INTRODUCTION

Top Abstract Introduction Appendix References

SURFACTANT REPLACEMENT THERAPY (SRT) is the standard treatment for premature neonates suffering from surfactant insufficiency of prematurity. Response is rapid, and complications such as cyanosis and bradycardia are minimal (45). Recent animal experiments have also shown that surfactant treatments are beneficial in some neonatal inflammatory lung diseases such as meconium aspiration (39) and some types of pneumonia (21). Surfactant has also been examined as a spreading agent for drug delivery (32). In view of the rapidly expanding role of surfactant instillation, it is increasingly important to understand the process by which a bolus of surfactant is dispersed through the lungs. Yet a comprehensive model or framework does not exist for describing how a surfactant bolus placed into the trachea is transported out to the periphery.¹

Recent modeling efforts have focused on transient spreading of localized surfactant monolayers by surface tension gradients. Profiles of liquid layer thickness and surfactant concentration have been reported, with the time for surfactant to reach the lung periphery predicted. In general, surface tension gradients induced by a nonuniform distribution of surfactant at the air-liquid interface drive liquid from regions of low surface tension to regions of high surface tension. The resulting flows cause liquid to accumulate at the leading edge of the monolayer with simultaneous thinning of the liquid layer at the deposition site. For a fixed amount of insoluble surfactant deposited over an initially surfactant-free interface, the spatial extent of the monolayer grows as follows: t^1/4 for a droplet (11), t^1/3 for a strip (9, 24), and t^1/2 for a front (1), where t is time.

Other studies have examined the case in which the surface-active agent is soluble in the liquid layer, primarily in the context of pulmonary drug delivery. For example, it has been shown that when the airway wall is a perfect absorber of the surfactant as it diffuses across the layer, axial gradients in surface tension decrease and the spreading rate is reduced (18). With no absorption at the wall, disturbances in the liquid layer increase, but spreading rates are not significantly altered from the insoluble situation (25). When a drug is modeled as a passive solute in the liquid layer, surface tension gradients sweep it along at one-half the surface velocity (27).

The time for surfactant to spread to the periphery has been estimated in two separate studies with seemingly contradictory results. Espinosa et al. (9), extrapolating from their findings in a single-tube model, estimated that surfactant could reach the periphery in as little as 12 s. Jensen et al. (26), using a model that accounted for real airway geometry, estimated the transit time from the trachea to airway generation 16 to be 15 min. Jensen et al. attributed the difference between their prediction and that of Espinosa et al. to the effects of surfactant dilution associated with the rapid increase in surface area in the direction of the periphery. Although these effects certainly influence transport rates, much of the difference is attributable to different values used for the liquid layer viscosity and thickness in the respective calculations. The viscosity (µ) used by Espinosa et al., µ = 0.01 g · s¹ · cm¹, was a factor of 10 lower than that of Jensen et al. When this lower value is used in the model of Jensen et al., the transit time is reduced from 15 to 1.5 min. Additionally, the thickness of the endogenous liquid layer was taken to be a uniform 10-µm layer in the model of Espinosa et al. but varied between 2 µm in the trachea and 0.2 µm at generation 16 in the calculations of Jensen et al. Because spreading time is inversely proportional to liquid layer thickness, this further contributes to the difference in predicted times. Last, Espinosa et al. modeled a finite amount of insoluble surfactant, leading to a decrease in surfactant concentration as the monolayer spread toward regions of higher surface tension. In contrast, Jensen et al. held surface concentration fixed at the trachea, essentially approximating a source of surfactant that maintained a surface tension gradient along the airway tree. This would reduce transit times; however, a direct comparison of the two predictions cannot easily be made. The effect of dilution, therefore, is difficult to discern but appears to be smaller than initially believed. Clearly, the choice of values for the physical variables can have a profound effect on the transit times predicted.

The behavior of these models is dramatically influenced, however, by the presence of endogenous surfactant; the accumulation of liquid at the leading edge is reduced (9, 24), and spreading rates diminish (15).

