Biomedical Engineering Reference
In-Depth Information
The pressure gradient is related to the characteristics of the airway duct via the
Moens-Korteweg relation for the wave velocity
c
0
, with
´
Eh
(
2
ρR(
1
c
0
=
´
(4.30)
ν
P
))
−
The model for wave propagation in function of the pressure
p
(kPa) for axial
w
(m/s) and radial
u
(m/s) velocities, for flow
Q
(l/s) and for the wall deformation
ζ
(%) at the axial distance
z
=
0 is given by the set of equations:
A
P
e
j(ωt
−
φ
P
)
p(t)
=
(4.31)
cos
ωt
−
ε
1
−
φ
P
+
ε
2
(y)
+
RA
P
ω
2
ρ
M
2
(y)
M
1
π
2
u(y, t)
=
·
(4.32)
c
0
´
sin
ωt
−
R
2
A
P
ω
´
M
0
(y)
δ
2
ε
1
2
−
φ
P
+
ε
0
(y)
+
π
2
w(y,t)
=
c
0
μ
√
M
1
·
(4.33)
sin
ωt
πR
4
μ
A
P
ω
c
0
√
M
1
M
1
δ
2
ε
1
2
−
π
2
=
+
φ
P
+
Q(t)
(4.34)
A
P
ζ(t)
=
cos
(ωt
−
φ
P
)
(4.35)
hE
R
2
ρ
wall
hω
2
−
with
2
R
E
1
R
2
−
ρ
wall
hω
2
h
A
P
=
(4.36)
−
ν
2
Eh
(
2
ρR(
1
c
0
=
(4.37)
−
ν
P
))
One should note that the model given by (
4.31
)-(
4.35
) is a linear hydrodynamic
model, adapted from Womersley [
168
]. This model has been used as basis for fur-
ther developments by numerous authors [
115
,
139
]. The assumption that air is in-
compressible and Newtonian has been previously justified and the equations are
axi-symmetric for flow in a circular cylinder. The boundary condition linking the
wall and pipeline equations (
4.31
)-(
4.35
) is the no-slip condition that assumes the
fluid particles to be adherent to the inner surface of the airway and hence to the mo-
tion of the elastic wall. Due to the fact that the wall elasticity is determined by the
cartilage fraction in the tissue, it is possible to consider variations in elasticity with
morphology, which in turn varies with pathology.
Generally, it is considered that if the Reynolds number
N
RE
is smaller than 2000,
then the airflow is laminar; otherwise it is turbulent [
165
]. Based on the airway
geometry and on an average inspiratory flow rate of 0.5 (l/s) during tidal breathing
