# In-Vitro Simulation of the Blood Flow in an Axisymmetric Abdominal Aortic Aneurysm
## Appendix A  
To quantify the WSS on an infinitesimal oriented surface element \(dS\) of the wall we considered the total stress \(\tau\) occurring on it i.e. WSS = 2 \(\tau\) -twhere \(\tau\) = \(T\mathbf{n}\), \(\mathbf{T}\) is the stress tensor, for a Newtonian fluid \(\mathbf{T}\) = 1/2 \(\mu\) [\(\
abla\mathbf{u}+\
abla\mathbf{u}^{T}\)], \(\
abla\mathbf{u}\) is the velocity gradient tensor (1/2 [\(\
abla\mathbf{u}+\
abla\mathbf{u}^{T}\)] represents then the rate of deformation tensor) and \(\mathbf{n}\) and \(\mathbf{t}\) are, respectively, the unit vectors normal and tangential to \(dS\)[18; 20]. Since all quantities need to be evaluated in correspondence of the wall surfaces, we need first to know the equation of its profiles. This equation was obtained through image analysis by following the steps listed below:  
* Images of the vessel model (healthy aorta + aneurysm bulge) were acquired (see Figure 11a). * Boundaries were reconstructed through image analysis. In particular, by means of a pre-processing procedure (background subtraction), the contrast between the walls and the background was emphasized, a subsequent edge detection allows for the extraction of pixels belonging to the walls. * The obtained discrete sets of pixels were then interpolated using sixth-order polynomials (see Figure 11b). Considering a 2D acquisition in the \(x\)-\(z\) plane, an equation \(x\) = \(f\) (\(z\)) is then obtained for both right and left wall profiles. In Figure 11, we reported an example of an acquired image, the reconstructed vessel boundaries, and the image resulting after the application of the mask. As sketched in Figure 11, the first derivative of this function provides the slope of the curve at each point, \(dx/dz=f^{\prime}(z)=tan\theta\) and the \(x\)-\(z\) components of the unit tangent and normal vectors are found, respectively, as \(n_{x}=cos\theta=t_{z}\); \(n_{z}\)= \(sin\theta=-t_{x}\). Once the flow field is measured on a plane by means of 2D image analysis, the velocity components \(u\) and \(w\) are obtained on each point of a 256 x 60 regular grid and the stress tensor can be calculated as:  
\[\mathbf{T}=\ \frac{1}{2}\mu\left[ \begin{array}{cc}2\frac{\partial u}{\partial x}& \left(\frac{\partial w}{\partial x}+\frac{\partial u}{\partial z}\right)\\ \left(\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x}\right)&2 \frac{\partial w}{\partial z}\end{array} \right]; \tag{11}\]the WSS is then:  
\[\begin{array}{c}\text{WSS}=\ \mu\left[ \begin{array}{cc}2\frac{\partial u}{ \partial x}&\left(\frac{\partial u}{\partial x}+\frac{\partial u}{\partial z} \right)\\ \left(\frac{\partial u}{\partial z}+\frac{\partial u}{\partial x}\right)&2\frac{ \partial u}{\partial z}\end{array} \right]\left[ \begin{array}{c}cos\theta\\ sin\theta\end{array} \right]\left[ \begin{array}{c}-sin\theta\\ cos\theta\end{array} \right]=\\ \mu\left[ \begin{array}{cc}2\frac{\partial u}{\partial x}&\left(\frac{\partial u }{\partial x}+\frac{\partial u}{\partial z}\right)\\ \left(\frac{\partial u}{\partial z}+\frac{\partial u}{\partial x}\right)&2\frac{ \partial u}{\partial z}\end{array} \right]\left[ \begin{array}{c}n_{x}\\ n_{z}\end{array} \right]\left[ \begin{array}{c}-n_{z}\\ n_{x}\end{array} \right]=\mu\left[\left(\frac{\partial u}{\partial x}+\frac{ \partial u}{\partial z}\right)\left(n_{x}^{2}-n_{z}^{2}\right)+2n_{x}n_{z}\left( \frac{\partial u}{\partial z}-\frac{\partial u}{\partial x}\right)\right] \tag{22}\]  
To give an estimation of the maximum possible shear stress on the surface, one can consider the following expression [36]:  
\[WSS_{max}=\ \frac{\tau_{2}-\tau_{1}}{2}=\ \mu(e_{2}-e_{1}) \tag{23}\]  
where \(\tau_{i}\) and \(e_{i}\) are, respectively, the eigenvalues of the stress and of the rate-of-deformation tensor. Below, the procedure considered to obtain the expression is shown. In a principal component reference system, the stress tensor can be written as  
\[\textbf{{T}}=\left[ \begin{array}{cc}\tau_{1}&0\\ 0&\tau_{2}\end{array} \right] \tag{24}\]  
\[\textbf{{T}}\textbf{{\cdot}}\textbf{{n}}=\left[