Appropriate Use and Limitation of Hagen-Poiseuille's Law for Wall Shear Stress Profiling in Coronary Arteries

Background: The Hagen-Poiseuille law is widely used to estimate wall shear stress (WSS, WSS_HP) in medicine. Only flow rate and local vessel diameter are needed as input data. This advantage is achieved at the cost of accuracy due to simplified assumptions concerning vessel shape. Calculation of WSS by numerical integration of the Navier-Stokes differential equations for incompressible fluids (CFD, WSS_CFD) in three-dimensional vessel reconstructions is more accurate, particularly in vessels of complex geometric shape as coronary arteries, but also more demanding and time-consuming. Methods: Vascular trees from 17 right (RCA) and 15 left (LCA) coronary arteries stratified with respect to luminal remodeling (controls (C), aneurysmatic (AnCAD) and non-aneurysmatic coronary artery disease (CAD)) were reconstructed from biplane angiograms. Patient specific flow simulations were performed by CFD. Results for WSS_CFD and WSS_HP were compared. Results: Mean WSS_CFD was 2.6 +/- 2.6 Pa and WSS_HP was 2.4 +/- 2.6 Pa. The mean difference was 0.23 (-0.09-0.58) was not significant and correlation was 0.942. There were major differences in distribution particularly in the low WSS range (minimum (min) WSS (r=0.494), percent area <0.4 Pa (A_.4) (r=0.811)), however. Min WSS is over-estimated (p=0.01) and A_0.4 WSS is under-estimated (p=0.01) significantly by applying Hagen Poiseuille's law. The estimation of maximum (max) WSS can be improved (r=0.933; r=0.816 with max WSS_HP) by estimation from mean WSS_HP using a factor of 2 in RCA and 3 in LCA due to a strong trend of max WSS_CFD with mean WSS_CFD (different slopes in RCA/LCA). Conclusion: Estimation of WSS by Hagen Poiseuille's law performs well for mean and maximum values, but is a poor estimate in advanced analysis particularly in low WSS areas.