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

This paper investigates the optimal distribution of wall shear stress (wss) in a bifurcation and its effect on the morphology of blood vessels. the optimal wss is obtained through minimization of energy loss due to friction and metabolic consumption. it is 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 are observed. the wss is shown to be constant for the laminar flow, while it is a function of the tube diameter for the turbulent flow, in which the exponent of diameter varies with the tube relative roughness. based on the optimal wss and conservation of mass, the optimal relationship between diameters of mother and daughters' vessels is obtained for different flow regimes. also, it is theoretically shown that the optimal distribution of wss in a bifurcation minimizes flow resistance as well as energy loss. in addition, it is demonstrated that the specific relationship between the length and diameters of a blood vessel and optimal relationship between diameters lead to the optimal wss distribution. finally, the numerical simulation is 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.