Zhu Yan

Qiang Guo

Fahmi Zaïri
email: fahmi.zairi@polytech-lille.fr

Ali Zaoui

Qifeng Jiang

Xiaobing Liu

Continuum-based modeling large-strain plastic deformation of semi-crystalline polyethylene systems: Implication of texturing and amorphicity

Keywords: Semi-crystalline polyethylene, Crystallographic texturing, Amorphicity, Oligo-cyclic loading, Viscoplastic-viscohyperelastic modeling crystallinity Low crystallinity Rigid amorphous Crystalline lamella

The present contribution examines the ability of a constitutive model to capture polyethylene response variation with crystallinity. The model considers the elastic-viscoplastic crystal deformation and anisotropy due to the crystallographic texturing along with the amorphous chains network elastic-viscoplastic-viscohyperelastic deformation. The coupling between the two phases is performed by means of a multi-scale homogenization-based approach in which the interfacial interaction is considered. The model is applied on semi-crystalline polyethylene systems containing a broad range of crystallinities. Both monotonic and oligo-cyclic tensile loading sequences are considered upon large-strain plastic deformation. The model is found able to correctly capture the gradual transition of the rate-dependent monotonic response from a thermoplastic-like behavior at high crystallinity to a nearly elastomeric-like behavior at low crystallinity. The model abilities to capture cyclically loaded polyethylene systems are critically discussed. The influential and decisive role of the amorphous chains network alignment/relaxation on the cyclic stretching is emphasized by analyzing local stresses and stretches.

Introduction

Semi-crystalline polyethylene systems have a large range of applications, in sectors like oil industry, automobile, aeronautic, robotic, biomechanics and civil engineering, in which they 2 may be subjected to large strains under in-service or manufacturing process. These materials present an extremely large variety of molecular architectures with a crystallinity index ranged from about seventy percent down to a few percent only, passing the morphology from well-organized crystalline lamellae to discrete crystalline lamellae randomly dispersed within an amorphous matrix [START_REF] Peacock | Handbook of Polyethylene: Structures, Properties, and Applications[END_REF]. The broad morphology range leads basically to an extremely large variety of responses from the plastic response characteristics of thermoplastics to the nonlinear elastic response characteristics of elastomers [START_REF] Ayoub | Effects of crystal content on the mechanical behaviour of polyethylene under finite strains: experiments and constitutive modelling[END_REF][START_REF] Abdul-Hameed | Large-strain viscoelastic-viscoplastic constitutive modeling of semi-crystalline polymers and model identification by deterministic/evolutionary approach[END_REF] with a profound influence of the amorphous and crystalline features such as density of stress transmitters in the amorphous layers and crystal dimensions [START_REF] Hillmansen | The effect of strain rate, temperature, and molecular mass on the tensile deformation of polyethylene[END_REF][START_REF] Argon | Rate mechanisms of plasticity in semicrystalline polyethylene[END_REF][START_REF] Kazmierczak | Plastic deformation of polyethylene crystals as a function of crystal thickness and compression rate[END_REF][START_REF] Bartczak | Plasticity of semicrystalline polymers[END_REF]. The development of continuum-based models is a prerequisite for designing and performance predicting polyethylene-based products whether in the form of fibers, films or massive parts.

Modeling response of semi-crystalline polyethylene systems is to relate to their complex organization hierarchy from the nano-sized lamellar structure to the macroscopic scale. Purely phenomenological approaches using theories such as viscoelasticity [START_REF] Zhang | Nonlinear mechanical response of high density polyethylene. Part II: Uniaxial constitutive modeling[END_REF] or viscoplasticity [START_REF] Drozdov | Constitutive equations of finite viscoplasticity of semicrystalline polymers[END_REF][START_REF] Colak | Modeling viscoelastic and viscoplastic behavior of high density polyethylene (HDPE)[END_REF][START_REF] Zaïri | Numerical modelling of elastic-viscoplastic equal channel angular extrusion process of a polymer[END_REF][START_REF] Khan | Amorphous and semicrystalline solid polymers: Experimental and modeling studies of their inelastic deformation behaviors[END_REF][START_REF] Hadj Hamouda | Viscoplastic behaviour of a medium density polyethylene (MDPE): Constitutive equations based on double nonlinear deformation model[END_REF][START_REF] Dusunceli | Modelling effects of degree of crystallinity on mechanical behavior of semicrystalline polymers[END_REF][START_REF] Drozdov | Multi-cycle deformation of semicrystalline polymers: Observations and constitutive modeling[END_REF] have the advantage of simplicity. Nonetheless, these models provide only a mathematical description of the different aspects of the material response, generally under low-strain and without obvious connection to the actual microstructure and its deformation-induced evolution. Over the years, continuum-based models have been developed with the concern to introduce microstructural specificities as precisely as possible in the aim to provide a deep understanding of the separate and synergic effects of key microstructural parameters governing the macroscopic response. The link with the microstructure strongly depends on the approach used for the constitutive representation and the observation scale from which the model starts. In the most physically consistent models, the microstructure approximation of semi-crystalline polymer systems is generally based on composite-type representations.

Following the pioneering work of [START_REF] Haward | The use of a mathematical model to describe isothermal stress-strain curves in glassy thermoplastics[END_REF], continuum-based models have been proposed to constitutively describe the large-strain material behavior of semi-crystalline polymers by the combination of resistances representing intermolecular and molecular network micro-mechanisms [START_REF] Ayoub | Modelling large deformation behaviour under loading-unloading of semicrystalline polymers: application to a high density polyethylene[END_REF][START_REF] Ayoub | Effects of crystal content on the mechanical behaviour of polyethylene under finite strains: experiments and constitutive modelling[END_REF][START_REF] Abdul-Hameed | Large-strain viscoelastic-viscoplastic constitutive modeling of semi-crystalline polymers and model identification by deterministic/evolutionary approach[END_REF][START_REF] Makki | Mullins effect in polyethylene and its dependency on crystal content: A network alteration model[END_REF][START_REF] Sepiani | Modeling short-and long-term time-dependent nonlinear behavior of polyethylene[END_REF][START_REF] Chen | Effect of crystalline content on ratchetting of ultra-high molecular weight polyethylene polymers: Experimental investigation and constitutive model[END_REF][START_REF] Deplancke | 1D strain rate-dependent constitutive model of UHMWPE: From crystalline network to fibrillar structure behavior[END_REF][START_REF] Qi | Constitutive modeling for the elastic-viscoplastic behavior of high density polyethylene under cyclic loading[END_REF][START_REF] Bernard | From rheological to original three-dimensional mechanical modelling of semi-crystalline polymers: Application to a wide strain rate range and large deformation of Ultra-High Molecular Weight PolyEthylene[END_REF]. The latter micro-mechanism describes the resistance to deformation of the amorphous molecular structure to simulate the molecular orientation/relaxation process during the strain-hardening stage of the stress-strain response [START_REF] Boyce | Constitutive model for the finite deformation stress-strain behavior of poly(ethylene terephthalate) above the glass transition[END_REF]. In the above-cited models, a simple averaging homogenization is used and only for the intermolecular resistance which does not allow to represent the crystallinity effect on the deformation-induced orientation. The active interaction between crystalline and amorphous domains is a first-order factor which requires to be taken into account using more sophisticated micromechanics-based approaches. For instance, micromechanics-based models using the Eshelby inclusion theory were proposed through the matrix-inclusion constitutive representation of the semi-crystalline polymer system [START_REF] Bédoui | Micromechanical modelling of isotropic elastic behaviour of semicrystalline polymers[END_REF][START_REF] Gueguen | A new three-phase model to estimate the effective elastic properties of semi-crystalline polymers: Application to PET[END_REF][START_REF] Anoukou | A micromechanical model taking into account the contribution of αand γ-crystalline phases in the stiffening of polyamide 6-clay nanocomposites: a closed-formulation including the crystal symmetry[END_REF][START_REF] Hachour | Experiments and modeling of high-crystalline polyethylene yielding under different stress states[END_REF]. These models were restricted to the elastic stiffness and yield strength predictions, and more recently extended to the small-strain post-yielding behavior [START_REF] Mesbah | How stress triaxiality affects cavitation damage in high-density polyethylene: experiments and constitutive modeling[END_REF].

Multi-scale homogenization-based constitutive models have been also developed by considering at the mesoscopic scale an aggregate of two-phase layered composite inclusions consisting in parallel crystalline lamellae and amorphous layers. Using this concept, the elastic-viscoplastic deformation behavior of high-density polyethylene was predicted at small strain levels [START_REF] Nikolov | A micro/macro constitutive model for the small-deformation behavior of polyethylene[END_REF][START_REF] Nikolov | Multi-scale constitutive modeling of the small deformations of semi-crystalline polymers[END_REF] and moderate strain levels by simulating the texture evolution (Lee et al., 1993a(Lee et al., , 1993b;;[START_REF] Van Dommelen | Micromechanical modeling of the elasto-viscoplastic behavior of semi-crystalline polymers[END_REF][START_REF] Agoras | Multi-scale homogenization-based modeling of semi-crystalline polymers[END_REF][START_REF] Uchida | Micro-, meso-to macroscopic modeling of deformation behavior of semi-crystalline polymer[END_REF][START_REF] Mirkhalaf | Micromechanical modeling of anisotropic behavior of oriented semicrystalline polymers[END_REF]. The models are generally identified using loading modes in which the crystal plasticity is a first-order phenomenon, such as compression, channel die compression and shear. While the local events involved in yielding due to tensile straining are not connected with clear crystal shearing but with the phenomena occurring in the amorphous phase and the relation to crystal thickness is not expected, rather to crystallinity degree, read amorphicity (Seguela et al., 1998a(Seguela et al., , 1998b;;[START_REF] Bartczak | Influence of molecular parameters on high-strain deformation of polyethylene in the plane-strain compression. Part I. Stress-strain behavior[END_REF][START_REF] Bartczak | Plasticity of semicrystalline polymers[END_REF][START_REF] Rozanski | Plastic yielding of semicrystalline polymers affected by amorphous phase[END_REF]. The latter is an influential and decisive parameter in tensile yielding, the strength and consistency of the amorphous phase being thus first-order factors [START_REF] Mesbah | How stress triaxiality affects cavitation damage in high-density polyethylene: experiments and constitutive modeling[END_REF].

