Tung Doan

Hung Le-Quang

Quy-Dong To
email: quy-dong.to@u-pem.fr.

Coupled molecular dynamics and micromechanics study of planar elastic properties of graphene with void defects

Keywords: Graphene, Molecular Dynamics, Eshelby problem, polygonal

In this paper, we employ both molecular dynamics (MD) and micromechanics approach to analyze the influence of void defects on the overall 2-dimensional (2D) elastic behavior of graphene. In the micromechanics model (MM), the edge boundary is assumed to have distinct longitudinal elastic stiffness property (Gurtin-Murdoch model) which can be identified by MD simulations of pristine graphene sheet. Both the Finite Element Method (FEM) and MD are used to study the Eshelby problem involving polygonal nanovoids periodically embedded in a graphene sheet. To characterize the heterogeneity effect due to a single nanovoid, we propose to use MD to compute the tensor C which is the first order expansion of the effective tensor C with respect to volume fraction f , related to the dilute scheme estimate. It is shown that the MM performs well for voids with large edges and predicts results consistent with the properties of edge structures. However, MM fails for small voids with short edges where the discrepancies are due to the corner effect which can not be accounted for in MM.

nanovoids, edge effect.

Introduction

Since the Nobel prize discovery of graphene [START_REF] Novoselov | Electric field effect in atomically thin carbon films[END_REF], the material has been found to possess many extraordinary properties and gained significant scientific attention over the years. With electrical resistivity of order 10 -6 Ωcm and thermal conductivity of 5000 W m -1 K -1 [START_REF] Balandin | Superior thermal conductivity of single-layer graphene[END_REF], it is the best conductive material at room temperature. Being mechanically 200 times stronger and 5 times harder than steel, it is also the strongest and lightest 2D material.

There have been theoretical and experimental studies on the mechanical properties of graphene. Among a few experimental methods currently available are atomic force microscope (AFM), instrumented nano-intender (INI), and Raman spectroscopy (RS), etc. The capacity of these methods is limited to the indirect estimation of the elastic modulus of a tiny graphene flake. Properties of larger scale graphene membrane and other mechanical properties, e.g. Poisson's ratio, edge elastic constants, residual stresses, are not accessible. In this case, numerical methods e.g. Ab-initio, Molecular Dynamics (MD), Density function theory (DFT), Molecular Mechanics (MM), Tight Binding (TB), etc. are useful and can provide more direct estimates. However, like experiments, the results reported in the literature are scattering, for example the Young modulus can vary from 0.4 to 4.2T P a [START_REF] Shen | Temperature-dependent elastic properties of single layer graphene sheets[END_REF][START_REF] Cao | Atomistic studies of mechanical properties of graphene[END_REF][START_REF] Memarian | Graphene Young's modulus: Molecular mechanics and DFT treatments[END_REF]. It is known that the numerical results depend considerably on the computation methods, the techniques used and simulation conditions which are not the same in the previous works. Another distinct feature of graphene is that there are two main directions Armchair (AC) and Zigzag (ZZ) (see Figure 1) and the elastic properties along those two directions can be different [START_REF] Scarpa | Effective elastic mechanical properties of single layer graphene sheets[END_REF][START_REF] Pei | Mechanical properties of methyl functionalized graphene: a molecular dynamics study[END_REF][START_REF] Gao | Mechanical properties of monolayer graphene under tensile and compressive loading[END_REF].

In general, atoms near the free boundary behave differently from the bulk due to the difference in mobility, coordination number, and potential energy, etc. At large scales, this surface effect is dominated by bulk behavior and is not observable. However, at the nano-scale, it has been proved by experiments and atomistic simulations that this effect is considerable [START_REF] Cammarata | Surface and interface stress effects on interfacial and nanostructured materials[END_REF][START_REF] Gibbs | The scientific papers of J. Willard Gibbs[END_REF][START_REF] Shuttleworth | The surface tension of solids[END_REF][START_REF] Nix | Indentation size effects in crystalline materials: a law for strain gradient plasticity[END_REF] and resulting in the size-dependent behavior. In the case of graphene, edge effects have been studied in several researches. Using MD simulations, [START_REF] Reddy | Edge elastic properties of defect-free single-layer graphene sheets[END_REF] reported the edge elastic constants are 147.1556 eV /nm for ZZ direction and 112.6304 eV /nm for AC direction. It is also stated in their paper that the edge elastic modulus E e can vary from -294 eV /nm to 353 eV /nm depending on the edge structures. By similar approach, [START_REF] Lu | Atomistic simulation and continuum modeling of graphene nanoribbons under uniaxial tension[END_REF] found the results are 52 eV /nm for ZZ edge and 23 eV /nm for AC edge. For AC direction only, [START_REF] Li | Periodically modulated size-dependent elastic properties of armchair graphene nanoribbons[END_REF] reported negative values of edge elastic modulus from -64N/m to -47N/m. Despite the wide range of existing results, currently there is no experiment method to directly measure this constant of graphene membranes to confirm those values. Like all materials, defects are unavoidable in the synthesis process of graphene. There are several kinds of structural defects can happen and they can be classified into three main groups: vacancy defects, adatoms, and atom reconstruct defects [START_REF] Banhart | Structural defects in graphene[END_REF]. [START_REF] Wang | Large-scale molecular simulations on the mechanical response and failure behavior of a defective graphene: cases of 5-8-5 defects[END_REF] studied the effects of 5-8-5 defects on graphene and stated that, comparing with the elastic modulus of pristine graphene, the stiffness of defected graphene in the AC direction is smaller, where that of graphene in ZZ direction becomes larger. By using the TB method, [START_REF] Dettori | Elastic fields and moduli in defected graphene[END_REF] studied the elastic moduli of graphene with vacancy and various reconstructed defects and concluded that the important factor that affects the elasticity of defected graphene is the changing in the absolute mass density, not the types of the reconstruction of the atoms. This conclusion is also supported by [START_REF] Ansari | Mechanical properties of defective single-layered graphene sheets via molecular dynamics simulation[END_REF]. In this work, we are interested in vacancy defects which is one of important factors that affect the mechanical properties of graphene.

At continuum scale, micromechanics method is known as an analytically accurate and efficient approach to predict the effective properties of heterogeneous materials in general and porous materials in particular. The theoretical framework of the approach is well founded in literature (see e.g [START_REF] Mura | Micromechanics of Defects in Solids[END_REF][START_REF] Milton | The theory of composites[END_REF][START_REF] Torquato | Random heterogeneous materials: microstructure and macroscopic properties[END_REF], and the references therein). One of the standard techniques in micromechanics, is to study the Eshelby inclusion problem [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion, and related problems[END_REF] in which a single inclusion (or void) is embedded in an infinite matrix and subjected to an uniform loading at infinity. From this elementary solution, estimates for mixtures of inclusions at finite volume fraction f can be derived using relevant schemes, e.g dilute, Mori-Tanaka, self-consistent schemes, etc.. Since the original Eshelby's inclusion problem is devoted to an ellipsoidal inclusion, for nanovoids, it is necessary to introduce the edge effect in the micromechanics model. In many cases, using the elastic Gurtin-Murdoch (GM) edge model [START_REF] Gurtin | A continuum theory of elastic material surfaces[END_REF] can produce satisfactorily these edge effects. The Eshelby inclusion problem involving 2D polygonal nanovoids like graphene and using GM model have been studied recently in [START_REF] Doan | Effective elastic stiffness of 2d materials containing nanovoids of arbitrary shape[END_REF]. While all those works are based on the continuum mechanics, it can not capture all the behavior at the atomic scale, especially when there is variation of atomic structure along the void edges or interaction of atoms across the edges etc... The heterogeneity effect due to those factors are complex and can affect the accuracy of stiffness estimates. Most literature works based on MD methods (see e.g [START_REF] Rafiee | Comparative study on predicting Young's modulus of graphene sheets using nano-scale continuum mechanics approach[END_REF][START_REF] Savvas | Determination of random material properties of graphene sheets with different types of defects[END_REF] are more concerned about the effective properties of the graphene with voids without relating to the local behavior of voids. This motivates us to investigate the Eshelby problems under the micromechanics framework and characterize the void defects of graphene using both MD and continuum based methods in the present paper.

The paper will be organized as follows. First, we describe the general framework and procedure to find the elastic properties of graphene, including the edge elastic constants (Sec. 2) and graphene sheet with voids using MD simulations. Next, in Sec. 3, we carry out coupled MD-MM studies on graphene with nanovoids. The MD results are compared with the MM results based on FEM analysis using the bulk and the edge elastic properties obtained from previous steps. The objective of the comparisons is to investigate the impact of edge effects on the overall properties. Finally, some concluding remarks are given in the last section.

Determination of bulk and edge stiffness of graphene

At atomic scale, MD is an efficient computation method for studying the motion of atoms and determining the macroscopic properties of materials by statistical averages. We use an open source code LAMMPS [START_REF] Plimpton | Fast parallel algorithms for short-range molecular dynamics[END_REF] to run our MD simulation due to its computational efficiency and parallel capacity. The Adaptive Inter-molecular Reactive Empirical Bond Order (AIREBO) potential by [START_REF] Schall | Reactive empirical bond-order potentials[END_REF] which is relevant to simulate Carbon system like graphene, was used in the study. As mentioned previously, due to various reasons, the values of the bulk and edge stiffness reported in literature are scattering. To keep the consistency, we use the same computation method and simulation conditions throughout the present paper and devote this section to determine those constants with the method. In addition to the simulation conditions, the definition of geometric boundary of graphene sheet can also affect the computed values and will be discussed.

