Abderrahman Tamoud

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

Amar Mesbah

Fahed Zaïri

Fahmi Zaïri

Fahed Zaïri

Modeling multiaxial damage regional variation in human annulus fibrosus

Keywords: Annulus fibrosus, Regional dependence, Multiscale structure, Multiaxial damage and failure

Introduction

The annulus fibrosus of the intervertebral disc is probably one of the most extraordinary composites that the nature produces. The annulus soft tissues exhibit an organized structural arrangement of collagen network at different scales of hierarchical organization [START_REF] Yu | Elastic fibre organization in the intervertebral discs of the bovine tail[END_REF][START_REF] Pezowicz | The structural basis of interlamellar cohesion in the intervertebral disc wall[END_REF][START_REF] Pezowicz | Mechanisms of anular failure resulting from excessive intradiscal pressure: a microstructuralmicromechanical investigation[END_REF][START_REF] Melrose | Aggrecan, versican and type VI collagen are components of annular translamellar crossbridges in the intervertebral disc[END_REF][START_REF] Schollum | ISSLS prize winner: microstructure and mechanical disruption of the lumbar disc annulus: part I: a microscopic investigation of the translamellar bridging network[END_REF]. The detailed information was only appreciated very recently [START_REF] Han | Multi-scale structural and tensile mechanical response of annulus fibrosus to osmotic loading[END_REF][START_REF] Yu | A detailed examination of the elastic network leads to a new understanding of annulus fibrosus organization[END_REF][START_REF] Vergari | Lamellar and fibre bundle mechanics of the annulus fibrosus in bovine intervertebral disc[END_REF][START_REF] Han | Optical coherence tomographic elastography reveals mesoscale shear strain inhomogeneities in the annulus fibrosus[END_REF][START_REF] Tavakoli | Structure and mechanical function of the inter-lamellar matrix of the annulus fibrosus in the disc[END_REF][START_REF] Tavakoli | The ultra-structural organization of the elastic network in the intra-and inter-lamellar matrix of the intervertebral disc[END_REF][START_REF] Tavakoli | The ultrastructural organization of elastic fibers at the interface of the nucleus and annulus of the intervertebral disk[END_REF][START_REF] Tavakoli | Elastic fibers: the missing key to improve engineering concepts for reconstruction of the nucleus pulposus in the intervertebral disc[END_REF]. The structural arrangement of collagen network varies throughout the disc which leads to a regional variation of its mechanics. In this regards, various damage and failure mechanisms operate in the different disc regions [START_REF] Osti | Annular tears and disc degeneration in the lumbar spine: a post-mortem study of 135 discs[END_REF][START_REF] Boos | Classification of age-related changes in lumbar intervertebral discs[END_REF][START_REF] Vernon-Roberts | The natural history of age-related disc degeneration: the pathology and sequelae of tears[END_REF]. In order to prevent all kinds of sudden disc injuries, it is necessary to understand the failure risks considering the regional variation under various static loading conditions. Fruitful experimental studies have been undertaken over the years allowing the assessment of the annulus intrinsic constitutive response in relation to disc region and mechanical loading [START_REF] Acaroglu | Degeneration and aging affect the tensile behavior of human lumbar anulus fibrosus[END_REF][START_REF] Ebara | Tensile properties of nondegenerate human lumbar anulus fibrosus[END_REF][START_REF] Fujita | Radial tensile properties of the lumbar annulus fibrosus are site and degeneration dependent[END_REF][START_REF] O'connell | Human annulus fibrosus material properties from biaxial testing and constitutive modeling are altered with degeneration[END_REF]. Modeling multiaxial damage regional variation in annulus can provide a better understanding of the failure onset and location in the disc, beyond experimental observations. Modeling mechanics of the healthy annulus was developed in several papers considering different approaches as physically realistic as possible, to name a few recently published [START_REF] Derrouiche | A chemo-mechanical model for osmo-inelastic effects in the annulus fibrosus[END_REF][START_REF] Kandil | Interlamellar-induced time-dependent response of intervertebral disc annulus: a microstructure-based chemoviscoelastic model[END_REF][START_REF] Zhou | Multiscale composite model of fiberreinforced tissues with direct representation of sub-tissue properties[END_REF][START_REF] Ghezelbash | Modeling of human intervertebral disc annulus fibrosus with complex multi-fiber networks[END_REF][START_REF] Kandil | Interlamellar matrix governs human annulus fibrosus multiaxial behavior[END_REF][START_REF] Castro | Numerical implementation of an osmo-poro-viscohyperelastic finite element solver: application to the intervertebral disc[END_REF][START_REF] Komeili | The importance of intervertebral disc material model on the prediction of mechanical function of the cervical spine[END_REF][START_REF] Tamoud | A microstructure-based model for time-dependent mechanics of multi-layered soft tissues and its application to intervertebral disc annulus[END_REF][START_REF] Zhou | Fiber engagement accounts for geometry-dependent annulus fibrosus mechanics: a multiscale, structure-based finite element study[END_REF]. The quantitative prediction of the annulus damage is less reported. Damage criteria were proposed to predict annulus failure under different loading conditions [START_REF] Qasim | Initiation and progression of mechanical damage in the intervertebral disc under cyclic loading using continuum damage mechanics methodology: a finite element study[END_REF][START_REF] Qasim | Damage accumulation location under cyclic loading in the lumbar disc shifts from inner annulus lamellae to peripheral annulus with increasing disc degeneration[END_REF][START_REF] Shahraki | Failure criteria for prediction of clinically relevant damage of annulus fibrosus[END_REF][START_REF] Subramani | Fatigue damage prediction in the annulus of cervical spine intervertebral discs using finite element analysis[END_REF]. Nevertheless, the interest of this approach is questionable because it is intrinsically decoupled from the intrinsic properties and disconnected from the real microstructure and its evolution. The progressive loss of collagen network integrity implies a damage accumulation in annulus [START_REF] Tamoud | A multiscale and multiaxial model for anisotropic damage and failure of human annulus fibrosus[END_REF]. Proper description of damage and failure requires to consider the latter that can be introduced in continuum-based models. Mengoni et al. [START_REF] Mengoni | Modelling the failure precursor mechanism of lamellar fibrous tissues, example of the annulus fibrosus[END_REF] developed a damage model for the non-fibrillar matrix restricted to the anterior annulus stretched in the radial direction. Later, Gao et al. [START_REF] Gao | An anisotropic multiphysics damage model with application to annulus fibrosus[END_REF] and Ghezelbash et al. [START_REF] Ghezelbash | On the modeling of human intervertebral disc annulus fibrosus: Elastic, permanent deformation and failure responses[END_REF] proposed a damage model considering only the collagen fibers failure as local damage event. Tamoud et al. [START_REF] Tamoud | A multiscale and multiaxial model for anisotropic damage and failure of human annulus fibrosus[END_REF] developed a damage model considering the multiscale failure due to network hierarchical organization. The chemical-induced volumetric effects on the structure-damage relation were also introduced to consider the hydration state effects. Nonetheless, our previous work in [START_REF] Tamoud | A multiscale and multiaxial model for anisotropic damage and failure of human annulus fibrosus[END_REF] disregards the regional variation while it is the prerequisite condition to any disc prediction. The directional effects due to the structural arrangement along with the regional dependence make damage prediction a challenging task that still remains largely an open issue. To date, no constitutive model has been proposed to predict the multiaxial mechanics of the annulus fibrosus while the regional dependence of the relation between structure and damage is taken into consideration.

The main objective of this work is to formulate a fully three-dimensional constitutive representation of the annulus fibrosus to capture the regional dependence of the structuredamage relation. In order to reproduce the multiaxial loadings as close as to those undergone in-vivo by the annulus fibrosus, the material representation considers the complex collagen network arrangement of each disc region. To consider the progressive deformation-induced changes in the tissue, the model introduces the stretch-induced network re-orientation as well as the network damage leading to final failure. The model is identified at the scale of single annulus lamellae stretched along the fibers main direction. The regional dependent model predictions are then analyzed under various loading modes at the scale of multi-lamellae and compared to experimental data when available. The human annulus constitutive model is finally used to predict the heterogeneous mechanics of a full disc model.

The outline of the present paper is as follows. The constitutive model is fully described in Section 2. Section 3 presents the constitutive model identification and predictions. Full disc model predictions are presented and discussed in Section 4. Finally, remarks and conclusions are given in Section 5.

The following notation is used throughout the text. Tensors and vectors are respectively denoted by normal boldfaced letters and italicized boldfaced letters, while scalars and individual components of vectors and tensors are denoted by normal italicized letters. The superposed dot designates the time derivative.

