Effects of surface tension on airway reopening DONALD
P. GAVER
III,
and viscosity
RICHARD
W. SAMSEL,
AND
JULIAN
SOLWAY
Section of Pulmonary and Critical Care Medicine, The University of Chicago, Chicago, Illinois 60637
GAVER,DONALD P.,III, RICHARD W. SAMSEL,ANDJULIAN SOLWAY. Effects of surface tension and viscosity on airway reopening. J. Appl. Physiol. 69(l): 74-85, 1990.-We studied airway opening in a benchtop model intended to mimic bronchial walls held in apposition by airway lining fluid. We measured the relationship between the airway opening velocity (U) and the applied airway opening pressure in thin-walled polyethylene tubes of different radii (R) using lining fluids of different surface tensions (y) and viscosities (p). Axial wall tension (T) was applied to modify the apparent wall compliance characteristics, and the lining fluid film thickness (H) was varied. Increasing p or y or decreasing R or T led to an increase in the airway opening pressures. The effect of 23 depended on T: when T was small, opening pressures increased slightly as H was decreased; when T was large, opening pressure was independent of H. Using dimensional analysis, we found that the relative importance of viscous and surface tension forces depends on the capillary number (Ca = &J/y). When Ca is small, the opening pressure is -8r/R and acts as an apparent “yield pressure” that must be exceeded before airway opening can begin. When Ca is large (Ca > 0.5), viscous forces add appreciably to the overall opening pressures. Based on these results, predictions of airway opening times suggest that airway closure can persist through a considerable portion of inspiration when lining fluid viscosity or surface tension are elevated. airway closure; regional
ventilation;
airflow
obstruction
CLOSURE occurs at low lung volumes in normal adults and may exist at larger lung volumes in patients with airflow obstruction (3, 10, 15, 20). Airway closure can lead to local hypoventilation and impaired gas exchange and thus may be physiologically important. Its effect on gas transport is determined by the period of time the airway remains closed. For example, if an airway closes at end expiration but reopens at the onset of inspiration, the influence on gas transport may be minimal. However, if the time course of reopening is long, so that the airway remains closed for a significant fraction of inspiration, hypoventilation of the affected respiratory pathway may be substantial. Airways may close in two ways: 1) meniscus formation, where a film of airway lining fluid obstructs the lumen of an otherwise air-filled bronchus [recently described by Kamm and Schroter (9) as “film collapse”]; and 2) airway collapse (termed “compliant collapse” in Ref. 9), where the bronchial walls collapse and are held in apposition by the adhesive properties of the airway lining fluid (6, 9, 11). Evidence in favor of the latter mechanism is provided by Hughes, Rosenzweig, and Kivitz (8), whose
AIRWAY
74
0161-7567/90 $1.50 Copyright
0
photomicrographs of an isolated canine lung clearly demonstrate the collapse of distal bronchi. Macklem (11) speculated that airway collapse might exist as a result of pressures induced by a film meniscus. Recently, Yager and colleagues (22) have suggested that fluid within airway interstices may amplify bronchoconstriction by precisely this mechanism. We have studied the reopening of collapsed airways in a benchtop airway model (Fig. 1). Our approach was to measure the relationship between the velocity of airway opening and the applied transmural pressure, one of the key determinants of the time scale for airway reopening. By using several lining fluids and tubes, we investigated how the physical characteristics of the lining fluid and of the airway itself influence the opening pressure-velocity relationship. Our results provide a conceptual framework by which airway reopening may be understood. THEORY
Opening of a closed pulmonary airway was analyzed using a benchtop model (Fig. 1) that was based on the description of a closed airway presented in Greaves, Hildebrandt, and Hoppin (6). The open region of a cylindrical airway closes into a ribbonlike region where the airway lining fluid fills the lumen between the two flexible walls and acts as a viscous adhesive. The open portion of the tube has radius R, and the film layer in the closed section has thickness H. The walls are assumed to be completely flaccid until the airway is fully inflated and axial tension is applied to modify the apparent wall mechanical characteristics (i.e., cross-sectional area-transmural pressure relationship). When positive pressure is applied to the gas phase, the air-liquid interface (meniscus) progresses down the tube and opens the airway. This construct is similar to that described by Macklem, Proctor, and Hogg (12), who found that when inflating a bronchiole, they “were gradually peeling apart the opposing walls, and the liquid remained in situ, presumably lining that part of the bronchiole that had opened.” As will be shown, the surface tension of the airliquid interface, viscosity and thickness of the lining fluid, radius of the tube, and axial wall tension all influence the velocity (U) of airway opening for a given applied transmural pressure. This model for the opening of a closed pulmonary airway contains features of fluid mechanics problems that have not heretofore been applied toward the understanding of physiological phenomena: two-phase dis-
1990 the American
Physiological
Society
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AIRWAY
75
REOPENING
Virc;;uadLining \I
6 1
Tension Film Thickness
m
(H) FIG. 1. Airway opening model. Flexible-walled tube has a radius R, upstream film thickness H, and imposed axial wall tension T. Positive pressure is applied to gas, opening tube at velocity U resulting in an opening angle 0.
/
/
/
’
/’
/’
,//
/
,, ’
/
/I ’ /
,/’ I
,’
,.’
I
, ,’
/
,/’
, ,/’
,, ’
/
,’
,/
/
I
T
)U
R
FIG. 3. Tape-peeling model. A flexible strip (tape) is held to a rigid surface by a viscous adhesive. Tape peels at velocity U by applying a tension T at an angle 0, while pressure in gas phase remains at atmospheric pressure. Lining fluid has a downstream depth H, viscosity p, and surface tension y. Local wall curvature creates pressure gradients in lining fluid, inducing flow that allows tape to peel.
placement in a Hele-Shaw cell and the peeling of a flexible strip attached by a viscous adhesive. A brief description of these phenomena, and their relation to airway opening, is presented below.
peels, the gasphase remains at atmospheric pressure and the meniscus is sucked forward by the subatmospheric pressure in the liquid phase, induced by the local curvature of the flexible wall. The pressure caused by wall curvature also drives fluid upstream (toward the meniscus) and modifies the film thickness in the vicinity of the meniscus. Still, this model does not completely describe the wall motion in airway opening, as 8 is imposed in the tape-peeling model, but is part of the solution in the airway opening model.
Hele-Shaw Flow
Airway Opening
’
/
/
/’ ,’ ’ ,,’ I’ ,’ / r’/ /
,I’
/
,/
,’
/
, /
,/’ ‘/,,’,,
FIG. 2. Hele-Shaw model. Positive pressure drives fluid from between 2 rigid plates separated fluid has viscosity p and surface tension y.
/’ ’ I /’
//
/”
,,I
/
is applied to gas and by distance H. Lining
Two-phase displacement in a Hele-Shaw cell (16-18) is the flow induced when an inviscid fluid (e.g., air) displaces a more viscous fluid (e.g., mucus) from the region between two narrowly spaced plates, as shown in Fig. 2. The pressure P needed to push the air at velocity U depends on capillary and viscous contributions. The capillary pressure is determined by the surface tension and geometry of the air-liquid interface. For example, the capillary pressure of a spherical bubble with radius R is, by the law of Laplace, Pcap = 27/R, where y is the surface tension. The viscous stress is proportional to the strain rate in the lining fluid, where the constant of proportionality is I-C,the fluid viscosity. Saffman and Taylor (19) showed that a single dimensionless parameter (“capillary number,” Ca = pU/y) represents the relative magnitudes of the viscous and capillary pressures in the Hele-Shaw cell. Solutions of Hele-Shaw cell fl~~ are complicated by the fact that the air-liquid interface is a free boundary and is thus part of the solution as well as a boundary condition. Adhesive Peeling The primary limitation of comparing two-phase displacement in a Hele-Shaw cell with airway opening is that it does not account for the airway’s flexible walls. A model that describes the wall behavior is that of McEwan and Taylor (14), who investigated the peeling of a flexible strip (tape) attached to a rigid wall by a viscous adhesive. As shown in Fig. 3, the tape is pulled with tension T at an angle 8. The lining fluid has viscosity p, surface tension y, and thickness H far upstream of the moving meniscus. This problem has two free boundaries: the flexible wall and the air-liquid interface. As the tape
Although a complete mathematical model of airway opening is beyond the scope of this paper, a brief discussion is warranted because it will create the framework by which our airway opening experiments will be analyzed. When a finger of air displaces the liquid between narrowly spaced flexible sheets, an air-liquid meniscus interface is created, as shown in Fig. 1. As described above, the pressure needed to displace the more viscous fluid by the meniscus has two components tiotal
= Pc*ap + P&s
(1)
where Pzap is the capillary pressure associated with the surface tension at the air-liquid interface and P&s is the contribution to the total pressure due to the viscous stress that occurs when the viscous fluid is displaced by the air. The superscript asterisk denotes a dimensional pressure, here and throughout the remainder of this paper. In Eq. 1, we have assumed that Pc*apand P&s are independent. Clearly this is not always true, since the viscous flow may modulate the meniscus shape, altering the capillary pressure and leading to a nonlinear interaction between surface tension and viscous forces. As can be seen from Fig. 1, two obvious length scales exist in this problem: R, the radius of the open tube, and H, the gap between the closed tube walls. To identify the dominant balances of this problem, we will reduce Eq. I by the following scaling PZap -- c Pcap
(2a)
C
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76
AIRWAY
REOPENING PNEUMOTACHOGRAPH I ;Totol
and P&S
.
=P
u 7 Pvis
w
where Pcap and Pvis are the dimensionless capillary and viscous pressures and L, and L, are their associated length scales. The capillary pressure scaling assumes that the magnitude of the capillary pressure is of the order of r/L,, which is the pressure needed to maintain a static meniscus with a radius of curvature equal to L,. Additionally, we assume that the magnitude of the viscous pressure is approximately pU/L,, where UIL, is a measure of the strain rate in the neighborhood of the meniscus. By use of this scaling, the dimensionless total pressure, Ptotal, is defined as Ptotal
Hotal = Y/L
f/q Lc c Pvis = Pcap + - I Y 1 v
(3)
The dimensionless quantity @/r is the capillary number, Ca, and can be considered as a dimensionless velocity that reflects the relative magnitudes of the surface tension and viscous forces in this problem. When Ca is small, surface forces dominate and the pressure needed to move the meniscus at velocity U is determined by the pressure jump across the interface. As Ca increases, the viscosity of the lining fluid begins to retard the motion of the meniscus. Appropriate selection of L, and L, should allow our dimensional pressure-velocity data for different experiments (changing R, H, y, and p) to be represented by Eq. 3, thus collapsing a large amount of data to a simple relationship. Assumptions Several assumptions have been made in the development of this model. First, we have used a model of a closed airway where the lumen closes into a ribbonlike shape, as proposed by Greaves, Hildebrandt, and Hoppin (6) and as suggested by the photomicrographs of closed canine bronchi taken by Hughes, Rosenzweig, and Kivitz (8). We have also assumed that the viscosity and surface tension in the airway are each constant, although we acknowledge that pulmonary airway lining fluid is probably not Newtonian and may have viscoelastic properties (5). Furthermore, as the airway opens, the air-liquid interface expands, possibly leading to variation of the surface tension. Additionally, the interaction of individual airways is not considered and may be important in determining the opening velocity of sequential generations of pulmonary airways. These assumptions may limit the quantitative estimation of pulmonary airway reopening from results gathered here, but the qualitative insights from our analysis should be pertinent to the lung. METHODS
Benchtop Experiments Measurements of the pressure-velocity relationship during the opening of a closed flaccid tube were conducted using the apparatus shown schematically in Fig. 4. The polyethylene tube lies on a horizontal stainless
(0) POLYETHYLENE
TUBING
H‘IGH IMPEDANCE
(FOR TENSION)
FLOW SOURCE
4. Schematic of experimental apparatus. Polyethylene tubing lies on a stainless steel table, with tension applied by a weight W. Tube opens by pressure applied to gas, measured upstream as tube opens with velocity U. FIG.
