The radius that costs least

An artery divides into two. So does the one after it, and the one after that, thirty times over, ending in capillaries a few micrometres across. Every one of those divisions had to choose a radius, and the radii that were chosen follow a rule.

The rule is not a rule about space. It is the answer to a question about cost, and it produces a statement so specific that it can be tested against a photograph: the cube of a parent vessel’s radius equals the sum of the cubes of its branches.

Two bills, pulling in opposite directions #

A vessel of radius $r$ and length $L$ carrying a volume flow $Q$ presents its owner with two costs, and they are the whole model.

Pumping. In laminar flow the pressure drop is Poiseuille’s, and the power required is

$$P_{\text{pump}} = Q\,\Delta p = \frac{8\mu L}{\pi}\,\frac{Q^2}{r^4}$$

The fourth power is the striking part. Halving a vessel’s radius at the same flow multiplies the pumping bill by sixteen, which is why a narrowed artery is such a serious thing and why a small constriction dominates a network. It is the same steepness that makes the pressure drop along a pipe such a sensitive function of its bore: a 10 per cent reduction in radius costs 52 per cent more pressure for the same flow.

Upkeep. Owning the vessel costs something proportional to its volume: the blood in it has to be made and maintained, the wall has to be built and fed. Write it as

$$P_{\text{met}} = k\,\pi r^2 L$$

The first falls with radius and the second rises, so the total has a minimum. Differentiating and setting to zero:

$$\frac{\mathrm{d}}{\mathrm{d}r}\!\left(\frac{8\mu L Q^2}{\pi r^4} + k\pi r^2 L\right) = 0 \quad\Longrightarrow\quad Q^2 \propto r^6 \quad\Longrightarrow\quad \boxed{Q \propto r^3}$$

That is Murray’s law. A vessel built to carry a flow $Q$ at least total cost has a radius proportional to the cube root of it.

Both terms deserve a word about where they come from, because one of them is a theorem and the other is an assumption dressed as one.

The pumping term is the exact laminar pipe solution — no correlation, no fitted constant, and none of the equivalent-sand-grain apparatus the turbulent case needs. The upkeep term is not derived from anything: it asserts that the cost of owning a vessel is proportional to the volume of it, with a constant $k$ nobody has measured. That is a modelling choice, and the essay’s most important defence of it is that the result which follows does not contain $k$ at all.

The invariant that is not obvious #

The stationarity condition contains something the derivation does not need and that turns out to be the most useful thing in the essay.

At the optimum, the two costs are in a fixed ratio. Setting the derivative to zero gives $4P_{\text{pump}}/r = 2P_{\text{met}}/r$, so

$$P_{\text{pump}} = \tfrac{1}{2}P_{\text{met}}$$

The pumping bill is exactly a third of the total, whatever the numbers are. It does not depend on the viscosity, the flow, the metabolic constant or the length; it depends only on the two exponents, $-4$ and $+2$, and on nothing else. The solver asserts it at $33.3333$ per cent, which is the assertion that would catch an implementation with a wrong exponent however carefully the constants had been chosen to make the answer look right.

That is a satisfying kind of result. A model with three free parameters produces a prediction with none in it.

What was computed, and what the assertions catch #

The minimum is found by golden-section search on the computed cost, not by the formula:

$$r_{\text{best}} = 1.0838521 \qquad \left(\frac{16\mu Q^2}{k\pi^2}\right)^{1/6} = 1.0838521$$

to eight figures. Then two further assertions run on every bifurcation drawn.

The cube law closes. $r_1^3 + r_2^3 - r_0^3$ comes out at $10^{-16}$.

The wall shear stress is the same in all three vessels. For Poiseuille flow the shear at the wall is $\tau = 4\mu Q/\pi r^3$, and if $Q \propto r^3$ then $\tau$ is a constant of the network. Across a four-to-one flow split, the parent and both branches come out at $1.273239545$ — identical to twelve decimal places, which is what an exact algebraic consequence looks like when it is computed rather than asserted.

The solver also refuses a junction whose radii satisfy no branching exponent between 1 and 6, which is what a set of radii from a badly-measured photograph looks like.

The shear stress, and why it matters that a wall can feel it #

The constant-shear result is the bridge between a cost function and a mechanism, and without it Murray’s law would be a curiosity.

