Murray's law passes the pulse where the pulse is viscous

Murray’s law is not the rule for a pulse compared two ways of sizing a branching vessel. Murray’s law, from 1926, sizes each junction so that the steady flow through the tree is carried at the least cost — the work of pushing blood through the vessels plus the metabolic cost of the blood they hold — and it says the cube of the parent’s radius equals the sum of the cubes of the daughters’. The heart’s flow is not steady, though, and a pressure pulse arriving at a junction is partly reflected unless the daughters together present the same admittance as the parent. That essay found the transparent rule for an even split to be a different exponent, set by how the wave speed changes with radius; found that a six-generation Murray tree reflects a third of the pulse at the heart rate; and found that viscosity in the smallest branches means no single area rule is transparent at every frequency.

It ended by dropping a symmetry every calculation there had kept. Real arteries do not split evenly. The aorta runs from the heart to the pelvis shedding branches as it goes — the coronaries, the arch vessels, the renal and mesenteric arteries — each a fraction of its own size, while it carries on only slightly narrower. The question it left was whether, at such a lopsided junction, the rule that passes the pulse is still an exponent, and what a tree of lopsided junctions looks like when it is built to pass the pulse well. This essay answers both, and the second answer turns the first essay’s conclusion about Murray’s law into something sharper.

Without viscosity, asymmetry changes nothing #

At a junction the pressure is continuous and the flow is conserved, and a wave arriving along the parent is reflected according to how the sum of the daughters’ admittances compares with the parent’s. The admittance of an elastic tube, without viscosity, is its area over its density times its wave speed. With the wave speed varying as a power $m$ of the radius, the admittance goes as $r^{2+m}$, and admittances in parallel add. So a junction reflects nothing when

$$r_p^{\,2+m} = r_1^{\,2+m} + r_2^{\,2+m},$$

whatever the two daughters’ relative sizes. The transparent rule for an even split was the exponent $2+m$; for a lopsided split it is the same exponent, exactly. With the wave speed the same at every radius, $m = 0$, it is 2: the daughters’ total area equals the parent’s.

The tree built this way inherits it. The calculation here builds asymmetric binary trees — every junction splits a parent of radius $r$ into a continuing branch of radius $rt$ and a side branch of radius $r\lambda t$, where $\lambda$ is the asymmetry and $t$ is fixed by the branching exponent $k$ — and with viscosity switched off, the exponent set to $2 + m$ and the leaves ending in their own characteristic impedance, the root reflects nothing, to three parts in $10^{16}$, at every asymmetry tried. The rule does not care how lopsided the junctions are.

With viscosity, one number decides #

Blood is viscous, and in a pulsing tube the flow’s profile does not keep up with the pressure in the same way at every size. Womersley’s solution for an oscillating flow in an elastic tube makes each segment’s characteristic impedance complex, and its dependence on radius changes with the Womersley number, $\alpha = r\sqrt{\omega/\nu}$: the radius over the depth viscosity reaches in one cycle of the pulse. With each segment treated that way, a junction can no longer be made exactly transparent, but for each asymmetry there is an exponent at which it reflects least.

The first figure plots that exponent against the parent artery’s radius, at the heart rate of 1.2 cycles a second. For an aorta-sized parent, 12 mm in radius, it is 2.06: close to area-preserving, whatever the asymmetry. For a 3 mm parent it has risen to between 2.25 and 2.7, depending on how lopsided the split is. For a 1 mm parent it is 2.96, and for 0.3 mm it is 3.00 at every asymmetry.

Three is Murray’s exponent. In the small arteries the rule that passes the pulse is exactly the rule for the cheapest steady flow. The reason is that in a vessel whose Womersley number is small, viscosity reaches across the whole bore within a cycle, the oscillating flow has the parabolic profile of steady flow at every instant, and the tube’s impedance to the oscillation is the same Poiseuille resistance that sets the cost of steady flow, which goes as the inverse fourth power of the radius and gives a characteristic impedance going as the inverse cube. Admittances going as the cube, added at a junction, give Murray’s law. The pulse in a small artery moves as the steady flow does, and the rule that is cheapest for one is transparent for the other.

