Biomedical Engineering Reference
In-Depth Information
Fig. 4.1
Schematic
representation of the
infinitesimal distance
dx
over
the transmission line and its
parameters
conditions, the Reynolds number can be calculated as
ρ
μ
N
RE
=
w
·
2
R
·
(4.38)
1075 (g/m
3
) the air density BTPS (Body Temperature and Pressure, Satu-
rated) and
μ
with
ρ
=
=
0
.
018 (g/m s) the air viscosity BTPS. We have verified the values for
the Reynolds number, which indeed indicated laminar flow conditions throughout
the respiratory tree, varying from 1757 in the trachea to 0.1 in the alveoli. Hence,
the assumption of laminar flow conditions during tidal breathing is correct.
4.2 Electrical Analogy
By analogy to electrical networks, one may consider voltage as the equivalent for
respiratory pressure
P
and current as the equivalent for airflow
Q
[
83
]. Electri-
cal resistances
R
e
may be used to represent respiratory resistance that occur as a
result of airflow friction in the airways. Similarly, electrical capacitors
C
e
may rep-
resent volume compliance of the airways which allows them to inflate/deflate. The
electrical inductors
L
e
may represent inertia of air and electrical conductances
G
e
may represent the viscous losses. These properties are often clinically referred to
as mechanical properties: resistance, compliance, inertance, and conductance. The
aim of this section is to derive them in function of airway morphology in case of
an elastic airway wall (
R
e
,L
e
,C
e
) and in the case of a viscoelastic airway wall
(
R
e
,L
e
,C
e
,G
e
).
Suppose the infinitesimal distance
dx
of a transmission line as depicted in
Fig.
4.1
. We have the distance-dependent parameters:
l
x
induction/m;
r
x
resis-
tance/m;
g
x
conductance/m;
c
x
capacity/m. We consider the analogy to voltage as
being the pressure
p(x,t)
and to current as being the airflow
q(x,t)
and we apply
the transmission line theory. We shall make use of the complex notation:
P(x)e
j(ωt
−
φ
P
)
q(x,t)
=
Q(x)e
j(ωt
−
φ
Q
)
=
p(x,t)
(4.39)
where
x
is the longitudinal coordinate (m),
t
is t
he
time (s),
ω
is the angular fre-
quency (rad/s),
f
is the frequency (Hz) and
j
=
√
−
1. The pressure and the flow are