Model presentation

Constitutive representation

The organization hierarchy of the semi-crystalline polyethylene structure depends on the amorphicity. As illustrated in Fig. 1, decreasing the crystallinity leads to a morphology transition from micro-sized interconnecting spherulites to individually crystalline lamellae randomly dispersed in the rubbery amorphous matrix. According to the microstructure approximation based on a composite-type description, the morphology may be regarded as a percolated crystalline matrix at high crystallinity and a rubbery matrix at low crystallinity. A constitutive representation unifying these views is illustrated in Fig. 2. At the mesoscopic scale, the representative volume element of the heterogeneous material is seen as a mixture of crystalline and amorphous domains organized as an aggregate of two-phase layered composite inclusions which are randomly oriented, the macroscopic response being obtained by the averaging homogenization. The relative thickness of the two layers determines the crystal concentrations, which allows to consider high-to-low crystalline structures. Both crystalline lamella and amorphous layer are homogeneous and continuous media. The crystalline lamella consists of regularly ordered chains resulting in an orthorhombic structure with a high elastic modulus in the chain direction while the amorphous layer consists of randomly oriented entangled chains and is thus supposed to be an isotropic medium.

Constitutive equations

In what follows, the three-dimensional constitutive equations are described for the different physical sources governing the macro-response. The model satisfies the continuum mechanics rules within the large inelastic deformation kinematics. In the latter framework, the deformation gradient tensor ( )

,t = ∇ F x X
is introduced to map a material point from its initial position X in the reference configuration to its actual position x in the deformed configuration. The inelastic deformation mechanisms within each phase are considered to be distinct, and are described using the multiplicative decomposition concept of the deformation.

The introduction of an intermediate configuration during a spontaneous elastic unloading allows the multiplicative deformation decomposition into elastic and inelastic parts:

Elastic-viscoplastic-viscohyperelastic polyethylene amorphous network

We assume that the behavior of the amorphous domain is the sum of an elastic-viscoplastic resistance that accounts for intermolecular interactions and a viscohyperelastic resistance that accounts for molecular network stretching and orientation process, the two physically distinct (1)

From the rheological point of view, the intermolecular resistance is represented by an elastic spring in series with a viscous dashpot (Fig. 2). The intermolecular Cauchy stress _ i n t e r a σ is constitutively coordinated with the corresponding strain by the following relationship:

( )

_ inter _ inter _inter 1 ln e a a a a J = σ K V (2)
where

_ inter _ inter det e a a J = F
is the elastic intermolecular volume change, ( )

_ inter ln e a V
is the Hencky elastic strain and a K is the fourth-order elastic stiffness tensor of the isotropic amorphous layers given by:

6 6 _ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 0 0 0 0 0 0 0 a ij K λ µ λ λ λ λ µ λ λ λ λ µ µ µ µ ×           =     +  +     +    (3)
in which λ and µ are the amorphous Lame's constants.

From the rheological point of view, the molecular network resistance is constituted by a nonlinear spring in series with a viscous dashpot (Fig. 2). The molecular network Cauchy stress _ n e t a σ is given by the [START_REF] Arruda | A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials[END_REF] eight-chain model of rubber elasticity introducing the extensibility limit of amorphous chains as a deformation measure: ( ) ( ) ( )

( ) 2 _ net _ net _ net _ net _ net _ n t 1 e 1 3 e a a e a e a h e a a C N J N λ λ λ -  = -       B I σ L (
1 2 2 3 1 x x x x - ≈ - - L .
The time derivative of the deformation gradient tensor writes F , are given by:

1 _ inter _ inter _ inter _ inter _ inter in e in e in a a a a a - = D F F F F & and 1 _ net _ net _ net _ net _ net in e in e in a a a a a - = F D F F F & (6)
The amorphous elastic deformation gradients are then extracted from the multiplicative deformation decomposition: D , are given by the following flow rules:

1 _ inter _ inter _ inter e in a a a - = F F F and 1 _ net _ net _ net e in
_ inter _ inter _ inter _ inter 2 a in a a a γ ′ = σ D σ & and _ net _ net _ net _ net 2 a in a a a γ ′ = σ D σ & (7)
where ( ) _ n e t a γ& are the amorphous intermolecular and network shear strain rates, respectively, given by [START_REF] Boyce | Constitutive model for the finite deformation stress-strain behavior of poly(ethylene terephthalate) above the glass transition[END_REF]:

_ inter _ int 2 er _ in 1 ter 2 a a a ′ ′ = σ σ σ and ( ) 2 _ net _ ne 1 t _ net 2
_ inter _ inter ,0 = exp 1 a a a G k s γ γ θ     ∆     - -         σ & & and _ net _ net _ net 1 = 1 a v a in a C γ λ       -   σ & (8)
The left formula of Eq. ( 8) captures barriers to molecular chain segment rotation in the amorphous layers. The term ,0 a γ& is the pre-exponential factor, G ∆ is the activation energy (which must be overcome for inelasticity begins), s is the shear strength controlling the amorphous chain segment rotation:

1 0 _ inter n n n a s s s n κ γ - = & & (9)
where 0 s is the initial shear strength, n is the intermolecular hardening coefficient and κ is a factor introduced to take into account the crystal-dependency of concentration of stress transmitters in the amorphous layers.

The right formula of Eq. ( 8) captures molecular relaxation process in the amorphous layers.

The term _ net

in a λ must be slightly higher than one at the beginning of the loading process to avoid singularity and ensure stability of the numerical algorithm. It is given by: ( )

1 2 _ net _ net 1 = trace 3 in a in a λ ς   +     B (10) in which _ net _ net _ net T a in in a a in = F F B
is the inelastic left Cauchy-Green tensor and ς= 0.01 is the added perturbation coefficient ensuring numerical stability.

The term v C in Eq. ( 8) is the amorphous network viscosity coefficient given by:

exp v Q C D Rθ   = -     (11)
in which D and Q are two molecular relaxation parameters and R is the universal gas constant.

Elastic-viscoplastic polyethylene crystals

The contribution of the polyethylene crystals may be represented by an elastic spring in series with a viscoplastic dashpot capturing barriers to crystallographic shear in the crystalline phase (Fig. 2). The crystalline stress tensor c τ is constitutively coordinated with the strain as: 3.28 1.13 3.28 9.92 2.14 1.13 2.14 315.92 3.62

1.62 3.19 0 0 c ij K ×           =             [GPa] (13)
The plastic deformation of polyethylene crystals occurs by crystallographic slip that is captured by the following relation to account for the s N distinct slip systems:

1 s N in c c α α α γ = = ∑ L R & , ( ) 1 = 2 α α α α α ⊗ + ⊗ R n m m n (14)
where α R is the nonsymmetric Schmid tensor with α n and α m are the "slip plane" normal vector and the "slip direction" vector in this plane, respectively, and ( )

1 ,0 = sign m c c g α α α α τ γ γ τ & & (15)
where ,0 c γ& is a reference strain rate, m is the strain rate sensitivity parameter and g α is the shear strength of the slip system α listed in Table 1. The term ( ) sign x is the signum function defined as:

( )

1 0 sign = 0 0 1 1 x x x x - <   =   >  (16)
The crystalline inelastic deformation gradient

( ) ( ) 1 _ in in in c n c inc c n t t + = F F F (17) in which _ in c inc

F

is the incremental inelastic deformation gradient for the time increment t ∆ which can be treated by a Padé approximation:

( ) ( ) 1 _ 2 2 in in in c inc c c t t - ≈ -∆ + ∆ F I L I L (18)
The crystalline elastic deformation gradient is then extracted from the multiplicative deformation decomposition:

1 e in c c c - = F F F .
The definition of the mechanical coupling of the crystalline lamella with its corresponding amorphous layer remains unchanged from that initially proposed by Lee et al. (1993aLee et al. ( , 1993b) ) and [START_REF] Van Dommelen | Micromechanical modeling of the elasto-viscoplastic behavior of semi-crystalline polymers[END_REF]. Therefore, only a summary of the concept is addressed in the next subsection since the details can be found in the paper abovementioned.

Volume-averaging

The mechanical coupling between the deformation modes in amorphous and crystalline domains is obtained using as ensemble-volume averaged homogenization procedure a volume-weighted average of the respective phases.

The local Cauchy stress tensor i I σ of the composite inclusion i is given by: ,0 ,0

= i i i i i I c c a a f f + σ σ σ (19)
i c c i c i i a a c c f J f f J f J = + and ,0 ,0 ,0 i a a i a i i a a c c f J f f J f J = + (20)
where ,0 i c f and ,0 i a f are the corresponding initial volume fractions.

The macroscopic Cauchy stress tensor σ in the semi-crystalline polymer is obtained by the volume-averaging of the I N inclusions in the aggregate:

1 = I N i i I I i f = ∑ σ σ , ,0 ,0 1 1 I i i I I i I N j j I I I j f J f N f J = = ∑ = (21)
The local inclusion-averaged deformation gradient tensor i I F is given by: ,0 ,0

= i i i i i I c c a a f f + F F F (22)
The latter is equal to the imposed macroscopic deformation of the aggregate of the I N inclusions, i.e.

i I = F F . In the latter Taylor assumption, compatibility of deformation is satisfied everywhere by considering the local deformation to be related to the global one in an affine sense. It is obvious that the underlying assumptions to relate local and global responses impact the physical consistency and predictability of the model. A realistic modeling allows to propose a framework avoiding (or at least limiting) the arbitrary inclusion of heuristic parameters. Although the Taylor assumption may be questionable, it has the advantage of being easy to implement compared to more sophisticated micromechanical approaches based on an Eshelby formulation [START_REF] Mesbah | How stress triaxiality affects cavitation damage in high-density polyethylene: experiments and constitutive modeling[END_REF], in particular, regarding the complex coupling existing between microstructure, finite-strain, time-dependency, non-linearity and external mechanical loading. Note that the resulting model, applied here to a broad range of crystallinities, involves a relatively low number of parameters with all a physical meaning.

