polar angle and ϕ c the azimuth angle, (a c , b c ) are positive real numbers. Considering that the ellipsoid to be checked and the existing valid ellipsoid have different coordinate systems, based on their standard ellipsoid equations, the coordinate transformation related to the orientation angles of the ellipsoid (α, β, γ) is definitely needed. The transformation matrix is defined from the coordinate system of the sample cube (e x , e y and e z ) to that of the ellipsoid to be checked (e xc , e yc and e zc ) as [M c ], and that from the sample cube (e x , e y and e z ) to the existing valid ellipsoid (e xe , e ye and e ze ) as [M e ] (see Fig. 1 and Fig. 2). It should be noticed that the matrices [M c ] and [M e ] can be derived from the definition of the ellipsoid as the orientation angles (Euler angles) are well known for every ellipsoid. The relation between the transformation matrix [M c ] or [M e ]

  ξ < 1, the point P on the surface of the ellipsoid to be checked is inside the existing valid ellipsoid. By setting the range -π 2 < θ < π 2 ,

Figure 5 :

 5 Figure 5: Evolution of the sample morphology in function of the number of pores (a) to (e) and adapted porous structure mesh (f).

  displacement. It consists to incrementally apply a displacement u=1.5mm during loading and unloading. Its horizontal displacement is assumed to be equal to zero. This defines a non-linear Initial and Boundary Value Problem based on the Updated Lagrangian description, which needs to be incrementally linearized in order to find the roots using an iterative implicit scheme. All the examples presented herein have been obtained with Abaqus/Implicit thanks to the use of the developed UMAT user subroutine describing the SMA thermos-mechanical behavior. Our main goal is to illustrate the capability of the proposed methodology to predict the superelastic response of a porous SMA structure (effective stiffness, volume fraction of martensite evolution, forces in both loading and unloading and the dissipated energy) during isothermal and quasi-static compression. The two rigid planes was modeled by two discrete rigid surfaces using 4-node rigid shell elements (R3D4). A surface-to-surface contact with friction (µ = 0.1) was introduced to model the interaction between the platen and the specimen. The porous NiTi SMA structure with different porosities has a dimension of 10mm×10mm×10mm and is discretized by deformable continuum linear tetrahedral finite elements (C3D4). The thermomechanical material parameters, detailed in[START_REF] Chemisky | Constitutive model for shape memory alloys including phase transformation, martensitic reorientation and twins accommodation[END_REF], are adopted to study the superelasticity of the porous SMA at the austenitic initial state. Therefore, the mechanical loading/unloading under a constant temperature of T = 30 • C is simulated by Abaqus Implicit.

Figure 6 :

 6 Figure 6: FEM model and boundary conditions

Figure 7 :Figure 8 :

 78 Figure 7: Force-displacement response.

Figure 9 :

 9 Figure 9: Studied pores morphology

4. 1 .

 1 Porosity effect of random foams 4.1.1. Open cell foam In this section, the predicted compression loading and unloading force for different porosity values, ρ = 0 (bulk), 40, 60, 70 and 80% respectively, is presented. In Fig. 10, some typical points named A, B, C and D are chosen

Figure 10 :

 10 Figure 10: Compressive force vs displacement for different porosities of open cells.

Figure 11 :

 11 Figure 11: The effect of porosity on the SMA open cell foam on (a) effective stiffness and (b) dissipated energy.

Figure 12 :

 12 Figure 12: Compressive force vs displacement for different porosities of closed cells.

Fig. 12

 12 Fig. 12 shows the predicted loading and unloading compression force-displacement curves for different random porosity values of closed cells, ρ = 0 (bulk), 20, 30, 40% respectively. It should be noticed that, according to the distribution theory, the maximum porosity of the random closed cell model is about 50% [35]. The obtained results imply that the porosity of the porous material affects the mechanical properties of foam structure in term of stiffness, flexibility and damping (see Table.5). It shows the similar tendencies observed with the open cells,

  [START_REF] Otsuka | Shape memory materials[END_REF]. It shows the similar tendencies observed with the open cells, which are: the decreasing of the effective stiffness, the forces in typical points (A, B, C and D) and the dissipated energy with the increasing of porosity.

( a )Figure 13 :

 a13 Figure 13: CAD structure of open (a) to (c) and closed (d) to (f) foams for three unit cell sizes of the long axis.

Figure 15 :

 15 Figure 15: Compressive force vs displacement for different size values of closed cells.

3 Figure 16 :

 316 Figure 16: The whole structure and the main view (in the X-Y plan) of three ratios of unit cell.

Figure 17 :

 17 Figure 17: Compressive force-displacement responses for different ratio of long and short axis of open cells.

4. 4 .