Model formulation

We propose here a two-step strategy consisting in single lamellae identification followed by multi-lamellae multiaxial predictions as illustrated in Fig. 1. The required model parameters are identified on axial tests performed along the micro-sized collagen fibers of single annulus lamellae extracted from the different disc regions (AO: Anterior-Outer, AI: Anterior-Inner, PI: Posterior-Inner, PO: Posterior-Outer). Beyond taking into account the regional effect, this identification strategy is a valuable difference with our previous work [START_REF] Tamoud | A multiscale and multiaxial model for anisotropic damage and failure of human annulus fibrosus[END_REF] in which the model identification had been performed at the scale of multi-lamellae. In this paper, after its identification at the scale of single lamellae, the model is then verified by comparing its prediction with the experimental data of multi-lamellae performed under multiaxial loading. Uniaxial (UA), simple shear (SS), biaxial (BA) and plane-strain compression (PSC) tests are all being considered.

Multiscale structure

The model considers the hierarchical organization of the solid phase starting from the nanoscale [START_REF] Tavakoli | The ultrastructural organization of elastic fibers at the interface of the nucleus and annulus of the intervertebral disk[END_REF][START_REF] Tavakoli | Elastic fibers: the missing key to improve engineering concepts for reconstruction of the nucleus pulposus in the intervertebral disc[END_REF] as well as its regional variation. As schematically represented in Fig. 2, the disc annulus is a multiple-network medium in which the organization in terms of orientation and distribution is regional dependent. The annulus consists in concentric lamellae (LM) reinforced by micro-sized type-I oriented collagen fibers (OCF). Two adjacent LM are connected by an inter-lamellar (ILM) matrix. A network of nano-sized elastic fibers (NEF) connects both two OCF and two adjacent LM between them [START_REF] Yu | Elastic fibre organization in the intervertebral discs of the bovine tail[END_REF][START_REF] Melrose | Aggrecan, versican and type VI collagen are components of annular translamellar crossbridges in the intervertebral disc[END_REF]. The network of OCF and NEF will be described geometrically with respect to the global benchmark of the disc, respectively, with the angle θ with respect to the circumferential direction and with the angle ψ with respect to the radial direction in the parallel plane of the fibers. Quantities and orientations of the constituents can be found in the Supplementary Information. The latter is based on well-documented papers for the four main disc regions [11-12, 17-18, 38-39].

Damaged free energy functions

The randomly oriented proteoglycan macromolecules of the extracellular matrix (ECM), the OCF and the (inter-fibrillar and inter-lamellar) NEF are all being considered to model the solid phase. Each individual component is assumed as isotropic, homogeneous, incompressible and hyperelastic body. During the mechanical loading, the initiation and propagation of damage occur through a continuous evolution over the entire loading history.

A damage variable d is introduced to provide a mathematical description of the local damage events occurring within the collagen network. Their progressive evolution is given by an expression following a two-parameter Weibull statistical distribution governed by the internal stress:

1 exp d α β       = - -        P (1)
( ) ( ) ( ) 1 1 1 mech ECM ECM ECM OCF W d W d W d W θ θ ψ ψ ψ ψ φ φ φ = - + - + - ∑ (2)
where The set of free energies must be prescribed for each component of the solid phase. The simple Neo-Hookean formulation is employed for the ECM free energy ECM W :

( )

1 1 3 2 ECM ECM W G I = - (3)
in which ECM G is the ECM shear modulus and ( )

1 tr mech I = C
is the first stretch invariant of the right Cauchy-Green deformation 1

T mech mech mech = C F .F .
The collagen response exhibits 'toe' and 'linear' regions as described in Fig. 3. The 'toe' region is due to the collagen straightening during stretching that is followed by the "linear"

( ) ( ) ( ) ( ) 2 * 1 , 4 , 2 4 , , , 2 1 1 exp 1 1 2 C W H I C I C θ ψ θ ψ θ ψ θ ψ θ ψ λ λ   = - - - <     (4) ( ) ( ) ( ) 2 * * 3 , 4 , 4 , 4 , , 4 , , , , 1 2
C W H I I I I θ ψ θ ψ θ ψ θ ψ θ ψ θ ψ θ ψ θ ψ θ ψ ζ ν λ λ   = - - + - ≥     (5)
where 1 C , 2 C and 3

C are material constants, ( )

H • is the Heaviside function, , θ ψ
ζ and , θ ψ ν are functions that ensure continuity between 'toe' and 'linear' regions:

( ) ( ) ( ) 2 * * , 1 4 , 2 4 , 1 exp 1 C I C I θ ψ θ ψ θ ψ ζ = - - (6) ( ) ( ) ( ) 2 * * 1 , 4 , 2 4 , 2 , exp 1 1 2
C I C I C θ ψ θ ψ θ ψ θ ψ ν ζ = - - - (7)
The fourth stretch invariants 4 I θ and 4 I ψ are given by:

( ) ( ) 0 tan tan y x λ θ θ λ = and ( ) ( ) ( ) 0 0 0 0 tan 90 tan 180 tan 90 z f z f λ ψ ψ λ ψ λ ψ ψ λ  ≤ °  =   °+ > °  (10)
where 0 θ and 0 ψ are the angles in the initial configuration, x λ , y λ and z λ are the principal stretches, respectively, in the circumferential (x), axial (y) and radial (z) directions, and f λ is the stretch in the OCF direction. The re-orientation allows to update the fourth invariant by considering the current fibers orientation in the deformed configuration [START_REF] O'connell | Human annulus fibrosus material properties from biaxial testing and constitutive modeling are altered with degeneration[END_REF][START_REF] Ghezelbash | Modeling of human intervertebral disc annulus fibrosus with complex multi-fiber networks[END_REF][START_REF] Karsaj | The modelling of fibre reorientation in soft tissue[END_REF].

The hydration state of the annulus soft tissues due to internal fluid content variation by osmosis has a considerable effect on the disc mechanics [START_REF] Derrouiche | Osmo-inelastic response of the intervertebral disc[END_REF][START_REF] Derrouiche | How pre-strain affects the chemo-torsional response of the intervertebral disc[END_REF][START_REF] Yang | Intervertebral disc swelling maintains strain homeostasis throughout the annulus fibrosus: a finite element analysis of healthy and degenerated discs[END_REF][START_REF] Feki | How osmoviscoelastic coupling affects recovery of cyclically compressed intervertebral disc[END_REF] and especially on the failure response [START_REF] Werbner | Bovine annulus fibrosus hydration affects rate-dependent failure mechanics in tension[END_REF]. The chemical free energy chem W takes the following form [START_REF] Tamoud | A microstructure-based model for time-dependent mechanics of multi-layered soft tissues and its application to intervertebral disc annulus[END_REF]:

( ) ( ) 2 1 2 _ 2 1 exp 1 2 chem fluid chem d K W K K φ ε = - (11)
where

1 fluid ECM OCF FCN φ φ φ φ = - - -
is the volume fraction of the fluid phase, 1 K and 2 K are material constants, 1 K and 1 2 K K being the initial and maximal volumetric stiffness values, respectively, and _ chem d ε is the damaged chemical-induced volumetric strain. The latter quantity must be connected to the damage accumulation due to progressive local failure events. The following relation is used [START_REF] Tamoud | A multiscale and multiaxial model for anisotropic damage and failure of human annulus fibrosus[END_REF]:

( )( ) _ 1 1 1 1 n chem d chem ECM d d d n θ ψ ψ ε ε   = - - -     ∑ (12)
with ( )

det 1 chem chem ε = - F .
In virtue of the chemo-mechanical coupling, the total free energy W can be additively split into a mechanical part mech W and a chemical part chem W :

mech chem W W W = +
. The multi-layered lamellar/inter-lamellar annulus model is constructed by considering the compatibility conditions to be respected between adjacent layers. Some details are provided in the Supplementary Information.

Simulation vs. experiments

In this section, after its identification using available experimental data, the predictability of the above-described model is discussed. The constitutive equations were coded in MATLAB software and reduced to different loading forms.

Single lamellae identification

The intrinsic constitutive response of the annulus under UA loading was experimentally determined by Skaggs et al. [START_REF] Skaggs | Regional variation in tensile properties and biochemical composition of the human lumbar anulus fibrosus[END_REF] and Holzapfel et al. [START_REF] Holzapfel | Single lamellar mechanics of the human lumbar anulus fibrosus[END_REF] on single annulus lamellae stretched along the fibers main direction. Fig. 4 shows the computed results for the identification of the OCF properties for the different disc regions. The identification was performed numerically using the MATLAB software to minimize the following objective function:

2 1 1 h h obj SLM SLM stress h h SLM P P F P ρ ρ =   - =     ∑ % % (13)
in which the letters with an overbar denote the experimental data with ρ the number of considered data points. Table 1 provides the obtained values for the OCF properties. Note that the regional dependence of the single lamellae stiffness is explained by a difference in the initial undulated state and the OCF content [START_REF] Skaggs | Regional variation in tensile properties and biochemical composition of the human lumbar anulus fibrosus[END_REF][START_REF] Holzapfel | Single lamellar mechanics of the human lumbar anulus fibrosus[END_REF]. This structural difference is reflected by distinct values. The other model parameters are extracted from [START_REF] Tamoud | A multiscale and multiaxial model for anisotropic damage and failure of human annulus fibrosus[END_REF] and are listed in Table 2.