The compatibility condition on the deformation gradients is introduced at the amorphous-crystalline interfaces: Additionally, the stress equilibrium condition at the interface between the crystalline lamella 1 Lamellar surfaces in polyethylene are of the {h0l}-type where the angle between the chain direction and the normal of interface varies between 20° and 40° [START_REF] Bassett | On the morphology of melt-crystallized polyethylene I. Lamellar profiles[END_REF]. The initial angle between chain direction and normal of interface is set to be 35°.

0 0 0 i i i i i i c a I ⋅ = ⋅ = ⋅ F F F x x x ( 23
and the amorphous layer writes:

i i i i i i c a I ⋅ = ⋅ = ⋅ σ σ σ n n n (24)
where i n is the unit normal vector at the amorphous-crystalline interface in the current configuration.

When the crystalline lamella is sheared, the plastic deformation is frozen and the homogeneously plasticized crystals can be then regarded as rigid bodies connecting the amorphous chains and having the possibility to rotate during the stretching of amorphous chains, as conceptually realized in [START_REF] Guo | A physically-based thermo-mechanical model for stretch-induced crystallizable rubbers: crystallization thermodynamics and chain-network crystallization anisotropy[END_REF] for crystallized rubber systems. The texture evolution is related to the orientation of the amorphous-crystalline interfaces which can be updated at each increment. Considering two independent vectors at time 0 t = , 0 1 i δ x and 02 i δ x , in the crystalline-amorphous interface, they become at time t : 1 0 1

i i i δ δ = F x x and 2 0 2 i i i δ δ = F x x
where i F can be either

T T i i i i i i i i i i i i i δ δ δ δ - - × = × F F F = F F F x x n n x x n (25)
with ( )

01 02 0 det T i i i i i i i δ δ - × = F F F F x x
n , the symbol × meaning the cross product of the vectors.

The model introduces a connection between the polyethylene response, the structure and the physics of the deformation modes of the crystalline and amorphous phases. The combination of amorphous and crystalline stiffness and plastic flow describes the small and moderate-strain response. For the large deformation behavior, two molecular orientation effects that take place as the strain-hardening stage proceeds are taken into account:

Amorphous chains network alignment/relaxation and crystallographic orientation. In what follows, the model is applied to the large-plastic deformation of polyethylene materials covering a wide spectrum of the crystallinity. was extracted from SAXS results and reported in Table 3. The term c t is the mean crystal lamella thickness and a t is the mean amorphous layer thickness (see Fig. 1). The long period p L was computed thanks to the Bragg relation from the scattering vector q obtained by azimuthal integration of the 2D-patterns:

Results and discussion

3

2 d p n L q π = (26)
where d n is the diffraction order.

The lamellae in HD are the longest and thickest ones as observed in Fig. 3. Note that the nearly same long period for LL and UL means distinct c t values as reported in Table 3. The 2 AFM images were obtained on 0.5 mm thick films using a Dimension 3100 apparatus from Digital Instruments operated in Tapping Mode. The films were slowly cooled from the melt with free upper surface in order to generate a natural crystallization-induced morphology.

3 SAXS experiments were carried out on the BM02 beamline of the European Synchrotron Radiation Facility (Grenoble, France) equipped with a 0.1 mm point-focusing collimation and a CCD camera (Princeton Instruments, Trenton NJ, USA) at a distance of 127 cm from the specimen, using a wavelength of 1.54 Å.

latter was estimated as follows:

c p cw c t L f ρ ρ = (27)
where ρ is the density of the whole material (

f cw f H f H ∆ = ∆ (29) in which ,0 f H ∆
is the melting enthalpy of a theoretically fully crystalline polyethylene [START_REF] Wunderlich | Macromolecular Physics, Crystal Melting[END_REF] taking into account the melting temperature dependence for every crystal concentration (see Table 3).

Large-plastic deformation

The intrinsic large-plastic deformation was obtained using video-controlled tensile mechanical tests. They were performed on an electromechanical Instron-5800 universal testing machine interfaced with a non-contact optical extensometer for the local strains measurement on dog bone shaped samples4 of 28 mm gauge length, 8 mm width and 1 mm thickness; The gauge part presents a curvature radius of about 120 mm in order to localize the necking (if any) while minimize the stress triaxiality effect. The local true axial strain rate was controlled to remain strictly constant throughout the tests in the smallest sample cross-section even after necking by regulating the cross-head speed of the testing machine. A series of mechanical tests were carried out to obtain data on the main features of the large-strain plastic deformation at room temperature. Different experiments have been undertaken in the plastic instability:

• Single loading steps up to very large strains for model identification: the sample is stretched to a prescribed strain level of about 1.8 at different constant true strain rates: 0.01 /s, 0.005 /s, 0.001 /s, 0.0005 /s and 0.0001 /s.

• Single loading-unloading steps for model verification: the strain is ramped to a prescribed level at a constant true strain rate of 0.001 /s and then ramped down to zero stress under the same absolute strain rate. Three maximum applied strains are considered: 0.9, 1.2 and 1.5.

• Oligo-cyclic stretching for model verification: The previous sequence is performed repeatedly to determine the progressive stress-softening at maximum applied strain and the accumulation of plastic strain at zero stress with increasing cycle number.

The responses are plotted in the form of true (Cauchy) axial stress5 versus true (Hencky) axial strain.

Model identification

The model parameters were identified to fit the constant true strain rate monotonic curves up to very large strains. The values are listed in Tables 4 and5 for HD, LL and UL. The three polyethylene systems have been represented by an aggregate of 125 two-phase layered composite inclusions. As the local interaction between the two phases reduces the segmental mobility of the amorphous chains in the vicinity of the crystalline lamella (see Fig. 1), the mean stiffness of the amorphous layer is expected to be severely modified from a material system to another. Both initial elastic and hardening moduli vary with the crystal content.

Besides, the difference in amorphous network properties is consistent to the difference in molar weights (see Table 2) and in chain entanglements that are rejected from crystals to amorphous phase. The higher the crystallinity the more entangled the amorphous phase, showing higher concentration of stress transmitters in the amorphous layers including tie molecules and entanglements (i.e. lesser mean chain length N ). The overcoming plastic resistances in the two phases are assumed to be nearly invariant with the crystal content; Only the interaction coefficient changes to consider the morphology effect. Fig. 5 presents the identification results for the different local true axial strain rates: 0.01 /s, 0.005 /s, 0.001 /s, 0.0005 /s and 0.0001 /s. The solid lines represent the model simulations while the symbols represent the experimental data. The model appears to be relatively effective to capture the influence of the crystallinity on various features of the nonlinear mechanical response along with the nonlinear rate-dependency. Moreover, it is able to reproduce the crystal content effect on the gradual polyethylene response transition from the thermoplastic-like behavior of the HD system to the elastomeric-like behavior of the UL system. However, the low-strain response is captured rather poorly by the model with a divergence that increases with the crystallinity decrease. That can be mainly the result of the elastic properties of the polyethylene crystal taken from the literature and introduced as direct inputs into the model for the three polyethylene systems. Despite the latter, the model adequately reproduces, as the crystal content decreases, the decrease in plastic features (yield strength and strain-hardening ability) along with the decrease in rate-dependency.

Model results

Microstructure evolution

Highly anisotropic behavior takes place during the strain-hardening stage 6 . In the model, the orientation-induced strain-hardening region is governed by both crystallographic orientation and molecular orientation/relaxation of the amorphous phase. The predicted results of the crystallographic texture evolution can be evaluated by means of pole figures. Illustrative examples are presented in Fig. 6 in the un-deformed state and in three precise deformed states for preferential directions. The initially isotropic material with randomly oriented lamellae transforms into an anisotropic material due to the reorientation mechanism of the crystalline lamellae towards the maximum principal stretch direction. The (010) poles spread around the transverse direction (Fig. 6a) while the inclusion interface normals are migrating towards the transverse direction (Fig. 6b). The microstructure evolution can be also analyzed using the local axial stresses and stretches plotted in Figs. 7 and8, respectively. In the early stage of deformation until yield point, the response is governed by the plastic deformation in the crystalline phase and by the intermolecular interactions in the amorphous phase; Their relative importance depends on the amorphicity. It is worth noticing that the interfacial interactions between crystalline and amorphous domains may form a region in the vicinity of crystalline lamella where the amorphous chain mobility is reduced (see Fig. 1). Due to the loss of chain segmental mobility, the glass transition temperature is expected to be severely modified leading to an enhancement of the intermolecular resistance of the amorphous layers. The latter resistance depends on the crystallinity. For the higher crystalline system, the small-strain tensile yielding, manifested by the initial elastic response followed by the rollover to yield, is predominantly associated with the crystallographic shear in the crystalline phase but also, in a certain extent, with the intermolecular barriers to chain-segment rotation in the minor amorphous phase. In the plastic regime, the extension of chains in the amorphous phase progressively governs the response manifested by the progressive strain-hardening and finally the dramatic strain-hardening at very large strains. For low-crystalline systems, the molecular orientation/relaxation of the amorphous phase dominates the material response in all strain structural modifications accompanied by widely diverse deformation mechanisms in the two constitutive phases.

