# How to understand the biophysics and math behind a 0.2 second delay between the pulse in my arm and my ankle?

I recently had a test with six blood pressure cuffs; 2 each on arms, ankles and big toes. During the test I could feel the pulses in my arms and legs, and noticed a distinct ~0.2 second delay between the pulse in my arms and ankles.

I used my pulse rate at the time and the fraction of the pulse period between arm and leg to get the 0.2 seconds, and I unscientifically give myself a +/- 20% error bar on that number.

I was at first quite surprised, because the speed of sound in water is roughly 1500 meters per second, and since water and presumably blood is mostly incompressible I couldn't figure out what could cause such a long delay.

Then I realized that arteries must be somewhat elastic, and that the propagation of the pulse must involve the mass/inertia of the blood and the restoring force of the artery, and so this was a biophysics problem of some kind.

It turns out that the results of the test include a report of the propagation velocity which was of the order of 1100 centimeters/second. I'm not sure how that connects with my estimate of 0.2 seconds as it implies a distance of 2.2 meters and there's no way that my elbow is 2.2 meters farther from my heart than my ankle (I'm ~1.7 meters tall).

Questions: How to understand the biophysics and math behind a 0.2 second delay between the pulse in my arm and my ankle?

1. Is there an equation that relates the delay between arm and ankle pulse to the elasticity of the arteries?
2. How is the velocity calculated from the measure delay in this case?
• Since @bob1 outpaced me with his answer, I will give just a short comment. He has correctly warned you that the $\text{PWV}$ is not equal to the sound propagation velocity. In the Moens-Korteweg equation, you can see that the $\text{PWV}$ depends strongly on the dimensions of the artery, its elastic properties, and also on the blood pressure (indirectly, by changing the elasticity $E$). From the physiological perspective, this means that the $\text{PWV}$ is not constant throughout the circulatory system. Furthermore, measuring $\text{PWV}$ has generally a large measurement error. Aug 16, 2021 at 10:01
• Finally, the measurements of $\text{PWV}$ are generally on the order of $\text{m/s}$ (with outliers up to approx. $25 \;\text{m/s}$). Your result ($11\; \text{m/s}$) seems to lie on the upper part of the distribution. Some graphs for the age and pressure dependence can be found here and here. Aug 16, 2021 at 10:05
• It's like the difference between voltage propagation down a wire and movement of the electrons themselves, if that helps Aug 16, 2021 at 13:14
• @Domen - I'm a virologist not a cardiology person at all, so I would love if you could provide a better/more complete answer. I just did some quick googling, so am sure to have missed some salient points.
– bob1
Aug 16, 2021 at 21:21

So it turns out that your estimate is not bad, by simple velocity calculation, I get it to be a little over 50% out (see below), but as you say, there is some error in your measurement. However, it turns out the pulse wave velocity is much more complicated than that, and subject to some debate in the literature.

The speed at which the impulse propagates is not the speed of sound, but rather the speed of the impulse from your heart, and varies according to aterial stiffness (thinner arteries are faster, larger slower as they are more floppy). The equations for calculation are similar to those used to calculate the velocity of sound in a medium. There are two of these, where $$P$$ is pressure and $$V$$ is volume and $$\rho$$ is density of the blood:

The Frank/Bramwell-Hill equation

$$PWV = \sqrt{\frac{V.dP}{\rho.dV}}$$ And the Moens-Korteweg equation ($$E_{inc}$$ = vessel wall elasticity, $$h$$ = wall thickness, and $$r$$ = radius

$$PWV = \sqrt{\frac{E_{inc}.h}{2.r.\rho}}$$

Now these are super complicated for the average person to measure and it seems a bit difficult to work with, so it can be simplified to a the classic $$velocity = distance/time$$.

If you look at the structure of the arteries in the body, all the major blood ones to distal portions of the body feed off (unsurprisingly) the same source, the aorta.

Now to estimate the difference in time you need to work out the time it would take to propagate to each point.

I'm not too far off you in size (~10 cm taller), and did some very rough estimates of how far my elbow is from my heart and about how far my heart is from my ankle, and I got ~40 cm for elbow and ~135 cm for ankle.

You gave a propagation speed of 1100 cm/s, so substituting that into the simple equation: $$\text{time} = \text{distance}/\text{velocity}$$ So 40 cm distance: $$\text{time} = 40/1100$$ $$\text{time} = 0.0364 \text{ seconds}$$ and 135 cm distance

$$\text{time} = 135/1100$$ $$\text{time} = 0.1227 \text{ seconds}$$

$$\text{Difference in time} = 0.1227-0.0364$$ $$\text{Difference} = 0.0863 \text{ seconds}$$