Stéphane Avril

Pierre Badel

Ambroise Duprey

Anisotropic and hyperelastic identification of in vitro human arteries from full-field optical measurements

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

It is well assessed that, despite biochemical and hemodynamical factors play a primary role in the development of most vascular disorders, solid mechanics models may contribute to understand their genesis and progression. The realism of models in solid mechanics depends significantly on the mechanical properties used as input parameters. Therefore, characterizing the biomechanical properties of arteries remains an essential issue.

In vivo measurements with ultrasounds or magnetic resonance imaging (MRI) techniques provide relevant information on the vascular behavior [START_REF] Slager | True 3-dimensional reconstruction of coronary arteries in patients by fusion of angiography and IVUS and its quantitative validation[END_REF], Masson et al., 2008, Avril et al., 2009] although they are not sufficient for a rigorous determination of arterial wall constitutive equations. To investigate the passive structural response of the arterial tissue, a large variety of in vitro experimental protocols are available [Humphrey, 2002].

Among them, inflation/extension of arterial segments are physiologically meaningful tests since the in vivo loading conditions may be reproduced and the native geometry is preserved [Hayashi, 1993].

Classically, for the analysis, it is assumed that the artery is a perfect cylinder and that the loading induces a homogeneous stress and strain distribution [Humphrey, 2002, Fung, 1993]. These assumptions provide a framework for deriving stress-strain curves and to fit them by appropriate constitutive models.

However, inverse approaches should generally be employed since stresses and strains are always heterogeneous, because of noncylindrical shape [Holzapfel, 2004, Humphrey, 1999], locally varying material properties or experimental artifacts like edge effects [START_REF] Arimitsu | A study of Saint-Venant's principle for composite materials by means of internal stress fields[END_REF].

The combination of 3-D deformation measurements [Rastogi, 1999, Foster, 1978, Viotti et al., 2008, Matthys et al., 1991, Genovese, 2007, Genovese, 2009, Sutton et al., 2007] and inverse approaches [START_REF] Avril | Identification from measurements of mechanical fields[END_REF] is now very common in solid mechanics but it is still under-employed for identifying the anisotropic hyperelastic properties of the arterial tissues [START_REF] Seshaiyer | [END_REF]Humphrey, 2003, Einstein et al., 2005]. Moreover, the Virtual Fields Method [START_REF] Grédiac | The virtual fields method for extracting constitutive parameters from full-field measurements: a review[END_REF], which is an inverse method specifically dedicated to full-field data, has never been used for the mechanical identification of arterial tissues although it has very relevant assets: insensitivity to the uncertainty of boundary conditions [START_REF] Grédiac | The virtual fields method for extracting constitutive parameters from full-field measurements: a review[END_REF], robustness [START_REF] Avril | Sensitivity of the virtual fields method to noisy data[END_REF], fast convergence [Avril and Pierron, 2007].

This paper attempts to give a new contribution for addressing the mechanical identification of arterial tissues by presenting an implementation of the Virtual Fields Method to full-field experimental data measured on the whole surface of an arterial segment during inflation/extension tests. The objective of the paper is to present the principle of the approach and to prove its feasibility on an example.

Materials and Methods

Materials

Results reported in this paper are obtained on a fairly straight segment of a human ascending aorta (initial length: L 0 35 mm, initial average radius: R 0 10 mm, initial average thickness: e 0 1.3 mm). This vascular segment has been obtained from a cadaveric 65 years old female donor. All procedures were carried out in accordance with the guidelines of the Institutional Review Board of the University Hospital Centre of Saint-Etienne, France. After gently cleaning the artery with physiological solution, two side branches were clamped with surgical suture threads (Fig. 1).

Experiments

This study results from a collaboration between the University of Basilicata (Italy) and Ecole des Mines de Saint-Etienne (France). The experimental setup was developed at the University of Basilicata and presented in a companion paper [Genovese, 2009]. The current paper focuses on the inverse approach and all the details about the experimental techniques may be found in [Genovese, 2009].

The sample was mounted on the in vitro rig and preconditioned via 8 pressurization cycles from p = 0 to p = 80 mmHg (10.5 kPa) and then 8 cycles of axial stretching from L/L 0 = 1 to L/L 0 = 1.4. Then, different sets of inflation/extension tests were performed.

Let Ψ t denote the geometrical deformation of the artery induced by the loading at time t, such as x(t) = Ψ t (x 0 ), where x(t) is the position vector of a material point at t and x 0 is the initial position vector of the same material point.

The vascular segment is covered with N (N 300) black spherical markers (Fig. 1). The markers were fixed with cyanoacrylate adhesive which was shown not to diffuse in the tissues [START_REF] Holzapfel | Layerspecific 3d residual deformations of human aortas with non-atherosclerotic intimal thickening[END_REF]. The optical measurement presented in [Genovese, 2009] provides a point-wise evaluation of Ψ t at all the points where a marker has been bonded, with an accuracy of 0.17 mm.

During all the inflation/extension tests, the artery remains in a bath of physiological solution. The optical technique was calibrated accounting for the bath of physiological solution around the artery [Genovese, 2009]. However the artery was not hydrated while we bonded the markers. This lasted less than one hour. This could be reduced in the future by using glue that activates with water.

Derivation of the field of Green-Lagrange strain tensors

The surface of the artery is meshed for interpolating Ψ t across the whole artery from the data measured at the marker position. Triangular elements with linear interpolation functions are used. The mesh is defined using a Delaunay triangularization algorithm applied onto the marker initial positions (actually the marker position means the contact position of the marker onto the surface of the artery). Each triangular element is labeled by an index denoted q. Let x q (t) denote the position vector of the center of gravity of each element at time t Eventually, Ψ t is known everywhere across the meshed area of the artery. Let Ω be this domain across which the deformation is measured. Ω is defined in the initial configuration because the deformation is Lagrangian. Let ω(t) be the counterpart of Ω in the deformed configuration at time t.

All the derivations presented in this paper are based on the theory of finite deformations [Ogden, 1997]. Let F(x 0 , t) denote the deformation gradient:

F(x 0 , t) ≡ ∇Ψ t (x 0 )
where ∇ is the gradient operator.

Then we define the right Cauchy-Green tensor:

C(x 0 , t) ≡ F T F
where F T is the transpose of F.

Eventually, we define the Green-Lagrange strain tensor:

E(x 0 , t) ≡ [C -I]/2
where I is the identity tensor.

