Poiseuille Flow in Tubes of Bipolar Cross Sections: An exact hemodynamic analysis for potential mechanisms of aortopathy in bicuspid aortic valve

Steady blood flow, or Poiseuille flow, through compressed or defective blood vessels is a critical issue in hemodynamics, particularly in cardiovascular studies. This research explores a tube with a bipolar cross-section, which simulates the geometry of a bicuspid aortic valve (BAV) during an oval systolic opening. The BAV, which typically features two cusps instead of the usual three found in normal tricuspid configurations, introduces unique hemodynamic challenges. This condition, being the most prevalent congenital heart defect, increases the risk of aortic dilation and dissection among patients. A bipolar cross-sectional analysis provides a more accurate geometric approximation for modeling flow through these atypical valve shapes, crucial for understanding the specific fluid dynamics associated with BAV. In this study, we derived an exact solution for the governing equations of Poiseuille flow within a bipolar cross-sectional tube. The results include the velocity field, flow rate, and wall shear stress (WSS). The velocity profiles for the BAV show a remarkable agreement with those obtained in previous studies using coherent multi-scale simulations. These profiles consistently demonstrate the presence of a jet-like flow structure within the fluid, a feature that is notably absent in the tricuspid aortic valve (TAV) scenarios. The analysis reveals that at the center of the entrance, the blood flow velocity for the bicuspid aortic valve (BAV) is significantly higher than for the tricuspid aortic valve (TAV). However, the BAV bood flow velocity decreases more rapidly towards the vessel wall, creating a steeper vertical velocity gradient. This results in higher wall shear stress for the BAV. Additionally, the WSS, which is inversely proportional to \(\:sin\left({\xi\:}_{*}\right)\), where \(\:{\xi\:}_{*}\) represents the bipolar coordinate at the wall boundary, exceeds that found in a circular cylindrical tube with an equivalent diameter. In cases of aortic valve stenosis, where \(\:{\xi\:}_{*}\) approaches \(\:\pi\:\), the WSS increases very rapidly. This elevated WSS, commonly observed in BAV patients, may detrimentally impact the aortic wall in these structurally abnormal valves, particularly within the ascending aorta.