ISEGORIA / MATH ENCYCLOPEDIA
Murray’s law: the cube rule of branching vessels
In 1926 the physiologist Cecil Murray asked what size a blood vessel ought to be. A narrow vessel makes the heart work hard, because Poiseuille’s law charges a pumping power that grows as the inverse fourth power of the radius. A wide vessel holds a lot of blood, and blood is expensive tissue to make and keep alive. Balancing the two prices for every vessel gives a startlingly simple answer: the flow through a vessel should be proportional to the cube of its radius. At a branch point the cubes then add up, \(r_0^3=r_1^3+r_2^3\), the shear stress on the vessel wall is the same everywhere in the network, and the branches leave at angles fixed by their radii alone. The same rule turns up in the arteries of mammals, the larger airways of the lung, the xylem of some plants and engineered microfluidic networks. These experiments price a single vessel, place a junction where the cost is least, and grow a whole tree to see what the cube law buys and what departing from it costs.
Before you begin: Fluid dynamics (Poiseuille flow), minimizing a function of one variable, vectors and forces in the plane
Predict, manipulate, then test yourself: every experiment ends with practice questions that are generated fresh at three levels. Graphs illustrate the mathematics; they do not replace a proof.
1. One vessel: the price of a radius
A vessel of radius \(r\) and length \(l\) carries a steady flow \(Q\) of blood with viscosity \(\mu\). Poiseuille’s law says the pressure drop is \(8\mu lQ/(\pi r^4)\), so pushing the flow through costs power \(8\mu lQ^2/(\pi r^4)\). The blood inside also has to be made, oxygenated and kept warm, at a price \(b\) per unit volume per second, so filling the vessel costs \(b\pi r^2 l\). One price falls with \(r\) and the other rises; their sum has a single minimum. Setting the derivative to zero gives Murray’s law, \(Q\propto r^3\). Drag the radius and watch both the cost curve and the flow profile inside the vessel.
Worked example. Take \(Q=1\) mL/s of blood with \(\mu=3.5\) mPa·s and Murray’s own estimate of the upkeep, \(b=19\,300\) erg per cm³ per second, which is 1930 W/m³. Then \(r^*=(16\mu Q^2/\pi^2b)^{1/6}=1.197\) mm. Per metre of vessel the pumping costs 4.34 mW and the upkeep 8.69 mW, exactly twice as much, 13.0 mW in all. The mean speed is \(Q/\pi r^{*2}=22.2\) cm/s and the wall shear stress is \(\sqrt{b\mu}=2.60\) Pa. A vessel of 0.8 mm carrying the same flow costs 25.6 mW/m, 1.97 times as much, almost all of it pumping, and its wall feels 8.70 Pa. Eight times the flow needs only twice the radius.
Watch out. Poiseuille’s law assumes steady laminar flow of a Newtonian fluid in a long rigid tube. Blood in the aorta and large arteries is pulsatile and its viscosity depends on the vessel size, so measured exponents there are often below 3; the cube law fits best in the middle-sized arteries and arterioles. The value of \(b\) is only an estimate, but \(r^*\) depends on it weakly, as \(b^{-1/6}\). The flow profile is drawn to scale across the vessel and schematic along it.
Why is the wall shear stress the same in every vessel that obeys Murray’s law?
Poiseuille flow gives a wall shear stress \(\tau_w=4\mu Q/(\pi r^3)\). Murray’s law sets \(Q=\frac{\pi}{4}\sqrt{b/\mu}\,r^3\), and substituting gives \(\tau_w=\sqrt{b\mu}\), with no \(r\) and no \(Q\) left. The cube in the law is exactly the cube in the shear stress, which is why the endothelium can use shear stress as the signal that tells a vessel to widen or narrow.
2. The junction: where should a vessel branch?
A parent vessel must feed two daughters at fixed places, sharing its flow \(s:1-s\). Each vessel costs \(w_i\) per metre, so the junction should sit where \(\sum w_i\,\ell_i\) is least. That is a weighted Fermat point, and at it the three vessels pull on the junction like weighted strings, with forces \(w_i\) along their own directions, that exactly balance. When every vessel has its Murray radius its cost per metre is \(\frac32 b\pi r^2\), so the pulls are proportional to \(r^2\), and the triangle of forces gives Murray’s second law of 1926 for the branching angles. Change the radius rule to see the prediction fail when the vessels are not optimal.
