# Endovascular Treatment of Intracranial Aneurysm: The Importance of the Rheological Model in Blood Flow Simulations

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Patient-Specific Model

#### 2.2. Governing Equations and Numerical Setup

**,**$D$= (∇$\overrightarrow{u}$+ ∇$\overrightarrow{u}$

^{T})/2, according to the relation:

_{0}= 0.056 kg/(m·s) is the viscosity at zero shear rate $\dot{\gamma}$, μ

_{∞}= 0.0035 kg/(m·s) is the viscosity for an infinite shear rate, λ = 3.313 s is the relaxation time, and n = 0.3568 the power-law index [27].

_{0}= 0.16 kg/(m·s), μ

_{∞}= 0.0035 kg/(m·s), λ = 8.2 s, a = 0.64, and n = 0.2128 [28].

^{3}. A physiological velocity boundary condition was imposed at the inlet of the model (Figure 2) [29]. The period of the velocity waveform was equal to 0.8 s, the maximum velocity occurred at the systolic peak instant t = 0.16 s, and the minimum velocity was observed at the diastolic instant t = 0.432 s. Further significant instants were considered for describing hemodynamics inside the aneurysm, as illustrated in Figure 2. The instants t = 0.096 s, t = 0.112 s, and t = 0.208 s referred to systolic phase, and the instant t = 0.64 s referred to diastolic phase.

_{∞}, the diameter D of the healthy artery at model inlet (D = 0.00427 m), and the velocity U assigned at the inlet. It had an averaged value of about Re

_{ave}≈ 134 (corresponding to the time-averaged velocity) and a maximum value Re

_{max}≈ 257, corresponding to the systolic peak velocity. Based on these values, a laminar blood flow was assumed.

#### 2.3. Mesh Sensitivity Analysis

^{6}elements in the absence of flow diverter stent and 5 × 10

^{6}elements in the presence of the device. These values were obtained using a tetrahedral element size equal to 0.12 mm in the lumen in both cases, and 0.07 mm in the proximity of the flow diverter stent if inserted.

^{6}elements. For the three considered time steps, the WSS at systolic peak and TAWSS, both of them averaged on the aneurysmal wall, were compared to determine the optimal time step. The difference between results obtained with the two smallest time steps was less than 0.7% for WSS and less than 1.5% for TAWSS. Thus, the time step size 0.008 s was selected for the numerical simulations, and 200 iterations were performed for each time step. Furthermore, the convergence criteria for the residuals of velocity components and continuity were set to 10

^{−5}at each time-step.

#### 2.4. Hemodynamic Parameters

## 3. Results

#### 3.1. The Effect of the Rheological Model on WSS

#### 3.2. The Effect of the Rheological Model on Intra-Aneurysm Blood Flow

#### 3.3. The Effect of the Rheological Model on Hemodynamic Parameters

#### 3.4. The Effect of FDS Placement on Intra-Aneurysm Blood Flow

## 4. Study Limitations

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**CTA image, aneurysm reconstructed model, and the model with virtual FDS placement. The red circle in the CTA image highlights the patient-specific aneurysm.

**Figure 2.**Pulsatile velocity waveform assigned at the inlet of the intracranial aneurysm model. The red points in the Figure indicate the instants of the cardiac cycle that were selected for the hemodynamic investigation.

**Figure 3.**Dimensionless TAWSS along the selected middle curve in the patient-specific aneurysm for four different mesh sizes (very coarse, coarse, medium, fine). (

**a**) Refers to the aneurysm with no FDS placement and with number of mesh elements approximately equal to 0.5 × 10

^{6}, 1 × 10

^{6}, 2 × 10

^{6}, and 3 × 10

^{6}. (

**b**) Refers to the aneurysm with FDS placement and number of mesh elements approximately equal to 1.6 × 10

^{6}, 3 × 10

^{6}, 4 × 10

^{6}, and 5 × 10

^{6}. (

**c**) Shows the plots of the mesh convergence. All values refer to the Carreau rheological model.

**Figure 4.**WSS values averaged on the wall of the aneurysmal dilatation at selected instants of the cardiac cycle obtained using different rheological models.

