Biomedical Engineering Reference
In-Depth Information
surface, during panting at the airway opening and the alveoli. The only part which
is not in agreement with this hypothesis is the pleural surface, which has tissue on
both sides and its pressure distribution cannot be predicted.
To summarize, the results are obtained under the following assumptions:
•
laminar flow for typical Reynolds number during quiet breathing less than 2000;
•
ducts are long enough (this assumption is not true, thus neglecting the entrance
effects);
•
the air is homogeneous and Newtonian;
•
the axial velocity component is zero at the airway wall;
•
linear (visco)elastic, uniform cylindrical duct (valid as approximation);
•
for linearization we have assumed the following simplifications:
-
ωR
c
10
−
5
−
δ
, for in respiration we have values between 3
.
5904
×
and
10
−
6
;
- the air velocity is small compared to the wave velocity; this is valid for most
of the airways; i.e. in trachea there may be velocities as high as 10 m/s, with a
wave velocity of 339 m/s;
- the values for
y
vary between 0
2
.
1542
×
−→ ±
1 (rigid pipe), although in reality it
+
ζ/R)
(viscoelastic pipe);
-the
E
modulus is dependent on the airway type (cartilage fraction);
varies between 0
−→ ±
(
1
•
thin-walled ducts; for the healthy respiratory system, the ratio
h/R
varies between
0.4625 in trachea, to 0.0896 in alveoli. When calculating the value for
c
0
,the
´
geometrical characteristics are introduced, modifying it accordingly.
4.4 Summary
In this chapter, the results based on the Womersley theory have been used to deter-
mine an electrical equivalent of the respiratory system and capture the mechanical
properties in (
4.62
)-(
4.64
). For the respiratory system, transmission line models are
mostly used within high frequency ranges (above 100 Hz) for sound analysis diag-
nosis [
59
]. However, for lower frequencies (0.1-50 Hz) the transmission line theory
can be applied in a simplified form, leading to the exact solution for pressure and
flow changes in normal breathing conditions. A study of the systemic circulation
has been employed in [
115
], leading to the same formula for the compliance (
4.61
).
Similarity exists between the derivation of the input impedance in the respiratory
tree in this study and modeling the smaller systemic arteries, since in both simu-
lations the symmetric structure is employed, along with laminar flow conditions,
incompressibility, Newtonian fluid, and the no-slip boundary condition. The input
impedance is extended to a more general tree in [
115
], by adding the equation of
crossing a bifurcation based on a law on which the geometry changes over the junc-
tion. Nevertheless, we may argue that our choice of choosing to model a completely
symmetric tree still reflects its essential behavior, and can be extended with the
asymmetry relations adapted from [
115
].