The Womersley number is the whole answer #

The second figure confirms that nothing but the Womersley number is at work. Plotted against the parent’s radius, the transparent exponent at the heart rate and at three times the heart rate — the pulse’s third harmonic — would be two different curves, since the same artery has a larger Womersley number at a higher frequency. Plotted against the Womersley number itself, they are one curve. It is 3 below a Womersley number of about two and close to 2 above about fifteen, and it crosses over where the oscillating flow’s viscous layer is as thick as the artery is wide.

That gives the first essay’s “somewhere between area-preserving and Murray” an exact form. It is not a compromise between two rules applied everywhere; it is a map. Each junction’s transparent rule is set by its own Womersley number, which falls from about eighteen in the ascending aorta to below one in the arterioles, and a tree that passes the pulse at every junction is area-preserving at the top and Murray’s at the bottom, with the change made over the arteries of one to five millimetres. Measurements of real arterial trees put their branching exponents between about two and three, rising towards three in the smaller vessels, which is the same map read from the anatomy.

Where the crossover sits in the body #

The map can be laid over the anatomy. At the heart rate a Womersley number of fifteen belongs to an artery about ten millimetres in radius and a Womersley number of two to one about 1.4 millimetres. Everything larger than the first is on the area-preserving side: the aorta and the first few generations of its great branches. Everything smaller than the second is on Murray’s side: the small muscular arteries and the arterioles, where the flow is too slow to have a profile of its own and follows the pressure as steady flow would. Between them lie the medium arteries — the femoral, the brachial, the coronaries — which are exactly where the transparent rule is changing fastest and where asymmetry moves it most.

That is also where the pulse does most of its travelling. The pulse that has to travel from the heart to the periphery covers most of its distance in the large and medium arteries, and a tree whose junctions reflect little there returns little of the pulse to the heart. The smallest vessels, where Murray’s rule is also the transparent one, are so viscous that any wave entering them is damped within a few of its own lengths; a mismatch there reflects little that could reach the heart in any case.

One junction, two rules #

The third figure shows what the map costs if it is ignored. The same lopsided junction, with a side branch half the continuing trunk, reflects least near an exponent of 2.06 when its parent is aorta-sized and near 2.96 when it is a millimetre across. Each minimum is deep. Built to the other’s rule, each reflects several per cent of the pulse: Murray’s exponent at the aorta’s junction reflects nearly seven per cent, and the aorta’s rule at the small artery’s, nine. A tree built to one exponent throughout is therefore reflecting somewhere, whichever exponent is chosen, and the first essay’s finding that no single area rule is transparent at every frequency is the frequency version of the same fact.

A lopsided tree reflects less #

The fourth figure moves from one junction to a whole tree, with the leaves ending in their own characteristic impedance so that everything reflected comes from the junctions. Every tree is best near an exponent of 2.15 — the aorta-sized root’s rule, pulled slightly towards three by the smaller vessels below it — and that is so at every asymmetry, confirming that asymmetry does not move the rule much where the pulse is inertial. What asymmetry does change is how much comes back. At the best exponent the even tree reflects 0.149 of the pulse at its root; a tree whose side branches are half the trunk, 0.042; at 0.3 of the trunk, less than a hundredth.

The reason is the trunk. In a lopsided tree most of the flow stays in a vessel that changes size only a little at each junction, so the wave travelling down it meets a series of small mismatches rather than a few large ones, and the reflections from the small side branches are small because the side branches carry little. An even split sends the whole wave through a large change of vessel at every generation. The arterial tree, which is lopsided along the aorta and its main branches, is the shape that reflects least, for any rule.

The trunk stays nearly whole #

The fifth figure answers the essay before’s last question, whether the best pulsing tree is still close to area-preserving along its trunk. At a junction built to the aorta’s rule, $k = 2.06$, the daughters’ total area stays within two per cent of the parent’s at every asymmetry, and the continuing trunk keeps most of it: 92 per cent when the side branch is 0.3 of the trunk, 98 per cent when it is 0.15. Under Murray’s rule the trunk keeps even more, 98 per cent at 0.3, while the daughters’ total area grows past the parent’s — by seven per cent at 0.3, and by a quarter for an even split. Down a trunk that sheds many small branches, either rule gives a gently tapering vessel; the aorta’s rule makes it taper by exactly the side branches’ area.

