The Physiological Principle of Minimum Work: I. The Vascular System and the Cost of Blood Volume
Optimal vessel branching follows cube law: r₀³ = r₁³ + r₂³
Murray's 1926 paper established the fundamental mathematical principle governing optimal branching in vascular networks. By deriving that parent and daughter vessel radii follow the cube law (r₀³ = r₁³ + r₂³), Murray showed that evolution had discovered an optimization principle balancing infrastructure costs against transport efficiency.
This work provides the biological foundation for understanding organizational span-of-control decisions. Just as blood vessels face trade-offs between wide (easy to pump, expensive to maintain) and narrow (cheap, high resistance) configurations, managers face trade-offs between wide spans (fewer layers, less oversight) and narrow spans (more layers, better attention).
Key Findings from Murray (1926)
- Optimal vessel branching follows cube law: r₀³ = r₁³ + r₂³
- The relationship emerges from minimizing total energy cost (blood volume + pumping)
- Mammalian vasculature closely follows this law across species from mice to elephants
- Optimal vessel radius minimizes sum of material cost (building vessels) + operating cost (moving fluid)
- Mathematical relationship: parent radius cubed equals sum of children radii cubed
- Each bifurcation optimally reduces radius by ~21%
- Principle applies universally - validated in trees, blood vessels, rivers, lungs
Used in 2 chapters
See how this research informs the book's frameworks:
Established fundamental mathematical principle for optimal branching - evolution discovered optimization balancing infrastructure cost vs. transport efficiency.
See optimal branching math →Established cube law (r³ parent = Σr³ children) verified across biological systems - physics drives convergent evolution to identical solutions.
See network optimization →