 4 Pore orientation effect of periodic foam 4.4.1. Open cell foam

  23) indicate that, in the case of closed cells, the effective stiffness and the forces in typical points are slightly affected by the orientation of unit cell pores. The distributions of the martensite volume fraction at the end of loading for the three orientations are also investigated and shown in Fig. 24. It can be observed from the figure that the regions of completed martensite transformation grow slightly with the increase of the orientation. The regions of uncompleted martensite transformation occur at the bottom of the samples and in the vicinity of the pores with the arrangement in the transverse direction. The sample with more completed martensite transformation regions shows higher stiffness, which can well explain that the curve associated to the orientation of 90 o is slightly above that of 45 o , and that of the orientation of 45 o is slightly above that of 0 o . In fact, this phenomenon can be also found in open periodic foams. The difference between the open cells and the closed cells is the obvious effect of the orientation on the open cells, while on the closed cells it is slight. 4.5. Pore distribution of periodic foam In this section, three cases of pore distributions of open unit cell are considered (see Fig. 25): -Evenly distribution (case (a)); -Unevenly distribution in transverse direction (case (b)); -Unevenly distribution in longitudinal direction (case (c)).

  carried out. The open or closed foam structure, its porosity (which can be controlled), is well built on the basis of the ellipsoid unit cell pores with random dimension, position and orientation. Constitutive equations of superelastic behavior are used to simulate the compression of porous shape memory alloys. The obtained results indicate that the porosity is a crucial parameter to characterize SMA foams for both closed and open cells. When the foam porosity increases, the effective stiffness, the forces in the start and the finish of forward and reverse transformation (between Austenite and Martensite) and the dissipation energy decrease. The effects of size, ratio of long axis and short axis, orientation and distribution of unit cell pores on the response of loading and unloading compression are studied numerically for open and closed foams. For open cells particularly,the effective stiffness, the forward and reverse transformation forces and the dissipated energy increase with the decrease of size, the increase of orientation and the decrease of ratio of long axis and short axis of ellipsoid pores. For the distributions, the situation shows different results. This might be due to the effects of 'the transverse direction' and 'the longitudinal direction'. However,

3 Figure 18 :

 318 Figure 18: Distribution of the stress at the end of loading for three ratios of unit cell.36

Figure 19 :

 19 Figure 19: Compressive force-displacement responses for different ratio of long and short axis of closed cells.

Figure 20 :

 20 Figure 20: Distribution of the martensite volume fraction at the end of loading for three ratios of unit cell.

Figure 21 :Figure 22 :

 2122 Figure 21: The whole structure and the main view (in the X-Y plan) of three orientations of unit cell.

Figure 23 :

 23 Figure 23: Compressive force-displacement responses for different orientation values of closed cells.

Figure 24 : 3 Figure 25 :

 24325 Figure 24: Distribution of the martensite volume fraction at the end of loading for three orientations of unit cell.

Figure 26 :

 26 Figure 26: Compressive force-displacement responses for different types of distribution of open cells.

Table 1 :

 1 Summary of the possible cases.

Table 2 :

 2 Summary of control equations for the SMA model

	Strain mechanisms:

Table 3 :

 3 Summary of the possible cases.

	CAD Porosity Discrete porosity Discrete porosity
		Open cell foam	Closed cell foam
	20%	18.23%	17.68%
		Error: 8.8%	Error: 11.61 %
	30%	27.08%	26.59%
		Error: 9.72%	Error: 11.37%

Table 4 :

 4 Important parameters for different porosities for open cells.

	Porosity	0	40% 60% 70% 80%
	Maximum displacement (mm)	0.65	0.8	1.5	1.5	1.8
	Effective stiffness k slope (kN/mm) 395.5 175.5 88.5 61.3 18.2
	Dissipated energy (N×m)	13.1	7.7	5.6	4.9	2.2
	Force in Point A (kN)	52.7	15.1	8.0	5.7	1.4
	Force in Point B (kN)	64.1	46.6 36.2 25.9 12.1
	Force in Point C (kN)	34.8	27.0 17.4 12.5	3.7
	Force in Point D (kN)	24.6	6.9	3.0	2.6	0.6

Table 5 :

 5 Important parameters for different porosities for closed cells.These first results concern the open or closed cell sample foams in which the size, orientation, distribution, and ratio of long axis and short axis are random.

	Porosity	0%	20%	30%	40%
	Maximum displacement (mm)	0.65	0.8	0.8	0.8
	Effective stiffness k slope (kN/mm) 395.5 280.0 213.3 175.0
	Dissipated energy (N×m)	13.1	11.2	9.1	7.6
	Force in Point A (kN)	52.7	30.7	19.3	15.1
	Force in Point B (kN)	64.1	60.3	56.2	42.3
	Force in Point C (kN)	34.8	32.1	29.6	22.7
	Force in Point D (kN)	24.6	12.5	8.2	7.1

Table 6 :

 6 Comparison of the effects for both open and closed cells.

	Parameter effects Porosity	Size	Orientation	Ratio	Distribution
	Open cells	Obvious Obvious	Obvious	Obvious	Obvious
	Closed cells	Obvious	Slight	Slight	Slight	Slight

Table . 6

 . summarizes all the observations on the effects of size, orientation, distribution, and ratio of long axis and short axis of unit cell into consideration for both closed cells and open cells. From the Table.6, it can be inferred clearly that the parametric effect of foam with open cells is significant, whereas with closed cells is less. This statement is of great interest because it clearly shows

Acknowledgements

This work is supported by the China Scholarship Council (CSC).

for closed cells, the effects of pore geometry, dimension and distribution are not significant on foam mechanical responses.

