*2.1. Experimental Study on Mechanical Properties of Grafts*

The study investigated the mechanical properties of Gore-Tex vascular grafts. These are most frequently used in surgical practice for shunt placement, including the modified Blalock–Taussig shunt operation. The Gore-Tex vascular graft is a tube made of a special material. It is a waterproof and vaporizable membrane. Constants of the strain energy density function for the hyperelastic material were obtained as a result of the experiments. The influence of the wall mechanical properties on the blood flow in an artificial vessel was evaluated considering the interaction between the vessel wall and the blood flow. The study employed the methods of computational fluid dynamics and mechanics. Calculation

results were obtained for elastic and hyperelastic walls. The research considered local hemodynamics in the aorta with regard to anisotropy and hyperelastic properties; FSI modeling methods were applied. The results were analyzed to deduce the main factors influencing the local blood flow. influencing the local blood flow. Gore-Tex shunts (W. L. Gore & Associates, Inc., Flagstaff, AZ, USA) of different sizes and thickness were used for tests (Figure 1). Currently, PTFE shunts are actively used in surgery. The test temperature was 37 °C. Experiments were performed under different

Blalock–Taussig shunt operation. The Gore-Tex vascular graft is a tube made of a special material. It is a waterproof and vaporizable membrane. Constants of the strain energy density function for the hyperelastic material were obtained as a result of the experiments. The influence of the wall mechanical properties on the blood flow in an artificial vessel was evaluated considering the interaction between the vessel wall and the blood flow. The study employed the methods of computational fluid dynamics and mechanics. Calculation results were obtained for elastic and hyperelastic walls. The research considered local hemodynamics in the aorta with regard to anisotropy and hyperelastic properties; FSI modeling methods were applied. The results were analyzed to deduce the main factors

*Materials* **2022**, *15*, x FOR PEER REVIEW 3 of 24

Gore-Tex shunts (W. L. Gore & Associates, Inc., Flagstaff, AZ, USA) of different sizes and thickness were used for tests (Figure 1). Currently, PTFE shunts are actively used in surgery. The test temperature was 37 ◦C. Experiments were performed under different loading to assess the effect of load rate on deformation. loading to assess the effect of load rate on deformation. Most human tissues behave nonlinearly under a load. Such materials are called hyperelastic. A strain energy density function is used to describe the behavior of hyperelastic materials under a load.

**Figure 1.** Meshes and boundary conditions of the aorta–shunt–pulmonary artery system: (**a**) boundary conditions, (**b**) solid mesh, (**c**) velocity and pressure profiles, (**d**) aorta fluid mesh model. **Figure 1.** Meshes and boundary conditions of the aorta–shunt–pulmonary artery system: (**a**) boundary conditions, (**b**) solid mesh, (**c**) velocity and pressure profiles, (**d**) aorta fluid mesh model.

Most human tissues behave nonlinearly under a load. Such materials are called hyperelastic. A strain energy density function is used to describe the behavior of hyperelastic materials under a load.

The materials are incompressible. The results of the experimental study can be described by the five-parameter Mooney–Rivlin model:

$$\mathbf{W} = \mathbf{c}\_{10}(\overline{\mathbf{I}\_{1}} - \mathbf{3}) + \mathbf{c}\_{01}(\overline{\mathbf{I}\_{2}} - \mathbf{3}) + \mathbf{c}\_{20}(\overline{\mathbf{I}\_{1}} - \mathbf{3})^{2} + \mathbf{c}\_{11}(\overline{\mathbf{I}\_{1}} - \mathbf{3})(\overline{\mathbf{I}\_{2}} - \mathbf{3}) + \mathbf{c}\_{02}(\overline{\mathbf{I}\_{2}} - \mathbf{3})^{2} + \frac{1}{\mathbf{D}\_{1}}(\mathbf{J} - \mathbf{1})^{2} \tag{1}$$

and the three-parameter Yeoh model:

$$\mathcal{W} = \sum\_{i=1}^{3} \mathbf{c}\_{i0} \left(\overline{\mathbf{I}\_{1}} - \mathbf{3}\right)^{\mathrm{i}},\tag{2}$$

where cij are material constants; I1, I<sup>2</sup> are the first and the second invariant of the deviatoric strain tensors:

$$
\mathbf{I}\_1 = \lambda\_1^2 + \lambda\_1^2 + \lambda\_{1'}^2 \tag{3}
$$

$$\mathbf{I}\_2 = \lambda\_1 \lambda\_2 + \lambda\_2 \lambda\_3 + \lambda\_1 \lambda\_3 \tag{4}$$

$$
\overline{\mathbf{I}}\_3 = \lambda\_1^2 \lambda\_1^2 \lambda\_1^2 \tag{5}
$$

where λ<sup>i</sup> are principal stretches in Equations (3)–(5). The materials are incompressible; I<sup>3</sup> = 1.

The constants in the strain energy density function (Equations (1)–(2)) were determined in the ANSYS Workbench software (Ansys Workbench 18, Ansys Inc., Canonsburg, PA, USA) using a curve fitting procedure based on the experimental tensile diagrams obtained for the sample. The five-parameter Mooney–Rivlin model (Equation (1), Equations (3)–(5)) and the three-parameter Yeoh model (Equations (2)–(5)) are used to describe the behavior of mechanical properties of grafts in this study.
