Biomedical Engineering Reference
In-Depth Information
=
√
−
where
c
denotes the complex velocity of wave propagation and
j
˜
1. Further
simplifications lead to the following system of equations:
C
1
c
y
e
jω(t
−
δj
3
/
2
J
1
δj
3
/
2
y
+
jωR
μ
2
A
P
ρ
z
˜
c
)
u
=
or
c
˜
˜
(4.8)
c
J
1
δj
3
/
2
y
e
jω(t
−
jωR
δj
3
/
2
R
2
ρ
z
˜
c
)
c
M
P
e
j(ωt
−
Φ
P
)
u
=
C
1
+
˜
˜
C
1
J
0
δj
3
/
2
y
e
jω(t
−
A
P
ρ
z
c
)
w
=
+
or
˜
c
(4.9)
C
1
J
0
δj
3
/
2
y
e
jω(t
−
M
P
ωρ
z
c
)
π
2
)
e
j(ωt
−
Φ
P
−
=
+
w
dp
dz
=
z
c
)
A
P
e
jω(t
−
M
P
e
j(ωt
−
Φ
P
)
p(t)
=
or
−
(4.10)
A
P
ρ
1
with
C
1
=−
J
0
(δj
3
/
2
)
,
A
P
the amplitude of the pressure wave,
J
0
the Bessel
function of the first kind and zero degree,
J
1
the Bessel function of the first kind and
first degree [
1
], and where
c
˜
dp
dz
=
jω
˜
z
˜
A
P
e
jω(t
−
c
)
M
P
e
j(ωt
−
Φ
P
)
−
=
(4.11)
c
such that
c
ω
M
P
e
j(ωt
−
Φ
P
−
π/
2
)
˜
z
c
)
A
P
e
jω(t
−
=
(4.12)
It is supposed that the movement of the (relatively short) elastic airway ducts is
limited to the radial movement
ζ(z,t)
of the tube, being dependent only on the
longitudinal coordinate and the time. This supposition is valid for short segments
(
wavelength of the pressure wave) in which the longitudinal movement is negli-
gible compared to the radial. The wavelength corresponding to the tracheal tube is
about 2.5 m long, much longer than the length of the tube itself; hence, the supposi-
tion is valid in our case. Although the inspiratory and expiratory movements of the
airways involve both radial as well as longitudinal movement, we restrict our analy-
sis to the radial elongation only. The Poisson coefficient is denoted by
ν
P
; it equals
0.45 [
85
]. The problem now contains four unknowns:
u(y, z, t)
,
w(y,z,t)
,
p(z,t)
,
and
ζ(z,t)
; therefore we need an extra equation in order to solve the system: the
pipeline equation. The movement equation of the wall follows from the dynamical
equilibrium of the forces applied on the wall, similar to the work reported in [
115
].
Denoting with
ζ
the elongation of the tube radius from
R
to
R
+
ζ
,wehavethe
dynamic equilibrium equation in the radial direction:
ζ)dθ dz
d
2
ζ
dt
2
E
ζ
R
dθ dz
p(R
+
ζ)dθ dz
+
h
=
hρ
wall
(R
+
(4.13)
ν
P
1
−
