The pulse that grows as it leaves the heart

Measure blood pressure at the upper arm, then again at the ankle, and the ankle’s systolic pressure is usually the higher of the two. Measure it by catheter in the ascending aorta and the arm, and the arm’s pulse — the difference between systolic and diastolic — is typically larger than the aorta’s by a fifth or more in a young adult. Blood flows from the aorta to the arm and the ankle, it loses energy doing so, and the mean pressure does fall. The pulse grows anyway.

The pulse that has to travel set up the calculation that explains this and deliberately left it unused. It put a pressure wave on an elastic tube, found its speed and its attenuation from the Womersley number, and measured the tube’s length in the wave’s own wavelengths. It had no ending on the tube, and so no reflection. Every real vessel ends.

A line with an end #

A tube of length $L$ carries each harmonic of the pulse as a wave travelling away from the heart. At the far end it meets a load — a branching, a narrowing, a bed of smaller vessels — whose impedance is not the tube’s own. Part of the wave is reflected, and the ratio of the reflected pressure wave to the arriving one is the reflection coefficient $\Gamma$. A load that resists flow more than the tube does reflects pressure with the same sign, $\Gamma > 0$; a load that resists it less reflects pressure inverted, $\Gamma < 0$.

With the entrance pressure given, the pressure and the flow at any station are the sum of the outgoing and returning waves:

$$P(x) = P_\text{in}\,\frac{e^{-i\kappa x} + \Gamma\,e^{-i\kappa(2L - x)}}{1 + \Gamma\,e^{-2i\kappa L}}, \qquad Q(x) = \frac{P_\text{in}}{Z_0}\,\frac{e^{-i\kappa x} - \Gamma\,e^{-i\kappa(2L - x)}}{1 + \Gamma\,e^{-2i\kappa L}}.$$

Everything about the wave is in $\kappa$ and $Z_0$, both from the Womersley number of each harmonic. The one thing to notice before anything is computed is the sign in front of $\Gamma$. The returning wave adds to the pressure and subtracts from the flow. A pressure wave travelling backwards carries flow backwards, so wherever a reflection raises the pressure it lowers the flow, and the other way round.

The pulse is ten harmonics of a 1.2 Hz waveform with a sharp systolic rise and a notch — a shape chosen, not measured — carried on a 50 cm tube of radius 1 cm and Moens–Korteweg speed 5 m/s, the aorta of the essay before.

The far end #

With a reflection coefficient of 0.6, the pressure pulse at the far end is 1.88 times the pulse at the entrance, and the flow pulse is 0.26 of it. The far-end pressure waveform is taller, narrower and later; the far-end flow waveform is flattened almost to a ripple. None of this needed a pump downstream. Less energy arrives than left, and it arrives as more pressure and less flow.

One number decides which grows #

Sweeping the reflection coefficient from −0.9 to 0.9 shows the whole behaviour in two curves. They cross near zero, where both pulses fall to about 95 per cent of their entrance value — the wave’s own viscous loss over half a metre. On the positive side the pressure pulse rises and the flow pulse falls: 1.25 and 0.56 at Γ = 0.3, 1.88 and 0.26 at 0.6, 4.1 and 0.06 at 0.9. On the negative side the roles swap: at Γ = −0.5 the pressure pulse is 0.45 of the entrance’s and the flow pulse 1.29.

The reflection coefficient’s sign decides which pulse grows, and its size decides by how much. That is the sharp statement the elastic tube makes, and it has an immediate physiological reading. The arterial tree towards the limbs narrows and stiffens, so its reflections are positive, so pressure pulses grow and flow pulses shrink towards the hands and feet. A vessel that branches into many daughters whose combined area is larger than its own reflects negatively, and there the flow pulse grows. The same tube, driven the same way, amplifies whichever quantity its ending favours.

Where along the tube the growth happens #

The growth is not uniform. For every positive reflection coefficient the pressure pulse stays close to its entrance value over the first fifth of the tube and rises most steeply in the middle, levelling off towards the end. At Γ = 0.6 it is 1.04 times the entrance pulse ten centimetres in, 1.56 at the midpoint and 1.88 at the end.

The reason is phase. At the reflecting end the incident and reflected waves meet with no path difference between them, and the pressure there is $1 + \Gamma$ times the incident wave. Moving back towards the entrance, the reflected wave has further to come and has fallen behind the incident one by twice the distance, so the two begin to interfere destructively. For the pulse’s higher harmonics, whose wavelengths are comparable with the tube, that phase slip is large within a few tens of centimetres; for the fundamental, an eighth of a wavelength long, it is small. So the fast harmonics are amplified only near the end and the slow ones everywhere, and the envelope of the whole pulse rises most where the fast harmonics start to add.

