# Influence of Aortic Valve Leaflet Material Model on Hemodynamic Features in Healthy and Pathological States

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^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Geometry and Meshing

#### 2.2. Fluid Properties

^{3}and dynamic viscosity of 0.004 Pa∙s. The average Reynolds number is equal to 3685, approving the turbulent nature of the fluid flow through the valve. Furthermore, the modeled flow is pulsatile, so turbulent zones are expected to occur in the downstream regions. Blood, a multiphase fluid comprising plasma, blood cells, and platelets, exhibits a shear-thinning behavior at low shear rates and near-Newtonian behavior at higher shear rates. Recently, blood flow was shown to be Newtonian in turbulent downstream flows [63].

#### 2.3. Mechanical Properties of Aortic Valve Leaflets

#### 2.3.1. Mechanical Properties of Aortic Valve Leaflets in the Healthy State

#### 2.3.2. Mechanical Properties of Calcified Aortic Valve Leaflets

#### 2.3.3. Mechanical Properties of AVNeo Aortic Valve Leaflets (Ozaki Operation)

#### 2.4. Mathematical Problem Statement

#### 2.5. Boundary Conditions

#### 2.6. FSI Problem

## 3. Results

#### 3.1. Velocity Field and Pressure

#### 3.2. Von Mises Stress

#### 3.3. Wall Shear Stress and Hemodynamic Indicators

## 4. Discussion

#### 4.1. Wall Shear Stress

#### 4.2. Deformations

#### 4.3. Hemodynamic Indicators

#### 4.4. Vortices

#### 4.5. Limitations

#### 4.5.1. Geometry

#### 4.5.2. Boundary Conditions

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) B-mode views are used to determine aortic valve dimensions and the time-dependent inlet velocity profile measured via pulsed Doppler mode [44]. (

**b**) Idealized 2D geometry with dimensions. (

**c**) Velocity profile applied at the inlet boundary. (

**d**) Boundary conditions.

**Figure 3.**Velocity field and streamlines for (

**a**) healthy state, (

**b**) diseased model, (

**c**) AVNeo model, and (

**d**) maximum velocity. Time points are shown with red dots on the cardiac cycle. Flow direction is demonstrated with streamlines.

**Figure 4.**Comparison between peak velocity from model and in vivo experimental data [78].

**Figure 5.**Transvalvular pressure gradient (TPG) for healthy, calcified, and neo-cuspidated aortic valves.

**Figure 6.**Von Mises stress and minimal distance between leaflets for (

**a**) healthy state, (

**b**) diseased model, (

**c**) AVNeo model, and (

**d**) maximum first principal stress and strain. Time points are shown with red dots on the cardiac cycle. Von Mises stress is shown on a logarithmic scale with maximum and minimum values.

**Figure 7.**Wall shear stress (WSS) distribution along the aortic valve leaflets for the healthy state, diseased, and AVNeo model during the cardiac cycle represented with red dots at key time moments.

**Figure 8.**(

**a**) Maximum wall shear stress with shear vectors along the boundary; (

**b**) minimal distance between leaflets; (

**c**) time-averaged wall shear stress (TAWSS).

Material Model (Energy Potential) | Parameters | Source | |
---|---|---|---|

Normal Aortic Leaflets | Linear elastic isotropic material | $E=2\mathrm{MPa},\nu =0.3$ | [44] |

Linear elastic orthotropic material | $\begin{array}{l}\begin{array}{ll}{E}_{circ}=6.885\mathrm{MPa},\hfill & {G}_{xy}=1.121\mathrm{MPa},\hfill \\ {E}_{rad}=1.624\mathrm{MPa},\hfill & {G}_{yz}=1.121\mathrm{MPa},\hfill \\ {E}_{long}=1.624\mathrm{MPa},\hfill & {G}_{xz}=0.56\mathrm{MPa},\hfill \end{array}\\ \begin{array}{lll}{\nu}_{xy}=0.106,\hfill & {\nu}_{yz}=0.106,\hfill & {\nu}_{xz}=0.45\hfill \end{array}\end{array}$ | [56] | |

$W=\frac{\mu}{\alpha}\left({\lambda}_{1}^{\alpha}+{\lambda}_{2}^{\alpha}+{\lambda}_{3}^{\alpha}-3\right)+\frac{1}{2}K{\left(J-1\right)}^{2}$ | $\alpha =12.275,\mu =75310\mathrm{Pa}$ | [45] | |

$W={C}_{10}\left(\overline{{I}_{1}}-3\right)+{C}_{01}\left(\overline{{I}_{2}}-3\right)+{C}_{11}\left(\overline{{I}_{1}}-3\right)\left(\overline{{I}_{2}}-3\right)$ | $\begin{array}{l}{C}_{10}=\mathrm{32,823}\mathrm{kPa},{C}_{01}=2955\mathrm{kPa}\\ {C}_{11}=\mathrm{585,790}\mathrm{kPa}\end{array}$ | [47,57] | |

$W={\displaystyle \sum _{p=1}^{N}\frac{{\mu}_{p}}{{\alpha}_{p}}}\left({\lambda}_{1}^{{\alpha}_{p}}+{\lambda}_{2}^{{\alpha}_{p}}+{\lambda}_{3}^{{\alpha}_{p}}-3\right)$ | ${\mu}_{1}=7.6\mathrm{Pa},{\mu}_{2}=570\mathrm{Pa},\hspace{1em}{\alpha}_{1}={\alpha}_{2}=26.26$ | [48] | |

$W={c}_{0}\left[{e}^{{c}_{1}{\left({I}_{1}-3\right)}^{2}+{c}_{2}{\left({I}_{4}-1\right)}^{2}}-1\right]+p\left(J-1\right)$ | ${c}_{0}=5\mathrm{kPa},{c}_{1}=10,{c}_{2}=20$ | [58,59,60] | |

