# The velocity of the arterial pulse wave: a viscous-fluid shock wave in an elastic tube

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Theoretical Biology and Medical Modelling

, 5:15

First Online: 29 July 2008Received: 11 April 2008Accepted: 29 July 2008DOI: 10.1186-1742-4682-5-15

Cite this article as: Painter, P.R. Theor Biol Med Model 2008 5: 15. doi:10.1186-1742-4682-5-15

Abstract

BackgroundThe arterial pulse is a viscous-fluid shock wave that is initiated by blood ejected from the heart. This wave travels away from the heart at a speed termed the pulse wave velocity PWV. The PWV increases during the course of a number of diseases, and this increase is often attributed to arterial stiffness. As the pulse wave approaches a point in an artery, the pressure rises as does the pressure gradient. This pressure gradient increases the rate of blood flow ahead of the wave. The rate of blood flow ahead of the wave decreases with distance because the pressure gradient also decreases with distance ahead of the wave. Consequently, the amount of blood per unit length in a segment of an artery increases ahead of the wave, and this increase stretches the wall of the artery. As a result, the tension in the wall increases, and this results in an increase in the pressure of blood in the artery.

MethodsAn expression for the PWV is derived from an equation describing the flow-pressure coupling FPC for a pulse wave in an incompressible, viscous fluid in an elastic tube. The initial increase in force of the fluid in the tube is described by an increasing exponential function of time. The relationship between force gradient and fluid flow is approximated by an expression known to hold for a rigid tube.

ResultsFor large arteries, the PWV derived by this method agrees with the Korteweg-Moens equation for the PWV in a non-viscous fluid. For small arteries, the PWV is approximately proportional to the Korteweg-Moens velocity divided by the radius of the artery. The PWV in small arteries is also predicted to increase when the specific rate of increase in pressure as a function of time decreases. This rate decreases with increasing myocardial ischemia, suggesting an explanation for the observation that an increase in the PWV is a predictor of future myocardial infarction. The derivation of the equation for the PWV that has been used for more than fifty years is analyzed and shown to yield predictions that do not appear to be correct.

ConclusionContrary to the theory used for more than fifty years to predict the PWV, it speeds up as arteries become smaller and smaller. Furthermore, an increase in the PWV in some cases may be due to decreasing force of myocardial contraction rather than arterial stiffness.

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Author: **Page R Painter**

Source: https://link.springer.com/