model 3.2.2. Pressure distribution

**Isotropic Hyperelastic Material**

*3.2. Results of FSI Simulation of the Blood Flow* As a result of solving the problem, the distributions of hemodynamic parameters were obtained from three patients, including blood flow velocity, pressure, wall shear stress, time-averaged wall shear stress, and other parameters. The mechanical properties Figure 7 shows the pressure distribution at the peak moment of systole. The distribution of pressure along the walls of the aorta and pulmonary artery is uneven. The highest values are concentrated on the walls of the ascending part of the aorta and its branches (left subclavian artery, left common artery, and brachiocephalic trunk), while the lowest values are observed on the walls of the pulmonary artery and shunt.

C30 = 1.67·10−10

of the aorta–pulmonary artery–shunt system are shown in Table 5 in the considered computational domain. The most important results from the hemodynamic point of view were obtained at t = 0.09 s, corresponding to the maximum blood flow velocity. Similar results In the shunt zone, the maximum values are concentrated in the area of the junction with the aorta, then the pressures are distributed evenly up to the pulmonary artery.

obtained for simple geometry (straight vessel) are presented in Supplementary Materials.

**The Aorta The Shunt**

**Isotropic Elastic** 

**Table 5.** Mechanical parameters for aorta and shunt used in the study.

**Anisotropic Hyperelastic Material)**

**Figure 6.** Velocity distribution with anisotropic properties of the aorta and hyperelastic properties **Figure 6.** Velocity distribution with anisotropic properties of the aorta and hyperelastic properties of the shunt: (**a**,**d**) central shunt; (**b**,**e**) right shunt; (**c**,**f**) left shunt.

### of the shunt: (**a**,**d**) central shunt; (**b**,**e**) right shunt; (**c**,**f**) left shunt. 3.2.3. Wall Shear Stress

The distribution of shear stresses is important in the study of systemic blood flow. In the literature, particular importance is given to the distribution of the shear-wall shear stresses. Most authors associate hypoplasia of the intima of the vascular bed with high shear stress [21].

The wall shear stress indicates two problems: lipids remain on the vessel wall at low values, and they damage the vessel wall at high values, which also increases the ability of lipids to linger on the damaged intima.

Figure 8 shows the distribution of wall shear stress. The highest values are localized in the area of the shunt, which can lead to its thrombosis. Additionally, large values of parietal shear stresses are concentrated in the pulmonary artery in the vortex, with stagnant blood flow on the branches of the aorta (left subclavian artery, left common artery, and brachiocephalic trunk). The minimum values are observed in the areas of the descending part of the aorta and the beginning of the right and left pulmonary arteries. Figure 7 shows the pressure distribution at the peak moment of systole. The distribution of pressure along the walls of the aorta and pulmonary artery is uneven. The highest values are concentrated on the walls of the ascending part of the aorta and its branches (left subclavian artery, left common artery, and brachiocephalic trunk), while the lowest values are observed on the walls of the pulmonary artery and shunt.

*Materials* **2022**, *15*, x FOR PEER REVIEW 13 of 24

3.2.2. Pressure distribution

**Figure 7.** Pressure distribution with anisotropic properties of the aorta and hyperelastic properties **Figure 7.** Pressure distribution with anisotropic properties of the aorta and hyperelastic properties of the shunt: (**a**,**d**) central shunt; (**b**,**e**) right shunt; (**c**,**f**) left shunt.

In the shunt zone, the maximum values are concentrated in the area of the junction

The distribution of shear stresses is important in the study of systemic blood flow. In the literature, particular importance is given to the distribution of the shear-wall shear

The wall shear stress indicates two problems: lipids remain on the vessel wall at low values, and they damage the vessel wall at high values, which also increases the ability of

with the aorta, then the pressures are distributed evenly up to the pulmonary artery.

of the shunt: (**a**,**d**) central shunt; (**b**,**e**) right shunt; (**c**,**f**) left shunt.

lipids to linger on the damaged intima**.**

3.2.3. Wall Shear Stress

shear stress [21].

