Between fixed ends the transparent tree is the quiet one

A tree transparent at every junction is not the quietest reached a conclusion with an uncomfortable shape. Each junction of an arterial tree has a branching rule that lets the heart’s pulse through it without reflection, set by the junction’s Womersley number: area-preserving in the aorta, where the pulse is inertial, and Murray’s cube law in the small arteries, where viscosity reaches across the bore in each beat. Yet a tree built that way, every junction at its own transparent rule, reflected more of the pulse at its root than a tree built to one compromise exponent from top to bottom. And when the heart’s oscillatory work was added up, Murray’s tree — the one built for steady flow and indifferent to the pulse — took the least.

That essay explained the first result as an antireflection coating, and the explanation is sound as far as it goes: a tree of slightly mismatched junctions sends back small echoes of its own that can cancel the echo from its ends. But it ended by naming the flaw in its own comparison. Its trees all grew from the same aorta through the same six generations, so a tree with a larger exponent, whose daughters shrink more slowly, ended in larger leaves. Murray’s tree finished in vessels several times the size of the area-preserving tree’s. The trees were feeding different beds, and the heart’s work was being compared between them.

This essay makes the comparison it asked for. The root is the same aorta. The ends are the same small vessels. Each rule takes as many generations as it needs to get from one to the other, and the leaves are given a frequency dependence of their own. With both ends fixed, the ranking turns over.

Trees grown to a size rather than a count #

Every tree here starts from an aorta 12 millimetres in radius and divides asymmetrically at each junction. A parent of radius r becomes a trunk and a side branch, the side branch half the trunk’s size, with radii $rt$ and $\lambda rt$ where $\lambda = 0.5$ and $t = (1 + \lambda^k)^{-1/k}$, so that the k-th power of the radius is conserved at every division. Every segment is thirty radii long and is a viscous elastic tube in Womersley’s sense, with a wave speed of 5 metres a second. Those are the earlier essay’s choices, kept so that the only change is the stopping rule.

The stopping rule is now a size. A branch keeps dividing until its radius falls below half a millimetre, and there it ends in a leaf. The side branches, halving at each division, reach that size in four or five generations whatever the rule. The trunk shrinks much more slowly — by the factor t at each junction, which is 0.894 for the area-preserving rule and 0.962 for Murray’s — and so the trunk path is where the rules part company.

The leaves are no longer a flat reflection. Each is a small Windkessel: a resistance in parallel with a compliance. The resistance is set so that, on its own, it reflects half of a steady wave arriving along the leaf’s vessel, as the earlier trees’ leaves did. The compliance is set so that the corner where it takes over is 2 hertz. Below the corner the leaf behaves as a resistance and reflects; above it the compliance shunts the oscillation and the leaf looks more and more like an open end. That is the behaviour peripheral vascular beds are measured to have, and it means the heart rate of 1.2 hertz meets reflecting leaves while its harmonics meet leaves that reflect progressively less, and with the opposite sign.

The impedance of each tree is computed by working up from the leaves. At every junction the two daughters’ input impedances combine in parallel, the parent’s reflection follows from that load and its own characteristic impedance, and the reflection is carried up the parent’s length with the complex propagation constant of a Womersley tube, which both delays it and attenuates it. Two subtrees whose top radii are equal under one rule are the same subtree, so the calculation is memoised on the radius. That is what makes a tree of eighty generations along its trunk and thirty-eight thousand leaves a matter of seconds.

The order turns over #

The first figure is the reflection at the aortic root, at the heart rate and its first four harmonics. At 1.2 hertz the Womersley-map tree, every junction at its own transparent exponent, reflects 0.028 of the pulse. The best single exponent from the earlier essay, 2.354, reflects 0.035. Murray’s tree reflects 0.083 and the area-preserving tree 0.21. The junction-by-junction tree is now the quietest of the four at the heart rate.

