# A Comparison of Newtonian and Non-Newtonian Models for Simulating Stenosis Development at the Bifurcation of the Carotid Artery

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Model

#### 2.1. The Lattice Boltzmann Method

#### 2.2. Non-Newtonian Simulations

#### 2.3. Boundary Conditions

- The LBM simulation is run until it reaches equilibrium using the healthy artery model.
- A single period of the flow is simulated.
- The haemodynamic parameter (TAWSS or O:WS) is monitored over the period at all wall-adjacent fluid sites.
- The site where TAWSS has a minimum value (or O:WS a maximum) is selected.
- The artery wall is moved into the fluid at the selected site by a distance of 0.3 of the grid length (this may move it beyond the fluid site, which will then become a wall site).
- Steps 1–5 are repeated.

## 3. Model Parameters

## 4. Results and Discussion

#### 4.1. Stenosis Development

#### 4.2. Near-Wall Haemodynamics

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

BGK | Bhatnagar–Gross–Krook |

CCA | Common Carotid Artery |

C-Y | Carreau–Yasuda |

ECA | External Carotid Artery |

ICA | Internal Carotid Artery |

LBM | Lattice Boltzmann Method |

OSI | Oscillatory Shear Index |

O:WS | OSI/TAWSS |

RFI | Reverse Flow Index |

RRT | Relative Residency Time |

TA | Time-Averaged |

TAWSS | Time-Averaged Wall Shear Stress |

WSS | Wall Shear Stress |

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**Figure 1.**Carotid artery velocity waveform, obtained from Holdsworth et al. [56].

**Figure 2.**Artery geometry. The healthy artery is depicted by the solid lines and the stenosed artery by the dotted lines. Also shown are sections ${S}_{1}$–${S}_{6}$, which are used to present the development of the stenosis.

**Figure 3.**The extrapolation scheme. The fluid sites are shown as filled circles and the wall sites as open circles. They are separated by a physical boundary represented by the solid line. Also shown are the sites ${\mathbf{x}}_{f}$,${\mathbf{x}}_{w}$ and ${\mathbf{x}}_{ff}$, which are used in the extrapolation boundary scheme, and the normalised distance ${\delta}^{*}$.

**Figure 4.**Layer development of the stenosis using TAWSS as a marker for (

**a**) the Newtonian model and (

**b**) the non-Newtonian C-Y model.

**Figure 5.**Layer development of the stenosis using O:WS as a marker for (

**a**) the Newtonian model and (

**b**) the non-Newtonian C-Y model.

**Figure 6.**Stenosis development using TAWSS as a marker for (

**a**) Newtonian model, (

**b**) non-Newtonian C-Y model and (

**c**) their comparison.

**Figure 7.**Stenosis development using O:WS as a marker for (

**a**) Newtonian model, (

**b**) non-Newtonian C-Y model and (

**c**) their comparison.

**Figure 8.**Area removed from (

**a**) the ECA and (

**b**) the ICA by the developing stenosis using TAWSS and O:WS as markers for both the Newtonian and Non-Newtonian C-Y models.

**Figure 9.**Change in artery diameter as the plaque is deposited and the stenosis develops along the sections (

**a**) ${S}_{1}$, (

**b**) ${S}_{2}$, (

**c**) ${S}_{3}$, (

**d**)${S}_{4}$, (

**e**) ${S}_{5}$ and (

**f**) ${S}_{6}$ using TAWSS and O:WS as a marker for both the Newtonian and non-Newtonian models.

**Figure 10.**Near-wall haemodynamics on the ECA over the development of stenosis using TAWSS as a marker: (

**a**,

**b**) $\overline{{u}^{\left(1\right)}}$; (

**c**,

**d**) $\overline{{u}_{t}^{\left(1\right)}}$; (

**e**,

**f**) OSI; (

**g**,

**h**) RRT; and (

**i**,

**j**) RFI, on the outer wall of the ECA, using a Newtonian model (left column) and non-Newtonian C-Y model (right column).

**Figure 11.**Near-wall haemodynamics on the ICA over the development of the stenosis using TAWSS as a marker: (

**a**,

**b**) $\overline{{u}^{\left(1\right)}}$; (

**c**,

**d**) $\overline{{u}_{t}^{\left(1\right)}}$; (

**e**,

**f**) OSI; (

**g**,

**h**) RRT; and (

**i**,

**j**) RFI, on the outer wall of the ICA, using a Newtonian model (left column) and non-Newtonian C-Y model (right column).

**Table 1.**Carotid artery parameters, based on Holdsworth et al. [56] and equivalent Boltzmann scaled parameters.

Parameter | Artery | Boltzmann (lu) |
---|---|---|

${u}_{0}$ | $1.07\mathrm{m}{\mathrm{s}}^{-1}$ | $0.08152$ |

D | $6.4\times {10}^{-3}\mathrm{m}$ | 36 |

$\nu ={\nu}_{\infty}$ | $3.5\times {10}^{-6}$ Pa · s | $0.0015$ |

T | $0.91\mathrm{s}$ | 66,987 |

$\alpha $ | $4.5$ | $4.5$ |

$R{e}_{\delta}$ | 307 | 307 |

**Table 2.**C-Y model parameters from Abraham et al. [67] and equivalent Boltzmann scaled parameters.

Parameter | SI | Boltzmann (lu) |
---|---|---|

${\mu}_{0}$ | $2.17\times {10}^{-4}$ Pa · s | $0.068364$ |

${\mu}_{\infty}$ | $3.5\times {10}^{-6}$ Pa · s | $0.0015$ |

$\lambda $ | $1.5\phantom{\rule{3.33333pt}{0ex}}{\mathrm{s}}^{-1}$ | 603,619 |

a | $0.64$ | $0.64$ |

n | $0.2128$ | $0.2128$ |

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**MDPI and ACS Style**

Stamou, A.C.; Radulovic, J.; Buick, J.M.
A Comparison of Newtonian and Non-Newtonian Models for Simulating Stenosis Development at the Bifurcation of the Carotid Artery. *Fluids* **2023**, *8*, 282.
https://doi.org/10.3390/fluids8100282

**AMA Style**

Stamou AC, Radulovic J, Buick JM.
A Comparison of Newtonian and Non-Newtonian Models for Simulating Stenosis Development at the Bifurcation of the Carotid Artery. *Fluids*. 2023; 8(10):282.
https://doi.org/10.3390/fluids8100282

**Chicago/Turabian Style**

Stamou, Aikaterini C., Jovana Radulovic, and James M. Buick.
2023. "A Comparison of Newtonian and Non-Newtonian Models for Simulating Stenosis Development at the Bifurcation of the Carotid Artery" *Fluids* 8, no. 10: 282.
https://doi.org/10.3390/fluids8100282