Numerical modeling of endovascular treatment of AAA requires knowing the biomechanical properties of arterial vessels throughout the path traveled by guidewire catheter device from the femoral to the abdominal aorta. Due to the lack of data on the mechanical properties due to the degree of calcification [25], we propose to perform mechanical testing of human arterial tissue that allows distinguishing calcified walls, pathological or healthy, and to integrate these mechanical characteristics according to the degree of calcification in numerical model.

Femoral, iliac and aortic samples of aged patients were tested after being stored and dissected Figure 1. The dissection is performed as a function of parietal quality: data scanners that contain indications as “grid level” on

the state of arterial tissue were translated in terms of mechanical properties.

For this purpose, an atomic force microscope (AFM), a tribometer and rheometer were used to characterize the mechanical behavior in compression, shearing and friction of different layers of arteries Figure 2 as function of the degree of calcification. A large number of samples (474) were tested in a liquid environment (Phosphate buffered saline PBS pH 7.4) at room temperature.

Figure 3(a) depicts the number of events as a function of adhesion force to verify the low bond strength hypothesis in the theory of contact. Thereafter, for each sample, we measured the indentation force as a function of depth for different speeds. Compression and decompression paths do not always overlap. This explains the effect of some visco-plastic arterial walls properties Figure 3(b). Only the elastic part is retained for the measurement of mechanical properties, using AFM tools, we measured the normal load as function of the indentation. However, different theories of elastic contact have been proposed to improve the description of contact between two solid bodies: Hertz [26,27] [1889], Sneddon [1965], Johnson, Kendall and Roberts (JKR) in [1971], Derjaguin, Muller and Toporov (DMT) in [1975]. The Hertz model, in a liquid medium with the assumption of weak Vander-Waals intermolecular adhesion forces (capillarity), was adapted to characterize the tip-sample contact:

Thus, the relationship Equation (1) between force F_n and indentation δ for a sphere of radius R is:

For a cone of half-tip angle ϕ, we find Equation (2):

E: Elastic modulus and ν: Poisson coefficient.

An example is shown below of test results on an internal and external layer of a fatty artery. We find that the outer layer is more rigid than the inner layer Figure 4(a), and that the viscoelastic effect at the nanoscale is negligible, because the modulus of elasticity is not strongly

affected by the indentation velocity Figure 4(b).

One of the test results concerns measuring tangential forces and deducing the friction coefficient depending on the degree of calcification for the internal layer of a highly calcified artery. This coefficient is the ratio between the normal force that is imposed by gravity and the tangential force that is measured. The variation of this coefficient is recorded with time. Figure 5 represents the surface topography of the involved sample. Thus, the friction curve shows positive and negative parts, each corresponding to an alternating displacement. The friction coefficient remains stable over time. It is

about 0.5 for very calcified areas. 

The assessment of tests on different human arterial samples has led to a determination of nano-mechanical properties of different layers as a function of the degree of calcification. This was done through preoperative images of the wall artery as shown in Figure 6. We connected each grid point to a mechanical property such as elastic modulus and friction coefficient.

• The guidewire In this section, our goal is to make a local characterization of the mechanical properties of the guidewire catheter device, as input to a numerical endovascular model predicting the deformation of the anatomical

structure under the action of this device. 

• The folding stent: Catheter This system contains the stent; it has a 0.889 mm internal diameter, that allows its navigation in interaction with the guide. It is also composed of several areas: the tip with variable diameter, the area containing the stent, the tubular area, and the sheath. It is of variable construction (platinum-iridium, nickel, nitinol, polyester, etc.). 

To create a model and to understand the mechanical behavior of the guidewire and catheter devices, tests were conducted for each region, considering the variation of mechanical properties according to various medical tools. The 3D image correlation was found to be the best tool to follow this variation. 

The tests were performed on the tensile machine. A speckled black on white is applied to the tested sample. This is then subjected to uniaxial quasi-static tension and is fixed between the two jaws of the LFPlus machine, which is connected to an image acquisition system, lighting and 3D correlation Figure 7.

The results were obtained through a local characterization (3D correlation and extensometer) and global one. The curves in Figure 8 show one example of mechanical tests results on the catheter tip. We measured stress versus strain to obtain value of Young’s modulus. Figure 9 is an example of a summary of mechanical properties (elastic modulus and Poisson’s ratio) as a function of

position on the endovascular system (guidewire and catheter).

We note a very significant stiffness gradient between the tip and the stiff zone of the guide, which is added gradually in the transition area. The tips of different guides used in the clinical setting have a stiffness between 700 to 2000 MPa. The purpose is to make easier the navigation of the guidewire in vessels without pain and improve the contact between the guide head and blood vessels, while the rigid part is in the range of 11,800 to 19,000 MPa to make the artery straight. 