Temperature is known to have effects on the elastic modulus of graphene. The Young modulus of graphene drops approximately 10% when the temperature increases from 1 to 1200K [START_REF] Zhang | Temperature and strainrate dependent fracture strength of graphynes[END_REF]. Under finite temperature, multiple complex phenomena associated to the out-of-plane behaviors like rippling, bending, buckling, vibration etc.. can appear and interact with the in-plane behavior. Consequently, the computation results obtained under such conditions have very high uncertainty and vary significantly. Due to the ripples, the tension and compression stiffness can be different [START_REF] Sgouros | Compressive response and buckling of graphene nanoribbons[END_REF] and depend on the sample size [START_REF] Los | Scaling behavior and strain dependence of in-plane elastic properties of graphene[END_REF]. It is suggested that those issues can be addressed based on out-of-plane flexible plate and membrane models (see e.g [START_REF] Fasolino | Intrinsic ripples in graphene[END_REF][START_REF] Wang | Wrinkling of monolayer graphene: a study by molecular dynamics and continuum plate theory[END_REF]. In the present study of planar elastic stiffness, we do not consider the rippling phenomenon and maintain the system in the plane (see e.g [START_REF] Arroyo | Finite crystal elasticity of carbon nanotubes based on the exponential cauchy-born rule[END_REF][START_REF] Saavedra Flores | Hyperelastic tension of graphene[END_REF][START_REF] Marenić | Homogenized elastic properties of graphene for small deformations[END_REF][START_REF] Lu | Nonlinear mechanics of single-atomic-layer graphene sheets[END_REF][START_REF] Jiang | Elastic and nonlinear stiffness of graphene: A simple approach[END_REF][START_REF] Wei | Nonlinear elastic behavior of graphene: Ab initio calculations to continuum description[END_REF][START_REF] Javvaji | Mechanical properties of graphene: Molecular dynamics simulations correlated to continuum based scaling laws[END_REF] at temperature as low as 10K. The thickness of the system in z direction is taken as 3.4 Å. In the strain control simulation, the system is stretched (direct strain 1 or 2 imposed) or tilted (shear strain 3 imposed) on its periodic boundaries. After applying a small strain level (order 0.1%), the system was relaxed for the next 8e5 timesteps of 0.001ps to achieve the equilibrium NVT state where time and sample average of system properties can be computed. For example, in this case, we are interested in the average potential energy of the system at each strain value. At small strain and temperature, the graphene can be considered as linear elastic material and the potential energy density U is quadratic function of strain. Here, we assume that at temperature as low as 10K, the strain energy can be assimilated to the potential energy and the variation of the latter govern the elastic behavior of the nanosystem. By fitting the potential energy with quadratic functions of strain components, stiffness constants can be derived. The method can be applied to compute the properties for any systems with periodic boundaries along x-and/or ydirections of pristine graphene or graphene with defects, as described in the following.

To compute the bulk stiffness constant of graphene, we use a sufficiently large graphene sheet with periodic boundary conditions along x-and y-directions and impose strain on the plane xy as described previously. We can write using the Voigt's notation:

U = U 0 + U 1 + U 2 , (1)
U 0 = cste, U 1 = σ 0 i i , U 2 = 1 2 C ij i j (2)
where σ 0 i is the residual stress, C ij the elastic constants and U 0 the potential energy at the reference strain free state. Specifically, the quadratic form of U 2 can be written explicitly as follows

U 2 = 1 2 (C 11 2 1 + 2C 12 1 2 + 2C 13 1 3 + C 22 2 2 + 2C 23 2 3 + C 33 2 3 ) (3)
For example, to find the elastic constant C 11 (or C 22 and C 33 ), it is sufficient to vary the strain 1 = and maintain the other strain components at zero value 2 = 3 = 0. In this case, U becomes As a result, if we have sufficient number of potential energy -strain ( , U ) points from MD simulations, it is possible to identify the elastic constants C 11 by parabola fit (see Fig. 2). To determine C 12 , we apply simultaneously the same amount of direct strain 1 = 2 = along both 1 and 2 direction.

U = 1 2 C 11 2 + σ 0 1 + U 0 (4)
In this case, Equation 3 becomes

U = 1 2 (C 11 + 2C 12 + C 22 ) 2 + (σ 0 1 + σ 0 2 ) + U 0 (5)
The procedure of computing the hybrid coefficients C 11 + 2C 12 + C 22 associated to 2 is the same as before. With C 11 and C 22 already known from the previous step, we can solve for C 12 .

When applying the above method to pristine graphene, we obtain the bulk elastic stiffness of graphene C b ij . The elastic modulus E b and the Poisson's ratio ν for isotropic materials are computed from:

E b = C b 11 (1 -ν 2 b ), ν b = C b 12 C b 11 . (6)
Results from MD simulations of system 96.7 Å x 96.3 Å containing 3680 atoms, give the values of C b 11 = C b 22 = 953 GPa and C b 12 = 323 GPa respectively. Consequently, the matrix elastic modulus E b and the Poisson's ratio calculated from Eq. 6 take the values 843.8 GP a and 0.338 respectively. These values are very close to literature results [START_REF] Pei | Mechanical properties of methyl functionalized graphene: a molecular dynamics study[END_REF][START_REF] Terdalkar | Nanoscale fracture in graphene[END_REF][START_REF] Tu | Single-walled and multiwalled carbon nanotubes viewed as elastic tubes with the effective Young's moduli dependent on layer number[END_REF].

To determine the edge stiffness E e of graphene, we need to consider a system with free surfaces. This is the case of graphene nanoribbon (GNR) where we apply strain along its length l, the periodic direction, and the width w direction is free of stress. It is necessary to account for both bulk energy density U b uniform for the whole volume wl and edge energy U e , i.e energy localized along the two edges of length l, as follows

wlU = wlU b + 2lU e (7)
The dependency of those two quantities on prescribed strain is given by

U b = 1 2 E b 2 + σ 0 1 + U b0 , U e = 1 2 E e 2 + τ 0 1 + U e0 , (8)
where σ 0 1 , τ 0 1 respectively are the residual stresses in the bulk and in the edges, and U b0 , U e0 are the potential energies at the reference zero strain state in the bulk and in the edges respectively.

One can find that the effective Young modulus of the GNR can be determined as

E = E b + 2 w E e .
Since E for a given GNR can be determined from U using the same procedure as before (replacing E for C ii ), doing tests with different width values w will generate data to find E e and E b . We note that the width w determined from the volume of the GNR is different from the geometric distance of extreme atoms belonging to the two opposite edges (see Fig. 3). While the latter definition can be used to analyze the behavior of pristine graphene, it is not relevant for the study of nanovoids where the void volume fraction f must be defined as well. To guarantee the consistent volume definition throughout the paper and the conservation of volume, we consider that each atom occupies the same volume and the volume of graphene objects is proportional to the number of atoms contained inside. For example, if the number density of graphene is denoted as n, the area (or volume) of a graphene sheet composed of N carbon atom is equal to V = nN and the width w = V /L where L is the length of the graphene sheet. This definition will affect the width of the graphene sheets and consequently the value of edge stiffness when fitting MD results with continuum models. In the GNR simulations, the length of the ZZ ribbons are 67.7 Å while the width is changing from 16.7 Å to 100.5 Å. For AC ribbons, the length is 83.76 Å and the width is from 14.5 Å to 53.19 Å. From Fig. 4, we find that E varies linearly with the inverse width w -1 but differently for AC and ZZ directions. The slopes can be used to compute the edge elastic constants E e in each direction and the intersections with the vertical axis give E b . The GNR simulations yield the values E b = 842 GP a for both directions, close to the value 843.8 GP a determined from the previous periodic systems. The edge elastic modulus E e is found 11.9 GP a.nm for ZZ direction and -17.9 GP a.nm for AC direction, or 25.62 eV /nm and -38.54 eV /nm respectively. Since the stiffness of graphene can be considered as the contribution of the bulk and the edge, the ZZ edges with positive stiffness tend to stiffen and AC edges with negative stiffness soften the graphene. For the same void shape and size, graphene with ZZ edges is stiffer than graphene with AC edges. From the ratio E e /E b , it is suggested that for objects of size 1-2 nm, the difference due to the free edge is of order 1-2%. The difference can be more pronounced for objects of sub nanometric size. We note that due to the lattice structure of graphene, the characteristic size of objects cut from a pristine graphene sheet is limited by the lower bound value 1.42 Å. Furthermore, the above determination of E e is done on infinitely long GNR, which may be not valid for systems containing short edges. In this case, the local interaction of atoms at corners may also be important and affect the global behavior of the system.

Characterization of void effect in graphene

To investigate the void size and shape effects on the stiffness of the graphene, we run MD simulations of graphene sheets with different types of voids and use the same method described in Section 2 to calculate the effective properties of the system. Results are then compared with the continuum models (with and without edge effect), solved by the FEM. Comparisons are performed on the problem of periodic cell with single void. Like the volume of graphene, the void volume V void is computed from the number density n of graphene and the number of missing carbon atoms N void via the relation V void = nN void . We also denote f = V void /V , as the volume fraction of the void (or porosity), an important quantity to evaluate the effective properties of porous material. In the case where f → 0, we shall recover the Eshelby problem with a void in an infinite matrix. In the FEM models (Fig. 5), the cell dimensions are the same as those in the MD model. The bulk is isotropic with E b = 842 GPa and ν b = 0.338 and the edge effect is modeled by a thin material layer with thickness t and elastic constant E layer = E e t [START_REF] Shenoy | Edgestress-induced warping of graphene sheets and nanoribbons[END_REF]. In the ideal case where t → 0, it can be shown theoretically that the Gurtin-Murdoch model can be recovered [START_REF] Benveniste | Imperfect soft and stiff interfaces in twodimensional elasticity[END_REF][START_REF] Hashin | Thin interphase/imperfect interface in elasticity with application to coated fiber composites[END_REF][START_REF] Benveniste | A general interface model for a three-dimensional curved thin anisotropic interphase between two anisotropic media[END_REF]. In practical FEM applications, t should not be too small to avoid mesh issues and chosen as t = 0.005 Å [START_REF] Doan | Effective elastic stiffness of 2d materials containing nanovoids of arbitrary shape[END_REF] In addition to the effective stiffness C of graphene, we are also interested in the quantity C