Constitutive equations

Arteries in vitro are usually considered as anisotropic visco-hyperelastic materials. Here, the viscoelastic properties are not considered and only the instantaneous and monotonic response is modeled, after preconditioning. Pseudo-elasticity can therefore be applied [Fung, 1993]. Hyperelastic models are considered.

In these models, the Cauchy stress tensor, denoted S, is deduced from the Green-Lagrange strain tensor such as:

S = F : dΦ/dE : F T + λI (1)
where Φ is the strain energy function depending only on E and on material parameters. The term λI comes from the incompressibility and λ is assessed by writing the plane stress condition (S 33 = 0).

Different strain energy functions for the anisotropic hyperelastic behavior of arteries may be found in the literature. Some of them are phenomenological [Fung, 1993, Sun andSacks, 2005] and others are based on the microstructure [START_REF] Holzapfel | A new constitutive framework for arterial wall mechanics and comparative study of material models[END_REF]. In this study, we consider 3 different models:

1. the Delfino model [START_REF] Holzapfel | A new constitutive framework for arterial wall mechanics and comparative study of material models[END_REF]:

Φ = 1 2 β [exp (α (I 1 -3)) -1] (2)
where β and α are material parameters and I 1 = tr(C).

2. the 2-D Fung model [Fung, 1993], which may be written as:

Φ = 1 2 β exp α 11 (E 11 ) 2 + α 22 (E 22 ) 2 + α 12 E 11 E 22 -1 = 1 2 β [exp(Q) -1] (3)
where β, α 11 , α 22 and α 12 are material parameters and E ij are the components of the E matrix in the material coordinate system (indice 1 refers to the circumferential direction and indice 2 refers to the axial direction in the local coordinate system).

3. the Holzapfel model [START_REF] Holzapfel | A new constitutive framework for arterial wall mechanics and comparative study of material models[END_REF]. This model assigns separate strain energy functions to the media layer and the adventitia layer. The simplest form of the strain energy function may be written:

Φ = c 2 (I 1 -1) + k 1 2k 2 exp k 2 (I 4 -1) 2 -1 + k 1 2k 2 exp k 2 (I 6 -1) 2 -1 (4)
where c and k 1 are stress-like material parameters, k 2 is a dimensionless material parameter;I 4 and I 6 are the squares of the stretch component in the two families of symmetric tissue fibers.

Hence, this model prescribes a fiber orientation angle φ for each layer, based usually on analysis of microscopy data. The fiber angles for the specimen used in this work were not known, and so the fiber angles are another material parameter to be determined from the experimental data.

Inverse approach

The Virtual Fields Method (VFM) is an approach dedicated to the identification of material parameters from full-field measurements [START_REF] Grédiac | The virtual fields method for extracting constitutive parameters from full-field measurements: a review[END_REF], Avril et al., 2004, Avril and Pierron, 2007, Promma et al., 2009]. It is based on the weak form of equilibrium equations.

Assuming quasi-static conditions and no body forces, the weak form of equilibrium in the deformed configuration may be written such as:

-

ω(t)
S :

E * dv + ∂ω(t) t.u * ds = 0 (5)
where ∂ω(t) is the boundary of the volume in the deformed configuration, vector t denotes the tractions applied onto this boundary (pressure forces and axial load), u * is a displacement virtual field which acts as a test function in Eq. 5). E * is the strain virtual field deduced from the gradient of u * :

E * = 1 2 ∇u * + ∇ T u * (6)
At position x where a marker is located, u * 1 and u * 2 take the following values:

         u * 1 = sin π z -z b z t -z b i u * 2 = z -z t z b -z t k (7)
where:

• vectors i and k are respectively the radial and longitudinal axes in the global cylindrical coordinate system defined in Fig. 2;

• z is the component of x along the longitudinal axis: z = x • k ;

• z b and z t are defined in Fig. 2.

The rationale behind this choice of virtual displacements is:

• one involves mainly the radial properties (u * 1 )

• one involves mainly the axial properties (u * 2 )

• both zero the action of unknown reaction forces or involve their resultant if the resultant is measured [START_REF] Grédiac | The virtual fields method for extracting constitutive parameters from full-field measurements: a review[END_REF].

Given the previous points, infinity of virtual deformation may be used. The rationale behind the current choice is that the best results are always obtained with the simplest functions. This was proved in different papers [START_REF] Grédiac | The virtual fields method for extracting constitutive parameters from full-field measurements: a review[END_REF], Avril et al., 2004, Avril and Pierron, 2007, Promma et al., 2009].

This can be interpreted like this: a virtual deformation is a test function. If the test function is very smooth, it will average out the randomly distributed errors contained in the experimental data.

Eq. 7 only gives the values of the virtual displacement at marker positions. Between the marker positions, the values of u * 1 and u * 2 are interpolated linearly.

Moreover, regarding the external virtual work, it can be written:

           ∂ω(t) t.u * 1 ds = ∂ω(t) k t .u * 1 p(t)ds ∂ω(t) t.u * 2 ds = ∂ω(t) k t .u * 2 p(t)ds + F(t) (8)
where k t is the unit outer normal vector at t, F(t) is the axial resultant load measured by a load cell.

As S is deduced from experimental data, Eq. 5 cannot be satisfied exactly and the principle of the inverse method [START_REF] Grédiac | The virtual fields method for extracting constitutive parameters from full-field measurements: a review[END_REF] is to minimize the squares of residuals, as defined in the following cost function:

ζ(β, α 11 , α 22 , α 12 ) = k m - ω(t m ) S : E * k dv + p(t m ) ∂ω(t m ) k t m .u * k ds + δ k2 F(t) 2 (9)
where m labels the time when a measurement is achieved and k labels the virtual field (k = 1, 2);

δ k2 =0 if k = 1 and δ k2 =1 if k = 2.
Considering the interpolation of all quantities with the triangular mesh, the previous integrals in the principle of virtual work may be changed into discrete sums.

ζ(β, α 11 , α 22 , α 12 ) = k m - q S(x q,m ) : E * k (x q,m )A(x q,m )e(x q,m ) + p(t m ) q k t (x q,m ).u * k (x q,m )A(x q,m ) + δ k2 F(t) 2 (10)
where A(x q,m ) denotes the area of triangular element q at time t m and e(x q,m ) denotes the thickness of the artery1 at x q at time t m . E * k is derived from the virtual displacements u * k using Eq. 6. The virtual displacements u * k are defined in Eq. 7. The value of p are experimental data and S is derived using Eq. 1. It must be noted that the model parameters are involved in S. All the incremental steps are taken into account as the squares of residuals of each incremental step, denoted m, are summed up.