At the harmonics it is not merely the quietest but nearly silent. It reflects 0.0007 at the second harmonic and no more than 0.0046 up to the fifth, where the single exponent reflects between 0.015 and 0.030 and Murray’s between 0.033 and 0.070. The area-preserving tree, so loud at the fundamental, falls to 0.003 to 0.026 at the harmonics, which is the one respect in which it resembles the map tree. Its big reflection at the heart rate comes from its many junctions below a few millimetres, where the pulse is viscous and an area-preserving division is badly mismatched. At a harmonic the Womersley number of every vessel is larger by the square root of the harmonic’s order, so the inertial regime, where conserving area is the transparent rule, reaches down into smaller vessels, and fewer of its junctions are wrong.

What happened to the antireflection coating? It is still there in principle. The best single exponent still sends back small echoes from its slightly mismatched junctions, and they are still tuned to cancel something. What changed is how much there is to cancel. A tree of six generations had 64 leaves, all within a few path lengths of the root, so their echoes came back strongly and a deliberate mismatch could earn its keep by opposing them. A tree grown to half a millimetre has thousands of leaves spread across path lengths from under a metre to several metres, and every path is long in units of the viscous decay. By the time a leaf’s echo has been attenuated down the trunk and back and scrambled against thousands of others of different delay, there is little coherent echo left to cancel. With little to cancel, the coating’s own mismatches are no longer an investment; they are just reflection. The tree that sends back nothing from its junctions wins.

The same two ends, and very different trees #

The second figure shows what fixing both ends does to the trees themselves, and the differences are not small. The area-preserving tree reaches half a millimetre in 29 generations along its trunk and ends in 1,173 leaves. Murray’s tree needs 81 generations and ends in 38,215. The map tree, which is area-preserving at the top and Murray’s at the bottom, is 54 deep with 6,936 leaves, and the best single exponent 42 deep with 3,966.

That spread is the reason the six-generation comparison could not be trusted. In a tree of six generations Murray’s rule had only six chances to shrink the trunk by 0.962, so it ended in vessels four-fifths the size of the aorta, and those wide vessels presented a low impedance that took little power for a given flow. Grown to the same small vessels, Murray’s tree must divide eighty times along its trunk, and its large-vessel advantage is spent on the path to the bed rather than being the bed.

The volume column says the same thing with blood. Murray’s tree holds 4.97 litres in its vessels, 6.1 times the area-preserving tree’s 0.81 and 4.1 times the map tree’s 1.22. A human body holds about five litres of blood in all, most of it in veins, so Murray’s tree with these segment proportions is not a body that could exist; it is the cube law followed with a slowly shrinking trunk until it reaches a size the body actually needs to feed. The map tree’s 1.22 litres is closer to the arterial volume a body has, and the area-preserving tree’s is closer still.

Wherever the tree is stopped #

One choice of terminal size is one data point, and half a millimetre is a convenient number rather than an anatomical one. So the third figure stops the trees at every size from 0.3 to 2 millimetres. Stopping them sooner shortens all four, and Murray’s most, since it has the most generations to lose.

The map tree reflects between 0.022 and 0.051 across the whole range, and it is the quietest of the four at every terminal size but one. At 1.4 millimetres it and the best single exponent are level at 0.051, the map tree louder by two parts in ten thousand — a tie in any honest reading, and a reminder that the leaves’ positions can line up to give a single exponent’s junction echoes something to cancel. Murray’s tree stays near 0.08 to 0.10 throughout. The area-preserving tree falls from 0.24 to 0.13 as it shortens, because a shorter tree has fewer mismatched small junctions to reflect from, but it is never close to the other three.

The robustness is not limited to the stopping size. Moving the leaves’ corner frequency to half a hertz or to five, or the terminal radius to a millimetre, keeps the map tree lowest, between 0.014 and 0.030 at the heart rate. The conclusion does not depend on a lucky choice of the two numbers that describe the bed.

Only strongly reflecting leaves reverse it #

The earlier essay’s ranking came from somewhere, and the fourth figure finds it. Here the leaves are made purely resistive — no compliance, so they reflect the same at every frequency — and their reflection is swept from zero to nine-tenths. The map tree’s root reflection barely moves, from 0.022 with absorbing leaves to 0.041 with leaves that reflect nine-tenths. Its junctions pass the pulse down and the leaves’ echoes back up, and the long attenuating paths make the echoes weak whatever the leaves do.

