# Computational Study of Hemodynamic Field of an Occluded Artery Model with Anastomosis

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Model Geometry

_{0}is the point for placing the respective geometric equation which depends on the position and geometry of the stenosis, x is the length point for the calculation of y coordinate point.

_{0}term had to be equal to 21 and 3 respectively for the creation of the two curves, so that F

_{1}(0) = 24 mm and F

_{2}(0) = 0 mm. The stenosis curves are presented in Figure 3.

#### 2.2. Governing Equations

_{1ε}= 1.42 and C

_{2ε}= 1.68.

_{eff}/μ and ${C}_{\nu}$ ≈ 100.

_{t}= ρk/(μω), R

_{k}= 6, ${\alpha}_{0}^{*}$ = β

_{i}/3 and β

_{i}= 0.072.

#### 2.3. Numerical Model

#### 2.4. Grid Sensitivity Analysis

#### 2.5. Boundary Conditions

_{blood}= 3.5 cP) and density of ρ = 1060 kg/m

^{3}. Numerical simulations were conducted assuming the same blood properties, with steady flow inlet conditions, with total flow rate of Q = 220 lt/h.

## 3. Results

#### 3.1. Qualitative Comparison of Flow Models

#### 3.2. Velocity Field in the Artery Anastomosis Region

_{by-pass}= 100%) led to the impingement of fluid on artery walls as it inserted the aorta via the bypass graft, while the existence of stagnant fluid located downwards the stenosis resulted in the creation of a wide recirculation zone, accompanied by correspondingly high levels of vorticity and shear of the flow field. Additionally, high values of mean longitudinal and radial velocities were observed in the impingement and infusion regions of the flui, respectivelyy.

_{by-pass}= 85% intense flow mixing was observed but with lower levels of vorticity and shear when compared to the previous case, still strong fluid impingement appeared on artery walls as it emerged from the bypass graft.

_{by-pass}= 70% the topology of the flow field showed significant change. In this scenario, there was no fluid impingement on artery walls because of the high amount of flow emerging from the stenosis region, occupying 25% of the artery cross-section area in the mixing region of the two flows. In this region, high levels of shear were observed due to the two flow interactions and subsequent acceleration of the fluid from the stenosis region, due to the reduction of the effective area in which the fluid flows.

_{by-pass}= 53%, the topology of the flow field is almost the same as in the previous case with the flow ratio of $\dot{m}$

_{by-pass}= 70% but with smaller levels of shear due to a higher ratio of the flow from the stenosis region that leads to a 50% occupation of the artery area in the mixing region of two flows.

#### 3.3. Cross Sectional Velocity Distribution

_{by-pass}= 100% and 85%, the flow formed a rolling up due to the fact that the flow field in both cases was formed mainly by the bypass graft. Moreover, as expected, in the case of $\dot{m}$

_{by-pass}= 100%, the flow rolling up was more intense than that of $\dot{m}$

_{by-pass}= 85%.

_{by-pass}= 70% and 53% the topology of the flow field was quite different. In these two cases, the flow from stenosis occupied a significant amount of the aorta area and down the anastomosis region, the flow twisted. More specifically, when the flow ratios of $\dot{m}$

_{by-pass}= 70% the topology change of the flow field in contrast with the two previous cases was related to the fact that twisting characteristics of the flow appear in the mixing area of the two flows. with respect to the axis of symmetry of the vessel, in which the transverse component of velocity takes the same measure but opposite sign. In the case of $\dot{m}$

_{by-pass}= 53%, flow twisting became more pronounced.

#### 3.4. Wall Shear Stresses (WSS)

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 4.**(

**a**) and (

**b**) grid sections with tetrahedral elements, (

**c**) grid segment with hexahedral elements.

**Figure 5.**Streamwise x velocity profiles at two downstream stations from the stenosis (

**a**) 0.5D, (

**b**) 8.0D.

**Figure 6.**Streamwise x velocity profiles for two condensed grids at two downstream stations from the stenosis (

**a**) 0.5D, (

**b**) 8.0D.

**Figure 10.**Comparison between computational and experimental visualization results in the anastomosis region (

**a**) Computational results of the longitudinal velocity component, (

**b**) experimental visualization results.

**Figure 11.**Identification of the cross sections for computational result presentation along with the artery model.

**Figure 12.**Streamwise velocity distribution in three cross sections with different stenosis/anastomosis flow rate ratios.

**Figure 13.**(

**A**) Cross-sectional schematic diagram of a blood vessel illustrating hemodynamic shear stress, τ

_{s}, the frictional force per unit area acting on the inner vessel wall and on the luminal surface of the endothelium as a result of the flow of viscous blood. (

**B**) Tabular diagram illustrating the range of shear stress magnitudes encountered in veins, arteries, and in low-shear and high-shear pathologic states. Adapted from Malek [52].

Grids | Number of Elements | Number of Nodes |
---|---|---|

Grid 1 | 7.65·10^{5} | 4.28·10^{5} |

Grid 2 | 1.15·10^{6} | 5.81·10^{5} |

Grid 3 | 2.45·10^{6} | 1.23·10^{6} |

Grid 4 | 3.17·10^{6} | 1.63·10^{6} |

Cases | Flow Rate Proportion | Calculation Positions | Velocity (m/s) | Re |
---|---|---|---|---|

1st | 47% | A | 0.063 | 482 |

53% | B | 0.072 | 544 | |

47% | C | 0.254 | 964 | |

100% | D | 0.135 | 1026 | |

2nd | 30% | A | 0.041 | 308 |

70% | B | 0.095 | 718 | |

30% | C | 0.162 | 616 | |

100% | D | 0.135 | 1026 | |

3rd | 15% | A | 0.02 | 154 |

85% | B | 0.115 | 872 | |

15% | C | 0.081 | 308 | |

100% | D | 0.135 | 1026 | |

4th | 0% | A | 0 | 0 |

100% | B | 0.135 | 1026 | |

0% | C | 0 | 0 | |

100% | D | 0.135 | 1026 |

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**MDPI and ACS Style**

Parissis, P.; Romeos, A.; Giannadakis, A.; Kalarakis, A.; Peroulis, M.
Computational Study of Hemodynamic Field of an Occluded Artery Model with Anastomosis. *Bioengineering* **2023**, *10*, 146.
https://doi.org/10.3390/bioengineering10020146

**AMA Style**

Parissis P, Romeos A, Giannadakis A, Kalarakis A, Peroulis M.
Computational Study of Hemodynamic Field of an Occluded Artery Model with Anastomosis. *Bioengineering*. 2023; 10(2):146.
https://doi.org/10.3390/bioengineering10020146

**Chicago/Turabian Style**

Parissis, Panagiotis, Alexandros Romeos, Athanasios Giannadakis, Alexandros Kalarakis, and Michail Peroulis.
2023. "Computational Study of Hemodynamic Field of an Occluded Artery Model with Anastomosis" *Bioengineering* 10, no. 2: 146.
https://doi.org/10.3390/bioengineering10020146