Theoretical and computational investigation of optimal wall shear stress in bifurcations: a generalization of Murray’s law

Abstract

In this study, the optimal distribution of Wall Shear Stress (WSS) in a bifurcation and its effect on the morphology of blood vessels were investigated. The optimal WSS was obtained through minimization of energy loss due to friction and metabolic consumption. It was shown that the optimal WSS is a function of metabolic rate, fluid properties, diameter, and flow regime. For fully developed laminar and turbulent flows different patterns of WSS were observed. For laminar flows WSS is constant but for turbulent flows WSS is a function of diameter such that the exponent of diameter varies by tube relative roughness. Based on the optimal WSS and conservation of mass, the optimal relationship between diameters of mother and daughters' vessels was obtained for different flow regimes. Also, it was theoretically shown that the optimal distribution of WSS in a bifurcation minimizes flow resistance as well as energy loss. In addition, it was demonstrated that the specific relationship between the length and diameters of a blood vessel and optimal relationship between diameters lead to optimal WSS distribution. Finally, the numerical simulation was used to investigate the effect of Reynolds number on the optimal WSS and flow resistance, and to verify the theoretical formula predictions, obtained in this work.