A tree transparent at every junction is not the quietest

Murray’s law passes the pulse where the pulse is viscous answered a question about one junction at a time. When an artery divides, a pressure pulse arriving along the parent is partly reflected unless the two daughters together present the same admittance to the wave as the parent. For a junction in the large arteries, where the pulse is inertial and the flow profile flat, the daughters’ areas should add up to the parent’s; for one in the small arteries, where viscosity reaches across the whole bore in each heartbeat, the daughters’ radii cubed should add up to the parent’s radius cubed, which is Murray’s law. In between, the rule that passes the pulse is set by one number, the parent’s Womersley number — the radius over the depth viscosity reaches in one cycle — and that essay drew it as a map.

That essay’s trees were all built to one exponent from the aorta down. Its closing question was the obvious next one: build the tree the map describes, every junction at its own transparent rule, and ask how much less it reflects than the best single-exponent tree, what it costs in Murray’s terms, and whether a heart pumping into it would do less work. The answer to the first question is that it reflects more, and the reason is worth the essay.

There is a reason to expect the question to matter only a little, and it comes from the symmetric tree. Murray’s law is not the rule for a pulse built a tree of identical junctions and put ends on it that reflect half the pulse, and found the ends dominating everything: at the heart rate a Murray tree reflected 0.49 at its root and the best single exponent 0.45. In a symmetric tree every path from the root to a leaf has the same length, so all 64 leaf echoes arrive back at the heart together and in phase, and no branching rule can do much about a reflection that large arriving all at once. The asymmetric tree is different in exactly that respect. A path down the trunk and a path down the side branches differ in length several times over, so the leaves’ echoes return spread out in time and partly cancel one another before the branching rule has done anything. What is left, 0.07 to 0.15 at the root, is small enough for the rule to decide — and so small enough for the difference between a local rule and a global one to show.

A tree built junction by junction #

The tree is the earlier essay’s: six generations of asymmetric branching from an aorta 12 millimetres in radius, every junction dividing a parent into a continuing trunk and a side branch half its size, every segment thirty radii long, every segment a viscous elastic tube in Womersley’s sense with a wave speed of 5 metres a second, and every one of the 64 leaves ending in a reflection of half the pulse on its own impedance — the peripheral beds that a real arterial tree ends in reflect a substantial fraction of what arrives. The trees differ only in how each junction divides its parent. For a branching exponent k the trunk and side branch take radii rt and rλt, with $t = (1 + \lambda^k)^{-1/k}$ so that $r^k$ is conserved.

The map tree chooses k at each junction separately: the exponent at which that junction alone, with its own parent radius, reflects least of a 1.2-hertz pulse. The others use one k everywhere: 2, the area-preserving rule; 3, Murray’s; and the single exponent at which the whole tree’s root reflects least at the heart rate, which is 2.354.

The first figure shows what the map builds. The aorta’s junctions sit at 2.06, close to area-preserving, and the trunk’s stay near there as it narrows slowly. The side branches, a half the size at each division, fall quickly into the range where the pulse is viscous, and their junctions’ exponents rise towards three, reaching it at the tree’s smallest parents, about a quarter of a millimetre in radius. The map tree is area-preserving where the blood is and Murray’s where the branches are fine, exactly as the earlier essay’s map said a tree that passes the pulse should be.

The reason the rule changes along the tree is the Womersley number, $\alpha = r\sqrt{\omega/\nu}$, the vessel’s radius over the depth that viscosity reaches in one cycle. Too fast for a profile found that above about ten the core of an oscillating flow moves as a plug and the viscous layer is a thin skin at the wall; in this aorta at the heart rate α is about eighteen, and it falls to seven by a radius of five millimetres. There the pulse is carried by the fluid’s inertia and the wall’s elasticity, and the pulse that has to travel showed that it then moves as a wave with an admittance proportional to the vessel’s area — so daughters whose areas add up to the parent’s present the same admittance and pass the pulse. Below α of about one the flow is quasi-steady Poiseuille flow at every instant, the admittance goes as the fourth power of the radius over the length, and the rule that matches it is the cube law. Between the two the change is gradual rather than sudden: where the parabola goes measured how slowly the parabolic profile gives way as α rises, and the map’s exponent climbs through the same range, which in these trees is the arteries of one to five millimetres.

It reflects more #

The second figure is the result, and it is not the expected one. At the heart rate the map tree reflects 0.130 of the pulse at its root. The best single-exponent tree reflects 0.0735 — little more than half as much. Murray’s tree reflects 0.146 and the area-preserving tree 0.138, so the map tree is better than either extreme, but a tree built to one compromise exponent from top to bottom beats one whose every junction is individually transparent.

