Page 122 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

RESPIRATORY MECHANICS AND GAS EXCHANGE 99

space is that portion of the lung that does not transfer CO from the capillaries, measured by the
2
Bohr method. While the methods used to determine these two values are different, normally they
3
yield essentially the same result, approximately 150 cm in an adult male. In some disease states,
however, the Bohr method may be affected by abnormalities of the ventilation-perfusion relation-
ship. The alveolar volume V , where gas exchange occurs, is the lung volume minus the dead space.
A
4.3.2 Air Flow and Resistance
3
A typical V of 500 cm at a rate of 15 bpm yields a total ventilation V of (0.5 L × 15 breaths/min) =
T
3
7.5 L/min. Assuming that air flows in the conducting airways like a plug, for each 500 cm breath
3
3
inspired the tail 150 cm fills the dead space while the front 350 cm expands the alveoli, where it
mixes with the alveolar gas previously retained at FRC. This makes the alveolar ventilation V A = (0.5 −
0.15) L × 15 bpm = 5.25 L/min. The plug flow assumption is not correct, of course, but for resting
ventilation it is a useful simplification. It fails, for example, in high-frequency ventilation, 3,24
3
where the mechanical ventilator can operate at 15 Hz with V ≤ 5 cm ; i.e., the tidal volumes can be
T
smaller than the dead space. Adequate gas exchange occurs under these circumstances, because of
the Taylor dispersion mechanism, 33 which is a coupling of curvilinear, axial velocity profiles with
5
radial diffusion applied to a reversing flow. The properties of airway branching, axial curvature, and
flexibility can modify the mechanism considerably. 6,8,14

The ventilatory flow rate V results from the pressure boundary conditions imposed at the trachea
and alveoli. In lung physiology, the flow details are often neglected and simply lumped into a resis-
tance to air flow defined as the ratio of the overall pressure drop ΔP to the flow rate; i.e., ΔP/V = Res.
A typical value is Res = 0.2 cmH O-s/L for resting breathing conditions. The appeal of defining air-
2
way resistance this way is its analogy to electrical circuit theory and Ohm’s law, and other elements
such as capacitance (see compliance, below) and inertance may be added. 7,27,32 Then an airway at
generation n has its own resistance, Res , that is in series with its n − 1 and n + 1 connections but in
n
parallel with the others at n.
The detailed fluid dynamics can be viewed as contributing to either the pressure drop due to
kinetic energy changes, ΔP , or that due to frictional, viscous, effects, ΔP , so that
K F
ΔP =ΔP +ΔP (4.1)
K F
For a frictionless system, ΔP = 0, Eq. (4.1) becomes the Bernoulli equation, from which we learn
F
2
that, in general, ΔP = 1 /2 r(u 2 out − u ), where u (u in ) is the outlet (inlet) velocity. For a given flow
out
in
K

rate V, the average air velocity at generation n is u = /A , where A is the total cross-sectional area
V
n n n
at generation n. Because A << A , air velocities diminish significantly from the tracheal value to the
0 23
distal airways. This implies that ΔP < 0 for inspiration while ΔP > 0 for expiration.
K K
The frictional pressure drop ΔP is always positive, but its value depends on the flow regime in
F
4
each airway generation. For fully developed, laminar tube flow (Poiseuille flow) ΔP = (128 mL/pd )V,
F
where m is the fluid (gas) viscosity, d is the tube diameter, and L is the axial distance. This
formula is helpful in understanding some of the general trends for R, like its strong dependence on
airway diameter, which can decrease in asthma. Airways are aerodynamically short tubes under
many respiratory situations, i.e., not long enough for Poiseuille flow to develop. Then the viscous
pressure drop is governed by energy dissipation in a thin boundary layer region near the airway wall
where the velocity profile is curvilinear. For this entrance flow, ΔP ~ V 3/2 , so resistance now is flow
F
2
dependent. The local Reynolds number is given by Re = u d /v, where v ª 0.15 cm /s is the
n n n
kinematic viscosity of air. For Re ≥ 2300, (see Table 4.1), the flow is turbulent and ΔP in those
F
n
airways has an even stronger dependence on ventilation, ΔP ~ V 7/4 smooth wall, ~ V 2 rough wall.
F
Expressing these three cases for a single tube (airway) in terms of the friction coefficient,
2
C =ΔP / 1 /2ru . The comparison is shown in Eq. (4.2):
F F
L ⎛ L ⎞ 12 / L L
/
/
C = 64 Re −1 C = 6 Re −12 C F = . 032 Re −14 C = k (4.2)
F
F
F
d ⎝ d ⎠ d d
Poiseuille flow entraanceflow turbulent flow w turbulent flow
smooth wall rough wall l