Page 99 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

76 BIOMECHANICS OF THE HUMAN BODY
for which the wave speed c is given by
∂
AP
2
c = (3.17)
ρ ∂ A
This suggests that blood pressure disturbances propagate in a wavelike manner from the heart toward
the periphery of the circulation with a wave speed c. For a thin-wall elastic tube (with internal radius
R and wall thickness h), which is made of a hookean material (with Young modulus E) and sub-
0
jected to a small increase of internal pressure, the wave speed c can be expressed as
Eh
2
c = (3.18)
2ρ R
0
This equation was obtained by Thomas Young in 1808, and is known as the Moens-Kortweg wave
speed. The Moens-Kortweg wave speed varies not only with axial distance but also with pressure.
The dominant pressure-dependent term in Eq. (3.18) is E, the modulus of elasticity; it increases with
increasing transmural pressure as the stiff collagen fibers bear more of the tension of the artery wall.
The high-pressure portion of the pressure wave therefore travels at a higher velocity than
the low-pressure portions of the wave, leading to a steepening of the pressure front as it travels
from the heart toward the peripheral circulation (Fig. 3.5). Wave speed also varies with age because
of the decrease in the elasticity of arteries.
The arteries are not infinitely long, and it is possible for the wave to reflect from the distal end
and travel back up the artery to add to new waves emanating from the heart. The sum of all such
propagated and reflected waves yields the pressure at each point along the arterial tree. Branching is
clearly an important contributor to the measured pressures in the major arteries; there is a partial
reflection each time the total cross section of the vessel changes abruptly.

FIGURE 3.5 Steepening of a pressure pulse with distance along an artery.

3.3.4 Pulsatile Flow
Blood flow in the large arteries is driven by the heart, and accordingly it is a pulsating flow. The sim-
plest model for pulsatile flow was developed by Womersley (1955a) for a fully developed oscillatory
flow of an incompressible fluid in a rigid, straight circular cylinder. The problem is defined for a
sinusoidal pressure gradient composed from sinuses and cosinuses,
ΔP it
ω
= Ke (3.19)
l
where the oscillatory frequency is w /2p. Insertion of Eq. (3.19) into Eq. (3.5) yields
2
∂ u 1 ∂u 1 ∂u K it
ω
+ − =− e (3.20)
∂r 2 r ∂r v ∂t μ
The solution is obtained by separation of variables as follows:
ω
⋅
ur t) = W r e it (3.21)
(,
()