The mechanical analysis from the microstructure of porous SMA material provide us a better understanding of this kind of structure, which is used for the design of some specified devices demanding the advantage of porous SMA.

Precisely, this article could provide good ideas for the designer in terms of porosity, size, orientation, shape and distribution for SMA open cell foams.

Concerning our future work, the effect of interaction between unit cells for porous SMA will be investigated in order to explain the difference of the size, orientation, distribution and ratio effects between open and closed cells. In addition, since the parameter effect study considered in this article focuses mainly on the periodic model, it could be an interesting topic to understand how the parameter effects act on the random porous SMA model, which will be also a part of our future work.


Céline Bourgier

Anne Auperin

MD Sofia Rivera

MD Pierre Boisselier

Benoit Petit

MD Philippe Lang

MD Nathalie Lassau

MD Patrice Taourel

Raphaël Tetreau

MD Celine Bourgier
email: celine.bourgier@icm.unicancer.fr

MD Raphael Tetreau

MD David Azria

MD Jean Bourhis

MD Eric Deutsch

PhD ; Marie-Catherine Vozenin

M.D Celine Bourgier

Pravastatin Reverses Established Radiation-Induced Cutaneous and Subcutaneous Fibrosis in Patients With Head and Neck Cancer: Results of the Biology-Driven Phase 2 Clinical Trial Pravacur

Keywords: curative anti-fibrotic treatment, radio-induced fibrosis, statins Oral communication

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

agent in patients with grade ≥2 cutaneous and subcutaneous fibrosis after HNSCC radiotherapy.

INTRODUCTION

Head and neck squamous cell carcinoma (HNSCC) can develop in four main locations: oral cavity, oropharynx, larynx and hypopharynx. In 2015, 932,614 new cases were reported worldwide, and HNSCC was the 7 th most common cancer and cause of death [START_REF] Fitzmaurice | Global, regional, and national cancer incidence, mortality, years of life lost, years lived with disability, and disability-adjusted life-years for 32 cancer groups, 1990 to 2015: A systematic analysis for the global burden of disease study[END_REF].

The standard of care is a multimodal treatment approach with surgery followed by adjuvant radiotherapy (RT), or RT as definitive treatment. The control of locally advanced HNSCC, which is common in patients with history of smoking and alcohol consumption, has been significantly improved by the combination of RT and chemotherapy (RCT), and by RT altered fractionation. However, this population still displays poor outcome [START_REF] Blanchard | Meta-analysis of chemotherapy in head and neck cancer (mach-nc): A comprehensive analysis by tumour site[END_REF]. Conversely, Human Papilloma Virus-positive and non-smoking related HNSCCs have a better prognosis [START_REF] Ang | Human papillomavirus and survival of patients with oropharyngeal cancer[END_REF].

Treatment intensification in HNSCC is associated with increased frequency and severity of radiation-induced toxicities, particularly radiation-induced fibrosis (RIF). Neck RIF is a substantial late toxicity. Indeed, at 3 years post-treatment, the risk of grade ≥2 RIF is 56% and 28% in patients who received RT after or not neck surgery, respectively; and 34% after RCT alone [START_REF] Hirota | Subcutaneous fibrosis after whole neck irradiation[END_REF]. The use of modern RT modalities, such as intensity-modulated radiation therapy (IMRT), has significantly reduced the incidence of acute and late toxicities [START_REF] Veldeman | Evidence behind use of intensity-modulated radiotherapy: A systematic review of comparative clinical studies[END_REF].

However, grade ≥2 RIF occurrence is still high even with IMRT. In a recent study, Nevens and colleagues observed grade ≥2 RIF in 29.2% of patients at 6 months with an increased occurrence in the case of upfront neck dissection (70.6% versus 18.1%) [START_REF] Nevens | Radiotherapy induced dermatitis is a strong predictor for late fibrosis in head and neck cancer. The development of a predictive model for late fibrosis[END_REF]. RIF functional consequences can lead to a decreased patients' quality of life (QoL) as well as to dysphagia, trismus, lymphedema and limited cervical range of motion [START_REF] Shaw | Valid and reliable techniques for measuring fibrosis in patients with head and neck cancer postradiotherapy: A systematic review[END_REF].

RIF usually occurs at least 4 months after RT completion and progresses over the years. The main manifestations of cutaneous and subcutaneous RIF are skin induration and thickening. RIF severity is graded using the Common Terminology Criteria for Adverse Events (CTCAE) scale. As this rating scale may be subjective, other tools have been developed, such as ultrasonography (US) quantification using a high-frequency transducer.