Figure 8 shows the distribution of wall shear stress. The highest values are localized in the area of the shunt, which can lead to its thrombosis. Additionally, large values of parietal shear stresses are concentrated in the pulmonary artery in the vortex, with stagnant blood flow on the branches of the aorta (left subclavian artery, left common artery, and brachiocephalic trunk). The minimum values are observed in the areas of the descend-

ing part of the aorta and the beginning of the right and left pulmonary arteries.

**Figure 8.** Distribution of wall shear stress with anisotropic properties of the aorta and hyperelastic **Figure 8.** Distribution of wall shear stress with anisotropic properties of the aorta and hyperelastic properties of the shunt: (**a**,**d**) central shunt; (**b**,**e**) right shunt; (**c**,**f**) left shunt.

## properties of the shunt: (**a**,**d**) central shunt; (**b**,**e**) right shunt; (**c**,**f**) left shunt. 3.2.4. Distribution of Time-Averaged Shear Stress

3.2.4. Distribution of Time-Averaged Shear Stress Figure 9 shows the time-averaged shear stress. The values of the shear stress at the peak moment of systole are highest in the shunt area, causing shunt thrombosis. Additionally, large values are located in the area of vortex movement of blood in the underly-Figure 9 shows the time-averaged shear stress. The values of the shear stress at the peak moment of systole are highest in the shunt area, causing shunt thrombosis. Additionally, large values are located in the area of vortex movement of blood in the underlying region of the pulmonary artery, as well as in local areas of the branches of the aorta, due to the special geometric characteristics of each geometry of patients.

due to the special geometric characteristics of each geometry of patients.

ing region of the pulmonary artery, as well as in local areas of the branches of the aorta,

**Figure 9.** Distribution of time-averaged wall shear stress with anisotropic properties of the aorta and hyperelastic properties of the shunt: (**a**,**d**) central shunt; (**b**,**e**) right shunt; (**c**,**f**) left shunt. **Figure 9.** Distribution of time-averaged wall shear stress with anisotropic properties of the aorta and hyperelastic properties of the shunt: (**a**,**d**) central shunt; (**b**,**e**) right shunt; (**c**,**f**) left shunt.

3.2.5. Displacement Distribution

the aorta.

3.2.5. Displacement Distribution Figure 10 shows the distribution of displacements occurring in the system. The displacement values at the peak moment of systole are highest in the area of the shunt and the lateral part of the aorta with the central and right location of the shunt. With the leftsided shunt position, the maximum values are distributed only along the lateral part of Figure 10 shows the distribution of displacements occurring in the system. The displacement values at the peak moment of systole are highest in the area of the shunt and the lateral part of the aorta with the central and right location of the shunt. With the left-sided shunt position, the maximum values are distributed only along the lateral part of the aorta.

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**Figure 10.** Distribution of displacements with anisotropic properties of the aorta and hyperelastic properties of the shunt: (**a**,**d**) central shunt; (**b**,**e**) right shunt; (**c**,**f**) left shunt. **Figure 10.** Distribution of displacements with anisotropic properties of the aorta and hyperelastic properties of the shunt: (**a**,**d**) central shunt; (**b**,**e**) right shunt; (**c**,**f**) left shunt.

3.2.6. Von Mises Stress Distribution 3.2.6. Von Mises Stress Distribution

Figure 11 shows the distribution of stresses arising in the system. The stress values at the peak moment of systole are highest in the areas of blood flow separation and have a non-uniform distribution pattern. Additionally, high values are located in local areas of the aorta, characterized by the unevenness of the walls of the system. Figure 11 shows the distribution of stresses arising in the system. The stress values at the peak moment of systole are highest in the areas of blood flow separation and have a non-uniform distribution pattern. Additionally, high values are located in local areas of the aorta, characterized by the unevenness of the walls of the system.

**Figure 11.** Distribution of stresses in the case of anisotropic properties of the aorta and hyperelastic properties of the shunt: (**a**,**d**) central shunt; (**b**,**e**) right shunt; (**c**,**f**) left shunt. **Figure 11.** Distribution of stresses in the case of anisotropic properties of the aorta and hyperelastic properties of the shunt: (**a**,**d**) central shunt; (**b**,**e**) right shunt; (**c**,**f**) left shunt.
