Murray's cubic branching law (=3) rests on a two-term cost function that neglects the metabolic cost of vessel-wall tissue. Incorporating the empirical wall-thickness law h (r) = c₀ rᵖ (p 0. 77) introduces a third cost term r^1+p and yields seven rigorous results. First, the optimal vessel radius exists and is unique for any positive parameter values. Second, a universal branching exponent exists if and only if the optimal radius is a power law in flow rate. Third, for any single maintenance term r^, the branching exponent is exactly = (4+) /2, recovering Murray (=2), the Da Vinci rule (=1), and the wall-cost limit (=1+p) as special cases---and providing a physical grounding for the abstract pricing exponent of Bennett's EPIC framework. Fourth, with both volume and wall costs present, no universal exponent exists; instead, a scale-dependent local exponent ^* (Q) satisfies (5+p) /2 < ^* (Q) < 3. Fifth, these bounds are independent of the flow-splitting asymmetry, proving non-universality is an inherent property of the cost function rather than junction geometry. Sixth, the wall cost rigorously breaks the degeneracy of Murray's law regarding the branching number, analytically selecting N=2 (bifurcation) as the unique energy-minimizing topology. Seventh, the optimal symmetric bifurcation angle is tightly bounded strictly between 74. 9^ and 80. 0^ for physiological parameters, generalizing Zamir's force-balance derivation. Parameter-free numerical evaluation yields ^* = 2. 90 for porcine coronary arteries, within 1 of the morphometric measurement 2. 7 0. 2, resolving a long-standing discrepancy without invoking pulsatile wave dynamics.
Riccardo Marchesi (Thu,) studied this question.