Biomedical Engineering Reference
In-Depth Information
where X is the angular frequency of the wave, B (XS/N)
1/2
r
is a dimensionless
parameter known as the Womersley number, and
c
0
(
Eh
/2S
r
)
1/2
is the classic
Korteweg-Moens velocity (33,34). The wave velocity,
c
, and therefore,
Z
, are
both complex functions of X, so the wave is attenuated and dispersed as it
propagates. The character of the wave depends critically on whether |B| is less
than or greater than 1. This can be seen explicitly in Eq. [4], where the behavior
of the Bessel functions changes from a power-series expansion for small |B| to
an expansion with oscillatory behavior when |B| is large. In humans, B has a
value of around 15 in the aorta, 5 in the arteries, 0.04 in the arterioles, and 0.005
in the capillaries. When B is large (>1), Eq. [4] gives
c
~
c
0
, which is a purely
real quantity, so the wave suffers neither attenuation nor dispersion, demonstrat-
ing that viscosity plays almost no role in these large vessels. In this region (large
vessels), Eq. [4] also gives
Z
~ S
c
0
/Q
r
2
, and the minimization of energy loss is
attained through impedance matching, which eliminates the reflection of pulse
waves at junctions, leading to area-preserving for the vessels, Q
r
k
2
= nQr
2
k
+1
, so
that C =
n
-1/2
.
For small vessels where |B| < 1, the role of viscosity becomes increasingly
important until it eventually dominates the flow. Eq. [4] gives
c
~ (1/4)
i
1/2
B
c
0
l
0, in quantitative agreement with observation (33,34). Because
c
now has a sig-
nificant imaginary part, the traveling wave is heavily damped, leaving an almost
steady oscillatory flow whose impedance is, from Eq. [4], given by the classic
Poiseuille formula,
Z
k
= 8N
l
k
/Q
r
k
4
. Unlike energy loss due to possible reflections
at branch points, energy loss due to viscous dissipative forces cannot be entirely
eliminated. However, it can be minimized using the classic method of Lagrange
multipliers to enforce the appropriate constraints (12,36,37). To sustain a given
metabolic rate in an organism of fixed mass with a given volume of blood,
V
b
,
the cardiac output must be minimized subject to a space-filling geometry. This
leads to C =
n
-1/3
. Thus, for small vessels area-preserving branching is replaced
by area-increasing branching, and blood slows down, allowing efficient diffu-
sion of oxygen from the capillaries to the cells. Branching, therefore, changes
continuously through the network, so that C is
not
independent of
k
but changes
continuously from
n
-1/2
at the aorta to
n
-1/3
at the capillaries. Consequently, the
network is not strictly self-similar, but within the two different regions (pulsatile
and Poiseuille) self-similarity is a reasonable approximation that is well sup-
ported by empirical data (33,34,38-41).
To derive allometric relations we need to connect the scaling of vessel size
within an organism to its body mass,
M
. A natural vehicle for this is the total
volume of blood in the network,
V
b
, which can be shown to depend linearly on
M
if cardiac output is minimized (12,37).
V
b
is given by
¯
N
N
(
n
HC
2
)
Nk
+
_1
1
(
n
HC
2
)
k
1
¡
°
,
[5]
V
=
N V
=
n
k
Q
r l
2
=
n V
N
<
+
<
¡
°
b
k
k
k
N
(
n
HC
21
)
1
(
n
HC
21
)
1
¡
°
¢
±
k
=
0
k
=
0
<
<