Changes in tissue echogenicity (mild, moderate and severe) are observed in function of RIF severity [START_REF] Deng | Preliminary evaluation of reliability and validity of head and neck external lymphedema and fibrosis assessment criteria[END_REF]. Older studies showed that measuring tissue deformation in response to an applied force assessed using an ultrasound probe is a quantitative method to monitor RIF, and correlates with symptoms and neck rotation restriction [START_REF] Leung | Quantitative measurement of post-irradiation neck fibrosis based on the young modulus: Description of a new method and clinical results[END_REF]. RIF is the results of a dysregulation of the wound-healing process and is characterized by trans-differentiation of fibroblasts into myofibroblasts and by excessive accumulation of extracellular matrix. Tissue injury, inflammation, and repair play a role in RIF development and progression. The first strategies to reduce or treat cutaneous and subcutaneous RIF were based on top-down approaches, such as the use of superoxide dismutase (SOD) [START_REF] Baillet | Treatment of radiofibrosis with liposomal superoxide dismutase. Preliminary results of 50 cases[END_REF] or of pentoxifylline combined with α-tocopherol (vitamin E) [START_REF] Delanian | Randomized, placebo-controlled trial of combined pentoxifylline and tocopherol for regression of superficial radiation-induced fibrosis[END_REF][START_REF] Delanian | Kinetics of response to long-term treatment combining pentoxifylline and tocopherol in patients with superficial radiation-induced fibrosis[END_REF]. A better understanding of RIF molecular mechanisms allowed defining potential therapeutic targets, such as transforming growth factor beta (TGF-β) that is activated by RT and is a fibrosis driver [START_REF] Citrin | Radiation-induced fibrosis: Mechanisms and opportunities to mitigate. Report of an nci workshop[END_REF]. Galunisertib, a TGF-β receptor type I kinase inhibitor, combined with a platelet-derived growth factor receptor inhibitor significantly decreased lung RIF in mice models [START_REF] Dadrich | Combined inhibition of tgfbeta and pdgf signaling attenuates radiation-induced pulmonary fibrosis[END_REF]. The safety of an anti-TGF-β antibody (GC1008) is currently assessed in early clinical trials in patients with idiopathic pulmonary fibrosis (NCT00125385). Besides TGF-β targeting, we and others reported that the Rho/ROCK/CTGF signaling pathway also is involved in RIF development and maintenance (15,16). Pharmacological modulation of this pathway using statins (i.e., HMG-CoA reductase inhibitors) limits and reduces RIF in vitro, ex vivo and in vivo in various preclinical models of normal tissue radiation-induced toxicity (17-20).

Based on these promising preclinical results, we designed a biology-driven phase II clinical trial to assess pravastatin efficacy in patients with delayed cutaneous and subcutaneous grade ≥2 RIF after HNSCC RT (NCT xxx).

METHODS

The protocol was approved by all local institutional review boards and was accepted by the ethics committee of xxx (file number 10-001).

Patients

This bicentric phase II clinical trial enrolled 61 patients with cutaneous and subcutaneous RIF (grade ≥ 2, NCI-CTCAE v4.0) diagnosed at least 6 months but less than 24 months after HNSCC treatment (i.e., adjuvant or exclusive RT, combined or not with concomitant chemotherapy). Patients were in complete remission at inclusion. Additional inclusion criteria were adequate kidney function (serum creatinine ≤130µM/L) and hepatic function (aspartate aminotransferase and alanine aminotransferase levels at least 2 times lower than the laboratory upper normal limit; bilirubin level 1.5 times lower or more than the laboratory upper normal limit).

Exclusion criteria were long-term steroid therapy, or current treatment with statins, fibrates or cyclosporine; history of severe heart failure; history of muscle toxicity during previous treatments with fibrates or statins; history of hereditary muscle diseases; and baseline muscle creatine kinase (MM-CK) levels 3 times higher than the laboratory upper normal limit.

At baseline, all patients underwent head and neck computed tomography (CT) to confirm HNSCC remission and high-frequency US (HF-US; at least 16-MHz linear transducer) to assess RIF thickness (in mm) compared with neighboring normal skin (upper part of the chest wall). The radiologists who performed the HF-US assessments (image collection and interpretation) were HF-US experts. As RIF is a dynamic process due to permanent extracellular matrix remodeling, RIF thickness was measured at its thickest point at baseline and at all the study time points.

Before inclusion, a written informed consent was obtained from all patients. Then, investigators from the two centers sent by fax the data required for the patient's registration to the xxx. After eligibility check and registration, a unique identification number was assigned to each patient and provided by fax to the investigator and to the pharmacist. Thus, patient registration was done independently from the study investigators. Pravastatin was given to the patients by the hospital pharmacist only after registration.

Treatment

Treatment (40 mg pravastatin/day per os for 12 months) began at inclusion. Dose adjustment was not permitted. Premature drug discontinuation was planned in the case of pravastatin-related toxicity (renal, hepatic or muscle problems), cancer progression, consent withdrawal, or major protocol violation. Patients who discontinued pravastatin were followed like all the other patients enrolled in the trial. Neck motion exercises or scar manipulation were not performed during the study because at that time they were not part of the standard of care for HNSCC in xxx.

