INTRODUCTION

TOP ABSTRACT INTRODUCTION BACKGROUND EXPERIMENTAL METHODS RESULTS DISCUSSION REFERENCES APPENDIX A APPENDIX B

PULMONARY AIRWAY CLOSURE can occur when the lung reaches very low volumes (20). Healthy individuals reopen these airways on the next inspiratory effort. However, these airways remain closed in premature infants suffering from respiratory distress syndrome (RDS) and individuals with asthma, emphysema, or cystic fibrosis. These conditions can result in atelectasis and/or local hypoventilation. Pulmonary surfactant insufficiency, loss of parenchymal tethering, and an increase in fluid viscosity have been implicated as possible mechanisms by which airways remain closed (2, 19).

Avery and Mead (2) were the first to observe a lack of pulmonary surfactant production in premature neonates suffering from RDS. The production of fetal surfactant begins during the 4th mo of gestation. However, this surfactant system (consisting of phospholipids and proteins) may not mature until the 7th mo (20). A mature surfactant system decreases lung inflation pressures by reducing the surface tension forces that oppose airway opening. Therefore, an immature surfactant system results in large lung inflation pressures, mechanical instability, and subsequent atelectasis.

One possible method to treat RDS involves mechanical ventilation. However, large ventilation pressures can result in airway wall damage. Specifically, mechanical stresses at the wall can peel apart epithelial cells (22). As a result, the epithelium becomes highly permeable to proteins that can inactivate surfactant (32). This inactivation further elevates reopening pressures and establishes a positive-feedback cycle in which significant airway wall damage can occur. To minimize this damage, neonates with RDS have been treated with positive end-expiratory pressure ventilation, high-frequency ventilation, and/or several surfactant replacement therapies (14). The advent of surfactant replacement therapy has resulted in a 60% reduction in the infant mortality rate for RDS since 1989 (13). However, RDS was still the fourth leading cause of infant death, with 1,368 fatalities in 1996 (13).

Three distinct types of surfactant replacements, natural, modified natural, and synthetic, have been developed. One concern with natural surfactant is that sensitive patients may develop an allergic reaction to the surfactant-associated proteins (33). However, because of an immature immune system, a severe allergic response in neonates is rare (43). Synthetic surfactants frequently do not perform as well as the natural surfactant system (14). This is primarily due to a synthetic surfactant's inability to mimic the chemical complexity of a natural surfactant.

We hypothesize that the effectiveness of a surfactant replacement depends on its dynamic surface tension properties. Most researchers have focused on the fact that an ideal surfactant system should promote alveolar stability by reducing the surface tension on compression (3, 12, 25) or stabilize the airway lining fluid by preventing Rayleigh-Taylor instabilities (15, 17). However, to limit epithelial cell damage during airway reopening, the surfactant must also minimize reopening pressures and airway wall stresses. The ability of a surfactant to perform adequately during interfacial expansion depends on its transport characteristics (e.g., diffusivity and interfacial adsorption/desorption rates) and surface activity. The interplay between these mechanisms determines the efficacy with which a surfactant can reduce airway reopening stresses. The goals of the present study are to 1) develop a method for determining the capability of a surfactant to reduce airway reopening pressures and 2) demonstrate this method by determining the efficacy of several surfactants (physiological and nonphysiological) to reduce the dynamic surface tension.

	BACKGROUND

TOP ABSTRACT INTRODUCTION BACKGROUND EXPERIMENTAL METHODS RESULTS DISCUSSION REFERENCES APPENDIX A APPENDIX B

Airway closure occurs when a liquid occlusion spans the airway and blocks airflow. This occlusion can be extensive (as occurs before an infant's 1st breath or in atelectasis) or consist of a short meniscus (e.g., a mucous plug). The research described here relates to reopening an extended region of closure, wherein a semi-infinite air bubble penetrates the liquid occlusion and separates the airway walls as it displaces the lining fluid. This scenario has been described previously (9, 10, 16, 27, 28) and results in two-phase interfacial flow between the bubble and the lining fluid. Suki and colleagues (38, 39) investigated the avalanche-type behavior that occurs when multiple generations of airways open sequentially. Recent studies have used "stochastic resonance" to take advantage of this type of behavior and, thus, open airways with reduced average pressures (37).

