# Application of Mathematical Models for Blood Flow in Aorta and Right Coronary Artery
## 2 Materials and Methods
### Geometric Models  
The two vessels considered in our study are very different from each other in terms of size: the aorta, being the main vessel of the human body, has a diameter that typically varies between 20 and 30 mm, while the right coronary artery has a diameter significantly smaller, usually between 2 and 5 mm. This difference in size reflects their different functions: the aorta is responsible for distributing oxygenated blood throughout the body, while the RCA is responsible for supplying blood to the heart muscle. Finally, the geometry and flexibility of the coronary artery may vary (both inter- and intra-patient) more pronouncedly than that of the aorta. The coronary arteries therefore have a smaller diameter, more pronounced curvature compared to the aorta, and greater anatomical variability, factors which could give rise to shear-thinning behavior (reduction in viscosity with increasing shear rate) and other typical effects of non-Newtonian fluids. In contrast, the aorta, being a vessel of large diameter and having a more laminar flow, tends to behave like a Newtonian fluid, whose viscosity remains constant and proportional to the shear stress. This difference in rheological behavior between coronary arteries and the aorta is crucial for understanding blood flow dynamics and the clinical implications associated with cardiovascular disease.  
* **Three-Dimensional Geometric Model of the Aorta**:  
The geometric model used, developed by [PERSON] et al. [12], represents a specific aorta of a 50-year-old woman (height 160 cm), obtained from a CT scan performed for clinical purposes. The DICOM image sequence included 512 slices in the axial plane, 416 in the sagittal plane, and 88 in the coronal plane, with an in-plane resolution of 0.9375 x 0.9375 mm and a slice thickness of 1.4 mm. Written patient consent was obtained for the use of the medical images in the study, and this patient was chosen because she did not present any aortic diseases, such as aneurysms or dissections. The aorta was reconstructed through a segmentation process using the open-source software 3D Slicer. Subsequently, a reverse engineering algorithm was applied to obtain a more accurate virtual 3D model, suitable for computational analysis. The final model (shown in Figure 1) includes the ascending aorta, the aortic arch, the major vessels, and the descending trunk with abdominal vessels, and extends to the iliac arteries. The diameter of the aorta in the descending trunk is approximately 21 mm at the level of the aortic arch and 16 mm at the level of the renal arteries.  
* **Three-Dimensional Geometric Model of the RCA**:  
The 3D model of the right coronary artery, developed by [PERSON] et al. [13], is specific to a 64-year-old male patient and was obtained using a series of angiograms acquired during a standard angiography performed for clinical purposes. Written consent was obtained for the use of the medical images. Since the coronary arteries are compressed by the contraction of the cardiac muscle during systole, myocardial perfusion primarily occurs during the diastolic phase, when the muscle relaxes. For this reason, the geometric model was reconstructed considering the end of diastole. The true curvature of the vessel was obtained by combining the centerline path from the LAO (Left Anterior Oblique) and RAO (Right Anterior Oblique) views. The variation in diameter was modeled assuming a circular shape. Moreover, in order to obtain the real anatomy of the vessel, the known caliber of the catheter (6 Fr) was used, so the others measurements were obtained by means of a proportion. The RCA model (shown in Figure 2) includes the proximal segment, the mid-segment, and the distal segment (collateral branches were neglected).  
Figure 1: Aorta - 3D geometric model.