Cecil D. Murray

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 $a{Q}^2/r^4+b{r}^2$ gives $r^6=2a{Q}^2/b$, hence $Q\propto r^3$.^[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.