**Figure 5.**2D streamlines in the patient-specific aneurysm with and without FDS insertion obtained with different rheological models for blood: (

**a**) refers to the systolic velocity peak (t = 0.16 s), (

**b**) refers to the diastolic velocity minimum (t = 0.432 s), (

**c**) refers to the late diastole (t = 0.64 s). The first column shows the results obtained when adopting the Newtonian model in the numerical simulation; the second and third columns refer to the Carreau and Carreau–Yasuda models, respectively.

**Figure 6.**TAWSS contours in the absence of FDS placement (

**first row**) and in the presence of virtual FDS placement (

**second row**). First box shows results obtained adopting the Newtonian model; second box shows those obtained with the Carreau and Carreau–Yasuda non-Newtonian models (I and II column, respectively).

**Figure 7.**OSI contours in the absence of FDS placement (

**first row**) and in the presence of virtual FDS placement (

**second row**). First box shows results obtained adopting the Newtonian model; second box shows those obtained with the Carreau and Carreau–Yasuda non-Newtonian models (I and II column, respectively).

**Figure 8.**ECAP contours in the absence of FDS placement (

**first row**) and in the presence of virtual FDS placement (

**second row**). First box shows results obtained adopting the Newtonian model; second box shows those obtained with the Carreau and Carreau–Yasuda non-Newtonian models (I and II column, respectively).

**Figure 9.**RRT contours in the absence of FDS placement (

**first row**) and in the presence of virtual FDS placement (

**second row**). First box shows results obtained adopting the Newtonian model; second box shows those obtained with the Carreau and Carreau–Yasuda non-Newtonian models (I and II column, respectively).

**Figure 10.**TAWSS OSI, ECAP, and RRT values averaged on the aneurysmal surface in both the presence and absence of endovascular treatment (green bars and blue bars, respectively).

**Figure 11.**3D streamlines and velocity magnitude contours on a longitudinal section of the aneurysm at the systolic peak instant t = 0.16 s, the diastolic minimum instant t = 0.432 s and the late diastole instant t = 0.64 s obtained with the Carreau rheological models. First column refers to the case of no FDS placement; second column shows results in the presence of FDS within the aneurysm. The streamlines are colored by the local velocity magnitude.

**Figure 12.**TAWSS, OSI, ECAP, and RRT contours in the absence of FDS placement and in the presence of virtual FDS placement obtained adopting the non-Newtonian Carreau model.

**Table 1.**TAWSS averaged on the middle curve of the aneurysm, for different numbers of mesh elements (values refer to Carreau rheological model).

No FDS Placement | FDS Placement | ||||
---|---|---|---|---|---|

Elements(approximate number) | TAWSS (Pa) | Variation(%) | Elements(approximate number) | TAWSS (Pa) | Variation(%) |

0.5 × 10^{6} | 0.110436 | 1.6 × 10^{6} | 0.0830499 | ||

1 × 10^{6} | 0.110909 | 0.43 | 3 × 10^{6} | 0.064836 | −21.93 |

2 × 10^{6} | 0.112039 | 1.09 | 4 × 10^{6} | 0.065274 | 0.68 |

3 × 10^{6} | 0.111874 | −0.15 | 5 × 10^{6} | 0.064842 | 0.66 |

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**MDPI and ACS Style**

Boniforti, M.A.; Vittucci, G.; Magini, R.
Endovascular Treatment of Intracranial Aneurysm: The Importance of the Rheological Model in Blood Flow Simulations. *Bioengineering* **2024**, *11*, 522.
https://doi.org/10.3390/bioengineering11060522

**AMA Style**

Boniforti MA, Vittucci G, Magini R.
Endovascular Treatment of Intracranial Aneurysm: The Importance of the Rheological Model in Blood Flow Simulations. *Bioengineering*. 2024; 11(6):522.
https://doi.org/10.3390/bioengineering11060522

**Chicago/Turabian Style**

Boniforti, Maria Antonietta, Giorgia Vittucci, and Roberto Magini.
2024. "Endovascular Treatment of Intracranial Aneurysm: The Importance of the Rheological Model in Blood Flow Simulations" *Bioengineering* 11, no. 6: 522.
https://doi.org/10.3390/bioengineering11060522