Biomedical Engineering Reference
In-Depth Information
from which the conductance per unit distance can be extracted:
ω
2
πR
3
(
1
ν
P
)
2
−
g
x
=
sin
ϕ
E
(4.69)
|
E
|
h
and the capacitance per unit distance is given by
2
πR
3
(
1
ν
P
)
2
−
c
x
=
cos
ϕ
E
(4.70)
|
E
|
h
Thus, from the geometrical
(R, h)
and mechanical
(E
∗
,ν
P
)
characteristics of the
airway tube, and from the air properties
(μ, ρ)
one can express the
r
x
,
l
x
,
g
x
and
c
x
parameters. In this way, the dynamic model can be expressed in an equivalent
transmission line defined by Eqs. (
4.57
), (
4.58
), (
4.69
), (
4.70
). Similar to the elastic
wall case, we can estimate that, over the length
of an airway tube, we have the
corresponding properties:
μδ
2
πR
4
M
1
R
e
=
r
x
=
sin
(ε
1
)
(4.71)
ρ
πR
2
cos
(ε
1
)
M
1
L
e
=
=
l
x
(4.72)
G
e
=
g
x
=
ω
2
πR
3
(
1
ν
P
)
2
−
sin
ϕ
E
(4.73)
|
E
|
h
ν
P
)
2
C
e
=
c
x
=
2
πR
3
(
1
−
cos
ϕ
E
(4.74)
|
E
|
h
4.2.3 Generic Recurrence in the Airways
Variations of the ratios of the mechanical parameters within consecutive airway lev-
els in a branch are given in Fig.
4.2
-left. The ratio for the resistance is supra-unitary,
hence the resistance increases with the branch. Similarly, a sub-unitary ratio for
compliance denotes a decreases in elasticity with each level. Both effects are due to
a decrease in the airway cross-section. Figure
4.2
-right depicts the same variation,
but with the airway level. Due to an increase in the total cross-sectional area from
one level to another, the total resistance decreases (sub-unitary ratio), whereas the
total compliance increases (supra-unitary ratio). Exponential changes for resistance,
inertance and elastance from level 10 onwards show that mechanical properties be-
come important towards the lower ducts (gas exchange).
Notice from Fig.
4.2
that from level 11 onwards, the variations of the ratios are
smaller (except the last two bifurcations: 22 and 23). Physiologically, level 11 cor-
responds to the bronchiole [
59
,
114
]. We can correlate these effects to the variations
in the airway radius and in the airway cross-sectional area, respectively. The ra-
dius changes from 8 mm to 1.75 mm, whereas the area varies from 254.5 mm
2
to