Thus previous studies (9, 11, 18, 24-27) have extensively examined the general character of surfactant-driven flow, with application to spreading in the lungs. Yet certain aspects present in SRT have not been addressed. The exchange of material between the liquid layer and the epithelium has been considered in the context of drug delivery (18, 25, 27); the effects on surfactant delivery of surfactant exchange between the liquid deposited in the airways during SRT and the air-liquid interface has not. Additionally, effects such as gravity and shear stress from airflow in a whole lung geometry have not been previously considered. Furthermore, these earlier models consider transport of surfactant through the lung as occurring only over endogenous liquid layers. They do not address the clinical setting of how an exogenous surfactant originating in the trachea and main stem bronchi is pushed through the airways with the following inspiration. No consideration has been given to how this aspect of SRT determines the quantity of surfactant reaching the periphery or the amount of surfactant remaining in the airways. This comprehensive model for examining exogenous surfactant transport through the lungs is developed in the context of SRT.

	CURRENT PROCEDURE FOR SRT

The most common protocol calls for instilling surfactant at 100 mg/kg body wt suspended in saline at 4 ml/kg body wt into the lungs for the treatment of neonatal respiratory distress syndrome (RDS). This is carried out by injecting four quarter-dose aliquots of 1 ml/kg each into the trachea, with the neonate positioned in one of four orientations for each quarter-dose: head up, left or right lateral, and head down, left or right lateral. The infant is hand ventilated for 30 s between each aliquot at a rate of 30 breaths/min, with tidal volumes near 7-9 ml/kg. After treatment and hand ventilation, the neonate is returned to the ventilator (40). Variations of this procedure have been used clinically as well (46). In the instance that a meniscus occludes the trachea and main stem bronchi (7), initial dispersal of the surfactant through the lungs relies solely on the positive pressure from the hand ventilator. Otherwise, surfactant is likely to deposit in the dependent lung according to gravity (3). No studies have addressed how the first inspiration affects distribution of a meniscus in the airways.

	CONCEPTUAL MODEL AND OBJECTIVE

A comprehensive framework is proposed that addresses how a surfactant bolus that fills the bronchi and trachea (6, 7) is dispersed through the lungs. Conceptually, the distribution process is modeled as follows.

Phase 1

View larger version (23K): [in this window] [in a new window]   Fig. 1.   Conceptual model of surfactant dispersal into diseased lungs (from Ref. 6). A: surfactant is instilled, here depicted as a meniscus in trachea. B: phase 1, where bolus is initially pushed into periphery. C: continued spreading and recruitment during phase 2, as surface tension gradient sweeps surfactant distally during recruitment.

Phase 2

To examine this framework of surfactant dispersal, independent models for phases 1 and 2 are developed. The model for phase 1 applies experimental results obtained for a viscous liquid being purged by air from a single capillary tube to a branching lung geometry by using a Weibel morphometric model (43). Liquid layer profiles along the airway tree and the volume of the liquid initially left in the airways and that deposited in the periphery (past generation 14) are presented for different bolus viscosities and inspiration rates. The liquid layer profile established during phase 1 provides the "initial condition" for spreading during phase 2. Phase 2 is examined in a theoretical model similar to those described earlier (9, 11, 18, 24-26) but with the additional capability of simulating surfactant exchange from the bulk liquid to the interface.

Time-dependent evolution equations for liquid layer thickness, surfactant surface concentration, and surfactant bulk concentration are solved in a symmetric Weibel geometry. Effects on transport of sorption kinetics, liquid viscosity, initial liquid layer thickness, initial surfactant penetration into the lungs, gravity, and shear stress due to airflow are considered. Profiles of liquid layer thickness and surfactant concentration are presented, and the amount of surfactant carried to the periphery as a function of time is computed.

	MODEL DEVELOPMENT

Bolus Convection Into the Lungs

The advance of a surfactant bolus by the first inspiration is modeled using experimental results for displacement of a viscous liquid in a small tube by a finger of air (Fig. 2). The liquid has a surface tension ; the meniscus advances with velocity U inside a uniform capillary tube of radius R_c. The radius of the air-liquid interface for the liquid left behind is R_i. After passage of the meniscus, the fraction of the cross section occupied by the liquid coating the tube wall

(1)

can be expressed as a function of the capillary number (Ca = µU/), a ratio of viscous effects to surface tension effects. The conditions that determine the functional relationship between Eq. 1 and Ca have been the focus of investigation since 1935. This problem was initially formulated to examine the motion of an air bubble through a small, liquid-filled capillary tube (2, 10, 41) and has more recently found relevance in tertiary oil recovery (33, 37, 38).

View larger version (23K): [in this window] [in a new window]   Fig. 2.   Capillary model for an advancing bolus. Air moves from left to right, pushing viscous liquid forward and leaving a thin layer of liquid behind. h, Liquid layer thickness; Ri, radial distance to interface; Rc, radial distance to capillary wall; U, meniscus velocity.