The best single exponent moves the other way and much further. With absorbing leaves it reflects 0.052, worse than the map tree. As the leaves begin to reflect, its root reflection falls — 0.047, 0.037, 0.023 — and at a leaf reflection of 0.8 it reaches 0.005. Its junctions’ echoes are cancelling the leaves’, and the more the leaves send back, the more there is for the junctions to cancel. The two curves cross where the leaves reflect about half: at 0.45 the map tree is still ahead, 0.031 against 0.034, and at 0.55 it is behind, 0.033 against 0.027.

So the earlier finding was not wrong, but it was a finding about strongly reflecting ends. Six generations from the aorta, with leaves reflecting half the pulse on their own impedance, put that tree in the regime where cancellation pays. Peripheral beds are resistive at the heart rate but not perfectly so, and their compliance makes them absorb more of the harmonics; with leaves like that, the junction-by-junction tree is the one the pulse passes through most cleanly.

The heart’s work, with both ends fixed #

The fifth figure answers the question the pulse-transparency argument was always about. For the same ten-harmonic flow pulse, the oscillatory power the heart supplies is the flow amplitude squared times the real part of the tree’s input impedance, summed over the harmonics and divided by what the same flow would cost into an aorta that reflected nothing. With 2-hertz leaves, Murray’s tree takes 1.14 of that matched value, the most of the four. The map tree and the best single exponent take 1.067 each, level to the third figure. And the area-preserving tree takes 0.892, the least, below the matched value.

Murray’s tree has gone from cheapest to dearest, and the reason is the one this essay set out to remove. In the six-generation comparison its wide leaves gave it a low input impedance; here its leaves are the same size as everyone else’s, and what is left is its reflection at the heart rate, 0.083, coming back with a phase that raises the real part of its impedance at the fundamental, where most of the pulse’s power is. The ranking holds across the leaves’ corner frequency from half a hertz to five: Murray’s tree takes between 1.139 and 1.145 throughout, and the map tree between 1.053 and 1.068.

The area-preserving tree’s figure below one needs saying carefully, because it looks like a free lunch. Its large reflection at the heart rate comes back with a phase that lowers the real part of its input impedance at the fundamental, so the heart does less oscillatory work into it than into a matched tube. That is the reflection doing the opposite of harm — and it is bought with the third of the comparison the figure does not show, the steady flow, which the area-preserving tree resists more than any other.

What each tree costs in steady flow #

Murray’s law is the rule for the cheapest steady flow: the balance between the work of pushing blood through small vessels and the metabolic cost of keeping blood in large ones. The sixth figure plots those two costs for the four trees, with every leaf open to the same pressure. Murray’s tree has the least steady resistance, 3.49 MPa·s/m³, and holds the most blood, 4.97 litres. The area-preserving tree has the most resistance, 16.7, and the least blood. The map tree sits between, at 10.0 and 1.22 litres; the best single exponent at 8.5 and 1.41.

So fixing both ends does not make the map tree cheap in Murray’s sense. It resists steady flow nearly three times as much as Murray’s tree. What it buys with that is a quarter of the blood volume, the quietest root and an oscillatory load no worse than the single exponent’s. Whether that is a good trade depends on a weighting between the steady work and the volume that the fluid mechanics cannot supply. The radius that costs least wrote Murray’s balance as a price per unit volume of blood, and on that price the cube law is optimal by construction. A body that also has to keep its arteries from echoing the heart is paying for something Murray’s accounting does not include.

The resistance numbers also say something about the model rather than the trees. A resting adult’s whole systemic circulation resists flow at roughly 8 MPa·s/m³, and most of that is in the arterioles below half a millimetre — vessels none of these trees contain. The map tree’s 10.0, for its conducting arteries alone, is already more than the body’s whole, which is the price of segments thirty radii long followed for dozens of generations. The trees are a fair comparison with one another and not a model of a person’s arteries.

