Biomedical Engineering Reference
In-Depth Information
detailed simulations in flow analysis studies. Such a detailed analysis, however, in-
volves complex numerical computations and the effort may be justified only by the
need for aerosol deposition models, etc. This is obviously out of the scope of this
topic.
From the zoo of literature reports on pulmonary function, one may distinguish
two mainstreams:
•
a symmetrical structure of the lung [
163
,
164
] and
•
an asymmetrical [
54
,
65
] representation of the airways in the respiratory tree.
In this topic, a symmetric flow bifurcation pattern is assumed in order to derive the
pressure-flow relationship in the airways. However, both symmetric and asymmetric
airway networks will be discussed in the next chapter, by means of their electrical
analogues.
Womersley theory has been previously applied to circulatory system analy-
sis, considering the pulsatile flow in a circular pipeline for sinusoidally varying
pressure-gradients [
168
]. Taking into account that the breathing is periodic with
a certain period (usually, for normal breathing conditions, around 4 seconds), we
address the airway dynamics problem making use of this theory. Usually, when si-
nusoidal excitations are applied to the respiratory system [
69
,
116
], they contain
ten times higher frequencies than the breathing, which permits analyzing oscillatory
flow. To find an electrical equivalent of the respiratory duct, one needs expressions
relating pressure and flow with properties of the elastic tubes, which can be done
straightforward via Womersley theory [
3
,
115
,
139
].
The periodic breathing can be analyzed in terms of periodical functions, such
as the pressure gradient:
∂p
∂z
=
−
M
P
cos
(ωt
−
Φ
P
)
, where
z
is the axial coordi-
nate,
ω
2
πf
is the angular frequency (rad/s), with
f
the frequency (Hz),
M
P
the modulus and
Φ
P
is the phase angle of the pressure gradient. Given its period-
icity, it follows that also the pressure and the velocity components will be periodic,
with the same angular frequency
ω
. The purpose is to determine the velocity in
radial direction
u(r, z, t)
with
r
the radial coordinate, the velocity in the axial direc-
tion
w(r,z,t)
, the pressure
p(r,z,t)
and to calculate them using the morphological
values of the lungs. In this study, we shall make use of the Womersley parameter
from the Womersley theory developed for the circulatory system, with appropri-
ate model param
ete
rs for the respiratory system, defined as the dimensionless pa-
rameter
δ
=
R
ωρ
μ
[
139
,
168
], with
R
the airway radius. The air in the airways is
treated as Newtonian, with constant viscosity
μ
=
=
10
−
5
1
.
8
×
kg/m s and density
1
.
075 kg/m
3
, and the derivation from the Navier-Stokes equations is done in
cylinder coordinates [
165
]:
ρ
∂u
ρ
=
v
2
r
u
∂u
v
r
∂u
∂θ
+
w
∂u
∂t
+
∂r
+
∂z
−
μ
1
r
r
∂u
∂r
∂
2
u
∂θ
2
−
∂
2
u
∂z
2
∂p
∂r
+
∂
∂r
u
r
2
+
1
r
2
2
r
2
∂v
∂θ
+
=−
ρF
r
+
−
(4.1)