As a brief review, the experimental results of Fairbrother and Stubbs (10) indicate that m ~ Ca^1/2 for 10⁴ < Ca < 10². Taylor (41) later confirmed and extended this finding to Ca ~ 10¹. For larger Ca, m tapers off more rapidly than Ca^1/2, asymptomatically approaching m = 0.56 near Ca = 2 (41). These findings (as Ca approaches 0) are in conflict with the theoretical prediction of Bretherton (2), who found that the fraction of liquid lining the tube scaled as Ca^2/3, indicating that less liquid should remain in the tube than was found experimentally. Betherton's experimental results were in good agreement with m  Ca^2/3 for Ca > 10⁴ but gave thicker liquid layers for smaller Ca, where his analysis should have been more accurate. He speculated that the presence of contaminants in the experiments might support a shear stress at the air-liquid interface and thereby explain this difference. After reanalyzing the problem by treating the interface as rigid, the limiting case with surface impurities present, Bretherton found that m actually decreased or slightly increased and, therefore, discounted the effects of surface-active agents. Schwartz et al. (38) reexamined this problem and observed a bubble length-to-diameter ratio (L/D) dependence. Short bubbles with L/D < 15 were in good agreement with the results of Bretherton, whereas long ones (L/D > 25) were more aligned with the experiments of Fairbrother and Stubbs (10) for 10⁵ < Ca < 10³. Ratulowski and Chang (37) showed that surfactants can generate surface tension gradients that increase m by a maximum of 4^2/3 over the result of Bretherton at small Ca (Ca ~ 10⁶). This difference diminishes as Ca increases, with their theory giving m  Ca^2/3 for larger Ca (Ca > 10⁴).

In this work the experimental findings of Fairbrother and Stubbs (10) and Taylor (41) are used to model phase 1 of surfactant dispersal on the basis that 1) the finger of air that advances the bolus distally can essentially be considered as a bubble of infinite length as it passes through the airways; 2) the extension of results of Fairbrother and Stubbs and those of Taylor from a single tube to a bifurcating network of airways was found to be approximately valid in an in vitro experiment (35); and 3) only Taylor provides results for large Ca, allowing treatment of the intermediate airways where Ca can approach 1.

Therefore, the fraction of liquid lining the airways will be described by the experimental results of Fairbrother and Stubbs (10) and Taylor (41). The two regimens for small and large Ca can be captured by the following expressions

(2)

where the latter is similar to that used by Halpern and Gaver (17). These results are used in a symmetric Weibel geometry (43) that is scaled down for a neonate (22).

By computing the local bolus Ca along the airway tree, the fraction of liquid lining the airway walls for a given generation can be estimated from Eq. 2. With viscosity being a property of the bolus, only the local velocity and surface tension of the advancing meniscus need to be determined. This is accomplished by assuming a steady flow rate () at the trachea and dividing it by the total cross-sectional luminal area at the site of the meniscus, R²_i2ⁿ, where n is the airway generation number. Letting R²_i = R²_c(1  m) from Eq. 1, the local Ca is

(3)

For example, depending on the flow rate at the trachea, the viscosity of the surfactant bolus, and the surface tension, the Ca can range from ~1.5 in the trachea ( = 10 ml/s, µ = 0.8 g · s¹ · cm¹,  = 37 dyn/cm) to 10⁴ in generation 14 ( = 2 ml/s, µ = 0.01 g · s¹ · cm¹,  = 22 dyn/cm). Here the value of surface tension was estimated using a dynamic surface tension model for lung surfactant (36). For a given flow rate at the trachea, the area expansion rate at the meniscus was related to the area cycling rate in the dynamic surface tension model. In this manner the dynamic surface tension was determined throughout the airway tree and used in Eq. 3.

Once m is known, R_i can be calculated and the liquid layer thickness (h) can be determined. Because the deposited liquid lining is much thicker than the initial endogenous liquid layer, the former is assumed to consist entirely of instilled liquid. The volume of liquid lining the airways is calculated by integrating the liquid layer thickness along the airway tree. With use of this volume and the assumption of an initial bolus volume, the net volume reaching the periphery during the first inspiration can be estimated. The volume of liquid left coating the airways is presented in RESULTS for different values of viscosity and flow rate.

