# The Effects of the Mechanical Properties of Vascular Grafts and an Anisotropic Hyperelastic Aortic Model on Local Hemodynamics during Modified Blalock–Taussig Shunt Operation, Assessed Using FSI Simulation

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## Abstract

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^{®}are used for the treatment of cyanotic heart defects (i.e., modified Blalock–Taussig shunt). Significant mortality during this palliative operation has led surgeons to adopt mathematical models to eliminate complications by performing fluid–solid interaction (FSI) simulations. To proceed with FSI modeling, it is necessary to know either the mechanical properties of the aorta and graft or the rheological properties of blood. The properties of the aorta and blood can be found in the literature, but there are no data about the mechanical properties of Gore-Tex

^{®}grafts. Experimental studies were carried out on the mechanical properties vascular grafts adopted for modified pediatric Blalock–Taussig shunts. Parameters of two models (the five-parameter Mooney–Rivlin model and the three-parameter Yeoh model) were determined by uniaxial experimental curve fitting. The obtained data were used for patient-specific FSI modeling of local blood flow in the “aorta-modified Blalock–Taussig shunt–pulmonary artery” system in three different shunt locations: central, right, and left. The anisotropic model of the aortic material showed higher stress values at the peak moment of systole, which may be a key factor determining the strength characteristics of the aorta and pulmonary artery. 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 stress on the aortic wall is observed. According to computations, the anisotropic model shows smaller values for the displacements of both the aorta and the shunt, which in turn may affect the success of preoperative predictions. Thus, it can be concluded that the anisotropic properties of the aorta play an important role in preoperative modeling.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Study on Mechanical Properties of Grafts

#### 2.2. Mechanical Properties of Aorta

#### 2.2.1. Ogden Model for Description of Isotropic Hyperelastic Behavior of Aorta

_{1}= 2, α

_{2}= −2 with the constraint condition λ

_{1}λ

_{2}λ

_{3}= 1).

#### 2.2.2. Holzapfel–Gasser–Ogden Model for Description of Anisotropic Hyperelastic Behavior of Aorta

_{10}, $\mathsf{\kappa}$, k

_{1}, and k

_{2}are temperature-dependent material parameters; N is the number of families of fibers (N ≤ 3); ${\overline{\mathrm{I}}}_{1}$ is the first deviatoric strain invariant; and ${\overline{\mathrm{I}}}_{4\left(\mathsf{\alpha}\mathsf{\alpha}\right)}$ are pseudo-invariants of $\overline{\mathrm{C}}$ and ${\overline{\mathrm{E}}}_{\mathsf{\alpha}}$.

#### 2.3. FSI Simulations of Blood Flow in the Aorta–Pulmonary Artery–Shunt System

#### 2.3.1. Problem Formulation

#### 2.3.2. Mathematical Problem Statement

_{f}is the fluid density, p is the pressure, u is the fluid velocity vector, and u

_{g}is the moving coordinate velocity. In the arbitrary Lagrangian–Eulerian (ALE) formulation, (u−u

_{g}) is the relative velocity of the fluid with respect to the moving coordinate velocity. Here, τ is the deviatoric shear stress tensor (Equation (15)). This tensor is related to the velocity through the strain rate tensor; in Cartesian coordinates it can be represented as follows:

^{3}; the dynamic viscosity is constant and equal to $\mathsf{\mu}=0.0035$ Pa·s. The velocity profile during the systolic and diastolic phases of the left ventricle was applied at the aortic root inlet (Figure 1). The left ventricular systole period is t = 0.22 s. The period of ventricular diastole is t = 0.28 s. The total cardiac cycle is t = 0.5 s. The peak velocity is 1.4 m/s. A time-dependent pressure profile was used as the boundary conditions at the aortic outlets. Constant pressure of P = 20 mm Hg was applied at the pulmonary artery outlets.

#### 2.3.3. Mesh and Convergence

## 3. Results

#### 3.1. Results of the Experimental Study

_{Y}was also determined (Table 3); its value increases as the specimen diameter increases (Figure 4). The influence of loading rate on tensile strength σ

_{Y}determination was analyzed. The tensile strength value remained practically unchanged when the load rate application changed from 50 to 250 mm/min. The shape of the tensile test curve for all specimens was the same. Stress–strain relationships were obtained as a result of tensile tests for two specimens (specimen no. 1, diameter of 5 mm, thickness of 0.5 mm, length of 20 mm; specimen no. 2, diameter of 3 mm, thickness of 0.35 mm, length of 20 mm). The constants for the strain density function were determined from the obtained dependences (Table 4) and stress–strain dependences were plotted (Figure 5).

#### 3.2. Results of FSI Simulation of the Blood Flow

#### 3.2.1. Velocity Distribution

#### 3.2.2. Pressure distribution

#### 3.2.3. Wall Shear Stress

**.**

#### 3.2.4. Distribution of Time-Averaged Shear Stress

#### 3.2.5. Displacement Distribution

#### 3.2.6. Von Mises Stress Distribution

## 4. Discussion

#### 4.1. Difference between Isotropic and Anisotropic Models

#### 4.2. Concluding Remarks

#### 4.3. Limitations

#### 4.4. Possible Future Clinical Application

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**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 2.**Maximum stress variations with respect to the number of mesh elements for solid domain: (

**a**) mesh dependency tests for maximum von Mises stress, (

**b**) mesh dependency tests for minimum von Mises stress, (

**c**) maximum and minimum values for aorta, shunt, and pulmonary artery.

**Figure 5.**Stress–strain diagrams: (

**a**) specimen 1, (

**b**) specimen 2. The solid line (1) is the experimental curve, the dashed line (2) is the five-parameter Mooney–Rivlin strain energy density function, and the dotted line (3) is the three-parameter Yeoh strain energy density function.