The simulated curves of the single lamellae are in good agreement with the experiments of Holzapfel et al. [START_REF] Holzapfel | Single lamellar mechanics of the human lumbar anulus fibrosus[END_REF]. The results are shown until the complete loss of OCF load-bearing capability in order to compare the ultimate points with the experimental data of Skaggs et al. [START_REF] Skaggs | Regional variation in tensile properties and biochemical composition of the human lumbar anulus fibrosus[END_REF]. Except PI, it can be observed that the simulated stress and strain thresholds fall within the experimental standard deviation intervals.

Multi-lamellae multiaxial predictions

Table 3 presents the range of mechanical paths used to check the predictive capacities of the multi-lamellae model. It includes UA stretching in the circumferential direction, UA stretching in the radial direction, SS loading, BA stretching and PSC loading 2 . Except the latter loading (in which the sample is confined in the radial direction), the samples are loaded in specific directions while the others are free. All the samples are considered to be immersed in a physiological solution at a concentration of 0.15 M NaCl. When experimental data are available, a quantitative comparison with the model results is presented. The dimensions of the annulus specimens are given in Table 3 for the different loading conditions and disc regions.

The multi-lamellae predictions are presented in Figs. 5 and6 for UA stretching along the circumferential direction. The simulated stress-stretch curves are compared with the typical experimental curves presented by Ebara et al. [START_REF] Ebara | Tensile properties of nondegenerate human lumbar anulus fibrosus[END_REF] in Fig. 5a. Considering the important variability in the experimental annulus response and that these experimental data are typical results (without statistical analysis), the model-experiments comparison can be considered as acceptable. In accordance with experiments, the outer regions have much higher stiffness than the inner regions, due mainly to regional variation in OCF angle. The simulations are conducted until stress drop corresponding to the failure stage. The simulated ultimate properties are reported in Fig. 5b and compared to the failure points with their standard deviation intervals extracted from the works of Acaroglu et al. [START_REF] Acaroglu | Degeneration and aging affect the tensile behavior of human lumbar anulus fibrosus[END_REF] and Ebara et al. [START_REF] Ebara | Tensile properties of nondegenerate human lumbar anulus fibrosus[END_REF]. The failure strain predictions fail within the standard deviation intervals. Although the predicted failure stress can be found outside the experimental intervals, the predictive trends provide a regional difference in agreement with the experimental data. This important result confirms that the model provides a quite proper description of the regional effects on the relationship 2 These different loadings are representative of the disc mechanics. UA stretching along the circumferential direction and PSC are representative of the disc axial compression [START_REF] Shah | The distribution of surface strain in the cadaveric lumbar spine[END_REF]. UA stretching along the radial direction may be related to the disc shearing whereas SS may be related to the disc axial twist. BA stretching may be caused for example by a disc bending (i.e. flexion, extension and lateral flexion) [START_REF] Costi | Direct measurement of intervertebral disc maximum shear strain in six degrees of freedom: motions that place disc tissue at risk of injury[END_REF].

between the actual microstructure, the damage events and the macroscopic ultimate properties. Recall that the model connects macroscopic response and hydration (volumetric) effects. The capacity of the model to capture the regional variation of the transverse responses is shown in Fig. 6. The LM-ILM interaction highlights the crucial role that the OCF regional orientations play in the transverse response [START_REF] Kandil | Interlamellar matrix governs human annulus fibrosus multiaxial behavior[END_REF]. In particular, regarding the possibility of an apparent auxeticity manifested by the positive transversal strains of the lamellae plane [START_REF] Baldit | Etude des interactions hydro-chimio-mécaniques dans les tissus biologiques: application à la nutrition du disque intervertébral[END_REF].

Thanks to the incorporation of the actual nanostructure materialized by the network of fibrils, the model can predict the regional behavior of the annulus in the radial direction. Fig. 7a shows the multi-lamellae stress-stretch responses under UA stretching along the radial direction. Again, the simulations are conducted until the appearance of the stress drop. The experimental data of Fujita et al. [START_REF] Fujita | Radial tensile properties of the lumbar annulus fibrosus are site and degeneration dependent[END_REF] are reported in Fig. 8 to further verify the model predictive capacities. Both moduli and ultimate properties fall within the experimental standard deviation intervals of Fujita et al. [START_REF] Fujita | Radial tensile properties of the lumbar annulus fibrosus are site and degeneration dependent[END_REF]. By taking into account accurately the connections between layers, the model allows also to predict the shearing resistance. Fig. 7b shows such predictions for which it can be observed a strong regional dependence. In addition to amount, type and orientation of the collagen network, the regional variation of the NEF arrangement leads to different predicted stiffness and failure thresholds according to the disc region. The connection of collagen network between lamellae from nanoscale made it possible to represent precisely the damage processes in the LM and ILM connections in the different annulus regions. The model shows that the different regional thresholds of radial and shear failures are caused by the regional arrangement of the NEF. These results are in agreement with the qualitative data analyzed in Fujita et al. [START_REF] Fujita | Radial tensile properties of the lumbar annulus fibrosus are site and degeneration dependent[END_REF]. Indeed, it was reported from image analysis that inter-layer connections such as NEF contribute to the annulus radial resistance [START_REF] Pezowicz | The structural basis of interlamellar cohesion in the intervertebral disc wall[END_REF][START_REF] Tavakoli | New findings confirm the viscoelastic behaviour of the inter-lamellar matrix of the disc annulus fibrosus in radial and circumferential directions of loading[END_REF].

The directional effects are quite well captured by the model, especially the much higher stretchability along the radial direction than along the circumferential direction. The BA stress-strain responses are presented in Fig. 9 for different strain ratios (= circumferential strain : axial strain, {1:1, 1:2 and 2:1}). In order to further evidence the directional effects, Fig. 10 presents the PSC stress-stretch responses for different strain ratios. The strain ratio has an important effect on the stiffness and the ultimate properties, with a stronger effect on the axial properties than the circumferential ones. The posterior zones show higher stiffness than the anterior zones. That can be justified by the fact that the OCF angles are close to 45°, and that the forces are well shared according to the two loading directions (circumferential and axial). Although the location was not indicated in the paper of O'Connell et al. [START_REF] O'connell | Human annulus fibrosus material properties from biaxial testing and constitutive modeling are altered with degeneration[END_REF], it is satisfactory to observe that the BA predictions of the inner parts are in acceptable agreement with their typical experimental curves. The model indicates that the damage first initiates in the posterior zone. This result is actually found for all examined loading modes. PO has indeed a weak circumferential stretchability compared to other regions, as shown in UA stretching along the circumferential direction (see Fig. 5) and in BA stretching (see Fig. 7).

That explains why PO exhibits a low ductility in PSC as observed in Fig. 10. Indeed, once PO reaches the circumferential failure, the axial failure occurs. As a consequence, PO shows a very low failure axial stress.

Full disc model

In this section, predictions of a full disc model are presented using the regional dependent constitutive representation of multi-lamellae and local damage mechanisms are clarified in order to get a basic insight about the starting areas of failure modes.

Disc construction and boundary conditions

The L1-L2 disc geometric contours were approximated using special mathematical functions considering especially non-symmetric thicknesses and heights (Fig. 11a). Note that they may be adapted to consider the shape and size-changing with the spinal level and age [START_REF] Amonoo-Kuofi | Morphometric changes in the heights and anteroposterior diameters of the lumbar intervertebral discs with age[END_REF]. The annulus was subdivided into 15 different lamellae (separated by 14 ILM zones) according to the L1-L2 disc dimensions provided by Holzapfel et al. [START_REF] Holzapfel | Single lamellar mechanics of the human lumbar anulus fibrosus[END_REF]. The annulus size is considered to be approximately 50% of the disc volume [START_REF] Violas | Objective quantification of intervertebral disc volume properties using MRI in idiopathic scoliosis surgery[END_REF]. It is worth noticing that a notable difference exists between in-vivo and in-vitro responses regarding fluid exudation. In-vivo, the healthy disc completely recovers its height and intradiscal pressure after long diurnal loading. Invitro, several studies reported losses in the disc height and intradiscal pressure [START_REF] Feki | How osmoviscoelastic coupling affects recovery of cyclically compressed intervertebral disc[END_REF]. The chemical-induced volumetric strain was then affected by a multiplying factor. Several numerical simulations were performed with different values of the multiplying factor ranged from 0 to 100%, and a value of 50% was retained in the present work. During the simulations, a vertical axial displacement was applied on the superior surface while the inferior surface was constrained (Fig. 11b). The simulations were performed at a displacement rate of 0.01 mm/s in order to maintain quasi-static loading conditions.