C = C b + f C + O(f 2 ), or C = lim f →0 f -1 (C -C b ) (9)
This tensor C, equivalent to the first order expansion of C with respect to f , is both useful for MD and continuum models for the evaluation of nanovoid contribution in the effective stiffness. For the latter case, it corresponds to the limit of dilute and Mori-Tanaka (MT) scheme at a small f , related to the localization tensor and the equivalent stiffness of the nanovoid (see Appendix Appendix A for more details). It is noted that the MT estimate is based on solutions of heterogeneity problems involving single nanovoid in infinite matrix. The solutions can be obtained using the Conformal Mapping techniques proposed by previous works [START_REF] Doan | Effective elastic stiffness of 2d materials containing nanovoids of arbitrary shape[END_REF]. Due to a large number of simulated systems in the present paper, we shall focus on the comparison First, we take the pristine graphene system of 3680 atoms and remove the atoms at the center to obtain the desired void size and shape. The results for each kind of voids with different sizes are shown in Tables 1 and2. In these tables, F EM 1 and F EM 2 are the results from FEM models with and without edge effects respectively. We can see the edge effects on the elastic stiffness of the voids and the graphene sheet. As expected, for the case of ZZ edge voids with positive edge elastic modulus, the graphene cell is stiffened, and for AC voids, due to the negative stiffness, the effective elastic modulus decreased. For mixed edges void, the edge elastic constant in AC and ZZ are in opposite signs and the overall edge effect can be considered canceling out each other. For the pure AC or ZZ edge structure, the continuum models with edge behavior are in excellent agreement with MD results and perform slightly better than models without edge behavior. The difference due to the edge effect is relatively small for large edge due to the small ratio of E e /E b as mentioned previously. Since the size of the cell is fixed, it can be seen that the effect of edge structures decreases when the graphene area decreases. This fact agrees with the statement concluded by [START_REF] Dettori | Elastic fields and moduli in defected graphene[END_REF] previously mentioned.

However, from the micromechanics viewpoint, the effective behavior can be dominated by the bulk which has significant volume fraction and masks the local behavior of the nanovoid. We note that due to the void shape and the atomic structure, in most of the simulated cases, the maximal achievable volume fraction f of voids is relatively small, order 0.2. This motivates us to investigate the contribution of the nanovoid only via the tensor C, which is independent of volume fraction f and only depends on the void size and shape. We are also looking into the cases of small nanovoids (short edge) and higher volume fraction (small cell) where strong edge and corner effects can be observable.

We carried out MD simulations where the void size is fixed while the cell size increases. Shown in the Tables 3 and4 are the results in comparison with FEM models. Like the previous test, we found good agreement between MD simulations and FEM models and MT estimates for most of the considered cases. However, we can see the larger errors between the MD and the continuum models, especially for the case of small void and high volume fraction (order 0.2) including diamond void 8/112 ZZ and hexagon void 24/112 AC. A careful inspection of the f -C 11 and f -C 22 slope near the origin f = 0 shows that the nanovoid coefficients C11 and C22 are considerably different between the MD and the continuum models. It implies that the micromechanics approach already fails to solve the heterogeneity problem with acceptable error. The C11 and C22 errors for the two mentioned cases are of order 20%. It confirms the previous suggestion that due to the small volume fraction, the discrepancies due to the nanovoid effect are masked by the bulk behavior. We note that all the coefficients C11 and C22 are negative showing that the voids have weakened the graphene, even when ZZ edges with stiffening effect are used. Theoretically, using the ZZ edge hole to stiffen the graphene only works for very tiny hole, for example one atom vacancy defect. This is the case of [START_REF] López-Polín | Increasing the elastic modulus of graphene by controlled defect creation[END_REF] who found experimentally that it is possible to increase the stiffness of graphene by removing a very small number of carbon atoms.

There is another interesting feature we can observe at small nanovoids. While the use of edge elastic model improves the prediction of micromechanics approach in most cases, the case 24/112 with AC edges shows a reverse trend.

There can be a strong corner effect can affect the global behavior. As mentioned previously, the edge model is based on the parameter E e obtained for infinite long edge. For small nanovoids, the edge is short, with few atoms, and the number of corner atoms is comparable with edge atoms. Furthermore, atoms belonging to different edges near the common corner can interact within the interaction range (see Fig. 9) and it is difficult to distinguish edge and corner atoms. This corner effect seems to stiffen the hole and cancel out the softening effect of AC. As a result, the overall behavior of graphene shows that the model without edge model performs better. To capture the corner effect with micromechanics approach, we need a special model devoted to the corner. We note that similar ideas based on cohesive crack or adhesive contact models accounting for van der Waals interaction forces at the interface can be useful in this case and will be investigated in the future. Another issue of nanovoid that limits the application of micromechanics model is when the vacancy defects are reduced to several atoms. It is difficult to detect the clear edge structure and construct the geometry of micromechanics model. In this case, the present MD based approach can also address the problem. Especially, the determination of the nanovoid tensor C is always feasible. This tensor, which is independent of the volume fraction, can provide insight into the local behavior of the nanovoid by including both corner and edge effects.

Void/Cell(atoms)

f M D F EM 1 F EM

Conclusions and perspectives

In this work, we study the effective stiffness of pristine graphene and graphene containing nanovoids. The elastic constants of the bulk and the free edges according to the Gurtin-Murdoch model are determined by MD simulation. It is found that the free edge stiffness depends on the edge structure, negative for armchair or positive for zigzag. The continuum-based micromechanics models constituted of graphene and vacancy defects are then constructed using the elastic parameters obtained by MD methods. The effective properties of the defected graphene are computed by different approaches FEM and compared with full MD simulations. In addition to the effective properties of graphene with voids C, the heterogeneity effects due to a single nanovoid are characterized via the tensor C, the first-order expansion of C with respect to f . The results issued from the comparisons show that the effective elastic stiffness obtained by micromechanics models and estimates is in good agreement with the MD results and consistent with the softening and stiffening effect of the edge structures. For small nanovoids, we observe larger discrepancies between MD and MM and further inspection of nanovoid coefficients C show that MM fails to capture the behavior of nanovoids including both edge and corner effects. In some cases, those considerable errors may be masked in the overall behavior due to the small volume fraction of the nanovoid. We conclude that it is necessary to develop models devoted to the sharp corners, not only for graphene but also for nanoporous materials in general, to fully capture the behavior of the nanopores.

Another interesting problem that merits future consideration is to investigate the out-of-plane behavior of graphene. While plate and membrane models can successfully reproduce the behavior of pristine graphene, the influence of the defect heterogeneity combined with the intrinsic ripples on the effective bending and stretching behaviors needs to be explored and understood.

Figure 1 :

 1 Figure 1: Zigzag and Armchair directions

Figure 2 :

 2 Figure 2: Potential energy U as function of strain 1 = . The parabola fitting of MD results is used to determine C b 11 .

Figure 3 :

 3 Figure 3: Continuum boundary and width w of GNR. Top: GNR with AC edges, Bottom: GNR with ZZ edges.

Figure 4 :

 4 Figure 4: Effective Young modulus of GNR as function of inverse width w -1 . The linear fitting is used to find E b and E e (a) AC direction (b) ZZ direction.

Figure 5 :

 5 Figure 5: Finite element meshes of graphene sheet containing different types of voids (a) Hexagon (b) Diamond.

Figure 6 :

 6 Figure 6: Typical void shapes and edge structures (a) Hexagon AC edge (b) Hexagon ZZ edge (c) Diamond AC edge (d) Diamond ZZ edge.

Figure 7 :

 7 Figure 7: Circular void shape with mixed edge structures.

Figure 8 :

 8 Figure 8: Cracked graphene with ZZ edge structures.

Figure 9 :

 9 Figure 9: Interaction between atoms around corner

Table 1 :

 1 Effective elastic constant C 11 in MPa -MD results and solutions from FEM (with and without elastic model)

	2

Table 2 :

 2 Effective elastic constant C 22 in MPa -MD results and solutions from FEM (with and without elastic model)

			f	M D F EM 1 F EM 2 M T
			Diamond void -ZZ edges
	C 11	8/1232	0.0065 940	939	937	939
		8/448	0.0179 919	915	910	915
		8/240	0.0333 890	884	876	885
		8/112	0.0714 823	815	800	814
	C11	-	-	-3376 -3671	-4087 -3650
			Diamond void -AC edges
	C 11	54/3680	0.015	915	914	915	915
		54/2800	0.019	905	905	906	905
		54/1232	0.044	849	849	853	846
		54/240	0.225	561	559	570	528
	C11	-	-	-2590 -2658	-2590 -2616
			Hexagon void -ZZ edges
	C 11	24/3680	0.0065 936	934	933	935
		24/2800	0.0086 931	929	927	929
		24/1232	0.0195 903	899	897	899
		24/240	0.100	728	722	713	716
	C11	-	-	-2662 -2870	-3005 -2826
			Hexagon void -AC edges
	C 11	24/1232	0.019	901	892	896	893
		24/448	0.054	825	798	808	794
		24/240	0.100	729	696	711	685
		24/112	0.214	553	506	529	482
	C11	-	-	-2656 -3129	-2914 -3102
			Crack void -ZZ edges
	C 11	46/3680	0.0125 926	925	924	926
	C11	-	-	-2137 -2210	-2308 -2189

Table 3 :

 3 Effective elastic constants C 11 and C11 in MPa-Comparisons between FEM (with and without edge effect), MD and Mori Tanaka (MT) estimates.

			f	M D F EM 1 F EM 2	M T
			Diamond void -ZZ edges
	C 22	8/1232	0.0065	911	912	11	917
		8/448	0.0179	852	825	851	859
		8/240	0.0333	781	780	778	789
		8/112	0.0714	637	637	634	651
	C22	-	-	-3322	-3361	-3399	-3255
			Diamond void -AC edges
	C 22	54/3680	0.015	860	869	868	865
		54/2160	0.025	841	819	819	812
		54/1232	0.044	784	738	737	727
		54/240	0.225	380	273	275	321
	C22	-	-	-6359	-5725	-5816	-5973
			Hexagon void -ZZ edges
	C 22	24/3680	0.0065	935	934	933	935
		24/2800	0.0086	930	929	927	929
		24/1232	0.0195	903	899	897	899
		24/240	0.100	732	722	713	716
	C22	-	-	-2760	-2870	-3067	-2760
			Hexagon void -AC edges
	C 22	42/1232	0.0341	857	848	857	848
		42/448	0.0938	725	706	724	698
		42/240	0.1750	580	557	585	542
		42/112	0.3750	343	321	353	295
	C22	-	-	-2824	-3068	-2829	-3085
			Crack void -ZZ edges
	C 22	46/3680	0.0125	811	814	812	813
	C22	-	-	-11374 -11152 -11309 -11187

Table 4 :

 4 Effective elastic constants C 22 and C22 in MPa-Comparisons between FEM (with and without edge effect), MD and Mori Tanaka (MT) estimates.