TABLE
1. Physical properties of lining fluids Lining
Fluid
Polyethylene glycol lOW30 motor oil 85W gear oil
Viscosity
(p),
g.s-‘.cm-1 1.1
1.5 9.9
Surface
Tension w, dyn/cm
42.4 24.7 25.0
steel table, with its open end connected to a high-impedance flow source, used to blow the air-liquid meniscus down the tube and reopen the closed airway. The opposite end of the tube was closed permanently, with plastic struts placed so as to retain a ribbonlike configuration. A weight was supported from this end to provide the tube with a known axial wall tension. The low-friction stainless steel table surface allowed the tube to slide, helping to maintain constant wall tension during the duration of each experiment. Various fluids were used to provide the viscous adhesion that closes the tube. By changing lining fluids we investigated how fluid properties (y, p) influence the tube opening behavior. Additionally, we modified the fluid film thickness (H), tube radius (R), and axial wall-tension (T) to determine their roles in tube opening. We used three different lining fluids to investigate how surface tension, y, and viscosity, p, influence the pressures needed to reopen the closed airways: lOW30 motor oil, 85W gear oil, and polyethylene glycol (see Table 1). Viscosity was measured with a bulb viscometer (CannonFenske-Ostwald type), and surface tension was measured with a Wilhelmy balance. These fluids were selected because of their large range of viscosity and surface tension and because they wetted our polyethylene tubing. To study the tube radius dependency of airway opening, we conducted our experiments on thin-walled (0.002 in.) polyethylene tubes with radii of 0.54, 1.72, and 3.64 cm. The film-thickness dependency of airway opening was studied with two film thicknesses, H = 0.28 and 0.83 mm. Axial wall tension, T, was applied to modify the apparent airway wall mechanical characteristics. As shown schematically in Fig. 4, the weight, W, applied tension to the tube wall. Since the force was distributed along the circumference of the tube wall, the tension had a magnitude of T = W/27& Weights were adjusted between experiments with tubes of different radii so that isopleths of tension could be investigated. Two tensions were studied, T = 1 x lo4 and 5 X lo4 dyn/cm. Each experiment was begun by setting a uniform film thickness (H) with a fixed gap gauge. The upstream
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AIRWAY
REOPENING
valve was opened, increasing the pressure in the gas phase and initiating tube opening. The meniscus velocity was determined from the time it took the meniscus to travel 40 cm (20 cm for the smallest tube, R = 0.54 cm); the pressure was continuously sampled (30 Hz) and digitally recorded during the period the meniscus traveled this distance. The tubing was -1.5 m long for the tubes of R = 3.6 and 1.7 cm and 1 m long for the tube of R = 0.54 cm. We began recording the pressure signal when the meniscus passed a point 60 cm from the upstream rigid attachment for the 1.5-meter tubes and 40 cm from the attachment for the l-m tube. Thus the measurements were made at a distance of more than 14 tube radii from the attachments, so end effects were assumed to be small. We continuously monitored the flow rate with a pneumotachograph to ensure that the flow rate was constant. In each individual experiment, the recorded pressure signal was averaged creating one pressure-velocity (P- U) data point. Nearly 500 experiments were conducted for this study. By linear regression of the pressure and flow signals of 35 randomly selected experiments, we found that the flow was nearly constant, with a variation from the mean of -0.03 t 0.2%. However, the total pressure consistently decreased during the course of an experiment, from a magnitude 5.5 t 4.4% greater to 5.5 & 4.4% less than the average pressure. Even though this result demonstrates that our experiments do not reflect strictly steady-state reopening conditions, we believe that the variation is relatively small, and our results can thus be used to assess quasi-steady reopening phenomena (see DISCUSSION). Because of gravity, the weight of the liquid lining fluid determines the minimum pressure (-pgH/Z) in the upstream (open) end of the tube, where p is the fluid density and g is the gravitational constant. As will be shown, the capillary pressure is -87/R; thus the gravitationally induced component of the applied pressure is ~13% of the capillary pressure for our smallest tube (0.54 cm). For the larger-tube experiments (3.64 cm), the gravitationally induced component of the pressure is equivalent to the capillary pressure. In this way, gravity simply determines the minimum reopening velocity that we can investigate.
77
RESULTS
Dimensional
Representation
Figure 5 shows the pressure-velocity relationship for a tube of radius R = 1.72 cm with fixed tension T = 1 x lo4 dyn/cm, film thickness H = 0.28 mm, and three lining fluids. The opening pressures for 85W gear oil were clearly larger than for either polyethylene glycol and lOW30 oil, implicating viscosity as a critical determinant of the opening pressure. Also, the pressures needed to open the polyethylene glycol-coated tube are uniformly larger than those needed to open the tube coated with lOW30 oil. Since the surface tension of polyethylene glycol is greater than that of lOW30 oil, but their viscosities are nearly identical, this result shows that opening pressures are determined partially by surface forces. Figure 6 illustrates the tube radius dependency of airway opening pressures in experiments conducted with lOW30 oil, T = 1 x lo4 dyn/cm, H = 0.28 mm, and in tubes with radii of R = 0.54, 1.72, and 3.64 cm. It is evident that the opening pressure increased as the tube radius decreased. We demonstrate the influence of wall tension on airway reopening pressures in Fig. 7, A and B, with data from experiments conducted on a tube of radius R = 1.72 cm coated with lOW30 motor oil of thicknesses H = 0.28 and 0.83 cm. Figure 7A demonstrates that when wall tension is small, T = 1 X lo4 dyn/cm, decreasing the film thickness leads to an increase in the airway reopening pressure. In contrast, Fig. 7B shows that when T is large, T = 5 x lo4 dyn/cm, reopening pressures are largely independent of film thickness. Figure 8 shows the dependency of tube reopening pressures on the imposed axial wall tension. These data are from experiments conducted in lOW30 motor oilcoated tubes of radius R = 1.72 cm, film thickness H = 0.28 mm, and wall tensions T = 1 X lo4 and 5 X lo4 dyn/ cm. Figure 8 shows that the opening pressure decreased as the wall tension increased. Dimensionless
Representation
In this section, the dimensional data shown above are scaled in the manner discussed in the section, THEORY.
0.7 n r,U
0.6
!
u
869
c-4 0.5
I
-
O8
O@ E
0
0.4
F 3 cn $
03.
8
80
-
eP
0.2 -
00
43
#&AA
@
L n1_ Ae 0.0
FIG. 5. Dimensional pressure-velocity relationship for tube opening with 3 lining fluids: 85W oil (0), lOW30 oil (o), and polyethylene glycol (A). R = 1.72 cm, H = 0.83 mm,T= 1 X lo4 dyn/cm.
@ alGil
’
I
I
0
5
IO
Velocity
g
15
(cm/s)
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AIRWAY
REOPENING
0
0
Q0
FIG. 6. Dimensional pressure-velocity relationship for opening of tubes with 3 different radii: R = 0.54 cm (0), 1.72 cm (o), and 3.64 cm (A). H = 0.28 mm, T = 1 X lo4 dyn/cm, and lOW30 lining fluid.
0
5
Velocity
10
15
(cm/s)
Here we will systematically investigate these data so as to elucidate the appropriate capillary and viscous length scales, L, and L,, thus providing insight into the physics of airway opening. Since L, and L, are likely to depend on the tube radius (R) or film thickness (H), for a given tube (i.e., R fixed) with a given film thickness (i.e., H fixed), the ratio L,/ L, is fixed. A single relationship (shown in Eq. 3) should then hold true for different fluids if we have indeed selected the correct physical mechanisms as defining the airway opening pressures. Figure 9 represents the data of Fig. 5 in dimensionless form, where we have arbitrarily chosen L, = L, = R. In dimensionless form, these data appear to fall on a single curve, suggesting that opening phenomena are dominated by a balance of surface tension and viscous forces. Additionally, Fig. 9 shows that the capillary number, Ca = pU/r, is the parameter that correctly represents this balance for a given ratio of LIL,. Having validated the capillary number dependency for fixed L,IL,, we can now use Eq. 3 to determine the appropriate length scales L, and L, by investigating opening pressures for groups of experiments where H and R were not held constant. By plotting the data in dimensionless form using each of the four possible combinations of length scales, we shall determine the appropriate combination (i.e., the combination that yields a single pressure-relationship for all data). The four combinations of length scales and corresponding dimensionless relationships from Eq. 3 are Case1: L, = R,
L, = H: P/(7/R)
vs. (pU/$*(R/H)
Case2: L, = R:
L, = R: P/(y/R)
vs. (pU/$
Case3: L, = H,
Lv = HI P/(7/H)
vs. (pU/r)
Case4: L, = H,
L, = R: P/(7/H)
vs. &U/r)*
(H/R)
Figure 10 shows tube-opening data from experiments using three tubes and three fluids, with H = 0.28 mm and T = 1 x lo4 dyn/cm. In each case, the data from each tube lie on one curve, confirming the capillary number dependency for fixed H/R. However, only the
dimensionless representation of case 2 exhibits a single relationship for data of different tube radii, suggesting that the dominant length scales for airway opening are L c = R and L, = R. The selection of R as L,, the capillary length scale, is interesting physically, since it implies that the leading order meniscus curvature depends on the tube radius, R, and not on the film thickness, H. This result suggests that the air-liquid meniscus does not project into the closed portion of the airway, because the radius of curvature would then be -H/2. Furthermore, L, = R means only that the meniscus curvature is proportional to the airway radius R. Our experiments show that the zero capillary number int$rcept actually occurs at a dimensionless pressure of Ptotal/(y/R) = 8. This result implies that the onset of airway opening occurs at a pressure that is about eight times larger than the pressure based on a meniscus curvature equal to the airway tube radius. The flexible wall evidently allows the meniscus to have a radius of curvature eight times smaller than R. The selection of the viscous length scale, L, = R, suggests that the strain rate is proportional to U/R; hence the viscous stress scales as pU/R. We found this surprising, since the stress in the open portion of the tube is clearly not an important component of the overall viscous stress. We suspect that the viscous stress is determined primarily by the amount of fluid directly downstream of the meniscus (toward the closed end), where shear stresses are largest. The opening of the tube causes pressure gradients in the film layer that suck the fluid upstream, so that it wells up in the vicinity of the meniscus (see below). These mechanisms are similar to those discussed in McEwan and Taylor’s tape-peeling model (14). If the steady-state film thickness in front of the meniscus is proportional to the tube radius R, this could explain our finding that L, = R. Figure 11 shows nonlinear regressions of data from all experiments, where we have used the length scales L, = L = R in the dimensionless representation. The data have been regressed to the form Pt*btall(ylR) = Pvis CaR + Pcap, by analogy with the theory of two-phase displacement flow in a rigid-walled Hele-Shaw cell (16).