A classic problem in the hydrodynamic literature related to pulmonary airway reopening involves semi-infinite bubble progression in a rigid capillary tube. This system was originally studied by Bretherton (4), Fairbrother and Stubbs (7), and Taylor (40). These studies formulated a relationship between the dimensionless interfacial pressure drop, P/(/R), and the capillary number, Ca = µU/, where P is the dimensional pressure drop across the air-liquid interface in the hemispherical bubble cap region,  is the surface tension, R is the radius of the tube, µ is the fluid viscosity, and U is the bubble speed. Ca is a dimensionless velocity that represents the relationship of viscous to surface tension stresses. These studies assumed a constant  and, therefore, neglected the influence of surfactant. However, surfactant modifies the flow field by altering the local surface tension and mechanical stress balance at the interface. Several investigators (11, 31, 35, 36) have studied the interaction between surfactant physicochemistry and fluid mechanics. In rigid systems, these interactions modify the interfacial shape and pressure drop and, therefore, affect airway reopening characteristics. Recently, Yap and Gaver (45) demonstrated the importance of surfactant physicochemistry in flexible-walled systems intended to emulate collapsible airways and predicted that surfactant uptake could significantly influence airway reopening.

In coupling surfactant physicochemistry and fluid mechanics, it is important to understand the mechanisms of surfactant action and transport. Surfactant can reside in two phases: a bulk phase (C, mol/vol) and a surface phase (, mol/interfacial area). The surface phase directly modifies the interfacial surface tension by  = f(); this relationship is referred to as the surfactant equation of state. In general, increasing  reduces ; however, regions exist where d/d is nearly zero (41). In a static system, an equilibrium relationship between C and  exists, with an associated equilibrium surface tension (_eq). The interface becomes saturated with surfactant as C increases, resulting in a maximum equilibrium surface concentration (_sat), which determines the minimum equilibrium surface tension in a static system (_sat). In a dynamic system,  (and thus ) can vary as a function of interfacial position. In addition,  can exceed _sat under dynamic conditions resulting from surface compression (as occurs in the alveoli in vivo or experimentally in a Langmuir trough), which can reduce  significantly.

Dynamic surfactant transport is modeled as a multistep process between two phases: the bulk and the interface. In the bulk phase, surfactants are transported by bulk convection and/or diffusion, resulting in C(x,t), where x defines the general spatial coordinate system. The bulk phase concentration in contact with the interface is the subphase concentration C_s(s,t), where s is a spatial coordinate along the interface. The rate of transport between the subphase and interface is modeled by an adsorption isotherm, where the rate of adsorption/desorption depends on the deviation from equilibrium between  and C_s. This step is balanced by diffusion in the bulk. Finally, interfacial convection and diffusion along the interface can influence the local . These transport interactions determine the distribution of surfactant in the bulk and along the interface. This distribution establishes a stress balance along the air-liquid interface through the equation of state and thus impacts airway reopening pressures and stresses.

To conceptualize this interaction, consider the flow field surrounding a bubble flowing down a liquid-filled tube (Fig. 1A). Surfactant exists in the bulk (C) and on the interface (). Streamlines are drawn in a bubble-fixed reference frame in which flow enters from the right and exits to the left in the thin film. A recirculating region near the bubble tip occurs at low velocities. As a result, the rate of interfacial expansion/compression will vary with interfacial position (Fig. 1B). As shown in Fig. 1, the interface can be divided into three regions: a nearly hemispherical bubble cap region, a thin film, and a transition region between the bubble cap and thin film. A small portion of the bubble cap region near the tip will experience interfacial compression, whereas the majority of interfacial expansion takes place in upstream portions of the bubble cap and transition regions. In addition, this recirculating flow field creates a converging stagnation point at the bubble tip and a diverging stagnation ring near the thin film (42). Surface convection increases  (reducing ) at the converging stagnation point and reduces  (increasing ) at the diverging stagnation points. Variation of  from _eq can alter the bubble pressure through two distinct, yet interrelated, mechanisms.

View larger version (2118K): [in this window] [in a new window]   Fig. 1.   A: flow field surrounding a bubble flowing down a liquid-filled tube. Streamlines are drawn for a reference frame fixed with bubble. M, Direction of Marangoni stress. B: typical profile of interfacial stretching as a function of interfacial position in a surfactant-free system. Negative stretching indicates interfacial compression; positive stretching indicates interfacial expansion. Capillary number (Ca) = 0.12.