Damage fields

The damage process in the disc core simulated by the model is recorded in Fig. 12 by differentiating the damage of the different solid constituents (ECM, NEF and OCF). That permits a better understanding of the origin of the damage mechanisms affecting the disc and existing at different scales, nanoscale for ECM proteoglycan macromolecules and for NEF network and microscale for OCF. A global view at the contour plots shows that damage progressively increases in intensity and in extent as the applied compression increases, which is accompanied by a disc volumetric change and an annulus-nucleus non-symmetric interaction. The different damage mechanisms may co-exist at the same time and initiate at different locations. Remind that, in the present model, damage initiation and propagation are dependent on structure-property regional variation but also on the variation in displacement fields due to local differences in disc height. The highest damage zones in OCF are mainly observed in the anterior outer/lateral anterior ring. Excessive damage accumulation in these zones suggests the possibility of tear creation that may lead to an anterior herniation as experimentally reported in [START_REF] Osti | Annular tears and disc degeneration in the lumbar spine: a post-mortem study of 135 discs[END_REF][START_REF] Schollum | The influence of concordant complex posture and loading rate on motion segment failure: a mechanical and microstructural investigation[END_REF][START_REF] Sapiee | Sagittal alignment with downward slope of the lower lumbar motion segment influences its modes of failure in direct compression: A mechanical and microstructural investigation[END_REF]. Besides, the ECM and NEF damage patterns are presented in the two layers in order to appreciate the specific role of the ILM matrix and compare it to the adjacent LM. For both ECM and NEF, two critical zones appear in the lamellae on either side of the annulus with very low intensities. With relatively higher intensities, critical damage spots in the ILM matrix (for both ECM and NEF) are firstly located in the anterior side of the annulus and are then progressively more likely located in the neighborhood of the anterior inner part of the annulus. Quite interestingly, a confined distribution of the ILM damage can be observed in the lateral anterior inner and posterior inner sides of the annulus. The distribution of the fibrils in LM and in ILM makes the damage anisotropic. To illustrative this model capacity, the damage of the fibrils is plotted in Fig. 13 for two localities taken in posterior and anterior sides of the inner annulus (close to the annulus-nucleus interface), respectively, in LM and in ILM. The contour plots are plotted in a dial (90° representing the radial direction) in such a way that the directional effects can be observed during continuous loading. Whereas the damage in LM is confined to small degrees around the radial direction, with an onset at relatively important applied displacements and a relatively low intensity, the damage in ILM is developed earlier, it is more diffused and it is more intense. The damage in ILM is often judged as one of the most dangerous disc failure modes with the highest risk of disc delamination and failure propagation in the radial direction of the disc as experimentally reported in [START_REF] Osti | Annular tears and disc degeneration in the lumbar spine: a post-mortem study of 135 discs[END_REF][START_REF] Schollum | The influence of concordant complex posture and loading rate on motion segment failure: a mechanical and microstructural investigation[END_REF][START_REF] Sapiee | Sagittal alignment with downward slope of the lower lumbar motion segment influences its modes of failure in direct compression: A mechanical and microstructural investigation[END_REF].

Discussion and concluding remarks

In the present paper, a novel human annulus model was proposed to constitutively relate the regional variation in structural arrangement of collagen network at different scales and the multiaxial mechanics till failure. The model provides a useful tool to predict a variation of the annulus stiffness and ultimate properties with both disc region and loading mode. The different loading modes were simulated and chosen to better understand the regional behavior until the failure of the solid components in interaction with the chemical-induced volumetric response. The capabilities of the model were analyzed and an acceptable agreement between predicted values and experimental data was highlighted. A full disc model was constructed using the regional dependent constitutive representation of multi-lamellae and damage fields in the compressed disc were analyzed to assess the areas where the risk of failure is the highest.

The present model can be considered as the stepstone for the long-term prediction of the disc dysfunctions. During different complex loading scenarios of the spine, the disc dysfunctions may occur either by: i) mechanical damage due to a high level of loads [START_REF] Acaroglu | Degeneration and aging affect the tensile behavior of human lumbar anulus fibrosus[END_REF][START_REF] Ebara | Tensile properties of nondegenerate human lumbar anulus fibrosus[END_REF] or relatively moderate loads repeated several times which cause damage accumulation [START_REF] Qasim | Initiation and progression of mechanical damage in the intervertebral disc under cyclic loading using continuum damage mechanics methodology: a finite element study[END_REF][START_REF] Subramani | Fatigue damage prediction in the annulus of cervical spine intervertebral discs using finite element analysis[END_REF][START_REF] Green | Tensile properties of the annulus fibrosus[END_REF][START_REF] Iatridis | Mechanical damage to the intervertebral disc annulus fibrosus subjected to tensile loading[END_REF], ii) by an age-related biological degeneration [START_REF] Thompson | Preliminary evaluation of a scheme for grading the gross morphology of the human intervertebral disc[END_REF][START_REF] Urban | Degeneration of the intervertebral disc[END_REF][START_REF] Cegoñino | A constitutive model for the annulus of human intervertebral disc: implications for developing a degeneration model and its influence on lumbar spine functioning[END_REF] or by the cellular disorders, including the cell death, the production of inflammatory mediators and a shift towards catabolism [START_REF] Bruehlmann | In situ intercellular mechanics of the bovine outer annulus fibrosus subjected to biaxial strains[END_REF][START_REF] Walter | Complex loading affects intervertebral disc mechanics and biology[END_REF][START_REF] Bloom | Tendon multiscale structure, mechanics, and damage are affected by osmolarity of bath solution[END_REF].

Nonetheless, it is difficult to identify or precisely model the disc failure taking into account the two causes of dysfunction at the same time. Since our study is based on the in-vitro annulus results, it cannot be directly extrapolated to the in-vivo disc response. Indeed, the understanding of the disc mechanobiology must be taken into account by the model considering especially nutrition and cellular activity along with degenerative aspects [START_REF] Bruehlmann | In situ intercellular mechanics of the bovine outer annulus fibrosus subjected to biaxial strains[END_REF][START_REF] Walter | Complex loading affects intervertebral disc mechanics and biology[END_REF][START_REF] Bloom | Tendon multiscale structure, mechanics, and damage are affected by osmolarity of bath solution[END_REF].

The model could be then a real help to provide solutions for remodeling and self-healing response of the disc tissues [START_REF] Likhitpanichkul | Fibrin-genipin adhesive hydrogel for annulus fibrosus repair: performance evaluation with large animal organ culture, in situ biomechanics, and in vivo degradation tests[END_REF][START_REF] Borde | Injectable, high-density collagen gels for annulus fibrosus repair: an in vitro rat tail model[END_REF][START_REF] Wan | Selfassembling peptide hydrogel for intervertebral disc tissue engineering[END_REF][START_REF] Peng | Genipin-crosslinked decellularized annulus fibrosus hydrogels induces tissue-specific differentiation of bone mesenchymal stem cells and intervertebral disc regeneration[END_REF]. Moreover, the near-field direct local interactions within

  ECM d , d θ and d ψ are the damage variables that degrade the internal stiffness of respective components, the chemical equilibrium state. The term chem J is the chemical-induced volumetric change.

.

  b are the unit vectors expressed in the Cartesian coordinates by: The two angles follow the following geometrical relations to consider stretch-induced re-orientation of the OCF and NEF networks:

Figure captions Figure 1 .

 captions1 Figure captions

Figure 2 .

 2 Figure 2. Multiscale view of the intervertebral disc annulus. The organization, in terms of orientation and distribution, of this multiple-network medium is regional dependent.

Figure 3 .

 3 Figure 3. Phenomenon of reorientation of collagen fibers.

Figure 4 .

 4 Figure 4. Single lamellae identification for the different disc regions: (a) stress-stretch responses (lines: model simulations, symbols: experimental data of Holzapfel et al. [39]), (b) ultimate properties (bars: model simulations, symbols with standard deviation intervals: experimental data of Skaggs et al. [38]).

Figure 5 .

 5 Figure 5. Multi-lamellae UA predictions along circumferential direction for the different disc regions: (a) stress-stretch responses (lines: model simulations, symbols: experimental data of Ebara et al.[START_REF] Ebara | Tensile properties of nondegenerate human lumbar anulus fibrosus[END_REF]), (b) ultimate properties (bars: model simulations, square symbols with standard deviation intervals: experimental data of Ebara et al.[START_REF] Ebara | Tensile properties of nondegenerate human lumbar anulus fibrosus[END_REF], circle symbols with standard deviation intervals: experimental data of Acaroglu et al.[START_REF] Acaroglu | Degeneration and aging affect the tensile behavior of human lumbar anulus fibrosus[END_REF]).

Figure 6 .

 6 Figure 6. Multi-lamellae transverse responses under UA circumferential stretching for the different disc regions (solid lines: fibers plane, dashed lines: lamellae plane): (a) model results, (b) experimental data of Baldit [50].

Figure 7 .

 7 Figure 7. Multi-lamellae stress-stretch responses under (a) UA radial stretching and (b) SS loading for the different disc regions (lines: model simulations, circle symbols: experimental data of Fujita et al. [19]).

Figure 8 .