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AIRWAY
A
79
REOPENING
0.7
s
T = 1xl O4 dynes/cm
0.6
c\l 1
0.5
E 0
0.4
t!?
0.3
z ;
0.2
k
f30 0 O
0
0 80
0.1 0.0
0
Velocity
B
FIG. 7. Dimensional pressure-velocity relationships for reopening of tubes coated with 85W oil of film thicknesses: H = 0,28 mm (0) and 0.83 mm (0). R = 1.72 cm, T = 1 x lo4 dyn/cm (A) and 5 X lo4 dyn/cm (B).
(cm/s)
0.7
T = 5x1 O4 dynes/cm
0.6 EY cv I
0.5
E 0
0.4
aJ L
0.3
3 v, cn 2
O**
cl-
15
10
5
0.1
0.0 0
I
I
5
10
Velocity
15
(cm/s)
0.7 n 0.6 0 CY 1 0.5 E 0
0.4
P 3 m m i!!
0.3
n-
FIG. 8. Dimensional pressure-velocity relationship for opening of tubes with T = 1 X IO4 dyn/cm (0) and 5 X lo4 dyn/cm (e). R = 1.72 cm, H = 0.28 mm, and 85W oil lining fluid.
0.2
0.1 0.0 0
10
5
Velocity
15
(cm/s)
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80
AIRWAY 40
REOPENING
r
FIG. 9. Dimensionless pressure-velocity relationship for opening of tubes lined with 85W oil (0), lOW30 oil (o), and polyethylene glycol (A). R = 1.72 cm, H = 0.83 mm,T= 1 x lo4 dyn/cm.
-
5
Dimensionless “Capilllary
Velocity, Number”
@J/y
The capillary number dependence (0.61 5 B 5 0.98) corresponds to the derived 0.67 power dependence of the pressure on capillary number in Hele-Shaw flow (see Table 2). This nonlinear dependence on the capillary number differs from the linear dependence shown in Eq. 3, because the assumptions made in formulating Eq. 1 did not include the inherent nonlinear interaction of the capillary and viscous forces, as discussed above. Isopleths of T = 5 x lo4 dyn/cm (Fig. 11) demonstrate that H has very little effect on the opening pressure when the axial wall tension is large. This result supports our previous conclusion that the leading order viscous length scale is based on the tube radius, not the film thickness. In contrast, isopleths of T = 1 x lo4 dyn/cm show that the opening pressure increases with a decrease in film thickness. Thus, although R is the dominant length scale, H appears to play an additional minor role in determinR = 0.54
cm (o), H = 0.28
60
Case
1 :
L, = R,
1.72 mm,
ing the pressure necessary to overcome the viscous stress. Two explanations may account for this H dependency. First, we have observed that flows are induced by the meniscus and airway wall motion as the meniscus progresses down the tube (data not shown). These flows downstream of the meniscus are contained between the flexible walls and have a viscous length scale of H, which may add a small contribution to the overall viscous stress. Second, when wall tension is small, the flows induced by separation of the flexible walls may not be sufficient to increase the film thickness at the meniscus to a depth proportional to R. Note that the driving force for the film flow toward the meniscus is a negative “suction” pressure proportional to To K, where K is the local wall curvature. Therefore, when T is small, the driving force may not be large enough to increase the film thickness in the neighborhood of the meniscus to an extent that
cm (@), 3.64 cm (A) T = 1 x lo4
dynes/cm
L, = H
A AA
OL 0
200
400 NJ/Y)
600
800
0
2
6
8
* w/‘-o =H,
L,=R
0
FIG. 10. Determination of capillary length scale (L,) and viscous length scale (&) using Eq. 3. Experiments were conducted with tubes of radii R = 0.54 cm (0), 1.72 cm (o), and 3.64 cm (A). H = 0.28 mm, T = 1 x lo4 dyn/cm.
of3
- 0
2
6
8
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AIRWAY
35
’
30
’
81
REOPENING
25 20
FIG. 11. Nonlinear regression of experiments with H = 0.28 mm, T = 1 X lo4 dyn/cm (-); H = 0.28 mm, T = 5 X lo4 dyn/cm (. . .); H = 0.83 mm, T = 1 X lo4 dyn/cm (-.-); H = 0.83 mm, T = 5 x lo4 dyn/cm (- -).
15
“0
1
2
3
Dimensionless “Capilllory TABLE
Velocity, Number”
4
pU/y
2. Nonlinear regression data Coefficients Experiment Pvis
B
Pcap
R2
T = 1 X lo4 dyn/cm H = 0.28 mm
10.3
0.8
7.9
0.93
x lo4 dyn/cm H = 0.28 mm
9.6
0.80
5.3
0.89
x lo4 dyn/cm H = 0.83 mm
6.1
0.98
9.8
0.93
7.9
0.69
7.4
0.90
7.7
0.82
8.3
T = 5
T = 1
T = 5
x
lo4 dyn/cm
H = 0.83 mm Combined
data
5
magnitude is proportional to To K. K, the local curvature of the tube wall, is defined as the spatial derivative of the wall slope; K = (a/&) tan??(x). Although K may decrease with an increase in T (as does 0), increasing T apparently increases the product To K. By increasing T. K, the downstream pressure decreases and provides an additional driving force for meniscus motion. This driving force is identical to that which drives the meniscus motion in the tape-peeling model discussed at the outset of this paper. Predictions of Airway Opening Times
Nonlinear regression of experimental data of the form: R) = Pvis. Cal’ + Pcap. See text for details.
0.75 P*total/(y/
the principal viscous resistance (at the meniscus) is proportional to p U/R. Figure 11 also demonstrates that as wall tension increases(with film thickness held constant), airway opening pressure decreases. The interaction of wall tension with airway opening is complicated. As can be seen by Fig. 12, A and B, the wall angle is not fixed but depends on the capillary number. We measured airway opening angles during experiments performed with H fixed but variable capillary number and wall tension. Stopped frames of video recordings of the opening airway were analyzed to estimate 0, the airway opening angle. As airway tension increases, the airway opening angle decreases (Fig. 13). However, the fivefold increase in wall tension in these experiments does not lead to a fivefold decrease in sin 19,the component of tension normal to the tube axis (i.e., the component that would pull the tube open). Overall, the increase in T, although leading to a decrease in 0, decreasesthe required airway opening pressure by pulling the tube walls apart, thus decreasing the pressure needed to push the meniscus forward. An alternative, but related, explanation for the decrease in airway opening pressure with increased T stems from the resulting drop in downstream pressure, whose
We calculated the time to reopen a series of sequential airways using the nonlinear regression results of our cumulated experimental data (Table 2) dotall(y/R)
= Pvis CaH + Pcap = 7.7 Ca0*82+ 8.3 (R2 = 0.75)
(4)
With this regression formula, the capillary number for a given airway and airway pressure is dJ Ca -=-= Y
(5) [Pt*otal* R
8.3) 1’8’3
For a given tube of length L and radius R with lining fluid characteristics y and p specified, the opening velocity is U = rCa/p, and the opening time is simply topen = L/U. We calculated the time to open generations 8-14 assuming a Weibel lung geometry (21). Generation 8 was selected as the largest obstructed airway based on Martin and Proctor’s (13) speculation that airway closure might occur in Z-mm airways, since their relatively large compliance (compared with more proximal airways) and small radius might make them unstable. We investigated the opening time in four cases: I) a “normal lung,” where Y = 25 dyn/cm, the equilibrium surface tension of pulmonary surfactant, 7eq(7), and p = 0.01 gos-l cm-l, the
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82
AIRWAY
REOPENING
5.0 Velocity,
Dimensionless “Capillary
@J/y
Number”
FIG. 13. Airway opening angle (0) vs. dimensionless velocity for T = lo4 dyn/cm (0) and 5 X lo4 dyn/cm (m). R = 1.72 cm, H = 0.28 mm, lOW30 lining liquid.
00 2
3
4
5 Pressure
6 (cm
7
8
9
10
H20)
14. Predictions of airway opening times, generations 8-14 using Weibel lung geometry (21). y = 25 dyn/cm, p = 0.01 g.s-‘*cm-’ (0); y = 25 dyn/cm, p = 1.0 g.s-‘.cm-’ (o), y = 40 dyn/cm; a = 0.01 g.s-1. cm-’ (A); y = 10 dyn/cm, p = 0.01 g. s-l. cm-’ (A). FIG.
FIG. 12. Photographs of tube opening with Ca = 0.5,0 = 4” (A) and Ca = 4,@ = 9” (B). R = 1.72 cm, H = 0.28 mm, T = 1 x lo4 dyn/cm, lOW30 oil.
viscosity of water, which is assumed to be similar to that of serous secretions; 2) y = 25 dyn/cm and p = 1.0 g. S-la cm-l, an elevated viscosity due, for example, to mucous secretions; 3) y = 40 dyn/cm, a value larger than yeq because of either insufficient surfactant production or nonequilibrium conditions (e.g., rapidly increasing surface area as the airway opens), and p = 0.01 g-s-‘. cm-l; and 4) y = 10 dyn/cm, smaller than +req due to nonequilibrium conditions, and p = 0.01 g. s-l. cm-‘. The velocity of opening in each generation is assumed to be independent, and we include no serial interaction or inertia in these calculations. As such, we appreciate that the time to open derived in this manner must be considered speculative.
