Beyond Murray's Law: Non-Universal Branching Exponents from Vessel-Wall Metabolic Costs

TL;DR

This paper explains why arterial branching exponents deviate from Murray's law by incorporating vessel-wall costs, showing non-universality and deriving bounds for the scaling exponent based on structural and metabolic considerations.

Contribution

It introduces a model accounting for vessel-wall thickness costs, demonstrating non-universality of branching exponents and deriving bounds consistent with empirical data.

Findings

The model predicts a scale-dependent exponent within bounds (2.90, 2.94).

Murray's law is a special degenerate case of a more general cost function.

Binary bifurcation is favored due to steric constraints, explaining physiological branching patterns.

Abstract

Murray's cubic branching law ($\alpha=3$) predicts a universal diameter scaling exponent for all hierarchical transport networks, yet arterial trees yield $\alpha \sim 2.7-2.9$. We show that this discrepancy has a structural origin: Murray's universality is an artifact of cost homogeneity, not a biological property. Incorporating the empirical vessel-wall thickness law $h(r)=c_0 r^p$ ($p \approx 0.77$) introduces a third metabolic cost term $\propto r^{1+p}$ that renders the cost function inhomogeneous with incommensurate scaling exponents. By Cauchy's functional equation, homogeneity is necessary and sufficient for a universal branching exponent to exist; its absence implies non-universality, and Murray's law is identified as a singular degeneracy of the cost-function family rather than a general principle. We prove that the resulting scale-dependent exponent satisfies the strict bounds $(5+p)/2 < \alpha^*(Q) < 3$ independently of flow asymmetry (Theorem 4, Corollary 5). The static wall-tissue mechanism bounds the symmetric bifurcation exponent to $\alpha_t \in [2.90, 2.94]$ from measured parameters, marking a first-order symmetry breaking from Murray's law that narrows the empirical gap by one-third. The remaining discrepancy with the cardiovascular mean ($\alpha_{exp} \approx 2.70$) is not a model failure but a mathematical necessity that signals the independent contribution of pulsatile wave dynamics. Additionally, the wall cost breaks Murray's topological degeneracy, bounding the optimal branching number to small finite integers; binary bifurcation emerges as the physiologically selected minimum under steric constraints.