It is determined by the stretching of chains in the amorphous phase occurring in the first place (involving inter-lamellar separation, inter-lamellar shear and lamellar rotation) and by the plastic deformation events in the crystalline lamellae followed by their fragmentation leading to the formation of a fibrillar microstructure at large strains.

range, the contribution of the discrete crystalline lamellae to the overall resistance to deformation being slight and they rather act as junction points in the amorphous matrix.

Non-monotonic conditions being one of the most severe tests for the verification of a constitutive model, its predictive capacities are discussed in the two next subsections using the model parameters previously identified in monotonic loading.

Loading-unloading

The model predictions under a loading-unloading path are depicted in Fig. 9 by comparing them with the experimental observations of the three polyethylene systems. A global view at these plots shows that the unloading path is strain dependent. A good agreement appears between model simulations and experimental data of the HD system. The plastic strain at zero stress is very well predicted by the model. For the LL system, it is satisfactory to observe that the model predicts the nonlinearity of the unloading path when the strain level is large. For more moderate strains, the model is unable to capture the nonlinear unloading path which leads to an overestimation of the plastic strain at zero stress. Indeed, the stiffness evolves during retraction which leads to a strong nonlinear unloading path. The abrupt initial unloading tangent does not allow to the model to reproduce the unloading response of the UL system. The increase in crystal content leads to a gradual transition from a linear unloading response dominated by the stiff crystalline lamellae to a drastic recovery upon unloading dominated by the rubbery amorphous layers. It is satisfactory to observe that the model reproduces, as the crystal content decreases, the decrease in residual strain at zero stress. The latter may be associated to the drastic recovery governed by the network elastic stretch during retraction (Fig. 8).

Oligo-cyclic loading

In order to check further the predictive capabilities of the constitutive model, the cyclic response is presented and discussed. The predicted results are compared to the cyclic experimental data in Fig. 10 for a maximum strain value of 1.5 and ten consecutive cycles.

The cyclic response exhibits a Mullins effect common with elastomeric-like materials characterized by a (i) progressive stress-softening with increasing cycle number, (ii) a hysteresis loop (i.e. a difference between unloading and reloading paths), (iii) an accumulation of plastic strain at zero stress [START_REF] Drozdov | Mullins' effect in semicrystalline polymers[END_REF][START_REF] Wang | Network alteration theory on Mullins effect in semicrystalline polymers[END_REF][START_REF] Makki | Mullins effect in polyethylene and its dependency on crystal content: A network alteration model[END_REF][START_REF] Guo | Characterization of the spherulitic deformation in equatorial region and cavitation in HDPE materials submitted to mixed-mode oligo-cyclic tensile loading[END_REF]. Their amounts depend on the amorphous phase fraction. The plastic strain evolution is plotted in Fig. 11 until one hundred consecutive cycles. Note that it is divided by the value at the first cycle. The instantaneous adjustment of the cross-head speed of the testing machine to maintain constant the local strain rate (especially during the necking and post-necking stages if any) induced serrated experimental curves. The observed inflection points on the theoretical evolutions are mainly due to the fact that they are not continuously obtained but are punctually extracted from the cyclic stress-strain curves. The accuracy in the data extraction affects thus the smooth character of the theoretical evolutions. The model ability to capture the plastic strain evolution depends on the amorphous phase fraction. It underestimates the HD and LL data in the first fifty cycles and then overestimates the data, whereas it always underestimates the UL data.

The deformation mechanisms associated to the Mullins effect can be discussed thanks to the plots of Figs. 7 and8. When the macroscopic strain is retracted to zero macroscopic stress, the local axial stress in the crystalline lamellae is negative in average whereas the local axial stress in the amorphous phase is positive. The further decomposition of the amorphous stress shows a negative intermolecular stress and a positive network stress. Because the previously sheared crystalline lamellae are in a frozen state, the crystalline stress is seen to be stabilized throughout the consecutive deforming cycles. By contrast, the network stress significantly decreases with increasing cycles whereas the intermolecular stress slightly decreases. The inelastic intermolecular stretch is permanent while the inelastic network stretch slightly increases with increasing cycles. In addition to the crystal texturing, a new molecular network configuration appears in the amorphous linking modifying the previous one at each new cycle of the cyclic loading.

It can be observed that prediction and experimental data are close to each other for the HD system but the predicted hysteresis loop diverges from the experimental data with the crystallinity increase. Recall that the elastic properties of the polyethylene crystal are 20 introduced as direct inputs into the model which leads to the abrupt reloading. Note that the main goal here is to observe the model limitations rather than propose improvements of modeling capabilities. Disregarding the micromechanical scale transition, the simplest way to improve the model capacities could be done by adapting the crystal stiffness for each fraction and/or by introducing a network alteration upon cyclic loading through an appropriate change of the number of entangled points. As some entanglements will become ineffective with increasing cycle number, h c C n kθ = will decrease and N will increase (the product c n N

remaining constant in order to conserve mass) while the other model parameters would remain unchanged. The evolution equations of c n and N would require supplementary parameters fitted on the hysteretic response evolution with increasing cycle number [START_REF] Makki | Mullins effect in polyethylene and its dependency on crystal content: A network alteration model[END_REF]. Nonetheless, our objective here is to limit the number of model parameters and to present the model plots for predictive purpose only introducing the crystal content as main variable input.

The quite good agreement with the experimental observations of the HD nonlinear cyclic response confirms the ability of the constitutive representation to describe high crystalline media. Fig. 12 compares the predicted and measured curves to present the strain level effect on different aspects of the HD cyclic response considering both one cycle with increasing maximum strain levels and oligo-cyclic loading with two strain levels. It can be observed that the model recaptures the same path as that corresponding to the monotonic one when the strain exceeds the maximum strain previously applied. The general trends of the oligo-cyclic stretching are adequately predicted by the model, in particular the increase in stress-softening magnitude with strain level.

Concluding remarks

In this paper, the highly nonlinear and rate-dependent mechanical response of semi-crystalline polyethylene systems covering a wide spectrum of the crystallinity was investigated within a multi-scale homogenization-based approach. The model ability to capture the large-strain response variation with the crystallinity was criticized. The model exhibited weaknesses in small-strain region but it was able to reproduce, within large-strain region, the rate-dependent monotonic response variation with crystallinity. The general trends of the high-crystalline material cyclic response were quantitatively well reproduced by the model while the deviation from experiments increased with amorphicity, which would require to take into account the amorphous network alteration.

In addition to molecular orientation effects that takes place as the deformation proceeds, the cavitation damage associated with progressive nucleation and growth of cavities in the amorphous layers will be presented in a future work.

Table 1. Slip systems and associated shear strength (Lee et al., 1993a x 1

x 2

x 1

x 2

x 1

x 2

x 1

x 2

x 1

x 2

x 1

x 2

x 1

x 2

x 1

x 2 Figure 7. Local axial stresses in the three polyethylene systems upon oligo-cyclic loading at a strain level of 1.5 and at a true strain rate of 0.001 /s.

  The aim of the present article is to examine the ability of a multi-scale homogenization-based constitutive model to capture the polyethylene large-strain tensile response variation with crystal concentration. The model is applied on a high-density polyethylene with 0.72 crystal content, a low-density polyethylene with 0.3 crystal content and an ultra-low-density polyethylene with 0.15 crystal content. The results of our simulations are criticized by analyzing the model ability to capture a series of experimental observations under monotonic, loading-unloading and oligo-cyclic stretching over a large strain range. The effect of the amorphous phase fraction on the microstructure evolution is finally discussed thanks to the model. The present paper is organized as follows. Section 2 presents the fully three-dimensional continuum-based constitutive theory. The model-experiments comparisons are presented and discussed in Section 3. Concluding remarks are finally given in Section 4. The following notation is used throughout the text. Tensors and vectors are denoted by normal boldfaced letters and italicized boldfaced letters, respectively, while scalars and individual components of vectors and tensors are denoted by normal italicized letters. The superposed dot designates the time derivative. The superscript T indicates the transpose quantity.

  in which the elastic and inelastic components are depicted by superscripts e and in, respectively. The two tensors can be further decomposed into stretch (right U or left V ) and rotation R movements using the polar decomposition: The subscripts c and a are used to differentiate the crystalline and amorphous phases, respectively.

  ). The Taylor assumption is used such that the deformation gradients in the intermolecular and network resistances are equal: with progressive nucleation and growth of cavities in the amorphous layers[START_REF] Zaïri | A physically-based constitutive model for anisotropic damage in rubber-toughened glassy polymers during finite deformation[END_REF][START_REF] Mesbah | How stress triaxiality affects cavitation damage in high-density polyethylene: experiments and constitutive modeling[END_REF] is neglected and the amorphous inelastic deformation is assumed to be isovolumetric, i.e.

  volume change, c σ is the crystalline Cauchy stress tensor, -Lagrange strain tensor and c K denotes the anisotropic fourth-order elastic stiffness tensor of the orthorhombic crystalline phase of polyethylene taken from the literature[START_REF] Tashiro | Calculation of three-dimensional elastic constants of polymer crystals. 2. Application to orthorhombic polyethylene and poly(vinyl alcohol)[END_REF]:

  crystalline shear strain rate constitutively coordinated to the resolved shear stress :

  volume fractions, respectively, expressed as: ,0

  the amorphous-crystalline interface in the reference configuration 1 .

  systems, differing by their molecular topology and molar masses, were ordered from Total Petrochemicals and DOW Chemicals: a Ziegler-Natta high-density ethylene-hexene copolymer (HD) from Total Petrochemicals, a linear low-density ethylene-octene copolymer (LL) and an ultra-low density ethylene-octene copolymer (UL) from DOW Chemicals, both issued from metallocene catalysis. The molecular characteristics of the polyethylene pellets determined by Gel Permeation Chromatography (GPC) are listed in Table2(provided by the manufacturers). The high co-unit concentrations for LL and UL interfere with the crystalline lamellae formation, and thus induce lower density and crystal degree than for HD. The crystal morphologies can be observed from the Atomic Force Microscopy (AFM) 2 pictures in Fig.3. Highly regular stacking of crystal lamellae for HD and wide thickness distributions of both the amorphous layers and the crystal lamellae for LL and UL have been evidenced from Small-Angle X

  measured by DSC at a scanning rate of 10°C/min under nitrogen flow with a Q100-apparatus from TA Instruments, which has been indium-calibrated. Melting endotherms and the crystallizing exotherms are provided in Fig.4

Figure 1 .Figure 2 .Figure 3 .