Cost function ζ figures the quadratic gap between the internal virtual work (IVW) and the external virtual work (EVW) where:

IVW k,m = e 0 q S(x q,m ) : E * k (x q,m )A(x q,0 ) EVW k,m = p(t m ) q k t (x q,m ).u * k (x q,m )A(x q,m ) + δ k2 F(t) (11)
Cost function ζ and IVW are driven by the choice of the unknown material parameters. Eventually, the cost function can be minimized through an iterative scheme using the Nelder-Mead algorithm [START_REF] Nelder | A simplex method for function minimization[END_REF]. This yields the unknown material parameters.

Results

A sample was tested for proving the feasibility of the approach (Fig. 1 and Fig. 2). We report data corresponding to 8 pressure levels distributed between 0 and 150 mmHg at L/L 0 = 1.1 (Fig. 3 and Fig. 4).

The et al., 2000et al., , Fung, 1993]]. Parameter β is usually around 10 kPa. It is interesting to notice the large difference between the value of α 11 and α 22 . It means that there is a large anisotropy in this specimen, which has already been reported in the literature [START_REF] Holzapfel | A new constitutive framework for arterial wall mechanics and comparative study of material models[END_REF].

It can be observed in Fig. 5 that the gap between the external virtual work and the internal virtual work is very low for the obtained material parameters.

Discussion

4.1 Application to the other constitutive models

The Delfino model

The material parameters of Eq. 2, namely α and β can be identified using only u * 1 . This means that only the circumferential response is used for the identification. The results obtained with the VFM for these two material parameters are: β = 1.5 kPa and α = 4.6.

However, it is interesting to notice in Fig. 6 that the gap between IVW and EVW remains very large for u * 2 when computed with the constitutive equations of Eq. 2 and the material parameters reported above. This indicates irrelevancy of the isotropic assumption. This justifies the choice of anisotropic laws.

The Holzapfel model

Here, having access only to measurements on the surface of the artery, it is only possible to identify the average strain energy function across the whole thickness, but not separately strain energy functions of the media layer and of the adventitia layer. If, as in [START_REF] Marra | Elastic and rupture properties of porcine aortic tissue measured using inflation testing[END_REF], the strain energy function is assumed homogeneous across the thickness, then a mean fiber angle and mean values of the other parameters of the Holzapfel model can be identified by the VFM. The obtained results are: c = 0.4 kPa, k 1 = 8.7 kPa, k 2 = 5.4 and φ = 27 • . These results show that parameters of a Holzapfel model may be retrieved by the VFM.

Fiber angles and material parameters in human arteries are usually different for the media and the adventitia. Only average values are reported here. A separate identification of media and adventitia constitutive equations would be possible by mechanical separation of the layers of the human artery, as

shown in [START_REF] Holzapfel | A new constitutive framework for arterial wall mechanics and comparative study of material models[END_REF].

Material heterogeneities

Arteries may also have properties varying along the axis. The VFM was already applied successfully to the identification of nonlinear heterogeneous behavior for metals [START_REF] Sutton | Identification of heterogeneous constitutive parameters in a welded specimen: Uniform stress and virtual fields methods for material property estimation[END_REF]. The implementation of a similar approach for arteries is possible but it will require that the spatial resolution is adapted to the length scale of local variations of the material properties. This may be achieved by employing 3-D digital image correlation, as shown in [START_REF] Sutton | Strain field measurements on mouse carotid arteries using microscopic three-dimensional digital image correlation[END_REF].

Discussions for α 12 in the Fung model

Parameter α 12 is the coupling factor between E 11 and E 22 in the exponential of the strain energy function.

It means that α 12 characterizes the effect of the circumferential strain onto the axial response, or viceversa. A value of 0.1 was found for parameter α 12 (Tab. 1). For obtaining this result, the algorithm was initialized according to values2 reported in [Fung, 1993].

However, it must be noted that the results regarding α 12 are significantly affected by the value input for initializing the optimization algorithm. This may indicate a lack of sensitivity of the cost function to α 12 , inducing the existence of a valley in the cost function.

The dependence of the VFM to the initializing values was not observed for the Holzapfel model.

Nevertheless, in the Holzapfel model, the exponential part is driven by only 3 material parameters, whereas the exponential part of the Fung model is driven by 4 material parameters. This supplementary material parameter in the Fung model does not affect sufficiently the value of cost function ζ in Eq. 10, inducing the lack of sensitivity to α 12 . If one would like to increase the sensitivity to α 12 , one would have to consider the response of the artery not only to incrementally varying pressures, but also to incrementally varying stretches in the axial direction, and to include these responses in the definition of cost function ζ. This will be considered in future studies.

Implementation of the identified model in a FE code

In [START_REF] Sun | Finite element implementation of a generalized Fung-elastic constitutive model for planar soft tissues[END_REF], the implementation of the Fung model in a FE code was discussed. In order to check the feasibility of utilizing this model with the parameters that we identified here (Tab. 1), computations were achieved on the geometry of the artery with the Abaqus R software. The geometry of the model is a 33.8 mm long cylinder with an initial diameter of 21.5 mm, and initial thickness of 1.3 mm.

It was meshed with 1276 membrane elements (M3D4R type in Abaqus R ). The Fung model is a built-in feature of Abaqus R , which requires the definition of a local coordinate system related to the anisotropy directions and the constitutive parameters reported in Tab. 1. No residual stress was incorporated in this model. One end of the cylinder was clamped. Regarding the other end, radial displacements were fixed and a longitudinal displacement was prescribed, corresponding to L/L 0 = 1.1. The different experimental pressure steps were applied successively. The resolution of the problem was performed using an implicit scheme accounting for large strains.

A comparison of the average radii computed by Abaqus R to the measured average radii is shown in Fig. 7. The comparison is made for the cross section located at z = (z b + z t )/2, i.e. at midspan. The results show a good agreement.

However, the radii computed by FE model are slightly larger than the experimental ones. It may be induced by the fact that, in the model, we neglected the pressure applied by the physiological bath on the external surface of the artery (about 5 mmHg 0.7 kPa at the midspan of the arterial segment).

Discrepancies may also be induced by a difference between the experimental boundary conditions and the actual ones at the ends. For the FE model, the boundary conditions were of clamping type at both ends, whereas the actual ones are less rigid.

