Annals of Biomedical Engineering, Vol. 9, pp. 489-499, 1 9 8 1
Printed in the USA. All rights reserved.
0090-6964/81/050489-11 $02.00/0
Copyright 9 1982 Pergamon Press Ltd.
MECHANISMS OF EXPIRATORY
FLOW LIMITATION
Robert E. Hyatt
Joseph R. Rodarte
Division of Thoracic Diseases and Internal Medicine,
Mayo Clinic and Mayo Foundation, Rochester, Minnesota
Theodore A. Wilson
Department of Aerospace Engineering and Mechanics,
University of Minnesota, Minneapolis, Minnesota
Rodney K. Lambert
Department of Chemistry, Biochemistry and Biophysics,
Massey University, Palmerston North, New Zealand
The role o f isovolume pressure flow curves in directing attention to expiratory flow limitation and in thedevelopment o f the flow volume curve is reviewed.
The approaches to modelling the flow-limiting mechanism are traced from the
equal pressure point concept to current concepts that suggest that there are two
basic mechanisms involved. One is the wave-speed mechanism resulting from the
coupling between airway compliance and the pressure drop due to convective
acceleration. The other is the coupling between airway compliance and viscous
losses in the flow. A computational model for a uniformly emptying lung is
presented. The model predicts the pressure distribution in the airways, isovolume pressure flow curves, and flow volume curves. The model tested well against
data obtained from excised human lungs. Potential limitations o f this model
are discussed, as are areas requiring further development.
MECHANISMS OF EXPIRATORY FLOW LIMITATION
It was long known that measurements made during the forced vital capacity maneuver (FVC) were useful in detecting obstructive lung disease.
However, the reasons for the value of this procedure were not initially
appreciated. Dayman (2) noted the importance of the lung's elastic recoil
in determining maximal flow. The crucial step leading to the concept of
expiratory flow limitation was made by Fry e t al. (6). They quantified
Address correspondence to R.E. Hyatt, Mayo Clinic, 200 First St., S.W., Rochester, Minnesota 55901.
Acknowledgment-This work was supported in part by Research Grant HL-21584 from the
National Institutes of Health, Public Health Service.
489
490
R.E. Hyatt, J.R. Rodarte, T.A. Wilson, and R.K. Lambert
pressure-flow relations at isovolume, the isovolume pressure-flow (IVPF)*
curve, and showed that expiratory flow became limited at modest, positive
transpulmonary pressures. This observation led directly to the description
of the maximal expiratory flow-volume (FV)* curve (10) which showed
that in a given subject there was at most lung volumes a limit to maximal
expiratory flow. Expiratory flow limitation was the reason why the FVC
maneuver had proved so useful. The functional relationships between transpulmonary pressure, respiratory gas flow, and lung inflation are illustrated
in Fig. 1 (left) where three isovolume pressure-flow curves from a normal
subject are plotted. These curves are obtained by having the subject breathe
repeatedly with increasing effort through the lung volume of interest. Flow
and transpulmonary pressure values at this volume are used to construct
the curves (10). Each curve relates transpulmonary pressure to flow at a
different, constant lung inflation. Curve V1 relates pressure to flow at a
volume of 0.8 L from total lung capacity (TLC). Curve V2 was measured
at 2.3 and curve V3 at 3.1 L from TLC, respectively.
The relations of pressure (P) to flow (l~) at given lung volumes are basic to
the interpretation of the FVC. The IVPF curves in Fig. 1 demonstrate the
important aspects of these relations:
1. Expiratory flow on PF curve VI, measured at high inflation, increases
as pressure increases. No defined limit to expiratory flow exists at
volumes near TLC. Maximal expiratory flow near TLC is highly dependent on the subject's effort.
2. Curves VI and V2, measured at lower volumes, have expiratory maxima. Flow increases with pressure until maxima are reached beyond
which further increases in pressure are associated with essentially no
change in flow. The values of flow at the maxima decrease with decreasing lung inflation. Maxima have been defined over approximately
the lower 80% of the vital capacity.
3. Evidence has been presented that expiratory flow on the plateau of
an isovolume PF curve is uniquely determined by the characteristics
of the flow and the physical properties of the lung (4). Maximum
expiratory flow over the range of lung volumes associated with PF
curve maxima requires less than maximal subject effort and represents
a limiting value that cannot be exceeded with but few exceptions
(7, 10).