The pulse grows and the power does not #

A pressure pulse that nearly doubles along a passive tube invites the suspicion that something is being created, and the energy account settles it.

The oscillatory power a harmonic carries past a station is half the real part of its pressure times the conjugate of its flow. Summed over the ten harmonics, the power reaching the far end of the tube is 91 per cent of the power entering it with no reflection, 84 per cent with a reflection of 0.6, and 83 per cent with a reflection of −0.5. In every case less arrives than left, as a passive tube with a viscous fluid in it requires.

What reflection changes is not the power but its split. Power is pressure times flow, and a positively reflecting end holds the flow down and lets the pressure swing, so the same power arrives as a large pressure oscillation driving a small flow oscillation. A negatively reflecting end does the opposite. A pressure gauge at the far end therefore reads a larger pulse, and a flow meter a smaller one, and neither is wrong: they are reading the two factors of a product that the reflection has re-apportioned.

The same re-apportioning is visible in a water-hammer line with a surge tank, where an open shaft is a negatively reflecting end: the tunnel’s pressure wave arriving at it is turned back inverted, the pressure swing at the shaft is small, and the flow into and out of the shaft is large. An artery ending in a narrowing is the same line with the opposite ending, and it trades the other way.

A quarter of a wavelength, and the resonance it makes #

Harmonic by harmonic, the amplification is a resonance. For a lossless tube with the entrance pressure held, the ratio of the far-end pressure to the entrance pressure is

$$\frac{P(L)}{P(0)} = \frac{1 + \Gamma}{e^{i\kappa L} + \Gamma\,e^{-i\kappa L}},$$

which at a quarter of a wavelength, $\kappa L = \pi/2$, has magnitude $(1 + \Gamma)/(1 - \Gamma)$ — four, at Γ = 0.6. The tube is a quarter-wave resonator for that harmonic: the reflected wave arrives back at the entrance exactly out of phase, the entrance is held at its imposed pressure by the source, and the far end swings as far as the reflection lets it. At half a wavelength the ratio is exactly one, whatever the reflection.

For the pulse at 1.2 Hz in a 50 cm aorta, the second harmonic sits at a quarter of a wavelength and is amplified 3.27 times — less than the lossless four, because at α = 21 the wave loses a fifth of itself in each wavelength. The fourth and eighth harmonics sit near a half and a whole wavelength and pass unamplified. The sixth and tenth are near the next resonances. The far-end pulse is taller and sharper than the entrance pulse because the tube boosts its second harmonic three-fold and leaves its fourth alone, and a waveform with more second harmonic and the same fourth has a higher, narrower peak.

The resonances also make the amplification sensitive to heart rate, which moves every harmonic’s wavelength. A tube a quarter of a wavelength long for the second harmonic at 72 beats a minute is a quarter-wave long for the fundamental at 144. A resonance that shapes a resting pulse’s peak therefore moves onto the pulse’s largest component during hard exercise, and the peripheral amplification of the pulse pressure changes with heart rate for a reason that has nothing to do with the heart.

The heart is a source of flow #

Holding the entrance pressure fixed is the right condition for asking what the tube does to a pulse that passes through it, and the wrong one for asking what the heart feels. A ventricle ejects a volume; it behaves much more like a source of flow than of pressure, and the pressure at the aortic root is whatever the aorta’s input impedance makes of that flow,

$$Z_\text{in} = Z_0\,\frac{1 + \Gamma\,e^{-2i\kappa L}}{1 - \Gamma\,e^{-2i\kappa L}},$$

harmonic by harmonic. Without reflection the input impedance is the tube’s own $Z_0$. With reflection it is larger or smaller depending on when the reflected wave comes back.

Stiffening the tube — raising its wave speed from 4 to 12 metres a second, roughly the change in an aorta between youth and old age — raises the heart’s pressure pulse twice over. The characteristic impedance grows in proportion to the wave speed, which alone would triple the pulse. And the reflected wave comes back sooner: at 4 m/s it takes a quarter of a second to go to the end of a 50 cm tube and back, and returns during diastole, adding 2 per cent to the pulse. At 12 m/s it takes 83 milliseconds and returns while the heart is still ejecting, adding 90 per cent. Together the pressure pulse at 12 m/s is 3.8 times the pulse at 5 m/s, for the same stroke.