It is worth noting here that this discussion about the boundary conditions at both ends does not affect the results obtained by the VFM. Indeed, the effect of unknown reaction forces and parasitic motions at both ends is filtered out by the VFM. This is an essential asset of the VFM, as shown in [START_REF] Grédiac | The virtual fields method for extracting constitutive parameters from full-field measurements: a review[END_REF].

Derivation of average stress/strain curves

It was shown that the identified Fung model can be used to derive stress/strain curves of the artery, both in the circumferential direction and in the axial direction. Results are presented in the appendices.

Conclusion

In conclusion, results presented in this work are promising regarding the application of the virtual fields method (VFM) for identifying the anisotropic hyperelastic properties of arteries. An innovative experimental device has been set up, calibrated and validated for characterizing arterial mechanical properties.

It is now important to carry out a large number of experiments for validating the approach with different sets of data and different tests. Other perspectives concern more complex loading conditions and more sophisticated models. More especially, next steps will consist in (i) considering the possible heterogeneity of the properties and geometry of the artery wall, (ii) processing data from different loading cases (including torsion), (iii) and considering independently the different layers of arteries and the residual stresses. Achievement of these steps will require optimal spatial resolution and accuracy of the measurement technique, employing for instance 3-D digital image correlation [START_REF] Sutton | Strain field measurements on mouse carotid arteries using microscopic three-dimensional digital image correlation[END_REF].

Acknowledgements

The authors would like to thank Professeur Jean-Pierre Favre and to his staff in the department of cardiovascular surgery at the University Hospital of Saint-Etienne (France), for their help in preparing the specimen tested in this study. The authors are also very grateful to Dr. Katia Genovese for the experimental work. This study is part of the Imandef project (Grant ANR-08-JCJC-0071) funded by the ANR (French National Research Agency).

Appendices 7.1 Appendix A

In this section, we show that the identified Fung model can be used to derive a stress/strain curve of the artery in the circumferential direction and that this curve is in agreement with the average stress/strain curve deduced from experimental data.

The average circumferential Green-Lagrange strain is defined by:

Ẽ11 (t) = 1 2 R R 0 2 -1 (12)
where R(t) is the average radius of the best fitting circle at time t at x 3 = (x b 3 + x t 3 )/2 and R 0 is R(t) at t = 0.

The circumferential stress component, denoted S11 , should satisfy the Laplace law everywhere:

S11 (t) = p(t) R(t) ē(t) (13)
where ē(t) is the average thickness at the midspan of the arterial segment.

We compare the value of S11 (t) and S11 (t), where S11 (t) is the value deduced from Ẽ11 by applying directly the Fung model of Eq. 3 to Ẽ11 with the assumption Ẽ22 = 0. The parameters for the Fung's model are the ones that are reported above in Tab. 1.

Results shown in Fig. 8a are in agreement. This means that the material parameters identified in the circumferential direction are consistent with a standard procedure (plotting stress/strain curves from pressure/diameter measurements). The advantage of the VFM is that the properties in the axial direction were also identified simultaneously.

Appendix B

In this appendix, we show that the identified Fung model can be used to derive a stress/strain curve of the artery subjected to a tensile test in the axial direction and that this curve is in agreement with the average stress/strain curve deduced from experimental data.

The axial Green-Lagrange strain is defined by:

Ẽ22 (t) = 1 2 L L 0 2 -1 (14)
where L(t) is the length of the arterial segment at time t.

The axial stress component, denoted S22 should satisfy:

S22 (t) = F(t) 2π R(t)ē(t) (15)
We compare the value of S22 (t) and S22 (t) , where S22 (t) is the value deduced from Ẽ22 by applying directly the Fung model of Eq. 3 to Ẽ22 with the assumption Ẽ11 = 0. The parameters for the Fung's model are reported in Tab. 1.

Results shown in Fig. 8b are in agreement. This means that the material parameters identified in the axial direction are in agreement with a classical analysis consisting in plotting stress/strain curves. The advantage of the VFM is that the properties in the axial direction were identified using only one stretch value, whereas a full tensile test was required in the axial direction for plotting the stress/strain curve with the standard analysis.

Figure 1 :

 1 Figure 1: Picture of the arterial segment used in the tests.

Figure 2 :Figure 3 :

 23 Figure 2: Meshing and schematic of the initial geometry: (a) 3D view (b) cross sectional view.

Figure 4 :Figure 5 :

 45 Figure 4: Plot of the circumferential components of the Cauchy stress tensor for different values of p. The plot is displayed in the deformed configuration. The Cauchy stress are calculated with the Fung model using the identified values reported in Tab.1

Figure 8 :

 8 Figure 8: Obtained stress/strain curves: (a) in the circumferential direction and (b) in the axial direction

Table 1 :

 1 results obtained with the VFM are reported in Tab. 1 for the Fung model. Convergence of the Results obtained with experimental data for the Fung model.

	Nelder Mead optimization routine was reached in nearly 10 minutes. The identified values for the Fung
	material parameters of the artery are consistent with the range orders reported in the literature [Holzapfel

  field 1 (a) and virtual field 2 (b) when an anisotropic model is identitied Comparison of the internal and external virtual work for virtual field 1 (a) and virtual field 2 (b) when an isotropic model is identitied

		with virtual field 1 (a)		with virtual field 2 (b)
				120 0.09
		experimental stress/strain curve		IVW experimental stress/strain curve
		EVW identified Fung model		0.08	EVW identified Fung model
				100
				0.07
				80 0.06
			virtual work S 22 (MPa)	60 0.04 0.05	experimental radii radii obtained by FE
				40 0.03
				0.02
	frag replacements S 11	pressure (MPa)	20 0.01	pressure (MPa) Pressure (MPa)
	E 11 S 22 E 22	0.01 Figure 7: Comparison of the deformed geometry between a FE model and experimental data 0.015 0.02 0.025 0 0.005 0.01 0.015 0.02 0.025 0 Figure 6: 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.1 0.2 0.3 0.4 E 11 0 0.1 0.2 0.3 0.4 0 0.5 E 22
			23

Due to the incompressibility assumption, A(x q,m )e(x q,m ) = A(x q,0 )e 0 , with x q,m = x q (tm).