One mechanism is related to the normal-stress boundary condition (law of Laplace), which states that the pressure difference across an interface is P = , where  is the local surface tension and  is the curvature. Under dynamic interfacial expansion, the effective dynamic surface tension in the bubble cap region can exceed the equilibrium surface tension ( > _eq). Therefore, the reopening pressure can exceed the pressure that would exist if  = _eq uniformly. We refer to this deviation as the nonequilibrium normal stress.

A second mechanism that increases the bubble pressure is related to the development of a surface tension gradient, d/ds. This gradient allows the interface to support a shear stress (Marangoni stress, _M) that is directed from regions of low  to regions of high  and opposes the surfactant-free flow field shown in Fig. 1A. Interfaces without a Marangoni stress are mobile, because they do not support a shear stress. Mobile interfaces occur when  is spatially constant, which can happen if  = 0 (clean interface),  is uniform, or  is nonuniform and d/d is small. Because the Marangoni stress opposes the basic flow field, the interface acts more like a rigid boundary, resulting in a larger bubble pressure.

Several theoretical models have explained the importance of surfactant transport properties on bubble motion in rigid tubes. Ratulowski and Chang (31) analytically modeled trace surfactant transport and evaluated several different scenarios related to hindered transport from the bulk to the interface. Fundamentally, two processes can hinder surfactant transport: diffusion limitation in the bulk and kinetic adsorption/desorption barriers from the subphase to the interface. The relative relationship between diffusion and convection in the bulk is governed by the Péclet number, Pe = U(²/R)/D, where D is the surfactant molecular diffusivity and the adsorption depth  = /C is a length scale related to the fluid thickness that contains sufficient surfactant molecules to bring the interfacial concentration to _eq. The relationship between surfactant adsorption and surface convection is governed by the Stanton number, St = k_aCR/U, where k_a is the adsorption rate constant.

The interface remains near equilibrium in the limit of Pe  0 and St  . The diffusion barrier can be reduced (small Pe) by diminishing  through an increase in C, a scenario analyzed by Stebe and Barthès-Biesel (35). Under these large C conditions, the adsorption barrier will be the only limitation to surfactant transport. Therefore, equilibrium behavior at large C indicates fast adsorption properties.

In summary, nonequilibrium normal and Marangoni stresses can significantly elevate reopening pressures when the transfer of surfactant to the interface is limited by bulk or sorptive transport processes. If bulk and sorptive transport processes are rapid, any surfactant deficiency on the interface is quickly replaced with surfactants from the bulk, resulting in a uniform . Under these conditions,   _eq and the normal stress approaches the equilibrium value. Furthermore, gradients in  (and, therefore, the Marangoni stress) are eliminated, and the interface is said to be "remobilized." Therefore, if adsorption is quick, the bubble motion is no longer hindered by the two mechanisms outlined above under large C conditions.

The main objective of the present study is to use these concepts to investigate how various pulmonary surfactant analogs affect the mechanics of a continually expanding interface, as occurs during airway reopening. We hypothesize that different surfactant systems will exhibit different capabilities for reducing reopening pressures and stresses owing to differences in physicochemical properties. Identifying these properties may be useful in recognizing and developing suitable surfactant replacements.

	EXPERIMENTAL METHODS

TOP ABSTRACT INTRODUCTION BACKGROUND EXPERIMENTAL METHODS RESULTS DISCUSSION REFERENCES APPENDIX A APPENDIX B

Several methods are available for measuring dynamic surface tension. Schürch et al. (34) used a captive bubble apparatus to measure surface tension as a function of interfacial concentration. Enhörning (6) developed a pulsating bubble surfactometer to mimic alveolar dynamics. In both systems, volume oscillations produce an oscillating surface area. Although these oscillatory techniques may model alveolar dynamics, they are not as useful in understanding airway reopening. The key aspect that differentiates airway reopening from alveolar dynamics is the continual expansion of an air-liquid interface into the bulk solution. In this situation, tangential flows create nonuniform interfacial concentrations, and a Marangoni stress is generated. To identify the importance of nonequilibrium normal and Marangoni stresses during airway reopening, we developed a simple experimental apparatus that mimics the continual interfacial expansion aspects of airway reopening.

