**4. Discussion**

## **4. Discussion** *4.1. Difference between Isotropic and Anisotropic Models*

*4.1. Difference between Isotropic and Anisotropic Models* The distribution of hemodynamic parameters in the anisotropic and isotropic models of materials in patients has the same distribution pattern throughout the system. The dynamics of blood flow is identical in all patients. The numerical values are also the same (Figures 12 and 13). The differences between the anisotropic and isotropic properties of the aorta and the pulmonary artery are noticeable only when analyzing the stress–strain The distribution of hemodynamic parameters in the anisotropic and isotropic models of materials in patients has the same distribution pattern throughout the system. The dynamics of blood flow is identical in all patients. The numerical values are also the same (Figures 12 and 13). The differences between the anisotropic and isotropic properties of the aorta and the pulmonary artery are noticeable only when analyzing the stress–strain state, i.e., in displacements and stresses arising in the aorta–shunt–pulmonary artery system.

state, i.e., in displacements and stresses arising in the aorta–shunt–pulmonary artery system. The opposite situation is seen with von Mises stress distribution (Figure 14). There is also a similar pattern of stress distribution throughout the system, with the exception of the central location of the shunt, where, according to the anisotropic model, increased stress values are mostly observed in the aorta. In addition, the anisotropic model of the material shows higher stress values than the isotropic model of the aortic material. The maximum stress on the wall of the anisotropic aorta is about 200 kPa; the maximum stress on the wall of the isotropic aorta is about 150 kPa.

*Materials* **2022**, *15*, x FOR PEER REVIEW 18 of 24

**Figure 12.** Velocity distribution: (**a**,**d**,**a1**,**d1**) central shunt; (**b**,**e**,**b1**,**e1**) right shunt; (**c**,**f**,**c1**,**f1**) left **Figure 12.** Velocity distribution: (**a**,**d**,**a1**,**d1**) central shunt; (**b**,**e**,**b1**,**e1**) right shunt; (**c**,**f**,**c1**,**f1**) left shunt. shunt.

shunt.

left shunt.

**Figure 13.** Wall shear stress distribution: (**a**,**d**,**a1**,**d1**) central shunt; (**b**,**e**,**b1**,**e1**) right shunt; (**c**,**f**,**c1**,**f1**) **Figure 13.** Wall shear stress distribution: (**a**,**d**,**a1**,**d1**) central shunt; (**b**,**e**,**b1**,**e1**) right shunt; (**c**,**f**,**c1**,**f1**) left shunt. **Figure 13.** Wall shear stress distribution: (**a**,**d**,**a1**,**d1**) central shunt; (**b**,**e**,**b1**,**e1**) right shunt; (**c**,**f**,**c1**,**f1**) left shunt.

on the wall of the isotropic aorta is about 150 kPa.

The opposite situation is seen with von Mises stress distribution (Figure 14). There is also a similar pattern of stress distribution throughout the system, with the exception of the central location of the shunt, where, according to the anisotropic model, increased stress values are mostly observed in the aorta. In addition, the anisotropic model of the material shows higher stress values than the isotropic model of the aortic material. The maximum stress on the wall of the anisotropic aorta is about 200 kPa; the maximum stress

**Figure 14.** Von Mises stress distribution: (**a**,**d**,**a1**,**d1**) central shunt; (**b**,**e**,**b1**,**e1**) right shunt; (**c**,**f**,**c1**,**f1**) left shunt. **Figure 14.** Von Mises stress distribution: (**a**,**d**,**a1**,**d1**) central shunt; (**b**,**e**,**b1**,**e1**) right shunt; (**c**,**f**,**c1**,**f1**) left shunt.

