Murray's law

Murray's law predicts $Q\propto R^3$ for an ideal blood vessel whose radius balances pumping power against the cost of maintaining its blood volume. Cecil D. Murray formulated this minimum-cost model in 1926. ^[ref]

Consider a cylindrical segment of fixed length $\ell$, prescribed positive volumetric flow rate $Q$ and constant dynamic viscosity $\mu$. The Hagen-Poiseuille equation gives pumping power $Q\Delta p=8\mu \ell Q^2/(\pi R^4)$. Adding a maintenance cost $b>0$ per unit volume per unit time givesDifferentiation givesThe pumping term diverges as $R\to 0$, while the maintenance term diverges as $R\to \infty$; the unique stationary radius is the minimum.

At an ideal bifurcation, conservation of mass for an incompressible flow gives $Q_0=Q_1+Q_2$. If the three segments have the same $\mu$ and $b$, their common proportionality constant cancels:Equal daughters therefore have $R_1=R_2=2^{-1/3}{R}_0$, approximately $0.794R_0$. This prediction depends on the model: pulsatile inertia, differing maintenance costs and radius-dependent apparent viscosity, including the Fahraeus--Lindqvist effect, can change the optimum.