 8 Figure 8. Multi-lamellae UA predictions along radial direction for the different disc regions: (a) moduli, (b) ultimate properties (bars: model simulations, circle symbols with standard deviation intervals: experimental data of Fujita et al. [19]).

Figure 9 .

 9 Figure 9. Multi-lamellae stress-stretch responses under BA stretching with various strain ratios for the different disc regions (lines: model simulations, circle symbols: experimental data of O'Connell et al.[START_REF] O'connell | Human annulus fibrosus material properties from biaxial testing and constitutive modeling are altered with degeneration[END_REF]).

Figure 10 .

 10 Figure 10. Multi-lamellae stress-stretch responses under PSC loading with various strain ratios for the different disc regions.

Figure 11 .

 11 Figure 11. Full disc model: (a) Passage from annulus constitutive model to anatomically accurate disc model including volumetric strain, regional and directional effects (disc dimensions at the L1-L2 level), (b) boundary conditions of the uniaxially compressed disc.

Figure 12 .

 12 Figure 12. Damage contour plots in the superior view of the uniaxially compressed disc for different applied displacement levels (0.5, 1.0 and 1.5 mm); ECM damage ECM d in LM and in ILM, NEF damage NEF d in LM and in ILM, OCF damage d θ in LM.

Figure 13 .

 13 Figure 13. Damage of the fibrils in different directions (90° is the radial direction) at two zones of the uniaxially compressed disc: (a) posterior (in LM) and (b) anterior (in ILM) sides of the inner annulus for different applied displacement levels (from zero in the dial center to 2 mm in the dial boundary).

Figure 1 .

 1 Figure 1. Two-step strategy consisting in annulus model identification using single lamellae extracted from different disc regions followed by multi-lamellae annulus model predictions for multiaxial elementary loading modes representative of the spine movement.

Figure 2 .

 2 Figure 2. Multiscale view of the intervertebral disc annulus. The organization, in terms of orientation and distribution, of this multiple-network medium is regional dependent.

Figure 3 .Figure 4 .Figure 5 .Figure 6 .Figure 7 .

 34567 Figure 3. Phenomenon of reorientation of collagen fibers.

Figure 8 .Figure 9 .

 89 Figure 9. Multi-lamellae stress-stretch responses under BA stretching with various strain ratios for the different disc regions (lines: model simulations, circle symbols: experimental data of O'Connell et al. [20]).

Figure 10 .

 10 Figure 10. Multi-lamellae stress-stretch responses under PSC loading with various strain ratios for the different disc regions.

Figure 11 .

 11 Figure 11. Full disc model: (a) Passage from annulus constitutive model to anatomically accurate disc model including volumetric strain, regional and directional effects (disc dimensions at the L1-L2 level), (b) boundary conditions of the uniaxially compressed disc.

Figure 12 .Figure 13 .

 1213 Figure 12. Damage contour plots in the superior view of the uniaxially compressed disc for different applied displacement levels (0.5, 1.0 and 1.5 mm); ECM damage ECM d in LM and in ILM, NEF damage NEF d in LM and in ILM, OCF damage d θ in LM.

Table captions Table 1 .

 captions1 OCF parameters.

Table 2 .

 2 ECM, NEF and swelling parameters.

Table 3 .

 3 Loading modes for the multi-lamellae.

Table 1 .

 1 OCF parameters.

	Constituents Parameters ECM ECM G ECM α ECM β NEF 1 C 2 C 3 C * ψ λ NEF α NEF β Swelling 1 K 2 K	Values [ M P a -[ ] [ M P a [ M P a -[ ] [ M P a -[ ] -[ ] [ M P a [ M P a 0.0075 0.01 2 5.5 12 0.2 1.5 1.7 6 400 0.14 [ ] -	] ] ] ] ] ]

Table 2 .

 2 ECM, NEF and swelling parameters.

			Sample size
		Loading modes	 	mm	3	 	Experimental data
			Anterior Posterior
	Uniaxial (UA)	Circ.	7×2×2		10×4×4	Ebara et al. [17] and Acaroglu et al. [18]
	Uniaxial (UA)	Radial	2×2×2			4×4×4	Fujita et al. [19]
	Simple shear (SS)		2×2×2			4×4×4	No available data
	Biaxial (BA)	Axial-circ.	7×7×2 10×10×4	O'Connell et al. [20]
	Plane-strain compression (PSC)	Axial-circ.	7×7×2 10×10×4	No available data

Table 3 .

 3 Loading modes for the multi-lamellae.

Acknowledgements

This work was financially supported by the PROFAS B+ Scholarships Program and the International Mobility Grants Program of the University of Lille.

A lot of finite element disc models were developed in the literature [26-27, 30-33, 52]. Beyond fundamental differences in the constitutive representation of the disc tissues, the time of model design and the time of calculation remain the main common weak points of all previous disc models. In the final objective to create time and cost-efficient patient-specific models of human discs, the present disc modeling strategy is purely analytical and again coded in MATLAB. The volume element of multi-lamellae, constitutively designed in the Cartesian space (x, y, z), was transferred to the Cylindrical space (r, θ, z) in order to consider the full disc geometry as illustrated in Fig. 11a. From the theoretical point of view, the constitutive equations were re-written within a finite-strain kinematic framework in Cylindrical coordinates. The reader is referred to the work of Holzapfel et al. [START_REF] Holzapfel | A new constitutive framework for arterial wall mechanics and a comparative study of material models[END_REF] for the general theory on tube-like solid mechanics. In the present paper, we restrict the loading to a uniaxial compression at the disc external boundary. The disc is thus seen as a cylindrical solid loaded in compression on the superior surface and with a pressure in the cylindrical internal wall (Fig. 11b). The latter simulates the nucleus swelling applied to the internal wall of the disc annulus; the applied vertical axial displacement was translated to an internal pressure by considering the nucleus bulk modulus. The regional variation in structure and properties of the disc annulus was taken into account using the constitutive representation of each human annulus region. In order to ensure smooth transitions of the microstructural features (ECM, NEF, OCF and fluid) and properties from a disc annulus region to another, with a radial and circumferential variation, linear interpolations were considered using the data provided in the Supplementary Information. An angular increment δθ of 1° was chosen for the spatial discretization in the circumferential direction. Note that the number of layers determines the radial discretization.

An anatomically accurate three-dimensional model of a lumbar disc at the L1-L2 level was constructed using quantitative anatomical data taken from the work of Holzapfel et al. [START_REF] Holzapfel | Single lamellar mechanics of the human lumbar anulus fibrosus[END_REF]. the collagen network and the ECM ought to be treated through either the incorporation of a supplementary energy term [START_REF] Guerin | Quantifying the contributions of structure to annulus fibrosus mechanical function using a nonlinear, anisotropic, hyperelastic model[END_REF] or a mean-field approach [START_REF] Saadedine | A micromechanicsbased model for visco-super-elastic hydrogel-based nanocomposites[END_REF].


Claire Bouleti
email: claire.bouleti@gmail.com

Morgane Michel

Antoine Jobbe Duval

Thibaut Hemery

Pierre-Philippe Nicol

Romain Didier

Floriane Zeyons

Oualid Zouaghi

Didier Tchetche

Clemence Delon

Clémence Delon

Maxence Delomez

Alain Dibie

David Attias

Hervé Le Breton

Bertrand Cormier

Jean- François Obadia

Christophe Tribouilloy

Emmanuel Lansac

Karine Chevreul

Nicole Naccache

Hélène Eltchaninoff

Martine Gilard

Bernard Iung

Current treatment of symptomatic aortic stenosis in elderly patients: Do risk scores really matter after 80 years of age?

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

Méthodes. -En septembre-octobre 2016, 1049 patients consécutifs ≥ 75 ans ont été inclus par 32 centres disposant du plateau technique pour TAVI et chirurgie. Le critère de jugement principal était la décision entre traitement médical, TAVI ou chirurgie.

Résultats. -L'âge moyen était de 84 ± 5 ans avec 53 % de femmes. Le risque opératoire était élevé (EuroSCORE II > 8 %) chez 18 % des patients, intermédiaire (EuroSCORE II = 4-8 %) chez 34 % et faible (EuroSCORE II ≤ 4 %) chez 48 %. Le TAVI était privilégié chez 71 % des patients, la chirurgie chez 19 % et le traitement médical chez 10 %. Le choix du TAVI était associé à un âge plus élevé (P < 0,0001) et un EuroSCORE II plus haut (P = 0,008). Cependant, le poids de l'EuroSCORE II diminuait nettement ≥ 80 ans, 77 % des patients étant orientés vers un TAVI, malgré un risque chirurgical considéré comme faible.

Conclusions. -L'impact des scores de risque opératoire sur la décision thérapeutique dépend de l'âge et diminue considérablement ≥ 80 ans, la majorité des patients étant orientés vers un TAVI indépendamment de leur risque chirurgical. Malgré des progrès, encore 10 % des patients sont récusés d'une intervention.