**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.

**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.

**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.

**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 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 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 12.**Velocity 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.

**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 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).

No. | Body Sizing, mm | Inflation | Number of Elements | Maximum Pressure, Pa | Maximum Velocity, m/s | ||
---|---|---|---|---|---|---|---|

Transition Ratio | Maximum Layers | Growth Ratio | |||||

1 | 0.95 | 0.5 | 3 | 1.2 | 80,353 | 17,634 | 3.85 |

2 | 0.8 | 0.4 | 5 | 1.4 | 159,379 | 17,952 | 4.38 |

3 | 0.63 | 0.3 | 7 | 1.3 | 349,926 | 17,892 | 4.47 |

4 | 0.5 | 0.35 | 8 | 1.6 | 709,578 | 18,470 | 4.71 |

5 | 0.38 | 0.32 | 10 | 1.3 | 1,544,745 | 18,509 | 4.75 |

Sample Number | E (MPa) | Diameter, d (mm) | Wall Thickness (mm) |
---|---|---|---|

1 | 7.41 | 4.32 | 0.34 |

2 | 9.8 | 3.4 | 0.4 |

3 | 10.3 | 4.5 | 0.35 |

4 | 11.1 | 5.5 | 0.48 |

5 | 43.5 | 5 | 0.53 |

Sample Number | σ_{Y} (MPa) | Diameter, d (mm) | Wall Thickness, (mm) | Loading Rate, (mm/min) |
---|---|---|---|---|

1 | 11.6 | 3.4 | 0.4 | 30 |

2 | 13.6 | 4.5 | 0.35 | 30 |

3 | 14.5 | 5 | 0.53 | 30 |

4 | 17.0 | 6.2 | 0.85 | 50 |

5 | 16.9 | 6.2 | 0.85 | 250 |

Strain Density Function | Constants, Specimen No. 1 (MPa) | Constants, Specimen No. 2 (MPa) |
---|---|---|

The five-parameter Mooney–Rivlin model | ${\mathrm{C}}_{10}=-1.64$, ${\mathrm{C}}_{01}=2.59$, ${\mathrm{C}}_{20}=4.46\times {10}^{-7}$, ${\mathrm{C}}_{11}=-2.39\times {10}^{-4}$, ${\mathrm{C}}_{02}=0.44$ | ${\mathrm{C}}_{10}=-2.2$, ${\mathrm{C}}_{01}=3.26$, ${\mathrm{C}}_{20}=3.86$, ${\mathrm{C}}_{11}=-8.6\times {10}^{-4}$, ${\mathrm{C}}_{02}=0.62$ |

The three-parameter Yeoh model | ${\mathrm{C}}_{10}=0.11$, ${\mathrm{C}}_{20}=-4.96\times {10}^{-6}$ ${\mathrm{C}}_{30}=1.67\times {10}^{-10}$ | ${\mathrm{C}}_{10}=0.20$ ${\mathrm{C}}_{20}=-6.73\times {10}^{-6}$ ${\mathrm{C}}_{30}=1.16\times {10}^{-10}$ |

The Aorta | The Shunt | ||
---|---|---|---|

Isotropic Hyperelastic Material | Anisotropic Hyperelastic Material) | Isotropic Elastic Material | Isotropic Hyperelastic Material |

Ogden model: ${\mathsf{\mu}}_{1}=1.274$ MPa ${\mathsf{\mu}}_{2}=-1.211$ MPa ${\mathsf{\alpha}}_{1}=24.074$ ${\mathsf{\alpha}}_{2}=24.073$ | Holzapfel–Gasser–Ogden model: ${\mathsf{\mu}}_{1}=2.363$ MPa ${\mathsf{\mu}}_{2}=0.839$ MPa ${\mathsf{\alpha}}_{1}=0.6$ $d=0.001$ MPa ^{−1} | E = 10.3 MPa $\mathsf{\mu}=0.49$ | Experimental data (Table 3) |

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**MDPI and ACS Style**

Kuchumov, A.G.; Khairulin, A.; Shmurak, M.; Porodikov, A.; Merzlyakov, A.
The Effects of the Mechanical Properties of Vascular Grafts and an Anisotropic Hyperelastic Aortic Model on Local Hemodynamics during Modified Blalock–Taussig Shunt Operation, Assessed Using FSI Simulation. *Materials* **2022**, *15*, 2719.
https://doi.org/10.3390/ma15082719

**AMA Style**

Kuchumov AG, Khairulin A, Shmurak M, Porodikov A, Merzlyakov A.
The Effects of the Mechanical Properties of Vascular Grafts and an Anisotropic Hyperelastic Aortic Model on Local Hemodynamics during Modified Blalock–Taussig Shunt Operation, Assessed Using FSI Simulation. *Materials*. 2022; 15(8):2719.
https://doi.org/10.3390/ma15082719

**Chicago/Turabian Style**

Kuchumov, Alex G., Aleksandr Khairulin, Marina Shmurak, Artem Porodikov, and Andrey Merzlyakov.
2022. "The Effects of the Mechanical Properties of Vascular Grafts and an Anisotropic Hyperelastic Aortic Model on Local Hemodynamics during Modified Blalock–Taussig Shunt Operation, Assessed Using FSI Simulation" *Materials* 15, no. 8: 2719.
https://doi.org/10.3390/ma15082719