= Cecil D. Murray
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<Cecil D. Murray> developed an optimization account of blood-vessel geometry by balancing the power needed to pump blood against the cost of maintaining blood <volume>. For a vessel segment of fixed length carrying fixed flow $Q$, <Hagen-Poiseuille equation> makes the pumping term proportional to $Q^2/r^4$, while a volume-maintenance term is proportional to $r^2$. Minimizing $aQ^2/r^4+br^2$ gives $r^6=2aQ^2/b$, hence $Q\propto r^3$.<The Physiological Principle of Minimum Work. I. The Vascular System and the Cost of Blood Volume by Cecil D. Murray>[1926, printed pp. 209–212, equations 1–7]{ref}
Flow conservation at a branch then gives <Murray's law>, $r_0^3=r_1^3+r_2^3$. The argument is valuable because it connects a physiological maintenance assumption to a testable geometric relation. It is a model of competing costs, not a claim that every real vessel obeys the optimum exactly.
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