The harmonics tell a more mixed story. At the third to fifth harmonics the map tree is the best of the four, reflecting 0.030, 0.038 and 0.024 against the single exponent’s 0.053, 0.048 and 0.047. Murray’s tree is remarkable at the second harmonic, reflecting almost nothing, and poor above it. No tree is best at every frequency, and the best single exponent is best only at the frequency it was chosen for.

The harmonics matter because the pulse is not a sine wave. The heart ejects in a short burst and then stops, so the flow into the aorta is carried by its fundamental and a series of harmonics falling off only slowly; the pulse used here, the one the earlier essays measured their trees with, has ten, and the second to fifth carry most of the shape that makes a pressure trace a pulse rather than a ripple. The reflected wave is the sum of each harmonic’s reflection with its own phase, so a tree that reflects less at the fundamental but more at the harmonics returns a smoother echo rather than a smaller one, and one that reflects less at the harmonics returns a smaller echo with the same slow swell. Which of those the heart would prefer is not settled by a reflection at one frequency, and the power calculation below is the attempt to fold all ten into a single number.

Why local transparency is not global #

A junction that reflects nothing lets the pulse through and sends nothing back — but the pulse it lets through reaches the leaves, and the leaves reflect half of it. That echo travels back up the tree, and a tree whose junctions are all transparent passes it back up as faithfully as it passed the pulse down. The root’s reflection is then mostly the leaves’ echo, attenuated by the viscous losses of the round trip and scrambled by the different lengths of the different paths.

A tree whose junctions are slightly mismatched sends back small reflections of its own, from every junction, with phases set by where each junction sits. If the mismatch has the right sign and size, those reflections arrive at the root out of phase with the leaves’ echo and partly cancel it. The best single exponent finds that mismatch: its first junction alone reflects 3.0 per cent of the pulse, four times the map tree’s 0.73, and the whole tree reflects half as much.

The figure tests the explanation directly by varying the leaves’ reflection. With leaves that absorb everything — nothing to cancel — the map tree is the best tree of all: it reflects 0.035 at the root against the best single exponent’s 0.042, re-chosen for absorbing leaves. As the leaves begin to reflect, the best single exponent pulls ahead, and by a leaf reflection of a quarter it is already better, 0.082 against 0.098. Making every junction transparent is the best design only when the ends absorb. When they reflect, the best design hides deliberate mismatches in its junctions, tuned to the echo.

An antireflection coating does the same #

The mechanism is familiar from optics. A lens’s surface reflects some of the light arriving at it because glass and air present different impedances to the wave. The remedy is not to make the glass match the air — impossible — but to add a thin coating whose own two surfaces reflect, a quarter of a wavelength apart, so that the two reflections arrive back out of phase and cancel. The coating adds mismatches to remove a reflection. The pulse that grows as it leaves the heart met the quarter-wave idea in an artery’s own length, as a resonance amplifying the pulse; here it appears in reverse, as a tree’s junctions arranged to cancel its ends’ echo. The best single exponent is an antireflection coating distributed through the tree, and like a coating it is tuned to one frequency, which is why it is worse than the map tree at the higher harmonics.

The same idea — accept a mismatch where it can be tuned, rather than seek a match that cannot be had everywhere — turns up elsewhere in the subject. The shock that passes without an echo found that a shock tunnel’s driver and driven gases can be tailored so that the reflected shock stops both at the same pressure and nothing comes back from their interface. That tailoring holds at one shock strength only, as the coating holds at one wavelength and the best single exponent at one heart rate. In each case the absence of an echo is not a property of any one boundary. It belongs to the whole arrangement at one condition, and it is lost as soon as the condition moves.

What the map tree costs #

Murray’s law is the rule for the cheapest steady flow: the one that balances the work of pushing blood through vessels, which small vessels make expensive, against the metabolic cost of the blood they hold, which large vessels make expensive. Measured against Murray’s tree built from the same aorta, the map tree holds 39 per cent less blood and resists steady flow 1.85 times as much. It is much closer to the area-preserving tree in both — which holds 43 per cent less and resists 2.1 times as much — because the large vessels that hold most of the blood and set most of the resistance are where the map says two. The best single-exponent tree sits between them, with 26 per cent less blood and 1.43 times the resistance.

So the tree that passes the pulse junction by junction is a costly tree in steady terms: it spends a great deal of resistance to save blood volume, which is the trade Murray’s law is written to refuse. Whether that trade is a good one depends on how the body weighs the two costs, and on how much of the heart’s work is the steady flow’s and how much the pulse’s.