4. Maxima do not occur on the inspiratory limbs of the isovolume PF
curves. Inspiratory flow depends primarily on the force the subject
can develop. Hence, in this sense maximal inspiratory flow is effortdependent, potentially quite variable, and will not be considered
further.
* I) would be preferable to F to represent flow and these curves could be designated IVPV and I~V
curves. However, since F was used originally and is most familiar, these designations for t h e curves
will be retained. Elsewhere Iy"is used for flow.
Expiratory Flow Limitation
491
Expir. ~/(L/S)
Expir. ~/(L/S)
6
/
--,o
--.o"
(-)
8;
40 80 120 160 200 (*)
Transpulmonary pressure
(cm H20 )
eO•
/
,8
-4
}
#
V3 V2
V1
Volume
Inspir. ~/
FIGURE 1. To the left are three isovolume pressure--flow curves measured at three volumes, Vz
being nearest total lung capacity. To the right the highest expiratory flow from each isovolume
pressure-flow curve is plotted against the volume at which the curve was measured (open circles).
From I V P F curves, a three-dimensional surface, which graphically describes the interrelationships among pressure, flow, and volume, can be
constructed (7) but this is tedious. Fortunately, the existence of expiratory
P F curve maxima provides a simplified approach to the evaluation of the
FVC. Since each isovolume P F curve relates flow to pressure at a given lung
volume, it is possible to plot the maximal flows from the P F curves against
the volumes at which they were measured. This has been done in Fig. 1
(right) for the three curves. If one recorded I V P F curves over a wide range
of lung inflations and plotted the expiratory flow and volume values in this
manner, the F V plot in Fig. 1 could be constructed. The dashed expiratory
line relates expiratory flow to volume when expiration is maximally forced,
as during the FVC, since pressures in excess of those occurring at the expiratory P F maxima are almost invariably developed (8, 10).
In practice one does not construct F V diagrams from isovolume P F
curves. Instead one has the subject breathe into a flowmeter-spirometer
system and flow is plotted as a function of volume during the FVC.
The phenomenon of expiratory flow limitation and the mechanisms
producing it have intrigued physiologists for years. Mead et al. (17) analyzed forced expiration in terms of the equal pressure point (EPP) concept.
Briefly, this approach said that, once flow was limited at a given lung volume, there was a site in the intrathoracic airways where intrabronchial and
intrapleural pressures were equal, the EPP. The driving pressure from alveolus to the EPP was the static recoil pressure of the lung (Pst). Airways
downstream (mouthward) of the EPP would be compressed while those upstream (alveolarward) would not be. Thus, an upstream resistance (Rus)
could be defined as Pst divided by Vmax, the maximal expiratory flow.
The factors determining Rus were analyzed in a very productive fashion and
492
R.E. Hyatt, J.R. Rodarte, T.A. Wilson, and R.K. Lambert
the importance of Pst in determining Vmax was clearly established. In a
series of demanding studies Macklem and associates (14, 15) defined the
anatomic location of the EPP in normal subjects and in patients with obstructive lung disease.
Pride et al. (21) likened flow limitation to the behavior of a Starling
resistor with the upstream driving pressure being Pst. They emphasized the
importance of the compressibility and tone of the flow limiting segment.
This approach has also been widely used.
However, the above analyses did not explain the mechanism of flow
limitation. Fry took a formal approach to the problem (4, 5) and outlined
a mathematical model for flow limitation. He pointed out that if [ 1 ] the
total cross-sectional area of the bronchial tree (A) could be defined as a
function of transpulmonary pressure (Ptp) and position along the tree (x),
A = f(Ptp, x)
(1)
and [2] the pressure gradient (dp/dx) in the airways could be described as
a function of area, position, and flow:
(2)
dp/dx = g(A,x; ~).
then for a given flow this coupled set of equation could be integrated from
the alveoli, where x = 0 and p = Pair (alveolar pressure), to the trachea. Fry
showed (5) that for some airway area-pressure curves, there is a maximum
value of expiratory flow for which a solution of these equations exists.