Initial values: β=29kPa, α 11 =2.5, α 22 =0.5 and α 12 =0.17

List of Figures


Ambroise Duprey
email: ambroise.duprey@gmail.com

Khalil Khanafer

Marty Schlicht

Stéphane Avril

David R Williams

Ramon Berguer

DAVID Matthew Williams

In Vitro Characterisation of Physiological and Maximum Elastic Modulus of Ascending Thoracic Aortic Aneurysms Using Uniaxial Tensile Testing

Keywords: Aorta, Elastic modulus, Biomechanical properties, Aneurysm, Bicuspid valve

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

Abstract

Objective Ascending thoracic aortic aneurysms (ATAA) are a life-threatening condition due to the risk of rupture or dissection. This risk is increased in the presence of a bicuspid aortic valve (BAV). The purpose of this study was to provide data on the elastic modulus of aortic wall of ATAA using uniaxial tensile testing in two different areas of the stress-strain relationship: physiological and maximum range of stresses. The influence of tissue location, tissue orientation and valve type on these parameters was investigated.

Materials and methods

Tissues freshly excised from ATAA with bicuspid or tricuspid aortic valve were obtained from greater and lesser curvature (GC and LC) and the specimens were tested uniaxially in circumferential (CIRC) and longitudinal (LONG) orientation. Maximum elastic modulus (MEM) was given by the maximum slope of the stress-strain curve before failure. Physiological modulus (PM) was derived from the Laplace law and from ranges of pressure of 80-120 mmHg. Means of each group of specimen were compared using Student's t-test to assess the influence of location, orientation and valve type on each mechanical parameter.

Results

PM was found to be significantly lower than the MEM (p < 0.001). The MEM and PM were significantly higher (p < 0.01) in the CIRC (n = 66) than in the LONG orientation (n = 42). The MEM was higher in the circumferential orientation in the BAV group (p < 0.001 in GC and p < 0.05 in LC). MEM and PM in GC specimens were higher in the longitudinal orientation than the LC specimens (p < 0.05).

Conclusion

This study demonstrates the anisotropy of the aortic wall in ATAA and provides data on the mechanical behaviour in the physiological range of pressure.

I. Introduction

The biomechanical properties of the aorta have been extensively studied over the past decades [START_REF] Choudhury | Local mechanical and structural properties of healthy and diseased human ascending aorta tissue[END_REF], [START_REF] Fung | Biomechanics. Mechanical properties of living tissues[END_REF], [START_REF] Mohan | Failure properties of passive human aortic tissue. I-uniaxial tension tests[END_REF], [START_REF] Okamoto | 3rd, Mechanical properties of dilated human ascending aorta[END_REF], [START_REF] Holzapfel | Determination of material models for arterial walls from uniaxial extension tests and histological structure[END_REF], [START_REF] Prendergast | Analysis of prolapse in cardiovascular stents: a constitutive equation for vascular tissue and finiteelement modelling[END_REF], [START_REF] Sokolis | Passive mechanical properties and structure of the aorta: segmental analysis[END_REF], [START_REF] Vorp | Effect of aneurysm on the tensile strength and biomechanical behavior of the ascending thoracic aorta[END_REF], [START_REF] Vande Geest | Age dependency of the biaxial biomechanical behavior of human abdominal aorta[END_REF] and [START_REF] Nichols | McDonald's blood flow in arteries[END_REF] and are still the subject of a growing interest to provide a better understanding of aortic dissection and aneurysm and predict the behaviour of stent grafts. These aortic diseases are associated with changes in the mechanical properties of the aortic wall. The characterisation of the biomechanical behaviour of the aortic wall is a potential tool for the prediction of growth and rupture of aneurysms and consequently may help clinicians in their decision-making process. In addition, the accuracy of in vitro experiments and computational analysis depends significantly on the mechanical properties assigned to the aortic wall. Mechanical testing of tissue specimen is one of the most widespread methods to gain this knowledge. Ascending thoracic aortic aneurysms (ATAA) are a life-threatening condition that could benefit from such a prediction tool. They affect approximately 59 of 100,000 people in the United States every year [START_REF] Bickerstaff | Thoracic aortic aneurysms: a population-based study[END_REF]. Biomechanical properties of ATAA have been studied by Choudhury et al. [START_REF] Choudhury | Local mechanical and structural properties of healthy and diseased human ascending aorta tissue[END_REF]. Okamoto [START_REF] Okamoto | 3rd, Mechanical properties of dilated human ascending aorta[END_REF], Vorp [8] and Peterson [START_REF] Peterson | Biaxial mechanical properties of dilated human ascending aortic tissue[END_REF] A large panel of different mechanical parameters, such as tensile strength, maximum elastic modulus, incremental modulus and others, has been investigated in the literature. Similarly, different stress-strain curve definitions have been proposed in the literature to study the elasticity of the aneurysmal wall. For example, Raghavan et al. [START_REF] Raghavan | Ex vivo biomechanical behavior of abdominal aortic aneurysm: assessment using a new mathematical model[END_REF] determined the mechanical properties of the abdominal aortic aneurysm as compared to non-aneurysmal aorta using uniaxial tensile testing and the stress-strain curve was plotted using true stress versus engineering strain. This method for the calculation of the elastic modulus was used in many other publications [START_REF] Vorp | Effect of aneurysm on the tensile strength and biomechanical behavior of the ascending thoracic aorta[END_REF], [START_REF] Gundiah | Determination of strain energy function for arterial elastin: experiments using histology and mechanical tests[END_REF], [START_REF] He | The composition and mechanical properties of abdominal aortic aneurysms[END_REF] and [START_REF] Thubrikar | Mechanical properties of abdominal aortic aneurysm wall[END_REF] although none of them explained the rationale for this choice. Another combination -engineering stress versus engineering strain [START_REF] Choudhury | Local mechanical and structural properties of healthy and diseased human ascending aorta tissue[END_REF] and [START_REF] Sokolis | Passive mechanical properties and structure of the aorta: segmental analysis[END_REF] -has also been used where only the initial dimensions of the tissue at rest are considered. Given that specimens undergo large deformation in uniaxial testing, the initial dimensions are only remotely related to the range of strain where tissue failure occurs. We propose that the true stress-true strain relationship is the only method that truly takes into account the actual variation of the dimensions of the specimen during tensile testing. This method that has never been used in the bioengineering literature is proposed here as the method of reference. Another issue found in the biomechanical engineering literature is that most studies deal with the failure behaviour of the aorta. The mechanical parameter given by this particular range of the stress-strain relationship is often referred to as the maximum elastic modulus. The pitfall of such analysis is the lack of data regarding the physiological range of the stress-strain relationship. Thus, it is possible to identify two areas of interest from the stress-strain relationship, which correspond to the physiological and failure range of the aorta. The purpose of this study was to determine the elastic modulus of the wall of ascending aortic aneurysms in these two areas using the true stress-true strain definition. Results are then compared between specific tissue locations (greater vs. lesser curvature) and tissue orientations (longitudinal vs. circumferential). The impact of a bicuspid aortic valve (BAV) which is thought to correlate with abnormal wall elasticity was also investigated.