Endpoints

The primary endpoint was RIF thickness reduction of at least 30% (compared with baseline) measured by cutaneous HF-US at 12 months.

The secondary efficacy endpoints were reduction of RIF severity (according to the CTCAE v.4.0) evaluated by clinical examination, and QoL changes using the selfadministered VQ-Dermato questionnaire [START_REF] Grob | Development and validation of a quality of life measurement for chronic skin disorders in french: Vq-dermato. The reseaudepidemiolo gie en dermatologie[END_REF]. The VQ-Dermato questionnaire is a valid and reliable dermatology-specific QoL instrument for chronic skin diseases, particularly for assessing the effect on QoL of therapeutic strategies [START_REF] Grob | Development and validation of a quality of life measurement for chronic skin disorders in french: Vq-dermato. The reseaudepidemiolo gie en dermatologie[END_REF]. It includes seven dimensions (selfperception, daily living activity, mood state, social functioning, leisure activity, treatmentinduced limitations, and physical discomfort) explored by 28 items.

Pravastatin tolerance and compliance were evaluated. Pravastatin-related toxicities were assessed by clinical examination and by blood testing (cholesterol and MM-CK level variations).

Follow-up

Follow-up visits with the investigator were planned each month during the first three months and then at 6, 12, 18 and 24 months after inclusion. RIF was evaluated clinically and by HF-US every 6 months after treatment start. Patients were asked to fill in the VQ-Dermato questionnaire at baseline and at 6, 12, 18 and 24 months after treatment start. Pravastatinrelated toxicities were assessed monthly during the first 3 months, and then at 6 and 12 months.

Ancillary study

Before enrollment, an ancillary study to assess pravastatin biological efficacy was proposed to each patient. After signature of the written informed consent for the ancillary study, a skin punch biopsy in the RIF area was carried out before pravastatin treatment initiation and at 12 months. Biopsies were processed using classical histopathological procedure. Hematoxylin-Eosin (HE) staining was used for histological analysis and Sirius Red staining for quantification of collagen infiltration, as previously described (22). Each patient was its own control.

STATISTICAL ANALYSIS

This single-stage phase II trial tested the null hypothesis that the success rate would be lower than 10% versus the alternative hypothesis that it would be higher than 30%. This required the inclusion of 40 evaluable patients (i.e., patients who received pravastatin for at least 11 months). If there were more than eight successes among the 40 evaluable patients, pravastatin would be considered interesting in this setting. With this design, the one-sided alpha error rate was 4.2% and the power was 94.5%. According to the expected rate of early discontinuation (30%), around 55 patients were needed for the trial.

The success rate (according to the HF-US assessment and CTCAE grading) was described with the 95% confidence interval. The paired t-test was used to compare the VQ-Dermato scores at baseline and at 12 months, and to compare RIF thickness and normal skin thickness at baseline. As the sample size was small, the Wilcoxon signed rank test was used when appropriate and gave similar results as the paired t-test.

This study was registered at ClinicalTrials.gov, number xxx.

ROLE OF THE FUNDING SOURCE

The study funder (xxx) had no role in the study design, in the data collection, analysis and interpretation, or in the report writing. The corresponding author had full access to all data and the final responsibility for the decision to submit for publication.

RESULTS

Pravastatin is well tolerated and shows anti-fibrotic effect in a subset of patients

From February 2011 to April 2016, 61 patients with grade ≥2 cutaneous and subcutaneous RIF after HNSCC were registered (Figure 1). One patient was not eligible because he was already treated with statins, and was excluded from the analyses. The patients' demographic and clinical characteristics at baseline are listed in Supplementary table 1. The total RT dose ranged from 50 to 70 Gy. Patients were mainly treated by 3D-conformal RT (61%), and the others by IMRT (37%). At baseline, 62% of patients had grade 2 RIF and 37% grade 3 RIF (Table 1). RIF was detected after a mean of 9.9 months post-RT. At baseline, skin thickness (measured by HF-US) was significantly increased in the RIF area compared with normal tissue (3.44 mm versus 1.97 mm; p=0.0004).