Appendix A. Relation between the void coefficient C and the heterogeneity problem with edge effects

Let us consider the heterogeneity problem where a single nanovoid (index v) of edge stiffness E e is embedded in an infinite matrix of bulk stiffness C b subject to remote strain E. Solving the problem will yield the localization tensor L v and equivalent elastic stiffness tensor C v of the nanovoid as follows

where σ v and ε v stand for the average stress and strain of the nanovoid and : for double contracted product.

From the results of the above heterogeneity problem, we can estimate the effective elastic stiffness tensor C of materials containing nanovoids of volume fraction f by Mori-Tanaka scheme

In the case where f → 0, we obtain the asymptotic behavior of dilute scheme

Using the definition of C in (9), we have .4)


Diddahally R Govindaraju

Hideki Innan

Reiner A Veitia
email: reiner.veitia@ijm.fr

The Muller's ratchet and aging

Keywords: Aging, senescence, mutation accumulation, somatic cell-lineages, Muller's ratchet, fitness decay, frailty, morbidity

Aging entails an irreversible deceleration of physiological processes, altered metabolic activities and a decline of the integrity of tissues, organs and organ systems.

The accumulation of alterations in the genetic and epigenetic spaces has been proposed as an explanation for aging. They result, at least in part, from DNA replication and chromosome segregation errors due to cell division during development, growth, renewal and repair. Such deleterious alterations, including epigenetic drift, irreversibly accumulate in a stepwise ratchet-like manner, and reduce cellular fitness, similar to the process known as Muller's ratchet. Here, we revisit the Muller's ratchet principle applied to aging of somatic cell populations and discuss implications for understanding the origins of senescence, frailty and morbidity.

Genetic and epigenetic somatic variation as a catalyst of senescence and aging

Aging entails an irreversible deceleration of physiological processes at the cellular level. It is accompanied by an altered metabolic activity and a decline of the integrity of tissues, organs and organ systems [START_REF] Kirkland | Cellular Senescence: A Translational Perspective[END_REF]. The cumulative impact of such deterioration is paralleled by a functional decline and affects viability and reproduction.

In principle, senescence and aging accompany differentiation, development and growth, a set of processes that are initiated with the formation of the zygote (See Glossary) [START_REF] Vijg | Aging of the genome: the dual role of the DNA in life and death[END_REF].

Thus, aging represents cumulative and deteriorative processes impacting on the architecture, development and growth of cell populations and individuals at all levels of the Genotype-Epigenetic-Phenotype (G-E-P) spaces with the passage of time [START_REF] Lewontin | The units of selection[END_REF].

From a gerontological perspective, however, aging is primarily restricted to postreproductive stages during which, many organs and organ systems of individuals show both physiological and functional decline, collectively termed frailty. Specifically, cellular frailty encompasses dynamic and stochastic processes of accumulation of deficits at the subcellular and cellular levels [START_REF] Mitnitski | Accumulation of deficits as a proxy measure of aging[END_REF][START_REF] Rockwood | Fifteen years of progress in understanding frailty and health in aging[END_REF]. Furthermore, independent or cumulative systemwide decline of physical and physiological factors contribute to the manifestation of frailty of the whole organism [START_REF] Van Deursen | The role of senescent cells in ageing[END_REF]. Thus, senescence, frailty and morbidities may share common origins [START_REF] Villacampa-Fernández | Frailty and multimorbidity: Two related yet different concepts[END_REF]. For clarity, although aging and senescence are often used interchangeably, we equate senescence to irreversible cessation of mitotic activities of normal somatic cells [START_REF] Melk | Cell senescence and its implications for nephrology[END_REF]. Similarly, because somatic mutations potentially start accumulating from the first mitotic division of the zygote, we define "aging as all possible changes between conception and death" [START_REF] Vijg | Aging of the genome: the dual role of the DNA in life and death[END_REF].

Several theories have been proposed to explain aging. One of them is based on the accumulation and amplification of inherited and acquired (somatic) genetic variants (mutation accumulation theory (MA) [START_REF] Szilard | ON THE NATURE OF THE AGING PROCESS[END_REF][10][11]). Another theory by Muller [START_REF] Muller | Our load of mutations[END_REF] emphasized the negative and cumulative effects of such accumulating mutations or mutation load (ML) that impose a burden, "measured in terms of reduced fitness, but felt in terms of death, sterility, illness, pain and frustration" [START_REF] Crow | Genetic Loads and the Cost of Natural Selection[END_REF]. Clearly, both MA and ML result from sequential cell division generated by growth and development, renewal and repair, which inevitably engenders DNA replication and chromosome segregation errors. Thus, the burden of somatic variants is time-dependent and increases with age.

Both MA and ML theories, subtleties aside, converge on the adverse effects that genomic variation exerts on somatic and reproductive tissues and, ultimately, on health and longevity. Accumulation of somatic mutations is initiated at the very first cell division, and numerically increase through successive mitotic divisions. Therefore, we assume that biological aging is initiated at the time of zygote formation, a view consistent with demographic and genetic principles [START_REF] Ricklefs | Life-history connections to rates of aging in terrestrial vertebrates[END_REF][START_REF] Innan | Genetic and epigenetic Muller's ratchet as a mechanism of frailty and morbidity during aging: a demographic genetic model[END_REF]. The MA process is a gradual and ubiquitous feature among all differentiating cell lineages within an organism [START_REF] Blokzijl | Tissue-specific mutation accumulation in human adult stem cells during life[END_REF] and is comparable to what happens in asexual or clonally-reproducing organisms [START_REF] Muller | Some genetic aspects of sex[END_REF]. Each additional de novo deleterious variant [START_REF] Goldmann | De Novo Mutations Reflect Development and Aging of the Human Germline[END_REF] may have both cumulative and multiplicative effects at various levels of organismic organization: genome instability, molecular heterogeneity, altered gene expression, impaired inter-cellular communication, tissue disintegration, vulnerability to stress, etc, which may influence both fitness [START_REF] Govindaraju | Evolutionary genetic bases of longevity and senescence[END_REF] and health span [20,[START_REF] Rattan | Biogerontology: research status, challenges and opportunities[END_REF].

Genetic variants roughly accumulate at a rate of 1-2 per cell division during embryogenesis [START_REF] Lee-Six | Population dynamics of normal human blood inferred from somatic mutations[END_REF]. Thus, cell division over decades entails the accumulation of billions of variants in every tissue and organ. The extent of this phenomenon can be appraised in an adult human who is composed of about 37 trillion cells [START_REF] Bianconi | An estimation of the number of cells in the human body[END_REF], all of which stem from a single cell -the zygote [START_REF] Evrony | One brain, many genomes[END_REF]. The Muller's ratchet principle suggests that genomic alterations accumulate in a stepwise ratchet-like manner (Figure 1a), especially in asexual lineages [START_REF] Felsenstein | The evolutionary advantage of recombination[END_REF][START_REF] Muller | THE RELATION OF RECOMBINATION TO MUTATIONAL ADVANCE[END_REF], and in non-recombining genomes or genomic regions. This process eventually leads to an increase in the average number of unfavorable variants present in the genome of each cell [START_REF] Felsenstein | The evolutionary advantage of recombination[END_REF] entailing a gradual decline of fitness of individual cells and tissues. Figure 1b shows a modified Muller plot [START_REF] Muller | Some genetic aspects of sex[END_REF], which is a convenient way to represent this process over time.

The epigenome also changes across cell generations [START_REF] Field | DNA Methylation Clocks in Aging: Categories, Causes, and Consequences[END_REF]. For instance, aging mammalian cells display generalized DNA hypomethylation, primarily in DNA repeats, and local hypermethylation concentrated at promoters. This epigenetic drift results from an incomplete restoration of normal epigenetic patterns after DNA replication or repair. Indeed, it is known that epigenetic marks in twins are very similar during infancy but change with age, impacting gene expression. The impact of epigenetic variation on aging has also to be considered along with that of genetic alterations because of its higher frequency and because epigenetic drift causes departures from the norm by many orders of magnitude [START_REF] Fraga | Epigenetic differences arise during the lifetime of monozygotic twins[END_REF][START_REF] Fraga | Epigenetics and aging: the targets and the marks[END_REF]. Loss and gain of methylation during replicative senescence are similar to those observed in cancer. Thus, the epigenome of senescent cells might promote malignancy, if such cells escape the proliferative barrier [START_REF] Cruickshanks | Senescent cells harbour features of the cancer epigenome[END_REF][START_REF] Day | Differential DNA methylation with age displays both common and dynamic features across human tissues that are influenced by CpG landscape[END_REF].