II. Materials and methods

I II I. .1 1. . H
Hu um ma an n a ao or rt ti ic c t ti is ss su ue e s sp pe ec ci im me en ns s All procedures were carried out in accordance with the guidelines of the Institutional Review Board of the University of Michigan. A segment of aortic wall was excised from surgical specimens obtained from patients undergoing elective surgical repair of their ATAA. These segments were cut with custom-designed tissue cutters and multiple test specimens were available when the surgical aortic specimen was large enough. Samples were obtained from two different locations, greater curvature (GC) and lesser curvature (LC). The maximum diameter of ATAA was known from the patient's preoperative Computed Tomography Angiography (CTA). The tissue specimens were stored in gauze wetted with saline and refrigerated at 4 °C. Tissue-testing was performed within 48 h. After equilibration at room temperature, connective and adipose tissue were removed from the surface of the adventitia and the samples were cut along either circumferential (CIRC) or longitudinal (LONG) orientation (Figure 1). The original thickness and width of the sample were measured at zero stress state with a digital caliper. 3 I II I. .2 2. . U Un ni ia ax xi ia al l t te en ns si il le e t te es st ti in ng g Experiments were carried out using a tensile testing machine (Instron ® model 5542, Norwood, MA, USA). To avoid damaging the tissue fine grit adhesive sandpaper was placed on the surface of the pneumatic grips to prevent slippage during the test. To ensure a free shear deformation and local narrowing of the tissue between the grips, it was found during preliminary testing that an aspect ratio of at least 2 was required (the aspect ratio is defined as the ratio of gauge length divided by the width of the specimen). Once the tissue was in place, the pressure line operating the grips was slowly raised (to prevent crushing the sample) to 20 psi. The tissue was kept wet by a spray of phosphate buffering solution. Each specimen was preconditioned by applying two cycles of a 1 N load at 10 mm min -1 to decrease the effect of relaxation in the mechanical response. Then the tensile testing was performed at 10 mm min -1 until failure. Load and stretch were continuously recorded by the data acquisition software Merlin ® provided by Instron ® . I II I. .3 3. . D Da at ta a a an na al ly ys si is s The load-stretch curve was derived to obtain the true stress-true strain relationship, which was considered as our reference method. This choice of method takes into account the large deformation of the specimen and subsequently the actual change of dimension of the specimen during testing. The aortic wall was assumed to be an incompressible material [START_REF] Carew | Compressibility of the arterial wall[END_REF]. In order to calculate the true stress and true strain, we need to define engineering stress and engineering strain which depend on the initial dimensions of the tissue. The engineering stress is:

E 0 F A  
where F is the load and A o is the initial cross-sectional area.

The engineering strain is:

 0 E L  
where λ is the stretch and L o the initial length.

Then we define the true stress:

F A   where A is the current cross-sectional area.
The true strain is defined by:

T dL d L  
where dL is the instantaneous stretch and L the current length of the specimen. The assumption of the arterial incompressibility implies a zero change of volume during the tensile testing:

A 0 0 x L A x L 
and then 0 0 0 0

(1 )

T E E E F FL L x A A L L          
 doi:10.1016/j.ejvs.2010.02.015

The true strain is defined as the sum of all the current engineering strains. Then

0 0 0 0 ln ln ln(1 ) L T E L dL L L d L L L             
To characterise the physiological modulus (PM) of the specimen, the Laplace law was derived to obtain the circumferential stress σ θ , assuming the aneurysm to be of nearly cylindrical shape:

P x R t   
where P is the pressure, R the radius of the aneurysm and t the thickness of the specimen. Similarly, the longitudinal stress σ l was given by

2 l P x R t  
Pressure range of 80-120 mmHg was chosen as the physiological range (Figure 2). Considering that the stress-strain curve is nearly linear within this short range, a linear fitting was applied yielding the PM calculated as the slope of the fitted line. The maximum elastic modulus (MEM) was taken from each curve as the maximum slope prior to failure (Figure 2). Data processing was performed on Excel 2007 ® (Microsoft Corporation). Statistical analysis was performed on SPSS Statistics 17.0 ® (SPSS Inc., Chicago, USA and supplied by the University of Michigan, USA). Results of each testing were averaged on a per patient basis.

Results of each group of specimens were given as the mean of the patients' data in order to account for the different number of testings from one patient to another. Student's t-test was performed to assess the influence of location, orientation and valve type on each mechanical parameter. Significance was assumed for a p value less than 0.05.

III. Results

One hundred and eight specimens were obtained from 12 patients with ATAA. Six patients had a BAV. Table 1 summarises the patient characteristics. Sixty-five testings were performed in GC and 43 in LC. Sixty-six specimens were tested circumferentially and 42 longitudinally.

In the tricuspid aortic valve (TAV) group, specimens were significantly thicker in LC than GC. Tissues from BAV were significantly thicker than TAV in GC location (Table 2). No outlier was found in each set of testing for each patient and each group (location, orientation and valve type), demonstrating the consistency of the results. In 26 cases, the data could not be processed for the calculation of the mechanical parameters because the failure point was not reached for mechanical reasons such as slippage of the tissue. Results are shown in Table 3.

The results of this study showed that the effect of location (GC vs. LC) on the MEM results was not significant in the CIRC orientation group. In the LONG orientation, the MEM was found significantly higher in GC than LC (p < 0.05 in the BAV group and p < 0.01 in the TAV group). Similarly, the PM was significantly higher in GC than LC in the LONG orientation group (p < 0.01 in the BAV group and p < 0.05 in the TAV group). The PM was also found to be higher in GC than LC circumferentially in the BAV group with a p = 0.045. When comparing orientations, the MEM was found to be significantly higher in the CIRC than in the LONG group (p < 0.01) ([Figure 3] and [Figure 4]). Similarly, the PM was significantly higher circumferentially regardless of location and valve type (p < 0.01) ([Figure 5] and [Figure 6]). The MEM was compared between the BAV and TAV groups in both locations and both orientations. In the CIRC group, the MEM was significantly higher in the BAV than in the TAV group (p < 0.001 in the GC group and p < 0.05 in the LC group) The PM was compared between the valve types as well and the only significant difference was found in the LC LONG group where TAV specimens were stiffer than their BAV counterparts (p < 0.05).