Eighteen patients stopped pravastatin before 11 months of treatment because of grade ≥2 pravastatin-related toxicities (n=8), consent withdrawal or patient refusal to continue treatment (n=5), or tumor relapse or death (n=5). Among the patients with pravastatin-related toxicities, six stopped treatment during the first 3 months after initiation because of: grade 2-3 myalgia (n=3; pravastatin treatment duration: 0.3 -3 months), grade 3 arthralgia (n=1; pravastatin treatment duration: 3 months), grade 3 abdominal pain (n=1; pravastatin treatment duration: 2.8 months), and grade 3 esophagitis (n=1; pravastatin treatment duration: 1.7 months). These toxicities led to the definitive discontinuation of pravastatin. The other two patients reported hypersensitivity reaction (n=1 after 6 months of pravastatin treatment), and grade 1 diarrhea and grade 1 erectile dysfunction (n=1 after 8.7 months of pravastatin treatment). In total, 42 patients took pravastatin for at least 11.6 months. Among them, 40 patients underwent HF-US assessment both at baseline and at 12 months. In these 40 patients, the mean RIF thickness was 4.04mm at baseline and 2.24mm at 12 months (reduction by 16.9% ± 38.8%) (Figure 2). A RIF thickness decrease of 30% or more was observed in 15 patients (Table 1), corresponding to a success rate of 35.7% (95% confidence interval: 21.6% -52.0%). Therefore, according to the hypotheses tested in this trial, the use of pravastatin as anti-fibrotic agent was successful. Besides RIF thickness decrease, pravastatin also reduced RIF severity (CTCAE) in 50% of patients (95% confidence interval = 34.2%; 65.8%). In eight patients, RIF thickness and severity were decreased at the 12-month follow-up visit. After pravastatin completion, no "rebound effect" was observed. RIF thickness, assessed by HF-US, was not significantly different at 12 months (mean=2.24 (SD=1.07); median=2.05 [range=1.00;7.60]), 18 months (mean=2.47 (SD=2.28); median=1.9 [range=1.10;15.0]) and 24 months (mean=2.92 (SD=5.96); median=1.7 [range=1.20;36.0]).

Although the VQ-Dermato questionnaire was not always fully completed by the patients, analysis of the score variations between baseline and the 12-month follow-up indicated that pravastatin treatment significantly improved the self-perception (p=0.027), mood state (p=0.010), social functioning (p=0.040) and global scores (p=0.002) (Supplementary Table 2).

Among the patients who received pravastatin for more than 11 months, compliance was excellent with a mean treatment duration of 365 days [min, max= 335 -365] and the drug was well-tolerated (Table 2).

Skin structure is improved and collagen infiltration decreased after 12-month treatment with pravastatin (ancillary study)

HE staining of skin punch biopsies collected before and after pravastatin treatment (n=19 patients) showed that after 12 months of treatment, the skin histopathological structure was improved in 14 patients, as indicated by the decreased infiltration of immune cells and normalization of the epidermis thickness (Figure 3A). In the other five patients, no modification was observed. Moreover, collagen infiltration (Sirius Red staining quantification) was decreased in 8/14 patients after pravastatin treatment (Figure 3B andC).

DISCUSSION

This phase II clinical trial confirmed previous preclinical data on pravastatin antifibrotic potential. To the best of our knowledge, this is the first study combining objective and subjective criteria (RIF thickness and severity, respectively) to provide evidence of pravastatin anti-fibrotic efficacy in patients. Although IMRT has significantly reduced the incidence of acute and late toxicities, grade ≥2 RIF still occurs and pravastatin could be proposed to such patients. Pravastatin was well-tolerated and only six patients (10%) experienced discomfort (arthralgia, myalgia) that led to treatment withdrawal during the first 3 months of the trial. This suggests that pravastatin could be better tolerated than the current anti-fibrotic treatment based on pentoxifylline/vitamin E that caused discomfort (hot flushes, nausea, epigastralgia, severe asthenia, headache, or vertigo) in 45% of patients [START_REF] Delanian | Randomized, placebo-controlled trial of combined pentoxifylline and tocopherol for regression of superficial radiation-induced fibrosis[END_REF], and definitive treatment discontinuation in 11% of patients (n=3/27) because of myalgia, diarrhea or nausea during the first 4 months of treatment [START_REF] Gothard | Phase ii study of vitamin e and pentoxifylline in patients with late side effects of pelvic radiotherapy[END_REF].

Tissue injury induced by RT can be managed by administering prophylactic agents (or radioprotectors) before RT or mitigators after RT completion, and by curative interventions, after the appearance of radiation-induced toxicities. Here, pravastatin was assessed as a curative approach (i.e., several months after RT completion and in the presence of established ≥ grade 2 fibrosis). This strategy presents several advantages, including the absence of interference with the anti-cancer treatments and its administration only to the patients who really need it. Curative treatment after RIF appearance are based on the idea that fibrotic tissue can be mobilized and the fibrogenic process reversed by normalizing tissue homeostasis. The biological basis of fibrosis reversion has been reviewed elsewhere [START_REF] Bentzen | Preventing or reducing late side effects of radiation therapy: Radiobiology meets molecular pathology[END_REF][START_REF] Yarnold | Pathogenetic mechanisms in radiation fibrosis[END_REF][START_REF] Montay-Gruel | Expanding the therapeutic index of radiation therapy by normal tissue protection[END_REF], and several pharmacological strategies have been proposed to achieve this (reviewed by ( 26)).

