**2. Methods**

The first step necessary to study blood flow consisted in obtaining the threedimensional geometry of the lumen of an idealized coronary artery as depicted in **Figure 1**. This geometry was previously adopted in other works by these authors [27, 28].

In the construction of the solid domain, a thickness of 0.8 mm was considered, according to an *in vivo* study [29].

Moreover, the fluid domain and the solid domain were discretized in 428,800 and 15,480 hexahedral elements, respectively, making a total of 444,280 elements for the FSI simulations. To ensure the quality of the mesh, the skewness and orthogonal quality were evaluated. The average skewness and orthogonal quality were approximately 8.3 *e*<sup>2</sup> and 0.98, respectively. These parameters prove that the mesh is reliable for the study presented [30] and it is worth mentioning that a previous mesh study was carried out for CFD simulations [16], which was then adapted for this study.

#### **2.1 Mathematical formulation**

Blood flow is governed by the incompressible Navier-Stokes and the continuity equations as described in Eqs. (1) and (2),

*Comparison of CFD and FSI Simulations of Blood Flow in Stenotic Coronary Arteries DOI: http://dx.doi.org/10.5772/intechopen.102089*

**Figure 1.** *Geometry and mesh of the coronary artery for both solid and fluid domain with 50% of stenosis.*

$$\nabla \cdot \boldsymbol{u} = \mathbf{0} \tag{1}$$

$$
\rho \left( \frac{\partial \mathbf{u}}{\partial t} + \boldsymbol{u} \cdot \nabla \right) = -\nabla p + \mu \nabla^2 \boldsymbol{u} \tag{2}
$$

where *u* is the velocity, *p* is the static pressure, *ρ* is the fluid density, and *μ* the dynamic viscosity [30, 31].

In the present study, blood was modeled as incompressible, laminar, and non-Newtonian fluid, having a density of 1060 kg/m<sup>3</sup> [32]. Although it is commonly assumed as a Newtonian fluid, the ability of red blood cells to deform and aggregate makes blood a non-Newtonian fluid [33]. The well-known Carreau model was used to simulate the shear-thinning blood behavior, and it is defined by Eq. (3) [30, 31]:

$$
\mu = \mu\_{\infty} + (\mu\_0 - \mu\_{\infty}) \left[ \mathbf{1} + \lambda \mathbf{y}^{\frac{1}{2}} \right]^{\frac{n-1}{2}} \tag{3}
$$

where *μ* is the viscosity, *μ*<sup>∞</sup> ¼ 0*:*00345 Pa∙s is the infinite shear viscosity, *μ*<sup>0</sup> ¼ 0*:*0560 Pa � s is the blood viscosity at zero shear rate, *γ*\_is the instantaneous shear rate, *λ* ¼ 3*:*313 s is the time constant and *n* ¼ 0*:*3568 is the power-law index, as previously applied by [27].

The governing equation for the solid domain is the equilibrium equation (Eq. (4)) [30, 31]:

$$
\rho\_s \frac{\partial^2 u}{\partial t^2} - \nabla \stackrel{\equiv}{\bar{\sigma}} = \rho\_s \vec{b} \tag{4}
$$

where *ρ<sup>s</sup>* is the solid density, *u* represents the solid displacements, *b* ! , the body forces applied on the structure, and *σ* <sup>¼</sup> is the Cauchy stress tensor. For an isotropic linear elastic solid, the stress tensor is represented by Eq. (5) [30, 31]:

*Applications of Computational Fluid Dynamics Simulation and Modeling*

$$\stackrel{\equiv}{\sigma} = \mathcal{D}\,\mu\_L \stackrel{\equiv}{\bar{\varepsilon}} + \lambda\_L \text{tr}\left(\stackrel{\equiv}{\bar{\varepsilon}}\right) I \tag{5}$$

where *λ<sup>L</sup>* and *μ<sup>L</sup>* are the first and second Lamé parameters, respectively, *ε* ¼ , the strain tensor, *tr*, the trace function, and *I*, the identity matrix. For compressible materials, Lamé parameters can be written as a function of Young's modulus, *E*, and Poisson's coefficient, *v*, as follows.

$$
\lambda\_L = \frac{vE}{(1+v)(2v-1)}\tag{6}
$$

$$
\mu\_L = \frac{E}{2\left(1+v\right)}\tag{7}
$$

The arterial wall was modeled as a linear elastic, incompressible, isotropic, and homogeneous material with Young's modulus of 3*:*77 MPa [34], a density of 1120 kg*=*m<sup>3</sup> [35], and a Poisson's ratio of 0*:*49 [21].