The MEM and PM results were compared in each group of tissue. The MEM values were significantly higher in every group with p < 0.001.

IV. Discussion

Cardiovascular diseases are one of the leading causes of death in the modern world. ATAA is one of the most serious conditions as it can cause death by rupture or dissection. Since the advent of powerful computational software to perform finite element studies, the aortic diameter criterion has become insufficient to accurately determine patients eligible for surgical repair, especially to predict the likelihood of rupture of smaller aneurysms. Computational assessment of aneurysms on a patient-specific basis is one of the major challenges in the field of vascular surgery. This technology can be applied to the evaluation of new surgical devices such as aortic endografts. Knowledge of aortic wall behaviour is critical in improving the accuracy of computational analysis.

The mechanical parameters of the aortic wall have been studied by many researchers using both uniaxial and biaxial tensile testing. The common calculation methods found in the literature are the true stress versus engineering strain relationship and the engineering stress versus engineering strain relationship. These methods are relevant only for strain lower than 10%. Aortic specimens in uniaxial testing commonly undergo much larger strain, up to 60-70%. The problem with these methods is that they do not give a true indication of the deformation characteristics of the human tissue since they are based entirely on the original dimensions of the specimen and these dimensions change continuously during the test. The actual variation of the current cross-sectional area and length of the tissue must be taken into account and hence, under the assumption of the incompressibility of the tissue, the true stress-true strain relationship was introduced in the present study. The results presented in this study highlight many points that are discussed ahead. I IV V. .1 1. . E Ef ff fe ec ct t o of f t th he e l lo oc ca at ti io on n The comparison of MEM between GC and LC locations showed no difference in the CIRC orientation whereas the difference was significant in the LONG group, where the GC specimens were stiffer than their LC counterpart. The effect of the location in MEM values of ATAA has been studied by Choudhury et al. [START_REF] Choudhury | Local mechanical and structural properties of healthy and diseased human ascending aorta tissue[END_REF] who reported that the MEM was stiffer in the GC than LC for both orientations but without significance. Our results confirm this trend in the longitudinally oriented specimens. Thubrikar [START_REF] Thubrikar | Mechanical properties of abdominal aortic aneurysm wall[END_REF] studied yield stress and strain of abdominal aortic aneurysms and observed regional variation in wall stiffness. Our results suggest that theoretically, under the same pressure, the LC is sustaining more elongation than its GC counterpart. Perhaps the 3D movement of the aorta from the root to the left subclavian artery is accounting for the local variation of the elasticity. As was proposed by Prehn [START_REF] Van Prehn | Toward endografting of the ascending aorta: insight into dynamics using dynamic cine-CTA[END_REF], the dynamic cine-CTA seems a promising technique to investigate the complex 3D movement of the ascending aorta. Clinical observations show that aortic dissection is usually present along the greater curvature of the ascending aorta. This may be explained by our finding that the GC has a higher MEM than the LC in longitudinal direction. As such, the GC will experience higher stresses than the LC and consequently will be a potential site for rupture or dissection. I IV V. .2 2. . C Co om mp pa ar ri is so on n b be et tw we ee en n B BA AV V a an nd d T TA AV V BAV is commonly related to abnormalities of the thoracic aorta, such as dilated aortic root, dissection and aortic coarctation [START_REF] Yap | Abnormal aortic elastic properties in adults with congenital valvular aortic stenosis[END_REF]. Aortic stiffness was found to be increased and distensibility decreased in BAV patients assessed by echocardiography [START_REF] Nistri | Bicuspid aortic valve: abnormal aortic elastic properties[END_REF]. The MEM was found significantly higher in the BAV group in the CIRC orientation and both locations. The difference in mechanical properties between BAV and TAV was studied by Choudhury et al. [START_REF] Choudhury | Local mechanical and structural properties of healthy and diseased human ascending aorta tissue[END_REF] and their results showed that the BAV specimens were stiffer, as was observed in our study in the CIRC group. I IV V. .3 3. . E Ef ff fe ec ct t o of f t th he e o or ri ie en nt ta at ti io on n The anisotropy of the aortic wall is still a debated issue and conflicting results are found in the literature. In the present study, both MEM and PM were found to be significantly higher in the CIRC group than the LONG one, suggesting that the aortic wall in ATAA is an anisotropic material. Vorp [START_REF] Vorp | Effect of aneurysm on the tensile strength and biomechanical behavior of the ascending thoracic aorta[END_REF] studied ATAA using uniaxial tensile testing and the true stressengineering strain definition. The MEM was 4.67 ± 0.42 MPa and 4.48 ± 0.59 MPa in the CIRC and LONG orientations respectively. Using the same stress-strain definition, we observed the same range of values in our study in the CIRC group. However, the MEM was significantly lower in the LONG group, regardless of the location and valve type. Peterson [START_REF] Peterson | Biaxial mechanical properties of dilated human ascending aortic tissue[END_REF]s studied ATAA using biaxial testing and reported one of six patients with stiffer specimen in the circumferential orientation. Tissues were found isotropic in biaxial testing performed by Choudhury et al. [START_REF] Choudhury | Local mechanical and structural properties of healthy and diseased human ascending aorta tissue[END_REF] Okamoto et al. [START_REF] Okamoto | 3rd, Mechanical properties of dilated human ascending aorta[END_REF] also studied ATAA using biaxial testing and found the tissue 'somewhat anisotropic'. Healthy thoracic aorta has been studied by Vorp [START_REF] Vorp | Effect of aneurysm on the tensile strength and biomechanical behavior of the ascending thoracic aorta[END_REF] and no difference was found between the two orientations. Ferraresi [START_REF] Ferraresi | One-dimensional experimental mechanical characterisation of porcine aortic root wall[END_REF] performed uniaxial testing of porcine aortic root tissue and reported the CIRC-oriented tissues to be stiffer than their LONG counterpart. The issue of arterial wall anisotropy has been thoroughly investigated by Zhou and Fung [22] They performed biaxial tensile testing on canine thoracic aorta and demonstrated a significant anisotropy. This issue has been addressed in abdominal aortic aneurysms (AAA) as well. Thubrikar [START_REF] Thubrikar | Mechanical properties of abdominal aortic aneurysm wall[END_REF] studied AAA using uniaxial testing in both orientations and reported that the CIRC-oriented specimens were stiffer than their LONG counterpart. Conversely, AAA were found isotropic by Raghavan [START_REF] Raghavan | Ex vivo biomechanical behavior of abdominal aortic aneurysm: assessment using a new mathematical model[END_REF]. Healthy abdominal aorta demonstrated different MEM according to Xiong, [START_REF] Xiong | Measurement and analysis of ultimate