However, not enough was known at that time about the flow and the airways to implement Fry's approach with confidence. Furthermore, one had
to deal with the fact that at flow limitation, an increase in pleural pressure
decreased transmural pressure and airway area by just the right amount to
compensate for the increased pressure difference driving the flow to maintain Vmax constant on the P F curve plateau. It seemed unlikely that these
two effects of pleural pressure would balance for all lung volumes in health
and disease.
Perhaps there might be localized mechanisms that were dominant in producing flow limitation. Simplified models were proposed. Pardaens et al.
(19), Lambert and Wilson (12), and Pedersen and Nielsen (20) postulated
that most of the frictional head loss occurred in the periphery and the convective acceleration pressure drop occurred primarily in the central airways.
These assumptions led to the following expression:
P =Palv - A P f
where p
P
1
2
A2 ,
= lateral airway pressure at a point in the central airways,
-- the frictional head loss,
= gas density, with the third term being the Bernoulli term.
(3)
Expiratory Flow Limitation
493
Combining this approach with measures of APf and central airway areapressure plots in excised human lungs, a graphical solution (see below) for
l~rnax was obtained. Agreement with measured maximal flows was quite
good at high and midvolumes but poor at low volumes, where it was suggested that other mechanisms dominated the flow-limiting process (11).
Indeed, it now appears likely that there are two basic flow-limiting mechanisms. One is the wave-speed mechanism that results from the coupling
between airway compliance and the pressure drop due to the inviscid convective acceleration of the flow. The other is the coupling between airway
compliance and viscous losses in the flow (27). In normal lungs the first
mechanism appears to dominate over approximately the upper two-thirds
of the vital capacity (VC) and the second at lower lung volumes. The relative
importance of these two mechanisms in abnormal lungs is probably quite
different.
WAVE SPEED LIMITATION
A major contribution to our present understanding of flow limitation was
made by Dawson and Elliott (1) when they recognized that the lung, like
other systems, could not carry a greater flow than the flow for which the
fluid velocity equals wave speed at some point in the system. The pertinent
wave speed is the speed at which a small disturbance travels in a compliant
tube filled with fluid. The wave speed c in a compliant tube with an area A
that depends on lateral pressure p, filled with a fluid of density P, is given by
the following equation:
c =(A/p dA~ 1/2
dp ]
(4)
'
where dA/dp is the slope of the area-pressure curve for the airway. Maximal
flow is the product of the velocity at wave speed and airway area, cA.
The equation for the wave speed and its significance in limiting flow can
be derived from the physical laws that govern flow. A mathematical analysis
in that style has been presented (1,9). The mechanism can also be described
graphically. A typical area-pressure curve of an airway is shown in Fig. 2a in
which the pressure axis describes lateral pressure relative to pleural pressure.
If convective acceleration were the only cause of a pressure drop in the flow,
the Bernoulli equation could be used to describe the relation between lateral
pressure and airway area for a given volume flow rate:
P = Palv -- (1/2) O ~
(5)
This equation can also be plotted in Fig. 2a for different values of I7. The
airway area and pressure that would occur in this airway at a given flow V
are the coordinates of the intersection of this curve with the bronchial
494
R.E. Hyatt, J.R. Rodarte, T.A. Wilson, and R.K. Lambert
//
(a)
/]
.~
(b)
<r
/I
2/
PRESSURE
~
PALV
PRESSURE
4
3
2
I
I
I
I
I
i//
p
I
I
p
PALV
FIGURE 2. Graphical representations of flow limitation at wave speed.
Airway area as a function of p, the difference between lateral airway and
pleural pressure, is shown in panel a. Neglecting dissipative pressure losses,
lateral pressure can be calculated by subtracting the Bernoulli term from
alveolar pressure. Since alveolar pressure is measured relative to pleural pressure, it is equivalent to static recoil pressure. In panel b airway aree--pressure
curves for generations 2, 3, and 4 and the Bernoulli curve for maximum flow
are shown [from Wilson e t al. (27)].
area-pressure curve. That point represents the simultaneous solution of the
airway and flow pressure-area relations. For larger flows, the pressure-area
curve of Eq. 5 shifts up to the left. It is clear that there is a maximum flow,
pictured as V2 in Fig. 2a, for which a simultaneous solution exists. At that
flow, the two curves are tangent at the common value of A and p. Since the
curves are tangent, they have the same slope at the tangency, point. If Eq. 5
is differentiated with respect to p, holding Palv, P, and V constant, the
following equation is obtained:
dA
1 = p
A 3 dp
(6)
The slope dA/dp is equal to the slope of the bronchial area-pressure curve.
