Murray's law and local junction dissipation at vascular bifurcations

Murray's law predicts a cubic relation between vessel radius and flow rate by balancing distributed Poiseuille dissipation against material cost. We extend this local variational framework for a single vascular bifurcation by adding a positive-definite junction-dissipation term associated with velocity mismatch and irreversible mixing. For prescribed parent flow and daughter-flow partition, minimization determines the stationary daughter-to-parent radius ratios. In the equal-length Poiseuille baseline, the reduced problem is governed by one dimensionless parameter η, measuring the relative strength of localized junction dissipation and distributed viscous dissipation. As η→0, the classical cubic Murray allocation is recovered; when junction dissipation dominates, the optimum approaches velocity matching, for which the total daughter area equals the parent area. Intermediate values produce a continuous crossover between these limits. Symmetric bifurcations are described by a single η-dependent effective exponent, whereas asymmetric bifurcations generally require distinct branchwise exponents depending on both η and the imposed flow partition. Analytical approximations and direct numerical minimization characterize this local correction to Murray's law. Retaining the complete pairwise closure, including daughter–daughter velocity mismatch, preserves the limiting solutions while shifting the finite-crossover asymmetric radius allocation. The resulting framework provides a local steady-flow variational extension of the Poiseuille–Murray model, in which junction dissipation shifts the optimal radius allocation continuously away from cubic scaling.