Murray's law is not the rule for a pulse

The pulse that grows as it leaves the heart carries a pressure pulse along one elastic tube to one reflecting end, harmonic by harmonic, and finds that a single number — the reflection coefficient at the end — decides whether the pressure pulse grows towards the wrist or the flow pulse does. The arterial system is not one tube. The aorta divides, and its branches divide again, a dozen times or more before the vessels are small enough for the flow in them to have stopped being a wave. Each division is a junction at which a wave arriving along the parent meets two daughters of different size, and some of it is reflected.

The radius that costs least derives the most famous rule for how those divisions are sized. Minimising the power to push a steady flow plus the cost of the blood and tissue that fill the vessel gives a flow proportional to the cube of the radius, and at a junction Murray’s law: the parent’s radius cubed equals the sum of the daughters’. It is a rule for steady flow. This essay asks what the same junctions do to the pulse, which rule would make them transparent to it, and what a whole tree does that a single junction does not.

A junction, and the area ratio that hides it #

A wave in an elastic tube is carried as a pressure $p$ and a flow $q$ related by the tube’s characteristic impedance, $p = Z_0 q$ for a wave travelling forward. In the inviscid limit, the one the earlier essay’s large Womersley numbers approach, $Z_0 = \rho c/A$, with $c$ the wave speed and $A$ the area: a stiffer or narrower tube has a higher impedance. At a junction the pressure is the same in all three vessels and the flow into the parent equals the flow out into the daughters, since mass has nowhere else to go. Those two conditions fix the reflection at the junction:

$$\Gamma = \frac{Y_p - \sum Y_d}{Y_p + \sum Y_d}, \qquad Y = \frac{1}{Z_0} = \frac{A}{\rho c}.$$

The junction reflects nothing when the daughters’ admittances add up to the parent’s. Whether they do depends on the areas and on the wave speeds, and the wave speed depends on the radius through the wall: the Moens–Korteweg speed is $\sqrt{Eh/2\rho r}$, which is the same in every vessel if the wall’s thickness is a fixed fraction of its radius and its stiffness does not change, and rises in smaller vessels if they are relatively stiffer, as peripheral arteries are. Writing $c \propto r^{-m}$, the admittance goes as $r^{2+m}$, and a symmetric junction is transparent when

$$r_p^{\,2+m} = 2\,r_d^{\,2+m}, \qquad \frac{2A_d}{A_p} = 2^{m/(2+m)}.$$

The first figure draws the reflection for three wave-speed laws. With a constant wave speed the transparent junction preserves area: the daughters together have exactly the parent’s cross-section. With a wave speed rising as the inverse square root of the radius it widens by 15 per cent; with one rising as the inverse of the radius, by 26 per cent — which is Murray’s ratio, $2^{1/3}$. Murray’s law is the transparent rule only if the wave speed doubles every time the radius halves. With a constant wave speed a Murray junction reflects −11.5 per cent of the incident pressure. The sign says it behaves like a partly open end: the daughters together are too wide, and the wave arriving from the parent finds more room than it expected.

Eleven and a half per cent of the pressure is 1.3 per cent of the power, because power goes as the square. A single mismatched junction costs the heart almost nothing. The question is what many of them do together.

A tree is not the sum of its junctions #

A symmetric tree six generations deep, built to Murray’s law from a root 12 millimetres in radius, has 64 terminal branches 3 millimetres in radius. Each segment is a length of the earlier essay’s elastic-tube line, thirty radii long, carrying Womersley’s viscous correction to the wave speed and impedance, and the root’s input impedance is found from the leaves inwards: each segment transforms its load along its length, and each junction’s load is its two daughters in parallel.

The second figure gives the whole tree’s reflection at its root with ends that reflect nothing, so that everything drawn is the branching’s own doing. Without viscosity, the Murray tree reflects −0.60 at low frequency — five times what one of its junctions does. The reason is plain once seen. A wave much longer than the tree does not see the junctions one at a time; it sees the tree’s total cross-section at its ends, and a Murray tree’s cross-section grows at every generation. After six generations the 64 branches together have four times the root’s area. The tree, to a slow wave, is a pipe that opens out fourfold, and it reflects as one.

At the heart rate, 1.2 hertz, the wavelength is four metres, and the path from the root to a leaf, seven segments, is 1.4 metres long — a third of a wavelength. The junction reflections no longer arrive all at once, and the tree’s reflection falls to 0.41 without viscosity and 0.30 with it. That is still nearly three times one junction. Only above about two hertz, where the wavelength is comparable to the depth of the tree, do the junctions’ reflections arrive spread out enough to partly cancel, and the curve drops to a ripple of 0.05 to 0.2.