KEYWORDS

Background

Randomized trials first validated transcatheter aortic valve implantation (TAVI) in patients contraindicated to or at high risk from surgery, and then confirmed its efficacy in patients at lower surgical risk [START_REF] Siontis | Transcatheter aortic valve implantation vs. surgical aortic valve replacement for treatment of symptomatic severe aortic stenosis: an updated meta-analysis[END_REF]. Surgical risk scores initially played an important role in patient selection, but the clinical judgement of the Heart Team was rapidly emphasised in guidelines [START_REF] Nishimura | AHA/ACC Guideline for the Management of Patients With Valvular Heart Disease: executive summary: a report of the American College of Cardiology/American Heart Association Task Force on Practice Guidelines[END_REF][START_REF] Vahanian | Guidelines on the management of valvular heart disease (version 2012)[END_REF]. The use of a surgical risk score -particularly EuroSCORE II -remains useful for identifying patients at low risk from surgery [START_REF] Baumgartner | ESC/EACTS Guidelines for the management of valvular heart disease[END_REF]. Surgical aortic valve replacement (SAVR) is indeed recommended in patients at low estimated risk (EuroSCORE II ≤ 4%), whereas for patients with an increased surgical risk, or with other risk factors (e.g. chest radiation), the decision between SAVR and TAVI should be made by the Heart Team, with TAVI being favoured in elderly patients (Class I recommendation) [START_REF] Baumgartner | ESC/EACTS Guidelines for the management of valvular heart disease[END_REF][START_REF] Nishimura | AHA/ACC Focused Update of the 2014 AHA/ACC Guideline for the Management of Patients With Valvular Heart Disease: A Report of the American College of Cardiology/American Heart Association Task Force on Clinical Practice Guidelines[END_REF]. Large registries have shown a decrease in estimated surgical risk over the past few years, whereas the mean patient age has remained unchanged [START_REF] Auffret | Temporal Trends in Transcatheter Aortic Valve Replacement in France: FRANCE 2 to FRANCE TAVI[END_REF]. Given the importance of age and surgical risk scores in decision making, we aimed to analyse the respective significance of these variables in the therapeutic choice in elderly patients.

The RAC survey was specifically designed: (1) to assess prospectively the indications for TAVI, SAVR and medical therapy in elderly patients referred for severe symptomatic aortic stenosis (AS); and (2) to analyse the respective contribution of surgical risk scores and age in decision making between the three therapeutic options.

Methods

Population

The RAC study was a national multicentre prospective survey that included consecutive patients aged ≥ 75 years who were hospitalized for severe symptomatic AS in 32 French centres with on-site TAVI and SAVR facilities [START_REF] Auffret | Temporal Trends in Transcatheter Aortic Valve Replacement in France: FRANCE 2 to FRANCE TAVI[END_REF]. Patients were included between 01 September and 31 October 2016 if they had severe AS according to current guidelines [START_REF] Nishimura | AHA/ACC Guideline for the Management of Patients With Valvular Heart Disease: executive summary: a report of the American College of Cardiology/American Heart Association Task Force on Practice Guidelines[END_REF][START_REF] Baumgartner | ESC/EACTS Guidelines for the management of valvular heart disease[END_REF], associated with New York Heart Association class II-IV dyspnoea, angina or syncope.

The choice of the type of prosthesis for TAVI and SAVR was left at the discretion of investigators, as was the approach route for TAVI. The complete list of investigators can be found in the Appendix.

The study protocol conformed to the ethical guidelines of the 1975 Declaration of Helsinki, as reflected in approval by the ethics committee of the French Society of Cardiology and by the National Commission for Data Protection and Liberties. All patients provided written informed consent. The trial is registered at https://www.clinicaltrials.gov (identifier: NCT03343314).

Primary endpoint

The primary endpoint was the Heart Team's therapeutic decision between TAVI, SAVR and medical management (including balloon aortic valvuloplasty).

Patient characteristics and risk scores

Baseline patient characteristics were collected for the 1049 patients with a therapeutic decision. We detailed comorbidities not included in the usual surgical risk scores: history of radiotherapy, cirrhosis, active neoplasia and the presence of coronary artery disease.

Multivariable surgical risk scores were analysed as recommended in the guidelines [START_REF] Nishimura | AHA/ACC Guideline for the Management of Patients With Valvular Heart Disease: executive summary: a report of the American College of Cardiology/American Heart Association Task Force on Practice Guidelines[END_REF][START_REF] Vahanian | Guidelines on the management of valvular heart disease (version 2012)[END_REF][START_REF] Baumgartner | ESC/EACTS Guidelines for the management of valvular heart disease[END_REF][START_REF] Nishimura | AHA/ACC Focused Update of the 2014 AHA/ACC Guideline for the Management of Patients With Valvular Heart Disease: A Report of the American College of Cardiology/American Heart Association Task Force on Clinical Practice Guidelines[END_REF].

EuroSCORE II was preferred because its calibration is better than that of the logistic EuroSCORE [START_REF] Baumgartner | ESC/EACTS Guidelines for the management of valvular heart disease[END_REF][START_REF] Nashef | EuroSCORE II[END_REF]. Patients with missing data that precluded calculation of the EuroSCORE were excluded. Risk scores were thus calculated for 956 patients.

Emergency intervention was defined as required within 24 hours, and patients were dichotomized between emergency or non-emergency procedures, as published previously [START_REF] Grant | Performance of the EuroSCORE models in emergency cardiac surgery[END_REF]. The population was divided into low, intermediate and high risk using validated cut-offs (EuroSCORE II ≤ 4%, 4-8% and > 8%, respectively) [START_REF] Nishimura | AHA/ACC Guideline for the Management of Patients With Valvular Heart Disease: executive summary: a report of the American College of Cardiology/American Heart Association Task Force on Practice Guidelines[END_REF][START_REF] Vahanian | Guidelines on the management of valvular heart disease (version 2012)[END_REF].

Data management

Data were recorded on a standardized electronic case report form and sent to a central secure database hosted by the French Society of Cardiology.

Statistical analysis

Continuous variables are expressed as mean and standard deviation, and categorical variables as number and percentage. Statistical differences between the three treatment groups were assessed by the Kruskal-Wallis test for continuous variables, and the χ 2 test for categorical variables. Graphical renditions of cumulative risk were used to represent the cumulative probability of each type of treatment depending on the patients' risk scores, overall and for different age groups (75-79, 80-84 and ≥ 85 years).

Multivariable multinomial logistic models were used to assess the association between the therapeutic decision and patient characteristics, taking into account all characteristics described previously and the treating centre.

An interaction variable between age and EuroSCORE II was introduced in a second set of models. In these models, for interpretation purposes, age was divided into three groups: ≤ 79, 80-84 and ≥ 85 years. The 956 patients with no missing data for the explanatory variables were included in the modelling.

All analyses were performed using SAS® software, version 9.4 (SAS Institute, Cary, NC, USA).

Results

Population characteristics

In the RAC survey, 1101 consecutive patients with severe symptomatic AS were screened in 32 centres; 1049 had a non-pending therapeutic decision and form the basis of the study (Fig. 1). The characteristics of these patients are detailed in Table 1, for the whole population and the three therapeutic option groups (SAVR, TAVI and medical treatment). Patients were on average 84 years of age, and 47% were men. Patients in the SAVR group were younger than in the other therapeutic option groups. Left ventricular ejection fraction was preserved in 86% of patients treated by SAVR, but in only 77% and 60% in the TAVI and medical treatment groups, respectively (P < 0.0001).

Consistently, systolic pulmonary artery pressure was lower in the SAVR group compared with the other groups.

Risk scores

The EuroSCORE was calculated for each of the 956 patients with no missing data, and the results are presented in Table 2. The Logistic EuroSCORE and EuroSCORE II were on average 18% and 6%, respectively, reflecting the advanced age of the population and the inherent burden of comorbidities.

The scores increased significantly from the SAVR group to the medical treatment group, with the TAVI group in between.

According to EuroSCORE II, 18% of the total population was considered to be at high risk (estimated in-hospital mortality > 8%), whereas 48% was at low risk (estimated mortality ≤ 4%).

Regarding the three treatment groups, low-risk patients represented 80% of patients in the SAVR group, 43% in the TAVI group and 27% in the medical treatment group (P < 0.0001) (Fig. 2A and Table 2).

Therapeutic decision

The therapeutic decision was TAVI in 748 patients (71%), SAVR in 199 (19%) and medical treatment in 102 (10%), including balloon aortic valvuloplasty in 23 (Table 1). Of the 102 patients who were denied TAVI and SAVR, the reasons cited by the responsible practitioner were functional impairment in 31 (30%), age in 29 (28%), cognitive impairment in 22 (22%), comorbidities in 19 (19%), patient refusal in 16 (16%), contraindication to TAVI (anatomical issues and life expectancy < 1 year) in 14 (14%) and other reasons in 22 (22%). In the medical treatment group, 15% of patients were aged > 80 years and 24% were at low estimated surgical risk, but only four patients combined these conditions. Moreover, seven patients had a history of chest radiation, one had cirrhosis and six had a history of cardiac surgery. Hence, TAVI could have been offered preferentially to these patients; however, seven patients refused intervention and four had a contraindication to the TAVI procedure. The 23 patients who underwent balloon aortic valvuloplasty differed significantly from those who received medical treatment only: they were more often male, and had a more severely impaired left ventricular ejection fraction and a higher EuroSCORE II (Table A.1).

