= Murray's law
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<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. <The Physiological Principle of Minimum Work. I. The Vascular System and the Cost of Blood Volume by Cecil D. Murray>[Printed pp. 209–212, equations 1–7]{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 gives
$$
P(R)=\frac{8\mu\ell Q^2}{\pi R^4}+b\pi R^2\ell.
$$
Differentiation gives
$$
P'(R)=-\frac{32\mu\ell Q^2}{\pi R^5}+2b\pi R\ell=0,
\qquad
Q=\frac{\pi}{4}\sqrt{\frac b\mu}\,R^3.
$$
The pumping term diverges as $R\to0$, 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:
$$
R_0^3=R_1^3+R_2^3.
$$
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.
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