A cost function is a thing an engineer computes. It is not a thing a blood vessel has access to: no cell knows the total metabolic budget of the organism, and no cell can differentiate a sum. But shear stress at the wall is a local, physical quantity — the drag the flowing blood exerts on the endothelium — and endothelial cells demonstrably sense it. They align with it, they change their gene expression in response to it, and vessels remodel their diameter over weeks in response to sustained changes in it.

So a very simple local rule — if the shear is above the set point, widen; if below, narrow — drives the whole network to Murray’s optimum without anything anywhere computing a cost. That is why the result is biologically credible rather than merely tidy.

What the network looks like, and what happens to the speed #

Apply the rule at every junction and the geometry follows. For a symmetric split, $r_1 = r_0 2^{-1/3} = 0.7937\,r_0$.

The consequence for the velocity is the point of the whole vascular system. Total area at each generation is $2^n \pi r_n^2 = 2^{n/3}\pi r_0^2$, which grows, so the mean velocity falls as $2^{-n/3}$. After thirty generations the area has grown by a factor of a thousand and the blood is travelling a thousand times more slowly.

The arithmetic is worth doing in full, because the numbers are startling. Human capillaries have a total cross-sectional area of the order of a square metre, against a couple of square centimetres for the aorta — a factor of several thousand — and the mean velocity falls by the same factor. A red cell that left the heart at 0.4 m/s spends about a second crossing a capillary a millimetre long, and one second is roughly what oxygen needs to diffuse across a few micrometres of plasma and tissue.

That is what a capillary bed is for. Exchange of oxygen and solutes is diffusion-limited, diffusion needs time, and the network’s geometry is the arrangement that supplies it. Blood leaves the heart at about 0.4 m/s and crawls through a capillary at under a millimetre per second, and the slowing is a direct consequence of the cube law rather than an independent design decision.

It also puts the whole network firmly in the regime where inertia has stopped mattering. A capillary at 5 µm diameter and 0.5 mm/s has a Reynolds number around 0.003. Poiseuille’s law is not an approximation there; it is exact, the flow is reversible, and the assumption the cost function was built on is at its most secure precisely where most of the vessels are.

Where the law is not obeyed #

The exponent that real networks show is not three, and the discrepancy is informative rather than embarrassing.

Three reasons, each of which identifies an assumption:

Large arteries are not laminar and not steady. The aorta has a Reynolds number in the thousands and a pulsatile flow, so the pressure drop is not Poiseuille’s — it depends on the flow to a power between one and two rather than to the first. Redo the minimisation with a turbulent friction law and the exponent falls towards 7/3. The measured 2.7 in large vessels sits between the laminar and turbulent predictions, which is what a network spanning both regimes should show.

Trees are not vessels. A tree’s xylem carries water under tension, its branches must also hold themselves up mechanically, and the mechanical cost of a cantilever goes as a different power of the radius from the metabolic cost of a tube. Leonardo’s rule — area preserved, exponent two — comes out of a purely mechanical argument, and real trees sit between the two.

The cost of the fluid is not the only cost. Murray’s $k$ lumps together the maintenance of the blood and of the wall, which scale differently, and in a large vessel the wall is a thick muscular structure whose cost goes as something nearer $r$ than $r^2$.

None of those makes the law wrong. They make it the laminar, steady, metabolically-dominated limit of a family of results, and the exponents observed in different parts of a real network are a map of which limit applies where.

An optimisation, in a field of conservation laws #

It is worth registering how different this essay’s method is from the rest of the field, because the difference is not cosmetic.

Every other result in this field is a conservation argument. Betz’s limit, the propulsive efficiency, the Borda–Carnot loss, the conjugate depths of a jump and Euler’s turbomachinery equation all come from writing down what crosses the faces of a box, and every one of them is a statement about what is possible. They say a machine cannot beat 16/27 and are silent about what any particular machine does.

Murray’s law is a variational argument, and it says something quite different: not what is possible, but what is best, given a stated notion of cost. That changes what the result is answerable to. A conservation result is wrong only if the physics is wrong. An optimisation result can be perfectly correct and still describe nothing, because the thing it optimises may not be the thing the world is optimising — or the world may not be optimising anything at all.

The defences available are therefore different too, and this essay uses all three. The result is parameter-free where it can be: the cube law and the one-third share contain none of the constants that went in. It has a local mechanism: the constant shear stress is something a wall can sense, which is what turns “least cost” from a claim about a designer into a claim about growth. And it is falsifiable against a photograph, since the cube law is a statement about measurable radii rather than about anything hidden.