The results of this exercise (Fig. 14) demonstrate several interesting features. First, there is an apparent “yield pressure” that must be exceeded before opening can occur. This yield pressure is simply the capillary pressure, Pc*ap, determined by the radius and surface tension of the smallest airway. Even in the normal lung this pressure is relatively large, -5.8 cmHzO. Any applied airway pressure greater than Pc*ap is utilized to overcome the viscous stress associated with displacing the viscous lining fluid. Figure 14 shows that in a lung with ru = 0.01 g . s-l. cm-l, the opening time is very small once the airway pressure exceeds Pc*ap. This behavior gives the appearance of the airway “popping” open at a pressure slightly greater than the yield pressure. If mucous secretions exist, the viscosity of the lining may increase by several orders of magnitude (5), and the opening time increases dramatically. Once the opening time becomes a significant fraction of the inspiration time, hypoventilation of the affected respiratory pathway may conceivably occur. DISCUSSION
In this study we have investigated the influence of airway lining fluid surface tension (y) and viscosity (p)
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AIRWAY
REOPENING
on the reopening of closed airways. We found that both the surface tension and viscosity, as well as the airway radius, play important roles in determining the characteristics of airway opening. The effect of surface tension is evident by a capillary pressure, or apparent yield pressure, that must be applied before airway opening will begin. Once the applied airway pressure exceeds the capillary pressure, the mechanics of opening depend on the relative magnitudes of viscous and surface tension forces, as reflected in the dimensionless capillary number, Ca = ,UU/r (where U is the velocity of airway opening). Together with airway length, viscous and surface tension forces determine the time required to reopen airways, therein setting the ventilatory impact of airway closure. Our paradigm for airway closure by apposition of bronchial walls (compliant collapse, Ref. 9) is shown in Fig. 1. Does this type of airway collapse really happen in the lung? Previous studies (9-11) have emphasized the role of surface meniscus formation due to surface film instability in the closure of pulmonary airways, a phenomena termed film collapse by Kamm (9). These studies primarily were based on the concept that the airway lining was an unsupported film, except at its ends. If this were true, the film would create a catenoid, a configuration that becomes unstable when the length-diameter ratio exceeds 2-l” (1). Instability of the catenoid film might lead to a meniscus occlusion of the airway. However, Macklem, Proctor, and Hogg (12) found no evidence of catenoid formation in excised feline lungs, suggesting that airway instability cannot be understood solely by the mechanics of an unsupported film layer. Nonetheless, Kamm (9) has shown that even a supported film coating a rigid-walled tube may become unstable and obstruct the airway by meniscus formation. Once meniscus formation occurs, it seems likely that airway collapse will ensue. In his analysis of airway closure by film collapse, Macklem (11) estimated the Laplace pressure difference across the meniscal air-liquid interface by assuming a spherical shape with a radius of curvature equal to that of the open airway (R). This would lower the airway transmural pressure (Ptm) in the meniscus-filled region, and thus decrease the airway cross-sectional area (A) according to its A-Ptm characteristic. As the airway diameter decreases, the pressure in the meniscus fluid will become even more negative due to the decrease in the meniscus’ radius of curvature, leading to further airway collapse and eventual buckling into a flattened profile. Indeed, our studies show that the local radius of curvature may fall to as little as R/8, as the meniscus projects partially into the flattened airway. With airway flattening, the fluid meniscus elongates longitudinally, extending the tendency to airway collapse proximally and distally along the airways. Experimental evidence supporting the airway collapse hypothesis is provided by Hughes et al. (8), who demonstrated airway closure due to collapse of distal canine bronchi, and by Macklem et al. (12), who observed that inflating closed bronchioli required the peeling apart of opposing walls. Thus our airway closure paradigm seemsjustified. One of the key determinants of airway opening is the
83
surface tension of the lining fluid (Fig. 5). Surface tension forces create a capillary pressure that must be exceeded before airway opening can proceed. The zero-velocity intercept of our experiments suggests the possibility of a yield-pressure behavior that could explain the observations of Macklem, Proctor, and Hogg (12), who found that meniscus motion in excised canine bronchioli began only after a “critical pressure” was applied. Although this pressure has generally been thought to be equal to twice the surface tension divided by the open airway radius (II), 27/R, we found apparent yield pressures approximately four times greater, Pc*ap = 87/R (Fig. 11). We believe that airway wall flexibility explains this marked discrepancy. Apparently, the local airway collapse decreases the local meniscus radius of curvature to a value that is much smaller than the open airway’s radius, creating a commensurate rise in the capillary pressure. This smaller radius is evident in the acute opening angles observed in the vicinity of the meniscus (Figs. 12 and 13). It is interesting that the thickness of the lining fluid in the collapsed portion of airway (H) has little effect on the capillary pressure, presumably because fluid upstream of the meniscus is drawn toward the meniscus, increasing the local fluid thickness to a value that depends mostly on airway radius. It is important to note that the apparent yield-pressure phenomenon discussed above arises from dynamic experiments, where we measured the pressures during nearly steady-state conditions. It is possible that the true “start-up” behavior (i.e., the pressure-velocity relationship immediately after reopening commences) may differ from the behavior extrapolated from our quasi-steadystate dynamic studies. Conceivably, the apparent yield pressure may reflect the presence of an undiscovered large slope in a dimensionless P-U relationship that passes through the origin in the neighborhood of Ca = PW = 0. Still, an apparent yield pressure would exist for all practical purposes during airway reopening, since the capillary number would most likely fall within the range of our study. Second, during the transient start-up period the fluid near the meniscus may not have reached its equilibrium depth, so that both pressure components, dependent on meniscus fluid depth, would possibly have a greater dependence on the upstream film thickness, H, than is evident during steady-state reopening. When H is small ( 0.5) due to rapid airway opening, elevated lining viscosity, or small surface tension, then viscous forces add appreciably to the overall opening pressures. If Ca is small, the opening pressures are mostly determined by the surface tension of the liquid lining and are Pc*ap = 87/R. It was surprising to us that the shear stress (reflected in viscous pressure) scaled primarily inversely with airway radius, rather than inversely with lining fluid thickness. This finding again probably relates to accumulation of lining fluid in the neighborhood of the meniscus, where shear rates are greatest. The driving force for the accumulation is suction provided by the product of wall tension and local wall curvature, T. K. Our results suggest that when T is large, the fluid depth in the neighborhood of the meniscus is proportional to R. When T is small, the suction is not large enough to increase the fluid depth to a magnitude proportional to R, and H then plays a small role in determining the viscous shear stress. In addition to its role in setting the length scales for viscous and capillary stress, the drop in downstream pressure due to To K provides an additional driving force, above the applied pressure, for meniscus motion. It does this by decreasing the pressure below atmospheric pressure directly downstream of the meniscus. This mechanism is identical to that which exists in the tape-peeling model presented earlier. The increased driving force is evident by the decrease in applied pressure needed to open an airway at a given velocity when wall tension is increased (Fig. 11). In our model, axial wall tension was utilized to modify the apparent wall characteristics and may be thought of as influencing the tube’s compliance characteristics. Thus increasing the airway stiffness, for example, by increasing lung volume, may promote airway reopening. Likewise, pulmonary fibrosis may lead to smaller airway closing volumes. Macklem (11) has suggested that airway constriction may protect against airway closure by narrowing the lumen and increasing the wall thickness, hence increasing wall stiffness. However, decreasing luminal diameter will increase the film lining thickness, possibly contributing to its instability, as suggested by Frazer, Stengel, and Weber (4). Furthermore,
if constricted airways were to close, their opening pressures would probably be larger than their unconstricted opening pressures due to an increase in the capillary pressure, Pc*ap = Q/R. As described above, dimensional analysis shows that the scales for both viscous and surface tension forces increase inversely with airway radius, R. This result suggests that the smallest airways are the most difficult to reopen, although generational variation in lining fluid secretion physical properties might partially offset this radial dependency. Using the results of our study, we predicted airway reopening times for a series of sequential airways (generations 8-14) to evaluate how airway reopening times may depend on both surface tension and viscosity (assumed constant along the airways). Figure 14 shows that surface tension defines the initial opening pressure but that the time to opening is determined by the viscosity of the lining fluid. If the fluid is relatively inviscid, the airway appears to pop open. However, if this fluid is viscous, as may happen in mucous plugging or cystic fibrosis, the opening times may be large and prevent airway opening during a normal tidal breath. We are appreciative of the technical assistance provided by Linda E. Alger and Robert Monk. This research was supported by National Heart, Lung, and Blood Institute Grants HL-02205, HL-07912, HL-41009, and HL-01857 and by the Whitaker Foundation. Present address and address for reprint requests: D. P. Gaver III, Dept. of Biomedical Engineering, Tulane University, New Orleans, LA 70118. Received
7 August
1989; accepted
in final
form
16 February
1990.
REFERENCES 1. BOYS, C. V. Soap Bubbles, Their Colours and the Forces Which Mould Them. New York: Dover, 1959. 2. CRAWFORD, A. B. H., D. J. COTTON, M. PAIVA, AND L. A. ENGEL. Effect of airway closure on ventilation distribution. J. Appl. Physiol. 66: 2511-2515, 1989. 3. ENGEL, L. A., A. GRASSINO, AND N. R. ANTHONISEN. Demonstration of airway closure in man. J. Appl. Physiol. 38: 1117-1125, 1975. 4. FRAZER, D. G., P. W. STENGEL, AND K. C. WEBER. Meniscus formation in airways of excised rat lungs. Respir. Physiol. 36: 121129, 1979. 5. FUNG, Y. C. Biomechanics. New York: Springer-Verlag, 1981. 6. GREAVES, I. A., J. HILDEBRANDT, AND F. G. HOPPIN, JR. Micromechanics of the lung. In: Handbook of Physiology. The Respiratory System. Mechanics of Breathing. Bethesda, MD: Am. Physiol. Sot., 1986, sect. 3, vol. III, pt. 1, chapt. 14, p. 195-216. 7. HILLS, B. A. The Biology of Surfactant. Cambridge, UK: Cambridge Univ. Press, 1988. 8. HUGHES, J. M. B., D. Y. ROSENZWEIG, AND P. B. KIVITZ. Site of airway closure in excised dog lungs: histologic demonstration. J. Appl. Physiol. 29: 340-344, 1970. 9. KAMM, R. D., AND R. C. SCHROTER. Is airway closure caused by liquid film instability? Respir. Physiol. 75: 141-156, 1989. 10. LEBLANC, P., F. RUFF, AND J. MILIC-EMILI. Effects of age and body position on “airway closure” in man. J. Appl. Physiol. 28: 448-451, 1970. 11. MACKLEM, P. T. Airway obstruction and collateral ventilation. Physiol. Rev. 51: 368-436, 1971. 12. MACKLEM, P. T., D. F. PROCTOR, AND J. C. HOGG. The stability of peripheral airways. Respir. Physiol. 8: 191-2010, 1970. 13. MARTIN, H. B., AND D. F. PROCTOR. Pressure-volume measurements on dog bronchi. J. Appl. Physiol. 13: 337-343, 1958. 14. MCEWAN, A. D., AND G. I. TAYLOR. The peeling of a flexible strip attached by a viscous adhesive. J. FZuid Mech. 26: 1-15, 1966.
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AIRWAY 15. MILIC-EMILI, J., J. A. M. HENDERSON, M. B. DOLOVICH, D. TROP, AND K. KENEKO. Regional distribution of inspired gas in the lung. J. Appl. Physiol. 21: 749-759, 1966. 16. PARK, C.-W., AND G. M. HOMSY. Two-phase displacement in Hele Shaw cells: theory. J. Fluid Mech. 139: 291-308, 1984. 17. REINELT, D. A. Interface conditions for two-phase displacement in Hele-Shaw cells. J. FZuid Mech. 183: 219-234, 1987. 18. REINELT, D. A., AND P. G. SAFFMAN. The penetration of a finger into a viscous fluid in a channel and tube. SIAM J. Sci. Stat. Comput. 6: 542-561, 1985.
REOPENING
85
19. SAFFMAN, P. G., AND G. I. TAYLOR. The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. R. Sot. Lond. A Math. Phys. Sci. 245: 312-329, 1958. 20. SUTHERLAND, P. W., T. KATSURA, AND J. MILIC-EMILI. Previous volume history of the lung and regional distribution of gas. J. Appl. Physiol. 25: 566-574, 1968. 21. WEIBEL, E. R. Morphometry of the Human Lung. New York: Academic, 1963. 22. YAGER, D., J. P. BUTLER, J. BASTACKY, E. ISRAEL, G. SMITH, AND J. M. DRAZEN. Amplification of airway constriction due to liquid filling of airway interstices. J. Appl. Physiol. 66: 2873-2884, 1989.