As the objective evaluation of fibrosis regression is a critical issue in clinical trials, we chose HF-US to assess pravastatin efficacy and to obtain quantitative measurement of RIF thickness. Delanian and colleagues measured the variation in length and width of the cutaneous fibrotic surface to monitor the efficacy of the pentoxifylline-vitamin E combination [START_REF] Delanian | Kinetics of response to long-term treatment combining pentoxifylline and tocopherol in patients with superficial radiation-induced fibrosis[END_REF]. However, this measure is operator-dependent and consists in the palpation of the fibrotic block edges. Recently, a systematic review by Shaw and colleagues on objective tools including CT (densitometry and perfusion), Cutometer and US found that US is a better objective measurement than palpation [START_REF] Leung | Quantitative measurement of post-irradiation neck fibrosis based on the young modulus: Description of a new method and clinical results[END_REF]. Moreover, 84.9% agreement of inter-rater reliability regarding RIF grade was reported when physical examination was associated with US (8); however, the value of US-based assessment of RIF modulation after anti-fibrotic treatment has never been evaluated. In our study, the decrease of both RIF thickness and severity was observed only in 8 patients. Our findings in normal tissue and RIF areas are in agreement with those of a pilot-study using HF-US for RIF assessment presented at the ESTRO 35 meeting [START_REF] Schack | Ep-1155: Radiation-induced morbidity evaluated by high-frequency ultrasound: A pilot study[END_REF]. Nevertheless, additional investigations are needed to confirm HF-US value for RIF assessment during anti-fibrotic treatment.

One of the major secondary outcomes of the present phase II clinical trial is that pravastatin efficiently reduced RIF severity and improved QoL. Specifically, 50% of patients displayed a RIF severity decrease of at least one point in the CTCAE grading system, resulting in a quality of life improvement, particularly self-perception, mood and social functioning. No other clinical trial assessing anti-fibrotic agents has focused on these secondary endpoints. Here, we found that pravastatin significantly improved three dimensions of the VQ-Dermato questionnaire (self-perception, mood state and social functioning) and consequently also the global score. Conversely, Gothard and colleagues did not observe any QoL improvement after 6 months of pentoxifylline-vitamin E treatment for breast RIF [START_REF] Gothard | Double-blind placebo-controlled randomised trial of vitamin e and pentoxifylline in patients with chronic arm lymphoedema and fibrosis after surgery and radiotherapy for breast cancer[END_REF].

Our results are consistent with those reported by Delanian and colleagues (pentoxifyllinevitamin E for at least 6 months), with a mean decrease of the SOMA score by 35% (±20%) and by 48% (±21%) at 6 months and 1 year, respectively [START_REF] Delanian | Striking regression of chronic radiotherapy damage in a clinical trial of combined pentoxifylline and tocopherol[END_REF].

Many preclinical studies assessed different anti-fibrotic agents that target TGF-β 1, inhibit collagen production, or deplete macrophages. Some clinical trials and pilot studies showed RIF reduction after low-dose interferon-gamma therapy (n=4) [START_REF] Gottlober | Interferon-gamma in 5 patients with cutaneous radiation syndrome after radiation therapy[END_REF], and higher range of motion after treatment with pirfenidone (n=6) [START_REF] Simone | Oral pirfenidone in patients with chronic fibrosis resulting from radiotherapy: A pilot study[END_REF]. Ongoing clinical trials are assessing different curative strategies: the Tocovid SupraBio-pentoxifylline combination in bowel radiation-induced disease after pelvic RT (NCT02230800); topical superoxide dismutase in skin RIF in patients after HNCC (NCT01771991). A phase I trial evaluates the efficacy of umbilical cord mesenchymal stem cells in established lung fibrosis (NCT02277145). Other clinical trials are assessing mitigation strategies, particularly for the prevention of lung fibrosis by administration of enalapril (NCT01754909), captopril (NCT00077064), or nicorandil (NCT02809456). To date, none of them has reported results comparable with those obtained here using a compound selected on the basis of its biological efficacy (18,20,[START_REF] Wang | Rhoa/rho-kinase triggers epithelial-mesenchymal transition in mesothelial cells and contributes to the pathogenesis of dialysis-related peritoneal fibrosis[END_REF].

Finally, the results of our ancillary study are consistent with the clinical data, although performed in a limited number of patients. They showed that pravastatin treatment induced structural improvement of the skin in 14 of the 19 patients, suggesting that increasing the treatment duration could lead to further improvement. Moreover, collagen infiltration was reduced in 8/14 patients. In these eight patients, RIF thickness also was reduced by more than 30% (by HF-US), or RIF severity grade was decreased. A specific evaluation of the Rho/ROCK/CTGF pathway would have been interesting, but was difficult to perform in skin punch biopsies due to their small size. These molecular analyses are more accurate in preclinical and experimental models in which more mechanistic studies can be conducted using both pharmacologic and genetic approaches.

