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

Riccardo Marchesi

Abstract

Murray's cubic branching law ($α=3$) predicts a universal diameter scaling exponent for all hierarchical transport networks, yet arterial trees yield $α\sim 2.7-2.9$. 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 $(5+p)/2 < α^*(Q) < 3$ independently of flow asymmetry (Theorem 4, Corollary 5). The static wall-tissue mechanism bounds the symmetric bifurcation exponent to $α_t \in [2.90, 2.94]$ 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 ($α_{exp} \approx 2.70$) 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.

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

Abstract

Murray's cubic branching law () predicts a universal diameter scaling exponent for all hierarchical transport networks, yet arterial trees yield . 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 () introduces a third metabolic cost term 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 independently of flow asymmetry (Theorem 4, Corollary 5). The static wall-tissue mechanism bounds the symmetric bifurcation exponent to 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 () 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.
Paper Structure (19 sections, 13 theorems, 27 equations, 1 figure, 3 tables)

This paper contains 19 sections, 13 theorems, 27 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Setup
  3. Empirical inputs (stated, not derived).
  4. Mathematical Results
  5. Structural stability of the non-universality result
  6. Perturbation 1: Generation-dependent wall scaling $p(g)$.
  7. Perturbation 2: Active smooth-muscle tone.
  8. Perturbation 3: Non-Newtonian viscosity in small vessels.
  9. Bifurcation Angles
  10. Empirical comparison.
  11. Optimal Branching Number
  12. Robustness to Non-Isometric Scaling
  13. Numerical Verification
  14. Parameters.
  15. Results.
  16. ...and 4 more sections

Key Result

Theorem 1

For every $Q>0$ and every $A,B,C,p>0$, the function $r\mapsto\Phi(r,Q)$ has a unique global minimum $r^*(Q)$ on $(0,+\infty)$.

Figures (1)

  • Figure 1: Scale-dependence of the local branching exponent $\alpha^*(Q)$. As flow $Q$ increases, the exponent monotonically approaches the Murray limit ($\alpha=3$) from below. Thicker walls (higher metabolic wall cost $m_w$) shift the entire curve toward the wall-dominated lower bound $(5+p)/2 \approx 2.885$. The shaded region marks the empirical morphometric range for porcine coronary arteries Kassab1993.

Theorems & Definitions (29)

  • Theorem 1: Existence and uniqueness of the optimal radius
  • proof
  • Lemma 2: Power law $\Leftrightarrow$ Murray's law on all trees
  • proof
  • Remark 1
  • Remark 2: Topological acyclic constraint
  • Theorem 3: Single-term classification
  • proof
  • Theorem 4: Strict bounds for the three-term case
  • proof
  • ...and 19 more