 123 Figure1. Two-scale view of the semi-crystalline structure; At the mesoscopic scale, the crystalline lamellae may be arranged in highly regular stacking to form spherulitic morphology for high-crystalline media or they may be randomly distributed in the amorphous matrix for low-crystalline media; Both phases are coupled in a complex manner.

Figure 4 .Figure 5 .Figure 6 .

 456 Figure 4. Melting endotherms and cooling exotherms of the three polyethylene systems. The HD heat flow is typical of a high-crystalline polymer. The very broad LL and UL melting endotherms show non-uniform chemical composition distributions due to high co-unit concentrations.

Figure 8 .Figure 9 .Figure 10 .Figure 11 .

 891011 Figure8. Local axial stretches in the three polyethylene systems upon oligo-cyclic loading at a strain level of 1.5 and at a true strain rate of 0.001 /s.

Table 2

 2 Table3provides the crystallinity characteristics at room temperature. The specific melting

	), c ρ =1.00 g/cm 3 is the density of the

Table 2 .

 2 ). Molecular characteristics and density of the three polyethylene systems: number-average molar weight

		Slip system	g α (MPa)
			(100)[001]	8
			(010)[001]	20
			{110}[001]	20
			(100)[010]	13.3
			(010)[100]	20
		{110}<11 0>	17.6
	Material	n M (kDa)	w M (kDa)	ξ (mole%) †	ρ (g/cm 3 ) *
	UL	75	150	12 (60)	0.870
	LL	50	104	5.0 (25)	0.902
	HD	14	174	0.1 (0.5)	0.959
	† hexene for the HD system; octene for the LL and UL systems; between brackets is the short chain branching per
	1000 C atoms in the backbone.

n M ; weight-average molar weight w M ; counit content ξ ; density ρ . * density of compression-molded sheets.

Table 3 .

 3 Physical characteristics of the three polyethylene systems: long period p

	L ; mean crystalline

Table 5 .

 5 UL and LL model parameters (all other parameters are listed in Table4).

	Material	λ (GPa)	µ (MPa)	κ	Ch (MPa)	N
	UL	1	2	0.28	0.24	42
	LL	16.6	33	1.4	0.8	42

Although no initial preferential orientation was expected, the samples were cut off in the same direction of 1 mm thick compression-molded sheets; The latter were molded from polyethylene pellets at 180°C for 10 min and slowly cooled to room temperature in order to avoid residual thermal stresses generated during cooling.

The stress was calculated by dividing the value of the force (registered by a standard load cell of 1 kN) with the actual specimen cross-section recorded by the optical extensometer.

From the earlier studies of[START_REF] Peterlin | Plastic deformation of polymers with fibrous structure[END_REF], the prevailing picture of the microstructure evolution due to tensile straining of a semi-crystalline polymer has been stated(Seguela et al., 1998a(Seguela et al., , 1998b;;[START_REF] Bartczak | Influence of molecular parameters on high-strain deformation of polyethylene in the plane-strain compression. Part I. Stress-strain behavior[END_REF][START_REF] Bartczak | Plasticity of semicrystalline polymers[END_REF][START_REF] Rozanski | Plastic yielding of semicrystalline polymers affected by amorphous phase[END_REF]; The tensile yielding results from massive


Olivier Delattre

Thomas Sellenet

Jose-Luis Barnay

Thomas Chevillotte

Marine De Tienda
email: marine.detienda@gmail.com

Transfer of distal peroneus longus tendon to tibialis anterior by retrograde fixation to treat spastic equinovarus foot in adults: Surgical Technique and Preliminary Results

Keywords: Equinovarus, Spasticity, Tendon Transfer, Fibular

We describe a retrograde transfer of the distal tendon of the peroneus longus (PL) onto the tibialis anterior (TA) tendon to treat spastic equinovarus foot (SEVF) in adults. The fact that the distal tendon insertions of the PL and TA are a mirror image makes them antagonists. The aim is to divert the distal tendon in front the inactive distal PL tendon, by fixing to the TA in the middle third of the lower leg. This transforms it into a dorsiflexor and reinforces its eversion ability. The suture level helps to avoid skin impingement when wearing shoes, and the complications inherent to transosseous fixation. In a preliminary case series of 10 patients, we found no complications at a mean follow-up of 4.7 years. The Foot Posture Index-6 improved by an average of 2.4 points. Four patients had regained active dorsiflexion. All patients reduced their use of orthotics. All patients improved according to Goal Attainment Scaling.

Introduction

Treatment of spastic equinovarus foot (SEVF) in adults aims to restore stable walking and a plantigrade foot position. Various tendon transfers have been used in the past to achieve this. The main ones are tibialis anterior (TA) transfer, hemi-TA transfer, and retrograde fixation of the distal peroneus brevis (PB) tendon onto the TA [START_REF] Denormandie | Surgical treatment of orthopedic deformities due to spasticity in the lower limb[END_REF]. Use of the peroneus longus (PL) has never been described in the literature. Yet, imbalance between TA hypertonicity and fibular muscle weakness is one of the main factors leading to forefoot supination. The goal of this work is to describe this transfer of the PL's distal tendon with retrograde fixation on the TA, to expose its anatomical and physiological bases and to evaluate the outcomes from a pilot series of cases.

Materials and Methods

Surgical technique

The patient is supine with a support under the buttocks. The first incision is 1 cm long, horizontal and 2 cm behind the base of the fifth metatarsal (Fig. 1). It is used to isolate the PL, which courses behind, then under the PB. Light traction is placed on the tendon to make it easier to palpate its proximal trajectory. A second vertical, lateral incision about 4-cm long is made about two-thirds of the way down the lower leg (15 cm above the tibiotalar joint line) and used to do a PL tenotomy as proximally as possible after having released any adhesions. The tendon is placed on the traction suture and easily retrieved through the first incision. The PL externalized distally is 18 to 20 cm long (Fig. 2,(2.1)). One then simulates its transfer as proximally as possible on the TA while holding the tendon outside the skin incision. This is done to determine the location of the third incision (longitudinal and about 5 cm long) on the lateral side of the tibial crest. The third incision is used to isolate the TA tendon. Long, blunt instruments (scissors, curved forceps) are used to develop an atraumatic subcutaneous tunnel, which is superficial relative to the extensor retinaculum.

The tendon on the traction suture is transferred subcutaneously distally to proximally.

If the Achilles tendon is retracted, it must be lengthened beforehand. If this procedure is not done at the start, it is vital to do it before setting the tension on the tendon transfer, as the latter is done with the foot held by the surgical assistant in 10° dorsiflexion and maximum eversion. Maximum traction is placed on the PL, which has been passed through the TA via a small longitudinal midline incision (Fig. 2.2). Two cross stitches of non-absorbable 3-0 suture are placed. This is supplemented by an edge-to-edge suture over about 3 cm to achieve a long and flat transfer area with two rows of three other cross stitches [START_REF] Brown | Mechanical strength of the side-to-side versus Pulvertaft weave tendon repair[END_REF]. Keeping maximum tension on the PL, while slightly relaxing the TA, results in a Y anastomosis (Fig. 3).

Tenotomy of the toe flexors is often added to the procedure. If the varus is not fully reducible, one can consider adding valgus-inducing osteotomy of the calcaneus.

Postoperatively, immobilization is done with a resin walking cast, with the foot in 10° dorsiflexion and -most importantly -maximum eversion for 6 weeks. Full weight bearing is allowed. The patients are seen again after 6 weeks to remove the walking cast. Rehabilitation is started immediately on an outpatient basis.

Clinical study

A single-center retrospective study was done of 10 patients operated by the same surgeon between January 2006 and April 2016.

The inclusion criteria were a varus and supination deformity due to TA hypertonicity, which was reducible, whether or not associated with equinus. TA hypertonicity was diagnosed with Strumpell's maneuver. This clinical sign reveals involuntary TA recruitment during resisted triple flexion of the hip, thigh, and knee.

Potential associated procedures were a valgus-inducing osteotomy of the calcaneus, Achilles tendon lengthening or tenotomy of the toe flexors; however, none were done in this case series on the TP.

The main endpoint was the Foot Posture Index (FPI)-6 as described by Keenan [START_REF] Keenan | The Foot Posture Index: Rasch Analysis of a Novel, Foot-Specific Outcome Measure[END_REF][START_REF] Mcbride | The relationship between supination resistance and the kinetics and kinematics of the foot and ankle during gait[END_REF]. This is a clinical assessment of the foot's position during weightbearing with items on the rear-foot, midfoot and forefoot. Satisfaction was evaluated with Goal Attainment Scaling (GAS) where the goal was defined by the patient, surgeon, and rehabilitation specialist during the preoperative consultation (5).

Results

Ten procedures were done on 10 patients with this technique. The patients had a mean age of 56.2 years (min-max 30-80). The mean follow-up was 4.7 years (min 18 months, max 11 years). There were 6 women and 4 men who were all ambulatory; 9 had sequelae of a stroke and 1 had cerebral palsy. Four patients had their gait recorded on video after the surgical procedure.