Solving Eq. 6 for l~, we obtain the following equation for maximal flow:
l~ = A
/P d p ]
(7)
This is identical to multiplying Eq. 4 by A.
At high lung volumes at which the recoil pressure is relatively large,
lateral pressure in the peripheral airways is positive and the airway areapressure curves are nearly horizontal. Also, in the periphery, the total cross
sectional area of the parallel flow paths is large and the line representing the
Bernoulli equation is nearly vertical if A is large. The two curves are far
from having common slopes in the periphery. As flow increases, a tangency
point will appear first in the central airways where the total cross sectional
area is small and the pressure has fallen to a value for which the airways are
Expiratory Flow Limitation
495
more compliant. Therefore, at high lung volume, flow is determined primarily by convective losses and the wave speed and area of the central
airways.
Figure 2b shows representative area pressure curves for the total cross
sectional area of the 2nd, 3rd, and 4th generations of the Weibel model of
the bronchial tree. The Bernoulli curve for maximal flow is also shown. The
pressures and areas in each generation are shown by X. The flow limiting
site, a choke point (CP) in wave speed terminology, is in generation 3 where
the tangency condition is satisfied. Additional pressure drops through the
bronchial tree occur because of additional compression of the airways downstream from the CP.
At lower lung volumes, the Bernoulli curves originate from a lower alveolar pressure. The Bernoulli curves in Fig. 2 are shifted to the left. Tangency
occurs at lower flows and the point of tangency may shift to a more peripheral generation.
Elliott and Dawson confirmed that the pressure distribution through a
compliant channel was consistent with wave-speed theory (3). Mink e t al.
(18) and Hyatt et al. (11) showed that properties of maximum flow at midand high-lung volumes in dog lungs and excised human lungs, respectively,
are consistent with this model. In these studies the site o f flow limitation
was identified by intrabronchial pressure measurements.
Equation 7 is not a complete predictive statement. Airway area, compliance and wave speed are functions of transmural pressure. Therefore, the
maximum flow that an airway can carry is a function of lateral pressure
within the airway. If convective acceleration were the only cause of a pressure drop in the flow and the mechanical properties of the entire bronchial
tree were.known, maximal flow could be predicted by finding the lowest
value of V for which values of p, A, and dA/dp at some point on the bronchial area-pressure curves satisfy Eqs. 5 and 7 simultaneously. In the real
situation, dissipative pressure losses contribute to determining the pressure
distribution, and the flow limiting site and the value of maximum flow cannot be predicted without simultaneously predicting the real pressure distribution. Nonetheless, the concept of flow limitation at wave speed is an
essential feature of a complete model and provides useful insights. For
example, by using wave speed theory and some modest assumptions about
the nature of the dissipative losses, it has been possible to deduce that the
viscosity dependence and density dependence of maximum flow must be
related in a particular way (25).
VISCOUS FLOW LIMITATION
At low lung volumes, the density dependence of maximum flow is small,
the viscosity dependence is large, and the predictive capability of the wave
speed concept is poor. Shapiro has described a purely viscous flow limitation
in a compliant tube (24). Attach a compliant tube with the area-pressure
properties shown in Fig. 3a to rigid supporting tubes at the ends, a distance
496
R.E. Hyatt, J.R. Rodarte, T.A. Wilson, and R.K. Lambert
(a)
(b)
0
d
u_
W
S
,
PRESSURE
Pl
PRESSURE DROP P,-P2
FIGURE 3. Viscous flow limitation. Area--pressure curve of small airway in
panel a. Panel b shows pressure--flow relations for Poiseuille flow. See text
for further details [from Wilson et aL (27)].
L apart. If the cross section o f the tube remained circular and if the tube
area and flow were small enough, the pressure drop in the tube would be
described by the Poiseuille equation,
dp
#1~
=
,
(8)
where a is a numerical constant and # = gas viscosity. By multiplying b o t h
sides by A 2 ( d x / a # ) and integrating from x = 0, the upstream end o f the tube
where the pressure is P l , to x = L, the downstream end where the pressure
is P2, the following expression for 12 is obtained:
P2
=
1 f
a/aL P1
A 2 dp.
