Page 97 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

74 BIOMECHANICS OF THE HUMAN BODY

For a uniform pressure gradient (ΔP) along a tube, we get the parabolic Poiseuille solution
Δ P
R − )
ur() =− ( 0 2 r 2 (3.8)
4μ l

2
The maximal velocity u = (R ) ΔP/4ml is obtained at r = 0.
max 0
The Poiseuille equation indicates that the pressure gradient ΔP required to produce a volumetric
flow Q = uA increases in proportion to Q. Accordingly, the vascular resistance R will be
defined as
Δ P
R = (3.9)
Q

2
5
3
If the flow is measured in cm /s and P in dyn/cm , the units of R are dyn . s/cm . If pressure is
3
measured in mmHg and flow in cm /s, resistance is expressed in “peripheral resistance units,” or
PRU.
The arteries are composed of elastin and collagen fibers and smooth muscles in a complex
circumferential organization with a variable helix. Accordingly, the arteries are compliant vessels,
and their wall stiffness increases with deformation, as in all other connective tissues. Because of their
ability to expand as transmural pressure increases, blood vessels may function to store blood volume
under pressure. In this sense, they function as capacitance elements, similar to storage tanks. The
linear relationship between the volume V and the pressure defines the capacitance of the storage
element, or the vascular capacitance:
dV
C = (3.10)
dP
Note that the capacitance (or compliance) decreases with increasing pressure, and also decreases
with age. Veins have a much larger capacitance than arteries and, in fact, are often referred to as
capacitance or storage vessels.
Another simple and useful expression is the arterial compliance per unit length, C , that can be
u
derived when the tube cross-sectional area A is related to the internal pressure A = A(P, z). For a
thin-wall elastic tube (with internal radius R and wall thickness h), which is made of a hookean
0
material (with Young modulus E), one can obtain the following useful relation,

dC 2π R 3
C ≡ ≈ 0 (3.11)
u
dz hE

3.3.3 Wave Propagation in Arteries
Arterial pulse propagation varies along the circulatory system as a result of the complex geometry
and nonuniform structure of the arteries. In order to learn the basic facts of arterial pulse character-
istics, we assumed an idealized case of an infinitely long circular elastic tube that contains a homoge-
nous, incompressible, and nonviscous fluid (Fig. 3.4). In order to analyze the velocity of propagation
of the arterial pulse, we assume a local perturbation, for example, in the tube cross-sectional area,
that propagates along the tube at a constant velocity c.
The one-dimensional equations for conservation of mass and momentum for this idealized case
are, respectively (Pedley, 1980; Fung, 1984),