The distribution of therapeutic decision across the patient age groups is illustrated in Fig. 2B for the whole cohort and in Fig. 2C for patients at low estimated surgical risk.

Patient characteristics associated with the therapeutic decision

There were significant differences in age and risk scores between the three treatment groups: patients treated with SAVR were younger and had lower risk scores. Indeed, compared with the SAVR group, the logistic EuroSCORE and EuroSCORE II both doubled in the TAVI group and then increased again by 30-50% in patients receiving medical treatment (Table 2).

The probability of being treated with SAVR varied greatly with age (Fig. 3A). Indeed, a patient aged < 80 years was likely to be treated with SAVR (> 50%), particularly if they were in the low-risk or intermediate-risk category. On the other hand, a patient aged ≥ 85 years had a probability of < 5% of having SAVR, regardless of risk scores. As for TAVI treatment, with increasing age, patients were more likely to be treated with TAVI, and this probability increased progressively with risk score (Fig. 3B). Regarding the medical treatment group, both elevated age and risk score accounted for the increased probability of being contraindicated for invasive therapeutic options (Fig. 3C). The increase in the probability of medical treatment with risk score was particularly marked in patients aged ≥ 85 years.

The multinomial regression models were performed using EuroSCORE II, as recommended in the European Society of Cardiology (ESC) guidelines [START_REF] Vahanian | Guidelines on the management of valvular heart disease (version 2012)[END_REF][START_REF] Baumgartner | ESC/EACTS Guidelines for the management of valvular heart disease[END_REF].

TAVI versus surgery

Of the 691 patients treated with TAVI, 296 (43%) were at low surgical risk according to risk scores.

Among those patients, six had cirrhosis, 20 had a history of radiation and 22 had cancer; these comorbidities are not considered in the EuroSCORE. However, 242 patients (82%) did not have any of these comorbidities to explain the choice of TAVI. We found that age, EuroSCORE II risk category, history of radiotherapy and history of cirrhosis were significantly associated with the decision to treat a patient with TAVI rather than SAVR (Table 3). In particular, patients at intermediate and high surgical risk were, respectively, 3 and 11 times more likely to be treated with TAVI.

In the model with an interaction between age and EuroSCORE II, we found that the probability of being treated with TAVI rather than SAVR increased particularly with surgical risk in patients aged < 80 years (Table A.2). The impact of surgical risk score on decision making decreased markedly over the age of 80 years and was no longer significant in patients aged ≥ 85 years, with as many as 90% of patients being treated by TAVI despite a low estimated surgical risk.

Medical treatment versus TAVI

Practitioners reported advanced age as an argument for medical treatment in 28% of patients, but age was no longer significant in the multivariable analysis. Only high surgical risk was significantly associated with the likelihood of medical treatment over TAVI, with a 4-fold increase (Table 3). The importance of high surgical risk over age was further confirmed in the model with interaction (Table

A.2).

Multivariable models considered an overall centre effect of the models (P = 0.006 in the model with interaction, and P = 0.02 in the model without interaction).

Discussion

In this survey, elderly patients with severe symptomatic AS were considered for TAVI or SAVR in 90% of cases, whereas 10% received medical treatment. About two-thirds of patients in the medical treatment group were considered at low or intermediate risk, and the main reasons to deny an interventional treatment were reduced autonomy, comorbidities and advanced age. The choice of TAVI over SAVR was associated with older age and higher EuroSCORE II, but the weight of risk assessment decreased markedly after the age of 80 years. Indeed, as many as 77% of patients aged ≥ 80 years were offered TAVI while at low estimated surgical risk.

Evolution of indications for TAVI over time

TAVI was first developed to treat patients at high risk from surgery, and surgical risk scores then played an important role in patient selection. Growing evidence showed the limitations of the predictive performance of these scores, in particular calibration drift [START_REF] Barili | Does EuroSCORE II perform better than its original versions? A multicentre validation study[END_REF][START_REF] Provenchere | Is the EuroSCORE II reliable to estimate operative mortality among octogenarians?[END_REF][START_REF] Bendiab | Performance of EuroSCORE II and Society of Thoracic Surgeons risk scores in elderly patients undergoing aortic valve replacement surgery[END_REF]. Nevertheless, risk scores are still taken into account in recent guidelines, which provide thresholds contributing to the choice between SAVR and TAVI [START_REF] Baumgartner | ESC/EACTS Guidelines for the management of valvular heart disease[END_REF][START_REF] Nishimura | AHA/ACC Focused Update of the 2014 AHA/ACC Guideline for the Management of Patients With Valvular Heart Disease: A Report of the American College of Cardiology/American Heart Association Task Force on Clinical Practice Guidelines[END_REF]. SAVR is indicated in low-risk patients, whereas a multifactorial approach by the Heart Team is preferred in patients with increased estimated surgical risk or comorbidities not considered in the EuroSCORE (chest radiation therapy, cancer, etc.) [START_REF] Baumgartner | ESC/EACTS Guidelines for the management of valvular heart disease[END_REF][START_REF] Nishimura | AHA/ACC Focused Update of the 2014 AHA/ACC Guideline for the Management of Patients With Valvular Heart Disease: A Report of the American College of Cardiology/American Heart Association Task Force on Clinical Practice Guidelines[END_REF]. In this survey, the most frequent therapeutic decision was TAVI in 71% of patients, although high-risk patients represented only 18% of the population. Current practices thus anticipated the extension of indications for TAVI to intermediate-risk patients, which was then confirmed in the 2017 guidelines [START_REF] Baumgartner | ESC/EACTS Guidelines for the management of valvular heart disease[END_REF][START_REF] Nishimura | AHA/ACC Focused Update of the 2014 AHA/ACC Guideline for the Management of Patients With Valvular Heart Disease: A Report of the American College of Cardiology/American Heart Association Task Force on Clinical Practice Guidelines[END_REF]. Indeed, intermediate-risk patients represented 38% of the present population treated by TAVI. Moreover, as many as 43% of patients in whom there was a decision to perform TAVI were at low estimated surgical risk. Among those patients, only 16% had a comorbidity that might explain the choice of TAVI, whereas the others should, in theory, have been orientated to surgery.

The choice of TAVI over SAVR in low-risk patients

In the RAC survey, TAVI was the therapeutic choice in 64% of low-risk patients, although only one randomized trial supported this strategy at that time [START_REF] Thyregod | Transcatheter Versus Surgical Aortic Valve Replacement in Patients With Severe Aortic Valve Stenosis: 1-Year Results From the All-Comers NOTION Randomized Clinical Trial[END_REF]. Moreover, patients aged ≥ 80 years were most often considered for TAVI, regardless of their surgical risk. In the very elderly, guidelines are therefore poorly followed regarding referral for surgery, and age outweighs the estimated surgical risk when decision making in clinical practice. This current trend towards an extension of indications for TAVI to low-risk patients, in particular in patients aged ≥ 80 years, is consistent with recent reports, despite no corresponding recommendation in guidelines [START_REF] Bekeredjian | Patients at low surgical risk as defined by the Society of Thoracic Surgeons Score undergoing isolated interventional or surgical aortic valve implantation: in-hospital data and 1-year results from the German Aortic Valve Registry (GARY)[END_REF][START_REF] Morrow | Assessing the safety and efficacy of TAVR compared to SAVR in low-to-intermediate surgical risk patients with aortic valve stenosis: An overview of reviews[END_REF][START_REF] Waksman | Transcatheter Aortic Valve Replacement in Low-Risk Patients With Symptomatic Severe Aortic Stenosis[END_REF]; it highlights the need for country-specific and updated cost analyses to assess the impact on healthcare expenses. This is of particular importance because of the expected increase in the number of elderly patients with AS in the coming decades. Several factors may account for this choice. Non-inferiority of TAVI versus SAVR regarding morbimortality may be an incentive to favour TAVI in elderly patients. Indeed, cardiac surgery is usually associated with a prolonged length of stay compared with TAVI [START_REF] Durand | Evaluation of length of stay after transfemoral transcatheter aortic valve implantation with SAPIEN 3 prosthesis: A French multicentre prospective observational trial[END_REF][START_REF] Sheng | In-Hospital Outcomes After Transcatheter Versus Surgical Aortic Valve Replacement in Octogenarians[END_REF][START_REF] Zhou | Cost-effectiveness of transcatheter aortic valve implantation compared to surgical aortic valve replacement in the intermediate surgical risk population[END_REF]. This is paramount in elderly patients, who are less likely to be discharged home, and are more likely to require rehabilitation and/or institutionalization after a long hospital stay. TAVI, compared with SAVR, also reduces the risk of atrial fibrillation, acute kidney injury and bleeding, with high-certainty evidence [START_REF] Kolkailah | Transcatheter aortic valve implantation versus surgical aortic valve replacement for severe aortic stenosis in people with low surgical risk[END_REF]. A recent review on patients' preferences regarding TAVI or SAVR further confirmed these data, with most participants choosing minimally invasive procedures, with a shorter hospital stay and recovery period [START_REF] Heen | Patient values and preferences on valve replacement for aortic stenosis: a systematic review[END_REF]. In addition, the potential concerns about TAVI, such as the consequences of pacemaker implantation and uncertainties regarding long-term valve durability, have a lower impact in patients with shorter life expectancy, although no negative signal appeared with TAVI prostheses [START_REF] Fauvel | Durability of transcatheter aortic valve implantation: A translational review[END_REF]. Choosing TAVI over SAVR in patients aged ≥ 80 years, regardless of their risk scores, could therefore be discussed.

