A Computational Investigation of Murray’s Law Using 3-D Navier Stokes Solutions and Derivative-Free Optimization

Murray’s law has been the subject of continual investigation since its derivation in 1926. In his original work Murray solved a simple analytical optimization problem to minimize power requirements, balancing flow rate and metabolic function to find the optimal relationship between the radius of a parent and daughter vessel in a bifurcation along with the bifurcation angle. The cost function used in his work was the sum of a pressure loss term and a metabolic cost term. Minimizing this cost function, Murray derived the following relationship between parent and daughter vessels, and an angle between the branches of 37.5°.