(9)
The flow is 1/alsL times the area under the curve o f A 2 vs p between Pl and
P2. If Pl is held fixed and P2 is decreased, l)increases, but i f A 2 approaches
zero fast enough as P2 becomes negative, the integral approaches a finite
limit as P2 becomes infinitely negative. Therefore, if Pl is held fixed and
P2 is reduced, I) approaches a limiting value. The curve o f V vs P2 corresponding to the tube properties and value o f Pl shown in Fig. 3a is shown
in Fig. 3b. If A 2 approaches zero fast enough for the integral to remain
finite as P2 becomes large and negative, then the wave speed limit on the
flow cA approaches zero as P2 becomes large and negative, and the wave
speed limit would be reached at some negative value of P2. However, if
the tube is small and the flows are small, flow plateaus at its m a x i m u m value
and becomes essentially independent o f P2 by the viscous mechanism at a
value of P2 m u c h higher than the value at which wave speed is eventually
reached. The limiting flow is basically established by the coupling b e t w e e n
viscous losses and tube compliance and limitation at wave speed is not the
significant mechanism.
Expiratory Flow Limitation
497
COMPUTATIONAL MODEL
A complete model includes convective acceleration, laminar and turbulent
dissipation in the flow, and airway compliance, and requires a computer to
predict from these the pressure distribution in the airways, isovolume pressure flow curves and maximum flow volume curves. Such a model has been
proposed by Lambert et al. (13). It consists of an equation for the pressure
gradient in the flow and airway area-pressure curves for 17 generations of
the bronchial tree. The pressure gradient in the flow is described by the
equation,
dp=
1
d
dx - ~ P ~
(___~2
\a! -.t
(10)
where the dissipation pressure loss f is described by an equation in the form
of Rohrer's equation:
f = ala " ~
+bp -~T
(11)
This expression and values for the constants a and b were obtained from
studies of flow through a cast of a canine bronchial tree (22). The choice of
airway area-pressure curves was constrained by data on human airway areapressure curves of the central airways (11) and Weibel's description of the
maximum area for all generations (26). For a given lung volume and a given
value of l?, Eq. 10 was integrated from the periphery to the trachea. Flow
was increased and the integration repeated until a maximum flow was
reached for which wave speed occurred at some point in the airways or the
pressure at the end of the trachea was -100 cm H20.
By making the peripheral airways quite compliant, it was possible to
match the average air F V curves of five human lungs. The predicted curves
for He---O2 and SF6 also matched the observed data for these lungs. General
features of the predictions, such as lung conductance, density, and viscosity
dependence of maximum flow, and pressure required to reach maximum
flow, were similar to published data (13).
By altering certain parameters of the model it is possible to mimic characteristics frequently seen in individual F V curves. For example, decreasing
the area of the third generation results in the flow limiting site remaining in
that generation to a lower lung volume and produces a knee in the F V curve.
Decreasing the area of the peripheral airways results in a decreased density
dependence, especially at lower lung volume.
Although promising, this model requires further study. The sensitivity of
the F V curve to each of the parameters of the model should be determined.
The uniqueness of the relationship between the model parameters and the
predicted F V curves should be investigated.
Finally, the objective is to model disease. This will be a difficult task
because inhomogeneities are expected. In modeling flow through an inhomo-
498
R.E. Hyatt, J.R. Rodarte, T.A. Wilson, and R.K. Lambert
geneous bronchial tree, a method must be developed for finding different
regional flows for which the pressures match at c o m m o n points in the
branching network. In addition, inhomogeneous emptying introduces time
as an independent variable; flow at a given lung volume can not be analyzed
independently of flow at preceding volumes.
The above model for normal lungs has potential limitations. It uses a
symmetric model of the airways, does not directly take into account bronchial-parenchymal interdependence (16) or viscoelastic behavior of the
airways (23), nor does it predict negative effort dependence. Nevertheless,
it provides a promising basis for modeling more complicated problems such
as flow limitation in abnormal lungs.
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Expiratory Flow Limitation
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