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, simulating the geometry of a bicuspid aortic valve (BAV) during an oval systolic opening. The BAV, typically featuring two cusps instead of the usual three found in normal tricuspid configurations, introduces unique hemodynamic challenges. As the most prevalent congenital heart defect, BAV increases the risk of aortic dilation and dissection. 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. We derived an exact solution for the governing equations of Poiseuille flow within a bipolar cross-sectional tube, including velocity field, flow rate, and wall shear stress (WSS). The velocity profiles for BAV show remarkable agreement with previous studies using coherent multi-scale simulations, consistently demonstrating a jet-like flow structure absent in tricuspid aortic valve (TAV) scenarios. Analysis reveals that at the center of the entrance, BAV blood flow velocity is significantly higher than TAV but decreases more rapidly towards the vessel wall, creating a steeper vertical velocity gradient and resulting in higher WSS for BAV. Additionally, the WSS, inversely proportional to sin({\xi}*), 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 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.