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P. GAVER
III,
and viscosity
RICHARD
W. SAMSEL,
AND
JULIAN
SOLWAY
Section of Pulmonary and Critical Care Medicine, The University of Chicago, Chicago, Illinois 60637
GAVER,DONALD P.,III, RICHARD W. SAMSEL,ANDJULIAN SOLWAY. Effects of surface tension and viscosity on airway reopening. J. Appl. Physiol. 69(l): 74-85, 1990.-We studied airway opening in a benchtop model intended to mimic bronchial walls held in apposition by airway lining fluid. We measured the relationship between the airway opening velocity (U) and the applied airway opening pressure in thin-walled polyethylene tubes of different radii (R) using lining fluids of different surface tensions (y) and viscosities (p). Axial wall tension (T) was applied to modify the apparent wall compliance characteristics, and the lining fluid film thickness (H) was varied. Increasing p or y or decreasing R or T led to an increase in the airway opening pressures. The effect of 23 depended on T: when T was small, opening pressures increased slightly as H was decreased; when T was large, opening pressure was independent of H. Using dimensional analysis, we found that the relative importance of viscous and surface tension forces depends on the capillary number (Ca = &J/y). When Ca is small, the opening pressure is -8r/R and acts as an apparent “yield pressure” that must be exceeded before airway opening can begin. When Ca is large (Ca > 0.5), viscous forces add appreciably to the overall opening pressures. Based on these results, predictions of airway opening times suggest that airway closure can persist through a considerable portion of inspiration when lining fluid viscosity or surface tension are elevated. airway closure; regional
ventilation;
airflow
obstruction
CLOSURE occurs at low lung volumes in normal adults and may exist at larger lung volumes in patients with airflow obstruction (3, 10, 15, 20). Airway closure can lead to local hypoventilation and impaired gas exchange and thus may be physiologically important. Its effect on gas transport is determined by the period of time the airway remains closed. For example, if an airway closes at end expiration but reopens at the onset of inspiration, the influence on gas transport may be minimal. However, if the time course of reopening is long, so that the airway remains closed for a significant fraction of inspiration, hypoventilation of the affected respiratory pathway may be substantial. Airways may close in two ways: 1) meniscus formation, where a film of airway lining fluid obstructs the lumen of an otherwise air-filled bronchus [recently described by Kamm and Schroter (9) as “film collapse”]; and 2) airway collapse (termed “compliant collapse” in Ref. 9), where the bronchial walls collapse and are held in apposition by the adhesive properties of the airway lining fluid (6, 9, 11). Evidence in favor of the latter mechanism is provided by Hughes, Rosenzweig, and Kivitz (8), whose
AIRWAY
74
0161-7567/90 $1.50 Copyright
0
photomicrographs of an isolated canine lung clearly demonstrate the collapse of distal bronchi. Macklem (11) speculated that airway collapse might exist as a result of pressures induced by a film meniscus. Recently, Yager and colleagues (22) have suggested that fluid within airway interstices may amplify bronchoconstriction by precisely this mechanism. We have studied the reopening of collapsed airways in a benchtop airway model (Fig. 1). Our approach was to measure the relationship between the velocity of airway opening and the applied transmural pressure, one of the key determinants of the time scale for airway reopening. By using several lining fluids and tubes, we investigated how the physical characteristics of the lining fluid and of the airway itself influence the opening pressure-velocity relationship. Our results provide a conceptual framework by which airway reopening may be understood. THEORY
Opening of a closed pulmonary airway was analyzed using a benchtop model (Fig. 1) that was based on the description of a closed airway presented in Greaves, Hildebrandt, and Hoppin (6). The open region of a cylindrical airway closes into a ribbonlike region where the airway lining fluid fills the lumen between the two flexible walls and acts as a viscous adhesive. The open portion of the tube has radius R, and the film layer in the closed section has thickness H. The walls are assumed to be completely flaccid until the airway is fully inflated and axial tension is applied to modify the apparent wall mechanical characteristics (i.e., cross-sectional area-transmural pressure relationship). When positive pressure is applied to the gas phase, the air-liquid interface (meniscus) progresses down the tube and opens the airway. This construct is similar to that described by Macklem, Proctor, and Hogg (12), who found that when inflating a bronchiole, they “were gradually peeling apart the opposing walls, and the liquid remained in situ, presumably lining that part of the bronchiole that had opened.” As will be shown, the surface tension of the airliquid interface, viscosity and thickness of the lining fluid, radius of the tube, and axial wall tension all influence the velocity (U) of airway opening for a given applied transmural pressure. This model for the opening of a closed pulmonary airway contains features of fluid mechanics problems that have not heretofore been applied toward the understanding of physiological phenomena: two-phase dis-
1990 the American
Physiological
Society
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AIRWAY
75
REOPENING
Virc;;uadLining \I
6 1
Tension Film Thickness
m
(H) FIG. 1. Airway opening model. Flexible-walled tube has a radius R, upstream film thickness H, and imposed axial wall tension T. Positive pressure is applied to gas, opening tube at velocity U resulting in an opening angle 0.
/
/
/
’
/’
/’
,//
/
,, ’
/
/I ’ /
,/’ I
,’
,.’
I
, ,’
/
,/’
, ,/’
,, ’
/
,’
,/
/
I
T
)U
R
FIG. 3. Tape-peeling model. A flexible strip (tape) is held to a rigid surface by a viscous adhesive. Tape peels at velocity U by applying a tension T at an angle 0, while pressure in gas phase remains at atmospheric pressure. Lining fluid has a downstream depth H, viscosity p, and surface tension y. Local wall curvature creates pressure gradients in lining fluid, inducing flow that allows tape to peel.
placement in a Hele-Shaw cell and the peeling of a flexible strip attached by a viscous adhesive. A brief description of these phenomena, and their relation to airway opening, is presented below.
peels, the gasphase remains at atmospheric pressure and the meniscus is sucked forward by the subatmospheric pressure in the liquid phase, induced by the local curvature of the flexible wall. The pressure caused by wall curvature also drives fluid upstream (toward the meniscus) and modifies the film thickness in the vicinity of the meniscus. Still, this model does not completely describe the wall motion in airway opening, as 8 is imposed in the tape-peeling model, but is part of the solution in the airway opening model.
Hele-Shaw Flow
Airway Opening
’
/
/
/’ ,’ ’ ,,’ I’ ,’ / r’/ /
,I’
/
,/
,’
/
, /
,/’ ‘/,,’,,
FIG. 2. Hele-Shaw model. Positive pressure drives fluid from between 2 rigid plates separated fluid has viscosity p and surface tension y.
/’ ’ I /’
//
/”
,,I
/
is applied to gas and by distance H. Lining
Two-phase displacement in a Hele-Shaw cell (16-18) is the flow induced when an inviscid fluid (e.g., air) displaces a more viscous fluid (e.g., mucus) from the region between two narrowly spaced plates, as shown in Fig. 2. The pressure P needed to push the air at velocity U depends on capillary and viscous contributions. The capillary pressure is determined by the surface tension and geometry of the air-liquid interface. For example, the capillary pressure of a spherical bubble with radius R is, by the law of Laplace, Pcap = 27/R, where y is the surface tension. The viscous stress is proportional to the strain rate in the lining fluid, where the constant of proportionality is I-C,the fluid viscosity. Saffman and Taylor (19) showed that a single dimensionless parameter (“capillary number,” Ca = pU/y) represents the relative magnitudes of the viscous and capillary pressures in the Hele-Shaw cell. Solutions of Hele-Shaw cell fl~~ are complicated by the fact that the air-liquid interface is a free boundary and is thus part of the solution as well as a boundary condition. Adhesive Peeling The primary limitation of comparing two-phase displacement in a Hele-Shaw cell with airway opening is that it does not account for the airway’s flexible walls. A model that describes the wall behavior is that of McEwan and Taylor (14), who investigated the peeling of a flexible strip (tape) attached to a rigid wall by a viscous adhesive. As shown in Fig. 3, the tape is pulled with tension T at an angle 8. The lining fluid has viscosity p, surface tension y, and thickness H far upstream of the moving meniscus. This problem has two free boundaries: the flexible wall and the air-liquid interface. As the tape
Although a complete mathematical model of airway opening is beyond the scope of this paper, a brief discussion is warranted because it will create the framework by which our airway opening experiments will be analyzed. When a finger of air displaces the liquid between narrowly spaced flexible sheets, an air-liquid meniscus interface is created, as shown in Fig. 1. As described above, the pressure needed to displace the more viscous fluid by the meniscus has two components tiotal
= Pc*ap + P&s
(1)
where Pzap is the capillary pressure associated with the surface tension at the air-liquid interface and P&s is the contribution to the total pressure due to the viscous stress that occurs when the viscous fluid is displaced by the air. The superscript asterisk denotes a dimensional pressure, here and throughout the remainder of this paper. In Eq. 1, we have assumed that Pc*apand P&s are independent. Clearly this is not always true, since the viscous flow may modulate the meniscus shape, altering the capillary pressure and leading to a nonlinear interaction between surface tension and viscous forces. As can be seen from Fig. 1, two obvious length scales exist in this problem: R, the radius of the open tube, and H, the gap between the closed tube walls. To identify the dominant balances of this problem, we will reduce Eq. I by the following scaling PZap -- c Pcap
(2a)
C
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76
AIRWAY
REOPENING PNEUMOTACHOGRAPH I ;Totol
and P&S
.
=P
u 7 Pvis
w
where Pcap and Pvis are the dimensionless capillary and viscous pressures and L, and L, are their associated length scales. The capillary pressure scaling assumes that the magnitude of the capillary pressure is of the order of r/L,, which is the pressure needed to maintain a static meniscus with a radius of curvature equal to L,. Additionally, we assume that the magnitude of the viscous pressure is approximately pU/L,, where UIL, is a measure of the strain rate in the neighborhood of the meniscus. By use of this scaling, the dimensionless total pressure, Ptotal, is defined as Ptotal
Hotal = Y/L
f/q Lc c Pvis = Pcap + - I Y 1 v
(3)
The dimensionless quantity @/r is the capillary number, Ca, and can be considered as a dimensionless velocity that reflects the relative magnitudes of the surface tension and viscous forces in this problem. When Ca is small, surface forces dominate and the pressure needed to move the meniscus at velocity U is determined by the pressure jump across the interface. As Ca increases, the viscosity of the lining fluid begins to retard the motion of the meniscus. Appropriate selection of L, and L, should allow our dimensional pressure-velocity data for different experiments (changing R, H, y, and p) to be represented by Eq. 3, thus collapsing a large amount of data to a simple relationship. Assumptions Several assumptions have been made in the development of this model. First, we have used a model of a closed airway where the lumen closes into a ribbonlike shape, as proposed by Greaves, Hildebrandt, and Hoppin (6) and as suggested by the photomicrographs of closed canine bronchi taken by Hughes, Rosenzweig, and Kivitz (8). We have also assumed that the viscosity and surface tension in the airway are each constant, although we acknowledge that pulmonary airway lining fluid is probably not Newtonian and may have viscoelastic properties (5). Furthermore, as the airway opens, the air-liquid interface expands, possibly leading to variation of the surface tension. Additionally, the interaction of individual airways is not considered and may be important in determining the opening velocity of sequential generations of pulmonary airways. These assumptions may limit the quantitative estimation of pulmonary airway reopening from results gathered here, but the qualitative insights from our analysis should be pertinent to the lung. METHODS
Benchtop Experiments Measurements of the pressure-velocity relationship during the opening of a closed flaccid tube were conducted using the apparatus shown schematically in Fig. 4. The polyethylene tube lies on a horizontal stainless
(0) POLYETHYLENE
TUBING
H‘IGH IMPEDANCE
(FOR TENSION)
FLOW SOURCE
4. Schematic of experimental apparatus. Polyethylene tubing lies on a stainless steel table, with tension applied by a weight W. Tube opens by pressure applied to gas, measured upstream as tube opens with velocity U. FIG.