That is the mechanism behind the rise in central pulse pressure with age, and the reason the systolic pressure a heart works against can rise faster than stiffness alone explains. It also completes the contrast with the entrance-pressure view. Seen from the periphery, reflection makes a pulse grow on the way out; seen from the heart, the same reflection, arriving early, makes the pulse grow at its source.

Why a cuff on the arm is not a catheter in the aorta #

The two views together explain the clinical puzzle the essay began with. The pressure a cuff measures at the arm is the aortic pressure carried along a positively reflecting tube, and so is amplified. How much it is amplified depends on the tube’s length in wavelengths, the reflection at its end and the harmonics in the pulse, and all three change with age, heart rate and drugs that dilate or constrict the small vessels.

In a young adult the amplification is large and the arm overstates the central pulse. In an old one the stiffened aorta carries its reflections back to the heart early, the central pulse is itself inflated, and the arm’s amplification over it is smaller. A drug that lowers the arm’s pressure by a given amount can therefore lower the central pressure by more or by less, depending on what it does to the reflections, and the arm alone cannot say which. That is why methods that estimate the central waveform from a peripheral one exist, and why they are built on a transfer function between the two — which is exactly the ratio $P(L)/P(0)$ above, inverted.

The harmonic picture also says what such a transfer function cannot be. It is not a single gain, because the amplification is different for every harmonic — 1.35 for the fundamental here, 3.27 for the second, 0.98 for the fourth — so it reshapes the waveform as well as scaling it. And it is not fixed, because every one of those gains depends on where the harmonics fall against the tube’s resonances, which moves with heart rate and with wave speed. A transfer function measured on one person at rest is an average over a family of lines, and the calculation above is the reason it works as well as it does and the reason it cannot work perfectly.

A ledger of the reflecting line #

The power rows are the check that the amplification is not an artefact: the pulse grows at the far end in every positively reflecting case while the power reaching it falls. The line is the one the previous essay checked against a characteristics march, which shares no code with the harmonic sums and agrees with them on the far-end pulse to half a per cent even at a reflection of 0.9. Two further checks are built into the numbers: with no reflection the far-end pressure over the entrance pressure is exactly the attenuation over the tube’s length, 0.961 for the fundamental, and the resonance at a quarter wavelength comes out below the lossless $(1 + \Gamma)/(1 - \Gamma)$ by the amount the wave’s own attenuation predicts.

What the single tube cannot show #

A tree. The arterial system is not one tube with one ending but a branching network whose reflections come from every junction, at every distance, with every sign. The effective reflection seen from the aorta is a smeared average of all of them, and its timing is not one round trip but a distribution of them.

A frequency-dependent load. The reflection coefficient here is a real number, the same at every harmonic. A vascular bed’s impedance falls with frequency, because its compliance matters more at high frequencies, so the real reflection is strongest for the fundamental and weaker for the harmonics — which moves the resonances and reduces the amplification of the sharp features.

Taper. A tapering tube reflects continuously along its length rather than at its end, and amplifies the pressure pulse even with a matched load.

Nonlinearity and wall viscoelasticity. Both, set aside in the essay before, matter more here: a pulse amplified to twice its size stiffens the wall more, and the viscoelastic wall damps precisely the high harmonics that give the far-end pulse its shape.

A measured waveform. The ten harmonics are chosen to give a recognisable shape, not taken from a measurement, so the numbers are illustrations of the mechanism at a realistic scale rather than predictions for a patient.

Who worked it out #

That the pulse pressure increases towards the periphery was measured in the nineteenth century, and Otto Frank understood by the 1900s that the Windkessel could not explain it and that waves and reflections could. Donald McDonald and John Womersley’s work in the 1950s gave the calculation used here, and McDonald’s book of 1960 set out the input impedance of the arterial tree as the object of study. The role of early wave reflection in raising central pressure with age was developed through the 1980s and 1990s, notably by Michael O’Rourke and colleagues, whose work led to the clinical measurement of central pressure from peripheral waveforms through a transfer function.

Still open: the tree instead of the tube #

The calculation above has one tube and one ending. The arterial system has a trunk that divides and divides again, with the reflection at each junction set by how the daughters’ impedances compare with the parent’s, and whether a junction is matched — reflecting nothing — depends on the areas and wave speeds of three vessels at once.

The next calculation builds a small symmetric tree from the same elastic-tube line, with the junction condition that pressure is continuous and flow is conserved, and asks two things of it: what effective reflection coefficient the whole tree presents at its root as a function of frequency, and whether there is a rule for the branching areas — the analogue of Murray’s law for steady flow — that makes every junction transparent to the pulse.