All but one patient improved on the FPI-6, with a mean of -3.5 postoperatively versus -5.9 preoperatively (Table 1). Heel-toe gait was restored in 7 patients (Fig. 4.1, 4.2). The equinus was improved overall and completely corrected in 4 patients. Four patients recovered active dorsiflexion (Fig. 5).

Two patients who had not gotten tenotomy of the toe flexors were hindered by claw toe deformity, thus required another surgical procedure. At the final assessment, none of the patients had excessive talus.

The use of orthotic devices was reduced in all patients and all of them improved their walking distance (Table 2). None of the patients reported rubbing of their surgical scars with footwear.

At the final assessment, there were no instances of suture failure or recurrence.

All patients improved their GAS by at least 1 point (Table 1), and all would recommend this procedure to a family member or close friend who had the same pathology.

Discussion

Three techniques have been described for SEVF in adults: transfer of hemi-TA to lateral side of foot (6-10), complete TA transfer with transosseous fixation on the cuboid [START_REF] Ponseti | The classic: observations on pathogenesis and treatment of congenital clubfoot[END_REF] and transfer of the distal tendon of the PB on the tibialis anterior, first described by Bardot then Curvale [START_REF] Bardot | Orthopedic surgical corrections of spastic disorders[END_REF][START_REF] Curvale | Treatment of spastic equinovarus foot in the hemiplegic adult by retrograde fixation of the peroneus brevis tendon onto the anterior tibialis tendon[END_REF].

Hence, PL transfer has several advantages and gets around some of the complications associated with the above techniques:

-It preserves the TA in its entirety and its strength. It is commonly accepted that a tendon transfer reduces the muscle's strength by one grade level (out of five) [START_REF] Omer | Evaluation and reconstruction of the forearm and hand after acute traumatic peripheral nerve injuries[END_REF]. This is even more problematic in patients experiencing spasticity and paresthesia. This technique allows us to get around any weakness secondary to TA transfers. Also, an appreciable loss of tension in the TA has been described when the entire tendon is transferred [START_REF] Ponseti | The classic: observations on pathogenesis and treatment of congenital clubfoot[END_REF].

-Because suture fixation is very proximal, it prevents rubbing of the skin scars with orthotic devices or shoes. This is one of the complications of the Bardot transfer with keloid scars over the tendon suture, which is subcutaneous, and impingement on the anterior side of the ankle when wearing shoes.

-It provides a good quality transplant because of the use of an entire tendon, and thus reduces the risk of suture failure associated with transfer of half-TA tendon.

-It gets around the mechanical complications of bone fixation typically found in 5% of cases in older patients with limited activity level and often osteoporosis (fracture, anchor failure, etc.) [START_REF] Tracy | Operative treatment of the plantar-flexed inverted foot in adult hemiplegia[END_REF][START_REF] King | Spastic foot and ankle deformities: evaluation and treatment[END_REF][START_REF] Asencio | Surgery of the foot in equinus deformity in hemiplegic adults[END_REF].

-Lastly, using the PL is a major anatomical asset. In fact, its distal insertion is made on the plantar side of the 1st metatarsal and the medial cuneiform. Thus, it has a plantar flexion action in addition to its eversion action. It is a perfect mirror image with the TA insertion, which occurs on the dorsal side of the 1st metatarsal and cuneiform.

After the transfer, when the TA is recruited, it became a dorsiflexor and foot evertor.

In some patients, parameters related to the midfoot and hindfoot were improved. This is probably because an action on the TA causes derotation of the calcaneopedal unit [START_REF] Seringe | Le concept de bloc calcanéo-pédieux[END_REF].

We only use Watkins' technique [START_REF] Watkins | Transplantation of the posterior tibial tendon[END_REF] of transposing the terminal tendon of the tibialis posterior muscle for purely posterior varus feet.

Conclusion

This technique yields good results on the varus and the equinus. It is a viable alternative to half-TA transfer, the procedure that is currently used the most as a surgical treatment of spastic feet. It has several advantages. With a follow-up that was more than 10 years in some patients, we found no loss of tension or recurrences. However, the benefit of this surgery is modest in hemiplegia cases.

The aim is not to restore normal walking ability but to reduce the use of orthotic devices. These findings must be confirmed in a larger study where the patients undergo gait analysis. A tenotomy is done through the second incision [START_REF] Brown | Mechanical strength of the side-to-side versus Pulvertaft weave tendon repair[END_REF]. The end of the tendon is retrieved distally (1) then transferred subcutaneously (a') to the TA (c) and sutured through a third incision (3). The foot is then held in eversion and dorsiflexion to suture the PL tendon under maximum tension. If needed, percutaneous Achilles tendon lengthening (d) can be added.

Éversion du pied = Foot eversion

Talar head palpation -1 0 -1 -1 -1 -1 -1 -1 X -1 -1 0 -1 -1 X X -1 -1 -1 - 1
Supra-and infra-lateral malleoli curvature

-2 -1 0 0 0 0 -1 -1 X -1 -1 -1 -1 -1 X X -1 -1 -1 -1 Calcaneal frontal plane position -2 -1 -1 0 -1 0 -2 0 -2 -1 -2 -1 -2 -1 -2 0 -2 -1 -2 0
Prominence in region of talonavicular joint

-1 0 -1 -1 -1 -1 -1 0 X -1 -1 0 -1 -1 X X -1 -1 -1 -1
Congruence of medial longitudinal arch

-1 0 -1 -1 -1 -1 -1 -1 X -1 -1 -1 -1 -1 X X -1 -1 -1 -1
Abduction/adduction of forefoot on rearfoot

Figure 1 .

 1 Figure 1. Surgical approach to the PL; tenotomy by the proximal lateral approach; intraoperative view

Figure 2 : 1 /Figure 3 .

 213 Figure2:1/ Distal tendon of the PL is externalized by the distal surgical approach; 2/ TA is sutured through the proximal and medial approach while the foot is held in 10° flexion and eversion. Intraoperative view Figure3. The first incision is submalleolar (1) to locate the PL tendon (a) behind the PB (b). A tenotomy is done through the second incision[START_REF] Brown | Mechanical strength of the side-to-side versus Pulvertaft weave tendon repair[END_REF]. The end of the tendon is retrieved distally (1) then transferred subcutaneously (a') to the TA (c) and sutured through a third incision (3). The foot is then held in eversion and dorsiflexion to suture the PL tendon under maximum tension. If needed, percutaneous Achilles tendon lengthening (d) can be added.

Figure 4 : 1 /Figure 5 .

 415 Figure 4: 1/ Preoperative equinovarus and supination; 2/ Postoperative results: Recruitment of graft during voluntary TA contraction Figure 5. Postoperative results: Recruitment of graft during voluntary TA contraction and active dorsiflexion

Table 1

 1 Clinical outcomes (FPI-6) determined before and after the surgery

	Results of Goal Attainment Scaling
	X: Missing data

Table 2

 2 Changes in technical and human aids for walking after this transfer surgery OS: orthopedic shoes

Table 1

 1

	P1		P2	P3	P4	P5	P6	P7	P8	P9	P10
	Pre	Post	Pre Post Pre Post Pre Post Pre Post Pre Post Pre Post Pre Post Pre Post Pre Post

Table 2

 2

		P1	P2	P3	P4	P5	P6	P7	P8	P9	P10
	reoperative assistance	OS Cane outsider OS Cane	OS Cane	Cane OS Orthotics Cane OS	OS	Cane	OS Cane	OS Orthotics outsider
	ostoperative assistance	OS	OS	None	None Orthotics	None	OS	None	OS	OS Cane
	ostoperative barefoot walking Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	No

Conflict of interest: none


MD Clara Vigneron

MD Julien Charpentier

MD Marie Wislez

MD Jean- Paul Paul Mira

MD Aurélie Lefebvre

MD Ludovic Fournel

MD Matthieu Jamme

MD Frédéric Pène
email: frederic.pene@aphp.fr

Prof. Frédéric Pène

Lung Cancer and Life-Threatening Complications

Keywords: Lung cancer, Intensive care, Outcome ICU (intensive care unit); SOFA (Sepsis-related Organ Failure Assessment);

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.

1

To the Editor, Lung cancer accounts for the second most frequent solid tumour and for about one in four deaths by cancer. Survival to lung cancer has improved since the 1990s owing to early tumor detection, therapeutic advances through improvements in surgical techniques and the development of immunotherapy and targeted therapy [START_REF] Siegel | Cancer statistics[END_REF] . However new treatments resulted in new life-threatening side effects likely to require intensive care unit (ICU) admission [START_REF] Kroschinsky | New drugs, new toxicities: severe side effects of modern targeted and immunotherapy of cancer and their management[END_REF][START_REF] Suresh | Immune Checkpoint Immunotherapy for Non-Small Cell Lung Cancer[END_REF] .

Patients with lung cancer may be considered as a particular group of interest in the ICU for the following reasons: i) lung cancer accounts for the leading malignancy among patients with solid tumors and unplanned ICU admission [START_REF] Vincent | In-hospital and day-120 survival of critically ill solid cancer patients after discharge of the intensive care units: results of a retrospective multicenter study-A Groupe de recherche respiratoire en réanimation en Onco-Hématologie (Grrr-OH) study[END_REF] ; ii) The majority of patients requiring unplanned ICU admissions have newly diagnosed cancer, and the acute episode is often related to a specific complication of cancer [START_REF] Soares | Intensive care in patients with lung cancer: a multinational study[END_REF][START_REF] Puxty | Factors associated with intensive care admission in patients with lung cancer: a population-based observational study of 26, 731 patients[END_REF] ; iii) short-term survival of critically ill lung cancer patients remains poor [START_REF] Vincent | In-hospital and day-120 survival of critically ill solid cancer patients after discharge of the intensive care units: results of a retrospective multicenter study-A Groupe de recherche respiratoire en réanimation en Onco-Hématologie (Grrr-OH) study[END_REF][START_REF] Soares | Intensive care in patients with lung cancer: a multinational study[END_REF] ; iv) data about their long-term outcomes are scarce.