Viscosity mismatches the small vessels #

The area-preserving tree in the same figure is transparent without viscosity — every junction matched, the ends matched, a reflection of exactly zero at every frequency, which is one of the checks on the calculation. With viscosity it is not. It reflects 0.69 at 0.05 hertz, 0.46 at 0.3 and 0.18 at the heart rate, and it is transparent only above a few hertz.

The third figure shows why. The inviscid impedance $\rho c/A$ assumes the flow moves as a plug, which it does in a tube whose Womersley number is large. In a tube whose Womersley number is near one the oscillating flow has time to develop most of a parabolic profile, and the tube resists it as Poiseuille’s law would. In the aorta at the heart rate it is 18, and the impedance is 4 per cent above the inviscid value. In a vessel 3 millimetres in radius it is 4.4 and the impedance is 17 per cent higher; at 1.5 millimetres it is 2.2 and the impedance is 46 per cent higher, because viscous drag now reaches most of the cross-section and the flow resists being oscillated. The wave in those vessels is slower and more heavily damped, and its impedance rises in a way no area can make up for, because the rise depends on frequency: an area chosen to match at one harmonic is mismatched at the next. The area-preserving tree’s smallest branches are therefore relatively too narrow, their impedance too high, and the tree reflects positively at low frequency — like a partly closed end.

So the two trees are wrong in opposite directions. Murray’s is too wide for a wave and reflects negatively; the area-preserving tree is too narrow once viscosity is counted and reflects positively. Between them there must be a branching exponent that does best.

The exponent a pulsing tree wants #

The fourth figure scans the exponent $k$ in $r_p^{\,k} = 2r_d^{\,k}$ from 1.8 to 3.4 and scores each tree by its reflection averaged over the ten harmonics of the earlier essay’s pressure pulse, each weighted by its share of the pulse’s power. With a constant wave speed the best exponent is 2.15, with a weighted reflection of 0.125 — just above the inviscid match of two, pushed up by the viscous rise in the small branches’ impedance, which a slightly faster narrowing than area-preserving compensates. With a wave speed rising as the inverse square root of the radius the inviscid match is 2.5 and the best exponent 2.60, with a weighted reflection of 0.076. Murray’s exponent of three costs a weighted reflection of 0.25 in the first case and 0.17 in the second.

The surprise is that this is a known answer to a different question. In 1997 Geoffrey West, James Brown and Brian Enquist built a theory of why an animal’s metabolic rate scales as the three-quarter power of its mass on exactly this division: in the large vessels, where the flow is a pulsing wave, the network branches to preserve area so that it does not reflect; in the small ones, where viscosity dominates and the flow is steady, it branches by Murray’s cubic law to minimise the cost of pushing blood. The calculation here arrives at the same split from the other end, as one number for a tree whose small branches are already viscous enough to shift it: the best exponent for the pulse sits just above area-preserving, and moves towards three as viscosity matters more. Measured arterial junctions have exponents scattered between about two and three, which a network compromising between the two costs would be expected to show.

What the reflection costs the heart #

A reflection coefficient of 0.3 sounds large, and 1.3 per cent of the power at one junction sounds small, and neither says what the heart pays. The heart is closer to a source of flow than of pressure: it ejects a stroke volume, and the pressure it has to develop to do so is the flow times the input impedance it pumps into. The power it spends on the oscillating part of the flow is the real part of that impedance, harmonic by harmonic, times the square of each harmonic of the flow.

Averaged over the pulse’s harmonics with their own weights, the best-exponent tree with non-reflecting ends has an input resistance equal to its root’s characteristic impedance to within a tenth of a per cent — it is, to the heart, an endless tube. The Murray tree’s is 1.46 times as large. At the fundamental alone it is 1.68 times: by 1.2 hertz the reflected wave from the junctions has turned in phase until it arrives back at the root pushing the pressure up while the flow is still going out, which is exactly the loading a heart would least want. With the ends reflecting half the pressure, as real ends do, both trees load the heart less at the fundamental and the ratio between them survives: the Murray tree needs 1.45 times the oscillatory power of the best-exponent tree to deliver the same flow pulse.

That is a real cost, but it is a cost on the smaller part of the bill. Most of the heart’s work goes into the steady flow — the mean pressure times the mean flow — and that part is exactly what Murray’s law minimises, with the pumping cost a fixed third of the total at the optimum. The oscillating part is usually estimated at a tenth or so of the whole. So a tree optimised for the steady flow alone would pay for it in pulsatile work, a tree optimised for the pulse would pay in steady pumping and in blood volume, and a tree built for both would sit between the two exponents — which is where arterial trees are found.

