# Application of Mathematical Models for Blood Flow in Aorta and Right Coronary Artery
## 4 Discussion  
The results of this study suggest that the Newtonian approximation is suitable for modeling flow in large-caliber vessels, such as the aorta. The Newtonian model showed the highest accuracy for aortic flow, with a relative error of 1.1%, and significantly shorter simulation times compared to more complex models. This confirms that the Newtonian approximation is adequate for large-caliber vessels. In contrast, while the Casson model produced a similar error to the Newtonian model, it required simulation times approximately five times longer. Viscoelastic models did not show any advantage in simulations of aortic flow, where the flow is more stable and the viscoelastic behavior of blood has a lesser impact. However, for smaller caliber vessels, such as the right coronary artery (RCA), having more variable geometry and flexibility and being subjected to less stable flow conditions, viscoelastic models proved more accurate. This aligns with findings from earlier work [6], which \begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline
**Parameter** & **True Value** & **Newtonian** & **Carreau** & **Casson** & **Oldroyd-B** & **Giesekus** & **FENE-P** \\ \hline Average velocity on volume & 0.34300 m/s & 0.118 & 0.110 & 0.109 & 0.110 & 0.110 & 0.108 \\ Average velocity on surface (in) & 0.29338 m/s & 0.008 & 0.012 & 0.014 & 0.008 & 0.008 & 0.012 \\ Average velocity on surface (out) & 0.39355 m/s & 0.023 & 0.028 & 0.035 & 0.022 & 0.022 & 0.021 \\ Average pressure on surface (out) & 17,731.83 Pa & 0.001 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \\ Qin & \(2.25\times 10^{-6}\) m\({}^{3}\)/s & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \\ Qout & \(2.25\times 10^{-6}\) m\({}^{3}\)/s & 0.025 & 0.000 & 0.000 & 0.006 & 0.006 & 0.007 \\ \hline
**Avg Relative Error**\({}^{1}\) & & & **0.026** & **0.025** & **0.026** & **0.024** & **0.024** & **0.025** \\ \hline \hline \end{tabular}  
* Highlighted in bold for each model, which serves as a key comparative indicator.  
\end{table} 
Table 3: Relative error—RCA. Figure 10: RCA models - outlet section. showed that viscoelastic models are better suited for capturing complex fluid dynamics in smaller vessels. For coronary flow, shear-thinning models represent a viable alternative, offering a good balance between computational efficiency and accuracy. These models provide valuable insights into the non-Newtonian nature of blood flow, supporting previous studies that have emphasized their utility in simulating real-world fluid dynamics. The shear-thinning behavior is particularly relevant in microvascular and pathological flow conditions, where blood exhibits significant changes in viscosity under varying shear rates. From the perspective of our working hypothesis, the findings confirm that model selection depends not only on the vessel size but also on the specific characteristics of the blood flow. Our results are consistent with the hypothesis that more complex models, while computationally demanding, yield more accurate predictions for smaller vessels and pathological states. The broader implications of these findings extend to clinical applications, where accurate flow models are crucial for simulating pathological conditions, such as atherosclerosis, and for planning personalized treatment strategies. Future research should focus on developing hybrid models that integrate the strengths of various approaches, such as combining Newtonian and non-Newtonian elements, to improve model accuracy while maintaining computational feasibility. While numerical errors such as discretization and truncation are inherent to any finite element-based simulation, the observed variation in relative errors across rheological models is not random. On the contrary, it reflects the degree of alignment between the model's theoretical assumptions and the hemodynamic conditions of the specific vascular region.