Model Description

View larger version (30K): [in this window] [in a new window]   Fig. 2.   Schematic of experimental apparatus. Syringe pump creates a semi-infinite air bubble that displaces surfactant solution in capillary tube. Gas phase pressure and infrared detector signals are monitored via a data acquisition system.

Surfactant Preparation Methods

SDS (>99% purity; Sigma Chemical, St. Louis, MO) was used without further purification. Because SDS is ionic, solubilization was achieved by simply adding a known quantity to the volume of pure water in the end tank and stirring for ~5 min.

L--Dipalmitoylphosphatidylcholine (DPPC) in powdered form (99% purity; Sigma Chemical) was also used without further purification. DPPC dispersions were prepared using the method of Chung et al. (5). Batch solutions of DPPC at high concentrations (>0.3 mg/ml) were created by adding a known quantity of DPPC to a given volume of calcium-buffered saline. The suspension was stirred at temperatures above the DPPC transition temperature (43°C) to liquify DPPC crystals. This milky-white solution was then sonicated in 5-min intervals in an ultrasonic cleaner (model 1210, Branson) at 50°C until the solution was clear. The number of 5-min sonication intervals required to achieve the desired clarity was not constant. Dilution of the batch DPPC solution was performed in the end tank and stirred for 5 min. The batch solutions of DPPC were stored at 5°C. Before addition to the end tank, a batch solution was reheated and sonicated until the solution became clear. Because of the hydrolysis of DPPC that occurs with age (18), fresh batch solutions were made every 2 days.

The exogenous replacement surfactant Infasurf was obtained from ONY (Buffalo, NY). Infasurf is a natural lung extract that contains DPPC, fatty acids, and low-molecular-weight surfactant-associated proteins SP-B and SP-C. Infasurf is prepared by a saline rinse lavage of a calf lung and subsequent hydrophobic extraction and, therefore, contains all natural compounds. Infasurf was obtained as a milky-white liquid suspension. The phospholipid content of these batch solutions is 35 mg/ml. Dilution of this solution was performed in the end tank with buffered saline and stirred for 5 min.

With these preparatory techniques, a wide range of bulk concentrations could be investigated. We determine the surfactant's surface activity by investigating equilibrium interfacial properties. We also determine a surfactant's ability to minimize the nonequilibrium normal and Marangoni stresses that oppose reopening by investigating dynamic properties.

Equilibrium Measurements

Dynamic Measurements

View larger version (21K): [in this window] [in a new window]   Fig. 3.   Bubble pressure (Pbub) and infrared detector signals acquired as a function of time.

Before any surfactant experiments, baseline equilibrium and dynamic measurements of the surfactant-free system (calcium-buffered saline solution) were obtained daily. The surfactant was then added to the end tank to achieve a desired concentration. An equilibrium measurement was made first, and then a set of dynamic measurements was obtained at nine different velocities. This procedure could be repeated for several concentrations by addition of aliquots of surfactant to the end reservoir. At the end of each set of experimental trials, the entire system was cleaned with a 2% Micro solution.

Determination of P_cap

(1)

View larger version (10K): [in this window] [in a new window]   Fig. 4.   Pressures that exist at steady state. P1, fluid pressure at bubble tip; P2, fluid pressure where Poiseuille flow begins; P3, fluid pressure at end of tube. Hydrostatic head pressure in end tank is used as reference pressure. Ddetector and Dend are known distances.

To extract P_cap, the following assumptions are made. First, under low capillary number (Ca_eq = µU/_eq, a dimensionless velocity) conditions, the error associated with neglecting P_np is insignificant. We use the results from a computational model, which indicates that P_np is <1% of P_cap for Ca < 10² (see APPENDIX A) to justify this assumption. To determine whether pressure losses associated with exit flow into the end tank (P_end) are significant, we used laminar head-loss coefficients determined by Rao (29), which show that P_end < 1% of P_cap. Finally, P_p(t) ~  * L_p(t), where  is a Poiseuille flow proportionality constant. Because L_np is small (APPENDIX A), L_p(t) is the distance from the bubble tip to the outlet and L_p(t) = U(t  t_end). Therefore, from Eq. 1

(2)

Therefore, P_cap can be calculated once the bubble tip velocity [U = D_detector/(t₂  t₁)], the slope () of the P_bub vs. t curve in the steady-state region, and the time at which the bubble exits the capillary tube [t_end = (D_end + D_detector)/U] are determined.