The FSI simulations were performed using the Arbitrary-Lagrangian-Eulerian (ALE) methodology for the fluid flow. Taking into account that the interface between the lumen and the wall deforms, the equations governing fluid flow have to be expressed in terms of the fluid variables relative to the mesh movement. The ALE-modified Navier-Stokes momentum equation for a viscous incompressible flow is described as follows in Eq. (8) [30, 31]:

$$
\rho\_f \left( \frac{\partial u}{\partial t} + ((u - u\_\S) \cdot \nabla)u \right) = -\nabla p + \mu \nabla^2 u \tag{8}
$$

where *ρ <sup>f</sup>* , *p*, *u*, and *ug* are the fluid density, the pressure, the fluid velocity, and the moving coordinate velocity, respectively. The term *u* � *ug* , in the ALE formulation, is added to the conventional Navier-Stokes equations to account for the movement of the mesh.

The displacement and equilibrium forces at the interface are represented by Eqs. (9) and (10) [30, 31]:

$$
\mathfrak{u}\_{f,\Gamma} = \mathfrak{u}\_{\mathfrak{s},\Gamma} \tag{9}
$$

$$
\overrightarrow{\mathbf{t}\_{f,\Gamma}} = \overrightarrow{\mathbf{t}\_{\mathbf{t},\Gamma}}\tag{10}
$$

where *u <sup>f</sup>*,*<sup>Γ</sup>* is the displacement of the fluid at the interface, *us*,*<sup>Γ</sup>*, the displacement of the solid at the interface, *t <sup>f</sup>*,*<sup>Γ</sup>* !, the forces of the fluid on the interface and *ts*,*<sup>Γ</sup>* !, the forces of the solid on the interface.

#### **2.2 Boundary conditions**

Regarding the boundary conditions used in this study, at the inlet, a physiologically accurate pulsatile velocity profile was set, which is depicted in **Figure 2**. At the outlet, a pressure of 80 mmHg was assumed [17, 36].

The solid and fluid wall-boundaries were defined as a fluid-structure interface, and the inlet/outlet adjacent solid boundaries were fixed in all directions.

#### **2.3 Numerical solution**

For the CFD simulations, the Ansys Fluent software was used which applies the finite-volume discretization method. In this method, the fluid domain is divided

*Comparison of CFD and FSI Simulations of Blood Flow in Stenotic Coronary Arteries DOI: http://dx.doi.org/10.5772/intechopen.102089*

**Figure 2.** *Velocity profile implemented in CFD and FSI simulations.*

into a finite number of control volumes, the conservation equations are applied to each control volume. Then, a system of algebraic equations for the variables is obtained. For the velocity-pressure coupling, the semi-implicit method for the pressure-linked equations (SIMPLE) scheme was used [37].

In FSI simulations, the same finite-volume method is applied in the fluid domain, and the computed forces in Fluent are transferred to the solid domain, through the interface. The finite element method (FEM) is used to solve the governing equations of the solid domain. Then, the computed displacements are transferred back to the fluid domain. This two-way coupling process was repeated until the difference of the displacements and forces for the last two iterations is below 1%. A time step of 0.01 s was used for every simulation.

#### **2.4 Hemodynamic parameters**

The formation of atherosclerotic lesions has been widely associated with hemodynamic parameters, such as the wall shear stress (WSS) and its indices, timeaveraged wall shear stress (TAWSS), and oscillatory shear index (OSI) [38, 39]. These have been very useful to predict and estimate disturbed flow conditions and the development of local atherosclerotic plaques [27, 40].

The spatial WSS, *τw*, is calculated by Eq. (11), being γ *:* , the deformation rate, and *μ* the dynamic viscosity.

$$
\pi\_w = \mu \frac{\partial u}{\partial \mathbf{y}} = \mu \,\dot{\mathbf{y}} \tag{11}
$$

The TAWSS index allows obtaining an average temporal evaluation of the WSS exerted during a cardiac cycle (*T*) [40]. This is calculated by Eq. (12):

$$TAWSS = \frac{1}{T} \int\_{0}^{T} |WSS|dt\tag{12}$$

The OSI index is the temporal fluctuation of low and high average shear stress during a cardiac cycle (*T*) and it is calculated by applying Eq. (13):

*Applications of Computational Fluid Dynamics Simulation and Modeling*

$$OSI = \frac{1}{2} \left( \mathbf{1} - \frac{\left| \int\_0^T WSSdt \right|}{\int\_0^T |WSS|dt} \right) \tag{13}$$

The formulation developed in this section describes the models that couple the fluid dynamics and the mechanical interaction with the arteries' wall which is treated as a deformable material. This methodology enables the computation of critical parameters for understanding the hemodynamics in the presence of a stenosis, such as the WSS. The advantages of this method are made evident by comparing it with a simple CFD analysis as detailed in the following section.
