# Results of Numerical Modeling of Blood Flow in the Internal Jugular Vein Exhibiting Different Types of Strictures

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

_{1}) and gap between valve leaflets (d

_{2}) are shown in Figure 2B. The geometry is discretized into free triangular components by considering all quality measures, which include optimal skewness, orthogonality, and aspect ratio (Figure 2C).

**u**= U

_{inlet}and p

_{outlet}= 0 Pa) were applied at the inlet and outlet boundaries of the modeled blood vessels. The whole domain was discretized into a further two subdomains, blood (Domain 1) and venous valve (Domain 2). The turbulent flow modeling was coupled with solid mechanics module using the fluid–structure interaction approach, as the Reynold number (Re) exceeds 2000 in all cases, where the constitutive relation for blood shear stress to shear strain was considered Newtonian. The nomenclature is described in Table 1.

^{−3}(density of blood at 37 °C [31,32]) and the dynamic viscosity of fluid 2.78 × 10

^{−3}Pa·s (viscosity of blood at 37 °C). The flow velocity was 16 ± 4 cm/s, which is a typical IJV velocity in humans in the supine body position [33]. The density of the valve leaflets was set at 1200 kg m

^{−3}[34]. To simulate fluctuations of flow velocity in the modeled vein in a living subject resulting from the pulsation of the adjacent carotid artery and respiratory movements, the sine function of velocity at the inlet (Boundary 1) of the model was applied (Figure 2A). This sine function plotted the period of fluid velocity against time, as shown in Equation (7). The initial velocity of the fluid was 12 cm/s, which increased up to 16 cm/s (mean velocity) at t = 1.5 s and finally reached 20 cm/s (maximum velocity of blood) at t = 3 s, whereas it dropped back to 12 cm/s at t = 6 s.

_{solid}= 0 cm/s condition, while the rest of the valve was allowed to move freely within the fluid regime. All simulations were continued until 6 s and the graphical representations of the flow were analyzed. All the computations were executed in the Intel-INSPIRON (Intel, Santa Clara, CF, USA) equipped with the Intel-R Core-TM i7-11 Gen processor and the Intel-R Iris-XR Plus graphic card, and it took around 11 h of computational time for one case.

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The internal jugular vein with stricture in its upper part (red arrow), blue arrow points to the location of the jugular valve.

**Figure 2.**The scheme of the 2-dimensional model of the internal jugular vein and symmetrical 2-leaflets hyperelastic valve downstream; flow is from left to the right; this idealized model resembles real internal jugular vein with tandem stenosis from Figure 1. (

**A**) without stenosis; (

**B**) with 60% rigid stenosis at the beginning of this blood vessel (d

_{1}) and flexible valve (d

_{2}) downstream; and (

**C**) free-triangular mesh with boundary layer.

**Figure 3.**Mesh independence study: maximum blood flow velocity against the number of mesh elements in the domain.

**Figure 4.**Models of the internal jugular vein and symmetric 2-leaflets valve with velocity profiles at t = 0.25 s: (

**A**) without upstream stenosis; (

**B**) with 30% upstream stenosis; (

**C**) with 60% upstream stenosis; and (

**D**) with 75% upstream stenosis.

**Figure 5.**Graphical representation of flow with velocity profiles at t = 1.5 s: (

**A**) without upstream stenosis; (

**B**) with 30% upstream stenosis; (

**C**) with 60% upstream stenosis; and (

**D**) with 75% upstream stenosis.

**Figure 6.**Graphical representation of flow with velocity profiles at t = 3 s: (

**A**) without upstream stenosis; (

**B**) with 30% upstream stenosis; (

**C**) with 60% upstream stenosis; and (

**D**) with 75% upstream stenosis.

**Figure 7.**Asymmetric bending of valve leaflets (red) by vortices in the model D (with 75% upstream stenosis).

**Figure 8.**Comparison of maximum velocity (20 cm/s) profiles against the length of the modeled internal jugular vein, recorded at t = 3 s.

Quantity (Symbol) | Unit | Quantity (Symbol) | Unit |
---|---|---|---|

Density of blood (ρ) | kg m^{−3} | Partial derivative $\left(\partial \right)$ | – |

Time (t) | s | Fluid velocity (u) | m s^{−1} |

Fluid pressure (p) | Pa | Turbulent intensity (I) | – |

Dynamic viscosity (μ) | Pa s | External forces (F) | N |

Turbulent kinetic energy (k) | J | Turbulent viscosity (μ_{T}) | kg^{2} s m^{−3} |

Specific dissipation rate (ω) | J kg^{−1} s^{−1} | Mean rotation-rate tensor (Ω_{ij}) | – |

Mean strain-rate tensor (S_{ij}) | – | Venous valve density (${\rho}_{s}$) | kg m^{−3} |

Displacement vector (u_{d}) | m | Deformation gradient (F) | – |

Stress tensor (S, σ) | N m^{−2} | Volume force (F_{V}) | N m^{−3} |

Elastic volume ratio (J_{el}) | – | Identity tensor (I) | – |

External stress tensor (S_{ext}) | N m^{−2} | Strain energy density (W_{s}) | J |

Lamé’s second parameter (μ) | MPa | Lamé’s first parameter (λ) | MPa |

Modulus of elasticity (E) | MPa | Poisson’s ratio (ν) | – |

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**MDPI and ACS Style**

Rashid, A.; Iqrar, S.A.; Rashid, A.; Simka, M.
Results of Numerical Modeling of Blood Flow in the Internal Jugular Vein Exhibiting Different Types of Strictures. *Diagnostics* **2022**, *12*, 2862.
https://doi.org/10.3390/diagnostics12112862

**AMA Style**

Rashid A, Iqrar SA, Rashid A, Simka M.
Results of Numerical Modeling of Blood Flow in the Internal Jugular Vein Exhibiting Different Types of Strictures. *Diagnostics*. 2022; 12(11):2862.
https://doi.org/10.3390/diagnostics12112862

**Chicago/Turabian Style**

Rashid, Anas, Syed Atif Iqrar, Aiman Rashid, and Marian Simka.
2022. "Results of Numerical Modeling of Blood Flow in the Internal Jugular Vein Exhibiting Different Types of Strictures" *Diagnostics* 12, no. 11: 2862.
https://doi.org/10.3390/diagnostics12112862