The heart’s work #

The last question was whether a heart pumping into the map tree would do less work. For the oscillatory part — the power spent on the pulse itself, for the same ten-harmonic flow pulse — the answer is no. Murray’s tree takes the least, 1.089 times what a non-reflecting tube would take; the best single exponent 1.111; the map tree 1.154; the area-preserving tree 1.183. The differences are a few per cent, and they follow the trees’ sizes more than their reflections: a tree of wider vessels presents a lower impedance and takes less power for the same flow, whatever it reflects.

That is worth stating plainly because the argument for a pulse-transparent tree is usually made in terms of the heart’s work, and in this comparison it does not hold. The trees are all grown from the same aorta through the same number of generations, so the wider trees end in wider leaves, and part of the comparison is simply that.

The power itself is computed plainly. For each harmonic of the flow pulse, the heart does work at the rate of the flow amplitude squared times the real part of the tree’s input impedance at that frequency; summed over the ten harmonics and divided by what the same flow would cost into a uniform aorta that reflected nothing, it is the number drawn. A reflection that returns in phase with the pressure raises the real part and costs work; one that returns out of phase lowers it. That is why the reflection and the power do not rank the trees the same way: the reflection’s size at the root says nothing about its phase, and for these trees the phases differ enough to reorder them. A comparison at fixed leaf size — the capillary beds a tree has to feed — would be the fairer one, and is not what was computed.

What was checked #

The map tree’s recursion was written afresh, so it was checked against the earlier essay’s asymmetric tree: with a constant exponent of 2, 2.4 or 3, at the heart rate and its third harmonic, the two give the same root reflection to the last bit. The steady resistance of a one-junction tree, built by walking the tree, matches the Poiseuille resistances in series and parallel written out by hand, exactly. The map tree’s own junction exponents come from the earlier essay’s junction calculation, which was checked there against its inviscid limit. The tests refuse a frequency of zero, a negative exponent and a tolerance of zero.

What the picture cannot show #

One asymmetry. Every junction splits in the same proportion. Real trees are more varied, and the cancellation the best single exponent achieves depends on the distribution of path lengths to the leaves; a more varied tree would scramble the echo more and leave less for any rule to cancel.

Leaves on their own impedance. Each leaf reflects half the pulse on its own characteristic impedance. Real peripheral beds have impedances with their own frequency dependence — resistive at low frequency, compliant above it — and their reflection varies with the harmonic, which would move the balance between the map and the best single exponent at each harmonic.

Six generations and a uniform wave speed. An arterial tree has twenty or more generations to the arterioles, and its wave speed rises in the stiffer peripheral arteries. The map tree’s advantage at absorbing ends and disadvantage at reflecting ones would survive both; the numbers would not.

A rule local to each junction, and a design that is not #

The broader point is about how trees are built. A tree grown by a rule applied at each junction, from what that junction can sense, can only be locally optimal, and the radius that costs least and the angle a junction chooses were rules of that kind. For steady flow local optimality is global, because each segment’s cost depends only on its own flow and size. For a wave it is not, because a wave reflected from one junction interferes with waves reflected from every other and from the ends, and the whole tree’s reflection is a sum of phases no single junction can see. The quietest tree has to be designed knowing where its echoes come from — or be grown by a rule that can sense them, which is a question about biology rather than fluid mechanics.

Who worked it out #

Murray’s law is from 1926. Womersley’s solution for the oscillating flow in an elastic tube is from 1955. The analysis of arterial branching as impedance matching — the idea that junctions should be transparent to the pulse — is Taylor’s and McDonald’s from the 1960s, and the measured branching exponents of real arteries, between two and three and rising towards three in the smaller vessels, come from casts and imaging of human and animal trees in the decades since. Antireflection coatings, the quarter-wave cancellation this tree reinvents, date from Rayleigh’s observation of 1886 that tarnished glass reflects less than clean.

Still open: a tree at the size it has to feed #

Every comparison here grew its trees from the same aorta, so trees with larger exponents ended in larger leaves and the heart’s work was compared between trees feeding different beds. The fair comparison fixes both ends: the aorta at the root and the capillary beds at the leaves, with the number of generations whatever each rule needs to get from one to the other. The next calculation builds the four trees between fixed ends, with leaves that are resistive at low frequency and compliant above it, and asks again which reflects least, which costs least in Murray’s terms, and which takes least of the heart’s work — the question the pulse-transparency argument was always meant to answer.