Computational and Mathematical Methods in Medicine

Volume 2016, Article ID 7412926, 16 pages

http://dx.doi.org/10.1155/2016/7412926

## Effects of Reynolds and Womersley Numbers on the Hemodynamics of Intracranial Aneurysms

Department of Mechanical and Aerospace Engineering, University at Buffalo, The State University of New York, Buffalo, NY, USA

Received 3 May 2016; Revised 29 July 2016; Accepted 10 August 2016

Academic Editor: Giancarlo Ferrigno

Copyright © 2016 Hafez Asgharzadeh and Iman Borazjani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The effects of Reynolds and Womersley numbers on the hemodynamics of two simplified intracranial aneurysms (IAs), that is, sidewall and bifurcation IAs, and a patient-specific IA are investigated using computational fluid dynamics. For this purpose, we carried out three numerical experiments for each IA with various Reynolds ( to ) and Womersley ( to ) numbers. Although the dominant flow feature, which is the vortex ring formation, is similar for all test cases here, the propagation of the vortex ring is controlled by both and in both simplified IAs (bifurcation and sidewall) and the patient-specific IA. The location of the vortex ring in all tested IAs is shown to be proportional to which is in agreement with empirical formulations for the location of a vortex ring in a tank. In sidewall IAs, the oscillatory shear index is shown to increase with and because the vortex reached the distal wall later in the cycle (higher resident time). However, this trend was not observed in the bifurcation IA because the stresses were dominated by particle trapping structures, which were absent at low in contrast to higher .

#### 1. Introduction

The rupture of intracranial aneurysms (IAs) is highly associated with mortality and morbidity [1]. Hemodynamics has a significant role in the growth and rupture of IAs [2–4]. Among the hemodynamic factors, vortical structures determine the complexity and stability of the flow pattern in an IA dome, which plays an important role in the rupture of IAs [3, 5–7]. Computational fluid dynamics (CFD) holds an important position in the investigation of hemodynamic factors in aneurysms because of its higher resolution near the walls relative to experimental methods such as laser Doppler velocimetry, particle image velocimetry, and magnetic resonance imaging, which is required to compute hemodynamic factors such as shear stress correctly [8, 9]. Many investigations have been carried out on the hemodynamics of IAs using experimental methods [10–12] and CFD [7, 13, 14].

Reynolds and Womersley numbers (explained in Section 3) are the only two nondimensional parameters required for full dynamic similarity in pulsatile internal flows [15]. Therefore, the investigation of their effects on the hemodynamics of IAs is strongly required. However, contradictory conclusions have been made in the literature on this topic. In fact, Jiang and Strother [16] concluded that increase of Womersley number can significantly increase the complexity of the flow pattern and vortex structures in two patient-specific intracranial aneurysms based on their CFD simulations. In contrast, Le et al. [17] stated that Womersley number does not affect the flow feature and structures based on their CFD simulations of an IA from a rabbit. Furthermore, Gopalakrishnan et al. [18] stated that while Womersley number does not change the vortex mode, but high Womersley number is associated with weak vortex rings in their simulation on abdominal aortic aneurysms. Bouillot et al. [10] concluded that has a negligible effect on the flow structures of an idealized sidewall intracranial aneurysm for a steady inflow according to their PIV measurements. A similar conclusion was made by Le et al. [17] and Cebral et al. [19] based on their CFD simulations of cerebral aneurysms in rabbits and humans, respectively. In contrast, Gopalakrishnan et al. [18] stated that, by increasing , the strength of the main vortex structure increases.

Previous works [7, 19] have shown the significant effect of aneurysm geometry on the aneurysm hemodynamics, which is not investigated further here. Our aim is to compare the effects of the Reynolds and Womersley numbers on the hemodynamics of both sidewall and bifurcations IAs by keeping other parameters, for example, inlet flow waveform and geometry, constant. Furthermore, their effect on the hemodynamics is formulated using dimensional analysis and compared with the literature.

#### 2. Governing Equations and the Numerical Method

In this section, Einstein’s tensor notation, where repeated indices imply summation, is used unless otherwise indicated (, , ). The governing equations are the , unsteady incompressible Navier-Stokes equations for a Newtonian fluid in curvilinear coordinates as follows [20]:where , , and are the contravariant velocity, curvilinear coordinate, and Cartesian coordinates components, respectively. and are the nondimensional pressure and time, respectively. is the Jacobian of the geometric transformation, , and are the metrics of the transformation. is the nondimensional Cartesian velocity, is the contravariant metric tensor, , and is the Reynolds number of the flow based on characteristic length and velocity scales.

We use curvilinear/immersed boundary (CURVIB) and overset grid methods, which are extensively described and validated [20]. A sharp-interface immersed boundary method is used to handle the , arbitrary complex boundaries (IA geometry in this study) inside the flow domain [21]. The nodes that are outside the flow domain are blanked out and do not affect the solution. These nodes are identified using an efficient ray-tracing algorithm [22]. The boundary conditions are reconstructed on the fluid nodes in the immediate vicinity of the immersed boundary along the normal direction to the boundary [21]. The method has been shown to be second-order accurate for a variety of flows [21, 23]. The overset grid approach is implemented to reduce wasted nodes in a domain, which are blanked out by the immersed boundary method [20]. In this approach, a complex flow domain is divided into several arbitrary subgrids with overlaps. To solve the governing equations at each subgrid, boundary conditions at the interfaces are constructed by interpolation from host subgrid. The details of the overset-CURVIB method can be found in [20]. The method has been validated against experimental and benchmark solutions [20, 24] and has been applied to a variety of problems such as cardiovascular flows [17, 25–28], aquatic swimming [23, 29], and rheology [30]. Furthermore, we have validated our method for flows inside an immersed body by comparing our results with the measurements of the pulsatile flow through a bend in Appendix. As shown in Appendix, the computational results are in excellent agreement with our previous simulations using body-fitted grids [20] and the experimental results [31].

We have assumed rigid walls similar to previous simulations [2, 14, 17, 19] because the displacement of aneurysm’s wall is typically small and the flow patterns of small distensible and rigid models in the carotid region are very similar [32]. In addition, we have assumed Newtonian fluid in our simulations because the non-Newtonian effects are negligible in larger (>500 *μ*m) arteries [33] and previous simulations of both Newtonian and non-Newtonian fluids have shown similar flow patterns [19].

#### 3. Description of Simulated Test Cases

Numerical simulations have been carried out on two simplified geometries (sidewall and bifurcation), which can be considered as simplified models of IA, and a patient-specific IA geometry. Figure 1 shows the configuration of the simplified models for (a) sidewall and (b) bifurcation IAs. The inlet and outlet of the geometries are constructed from a pipe with the diameter . The aneurysm dome is modeled by a cone-like shape with elliptical base and locus, whose radiuses are , , and .