Sailing’s velocity made good is the field’s other optimisation, and it passes the same tests for the same reasons — a closed form with no boat in it, and an angle anybody can measure.

Three quantities that come out equal in every generation #

The cube law fixes the radii and says nothing about the lengths, so the network’s behaviour as a whole needs one more assumption. Take the simplest available — that a vessel’s length scales with its radius, so the tree is geometrically self-similar — and three results drop out at once, all of them exact.

A generation holds $2^n$ vessels, each of radius $r_0 2^{-n/3}$ and length $L_0 2^{-n/3}$. One vessel’s resistance goes as length over the fourth power of radius, which grows as $2^{n}$; putting $2^n$ of them in parallel divides that back out again. Every generation has the same resistance. The same cancellation run on volume — $2^n$ vessels of cross-section $r_n^2$ and length $L_n$ — gives $2^{n-n}$, so every generation holds the same volume of fluid. And since the whole flow passes through each generation in turn, the power dissipated is that flow squared times the resistance: every generation burns the same power and drops the same pressure.

An architecture in which every scale costs the same is a strong statement, and it is exactly the kind of statement a self-similar structure makes. It is also testable, and it fails. In the systemic circulation over half the pressure drop occurs across the arterioles alone, which is one narrow band of generations rather than an even spread over thirty of them.

The failure is informative in the way the exponent measurements are. The radii are not what is wrong — the cube law survives it — but the assumption that length follows radius does. Arterioles are long for their bore, and being long for their bore is precisely how a vessel becomes the place a network puts its resistance, which is what a body wants of a vessel it intends to use as a valve.

Where the model stops #

No pulsatility. The heart is a pump with a beat, arteries are elastic, and the resulting wave propagation is a subject of its own. The cost function has a steady flow in it.

No blood rheology. Blood is not Newtonian, and in vessels comparable to a cell’s diameter it stops behaving as a continuum at all — red cells travel in single file through capillaries, and the effective viscosity depends on the vessel’s diameter in a way that has its own name. Poiseuille’s law is used here as though the viscosity were a constant, and in the smallest vessels it is not.

No branching angles. Murray’s original work also derived optimal angles from the same cost argument, and those are not computed here.

And nothing here is about biology. The scope of this essay is a flow and a cost. The mechanism by which a vessel remodels, the signalling that carries the shear stress into a change of diameter, and what goes wrong when it fails are all outside it — named as the reason the result is credible, and not modelled.

There is one further absence worth naming because it is a limit on the method rather than on the model. The cost function has a single vessel in it, minimised independently. A real network has to deliver flow to a territory, so the lengths are not free and the branching angles interact with the radii, and the true problem is a global optimisation over a tree rather than a one-dimensional minimisation repeated. Murray’s own papers took the first step into that — the optimal angle at a junction falls out of the same cost with the geometry allowed to vary — and the full problem is a different subject. What the cube law survives is that the radius condition is local: it depends only on the flow that vessel carries, whatever the rest of the tree is doing, which is why it can be applied at every junction independently and why a simple local rule can reach it.

Who found it, and when #

Cecil D. Murray published the law in 1926, in two short papers in the Proceedings of the National Academy of Sciences, deriving both the cube law and the optimal branching angles from the cost argument above. He was a physiologist, and the argument is one of the earliest clean uses of an optimisation principle in biology.

Leonardo da Vinci’s rule — that the total cross-sectional area is preserved through a branching — is four centuries older, appears in his notebooks on trees, and is an observation rather than a derivation. That the two disagree, and that real networks sit between them, is the most useful thing about having both.

The shear-stress connection came much later, from the 1970s onwards, when it became possible to measure endothelial responses. Murray’s derivation is a cost argument with no mechanism in it; the discovery that its consequence is a locally-sensible quantity turned it from an elegant calculation into an explanation.

Where the field goes #

This anchor stands alone in the field, and it is the clearest case of the method being applied to something that grew rather than something that was built. What made it possible is that the flow regime is simple — laminar, steady, low Reynolds number — so the pressure-drop law is exact, and the entire content is in the optimisation.

The field’s last two anchors return to a body in a stream. A propeller blade or a hydrofoil moving fast enough through water pulls the pressure at its own surface below the vapour pressure of the liquid, and the liquid boils at room temperature. Where that happens is computable from the site’s own inviscid solver — the condition is simply that the cavitation number equals minus the minimum pressure coefficient — and the calibration is the circular cylinder, whose answer is exactly three.