Severe symptomatic AS and medical treatment

In the RAC survey, an intervention was denied in 10% of patients. Single-centre series in early experience with TAVI reported higher rates of 16-26% [START_REF] Bouleti | The impact of the development of transcatheter aortic valve implantation on the management of severe aortic stenosis in high-risk patients: treatment strategies and outcome[END_REF][START_REF] Pilgrim | Long-term outcome of elderly patients with severe aortic stenosis as a function of treatment modality[END_REF]. The development of TAVI has led to a decrease in the percentage of patients who are denied intervention, compared with when SAVR was the only treatment [START_REF] Iung | Contemporary Presentation and Management of Valvular Heart Disease: The EURObservational Research Programme Valvular Heart Disease II Survey[END_REF][START_REF] Malaisrie | Transcatheter aortic valve implantation decreases the rate of unoperated aortic stenosis[END_REF]. In the present study, patients treated medically exhibited significantly higher risk scores than others, which contrasts with older series, in which patients allocated to TAVI or medical treatment exhibited similar baseline characteristics [START_REF] Wenaweser | Clinical outcomes of patients with severe aortic stenosis at increased surgical risk according to treatment modality[END_REF]. The main reason for conservative treatment during that period was the refusal to undergo intervention by almost half of the patients, whereas only 16% of patients refused any intervention in our study [START_REF] Wenaweser | Clinical outcomes of patients with severe aortic stenosis at increased surgical risk according to treatment modality[END_REF]. These results suggest better selection of patients referred for severe symptomatic AS over time.

However, in 28% of cases, the reason cited by practitioners for not considering any intervention was the advanced age of the patient. Considering remaining life expectancy, one may wonder whether this is an effective strategy, regarding medical and economic facts (rehospitalization because of untreated AS) [START_REF] Wong | You're never too old[END_REF]. The benefit/risk ratio must be weighed carefully in elderly patients, given their burden of comorbidities, with frequent multifactorial dyspnoea and increased risks of procedural complications. Personalized evaluation of this particular population is of utmost importance, to better select patients who could really benefit from TAVI, and to avoid futile interventions. On the other hand, patients treated conservatively exhibit a poor outcome, and TAVI is known to lower mortality, improve quality of life and limit iterative hospitalizations, including in elderly patients [START_REF] Leon | Transcatheter aortic-valve implantation for aortic stenosis in patients who cannot undergo surgery[END_REF].

Study limitations

Frailty was not assessed in the present analysis, although it contributes to the therapeutic choice, particularly in elderly patients. Frailty is indeed common in older adults and was associated with both major bleeding after aortic valve replacement and higher mid-term mortality in a recent series, thus confirming its prognostic impact [START_REF] Bendayan | Frailty and Bleeding in Older Adults Undergoing TAVR or SAVR: Insights From the FRAILTY-AVR Study[END_REF]. However, the prevalence of frailty substantially differs according to the scale used, and its contribution is therefore difficult to evaluate in the absence of a standardized assessment [START_REF] Afilalo | Frailty in Older Adults Undergoing Aortic Valve Replacement: The FRAILTY-AVR Study[END_REF]. Only patients from centres with both on-site TAVI and SAVR were included, which may constitute a referral bias, but avoided a potential treatment allocation bias. The analyses were performed with EuroSCORE II, using the thresholds for low and high risk provided in the ESC guidelines [START_REF] Baumgartner | ESC/EACTS Guidelines for the management of valvular heart disease[END_REF]. Although EuroSCORE II was not specifically designed for very elderly patients, its predictive performance remains correct in octogenarians [START_REF] Provenchere | Is the EuroSCORE II reliable to estimate operative mortality among octogenarians?[END_REF]. We chose not to use the STS PROM score in this prospective survey, as its many variables make it rather tedious in clinical practice, and they are not collected as standard of care in our country.

Conclusions

In patients aged ≥ 75 years presenting with severe symptomatic AS, the therapeutic decision was TAVI in > 70% of cases. Age played a major role in the choice between TAVI and SAVR, and superseded the surgical risk assessment after the age of 80 years. Given the many advantages of TAVI in elderly patients, we endorse the current practice of choosing TAVI over SAVR after the age of 80 years, regardless of risk scores. Medical treatment was still decided upon in 10% of patients, with more than one out of four practitioners citing advanced age for this decision, which may be debatable. f Four emergency balloon aortic valvuloplasties.

  Abbreviations: AS, aortic stenosis; ESC, European Society of Cardiology; SAVR, surgical aortic valve replacement; TAVI, transcatheter aortic valve implantation.

Figure 1 .

 1 Figure 1. Flowchart of the RAC survey.

Figure 2 .

 2 Figure 2. A. Histogram of the distribution of risk categories according to the allocated treatment. B. Histogram of the distribution of allocated treatment according to age categories in the whole cohort. C. Histogram of the distribution of allocated treatment according to age categories in the subgroup of patients at low surgical risk. SAVR: surgical aortic valve replacement; TAVI: transcatheter aortic valve implantation.

Figure 3 .

 3 Figure 3. Probability of treatment decision based on age and EuroSCORE II. A. The probability of being treated with surgical aortic valve replacement (SAVR) varied greatly with age and to a lesser extent with risk score. B. The probability of being treated with transcatheter aortic valve implantation (TAVI) increased with age and risk score, the importance of risk score varying with age category. C. The probability of being treated medically increased with both age and risk score.

a

  Patients for whom data were available: total (n = 1001); SAVR (n = 184); TAVI (n = 721); medical treatment (n = 96). b Patients for whom data were available: total (n = 975); SAVR (n = 180); TAVI (n = 700); medical treatment (n = 95). c Patients for whom data were available: total (n = 1031); SAVR (n = 195); TAVI (n = 739); medical treatment (n = 97). d Patients for whom data were available: total (n = 941); SAVR (n = 169); TAVI (n = 681); medical treatment (n = 91). e Patients for whom data were available: total (n = 1001); SAVR (n = 188); TAVI (n = 717); medical treatment (n = 96).

Table 1

 1 Baseline characteristics of the 1049 patients.

	Total	SAVR	TAVI	Medical	P
				treatment
	(n = 1049)	(n = 199)	(n = 748)	(n = 102)

TAVI: transcatheter aortic valve implantation.

Table 2

 2 Risk scores of the 956 patients.Data are expressed as mean ± standard deviation or number (%). SAVR: surgical aortic valve replacement; TAVI: transcatheter aortic valve implantation.

		Total	SAVR	TAVI	Medical	P
					treatment
		(n = 956)	(n = 176)	(n = 691)	(n = 89)
	Logistic EuroSCORE (%)	18.0 ± 13.0	9.3 ± 5.3	19.3 ± 12.5	25.0 ± 18.4	< 0.0001
	EuroSCORE II (%)	5.9 ± 6.1	3.0 ± 2.3	6.2 ± 5.9	9.2 ± 9.5	< 0.0001
	EuroSCORE II					< 0.0001
	High risk	175 (18.3)	9 (5.1)	135 (19.5)	31 (34.8)
	Intermediate risk	321 (33.6)	27 (15.3)	260 (37.6)	34 (38.2)
	Low risk	460 (48.1)	140 (79.6)	296 (42.8)	24 (27.0)

Table 3

 3 Multivariable multinomial logistic regression model of patient characteristics associated with the therapeutic decision.

	Odds ratio (95% CI)
	TAVI (versus SAVR)	Medical treatment (versus TAVI)

CI: confidence interval; SAVR: surgical aortic valve replacement; TAVI: transcatheter aortic valve implantation.

Acknowledgements

We thank Isabelle Keating for English editing and proofreading

Sources of funding

The RAC survey was funded by an unrestricted grant from the European Association of Percutaneous Cardiovascular Interventions (EAPCI valve for life). This work was supported by the French Government and managed by the National Research Agency (ANR) under the ''Programme d'investissements d'avenir'', with the reference ANR-16-RHUS-0003.

The other authors declare that they have no conflict of interest concerning this article.