What a reflected pulse costs the heart #

The reason to care about a junction’s reflection is where the reflected wave goes. It travels back up the tree and arrives at the heart while the heart is still ejecting, adding to the pressure the heart must push against in the late part of each beat. The heart then does more work for the same flow, and the wall of the aorta carries a larger pressure swing. A young, compliant arterial tree times its reflections to arrive after the aortic valve has closed, where they help fill the coronary arteries in diastole instead; a stiff one returns them early, which is part of why stiff arteries raise the load on the heart. A tree built to reflect little at its junctions leaves the timing to its terminal beds and its taper, which is where the timing is controlled.

For a pulse in a small artery the cost is different in kind. There the flow is, in the language of the world with no inertia, nearly a Stokes flow at every instant, the wave barely propagates, and the question of reflection gives way to the question of resistance — which is Murray’s question, and the reason the two rules meet.

Why the steady and the oscillating rules can agree #

It is worth saying why Murray’s law and the pulse’s rule coincide in the small arteries and not by accident. The angle a junction chooses and Murray’s law both come from minimising a steady cost, and the cost of pushing steady flow through a tube is its Poiseuille resistance. The pulse’s reflection at a junction is decided by impedances, and in a tube narrow enough for viscosity to reach its axis within a cycle the impedance to the oscillation is that same resistance, with nothing added by the fluid’s inertia. The two questions reduce to the same arithmetic in the same vessels. In the aorta, where the oscillation’s inertia dominates its resistance, they are different questions and get different answers, and the steady optimum is not the one the pulse that grows as it leaves the heart needs.

What was checked #

The sixth figure is the ledger. With the asymmetry set to one, the asymmetric tree’s root impedance equals the symmetric tree’s of the essay before, computed by a different routine, to six parts in $10^{16}$. Without viscosity, built to the exponent $2 + m$ with matched leaves, it reflects nothing at three asymmetries and two wave-speed exponents, to three parts in $10^{16}$. And at an inviscid lopsided junction the reflected and transmitted powers add up to the incident power to machine precision.

What the lopsided tree leaves out #

Real geometry. Every junction here has the same asymmetry and every segment a length proportional to its radius; the aorta’s side branches leave at particular places and their lengths do not scale with their radii.

Stiffening with size. The wave speed is taken the same at every radius. Smaller arteries are stiffer, which raises the inviscid exponent above 2 and moves the whole map up.

The leaves. The capillary beds at the ends of the tree reflect strongly, and a real tree’s reflection at the heart is dominated by them and by the aorta’s own taper as much as by its junctions; the leaves were matched here to isolate what the junctions do.

Blood is a suspension. Its viscosity is taken as a constant, and it is not one: blood is a viscosity made of particles, its red cells crowd towards the axis of a narrow vessel, and its effective viscosity falls in tubes below a few hundred microns — the Fåhræus–Lindqvist effect. That raises the smallest vessels’ Womersley numbers a little and moves the lower end of the map; it does not move the crossover in the medium arteries, where blood behaves as a continuum.

The heart’s other harmonics. The figures are at the heart rate and its third harmonic. The pulse carries energy in the first ten or so, and a rule that passes the first may reflect the fifth.

The convention the numbers depend on #

The branching exponent $k$ is defined by $r_p^k = r_1^k + r_2^k$; the asymmetry $\lambda$ is the side branch’s radius over the continuing trunk’s. The Womersley number is the parent’s radius times the square root of the angular frequency over blood’s kinematic viscosity, 3.5 mm²/s. Reflection coefficients are of pressure, magnitude only. The wave speed is 5 m/s at every radius, blood’s density 1050 kg/m³, and segments are thirty radii long.

Who found it, and when #

Murray derived his law in 1926. Womersley gave the oscillating flow in an elastic tube in 1955 and the Womersley number its name. The impedance-matching argument for arterial junctions goes back to the wave-reflection studies of the 1960s and 70s, and Zamir’s measurements and analysis of arterial branching from the 1970s onwards documented the exponents of real trees and their asymmetry; the structured-tree models of Olufsen and others built lopsided trees of Womersley tubes for simulation from the late 1990s.

Still open: the tree the heart would build #

Every tree here is built to one exponent from top to bottom. The map says a tree that passes the pulse should change its exponent with its Womersley number, area-preserving at the top and Murray’s at the bottom. The next calculation builds that tree — each junction sized to its own transparent rule at the heart rate — and asks how much less it reflects than the best single-exponent tree, how much it costs in blood volume and pumping work against Murray’s steady optimum, and whether its reflection at the higher harmonics is better or worse, which is the question of whether a heart pumping into it would do less work.