After creating a functional relation for the mechanical properties of biological tissues and surgical tools, a simulation is possible to predict all steps of endovascular treatment using patient specific data and the behavior of arterial walls under the action of the guidewire and the catheter system. 

Creating a model of the complete system guidewire/ catheter/aortic/femoral/iliac/aneurysm/environment of the artery requires the separate modeling of its different components. The originality of this study is as follows:

Consideration of a realistic geometry of the biological system reconstructed from preoperative clinical images on a specific group of patients with high tortuosity and calcification.

Accounting for the environment of the artery, the degree of calcification, the reference geometry with preload, and the nonlinear behavior of materials and geometry. 

Considering frictional contact behavior between arterial walls and the surgical tools, and between the tools themselves. 

A composite model for biological tissue including fiber orientation.

The computer code used is Abaqus 6.10-1. We used Updated Lagrangian description due to large deformations. The arterial walls were modeled using thin shell theory with Kirchhoff approach. The delivery system was modeled using Timoshenko beam elements. 

Considering the strong nonlinearity of the problem, the explicit time integration scheme is a relatively easy numerical scheme to implement and less expensive in terms of computation time [28,29]. Thus we adopted the method β-2, which allows the introduction of a numerical damping coefficient that removes the high frequencies [30]. The contact between the tools and biological tissues was managed by unilateral penalty conditions, which consist of increasing the functional of total energy by a penalty function [31]. This contact represents the interaction between tools and soft tissues by a spring with a stiffness depending on the penetration. Femoral, iliac, and aortic wall thickness models have been introduced, varying from 1 to 2 mm.

The models were created using preoperative individual patient data. A three-dimensional scan of the patient is performed with a contrast injection to reveal the arrangement of blood vessels located around the area of interest Figure 10. 

Initially, a manual step is necessary. It consists of extracting the vessel centerline, the lumen of the artery by splines, and the planes orthogonal to the neutral line depending on the degree of curvature of the artery wall.

Then a reconstruction of the aneurysm is done, including bifurcations. The iliac and tortuous femoral artery is the final geometry to be used for  simulation Figure 11.

This technique has the advantage of generating a geometry which is very close to the real one. 

In order to overcome convergence difficulties due to the complex geometry, and problems arising from the contact between the endovascular tools and the arterial walls; only the aorta, femoral and internal iliac used for endovascular instruments navigation are included in the analysis. The external iliac and secondary vessels are taken into account using appropriate boundary conditions. 

- The artery was modeled using composite materials. It is considered as a matrix material with three layers that are reinforced by two families of fibers. Therefore its mechanical properties depend on both directions of the fibers. The orientation of collagen fibers was observed by microscopy in the layers of the human aorta, on the order of 10˚ on average for the collagen fibers in the

media, and 40˚ in the adventitia have been found [32]. These results are in accordance with the histologic findings of a general human abdominal aortic media.

- Environmental information of the artery was taken into from the data of the scanner images. We extracted the list of the nearest points of the spline, the centerline and secondary branches of the artery. Thus, we have represented this environment by elastic supports Figure 12(a).

- Information on the condition of the arterial wall was described in the numerical model. Information related to the distribution of the degree of calcification, which can differentiate healthy areas from aneurysmal and calcified regions was projected in the numerical model Figure 12(b).

Environmental information of the artery was taken into

- Information on the condition of the arterial wall was described in the numerical model. Information related to the distribution of the degree of calcification, which can differentiate healthy areas from aneurysmal and calcified regions was projected in the numerical model Figure 12

Healthy and aneurysmal areas were modeled as a hyper-elastic material with Mooney-Rivlin constitutive Equation (3), while calcified areas have been considered as linear elastic Figure 13:

where

with W is the energy density of Mooney-Rivlin, I₁, I₂, I₃ are the strain invariants, I₄₁ and I₄₂ describe the square

stretching fibers, characterized by the directions a₁ and a₂ (anisotropy terms), a₀₁ and a₁₀ are the constants parameters of Mooney-Rivlin, [C] Cauchy Green tensor, k₁ and k₃ are material parameters, k₂ and k₄ are dimensionless parameters, generated by two families of fibers and k is the bulk modulus.

The Mooney-Rivlin anisotropy terms present the energy stored in the fibers, they are described by exponential functions.

In their normal physiological condition, the arteries are in a state of longitudinal elongation and generally subject to internal pressure. This means that the stress straint is non-zero when the vascular smooth muscle is inactive. This hypothesis was justified by several experimental studies on normal and decellularized arteries and revealed the existence of significant residual stresses within the arterial wall [33,34]. In our model, we considered the reference state of the artery corresponding to the nonzero stress state when the intravascular pressure is zero. This pre-stress was estimated from a state of unknown geometry, subjected to a positive pressure that can reproduce the real reference geometry extracted from the preoperative data.