From the perspective of the G-E-P map [32,33], it is reasonable to suggest that the lifelong accumulation of genetic and epigenetic deficits in cells and tissues alters physiological homeostasis and ultimately leads or at least contributes to aging [START_REF] Clegg | Frailty in elderly people[END_REF].

Along with the Muller's ratchet principle applied to aging, we postulate that the frailty phenotypes that arise with age are a function of the load of harmful genetic and epigenetic variants carried by the individual and its components. In short, in a demographic sense, the degree of frailty is linked to the (epi-) mutational burden carried by the various cell lineages in relation to their age and stage of development [START_REF] Muller | Our load of mutations[END_REF][START_REF] Crow | The origins, patterns and implications of human spontaneous mutation[END_REF].

The mechanisms by which mutations of clonal origin influence aging and disease remain obscure [36]. Thus, ideas from evolutionary genetics can be extended to gain insights on some of the enduring questions in aging and human health. Although (epi-) genetic bases of the relationship among cellular aging, cancer and Muller's ratchet has been suggested by various investigators, detailed population and demographic genetic analysis of this phenomenon in terms frailty and morbidity are still required. Thus, we have extended here the Muller's ratchet principle to the biological aging process in somatic tissues [START_REF] Innan | Genetic and epigenetic Muller's ratchet as a mechanism of frailty and morbidity during aging: a demographic genetic model[END_REF] and discuss its implications for understanding the origins of frailty, senescence and morbidity.

Muller's ratchet model applied to aging

The Muller's ratchet in relation to cell proliferation and tissue formation can be modeled as Moran birth-death demographic process; which describes stochastic birth and death of individuals in finite populations. Accordingly, de novo genetic variants, with varying levels of deleterious effects, differentially accumulate among cells and tissues during the course of development. The Moran model allows us to explore the health (viability) consequences of such mutation-bearing clonal lineages. As the individual is a mosaic of cells and lineages of cell populations with different sets of somatic variants, heterogeneity among them also increases [37] leading to a spatial structuration [START_REF] Kryazhimskiy | Population subdivision and adaptation in asexual populations of Saccharomyces cerevisiae[END_REF]. By extension, we postulate that each specific cell population could show a range of viability fitness in adult tissues [START_REF] Arrojo E Drigo | Age Mosaicism across Multiple Scales in Adult Tissues[END_REF], which might have a translation in the existence of "ageotypes" at the population level [START_REF] Ahadi | Personal aging markers and ageotypes revealed by deep longitudinal profiling[END_REF]. Although this assumption is reasonable, experimental evidence for fitness differentials among cell lineages in relation to the accumulation of genetic variants in humans is yet to be established [41].

Recent studies, however, point toward such a possibility [START_REF] Hernando-Herraez | Ageing affects DNA methylation drift and transcriptional cell-to-cell variability in mouse muscle stem cells[END_REF][START_REF] Yamamoto | Large-Scale Clonal Analysis Resolves Aging of the Mouse Hematopoietic Stem Cell Compartment[END_REF].

The formal principles behind the model can be outlined as follows. We can reasonably consider that the cell is the unit of differentiation, development, growth and regeneration [START_REF] Jaenisch | Stem cells, the molecular circuitry of pluripotency and nuclear reprogramming[END_REF][START_REF] Brenner | Sequences and consequences[END_REF] and that tissues and organs are mosaics of cells carrying different mutation loads. Our model starts from a single "stem" cell through differentiation, development and growth into a mature individual. Thus, a population of N cells at any given time point results from a single cell. The model also encompasses gradual physiological decline. Each cell can be characterized by three time-dependent parameters, its rate of division (b), its functionality or healthiness (f) and its death rate (d), which depend on the number and impact of (epi-)genetic variants that the cell has accumulated over its entire life. However, keep in mind that mutations can occur not only through cell division but also replication-independently (BOX I). These three parameters may be affected by developmental stages and the environment. If we consider the negative effects of accumulating deleterious variants [START_REF] Ohta | Slightly deleterious mutant substitutions in evolution[END_REF]47], f and b are supposed to decrease and d increase with time. Assuming that f0 is the parameter at birth, the effect of mutation accumulation is given by f

= f0 (1-δf1)(1-δf2)…(1-δfi).(1-δfn),
where δfi is the reduction in the functionality caused by the i-th mutation. A similar reasoning applies to the other parameters.

After a lag-phase, the number of cells (N) rapidly increases up to a value K, representing the adult population size (growth phase). The cell population can maintain this size K when cells are healthy enough (so that b>d, stationary phase). As aging proceeds, the cell population becomes unhealthy and N declines (i.e., K cannot be maintained anymore). This decline phase appears when cell birth rate (b) becomes smaller than the population death rate (d). Throughout the whole process, the average of the number of accumulated mutations per cell (Save) increases, especially rapidly in the growth phase. BOX I provide further details on the principles and consequences of the model.

What the model says

The growth of a tissue from pluripotent stem cells to its adult size, relies on a large number of cell divisions, which are associated with the accumulation of numerous potentially deleterious variants among cells. Damage at the molecular level includes both benign and pathogenic variants that co-exist with an array of inherited variants [48]. The latter (constitutional) ones exert in most cases more powerful effects relative to somatic variants as they are already present since conception and affect all the cells of the individual. Somatic variants, in concert with constitutional variants will influence health, cancer and the course of hereditary diseases [49,[START_REF] Jones | Distinct DNA methylation patterns of cognitive impairment and trisomy 21 in Down syndrome[END_REF].

Our model and simulations suggest that the cell birth and death rates are crucial to determine the pattern of aging. In tissues like brain, where cells experience a relatively low death rate, genetic variants mainly accumulate during early stages of development, after which replication-independent mutations are the main contributors.

In contrast, in a tissue with a relatively high rate of cell turnover, (epi-)genetic variants keep accumulating all the way even during the stationary phase, so aging proceeds during maturity. These differences are illustrated in Figure I, which shows typical patterns of aging in normal tissues, together with an abnormal case. In both cases, the cell population size rapidly increases up to the adult size, arbitrarily set at K=100. Through the growth phase, the average number of accumulated somatic variants, Save, increased to ~4000 in both cases, resulting in a decline of cell healthiness (f, represented by the yellow-to-red transition in Figure I). The major difference between the two cases displayed is the death rate (d). In the normal aging case, with d=0.5b (Figure Ia), turnover of cells occurs quite often, leading to accumulation of new somatic variants after maturity (i.e., stationary phase). In the normal-aging case, Save keeps increasing in the stationary phase, resulting in further decrease of b and f and increase of d. Eventually d exceeds b. After this time point (denoted by TB=D), the system enters in the decline phase. In progeroid syndromes, such as the Hutchinson-Gilford syndrome [START_REF] Ahmed | Hutchinson-Gilford Progeria Syndrome: A Premature Aging Disease[END_REF], TB=D is small. Figure Ib shows a simulation where the curves cross around 25 years of age, which closely agrees with the average life expectancy of progeria patients [START_REF] Merideth | Phenotype and course of Hutchinson-Gilford progeria syndrome[END_REF]. In sum, a cell population can maintain a stable size when b>d but once d exceeds b, the population size starts declining and cells are no longer able to fill in all the empty space left by dead ones. In this decline phase, the aging process is accelerated, accompanied by a decline in cell population size. Thus, according to our model, b and d are key factors determining the pattern of aging among cell populations.

There is some experimental data supporting this concept in telomere dysfunctional mice. Indeed, both a slowdown in cell birth rates as well as an increase in cell death rates aggravate organism aging in the context of short telomeres [START_REF] Choudhury | Cdkn1a deletion improves stem cell function and lifespan of mice with dysfunctional telomeres without accelerating cancer formation[END_REF]54].

Our simple model recapitulates growth, stability and decline of cell populations both within organs and the organism. It is in agreement with molecular data [START_REF] Lee-Six | Population dynamics of normal human blood inferred from somatic mutations[END_REF]55] as well as with other models of aging [START_REF] Engelman | A Quiescent Phase in Human Mortality? Exploring the Ages of Least Vulnerability[END_REF][START_REF] Levitis | Before senescence: the evolutionary demography of ontogenesis[END_REF]. Cell deterioration closely tracks the number of "clicks" of the ratchet because cell division is its driving force. Not surprisingly, genetic variants appeared during early stages of development play a critical role in the aging of terminally differentiated tissues with minimum renewal. This is consistent with recent data suggesting that variants that appear before birth in the brain are subsequently amplified and can either cause or influence neurodegeneration [55,[START_REF] Bae | Different mutational rates and mechanisms in human cells at pregastrulation and neurogenesis[END_REF]. Indeed, in agreement with the Muller's ratchet principle, Lodato et. al 2018 reported the presence of approximately a few thousand single nucleotide variants by 5 months, steadily increasing after 40 and exponentially after 80 [55]. In contrast, tissues such as skin, blood or small intestine, with a high renewal rate, accumulate genetic variants all the way from early development to death. In concordance with these observations, clonal hematopoiesis as a factor of aging and disease has been proposed [36]. Our model also explains the almost star-shaped tree (i.e., with many short branches connected at an internal node) of hematopoietic stem cells [START_REF] Lee-Six | Population dynamics of normal human blood inferred from somatic mutations[END_REF], which grows rapidly, reaches a plateau while accumulating genetic variants by producing additional progenitor cells. As a rule, variants that arise very early during development have greater influence on aging than those appearing later in life because the former, in concert with inherited variants, can affect most tissues and organs. Accordingly, specific cell populations may accumulate an excessive number of variants and diverge with time, contributing to reduced viability fitness.