Identification of Noninertial Component (P_cap)_Stokes

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Dimensional analysis indicates that _cap = f(Ca,Wb), where Wb (Weber number) = U²/(/R) relates inertial to surface tension effects. As shown in APPENDIX B, surfactant-free experiments conducted with ethylene glycol (µ = 2.1 × 10² kg · m¹ · s¹,  = 48 mN/m) reproduce the Ca^2/3 behavior over 10⁸  Wb  10⁴, validating the data analysis approach described above. However, surfactant experiments were conducted in an aqueous solution in which inertia is potentially significant (10⁵  Wb  10¹). By comparing the Stokes flow behavior of the ethylene glycol system with the moderate Wb behavior of the aqueous system (see APPENDIX B), we can estimate the inertial and Stokes components of P_cap

(3)

where
The inertial pressure (P_inertial) is estimated by comparing aqueous surfactant-free measurements with ethylene glycol measurements by use of a least squares regression of both data sets to determine coefficients A and B (see APPENDIX B). Eight different determinations of A and B (based on 8 different surfactant-free data sets) varied by <7% (A = 1.24 ± 0.03, B = 0.565 ± 0.03). Errors associated with this approximation are most significant at higher velocities and are insignificant at the lowest velocities studied.

For a surfactant-free system,  in Eq. 3 is the surface tension. However, when surfactant is incorporated,  is a representative dynamic surface tension (), which we will show is not necessarily equal to _eq (see Data Analysis). The will be estimated from regression analysis presented below, thus allowing us to estimate P_inertial and thus (P_cap)_Stokes. From this, the dimensionless Stokes pressure is calculated based on _eq: (_cap)_Stokes = (P_cap)_Stokes/(_eq/R).

	RESULTS

TOP ABSTRACT INTRODUCTION BACKGROUND EXPERIMENTAL METHODS RESULTS DISCUSSION REFERENCES APPENDIX A APPENDIX B

Figure 5 shows the results of _eq measurements for the three different surfactant systems under investigation. As described in BACKGROUND, increasing C increases _eq until the interface is saturated. Because _eq is inversely related to , this results in _eq decreasing with increasing C until a critical bulk concentration (C_CBC) is reached. For C > C_CBC, the interface is saturated (_{eq  sat}), and _{eq  sat}. Large C behavior indicates that Infasurf is the most surface active of the three surfactants studied, with _sat ~ 27 mN/m.

View larger version (19K): [in this window] [in a new window]   Fig. 5.   Equilibrium surface tension as a function of bulk surfactant concentration for SDS (), L--dipalmitoylphosphatidylcholine (DPPC, ), and Infasurf (). CCBC, critical bulk concentration. Error bars are based on SD of 3 measurements.

To demonstrate how the nondimensional pressure, (_cap)_Stokes, helps identify the magnitude of the nonequilibrium normal and Marangoni stresses, we present dimensional and nondimensional pressure vs. velocity data for each surfactant in Figs. 6 and 7, respectively. For each C, (P_cap)_Stokes is determined over a range of U (Fig. 6). For a given C, (P_cap)_Stokes increases slightly with U. However, as C is increased, (P_cap)_Stokes vs. U curves shift downward. This downward shift is directly related to the decrease in _eq that occurs with increasing C (Fig. 5).

View larger version (13K): [in this window] [in a new window]   Fig. 6.   Dimensional Stokes bubble cap pressure jump as a function of bubble tip velocity. A: SDS at 0.0 mg/ml (), 0.577 mg/ml (), 1.73 mg/ml (), 2.31 mg/ml (), 2.88 mg/ml (), and 3.75 mg/ml (). B: DPPC at 0.0 mg/ml (), 0.0601 mg/ml (), 0.0940 mg/ml (), 0.220 mg/ml (), 0.416 mg/ml (), 0.725 mg/ml (), and 3.84 mg/ml (). C: Infasurf at 0.0 mg/ml (), 0.025 mg/ml (), 0.050 mg/ml (), 0.070 mg/ml (), 0.10 mg/ml (), 0.50 mg/ml (), and 3.0 mg/ml ().