$\begin{array}{l}W={C}_{10}\left\{\mathrm{exp}\left[{C}_{01}\left({\overline{I}}_{1}-3\right)\right]-1\right\}+\\ +\frac{{k}_{1}}{2{k}_{2}}{\displaystyle \sum _{i=1}^{2}\left[\mathrm{exp}\left\{{k}_{2}{\left[{\overline{I}}_{1}+\left(1-3\kappa \right){\overline{I}}_{4i}-1\right]}^{2}\right\}\right]}+\\ +\frac{1}{D}{\left(J-1\right)}^{2},i=1,2\end{array}$ | $\begin{array}{l}\begin{array}{ll}{C}_{01}=61.303\mathrm{kPa},\hfill & {k}_{1}=9.295,\hfill \\ {C}_{10}=0.285\mathrm{kPa},\hfill & {k}_{2}=99.684,\hfill \end{array}\\ \begin{array}{lll}\kappa =0.333,\hfill & \theta ={0}^{\xb0},\hfill & D=5\times {10}^{-4}\hfill \end{array}\end{array}$ | [51,61] | |

Bovine and Porcine Valve | $W=\frac{{c}_{0}}{{c}_{1}}\left[{e}^{{c}_{1}\left({\overline{I}}_{1}-3\right)}-1\right]+\frac{{c}_{2}}{2{c}_{3}}\left[{e}^{{c}_{3}{\left(\kappa {\overline{I}}_{1}+\left(1-3\kappa \right){\overline{I}}_{4}-1\right)}^{2}}-1\right]$ | $\begin{array}{l}\begin{array}{ll}{c}_{0}=4\mathrm{kPa},\hfill & {c}_{2}=128\mathrm{kPa},\hfill \\ {c}_{1}=9.6,\hfill & {c}_{3}=29.4,\hfill \end{array}\\ \theta ={13.1}^{\xb0}\end{array}$ | [62] |

$W={\displaystyle \sum _{i=1}^{N}\frac{2{\mu}_{i}}{{\alpha}_{i}^{2}}}\left({\lambda}_{1}^{{\alpha}_{i}}+{\lambda}_{2}^{{\alpha}_{i}}+{\lambda}_{3}^{{\alpha}_{i}}-3\right)$ | $\begin{array}{ll}{\mu}_{1}=19.58\mathrm{kPa},\hfill & {\alpha}_{1}=67.74,\hfill \\ {\mu}_{2}=260.56\mathrm{kPa},\hfill & {\alpha}_{2}=27.47\hfill \end{array}$ | [52] | |

TAV/BAV | $W=\frac{{c}_{0}}{2}\left({\overline{I}}_{1}-3\right)+\frac{{c}_{1}}{2}\left({e}^{{c}_{2}\left({\overline{I}}_{1}-3\right)}-1\right)$ | $\begin{array}{l}{c}_{0}=0.5\mathrm{MPa},{c}_{1}=0.02\mathrm{MPa}\\ {c}_{2}=100\end{array}$ | [31,46] |

$W=\frac{{c}_{0}}{2}\left({\overline{I}}_{1}-3\right)+\frac{{c}_{1}}{2}\left(\omega {e}^{{c}_{2}{\left({\overline{I}}_{1}-3\right)}^{2}}+\left(1-\omega \right){e}^{{c}_{3}{\left({\overline{I}}_{4}-1\right)}^{2}}-1\right)$ | $\begin{array}{ll}{c}_{0}=117.137\mathrm{kPa},\hfill & {c}_{3}=132.455\mathrm{kPa},\hfill \\ {c}_{1}=41.435\mathrm{kPa},\hfill & \omega =0.988,\hfill \\ {c}_{2}=109.742\mathrm{kPa},\hfill & \theta ={45}^{\xb0}\hfill \end{array}$ | [33] |

Normal Aortic Valve Model | Calcified Aortic Valve Model | AVNeo Model | ||||
---|---|---|---|---|---|---|

Peak velocity (m/s) | 1.7 | 1.1–1.7 [82] * | 2.75 | 2.5–2.9 [83] * | 1.53 | – |

Mean TPG (mmHg) | 6.94 | 6.47 [78] * | 17.84 | 15 ± 4 [84] * | 6.18 | 6.36 [78] * |

Maximum 1st principal stress (kPa) | 85.5 | – | 77.8 | – | 83.7 | – |

Maximum 1st principal strain (%) | 21.5 | 17 [56] ** | 17.5 | 20 [56] ** | 52.9 | 37 [56] ** |

Average WSS (Pa) | 20.4 | 15–21.3 [85] * | 51.6 | 25 [49] ** | 23.8 | – |

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**MDPI and ACS Style**

Pil, N.; Kuchumov, A.G.; Kadyraliev, B.; Arutunyan, V.
Influence of Aortic Valve Leaflet Material Model on Hemodynamic Features in Healthy and Pathological States. *Mathematics* **2023**, *11*, 428.
https://doi.org/10.3390/math11020428

**AMA Style**

Pil N, Kuchumov AG, Kadyraliev B, Arutunyan V.
Influence of Aortic Valve Leaflet Material Model on Hemodynamic Features in Healthy and Pathological States. *Mathematics*. 2023; 11(2):428.
https://doi.org/10.3390/math11020428

**Chicago/Turabian Style**

Pil, Nikita, Alex G. Kuchumov, Bakytbek Kadyraliev, and Vagram Arutunyan.
2023. "Influence of Aortic Valve Leaflet Material Model on Hemodynamic Features in Healthy and Pathological States" *Mathematics* 11, no. 2: 428.
https://doi.org/10.3390/math11020428