Throughout, we have assumed that accumulation of mutations with varying degrees of pathogenicity, from benign to clearly damaging, influence a range of cellular processes. However, rare 'advantageous' mutations arising in a few senescent cells could confer 'greater' fitness [START_REF] Gyllenberg | On fitness in structured metapopulations[END_REF][START_REF] Eyre-Walker | The distribution of fitness effects of new mutations[END_REF]. Such cells may consume large amounts of glucose for growth, survival, proliferation and long-term maintenance and turn into cancer cells [START_REF] Liberti | The Warburg Effect: How Does it Benefit Cancer Cells?[END_REF]. These plausible sequences indicate that cellular senescence and cancer may represent two Janus Faces of the same coin. Thus, an effective management of aging processes could perhaps help prevent cancer.

Concluding remarks and perspectives

The Muller's ratchet principle provides a theoretical foundation to study the origin and augmentation of vulnerability, frailty, morbidity and mortality of cells, tissues, organs and organ systems with age. This principle focuses on predictable effects of the accumulation of (epi-)genetic variants in line with the mutational, molecular, and cellular clocks that represent mutation rates, telomere shortening and the Hayflick process [62]. Moreover, it is not surprising that chromatin modifications, which link the G-P spaces have been shown to predict biological aging and longevity [START_REF] Field | DNA Methylation Clocks in Aging: Categories, Causes, and Consequences[END_REF][START_REF] Horvath | DNA methylation age of human tissues and cell types[END_REF]. Therefore, we postulate a predicable link between the pace of clicks and ticks of the Muller's ratchet and aging.

Clinical frailty involves a cumulative decline in anatomical and physiological functions [64]. It is apparent at multiple levels of biological organization: genome, epigenome, tissues, organs and the organism. Frailty is attributed to a "lifelong accumulation of molecular and cellular damage" [START_REF] Rockwood | Fifteen years of progress in understanding frailty and health in aging[END_REF][START_REF] Clegg | Frailty in elderly people[END_REF] leading to dysfunction of organs and the organism. Clearly, although frailty is primarily recognized as a geriatric condition, it can affect multiple components of the human body during the entire lifespan of individuals [START_REF] Buchman | The community of the self[END_REF].

How could the information discussed here be used in a clinical setting? It has become routine to classify genetic variants into a finite number of classes such as benign, pathogenic, uncertain, actionable, etc [START_REF] Adzhubei | A method and server for predicting damaging missense mutations[END_REF]. Instead, in some cases, and in accordance with the Haldane-Muller principle, which suggests that the average fitness of a population depends on the cumulative effect of the entire spectrum of genetic variants and not on the severity of individual ones. Accordingly, guidelines developed for clinical diagnostic purposes solely based on the effect of individual variants, needs to be reevaluated in the context of aging and genetic disorders that influence healthy aging. Detailed investigations into the origins, distribution and effects of somatic variants could shed light on many medically unexplained symptoms and even new disorders on the basis of their interaction with inherited variants and the degree of mutation load [START_REF] Calimport | To help aging populations, classify organismal senescence[END_REF].

From the perspective of precision medicine, this approach holds promise to effectively identify individuals at higher risk of developing single or multiple morbidities and for devising new preventive measures such as tissue-and organ-specific senolytics [START_REF] Kirkland | Cellular Senescence: A Translational Perspective[END_REF][START_REF] Lehmann | Senolytic drugs target alveolar epithelial cell function and attenuate experimental lung fibrosis ex vivo[END_REF].

Clearly, some concepts of evolutionary genetics have the potential not only to illuminate enduring questions in human health and longevity but also in public health (see also To gain insights into the principles and consequences of the model, let us first focus on the normal aging case. A simulated pattern of aging can be generated by assuming N=100, b=0.1, d=0.05, a per-genome mutation rate = 1 and replicationindependent mutations are ignored for simplicity. Initially, during development and growth (growth phase, Figure Ia), when b>>d, the global contribution of cell death d is small and cell division prevails. This leads to an increase of population size (N). Then, the system reaches a stationary phase, becoming mature at around 18 years old.

Throughout this process, we assume that when a cell dies, the dead cell is removed. This is supposed to be followed by a cell division to fill in the 'empty space', as long as b>d. Along with the accumulation of mutations, the functionality or healthiness (f) decreases, which is presented by the red color in the topmost panel. However, as cell accumulates deleterious (epi-)genetic variants through division, b would decrease and d increases. All the way up to the stationary phase, a process analogous to a Muller's ratchet in asexual populations is taking place [START_REF] Muller | Some genetic aspects of sex[END_REF], especially when the cell population size (N) is small. The ratchet proceeds with age and, at some time point (Tb=d), b becomes as smaller as d. After this point, the empty space left by a dead cell may not be filled by a new one (with a probability 1-b/d). As a consequence, the size of the cell population decreases (i.e., becomes < K), leading to the formation of interstitial spaces and eventual tissue disintegration (decline phase) [START_REF] Haubold | Acute effects of desmin mutations on cytoskeletal and cellular integrity in cardiac myocytes[END_REF][START_REF] Rodier | Four faces of cellular senescence[END_REF]. The average number of division-linked variants (Save) is expected to dramatically increase during the rapidgrowth phase required for development, whereas during the stationary phase, it increases linearly. Rapid aging can be modeled by setting d at 0.06. In this case, the pattern drastically changes and the stationary phase is shortened to about 25 years implying a rapid decline of the cell population (Figure Ib). In the above model, we assumed all mutations occur through cell division to obtain Save(t), and ignored replication-independent mutations. To take the latter into account, let v be the rate of production of such mutations. Thus, it is straightforward to derive S'ave(t) that incorporates both types of mutations as S'ave(t) = Save(t) + vt.

FiguresFigure 1 .

 1 Figures Figure 1. Accumulation of (epi-)genetic alterations upon cell division during development and subsequent stages of life. (a) Production of cell lineages with different mutation loads. (b) Modified Muller plot describing the same process in time. It combines information on the succession of different genotypes/phenotypes (horizontal axis) and their abundances (vertical axis). Different cell genotypes are represented by areas with different colors. Each genotype originates by mutation of the parental one.

Figure I .

 I Figure I. Typical patterns of aging obtained by mathematical simulation. (a) Normal aging was simulated assuming N=100, b=0.1, d=0.05 and a per-genome mutation rate = 1. During the growth phase (b>>d), cell death d is small and cell division prevails leading to an increase of N (green rectangle). Afterwards, the system reaches a stationary phase (yellow rectangle). When b d, a dead cell is removed and replaced by cell division. This process entails accumulation of deleterious (epi-)genetic variants (thus, b decreases and d increases). After Tb=d (vertical red arrow), the empty space left by a dead cell is not be filled in by a new one and the size of the cell population decreases (decline phase, orange). The functionality of the tissue, coded in red, in the topmost panel decreases with time. (b) In the case of fast aging (i.e., d=0.06) the span of the stationary phase is shortened, leading to an earlier decline of the cell population.

ACKNOWLEDGEMENTS:

We dedicate this paper to Richard Lewontin and Tomoko Ohta and for their prescient ideas on units of selection and mutational effects, respectively. DRG thanks Peter Ellison, Nir Barzilai, Jan Vijg for support, and Drs.

Caleb Finch, Robert Perlman, Annapurna Poduri, Sri Raj and Alan Templeton, for suggestions. RAV thanks the Centre national de la recherche Scientifique and Université Paris Diderot for financial support. We thank the three reviewers for many thoughtful comments and suggestions. We apologize for our inability to cite many relevant publications due to space limitations.

Glossary

Clonal hematopoiesis: Stem or progenitor cells in the bone marrow that give rise to genetically distinct subpopulations of blood cells. Such subpopulations are supposed to be "clonally" derived from single founder cells. It may occur in healthy people but it is often associated with hematologic and other diseases. Phenotype (G-E-P) map (spaces): metaphor used to describe the architecture and interaction between genotype and phenotype of traits via epigenetic modulation that can be influenced by the environment.

Genotype-Epigenetic-

Haldane-Muller principle:

The reduction in the average fitness of a population due to recurrent mutation depends only on mutation rate and is independent of the mild or harmful effect of individual variants. T b=d T b=d


Guillaume Laneelle

Mahé

Guilcher

; Tollenaere

; Métairie

pressure measurement readings using Automated ankle and toe measuring devices for diagnosis of lower extremity peripheral artery disease

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

ABSTRACT:

OBJECTIVE: Pressure measurement is a key component in the diagnosis of lower extremity peripheral artery disease (PAD) but is technically challenging and time-consuming for nonvascular specialists, thus hindering its wider implementation. The aim of this study was to assess the proficiency of students at obtaining satisfactory ankle or toe pressure readings for PAD diagnosis using two automated devices. METHODS: Medical students followed a training session after which they performed ankle and toe pressure measurements to calculate the ankle-brachial index (ABI) using the MESI ABP MD® device, and the toe-brachial index (TBI) using the SYSTOE® device. Blinded vascular specialists took the same measurements. Use of the automated devices was considered satisfactory when a valid reading was measured in as few attempts as possible. A comparison was made of each student's proficiency at performing valid ankle and toe pressure measurements. The secondary objective was to compare the readings taken by the vascular specialists with those of the students. RESULTS: Forty-three medical students were included. Mean number of attempts was 1.23 +/-0.48 with the MESI ABP MD® device and 1.44 +/-0.55 with the SYSTOE® device (p = 0.04). There was no statistically significant difference between ABI readings taken by the students and those taken by the vascular specialists, 1.17 [0.90; 1.39 ] versus 1.18 [0.86; 1.39 ] (p = 0.33), contrary to TBI readings 0.70 [0.22; 1.74] versus 0.72[0.23; 1.16] (p = 0.03).