View larger version (222222K): [in this window] [in a new window]   Fig. 7.   A: dimensionless Stokes bubble cap pressure jump as a function of equilibrium capillary number for SDS. B: dimensionless Stokes bubble cap pressure jump as a function of equilibrium capillary number for DPPC. C: dimensionless Stokes bubble cap pressure jump as a function of equilibrium capillary number for Infasurf. See Fig. 6 legend for concentrations of SDS, DPPC, and Infasurf.

In contrast, consider the dimensionless relationship (_cap)_Stokes vs. Ca_eq for SDS in Fig. 7A. The solid and dashed curves represent the data regression described in Data Analysis. This representation normalizes (P_cap)_Stokes based on _eq. Therefore, if the interface approaches equilibrium, the (_cap)_Stokes vs. Ca_eq curve for a given C will resemble the C = 0 data (solid circles in Fig. 7A), because the nonequilibrium normal and Marangoni stresses will be insignificant. At small C, the data in Fig. 7A initially shift upward with increasing C. This indicates that the interface is not in equilibrium and that the magnitudes of the nonequilibrium normal and Marangoni stresses are increasing with increasing C. However, as C increases further, the curves begin to shift downward toward C = 0. This downward shift indicates that the SDS system can approach (but not completely obtain) equilibrium at high C, which reduces the nonequilibrium normal stress and significantly remobilizes the interface.

Figure 7, B and C, demonstrates (_cap)_Stokes vs. Ca_eq relationships for different pulmonary surfactant analogs. In the DPPC system (Fig. 7B), (_cap)_Stokes is larger than the SDS values over all ranges of C. At low C, (_cap)_Stokes increases dramatically to (_cap)_Stokes ~ 3, even at the smallest velocities. As C is increased beyond 0.416 mg/ml, a minimal downward shift of (_cap)_Stokes is observed. However, increasing C does not return the system to the equilibrium behavior, even at the largest C (3.84 mg/ml). At these large concentrations the interface would be saturated in equilibrium. The large magnitude of (_cap)_Stokes indicates that nonequilibrium normal and Marangoni stresses are large in this system, whereas the minimal downward shift indicates that DPPC is not capable of significant nonequilibrium normal stress reduction or interfacial remobilization under the velocities studied.

Large (_cap)_Stokes values are also observed in the Infasurf system (Fig. 7C) for small C. These data show that at small C the combined effect of nonequilibrium normal and Marangoni stresses can also be significant in the Infasurf system. However, for C > 0.07 mg/ml, a significant downward shift of (_cap)_Stokes is observed, indicating that Infasurf contains constituents other than DPPC that allow this surfactant system to approach equilibrium under dynamic conditions.

Data Analysis

(4)

Clearly, when _eq is used as the surface tension scale, Eq. 4 cannot adequately describe the data in Fig. 7. This inadequacy is a direct result of the constant _eq bubble cap surface tension assumption. In reality, surface convection coupled with transport limitations can create a distribution of surfactant along the interface in the bubble cap region, as illustrated in Fig. 1A, resulting in an effective bubble cap surface tension   _eq in the dynamic system. Only if diffusive barriers are removed and the adsorption rate is fast (St = k_aCR/U  ) will  _eq. This limit is observed only during our equilibrium measurements, where U is identically zero. However, because the time to reach equilibrium can be ~100 s for DPPC at C = C_CBC, it is not surprising that the dynamic surface tension deviates from _eq.

To estimate the representative dynamic surface tension, we simplistically assume that depends only on the given bulk concentration for a given surfactant in the dynamic system and is independent of velocity over the range of velocities investigated (static system excluded). This assumption will be explained below. We define the surface tension ratio

(5)

where is a measure of the relative surface tension deviation from equilibrium and thus is a measure of the nonequilibrium normal stress. From Eq. 3 with as the dynamic surface tension scale

(6)

where

To estimate the magnitudes of nonequilibrium normal and Marangoni stresses, for each experiment, we calculate _cap, Ca_eq, and Wb_eq. A least squares regression is performed with Eq. 6, with _cap as the dependent variable and Ca_eq and Wb_eq as the independent variables at each bulk concentration. During each regression analysis, the concentration is constant, and, therefore, variations in Ca_eq and Wb_eq are governed only by variations in U. The best-fit relationships faithfully reproduce the experimental behavior at each concentration, as shown in Fig. 7, with R² > 0.98. As a result, and  are determined for each surfactant at each concentration (A and B are known from APPENDIX B). Once these parameters are known for a given concentration, P_inertial can be calculated, and Eq. 3 can be used to convert the P_cap data to (P_cap)_Stokes data. Because we are primarily concerned with noninertial or capillary effects, we have presented only (P_cap)_Stokes and (_cap)_Stokes data in Figs. 6 and 7, respectively.