Along with the hemodynamic parameters, the parameters of the stress–strain state, such as displacements and von Mises stress, also affect the success of a surgical interven-Along with the hemodynamic parameters, the parameters of the stress–strain state, such as displacements and von Mises stress, also affect the success of a surgical intervention [22].

tion [22]. It was shown that the anisotropic model of the aortic material shows higher stress values at the peak moment of systole, which in turn may be a key factor in determining the strength characteristics of the aorta and pulmonary artery, all other things being equal. Additionally, this mechanical parameter is important when installing a central shunt, since it is in the area of the central anastomosis that an increase in stresses on the aortic It was shown that the anisotropic model of the aortic material shows higher stress values at the peak moment of systole, which in turn may be a key factor in determining the strength characteristics of the aorta and pulmonary artery, all other things being equal. Additionally, this mechanical parameter is important when installing a central shunt, since it is in the area of the central anastomosis that an increase in stresses on the aortic wall is observed.

wall is observed. Displacement distribution is also important. According to the computations, the anisotropic model shows smaller values of the displacements of both the aorta and the shunt, which in turn may affect the success of preoperative prediction. Thus, it can be concluded that the anisotropic properties of the aorta play an important role in preoperative model-Displacement distribution is also important. According to the computations, the anisotropic model shows smaller values of the displacements of both the aorta and the shunt, which in turn may affect the success of preoperative prediction. Thus, it can be concluded that the anisotropic properties of the aorta play an important role in preoperative modeling.

ing. The time dependences of the volumetric flow rate of blood flow inside the shunt show that, for all locations of the shunt and taking into account the hyperelasticity of the shunt, the results are almost identical (Figure 15). However, in the case of the central position of the shunt, when the aortic walls were considered anisotropic material, the volumetric flow of blood within the shunt was different. The maximum deviation of this value was 12%, at 0.2 s of the cardiac cycle. For the elastic shunt, the volumetric flow rate throughout almost the entire time exceeded the analogous value for the hyperelastic shunt. This difference is essential for the further correct formation of pulmonary blood flow.

flow.

**Figure 15.** Flow rate through the shunt: (**a**) central shunt case (isotropic aortic wall), (**b**) central shunt case (anisotropic aortic wall), (**c**) left shunt case (isotropic aortic wall), (**d**) left shunt case (anisotropic aortic wall), (**e**) right shunt case (isotropic aortic wall), (**f**) right shunt case (anisotropic aortic wall). **Figure 15.** Flow rate through the shunt: (**a**) central shunt case (isotropic aortic wall), (**b**) central shunt case (anisotropic aortic wall), (**c**) left shunt case (isotropic aortic wall), (**d**) left shunt case (anisotropic aortic wall), (**e**) right shunt case (isotropic aortic wall), (**f**) right shunt case (anisotropic aortic wall).

The time dependences of the volumetric flow rate of blood flow inside the shunt show that, for all locations of the shunt and taking into account the hyperelasticity of the shunt, the results are almost identical (Figure 15). However, in the case of the central position of the shunt, when the aortic walls were considered anisotropic material, the volumetric flow of blood within the shunt was different. The maximum deviation of this value was 12%, at 0.2 s of the cardiac cycle. For the elastic shunt, the volumetric flow rate throughout almost the entire time exceeded the analogous value for the hyperelastic shunt. This difference is essential for the further correct formation of pulmonary blood

Based on the same dependences of volumetric blood flow on time for the aortic wall, and taking into account its anisotropy, it can be concluded that this is reflected only in the Based on the same dependences of volumetric blood flow on time for the aortic wall, and taking into account its anisotropy, it can be concluded that this is reflected only in the case of a central shunt, when the shunt is an elastic material. Susequently, there is an excess of volumetric flow in 12% of cases with a hyperelastic shunt. Thus, taking into account the hyperelasticity of the shunts allows for obtaining more realistic results, reducing the possible negative consequences of operations.

Hemodynamic parameters are very important for the assessment of shunting [23]. To evaluate the effectiveness of shunting, it is worth considering indicators that can describe the probable risk of shunt thrombosis. Such indicators are wall shear stress and timeaveraged wall shear stress.

As a result of the analysis of the distribution of indicators, high values of wall shear stress were revealed, and, consequently, time-averaged wall shear stress in the anastomotic region in all models. This, in turn, may indicate the risk of developing thrombosis [24–26]. It is also supported by the clinical and literature data [27–29].