Worked example. For a symmetric split the cube law gives \(r_1=r_2=2^{-1/3}r_0=0.794\,r_0\), so the pulls are in the ratio \(1:2^{-2/3}:2^{-2/3}=1:0.630:0.630\). Then \(\cos\theta=r_0^2/(2r_1^2)=2^{-1/3}=0.794\) and each daughter leaves at \(\theta=37.5^\circ\), \(74.9^\circ\) between them. With the parent at \((0,-4)\) and the daughters’ ends at \((\pm3.2,3)\), the junction sits at \((0,-1.18)\). For an 80 : 20 split the radii are \(0.928\,r_0\) and \(0.585\,r_0\), and the large branch bends only \(19.4^\circ\) while the small one leaves at \(56.8^\circ\): a big branch nearly continues the trunk, a small one leaves almost sideways.
Watch out. The model puts all three vessels in one plane and ignores the extra cost of the junction itself and any constraint from the surrounding tissue. Real branching angles scatter widely around Murray’s prediction. The cost bowl is drawn to scale in its shape, but its height is rescaled to fit the figure, and the tube radii are drawn larger than the geometry would make them.
Why does a small side branch leave the trunk almost at a right angle?
Its pull, proportional to \(r_2^2\), is tiny, so it barely deflects the trunk: the parent and the large daughter pull almost head to head and must nearly cancel, which leaves the small branch to point wherever the small remainder points, at right angles to the trunk. In the formula, as \(r_2\to0\) the cube law gives \(r_1^4\approx r_0^4-\frac43r_0r_2^3\), so \(\cos\theta_2\approx\frac23r_2/r_0\to0\) and \(\theta_2\to90^\circ\).
3. The tree: one law, all the way down
Apply the rule at every junction and a whole tree follows. In a symmetric tree the radius shrinks by \(2^{-1/3}\) at each generation, the number of vessels doubles, and the total cross-section grows by \(2^{1/3}\), so blood slows as it spreads out. The wall shear stress \(\tau\propto Q/r^3\) is the same in every vessel, which the coloring shows as a single tone. Try other rules \(r\propto Q^{1/k}\): even with the overall scale chosen as well as possible, every exponent other than 3 costs more, and the shear stress drifts up or down towards the smallest vessels.
Worked example. After 7 symmetric generations there are 128 vessels of radius \(2^{-7/3}r_0=0.198\,r_0\) and their total cross-section is \(2^{7/3}=5.04\) times the root’s, so the mean speed has dropped to 0.198 of the root’s. With lengths proportional to radius, as drawn here, the best tree with area-preserving branching, \(k=2\), costs 29.7% more than the cube law; \(k=2.5\) costs 4.5% more, \(k=3.5\) 2.3% and \(k=4\) 7.1%. Going from a 12.5 mm aorta down to 4 µm capillaries by symmetric cube-law halving would take \(3\log_2(12.5/0.004)=34.8\) generations.
Watch out. A real vascular tree is far from a symmetric binary tree: branching is asymmetric, lengths vary, and the largest arteries do not follow the cube law. The tree here is an idealization in which every junction is optimal and alternate generations branch in perpendicular planes. The radii are drawn relative to the root, and the moving dots travel at each vessel’s mean speed on a common, slowed time scale.
Why does blood slow down as it goes deeper into a cube-law tree?
The mean speed is \(Q/(\pi r^2)\), and the cube law makes it proportional to \(r\): \(\bar u=\frac14\sqrt{b/\mu}\,r\). Smaller vessels therefore carry slower blood. Equivalently, at each symmetric split the total cross-section grows by \(2\cdot2^{-2/3}=2^{1/3}\), and the same total flow through a larger total area moves more slowly, which gives the time needed for exchange in the capillaries.