Where the tree stops carrying a wave #

The whole argument has a lower edge. A vessel carries the pulse as a wave only while its Womersley number is well above one; below that the oscillating flow is quasi-steady, viscosity sets its profile everywhere across the tube, and the flow is governed by Poiseuille’s law at each instant rather than by a wave equation. At the heart rate the Womersley number falls to one at a radius of 0.68 millimetres. In a tree built to Murray’s law from a 12-millimetre root, that is about twelve generations down; in an area-preserving tree about eight, because the radius falls faster. For the fifth harmonic the edge is at 0.30 millimetres, sixteen Murray generations down.

Below that edge the reasons for preferring one branching rule over another change completely. There is no wave to reflect, so there is nothing to match, and the only cost left is the steady one, which Murray’s law is built to minimise. Above it the wave dominates and matching the impedance is what saves the heart work. A tree that is good at both would change its exponent somewhere near the generation where the Womersley number passes through one — from close to two above it to close to three below — which is precisely the switch West, Brown and Enquist built into their theory, arrived at here by asking where the wave ends.

What the ends do #

The ends of a real arterial tree are not transparent. The small arteries and arterioles present a high resistance, and a wave arriving at them is reflected strongly and positively — the single-tube essay’s reflection coefficient of 0.5 or more, which made the pressure pulse grow towards the periphery.

The fifth figure puts the ends back, reflecting half the pressure, and the picture changes. Near the heart rate the two trees return nearly the same reflection at their root, 0.49 for Murray’s and 0.45 for the best exponent: the ends dominate, and the branching rule shifts the answer by a few per cent. At higher harmonics the ends’ reflection is worn down on its way back through the narrowing viscous branches, to 0.3 at 2.4 hertz and 0.24 to 0.3 at 6, and what remains of the difference between the trees is the ripple the Murray tree’s junctions add on top. So in a tree with reflecting ends — which is every real one — the branching rule decides the fine structure of the reflected wave rather than its size, and the size is set where the single-tube essay found it: at the ends.

What was checked #

The sixth figure is the ledger, and its last row is the heart’s bill from the section above. The matched area ratio reflects nothing for each of the three wave-speed laws, and at every junction tested the transmitted and reflected power add up to the incident, $1 - \Gamma^2$, to rounding. The whole-tree recursion was checked two ways that do not use it: an inviscid tree matched at every junction, with ends reflecting 0.5, returns a reflection of exactly 0.5 at its root at every frequency, as it must when nothing in between reflects; and a tree of one junction gives the reflection computed directly from the daughters’ impedance in parallel.

What the picture cannot show #

Linear waves, and no transients. A water main’s junctions reflect a surge by the same admittance rule, and a closing valve’s wave is the large-amplitude case; the pulse here is small and periodic.

A symmetric tree. Real junctions are asymmetric — a large trunk giving off a small side branch — and the matching condition is then a condition on both daughters at once. The aorta also tapers between branches, which reflects continuously along its length.

Linear, one-dimensional waves in straight tubes. The junction is a point; its angle, and the angle a junction chooses, do not enter. The walls are elastic, not viscoelastic, and there is no nonlinearity.

A stated wave-speed law. Two laws are computed, and a real arterial tree’s wave speed roughly doubles from the aorta to the peripheral arteries, which is between them over part of the tree.

Ends with one reflection coefficient. Real ends are resistances with compliance, and their reflection depends on frequency.

The convention the numbers depend on #

The reflection coefficient is for pressure, forward to backward, so a negative value is an open-end-like reflection and a positive one a closed-end-like one. The branching exponent $k$ is defined by $r_p^{\,k} = 2r_d^{\,k}$ at a symmetric junction; $k = 2$ is area-preserving and $k = 3$ is Murray’s law. The root is 12 millimetres in radius with a wave speed of 5 metres per second, segments are thirty radii long, the tree is six generations deep, and blood is a Newtonian fluid of density 1050 kilograms per cubic metre and kinematic viscosity $3.5\times10^{-6}$ square metres per second. The pulse-weighted reflection weights each harmonic by the square of its amplitude in the stated pressure pulse.

Who worked it out #

Cecil Murray derived his law in 1926. John Womersley’s theory of the pulse in an elastic tube dates from 1955 to 1957, and the treatment of an arterial junction as a transmission-line junction, with reflections set by the admittances, was developed by Donald McDonald, Michael Taylor and others through the 1950s and 1960s. West, Brown and Enquist’s allometric theory, with its area-preserving large vessels and cubic small ones, was published in 1997.

Still open: an asymmetric tree #

The next calculation drops the symmetry. At an asymmetric junction the parent’s reflection depends on both daughters’ admittances and on where along each the next junction lies, and the rule that makes it transparent becomes a relation among three radii rather than an exponent. Building a tree whose side branches are a stated fraction of the trunk at each junction, as the aorta’s are, and scanning that fraction and the exponent together, would say whether the best pulsing tree is still close to area-preserving along its trunk, and whether its side branches — which take a small share of the flow each time — are where Murray’s cubic rule takes over.