The leaf, the memo and the cube law, checked #

The last figure is the ledger. The leaf’s impedance at zero frequency equals its resistance, and at its corner frequency has the magnitude of the resistance over the square root of two, as a resistance in parallel with a compliance must. The memoised recursion, which treats every subtree of one radius as the same subtree, was checked against a full walk of every branch on a tree stopped at 3 millimetres, small enough to walk, and agreed exactly: the memo changes the cost of the calculation and not a single bit of its answer. Murray’s tree, walked branch by branch down to 2 millimetres and summed over its 590 leaves, conserves the cube of the radius: the leaves’ cubes add up to the aorta’s to five parts in a thousand million million. The model refuses a frequency of zero, a negative exponent and a rule that is not a number.

What thirty-radius segments leave out #

A path that is too long. Every segment is thirty radii long and every junction divides in the same proportion, so the trunk path of the map tree is 4.4 metres from the aorta to half a millimetre, and Murray’s is nine. A human arterial path from the heart to the foot is about a metre and a half. Longer paths attenuate echoes more, and that attenuation is part of why the junctions’ own reflections decide the ranking here. A tree with realistic lengths would carry stronger leaf echoes and give the single exponent more to cancel; the crossover in the fourth figure says where that would start to matter.

One wave speed. The wave speed is 5 metres a second everywhere. Real peripheral arteries are stiffer and carry the pulse faster, which shortens their electrical length and moves the junctions’ echoes in phase.

Junctions with no losses of their own. The junctions are points where impedances meet. Real branchings have angles and lose some of the pulse to separation at the branch, which the angle a junction chooses treated for steady flow; neither is in these trees.

Why fixing the ends mattered #

The general lesson is about comparing designs. The six-generation trees were a natural first experiment because they held the topology fixed: the same number of junctions, the same number of leaves, a change in one exponent. But the thing a body has to do is fixed at the other end, by the beds to be fed, and holding the topology fixed meant letting the beds float. A comparison that holds the wrong thing constant can reverse a ranking, and here it reversed two.

Murray’s law passes the pulse where the pulse is viscous drew the map that says which exponent passes the pulse at each Womersley number, and Murray’s law is not the rule for a pulse showed that in a symmetric tree the ends dominated everything. Between those essays and this one, the map has gone from a statement about one junction to a design for a whole tree, and the design has passed the test that matters: built to the same ends as its rivals, it is the quietest and among the cheapest for the pulse. The Womersley number that too fast for a profile introduced as a statement about a velocity profile has become the rule a branching tree should follow to keep the heart’s pulse from coming back.

Two older pieces of the subject carry the same moral about what a match is matched to. The pulse that has to travel found that an elastic tube carries a wave only when it is long in wavelengths, so whether a junction’s echo matters at all depends on where along the tree it sits — the attenuation that silenced the leaves here is that essay’s second number at work. And the shock that passes without an echo found that a shock tunnel’s interface sends nothing back at one incident strength only, found by asking the reflected shock to stop two gases at the same pressure. A matched condition is always a condition matched at something, and the choice of that something is the design decision.

From Frank’s Windkessel to fixed-end trees #

Murray’s law is from 1926 and Womersley’s oscillating-flow solution from 1955. Treating the peripheral beds as a resistance with a compliance is Otto Frank’s Windkessel of 1899, put in parallel at the leaves as later three-element models do. Taylor’s and McDonald’s analyses of arterial branching as impedance matching are from the 1960s, and the measured branching exponents of real arteries — near two in the aorta, rising towards three in the smaller vessels — come from casts and imaging in the decades since. The fixed-end comparison is the standard one in vascular optimisation; what it adds here is the reversal of the six-generation result.

Still open: a tree whose paths have real lengths #

The trees here have paths several times longer than a body’s, because every segment is thirty radii long whatever its generation. Real arteries grow shorter relative to their radius as they branch, and a tree from the aorta to the arterioles fits within a metre and a half. The next calculation gives each generation a measured length-to-radius ratio, so the paths are the body’s length, and asks whether the stronger leaf echoes that shorter paths carry are enough to bring the antireflection coating back — whether, at physiological lengths, the junction-by-junction tree still wins, or whether the crossover in leaf reflection moves below the half where real beds sit.