During dynamic conditions, reflects the magnitude of the nonequilibrium normal stress near the bubble cap region. In Fig. 8A, vs. C/C_CBC is presented. The  ~ 1 for SDS, indicating that the nonequilibrium normal stress in the cap region is minuscule. In contrast, DPPC and Infasurf exhibit much larger nonequilibrium normal stresses, with  ~ 1.5, even when C ~ C_CBC. As discussed in BACKGROUND, increasing C decreases the adsorption depth (); therefore, the major transport limitation that remains is due to interfacial adsorption. Therefore, the magnitude of at large C is directly related to the surfactant's adsorption properties and equation of state. The data in Fig. 8A suggest that SDS adsorption rates are rapid enough for  _eq uniformly at high concentrations. In contrast, the slight reduction of at large C in the DPPC system indicates that an adsorption limitation exists for this surfactant. The pulmonary replacement surfactant Infasurf apparently adsorbs more rapidly than DPPC and thus suffers only modest adsorption limitation. Evidently, constituents of Infasurf other than DPPC (the primary component) improve adsorption to the moving interface.

View larger version (1315K): [in this window] [in a new window]   Fig. 8.   A: coefficient values as a function of dimensionless bulk concentration for SDS (), DPPC (), and Infasurf (). B: coefficient  values as a function of dimensionless bulk concentration for SDS (), DPPC (), and Infasurf (). Horizontal line, surfactant-free (clean) system value.

The term ^1/3 represents viscous contributions to (_cap)_Stokes. In the absence of Marangoni stress,  = 5.72 (ideally; see APPENDIX B). The presence of a Marangoni stress in the region between the bubble cap and thin film (transition region) can increase . In Fig. 8B the values of  are plotted vs. C/C_CBC. The experimentally determined constant surface tension value for  (5.25) is shown as a horizontal line (see APPENDIX B). The values of  in the SDS system initially increase to a maximum of   20 and then decrease slightly after C_CBC. The magnitude of  in the DPPC and Infasurf systems increases with C and does not show any remobilization trends. This indicates that the Marangoni stress in the transition region remains large in DPPC and Infasurf systems, even at high bulk concentrations.

Justification of regression model. In the regression analysis, is implicitly assumed to be constant for each concentration over the range of velocities investigated. This violates the fact that at U = 0,  = 1, and thus the relationships obviously cannot extend to a static system (Ca_eq identically equal to 0). Several questions naturally arise: For a given concentration, is it likely that is independent of velocity? Is it justifiable to assume that nonequilibrium behavior exists, even at the exceedingly small velocities spanned in the present study?

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	DISCUSSION

TOP ABSTRACT INTRODUCTION BACKGROUND EXPERIMENTAL METHODS RESULTS DISCUSSION REFERENCES APPENDIX A APPENDIX B

The data presented above demonstrate that SDS, DPPC, and Infasurf significantly modify the interfacial pressure drop associated with bubble motion through a capillary tube, which is intended to model interfacial flows similar to those that exist during pulmonary airway reopening. We will attempt to further interpret these data to more completely distinguish the relative contributions of nonequilibrium normal and Marangoni stresses on the system. The complete identification of the magnitudes of these contributions must await the full analysis of convection, diffusion, and adsorption dynamics in this system, which is beyond the scope of this study.

Physical Interpretation

The magnitude of the nonequilibrium normal stress () is related to the effective bubble cap surface tension and is thus a function of adsorption dynamics in this region. The magnitude of the Marangoni stress () is primarily a function of adsorption dynamics in the transition region, where interfacial stretching is large. A reduction of indicates that the bubble cap region is in equilibrium, whereas a reduction in  would indicate that the transition region is in equilibrium. Therefore, as the adsorption rate is increased, we expect to observe a reduction in