INTRODUCTION

Lower extremity peripheral artery disease (PAD) is a common medical condition affecting almost 202 million people worldwide [START_REF] Fowkes | Comparison of global estimates of prevalence and risk factors for peripheral artery disease in 2000 and 2010: a systematic review and analysis[END_REF]. The European Society of Cardiology (ESC) and the American Heart Association (AHA) recommend ankle-brachial index (ABI) pressure measurement as a first-line non-invasive examination technique to investigate and establish diagnosis of PAD [START_REF] Gornik | AHA/ACC Guideline on the Management of Patients With Lower Extremity Peripheral Artery Disease: A Report of the American College of Cardiology/American Heart Association Task Force on Clinical Practice Guidelines[END_REF][START_REF] Aboyans | ESC Guidelines on the Diagnosis and Treatment of Peripheral Arterial Diseases, in collaboration with the European Society for Vascular Surgery (ESVS): Document covering atherosclerotic disease of extracranial carotid and vertebral, mesenteric, renal, upper and lower extremity arteriesEndorsed by: the European Stroke Organization (ESO)The Task Force for the Diagnosis and Treatment of Peripheral Arterial Diseases of the European Society of Cardiology (ESC) and of the European Society for Vascular Surgery (ESVS)[END_REF].

Although pressure measurement plays a major role in establishing diagnosis of PAD, little research has focused on how this measurement technique is taught(4) and the knowledge of French medical students on this subject leaves much to be desired [START_REF] Stivalet | Noninvasive Peripheral Artery Disease Screening Tools: A Deficient Knowledge among French Vascular Residents from 4 Medical Schools[END_REF][START_REF] Lanéelle | Vascular medicine residents lack adequate training for limb pressure measurement: A nationwide survey in France[END_REF] .

Wider implementation of this technique however is hampered, especially among non-vascular physicians, by its high requirements in terms of technical and learning skills. A previous study revealed that even after following a practical training course, only 11 of 20 students succeeded in measuring systolic pressure and even then had lost their proficiency six months later through lack of practice in the intervening period [START_REF] Donnou | Medical students' proficiency in performance of the resting ankle-brachial index is not sustained at 6 months[END_REF][START_REF] Donnou | How to become proficient in performance of the resting ankle-brachial index: Results of the first randomized controlled trial[END_REF].

In terms of screening, follow-up and overall coordination primary care physicians are key players in the management of PAD, which represents a serious public health concern with regard to morbidity and mortality rates and financial impact [START_REF] Diehm | Mortality and vascular morbidity in older adults with asymptomatic versus symptomatic peripheral artery disease[END_REF][START_REF] Hirsch | National health care costs of peripheral arterial disease in the Medicare population[END_REF]. And yet wider implementation of ABI measurement in primary care settings is severely hindered by timerestricted primary care consultations, insufficient clinical criteria for the diagnosis of PAD [START_REF] Khan | Does the clinical examination predict lower extremity peripheral arterial disease?[END_REF], and the technical and practical skills required to ensure reliable continuous-wave Doppler readings.

Several automated methods for the detection of PAD have been discussed [START_REF] Wohlfahrt | A novel oscillometric device for peripheral arterial disease screening in everyday practice. The Czech-post MONICA study[END_REF], with reference to certain factors such as diagnostic thresholds or particular populations (for example patients with diabetes) that are liable to affect their validity [START_REF] Herráiz-Adillo | Factors affecting the validity of the oscillometric Ankle Brachial Index to detect peripheral arterial disease[END_REF][START_REF] Mayr | A randomized cross-over trial in patients suspected of PAD on diagnostic accuracy of ankle-brachial index by Dopplerbased versus four-point oscillometry based measurements[END_REF]. These methods would, however, make it much quicker and easier for non-specialists to measure systolic pressure and would open up new prospects for wider implementation of screening for PAD. This is especially true given that the training methods for these techniques are potentially far simpler than those of previously validated systolic pressure techniques. Moreover, medical students as budding physicians are an ideal population to target for greater awareness of ABI measurement and its training programs [START_REF] Chaudru | Ankle brachial index teaching: A call for an international action[END_REF]. On another point, training resources for these automated methods have yet to be investigated.

The main aim of the present study was to assess the proficiency of medical students at obtaining satisfactory pressure readings (i.e. valid reading obtained in as few attempts as possible) for diagnosis of PAD using two automated devices: the MESI ABP MD® and the SYSTOE®. Secondary aims were to assess i) measurement duration, and ii) concordance of readings taken by the vascular specialists with those taken by the students with the various devices

MATERIALS AND METHODS

Study Design

This was a single center controlled prospective study conducted at the Vascular Medicine Unit of the university hospital of Rennes in France from December 2018 to February 2019. It was approved by the ethics committee of Rennes Hospital (no. 16.150). Second to 6th year medical students with no prior training or knowledge of the devices involved were invited by electronic mail to participate in the trial having first provided written consent. Patients with scheduled Doppler ultrasound appointments were included and oral consent was provided.

Patients with wounds were excluded.

Initial training session

The initial hour-long training session, dispensed by a vascular specialist, was composed of an introduction to the pathophysiology of PAD, an update on PAD guidelines and introduction to standard Doppler ABI measurement. The students then viewed a video presentation of how to use each device. The MESI ABP MD® simultaneously records systolic ankle and brachial pressures using three color-coded cuffs (one on the arm and the other two on the ankles with the tubes pointing upwards) [START_REF] Špan | Detection of peripheral arterial disease with an improved automated device: comparison of a new oscillometric device and the standard Doppler method[END_REF]. Systolic pressure is calculated within minutes and does not require pre-measurement resting. The SYSTOE® employs photopletysmography to record systolic pressure in the hallux (TBI) after first taking the temperature of the toe with an infrared thermometer [START_REF] Špan | Detection of peripheral arterial disease with an improved automated device: comparison of a new oscillometric device and the standard Doppler method[END_REF][START_REF] Pérez-Martin | Validation of a fully automatic photoplethysmographic device for toe blood pressure measurement[END_REF]. Indeed, when toe temperature is low, the measured pressure may be falsely low [START_REF] Sawka | Effect of temperature on digital systolic pressures in lower limb in arterial disease[END_REF]. A cuff is placed at the base of the toe with the tube pointing downwards. The sensor is positioned at the top of the toe by means of an adhesive pad and pressure is applied to the sensor by the operator during deflation then released. The diagnostic threshold adopted for PAD was 0.70 [START_REF] Høyer | The toe-brachial index in the diagnosis of peripheral arterial disease[END_REF] regarding TBI and 0.90 regarding ABI [START_REF] Aboyans | Measurement and interpretation of the ankle-brachial index: a scientific statement from the American Heart Association[END_REF]. In the absence of any research validating the SYSTOE® threshold, a value of 0.70 was adopted.

Real-life patient assessment:

Each student was called individually to measure patient's pressure with both devices within one week to one month of the initial training session.

The procedure consisted of obtaining one pressure reading with the MESI ABP MD® and another with the SYSTOE® from each foot. Brief written instructions were provided to facilitate the smooth running of the procedure. The stopwatch used to time measurement acquisition was started once the first cuff had been put in place. The SYSTOE® was used in semi-automatic mode (in view of the fact that automatic mode requires applying two cuffs and this can be problematic depending on the specific features of certain toes) and the student was responsible for determining the inflection point of the curve. Arterial pressure was measured by an automatic device [START_REF] Stivalet | Exercise testing criteria to diagnose lower extremity peripheral artery disease assessed by computed-tomography angiography[END_REF].

A blinded vascular specialist present in the unit then took measurements following the same procedure and concluded by taking Doppler pressure measurements in line with standard procedure (i.e. right brachial artery, right posterior tibial artery, right anterior tibial artery, left posterior tibial artery, left anterior tibial artery, left brachial artery and right brachial artery) [START_REF] Aboyans | Measurement and interpretation of the ankle-brachial index: a scientific statement from the American Heart Association[END_REF][START_REF] Chaudru | Training to Perform Ankle-Brachial Index: Systematic Review and Perspectives to Improve Teaching and Learning[END_REF]. The highest brachial and ankle pressure values were used to calculate the ABI.

Patient characteristics:

During the 10 minute pre-measurement rest period, the following information was gathered: cardiovascular risk factors, previous history of kidney failure or cardiovascular disease, anticoagulant treatment (vitamin K antagonists or direct oral anticoagulants), weight and height [START_REF] Sadler | The effect of pretest rest time on automated measures of toe systolic blood pressure and the toe brachial index[END_REF].

Statistical Analysis:

Descriptive analysis was used to investigate each variable collected. Quantitative variables were expressed as mean, standard deviation, median, and maximal and minimal values.

Qualitative variables were expressed as counts and percentages. Systolic pressure readings from the various devices and time taken by each operator were compared using the Student's t-test for paired data that followed a Gaussian distribution and the Wilcoxon signed-rank test for paired data where this was the opposite. A linear mixed effects model was used to assess time spent on each device and operator status. Measurement concordance was summarized using the Pearson correlation coefficient, the Bland-Altman plot(24) and Lin's concordance correlation coefficient [START_REF] Lin | A Concordance Correlation Coefficient to Evaluate Reproducibility[END_REF]. The latter simultaneously calculates correlation with and deviation from perfect agreement between the two measuring devices. Statistical analysis was conducted using R software (version 3.5.3) and the Tidyverse packages (R core Team 2019, Hadley Wickham 2017). A p-value of p < 0.05 was regarded as statistically significant.

RESULTS

Forty-three medical students participated in measuring systolic pressure (37.2% 2nd year students, 20.9% 3rd year students, 16.3% 4th year students, 7.0% 5th year students and 18.6% 6th year students).

Forty-three patients (67% male) were included. Their characteristics are shown in Table 1.

Prevalence of lower extremity PAD in the 43 subjects as measured by the MESI ABP MD®, the SYSTOE® and Doppler ultrasound was 4.6 %, 55.0 % et 11.0% respectively (according to vascular specialist data).

