# Impact of Multi-Grade Localized Calcifications on Aortic Valve Dynamics under Helical Inflow: A Comparative Hemodynamic Study

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Mathematical Modeling

**x**$=(x,y,z)\in \Omega $. The initial coordinate system for the structure is denoted as $\mathit{U}\subset {\mathbb{R}}^{3}$, and the curvilinear Lagrangian coordinates are represented as $(q,r,s)\in U$. At a given time t, the position of a material point $\mathbf{X}$ is described by $\chi (\mathbf{X},t)$. Therefore, the fluid and the structure occupy the domains ${\Omega}_{f}\left(t\right)=\Omega \setminus {\Omega}_{s}$ and $\chi (\mathbf{X},t)={\Omega}_{s}$, respectively. The governing equations for the coupled system are as follows:

#### 2.2. Material Model

#### 2.3. Geometric Model

- Grade 1 corresponds to the absence of calcification;
- Grade 2 associates with specific regions displaying elevated reflectivity (calcification) but no zones of intense calcification;
- Grade 3 refers to a substantial enhancement in one leaflet’s reflectivity, with other leaflets showing comparable or less pronounced changes than those in Grade 2;
- Grade 4 characterizes by a significant increase in reflectivity in two leaflets, with the third leaflet exhibiting equivalent or lesser modifications than those noted in Grade 2;
- Grade 5 signifies a moderate enhancement in reflectivity across all leaflets;
- Grade 6 corresponds to a severe increase in reflectivity throughout all leaflets.

#### 2.4. Numerical Schemes and Boundary Conditions

#### 2.4.1. Fluid and Structure Domains Mesh

#### 2.4.2. Constraints and Boundary Conditions

#### 2.4.3. Numerical Implementation

^{®}-based network, AMD EPYC 7742 2.24 GHz CPUs, and 256 GB of RAM. Each typical simulation took approximately 240 h of wall clock time on one node with 128 cores, equivalent to roughly 30,000 CPU hours.

#### 2.5. Comprehensive Quantification of Transvalvular Hemodynamic, Wall Shear Stress-Based Indices, and Helicity Descriptors

## 3. Results

#### 3.1. Valve Kinematics

#### 3.2. Hemodynamic Flow Patterns in the Presence of Aortic Valve Calcification

#### 3.3. Transvalvular Hemodynamic Indices

#### 3.4. WSS-Based Hemodynamic Indices

#### 3.5. Quantification of the Helicity

#### 3.6. Vortex Dynamics

## 4. Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Cross-sectional view of the computational mesh for the fluid domain and native aortic valve model, demonstrating the inner structure and housing.

**Figure 3.**Detailed view of structural meshes for aortic valves with varying degrees of calcification and housing: (

**a**) mesh for Grade 3; (

**b**) mesh for Grade 4; (

**c**) mesh for Grade 5; (

**d**) cut-away view of the reference housing with no calcification; (

**e**) mesh for Grade 6.

**Figure 4.**The through-plane and in-plane inflow velocity profiles for the healthy case and the corresponding aortic pressure [38].

**Figure 5.**Mesh independence analysis; horizontal axis is non-dimensionalized with the maximum radial length of the Valsalva sinus: (

**a**) average velocity measurements at arbitrary selected locations ($\mathrm{I}=0.07\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$ and $\mathrm{II}=0.08\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$); (

**b**) for average velocity magnitude profiles over a line at $y=0.07\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$; (

**c**) for average velocity magnitude profiles over the line at $y=0.08\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$.

**Figure 6.**Evolution of valve kinematics over a cardiac cycle for different grades of aortic valve calcification under uniform flow condition.

**Figure 7.**Evolution of valve kinematics over a cardiac cycle for different grades of aortic valve calcification under helical flow condition.

**Figure 8.**Three-dimensional velocity streamlines illustrating uniform blood flow in aortic valves across calcification grades.

**Figure 9.**Three-dimensional velocity streamlines illustrating helical blood flow in aortic valves across calcification grades.

**Figure 10.**Velocity magnitude vectors for various calcification cases under both uniform and helical flow conditions, shown at late systole with $t=0.109\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$.

**Figure 11.**Temporal variation of GOA for non-calcified and calcified aortic valves over a cardiac cycle for uniform and helical inflow.