The aim of our study was to analyse the admission patterns and the eventual short-term and long-term prognosis of critically ill lung cancer patients requiring unplanned ICU admission.

Methods

We enrolled adult patients with a diagnosis of lung cancer (before or during the ICU stay) hospitalized in a medical ICU located inside a tertiary care hospital with oncology and thoracic surgery departments, over a 12-year period (2007-2018). Post-operative management of patients with elective thoracic surgery is provided in surgical ICU. Exclusion criteria were the following: cancer cured for more than 5 years, planned admissions following elective surgery and admissions for securing a procedure. Continuous variables were expressed as median (interquartile range) and categorical variables as counts (percentages), and were compared over time by using Cuzik test and chi-square test for trend, respectively. Determinants of vital status were analysed through a multivariate time-dependent Cox causespecific proportional hazard model. The study was approved by the ethics committee of the French Intensive Care Society (#CE-SRLF-17-03)

Results

The study included 379 patients. Non-small cell lung cancer accounted for the large majority (75%) of cancer subsets (Table 1). The oncologic characteristics dramatically changed over the study period. Proportions of patients with new-onset disease decreased while patients presented with more advanced stages of cancer in the recent years, as revealed by extended time between diagnosis and ICU admission and increasing proportions of metastatic diseases.

Patients from the recent period were more likely to be under immunotherapy or targeted

Discussion

We observed a remarkable raise in ICU admissions over the 12-year study period, albeit linked in part to the development of a thoracic oncology center in our hospital in the early 2010s. In addition, we have long been promoting a broad ICU admission policy for oncohematology patients, with indications driven by the emergence of new therapeutic strategies. Some patients with metastatic diseases, considered without realistic therapeutic options a few years ago, became eligible to advanced ICU management with new strategies based on immunotherapy or targeted therapy [START_REF] Suresh | Immune Checkpoint Immunotherapy for Non-Small Cell Lung Cancer[END_REF] . This is highlighted by an overall increase in stages of malignancy during the study period, probably explaining why we did not observe any continuous improvements in outcomes [START_REF] Slatore | Intensive Care Unit Outcomes Among Patients With Lung Cancer in the Surveillance, Epidemiology, and End Results-Medicare Registry[END_REF][START_REF] Zampieri | Trends in clinical profiles, organ support use and outcomes of patients with cancer requiring unplanned ICU admission: a multicenter cohort study[END_REF] .

Specific complications accounted for one quarter of ICU admissions. Procedural interventions such as pleurocentesis, laser airway desobtruction, tracheobronchial or intravascular prosthesis insertion aim at rapidly restoring organ functionalities, pending histologic documentation and prompt initiation of antitumoral treatment whenever possible. Only cytostatic chemotherapy, and to a lesser extent targeted therapies such as kinase inhibitors in patients with adenocarcinoma and actionable oncogenic mutations, are likely to result in fast tumor reduction required in the setting of life-threatening acute respiratory failure. In contrast, the effect of immunotherapy on tumor volume is delayed and may occasionally result in paradoxical transient flare-up.

With respect to short-term mortality rates ranging from 22% to 31% and long-term 6-month and one-year mortality rates ranging from 32% to 55% in previous studies performed elsewhere [START_REF] Soares | Intensive care in patients with lung cancer: a multinational study[END_REF][START_REF] Slatore | Intensive Care Unit Outcomes Among Patients With Lung Cancer in the Surveillance, Epidemiology, and End Results-Medicare Registry[END_REF][START_REF] Toffart | Use of Intensive Care in Patients With Nonresectable Lung Cancer[END_REF] , differences in underlying oncologic characteristics make them hardly comparable with those observed in our own cohort. Besides common and largely described determinants of ICU mortality, identification of prognostic factors for one-year survival is a hallmark of this study. Impaired functional status prior to the acute complication, decisions to forgo life-sustaining therapies and the ability to receive optimal oncologic treatment thereafter are intertwined determinants of long-term prognosis. The decision-making process for ICU admission should be able to take into account the eventual realistic therapeutic options in ICU survivors.

Conclusions

ICU admissions in lung cancer patients shifted to more advanced malignancy stages over the study period, albeit without impairing both short-term and long-term survival rates.

Performance status prior to the acute complication and the continuation of oncologic treatment after ICU discharge are major prognostic factors of one-year survival.

  therapy. ICU admissions linked to drug-related and procedure-related adverse events increased over time from 4.0% in 2007-2008 to 11.6% in 2017-2018 (p=0.02). The severity of illness at ICU admission, as assessed by the Sequential Organ Failure Assessment (SOFA) score, as well as requirements of ventilatory and circulatory supports did not change over time. Decisions to forgo life-sustaining therapies were taken in 38.5% of patients. ICU, hospital and one-year mortality rates were 32.2%, 53.3% and 68.9% and did not change over the study period. Independent factors associated with ICU mortality were a Performance Status (PS) ≥ 3 prior to the acute complication (cause specific hazard (CSH) 2.04 [1.32-3.15], p=0.001), and the requirements for invasive mechanical ventilation (CSH 8.28 [1.78-38.5], p=0.007) and non-invasive ventilation (CSH 16.4 [3.87-69.1], p<0.001). When adjusted with admission SOFA score, one-year mortality in ICU survivors was independently associated with advanced metastatic stage (CSH 2.10 [1.38-3.18], p<0.001), prior PS 3-4 (CSH 2.60 [1.75-3.87], p<0.001), ICU admission for specific complications (CSH 1.53 [1.07-2.19], p=0.020), cancer treatment < 3 months (CSH 1.36 [1.02-1.93], p=0.045), decisions to forgo life-sustaining therapies (CSH 2.63 [1.74-3.95], p<0.001), and inability to receive oncologic treatment at ICU discharge (treated as reference for ability to receive oncologic treatment (CSH 0.23 [0.13-0.42], p=0.001) and for absence of indication (CSH 0.15 [0.05-0.48], p=0.002) (Figure 1).

Figure legend Figure 1 .Figure 1 .

 legend11 Figure legend

Table 1 .

 1 Characteristics of patients.

	Characteristics	2007-2008 (n=50)	2009-2010 (n=55)	2011-2012 (n=50)	2013-2014 (n=60)	2015-2016 (n=69)	2017-2018 (n=95)	p
	Age (years)	66 [56-73]	68 [59-75]	69 [59-72]	64 [56-73]	66 [61-72]	68 [62-74]	0.23
	Male gender	35 (70.0)	36 (65.5)	39 (78.0)	41 (68.3)	46 (66.7)	65 (68.4)	0.76
	Type of cancer							0.003
	Small cell lung cancer	6 (12.0)	10 (18.2)	4 (8.0)	6 (10.0)	12 (17.4)	7 (7.4)
	Non-small cell lung cancer	36 (72.0)	31 (56.4)	40 (80.0)	46 (76.7)	51 (73.9)	83 (87.4)
	Unknown	8 (16.0)	14 (25.5)	6 (12.0)	8 (13.3)	6 (8.7)	5 (5.3)
	Delay diagnosis-ICU admission (days)	67 [10-302]	101 [0-302]	122 [30-334] 148 [15-303] 136 [43-461] 183 [53-388]	0.01
	Stage							0.03
	Localized	10 (20.0)	8 (14.5)	6 (12.0)	9 (15.0)	13 (18.8)	11 (11.6)
	Advanced	13 (26.0)	19 (34.5)	10 (20.0)	14 (23.3)	13 (18.8)	16 (16.8)
	Metastatic	25 (50.0)	25 (45.5)	34 (68.0)	37 (61.7)	43 (62.3)	68 (71.6)
	Status							0.01
	Newly diagnosed	27 (54.0)	29 (52.7)	20 (40.0)	22 (36.7)	28 (40.6)	33 (34.7)
	Partial remission	3 (6.0)	1 (1.8)	7 (14.0)	9 (15.0)	11 (15.9)	19 (20.0)
	Complete remission	6 (12.0)	8 (14.5)	2 (4.0)	9 (15.0)	5 (7.2)	9 (9.5)
	Progressive	14 (28.0)	17 (30.9)	21 (42.0)	20 (33.3)	25 (36.2)	34 (35.8)
	Treatment within last 3 months
	Antitumoral drugs	17 (34.0)	20 (36.4)	26 (52.0)	29 (48.3)	41 (59.4)	51 (53.7)	0.009
	Immuno-and targeted therapy	5 (10.0)	1 (1.8)	5 (10.0)	9 (15.0)	10 (14.5)	21 (22.1)	0.001
	First-line treatment	11 (22.0)	13 (23.6)	14 (28.0)	22 (36.7)	27 (39.1)	27 (28.4)	0.15
	Radiotherapy	2 (4.0)	4 (7.3)	5 (10.0)	6 (10.0)	6 (8.7)	7 (7.4)	0.58
	Surgery	6 (12.0)	3 (5.5)	3 (6.0)	7 (11.7)	3 (4.3)	5 (5.3)	0.24
	Performance status 3-4	0/7(0)	10/32 (31.2) 16/44 (36.3) 14/57 (24.5) 24/64 (37.5) 28/93 (30.1)	0.57
	Main causes of ICU-admissions
	Specific complications	19 (38.0)	24 (43.6)	13 (26.0)	30 (50.0)	19 (27.5)	37 (38.9)	0.79
	Airway obstruction	5 (10.0)	4 (7.3)	0 (0.0)	1 (1.7)	0 (0.0)	0 (0.0)
	Pleural effusion	1 (2.0)	3 (5.5)	3 (6.0)	6 (10.0)	1 (1.4)	5 (5.3)
	Metabolic disorders	1 (2.0)	3 (5.5)	0 (0.0)	1 (3.3)	3 (4.3)	1 (1.1)
	Other	12 (24.0)	15 (27.3)	10 (20.0)	21 (35.0)	16 (23.2)	23 (24.2)	0.16
	Non-specific complications	29 (58.0)	30 (54.5)	34 (68.0)	28 (46.7)	45 (65.2)	55 (57.9)
	Infection	19 (38.0)	12 (21.8)	17 (34.0)	15 (25.0)	25 (36.2)	25 (26.3)
	Venous thromboembolism	2 (4.0)	4 (7.3)	4 (8.0)	2 (3.3)	3 (4.3)	3 (3.2)
	Miscellaneous	8 (16.0)	14 (25.5)	13 (26.0)	11 (18.3)	17 (24.6)	19 (20.0)
	Adverse event complications	2 (4.0)	1 (1.8)	3 (6.0)	2 (3.3)	5 (7.2)	11 (11.6)	0.02
	Drug-related	0 (0.0)	1 (1.8)	2 (4.0)	1 (1.7)	3 (4.3)	3 (3.2)
	Procedure-related	2 (4.0)	0 (0.0)	1 (2.0)	1 (1.7)	2 (2.9)	8 (8.4)
	SOFA	7 [4-12]	5 [4-9]	5 [4-8]	4 [3-6]	6 [4-8]	5 [4-8]	0.35
	Organ support requirements
	Invasive mechanical ventilation	27 (54.0)	29 (52.7)	25 (50.0)	21 (35.0)	30 (43.5)	40 (42.1)	0.07
	Non-invasive mechanical ventilation	6 (12.0)	7 (12.7)	6 (12.0)	13 (21.7)	9 (13.0)	21 (22.1)	0.09
	Vasopressors/inotropes	27 (54.0)	19 (34.5)	13 (26.0)	13 (21.7)	27 (39.1)	32 (33.7)	0.15
	Renal replacement therapy	11 (22.0)	9 (16.4)	7 (14.0)	5 (8.3)	6 (8.7)	7 (7.4)	0.006
	Outcome
	ICU mortality	22 (44.0)	17 (30.9)	10 (20.0)	19 (31.6)	23 (33.3)	31 (32.6)	0.59
	Hospital mortality	29 (58.0)	29 (52.7)	26 (52.0)	31 (51.7)	38 (55.1)	49 (51.6)	> 0.9
	One-year mortality	35 (70.0)	35 (63.6)	37 (74.0)	40 (66.7)	50 (72.5)	64 (67.4)	0.63