TABLE
1. Physical properties of lining fluids Lining
Fluid
Polyethylene glycol lOW30 motor oil 85W gear oil
Viscosity
(p),
g.s-‘.cm-1 1.1
1.5 9.9
Surface
Tension w, dyn/cm
42.4 24.7 25.0
steel table, with its open end connected to a high-impedance flow source, used to blow the air-liquid meniscus down the tube and reopen the closed airway. The opposite end of the tube was closed permanently, with plastic struts placed so as to retain a ribbonlike configuration. A weight was supported from this end to provide the tube with a known axial wall tension. The low-friction stainless steel table surface allowed the tube to slide, helping to maintain constant wall tension during the duration of each experiment. Various fluids were used to provide the viscous adhesion that closes the tube. By changing lining fluids we investigated how fluid properties (y, p) influence the tube opening behavior. Additionally, we modified the fluid film thickness (H), tube radius (R), and axial wall-tension (T) to determine their roles in tube opening. We used three different lining fluids to investigate how surface tension, y, and viscosity, p, influence the pressures needed to reopen the closed airways: lOW30 motor oil, 85W gear oil, and polyethylene glycol (see Table 1). Viscosity was measured with a bulb viscometer (CannonFenske-Ostwald type), and surface tension was measured with a Wilhelmy balance. These fluids were selected because of their large range of viscosity and surface tension and because they wetted our polyethylene tubing. To study the tube radius dependency of airway opening, we conducted our experiments on thin-walled (0.002 in.) polyethylene tubes with radii of 0.54, 1.72, and 3.64 cm. The film-thickness dependency of airway opening was studied with two film thicknesses, H = 0.28 and 0.83 mm. Axial wall tension, T, was applied to modify the apparent airway wall mechanical characteristics. As shown schematically in Fig. 4, the weight, W, applied tension to the tube wall. Since the force was distributed along the circumference of the tube wall, the tension had a magnitude of T = W/27& Weights were adjusted between experiments with tubes of different radii so that isopleths of tension could be investigated. Two tensions were studied, T = 1 x lo4 and 5 X lo4 dyn/cm. Each experiment was begun by setting a uniform film thickness (H) with a fixed gap gauge. The upstream
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AIRWAY
REOPENING
valve was opened, increasing the pressure in the gas phase and initiating tube opening. The meniscus velocity was determined from the time it took the meniscus to travel 40 cm (20 cm for the smallest tube, R = 0.54 cm); the pressure was continuously sampled (30 Hz) and digitally recorded during the period the meniscus traveled this distance. The tubing was -1.5 m long for the tubes of R = 3.6 and 1.7 cm and 1 m long for the tube of R = 0.54 cm. We began recording the pressure signal when the meniscus passed a point 60 cm from the upstream rigid attachment for the 1.5-meter tubes and 40 cm from the attachment for the l-m tube. Thus the measurements were made at a distance of more than 14 tube radii from the attachments, so end effects were assumed to be small. We continuously monitored the flow rate with a pneumotachograph to ensure that the flow rate was constant. In each individual experiment, the recorded pressure signal was averaged creating one pressure-velocity (P- U) data point. Nearly 500 experiments were conducted for this study. By linear regression of the pressure and flow signals of 35 randomly selected experiments, we found that the flow was nearly constant, with a variation from the mean of -0.03 t 0.2%. However, the total pressure consistently decreased during the course of an experiment, from a magnitude 5.5 t 4.4% greater to 5.5 & 4.4% less than the average pressure. Even though this result demonstrates that our experiments do not reflect strictly steady-state reopening conditions, we believe that the variation is relatively small, and our results can thus be used to assess quasi-steady reopening phenomena (see DISCUSSION). Because of gravity, the weight of the liquid lining fluid determines the minimum pressure (-pgH/Z) in the upstream (open) end of the tube, where p is the fluid density and g is the gravitational constant. As will be shown, the capillary pressure is -87/R; thus the gravitationally induced component of the applied pressure is ~13% of the capillary pressure for our smallest tube (0.54 cm). For the larger-tube experiments (3.64 cm), the gravitationally induced component of the pressure is equivalent to the capillary pressure. In this way, gravity simply determines the minimum reopening velocity that we can investigate.
77
RESULTS
Dimensional
Representation
Figure 5 shows the pressure-velocity relationship for a tube of radius R = 1.72 cm with fixed tension T = 1 x lo4 dyn/cm, film thickness H = 0.28 mm, and three lining fluids. The opening pressures for 85W gear oil were clearly larger than for either polyethylene glycol and lOW30 oil, implicating viscosity as a critical determinant of the opening pressure. Also, the pressures needed to open the polyethylene glycol-coated tube are uniformly larger than those needed to open the tube coated with lOW30 oil. Since the surface tension of polyethylene glycol is greater than that of lOW30 oil, but their viscosities are nearly identical, this result shows that opening pressures are determined partially by surface forces. Figure 6 illustrates the tube radius dependency of airway opening pressures in experiments conducted with lOW30 oil, T = 1 x lo4 dyn/cm, H = 0.28 mm, and in tubes with radii of R = 0.54, 1.72, and 3.64 cm. It is evident that the opening pressure increased as the tube radius decreased. We demonstrate the influence of wall tension on airway reopening pressures in Fig. 7, A and B, with data from experiments conducted on a tube of radius R = 1.72 cm coated with lOW30 motor oil of thicknesses H = 0.28 and 0.83 cm. Figure 7A demonstrates that when wall tension is small, T = 1 X lo4 dyn/cm, decreasing the film thickness leads to an increase in the airway reopening pressure. In contrast, Fig. 7B shows that when T is large, T = 5 x lo4 dyn/cm, reopening pressures are largely independent of film thickness. Figure 8 shows the dependency of tube reopening pressures on the imposed axial wall tension. These data are from experiments conducted in lOW30 motor oilcoated tubes of radius R = 1.72 cm, film thickness H = 0.28 mm, and wall tensions T = 1 X lo4 and 5 X lo4 dyn/ cm. Figure 8 shows that the opening pressure decreased as the wall tension increased. Dimensionless
Representation
In this section, the dimensional data shown above are scaled in the manner discussed in the section, THEORY.
0.7 n r,U
0.6
!
u
869
c-4 0.5
I
-
O8
O@ E
0
0.4
F 3 cn $
03.
8
80
-
eP
0.2 -
00
43
#&AA
@
L n1_ Ae 0.0
FIG. 5. Dimensional pressure-velocity relationship for tube opening with 3 lining fluids: 85W oil (0), lOW30 oil (o), and polyethylene glycol (A). R = 1.72 cm, H = 0.83 mm,T= 1 X lo4 dyn/cm.
@ alGil
’
I
I
0
5
IO
Velocity
g
15
(cm/s)
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AIRWAY
REOPENING
0
0
Q0
FIG. 6. Dimensional pressure-velocity relationship for opening of tubes with 3 different radii: R = 0.54 cm (0), 1.72 cm (o), and 3.64 cm (A). H = 0.28 mm, T = 1 X lo4 dyn/cm, and lOW30 lining fluid.
0
5
Velocity
10
15
(cm/s)
Here we will systematically investigate these data so as to elucidate the appropriate capillary and viscous length scales, L, and L,, thus providing insight into the physics of airway opening. Since L, and L, are likely to depend on the tube radius (R) or film thickness (H), for a given tube (i.e., R fixed) with a given film thickness (i.e., H fixed), the ratio L,/ L, is fixed. A single relationship (shown in Eq. 3) should then hold true for different fluids if we have indeed selected the correct physical mechanisms as defining the airway opening pressures. Figure 9 represents the data of Fig. 5 in dimensionless form, where we have arbitrarily chosen L, = L, = R. In dimensionless form, these data appear to fall on a single curve, suggesting that opening phenomena are dominated by a balance of surface tension and viscous forces. Additionally, Fig. 9 shows that the capillary number, Ca = pU/r, is the parameter that correctly represents this balance for a given ratio of LIL,. Having validated the capillary number dependency for fixed L,IL,, we can now use Eq. 3 to determine the appropriate length scales L, and L, by investigating opening pressures for groups of experiments where H and R were not held constant. By plotting the data in dimensionless form using each of the four possible combinations of length scales, we shall determine the appropriate combination (i.e., the combination that yields a single pressure-relationship for all data). The four combinations of length scales and corresponding dimensionless relationships from Eq. 3 are Case1: L, = R,
L, = H: P/(7/R)
vs. (pU/$*(R/H)
Case2: L, = R:
L, = R: P/(y/R)
vs. (pU/$
Case3: L, = H,
Lv = HI P/(7/H)
vs. (pU/r)
Case4: L, = H,
L, = R: P/(7/H)
vs. &U/r)*
(H/R)
Figure 10 shows tube-opening data from experiments using three tubes and three fluids, with H = 0.28 mm and T = 1 x lo4 dyn/cm. In each case, the data from each tube lie on one curve, confirming the capillary number dependency for fixed H/R. However, only the
dimensionless representation of case 2 exhibits a single relationship for data of different tube radii, suggesting that the dominant length scales for airway opening are L c = R and L, = R. The selection of R as L,, the capillary length scale, is interesting physically, since it implies that the leading order meniscus curvature depends on the tube radius, R, and not on the film thickness, H. This result suggests that the air-liquid meniscus does not project into the closed portion of the airway, because the radius of curvature would then be -H/2. Furthermore, L, = R means only that the meniscus curvature is proportional to the airway radius R. Our experiments show that the zero capillary number int$rcept actually occurs at a dimensionless pressure of Ptotal/(y/R) = 8. This result implies that the onset of airway opening occurs at a pressure that is about eight times larger than the pressure based on a meniscus curvature equal to the airway tube radius. The flexible wall evidently allows the meniscus to have a radius of curvature eight times smaller than R. The selection of the viscous length scale, L, = R, suggests that the strain rate is proportional to U/R; hence the viscous stress scales as pU/R. We found this surprising, since the stress in the open portion of the tube is clearly not an important component of the overall viscous stress. We suspect that the viscous stress is determined primarily by the amount of fluid directly downstream of the meniscus (toward the closed end), where shear stresses are largest. The opening of the tube causes pressure gradients in the film layer that suck the fluid upstream, so that it wells up in the vicinity of the meniscus (see below). These mechanisms are similar to those discussed in McEwan and Taylor’s tape-peeling model (14). If the steady-state film thickness in front of the meniscus is proportional to the tube radius R, this could explain our finding that L, = R. Figure 11 shows nonlinear regressions of data from all experiments, where we have used the length scales L, = L = R in the dimensionless representation. The data have been regressed to the form Pt*btall(ylR) = Pvis CaR + Pcap, by analogy with the theory of two-phase displacement flow in a rigid-walled Hele-Shaw cell (16).