**Figure 12.**Maximum fluid velocities through the orifice area in uncalcified and calcified conditions, under uniform and helical inflow regimes. Velocities measured at 0.068 cm from the inlet.

**Figure 13.**Variation in kinetic energy in the Valsalva sinus and ascending aorta across different calcification severities for uniform and helical inflow.

**Figure 14.**Energy dissipation rate in aortic valves with varying calcification under uniform and helical inflow conditions over a cardiac cycle.

**Figure 15.**Vorticity magnitude in aortic valves with different calcification levels under uniform and helical inflow during a cardiac cycle.

**Figure 16.**Distribution of TAWSS [Pa] on aortic valve leaflets, comparing varying degrees of calcification under uniform and helical inflow conditions.

**Figure 17.**OSI contours on aortic and ventricle sides for varying grades of calcification and inflow patterns (uniform and helical). Unit is dimensionless.

**Figure 18.**Distribution of RRT [${\mathrm{Pa}}^{-1}$] across leaflets for varying grades of calcification and inflow types (uniform and helical).

**Figure 19.**Distribution of transWSS [$\mathrm{Pa}$] values across different grades of leaflet calcification and flow types, highlighting increased directional fluctuations in calcified cases and under helical flow conditions.

**Figure 20.**Iso-surfaces of the instantaneous Q-criterion for the iso-value of 100,000 ${\mathrm{s}}^{-2}$ under uniform blood inflow.

Part | Material Model | Parameters |
---|---|---|

Aortic valve | Neo-Hookean, $\mu =140\phantom{\rule{0.166667em}{0ex}}\mathrm{kPa}$ | $\lambda =17\phantom{\rule{0.166667em}{0ex}}\mathrm{MPa}$ |

Calcifications | Linear elastic, $E=2\phantom{\rule{0.166667em}{0ex}}\mathrm{MPa}$ | $\nu =0.45$ |

Helicity Descriptors | Uniform | Helical | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

No Calcification | Grade 3 | Grade 4 | Grade 5 | Grade 6 | No Calcification | Grade 3 | Grade 4 | Grade 5 | Grade 6 | |

${h}_{1}$ | −0.204 | 0.352 | −1.381 | 0.989 | 1.206 | 14.550 | 9.368 | 3.816 | 6.803 | 5.372 |

${h}_{2}$ | 0.418 | 1.521 | 1.606 | 1.866 | 1.398 | 71.374 | 82.921 | 86.552 | 75.592 | 95.855 |

${h}_{3}$ | −0.488 | 0.231 | −0.860 | 0.530 | 0.863 | 0.204 | 0.113 | 0.044 | 0.090 | 0.056 |

${h}_{4}$ | 0.488 | 0.231 | 0.860 | 0.530 | 0.863 | 0.204 | 0.113 | 0.044 | 0.090 | 0.056 |

${h}_{5}$ | 0.996 | 1.024 | 0.983 | 1.025 | 1.013 | 1.261 | 1.101 | 1.051 | 1.039 | 0.955 |

${h}_{6}$ | −0.994 | −1.000 | −0.977 | −1.067 | −1.021 | −0.883 | −0.928 | −0.877 | −0.934 | −1.025 |

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**MDPI and ACS Style**

Daryani, R.; Ersan, E.C.; Çelebi, M.S.
Impact of Multi-Grade Localized Calcifications on Aortic Valve Dynamics under Helical Inflow: A Comparative Hemodynamic Study. *Appl. Sci.* **2023**, *13*, 12983.
https://doi.org/10.3390/app132412983

**AMA Style**

Daryani R, Ersan EC, Çelebi MS.
Impact of Multi-Grade Localized Calcifications on Aortic Valve Dynamics under Helical Inflow: A Comparative Hemodynamic Study. *Applied Sciences*. 2023; 13(24):12983.
https://doi.org/10.3390/app132412983

**Chicago/Turabian Style**

Daryani, Reza, Emre Cenk Ersan, and Mustafa Serdar Çelebi.
2023. "Impact of Multi-Grade Localized Calcifications on Aortic Valve Dynamics under Helical Inflow: A Comparative Hemodynamic Study" *Applied Sciences* 13, no. 24: 12983.
https://doi.org/10.3390/app132412983