## 2 Materials and Methods
### Applied Methodology  
#### 2.2.1 Numerical Model  
Along with the observation of the CoA on the reconstructed 3D model, a numerical evaluation of the influence of this defect in the bloodstream was investigated. For this reason, CFD analyses were performed, utilizing the ANSYS(tm) Fluent module, in order to extract the pressure and velocity contours that quantify the blood flow in the cardiovascular system of the examined patient. In the context of this study, two different CFD analyses were conducted; the first one takes into account the reconstructed 3D model with the CoA, and in the second case, the examined aorta was modified through eliminating the defect and normalizing the aorta's walls correspondingly with the surfaces before and after the defect. It is worth noting that the internal volumes were used as control volumes for the bloodstream. Taking into account the existing literature [29], the conditions concerning the bloodstream inside the aorta were selected with the following assumptions. First, the blood was assumed to be an incompressible Newtonian fluid with a constant density (\(\rho\)) of 1060 kg/m\({}^{3}\) and a constant viscosity (\(\mu\)) of 0.004 Pa\(\cdot\)s (4 cP) [30; 31]. Furthermore, the boundary conditions regarding the arterial walls were set as stationary and rigid, with their elasticity and thickness set to be negligible. Regarding the initial boundary conditions, the inlet of the blood stream was applied on the ascending aorta with an initial velocity of 1.37 m/s, which is the mean blood velocity for an aorta according to published studies [32]. In addition, at the outlets, the initial pressure gauge (vessels and descending aorta) was evaluated at 100 mmHg (13,332 Pa) [33]. The ANSYS solver employed the Navier-Stokes formulation for the conservation of momentum of an incompressible flow in the three dimensions (\(R^{3}\)), as it is listed below [31], where \(u\) represents the velocity, \(p\) is the pressure, \(t\) is the time, and \(\mu\) is the viscosity. \[x:\ \frac{\partial u_{x}}{\partial t}+u\
abla u_{x}=-\frac{1}{\rho}\frac{ \partial p}{\partial x}+\mu\Big{(}\
abla^{2}u_{x}\Big{)}\]  
\[y:\ \frac{\partial u_{y}}{\partial t}+u\
abla u_{y}=-\frac{1}{\rho}\frac{ \partial p}{\partial y}+\mu\Big{(}\
abla^{2}u_{y}\Big{)} \tag{4}\]  
\[z:\ \frac{\partial u_{z}}{\partial t}+u\
abla u_{z}=-\frac{1}{\rho}\frac{ \partial p}{\partial z}+\mu\Big{(}\
abla^{2}u_{z}\Big{)}\]  
Figure 4: The segmented 3D model of the aorta, exhibiting its coarctation. Furthermore, the \(k\)-\(\epsilon\) turbulence model was utilized for the turbulence flow simulation, where \(k\) is the turbulence kinetic energy, \(\epsilon\) represents the rate of dissipation, \(\mu_{t}\) is the turbulence (known as eddy) viscosity, \(G_{k}\) represents the generation of turbulence kinetic energy due to the mean velocity gradients, \(G_{b}\) is the generation of turbulence kinetic energy due to buoyancy, and \(Y_{M}\) represents the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate. Furthermore, the flow coefficients coupled with their values are as follows: \(C_{\mu}=0.09\), \(\sigma_{k}=1\), \(\sigma_{\epsilon}=1.3\), \(C_{1\epsilon}=1.44\), \(C_{2\epsilon}=1.92\), and \(C_{3\epsilon}=1\)[31]. These values have been determined from experiments for fundamental turbulent shear flows including homogeneous shear flows and decaying isotropic grid turbulence according to ANSYS Fluent documentation [34]. They have been found to work fairly well for a wide range of wall-bound and free shear flows.