Acknowledgments

Guarantor statement FP takes responsibility for the content of the manuscript, including the data and analysis

Author contributions

CV and FP designed the study and drafted the manuscript.

CV and JC collected the data.

MJ performed the statistical analysis. JC, MJ, MW, JPM, AL, LF contributed to data interpretation and analysis, and revised the manuscript for important intellectual content. Financial/nonfinancial disclosures FP: ALEXION PHARMA (institutional grant), GILEAD SCIENCES (consulting and teaching personal fees); JPM: ROCHE, AM-PHARMA, FRESENIUS (personal fees); MJ: ALEXION PHARMA (grant); CV, JC, MW, AL, LF: none Funding None

Conflict of interest: FP: ALEXION PHARMA (institutional grant), GILEAD SCIENCES (consulting and teaching personal fees); JPM: ROCHE, AM-PHARMA, FRESENIUS (personal fees); MJ: ALEXION PHARMA (grant); CV, JC, MW, AL, LF: none


MD Marie Bensa
email: marie.bensa@chru-strasbourg.fr

MD Fred Xavier
email: fredxavierfx7@gmail.com

MD Sybille Facca
email: sybille.facca@chru-strasbourg.fr

MD Philippe Liverneaux
email: philippe.liverneaux@chru-strasbourg.fr

Iatrogenic hand trauma in institutionalized brain-injured patients: About 2 cases

Keywords: Fracture, dislocation, elderly, geriatric, iatrogenesis

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

Introduction

The prevalence of patients suffering from a brain injury is on the rise [START_REF] Mirski | Impact of a neuroscience intensive care unit on neurosurgical patient outcomes and cost of care: evidence-based support for an intensivist-directed specialty ICU model of care[END_REF]. There are no less than 1.4 million institutionalized elderly people in 2020 in the USA alone, of which, 1/5 would develop spasticity [START_REF] Gill | Prevalence of Spasticity in Nursing Home Residents[END_REF]. Among the latter, fractures caused by falls are very frequent [START_REF] Haga | Falls in the institutionalized elderly in Japan[END_REF]. The main location of these fractures is the femur [START_REF] Gifre | Incidence of skeletal fractures after traumatic spinal cord injury: a 10-year follow-up study[END_REF]. After a stroke, upper limb fractures represent 27-36% of all fall-related injuries [START_REF] Ramnemark | Fractures after stroke[END_REF], including 7.2% in the wrist and hand [START_REF] Haga | Falls in the institutionalized elderly in Japan[END_REF]. However, no articles in the literature report the slightest case of iatrogenic dislocation of the fingers in institutionalized brain-injured patients.

With these two clinical cases, the objective of this work was to draw the attention of caretakers to the risk of iatrogenic trauma related to improper handling of the fingers when opening patients' hands.

Clinical Cases

The first clinical case is that of a 97-year-old patient, living in a nursing home and dependent on others for the basic activities of daily living (ADLs).

The patient was presented to the hand emergency department for a "chronic wound" of the left thumb on a nonfunctional and retracted hygienic hand. Upon questioning the patient and the nursing home nurse, the date of onset and the mechanism of the injury could not be specified.

On examination, there was a skin defect from the volar aspect of the interphalangeal (IP) joint of the thumb, with an oozing, painless, proliferative scariform aspect (Fig. 1a &b). The X-ray showed a dorsal dislocation of the IP joint (Fig. 1c). A bacteriological sample was taken, which revealed a multisensitive Staphylococcus aureus.

It was a chronic open dorsal dislocation of the left thumb's IP joint. The nursing home medical records did not mention any trauma or fall. We, therefore, assumed that the mechanism of injury was iatrogenic and consisted of a forced hyperextension of the distal phalanx of the thumb to open the patient's hand during grooming.

From a therapeutic point of view, several options were considered: shortening arthrodesis, tenotomy of the flexors, and trans-IP joint amputation. Given the septic risk and the lack of functional use of the hand, it was decided, after a mutual agreement with the patient, to perform a trans-IP amputation of the thumb. The postoperative consequences were simple.

The second clinical case is that of a 64-year-old tetraparetic female who is institutionalized in a functional rehabilitation center. The patient suffered a ruptured intracranial aneurysm in 2013, which left her in a chronic vegetative state and sequelae of spasticity treated by a Baclofen® pump.

The patient was seen at a hand surgery consultation to discuss a surgical indication for hygienic purposes on a non-functional and retracted left hand.

On examination, there was a dry and painless skin defect with bone exposure on the radial aspect of the distal interphalangeal (DIP) joint of the fifth finger (Fig. 2a &b).

It should be noted that, during the preoperative assessment, the patient had an old bilateral femoral neck fracture that had gone unnoticed (Fig. 2c).

It was a chronic open ulnar dislocation of the DIP joint of the left fifth finger. The medical records did not mention any trauma or fall. We, therefore, assumed that the mechanism of injury was iatrogenic and consisted of forced manipulation of the distal phalanx of the finger to open the hand during washing.

Therapeutically, we performed a tenotomy of the extrinsic flexors of the fingers and wrist, as well as a trans-DIP joint amputation of the 5th finger. The postoperative consequences were simple.

Discussion

Fractures in brain-injured patients are common for two reasons: spasticity and osteoporosis.

Regarding the first one, in parallel with the increase in life expectancy linked to the improvement of their emergency care, the quality of life of brain-injured patients decreases because of spasticity, which causes gait disturbances and difficulty with hygiene care [6 Olivier]. In the limbs, spasticity mainly affects the elbow (79%), the wrist (66%), and the ankle (66%) [START_REF] Wissel | Early development of spasticity following stroke: a prospective, observational trial[END_REF].

There are few publications on the treatment of non-functional upper limbs in braininjured patients for several reasons: lack of knowledge of the therapeutic arsenal, difficulty in evaluating the benefit-risk balance of surgery, need for multidisciplinary management, and isolation of patients in institutions [START_REF] Facca | Braun's flexor tendons transfer in disabled hands by central nervous system lesions[END_REF].

Regarding the second reason, brain injury patients have a higher prevalence of osteoporosis than in the general population for several reasons: loss of mobility, vitamin D deficiency, and undernutrition. Osteoporosis increases the risk of fracture, especially in minor trauma such as bed-chair transfers [START_REF] Sezer | Chronic complications of spinal cord injury[END_REF]. There is no recommendation for the prevention and treatment of osteoporosis in these patients, although taking bisphosphonates once a week, started early after spinal cord injury, would prevent bone demineralization. [START_REF] Gilchrist | Alendronate prevents bone loss in patients with acute spinal cord injury: a randomized, double-blind, placebo-controlled study[END_REF].

Two others causes, quite similar, could be considered: an involuntary self-injury against the uprights or protections of the bed and an involuntary self-injury due to a forced grasping and/or tightening try.

While fractures of the lower limb in institutionalized brain-injured patients are regularly the subject of publications [START_REF] Elhassan | Bilateral knee extensor disruption in severe crouch gait[END_REF][START_REF] O'sullivan | The knee kinematic pattern associated with disruption of the knee extensor mechanism in ambulant patients with diplegic cerebral palsy[END_REF], this is not the case with the upper limb and in particular the hand. These two clinical cases allow attention to be drawn to the risk of iatrogenic trauma to the fingers during hand opening maneuvers in these

Figures legendFigure 1 .-

 1 Figures legend

Figure 2 .

 2 Figure 2.-Chronic open dislocation of DIP joint of the left fifth finger in a 64-year-old