mechanical properties, stress-strain curve fit, and elastic modulus formula of human abdominal aortic aneurysm and nonaneurysmal abdominal aorta[END_REF] but no statistical analysis was available to assess the significance. The histological structure of the wall, especially the orientation and integrity of elastin and collagen fibres, is related to the question of anisotropy and was studied by He and Roach [START_REF] He | The composition and mechanical properties of abdominal aortic aneurysms[END_REF] in aneurysmal and healthy abdominal aortas. They found that in a healthy aorta, the media is organised in lamellar units composed of elastin layers while in AAA the media lamellar units were damaged and the elastin fragmented. These findings suggest that healthy tissue is anisotropic and that, theoretically, the anisotropy is partially or fully lost in aneurysmal tissue. Fibres are known to be arranged circumferentially mainly [START_REF] Ferraresi | One-dimensional experimental mechanical characterisation of porcine aortic root wall[END_REF]. Orientation of collagen fibres has been investigated in uniaxially stretched arterial tissue. Holzapfel [START_REF] Holzapfel | Determination of material models for arterial walls from uniaxial extension tests and histological structure[END_REF] found that collagen fibres have angles with the circumferential axis of 18.8°, 37,8° and 58.9° in the intima, media and adventitia respectively. Using X-ray scattering technique to measure fibre orientation in the adventitia, Schmid [START_REF] Schmid | In situ tensile testing of human aortas by time-resolved small-angle X-ray scattering[END_REF] observed a predominant circumferential alignment of collagen fibres implying an increased circumferential stiffness. In our study, the aortic wall was significantly anisotropic and the tissue was stiffer circumferentially. As described earlier, our results are consistent with many of the previous studies. From a mechanical point of view, the predominant deformation of aortic wall in physiological condition is taking place circumferentially whereas longitudinally, the wall is pre-stretched and tethered and the elongation under pressure change is low. In addition, if the aorta is considered as a cylindrical shell of incompressible wall material, the derived Laplace law yields a longitudinal stress, which is half of the circumferential stress. It can then be considered that the anisotropy of the aortic wall meets these mechanical requirements. I IV V. .4 4. . P Ph hy ys si io ol lo og gi ic ca al l e el la as st ti ic c m mo od du ul lu us s It is widely admitted that the arterial wall is non-linear material. Besides the failure conditions, the elasticity of the aortic wall has to be studied in an additional region of interest corresponding to the physiological range of stress. The proposed method represents an attempt to better understand the mechanical behaviour of ATAA wall in the physiological range and aims to provide an actual value of elastic modulus for computational analysis and hence, a gain of accuracy. It can also help in the choice of the appropriate material when designing aortic bench-top models. Sokolis [START_REF] Sokolis | Passive mechanical properties and structure of the aorta: segmental analysis[END_REF] studied pig aortas using uniaxial tensile testing and the engineering stress-engineering strain definition. The stress-strain curve was divided into three parts where part II was considered as the physiological range, corresponding to the slope transition. The physiological range was also observed in the same region of the curve in our results. This nearly linear part was then fitted as was done in the present study. However, the physiological range of stress was set at 20-100 kPa, which is well below physiological aortic pressures. Our study calculated the stress taking into account the dimensions of the tissue and the aneurysm and a physiological variation of pressure. These results can be compared to the values obtained from the incremental modulus method, as described by Thubrikar [START_REF] Thubrikar | Mechanical properties of abdominal aortic aneurysm wall[END_REF]. In his study, the incremental modulus was defined as the differentiate function of the stress-strain relationship. A pressure of 100 mmHg was then chosen to calculate the elastic modulus in both orientations, yielding values of 4 MPa in the CIRC and 1.5 MPa in the LONG. These results are in the same order of magnitude as the values found in our study. I IV V. .5 5. . L Li im mi it ta at ti io on ns s One of the issues associated with in vitro characterisation of biomechanical behaviour of human tissue is the effect of the strain rate on the tensile properties of tissues. To date, this study and the majority of experimental investigations on arterial mechanics have used nonphysiological strain rates (i.e., 10 mm min -1 ) when determining the mechanical properties of human tissues in a tensile testing machine. For better assessment of the mechanical properties, tensile stress machines should be able to operate at a much higher strain rate corresponding to the physiology of aortic wall deformation. The second problem with in vitro tissue-testing is the absence of the internal pressure of the human body surrounding the vessel and the simplification of the forces being applied to the tissue in uniaxial (generally) or biaxial (rarely) direction. Zanchi et al. [START_REF] Zanchi | Differences in the mechanical properties of the rat carotid artery in vivo, in situ, and in vitro[END_REF] showed that the mechanical properties of the rat carotid artery substantially differ in vivo, in situ and in vitro. Another limitation in the present study is the small sample size. No comparison was possible on a per patient basis. In order to achieve statistical power, it was necessary to pool data from different patients within the same group. Specimen slippage is the main cause of data loss as 24% of testing failed because of tissue slippage. Tensile mechanical testing technique has to be improved in the future to allow data comparison within one particular patient in order to minimise this bias. In addition, slippage of the tissue was detected visually or on the loadstretch relationship. This method appeared to us as efficient but the lack of a video tracking device can be considered as a bias in detecting slippage.

V. Conclusion

We studied the physiological and MEM of the aortic wall of ATAA using uniaxial tensile testing. We proposed the use of the true stress-true strain definition for the calculation of the elastic modulus in human aortas: it takes into account the changes in cross-sectional area and length that occur during tensile testing. The aortic wall was significantly anisotropic with the CIRC-oriented specimens being the stiffest. Interestingly, the aorta was stiffer longitudinally in the GC than in the LC. This observation can correlate with the physiological movement of the arch. The tissues from BAV were stiffer than in the TAV group in the CIRC orientation. This study provides data on the mechanical behaviour in the physiological range for more accurate computational and experimental analysis.

Figures 10 F

 10 Figures

Acknowledgements

European Society for Vascular Surgery which is the major funding source of this study. Frankel Research Fund which partly funded this study. No assistance was received from both sponsors in the study design, in the collection, analysis and interpretation of data; in the writing of the manuscript and in the decision to submit the manuscript for publication.