In conclusion, this biology-driven phase II clinical trial shows a curative efficacy and good tolerance of pravastatin in patients with established cutaneous and subcutaneous grade ≥2 RIF in the neck after RT for HNSCC. These results need to be confirmed in a phase 3 randomized trial. However, as very few anti-fibrotic strategies can be proposed to patients in the clinical practice, statins could be already used as secondary treatment in patients with severe cutaneous RIF. More studies are required to investigate the biological differences between responders and non-responders to better target the patients who will benefit from this strategy.

FIGURES LEGENDSFigure 1 .Figure 2 :

 12 FIGURES LEGENDS

Figure 3 : 2 Figure 1 Figure 2

 3212 Figure 3: Histological analysis of a skin punch biopsy from a patient with RIF who responded

Table 1 :

 1 Evaluation of fibrosis at baseline

Pravastatin for <11 months (N=18) Pravastatin for ≥11 months (N=42) Total N=60 TIMING OF RIF APPEARANCE (after RT completion, in months)

	Mean (SD)]	10.7 (8.5)	9.6 (9.8)	9.9 (9.4)
	TIMING OF PRAVASTATIN INITIATION AFTER RIF ONSET (in months)
	Mean (SD)	17.7 (14.1)	16.9 (10.5)]	17.1 (11.6)
	BASELINE RIF SEVERITY GRADE (clinical assessment according to the CTCAE v.4.0)
	2	11 (61%)	26 (62%)	37(62%)
	3	7 (39%)	15 (36%)	22 (37%)
	4	0	1 (2%)	1 (2%)
	BASELINE RIF AND HEALTHY SKIN THICKNESS (in mm, by HF-US)
	RIF thickness (Mean (SD);	2.30 (0.96)	3.92 (4.42)	3.44 (3.80)
	median [range])	2.45 [0.70 -4.40]	2.40 [0.70 -20]	2.45 [0.70 -20]
	Healthy skin thickness (Mean (SD);	1.47 (0.52)*	2.17 (3.53)*	1.97 (2.99)**
	median [range])	1.60 [0.40 -2.40]	1.60 [0.70 -24]	1.60 [0.40 -24]
	Thickness difference between RIF and
		0.71 (0.86)*	1.38 (2.78)*	1.18 (2.39)**
	healthy skin (Mean (SD);
		0.70 [-1.4 ; 2.0]	0.70 [-4.0 ; 15.4]	0.70 [-4.0 ; 15.4]
	median [range])
	PRIMARY ENDPOINT -

RIF THICKNESS DECREASE (between baseline and the 12-month assessment)

	Unknown -HF-US not done	14 (77.8%)	2 (4.8%)	-
	SECONDARY ENDPOINT -
	Success (i.e., thickness decrease of 30%	0 (0%)	15 (35.7%)
	or more compared with baseline) [95%	[0% -18.5%]	[21.6% -52.0%]	-
	confidence interval]
	Failure (i.e., thickness decrease lower	4 (22.2%)	25 (59.5%)
				-
	than 30% compared with baseline)

RIF CTCAE GRADING MODIFICATION (12-month assessment compared with baseline)

  ** two patients received pravastatin for 1.7 months and one patient for 8.7 months $ No significant difference in healthy skin thickness between baseline and 12 months

	Decrease of severity (-2 points)	0 (0%)	2 (5%)	-
	Decrease of severity (-1 point)	3 (16.7%)	19 (45%)	-
	Stable (same score)	5 (27.8%)	17 (40%)	-
	Increase of severity (+1 point)	0 (0%)	3 (7%)	-
	No clinical evaluation at 12 months	10 (55.6%)	1 (2%)	-
	Success (grade decrease)	3*** (16.7%)	21 (50.0%)
				-
	[95% confidence interval]	[3.6% -41.4%]	[34.2% -65.8%]
	RIF AND HEALTHY SKIN THICKNESS AT 12 MONTHS (in mm, by HF-US)
	RIF thickness (Mean (SD);		2.24 (1.07)
	median [range])		2.05 [1.00 -7.60]
	Healthy skin thickness (Mean (SD);		1.33 (0.42) $
	median [range])		1.40 [0.50 -2.00]
	* data missing for one patient;
	** data missing for two patients;

*

Table 2 :

 2 Pravastatin tolerance in the 42 patients who received pravastatin for more than 11 months

	ARTHRALGIA	N (%)
	None	39 (93%)
	Grade 1	1 (2%)
	Grade 2	2 (5%)
	MYALGIA
	None	32 (76%)
	Grade 1	8 (19%)
	Grade 2	1 (2%)
	Grade 3	1 (2%)
	CRAMPS
	None	34 (81%)
	Grade 1	8 (19%)
	FATIGUE
	None	39 (93%)
	Grade 1	3 (7%)
	DYSPHAGIA -ESOPHAGITIS
	None	39 (93%)
	Grade 1	1 (2%)
	Grade 2	2 (5%)

This work was supported by grants from the French National Hospital Clinical Research Program (PHRC) obtained in 2010 Disclosure: The authors declare no conflict of interest. The authors would like to thank Elisabetta Andermarcher for English editing, Marc Bollet (Hartmann Hospital) and Xavier Cuenca