Biomedical Engineering Reference
In-Depth Information
c
x
in (
4.61
) is independent of the frequency, while both
r
x
(
4.57
) and
l
x
(
4.58
)are
dependent on frequency trough the
δ
parameter, present also in
M
1
. Because we are
interested only in the input impedance, we can disregard the effects introduced by
the reflection coefficient. Hence, for
|
|
1, one can estimate that over the length
of an airway tube, we have the corresponding properties [
73
]:
γ
R
e
=
r
x
=
μδ
2
πR
4
sin
(ε
1
)
M
1
(4.62)
ρ
πR
2
cos
(ε
1
)
M
1
L
e
=
l
x
=
(4.63)
C
e
=
c
x
=
2
πR
3
(
1
ν
P
)
−
(4.64)
Eh
4.2.2 Viscoelastic Tube Walls
Viscoelasticity is introduced assuming a complex function for the elastic modulus,
yielding a real and an imaginary part [
8
,
23
,
143
]. This can then be written as a
corresponding modulus and phase:
E
∗
(j ω)
=
E
S
(ω)
+
jE
D
(ω)
=|
E
|
e
jϕ
E
(4.65)
The complex definition of elasticity will change the form of the wave velocity from
(
4.37
)into
|
E
|
he
jϕ
E
2
ρR(
1
|
E
|
h
2
ρR(
1
ν
P
)
e
j
ϕ
2
c
0
=
ν
P
)
=
(4.66)
−
−
The viscoelasticity of the wall is determined by the amount of cartilage fraction in
the tissue, as the viscous component (collagen), respectively by the soft tissue frac-
tion in the tissue as the elastic component (elastin) [
8
]. The equivalent of (
4.65
)is
the ratio between stress and strain of the lung parenchymal tissue. The Young mod-
ulus is then defined as the slope of the stress-strain curve. With the model given by
the above described equations, it is possible to consider variations in viscoelasticity
with morphology and with pathology. This will be discussed in the next chapter.
For a
viscoelastic pipeline
, the characteristic impedance is given by
ρ
πR
2
1
|
h
2
ρR
E
|
1
√
M
1
e
−
j(
ε
2
+
ϕ
2
)
Z
0
=
(4.67)
ν
P
1
−
and the transversal impedance is given by
1
ω
2
πR
3
(
1
e
j(
2
−
ϕ
E
)
Z
0
ν
P
)
2
−
1
g
x
+
Z
t
=
jωc
x
=
Z
l
=
(4.68)
|
E
|
h
