## 2 Mathematical Model and Analysis
### Geometry and Mesh Details  
A careful search for a suitable physical configuration has been made after the formulation of the problem statement. An arteriole is chosen as an ideal geometrical object as arterioles are smaller than the arteries and these vessels also undergo pathological changes such as atherosclerosis and aneurysm. In addition, the study and modeling of arteriole stenosis and aneurysm have not been studied by any researcher previously. An arteriole is a blood vessel of smaller dimensions likely to experience the microcirculation that extends and spreads out from an artery connecting to it. The microcirculation is directed to capillaries. The network of blood vessels is seen in Figure 1. A segment of an arteriole of length 2 mm and diameter of 0.3 mm is selected for the present investigation, as seen in Figure 2. Into the flow field of the arteriole segment, a thin film of graphene is introduced. The layer of graphene is considered with a length of 9 \(\upmu\)m and a width of 0.3 mm is plugged into the arteriole flow path. The permeability of the graphene layer is maintained at 0.001 m\({}^{2}\). A fine mesh is generated on the geometry as a consequence of the mesh independence test. Figure 3 depicts the fine mesh generated over the geometry of interest. The quality of the mesh generated can be seen in Figures 4 and 5. The deposition of lipids along the arteriole wall results in the narrowing of the vessel causing the medical condition of atherosclerosis. The stenosed version of the arteriole segment is shown in Figure 6. The meshed version of the stenosed arteriole segment is seen in Figure 7 and the quality of meshing is presented in Figures 8 and 9. Figure 7 encloses the mesh elements of different shapes generated over the geometry constructed. The flow domain is covered with triangular shapes whereas the walls are defined using quadrilateral mesh elements. A neat triangular pattern developed from the graphene layer on either side of the layer can be seen in this figure. The features of a mesh make it possible to run a numerical PDE simulation accurately, efficiently, and faithfully to the physics involved. Figure 8 illustrates this aspect of the meshing produced. In the quality measure of choice, one stands for an element with complete regularity whereas zero stands for an element that has deteriorated. The stress exerted by the flowing blood on the vessel wall can result in the protrusion of the arteriole wall leading to a medical condition of an aneurysm. The aneurysm arteriole geometry, the mesh generated, and the quality of meshing can be seen in Figures 10-13, respectively. The fine mesh details are entailed in Table 1. The FEM solver is set to use the PARDISO solver to solve the system of linear equations due to its high-performing advantages [43]. The preordering algorithm used is a nested dissection multithread method and relative tolerance of 0.001 is applied to assure the convergence of the solution.  
Figure 1: Network of blood vessels (credits: Wikipedia).  
Figure 4: Mesh quality of the fine mesh generated.  
Figure 3: Meshing near the graphene layer in an arteriole segment.  
Figure 2: Geometry of the arteriole segment.  
Figure 5: Mesh quality near the graphene layer.  
Figure 6: Geometry of arteriole segment with stenosis in the post-graphene region.  
Figure 7: Meshing near the graphene layer and the stenosis.  
Figure 8: Meshing near the graphene layer and stenosis.  
Figure 10: Geometry of arteriole segment with aneurysm sac in the post-graphene region.  
Figure 9: Meshing near the stenosis wall.  
Figure 11: Meshing near the graphene layer and the aneurysmic sac.  
Figure 12: Mesh quality near the graphene layer and sac.  
Figure 13: Mesh quality near the sac.