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AIRWAY
A
79
REOPENING
0.7
s
T = 1xl O4 dynes/cm
0.6
c\l 1
0.5
E 0
0.4
t!?
0.3
z ;
0.2
k
f30 0 O
0
0 80
0.1 0.0
0
Velocity
B
FIG. 7. Dimensional pressure-velocity relationships for reopening of tubes coated with 85W oil of film thicknesses: H = 0,28 mm (0) and 0.83 mm (0). R = 1.72 cm, T = 1 x lo4 dyn/cm (A) and 5 X lo4 dyn/cm (B).
(cm/s)
0.7
T = 5x1 O4 dynes/cm
0.6 EY cv I
0.5
E 0
0.4
aJ L
0.3
3 v, cn 2
O**
cl-
15
10
5
0.1
0.0 0
I
I
5
10
Velocity
15
(cm/s)
0.7 n 0.6 0 CY 1 0.5 E 0
0.4
P 3 m m i!!
0.3
n-
FIG. 8. Dimensional pressure-velocity relationship for opening of tubes with T = 1 X IO4 dyn/cm (0) and 5 X lo4 dyn/cm (e). R = 1.72 cm, H = 0.28 mm, and 85W oil lining fluid.
0.2
0.1 0.0 0
10
5
Velocity
15
(cm/s)
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80
AIRWAY 40
REOPENING
r
FIG. 9. Dimensionless pressure-velocity relationship for opening of tubes lined with 85W oil (0), lOW30 oil (o), and polyethylene glycol (A). R = 1.72 cm, H = 0.83 mm,T= 1 x lo4 dyn/cm.
-
5
Dimensionless “Capilllary
Velocity, Number”
@J/y
The capillary number dependence (0.61 5 B 5 0.98) corresponds to the derived 0.67 power dependence of the pressure on capillary number in Hele-Shaw flow (see Table 2). This nonlinear dependence on the capillary number differs from the linear dependence shown in Eq. 3, because the assumptions made in formulating Eq. 1 did not include the inherent nonlinear interaction of the capillary and viscous forces, as discussed above. Isopleths of T = 5 x lo4 dyn/cm (Fig. 11) demonstrate that H has very little effect on the opening pressure when the axial wall tension is large. This result supports our previous conclusion that the leading order viscous length scale is based on the tube radius, not the film thickness. In contrast, isopleths of T = 1 x lo4 dyn/cm show that the opening pressure increases with a decrease in film thickness. Thus, although R is the dominant length scale, H appears to play an additional minor role in determinR = 0.54
cm (o), H = 0.28
60
Case
1 :
L, = R,
1.72 mm,
ing the pressure necessary to overcome the viscous stress. Two explanations may account for this H dependency. First, we have observed that flows are induced by the meniscus and airway wall motion as the meniscus progresses down the tube (data not shown). These flows downstream of the meniscus are contained between the flexible walls and have a viscous length scale of H, which may add a small contribution to the overall viscous stress. Second, when wall tension is small, the flows induced by separation of the flexible walls may not be sufficient to increase the film thickness at the meniscus to a depth proportional to R. Note that the driving force for the film flow toward the meniscus is a negative “suction” pressure proportional to To K, where K is the local wall curvature. Therefore, when T is small, the driving force may not be large enough to increase the film thickness in the neighborhood of the meniscus to an extent that
cm (@), 3.64 cm (A) T = 1 x lo4
dynes/cm
L, = H
A AA
OL 0
200
400 NJ/Y)
600
800
0
2
6
8
* w/‘-o =H,
L,=R
0
FIG. 10. Determination of capillary length scale (L,) and viscous length scale (&) using Eq. 3. Experiments were conducted with tubes of radii R = 0.54 cm (0), 1.72 cm (o), and 3.64 cm (A). H = 0.28 mm, T = 1 x lo4 dyn/cm.
of3
- 0
2
6
8
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AIRWAY
35
’
30
’
81
REOPENING
25 20
FIG. 11. Nonlinear regression of experiments with H = 0.28 mm, T = 1 X lo4 dyn/cm (-); H = 0.28 mm, T = 5 X lo4 dyn/cm (. . .); H = 0.83 mm, T = 1 X lo4 dyn/cm (-.-); H = 0.83 mm, T = 5 x lo4 dyn/cm (- -).
15
“0
1
2
3
Dimensionless “Capilllory TABLE
Velocity, Number”
4
pU/y
2. Nonlinear regression data Coefficients Experiment Pvis
B
Pcap
R2
T = 1 X lo4 dyn/cm H = 0.28 mm
10.3
0.8
7.9
0.93
x lo4 dyn/cm H = 0.28 mm
9.6
0.80
5.3
0.89
x lo4 dyn/cm H = 0.83 mm
6.1
0.98
9.8
0.93
7.9
0.69
7.4
0.90
7.7
0.82
8.3
T = 5
T = 1
T = 5
x
lo4 dyn/cm
H = 0.83 mm Combined
data
5
magnitude is proportional to To K. K, the local curvature of the tube wall, is defined as the spatial derivative of the wall slope; K = (a/&) tan??(x). Although K may decrease with an increase in T (as does 0), increasing T apparently increases the product To K. By increasing T. K, the downstream pressure decreases and provides an additional driving force for meniscus motion. This driving force is identical to that which drives the meniscus motion in the tape-peeling model discussed at the outset of this paper. Predictions of Airway Opening Times
Nonlinear regression of experimental data of the form: R) = Pvis. Cal’ + Pcap. See text for details.
0.75 P*total/(y/
the principal viscous resistance (at the meniscus) is proportional to p U/R. Figure 11 also demonstrates that as wall tension increases(with film thickness held constant), airway opening pressure decreases. The interaction of wall tension with airway opening is complicated. As can be seen by Fig. 12, A and B, the wall angle is not fixed but depends on the capillary number. We measured airway opening angles during experiments performed with H fixed but variable capillary number and wall tension. Stopped frames of video recordings of the opening airway were analyzed to estimate 0, the airway opening angle. As airway tension increases, the airway opening angle decreases (Fig. 13). However, the fivefold increase in wall tension in these experiments does not lead to a fivefold decrease in sin 19,the component of tension normal to the tube axis (i.e., the component that would pull the tube open). Overall, the increase in T, although leading to a decrease in 0, decreasesthe required airway opening pressure by pulling the tube walls apart, thus decreasing the pressure needed to push the meniscus forward. An alternative, but related, explanation for the decrease in airway opening pressure with increased T stems from the resulting drop in downstream pressure, whose
We calculated the time to reopen a series of sequential airways using the nonlinear regression results of our cumulated experimental data (Table 2) dotall(y/R)
= Pvis CaH + Pcap = 7.7 Ca0*82+ 8.3 (R2 = 0.75)
(4)
With this regression formula, the capillary number for a given airway and airway pressure is dJ Ca -=-= Y
(5) [Pt*otal* R
8.3) 1’8’3
For a given tube of length L and radius R with lining fluid characteristics y and p specified, the opening velocity is U = rCa/p, and the opening time is simply topen = L/U. We calculated the time to open generations 8-14 assuming a Weibel lung geometry (21). Generation 8 was selected as the largest obstructed airway based on Martin and Proctor’s (13) speculation that airway closure might occur in Z-mm airways, since their relatively large compliance (compared with more proximal airways) and small radius might make them unstable. We investigated the opening time in four cases: I) a “normal lung,” where Y = 25 dyn/cm, the equilibrium surface tension of pulmonary surfactant, 7eq(7), and p = 0.01 gos-l cm-l, the
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82
AIRWAY
REOPENING
5.0 Velocity,
Dimensionless “Capillary
@J/y
Number”
FIG. 13. Airway opening angle (0) vs. dimensionless velocity for T = lo4 dyn/cm (0) and 5 X lo4 dyn/cm (m). R = 1.72 cm, H = 0.28 mm, lOW30 lining liquid.
00 2
3
4
5 Pressure
6 (cm
7
8
9
10
H20)
14. Predictions of airway opening times, generations 8-14 using Weibel lung geometry (21). y = 25 dyn/cm, p = 0.01 g.s-‘*cm-’ (0); y = 25 dyn/cm, p = 1.0 g.s-‘.cm-’ (o), y = 40 dyn/cm; a = 0.01 g.s-1. cm-’ (A); y = 10 dyn/cm, p = 0.01 g. s-l. cm-’ (A). FIG.
FIG. 12. Photographs of tube opening with Ca = 0.5,0 = 4” (A) and Ca = 4,@ = 9” (B). R = 1.72 cm, H = 0.28 mm, T = 1 x lo4 dyn/cm, lOW30 oil.
viscosity of water, which is assumed to be similar to that of serous secretions; 2) y = 25 dyn/cm and p = 1.0 g. S-la cm-l, an elevated viscosity due, for example, to mucous secretions; 3) y = 40 dyn/cm, a value larger than yeq because of either insufficient surfactant production or nonequilibrium conditions (e.g., rapidly increasing surface area as the airway opens), and p = 0.01 g-s-‘. cm-l; and 4) y = 10 dyn/cm, smaller than +req due to nonequilibrium conditions, and p = 0.01 g. s-l. cm-‘. The velocity of opening in each generation is assumed to be independent, and we include no serial interaction or inertia in these calculations. As such, we appreciate that the time to open derived in this manner must be considered speculative.