Proficiency of students at obtaining satisfactory ABI and TBI readings:

Mean number of attempts was 1.23+/-0.48 regarding the MESI ABP MD® and 1.44+/-0.55 (p = 0.04) regarding the SYSTOE® (Figure 1). Forty students (93%) used the MESI ABP MD® correctly and 31 (72%) used the SYSTOE® correctly (p = 0.28). Errors in relation to the MESI ABP MD® occurred due to incorrect placement (such as cuff tubes pointing in the wrong direction, incorrect cuff placement or incorrect cuff size). As for the SYSTOE® device, problems essentially arose from the positioning of the sensor and cuffs applied back to front.

Ninety-three percent of the students took toe temperature before using the SYSTOE®. The MESI ABP MD® screen displayed a total of 8 readings indicating PAD and 2 messages indicating inflation error (error E2).

Comparison of measurement duration on each device:

The different duration of the measurements are presented in figure 2. As determined by a multiple regression model, MESI ABP MD® measurement took 3.7 times less time than Doppler ultrasound (-3.713+/-0.170) and SYSTOE® measurement took 2.9 times less time (-2.913+/-0.171), demonstrating statistical significance. The vascular specialists on the whole took 1.2 times less time than the students irrespective of the device used (-1.238+/-0.130).

Comparison of systolic pressure readings

No statistically significant difference was found between ABI readings obtained by the students and those obtained by the vascular specialists with the MESI ABP MD®: 1.17 This study highlights the fact that the learning benefits of a single training session are not the same from one automated measuring method to the next. For example, the MESI ABP MD® system entailed less attempts at obtaining a satisfactory reading and was found to be more user-friendly than the SYSTOE®. Thus, a single hour-long session combined with a video presentation is not enough to acquire the skills required for toe measurement, unlike the MESI ABP MD®. Donnou et al. illustrated that students failed to obtain systolic pressure readings using continuous-wave Doppler ultrasound after a single theoretical session combined with a video presentation [START_REF] Donnou | How to become proficient in performance of the resting ankle-brachial index: Results of the first randomized controlled trial[END_REF]. Mahe demonstrated that the teaching of systolic pressure measurement in medical faculties was predominantly based on theoretical training [START_REF] Mahé | Ankle-brachial index measurement: Methods of teaching in French medical schools and review of literature[END_REF]. Lecturers in medicine keen for students to learn how to obtain reliable pressure readings should introduce the measuring devices into their theoretical training sessions. Automated measuring devices have greater benefits than standard Doppler techniques, whose measurement duration produced statistically lower values irrespective of the operator. The vascular specialists, however, outperformed the students in terms of measurement duration, emphasizing that practice is required. This is an important point in the light of increasing patient numbers and time-restricted consultations in general practice (mean time of 16.4 minutes) [START_REF] Deveugele | Consultation length in general practice: cross sectional study in six European countries[END_REF][START_REF] Iacobucci | GP appointments last less than five minutes for half the world's population[END_REF]. And yet it is precisely during these consultations that patients should have access to this type of screening. Moreover, correlation between the students and the specialists was only 0.56 with the MESI ABP MD® and 0.34 with the SYSTOE®. This may result from lack of experience, especially when using the SYSTOE® which requires a certain level of technical skill to obtain a reading and identify the point of inflection that determines the pressure value. This is moreover corroborated by Bland-Altman results that were indicative of lower measurement variability between the students and the vascular specialists with the MESI ABP MD®.

A further point of importance aside from learning is the reliability of the readings obtained for the diagnosis of PAD. The MESI ABP MD® and the SYSTOE® have already been endorsed in clinical trials over corresponding gold standard measurement methods, namely continuouswave Doppler ABI measurement and laser Doppler velocimetry respectively [START_REF] Chaudru | Ankle brachial index teaching: A call for an international action[END_REF][START_REF] Špan | Detection of peripheral arterial disease with an improved automated device: comparison of a new oscillometric device and the standard Doppler method[END_REF]. In the present study, only the performance of the MESI ABP MD® was compared with continuouswave Doppler ultrasound, yielding an extremely low Lin concordance coefficient of 0.18. One possible explanation is that MESI ABP MD® measurement was done first and Doppler measurement last. There was therefore a time lapse of at least 15 minutes between the two readings. Furthermore, it is important to note that our population had no links with PAD screening program but was a pre-selected, hospital population. In fact the present study accentuates the importance of selecting a method of diagnosis adapted to the population in question. The discrepancy found between PAD prevalence determined by the SYSTOE® and that determined by Doppler ultrasound may be attributed to the participation of 11 diabetic patients and 16 renal insufficiency patients (of whom 2 were patients with both diabetes and renal failure), meaning 25 patients (58%) from our population were potentially at risk of increased arterial stiffness. Hence the possibility of falsely normal systolic pressure readings in these patients.

The present study also indicated a slight upward bias in the systolic pressure readings obtained from the MESI ABP MD® versus Doppler ultrasound (mean ABI 1.18+/-0.12 versus 1.15+/-0.20). These results are consistent with those found in the literature and may stem from observer bias where Doppler ultrasound is concerned, such as the time required to perceive the signal and stop deflating the cuff [START_REF] Sawka | Effect of temperature on digital systolic pressures in lower limb in arterial disease[END_REF][START_REF] Iacobucci | GP appointments last less than five minutes for half the world's population[END_REF]. This raises the issue of whether lower extremity PAD detection thresholds using these automated tools should be redefined, as previously suggested by other authors [START_REF] Špan | Detection of peripheral arterial disease with an improved automated device: comparison of a new oscillometric device and the standard Doppler method[END_REF]. A meta-analysis by Verberk et al. [START_REF] Verberk | Automated oscillometric determination of the ankle-brachial index: a systematic review and meta-analysis[END_REF] expresses a preference for a cut-off in the region of 1 for the purposes of improving the PAD detection sensitivity in these devices.

Unlike Doppler ultrasound, the MESI ABP MD® provides no precise details on the artery used for recording, and placing the cuff on the right arm may be questionable in the event of subclavian artery stenosis, whose prevalence is estimated to be between 2% and 5% in the general population [START_REF] Shadman | Subclavian artery stenosis: prevalence, risk factors, and association with cardiovascular diseases[END_REF]. The scope of both the MESI ABP MD® and continuous-wave Doppler ultrasound is also limited in certain patient groups (such as diabetic, elderly or renal failure patients) where readings may be falsely reassuring, contrary to the SYSTOE® whose measuring technique is unaffected by increased arterial stiffness. This is exemplified by disparity in prevalence from one measuring tool to another.

Limitations

This study has several limitations. Firstly, each system involves different pressure measurement sites. Toe-brachial pressure is undoubtedly more difficult to measure for the inexperienced, even with an automated device. Our aim was to assess the training methods for these automated tools. As such, it was found that following an hour-long training session, the students were more proficient at measuring ABI. A further limitation is that the participating students all came from the same faculty of medicine, although we made very sure that they were novices in measurement to avoid any previous training interfering with the results. The level of study of the student cohort taking the measurements was diverse and it was impossible to assess any influence of this diversity on proficiency at taking measurements. On another note, it was unfeasible to randomize the order of assessment of the measuring tools. It is impossible to rule out any potential impact of this factor on the results. However, the students were tested first on the easier MESI ABP MD®, conceivably making them feel more confident about measuring with the SYSTOE® second. A final limitation accounting for variation in the readings was inter-operator variability regarding the vascular specialists when measuring ABI and TBI.

CONCLUSION

Following an hour-long theoretical training session, the medical students were more proficient at obtaining ABI readings using an automated device than at obtaining TBI readings and were able to execute both far more quickly than with the gold standard Doppler ultrasound method used by the vascular specialists. This research project has demonstrated that even where automated devices are concerned it is of fundamental necessity to adopt specific training methods for pressure measurement, without disregarding the importance of practical experience. The benefits of implementing routine use of these tools in primary care practice have yet to be assessed.

Table 1 legend: Hypertension was defined as patients receiving antihypertensive drugs or an arterial pressure >140/90 mmHg. Dyslipidemia was defined as patients receiving statins.

Diabetes was defined as patients receiving antidiabetic drugs. Renal insufficiency was defined as creatinine clearance <60ml/min. Cardiovascular diseases was defined as a history of coronary artery disease or peripheral artery disease.

  [0.90; 1.39] versus 1.18 [0.86; 1.39] (p = 0.33), unlike TBI readings with the SYSTOE®: 0.70 [0.22; 1.74] versus 0.72[0.23; 1.16] (p = 0.03). There were 20 missing ABI values in student data (23.3%) versus 22 (25.6%) in specialist data and 14 missing TBI values (16.3%) versus 15 (17.4%). The correlation coefficient (r) between the readings obtained by the students and those obtained by the vascular specialists was 0.56 using the MESI ABP MD® and 0.34 using the SYSTOE® p < 0.05. The Lin concordance correlation coefficient was 0.55 [0.36; 0.70] and 0.34 [0.12; 0.53] respectively. Correlation between systolic pressure readings obtained by the specialists and students per device is shown in Figure 3. With respect to the vascular specialists, the correlation coefficient between the MESI ABP MD® and Doppler ultrasound was 0.2 (p = 0.11) and the Lin concordance correlation coefficient was 0.18 [-0.04; 0.39]. DISCUSSION This is the first research project to focus on the efficacy of training in ABI and TBI measurement using automated measuring devices and its subsequent implementation by medical students in the diagnosis of PAD.

Figure 1 Figure 2

 12 Figure 1 Title: Proficiency of students at obtaining satisfactory Ankle brachial index (MESI device) and Toe Brachial index (SYSTOE) readings

Figure 3

 3 Figure 3 Title: Bland and Altman representation between systolic pressure readings obtained by the specialists and students per device.

Table 1 :

 1 Patients' characteristics

	Clinical characteristics

Acknowledgement: The authors thank the MESI company which loan the MESI Device. The MESI company had no role in the design and interpretation of the results of the study.

Funding: This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Conflict of interest: None