The results of this exercise (Fig. 14) demonstrate several interesting features. First, there is an apparent “yield pressure” that must be exceeded before opening can occur. This yield pressure is simply the capillary pressure, Pc*ap, determined by the radius and surface tension of the smallest airway. Even in the normal lung this pressure is relatively large, -5.8 cmHzO. Any applied airway pressure greater than Pc*ap is utilized to overcome the viscous stress associated with displacing the viscous lining fluid. Figure 14 shows that in a lung with ru = 0.01 g . s-l. cm-l, the opening time is very small once the airway pressure exceeds Pc*ap. This behavior gives the appearance of the airway “popping” open at a pressure slightly greater than the yield pressure. If mucous secretions exist, the viscosity of the lining may increase by several orders of magnitude (5), and the opening time increases dramatically. Once the opening time becomes a significant fraction of the inspiration time, hypoventilation of the affected respiratory pathway may conceivably occur. DISCUSSION
In this study we have investigated the influence of airway lining fluid surface tension (y) and viscosity (p)
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AIRWAY
REOPENING
on the reopening of closed airways. We found that both the surface tension and viscosity, as well as the airway radius, play important roles in determining the characteristics of airway opening. The effect of surface tension is evident by a capillary pressure, or apparent yield pressure, that must be applied before airway opening will begin. Once the applied airway pressure exceeds the capillary pressure, the mechanics of opening depend on the relative magnitudes of viscous and surface tension forces, as reflected in the dimensionless capillary number, Ca = ,UU/r (where U is the velocity of airway opening). Together with airway length, viscous and surface tension forces determine the time required to reopen airways, therein setting the ventilatory impact of airway closure. Our paradigm for airway closure by apposition of bronchial walls (compliant collapse, Ref. 9) is shown in Fig. 1. Does this type of airway collapse really happen in the lung? Previous studies (9-11) have emphasized the role of surface meniscus formation due to surface film instability in the closure of pulmonary airways, a phenomena termed film collapse by Kamm (9). These studies primarily were based on the concept that the airway lining was an unsupported film, except at its ends. If this were true, the film would create a catenoid, a configuration that becomes unstable when the length-diameter ratio exceeds 2-l” (1). Instability of the catenoid film might lead to a meniscus occlusion of the airway. However, Macklem, Proctor, and Hogg (12) found no evidence of catenoid formation in excised feline lungs, suggesting that airway instability cannot be understood solely by the mechanics of an unsupported film layer. Nonetheless, Kamm (9) has shown that even a supported film coating a rigid-walled tube may become unstable and obstruct the airway by meniscus formation. Once meniscus formation occurs, it seems likely that airway collapse will ensue. In his analysis of airway closure by film collapse, Macklem (11) estimated the Laplace pressure difference across the meniscal air-liquid interface by assuming a spherical shape with a radius of curvature equal to that of the open airway (R). This would lower the airway transmural pressure (Ptm) in the meniscus-filled region, and thus decrease the airway cross-sectional area (A) according to its A-Ptm characteristic. As the airway diameter decreases, the pressure in the meniscus fluid will become even more negative due to the decrease in the meniscus’ radius of curvature, leading to further airway collapse and eventual buckling into a flattened profile. Indeed, our studies show that the local radius of curvature may fall to as little as R/8, as the meniscus projects partially into the flattened airway. With airway flattening, the fluid meniscus elongates longitudinally, extending the tendency to airway collapse proximally and distally along the airways. Experimental evidence supporting the airway collapse hypothesis is provided by Hughes et al. (8), who demonstrated airway closure due to collapse of distal canine bronchi, and by Macklem et al. (12), who observed that inflating closed bronchioli required the peeling apart of opposing walls. Thus our airway closure paradigm seemsjustified. One of the key determinants of airway opening is the
83
surface tension of the lining fluid (Fig. 5). Surface tension forces create a capillary pressure that must be exceeded before airway opening can proceed. The zero-velocity intercept of our experiments suggests the possibility of a yield-pressure behavior that could explain the observations of Macklem, Proctor, and Hogg (12), who found that meniscus motion in excised canine bronchioli began only after a “critical pressure” was applied. Although this pressure has generally been thought to be equal to twice the surface tension divided by the open airway radius (II), 27/R, we found apparent yield pressures approximately four times greater, Pc*ap = 87/R (Fig. 11). We believe that airway wall flexibility explains this marked discrepancy. Apparently, the local airway collapse decreases the local meniscus radius of curvature to a value that is much smaller than the open airway’s radius, creating a commensurate rise in the capillary pressure. This smaller radius is evident in the acute opening angles observed in the vicinity of the meniscus (Figs. 12 and 13). It is interesting that the thickness of the lining fluid in the collapsed portion of airway (H) has little effect on the capillary pressure, presumably because fluid upstream of the meniscus is drawn toward the meniscus, increasing the local fluid thickness to a value that depends mostly on airway radius. It is important to note that the apparent yield-pressure phenomenon discussed above arises from dynamic experiments, where we measured the pressures during nearly steady-state conditions. It is possible that the true “start-up” behavior (i.e., the pressure-velocity relationship immediately after reopening commences) may differ from the behavior extrapolated from our quasi-steadystate dynamic studies. Conceivably, the apparent yield pressure may reflect the presence of an undiscovered large slope in a dimensionless P-U relationship that passes through the origin in the neighborhood of Ca = PW = 0. Still, an apparent yield pressure would exist for all practical purposes during airway reopening, since the capillary number would most likely fall within the range of our study. Second, during the transient start-up period the fluid near the meniscus may not have reached its equilibrium depth, so that both pressure components, dependent on meniscus fluid depth, would possibly have a greater dependence on the upstream film thickness, H, than is evident during steady-state reopening. When H is small ( 0.5) due to rapid airway opening, elevated lining viscosity, or small surface tension, then viscous forces add appreciably to the overall opening pressures. If Ca is small, the opening pressures are mostly determined by the surface tension of the liquid lining and are Pc*ap = 87/R. It was surprising to us that the shear stress (reflected in viscous pressure) scaled primarily inversely with airway radius, rather than inversely with lining fluid thickness. This finding again probably relates to accumulation of lining fluid in the neighborhood of the meniscus, where shear rates are greatest. The driving force for the accumulation is suction provided by the product of wall tension and local wall curvature, T. K. Our results suggest that when T is large, the fluid depth in the neighborhood of the meniscus is proportional to R. When T is small, the suction is not large enough to increase the fluid depth to a magnitude proportional to R, and H then plays a small role in determining the viscous shear stress. In addition to its role in setting the length scales for viscous and capillary stress, the drop in downstream pressure due to To K provides an additional driving force, above the applied pressure, for meniscus motion. It does this by decreasing the pressure below atmospheric pressure directly downstream of the meniscus. This mechanism is identical to that which exists in the tape-peeling model presented earlier. The increased driving force is evident by the decrease in applied pressure needed to open an airway at a given velocity when wall tension is increased (Fig. 11). In our model, axial wall tension was utilized to modify the apparent wall characteristics and may be thought of as influencing the tube’s compliance characteristics. Thus increasing the airway stiffness, for example, by increasing lung volume, may promote airway reopening. Likewise, pulmonary fibrosis may lead to smaller airway closing volumes. Macklem (11) has suggested that airway constriction may protect against airway closure by narrowing the lumen and increasing the wall thickness, hence increasing wall stiffness. However, decreasing luminal diameter will increase the film lining thickness, possibly contributing to its instability, as suggested by Frazer, Stengel, and Weber (4). Furthermore,
if constricted airways were to close, their opening pressures would probably be larger than their unconstricted opening pressures due to an increase in the capillary pressure, Pc*ap = Q/R. As described above, dimensional analysis shows that the scales for both viscous and surface tension forces increase inversely with airway radius, R. This result suggests that the smallest airways are the most difficult to reopen, although generational variation in lining fluid secretion physical properties might partially offset this radial dependency. Using the results of our study, we predicted airway reopening times for a series of sequential airways (generations 8-14) to evaluate how airway reopening times may depend on both surface tension and viscosity (assumed constant along the airways). Figure 14 shows that surface tension defines the initial opening pressure but that the time to opening is determined by the viscosity of the lining fluid. If the fluid is relatively inviscid, the airway appears to pop open. However, if this fluid is viscous, as may happen in mucous plugging or cystic fibrosis, the opening times may be large and prevent airway opening during a normal tidal breath. We are appreciative of the technical assistance provided by Linda E. Alger and Robert Monk. This research was supported by National Heart, Lung, and Blood Institute Grants HL-02205, HL-07912, HL-41009, and HL-01857 and by the Whitaker Foundation. Present address and address for reprint requests: D. P. Gaver III, Dept. of Biomedical Engineering, Tulane University, New Orleans, LA 70118. Received
7 August
1989; accepted
in final
form
16 February
1990.
REFERENCES 1. BOYS, C. V. Soap Bubbles, Their Colours and the Forces Which Mould Them. New York: Dover, 1959. 2. CRAWFORD, A. B. H., D. J. COTTON, M. PAIVA, AND L. A. ENGEL. Effect of airway closure on ventilation distribution. J. Appl. Physiol. 66: 2511-2515, 1989. 3. ENGEL, L. A., A. GRASSINO, AND N. R. ANTHONISEN. Demonstration of airway closure in man. J. Appl. Physiol. 38: 1117-1125, 1975. 4. FRAZER, D. G., P. W. STENGEL, AND K. C. WEBER. Meniscus formation in airways of excised rat lungs. Respir. Physiol. 36: 121129, 1979. 5. FUNG, Y. C. Biomechanics. New York: Springer-Verlag, 1981. 6. GREAVES, I. A., J. HILDEBRANDT, AND F. G. HOPPIN, JR. Micromechanics of the lung. In: Handbook of Physiology. The Respiratory System. Mechanics of Breathing. Bethesda, MD: Am. Physiol. Sot., 1986, sect. 3, vol. III, pt. 1, chapt. 14, p. 195-216. 7. HILLS, B. A. The Biology of Surfactant. Cambridge, UK: Cambridge Univ. Press, 1988. 8. HUGHES, J. M. B., D. Y. ROSENZWEIG, AND P. B. KIVITZ. Site of airway closure in excised dog lungs: histologic demonstration. J. Appl. Physiol. 29: 340-344, 1970. 9. KAMM, R. D., AND R. C. SCHROTER. Is airway closure caused by liquid film instability? Respir. Physiol. 75: 141-156, 1989. 10. LEBLANC, P., F. RUFF, AND J. MILIC-EMILI. Effects of age and body position on “airway closure” in man. J. Appl. Physiol. 28: 448-451, 1970. 11. MACKLEM, P. T. Airway obstruction and collateral ventilation. Physiol. Rev. 51: 368-436, 1971. 12. MACKLEM, P. T., D. F. PROCTOR, AND J. C. HOGG. The stability of peripheral airways. Respir. Physiol. 8: 191-2010, 1970. 13. MARTIN, H. B., AND D. F. PROCTOR. Pressure-volume measurements on dog bronchi. J. Appl. Physiol. 13: 337-343, 1958. 14. MCEWAN, A. D., AND G. I. TAYLOR. The peeling of a flexible strip attached by a viscous adhesive. J. FZuid Mech. 26: 1-15, 1966.
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AIRWAY 15. MILIC-EMILI, J., J. A. M. HENDERSON, M. B. DOLOVICH, D. TROP, AND K. KENEKO. Regional distribution of inspired gas in the lung. J. Appl. Physiol. 21: 749-759, 1966. 16. PARK, C.-W., AND G. M. HOMSY. Two-phase displacement in Hele Shaw cells: theory. J. Fluid Mech. 139: 291-308, 1984. 17. REINELT, D. A. Interface conditions for two-phase displacement in Hele-Shaw cells. J. FZuid Mech. 183: 219-234, 1987. 18. REINELT, D. A., AND P. G. SAFFMAN. The penetration of a finger into a viscous fluid in a channel and tube. SIAM J. Sci. Stat. Comput. 6: 542-561, 1985.
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19. SAFFMAN, P. G., AND G. I. TAYLOR. The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. R. Sot. Lond. A Math. Phys. Sci. 245: 312-329, 1958. 20. SUTHERLAND, P. W., T. KATSURA, AND J. MILIC-EMILI. Previous volume history of the lung and regional distribution of gas. J. Appl. Physiol. 25: 566-574, 1968. 21. WEIBEL, E. R. Morphometry of the Human Lung. New York: Academic, 1963. 22. YAGER, D., J. P. BUTLER, J. BASTACKY, E. ISRAEL, G. SMITH, AND J. M. DRAZEN. Amplification of airway constriction due to liquid filling of airway interstices. J. Appl. Physiol. 66: 2873-2884, 1989.
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