Recruitment Model

View larger version (17K): [in this window] [in a new window]   Fig. 3.   Plot of initial conditions. A: liquid layer thickness (h/h0) along airways showing Hbol. B: surface concentration in equilibrium with bulk concentration (/0). Exogenous surfactant extends through generation nbol, with endogenous surfactant surface concentration (0/ref) distal to nbol. C: bulk concentration over nbol with a cosine square transition to endogenous concentration (c0/cref). Generation numbers (1-14) are shown along top of plots. h0 = 103 cm.

A summary of the governing equations that describe transport in a symmetric Weibel geometry is detailed in the APPENDIX. They consist of 1) a force balance between pressure gradients, surface tension gradients, airflow shear stress, wall shear stress, and gravity, where pressure in the liquid is determined from Laplace's law, 2) conservation of liquid, relating temporal changes in liquid layer thickness to the net flux of liquid along the airway tree, 3) conservation of surfactant, relating temporal changes in surface and bulk concentrations to the longitudinal gradient in surfactant flux at the surface and in the bulk and to the exchange rate at the air-liquid interface due to sorption, 4) a sorption model that simulates surfactant exchange between the interface and the bulk, 5) an equation of state, relating surface concentration to surface tension, 6) a model for applied shear stress at the air-liquid interface, valid for airflow through an airway network (4), and 7) a ventilation function that provides symmetric and asymmetric patterns with respect to inspiration and expiration times.

In this model, several assumptions are made. 1) Inertial effects in the liquid layer are neglected, because the Reynolds number (Re), a ratio of inertial to viscous forces, is much less than 1. That is

(4)

where  is density, u is the characteristic liquid velocity along the airways, _max is the maximum surface tension, h₀ is the reference liquid layer thickness, and L₀ is the length of the airway tree for a neonate [approximately one-third that of an adult (22)]; 2) effects of cilia are neglected; 3) transepithelial flux of liquid is neglected; 4) longitudinal transport of surfactant in the bulk and at the surface is entirely by convection, because the Peclet number (Pe), a ratio of convective to diffusive transport, is on the order of 10⁵; and 5) exchange of surfactant at the interface is sorption controlled, which is true when transport to the interface by diffusion is rapid compared with sorption to or from the surface.

The governing equations are cast in dimensionless form to reduce the number of parameters and to allow generalization of the numerical results to additional physical situations. The liquid layer thickness, h, surfactant surface concentration (), bulk concentration (c), axial position along the airway tree (x), and time scales are scaled by the respective reference quantities h₀, _ref, c_ref, L₀, and _D. L₀ and _D are the total length of the neonatal conducting airways and desorption time scale, respectively.

From this scaling, six dimensionless parameters are obtained. Two describe sorption: _A/_D = k₂/k₁c_ref, a ratio of surfactant adsorption to desorption time, and _v/_D, a ratio of viscous spreading to desorption time. Here _A = 1/k₁c_ref, _D = 1/k₂, and _v = µL²₀/h_0max, where k₁ and k₂ are the adsorption and desorption constants, respectively. The importance of gravity with respect to surface tension is examined through the Bond number (Bo = gh₀L₀/_max, where g is gravity). For a given tidal volume and breathing period, the effect of shear stress from airflow is measured by f, a ratio of the time for inspiration to the time for a complete breath. Last, two parameters enter from the initial conditions on the liquid layer thickness and surfactant concentration distributions (Fig. 3). H_bol = h/h₀ is the maximum liquid layer thickness in the trachea resulting from the initial dispersal process, and n_bol is the airway generation through which the surfactant has initially penetrated, providing a means of examining heterogeneities in the lungs.

Together, _A/_D, _v/_D, H_bol, n_bol, n_bol, Bo, and f comprise the set of parameters that characterize the process of recruitment. As described earlier, the initial profile for the liquid layer thickness was obtained from the bolus dispersal model (phase 1), with an arbitrary distribution for the exogenous bulk concentration assigned. The surface concentration is initially assumed to be in equilibrium with the bulk.

	RESULTS

The simulations examine how a bolus occluding the trachea and main stem bronchi reaches the periphery during SRT. The two models developed explore different phases of the transport process and are joined to provide a comprehensive picture of surfactant dispersal. First, a family of curves examining the effects of viscosity and ventilatory flow rate on liquid layer thickness (h/h₀), Ca, meniscus surface tension (*), liquid layer thickness-to-airway radius (h/R_c), and accumulated volume (V_acc) vs. distance along the airway tree are plotted. These results in phase 1 are employed as an initial condition for phase 2. Transient simulations provide time evolution profiles of liquid layer thickness, surfactant surface concentration, and surfactant bulk concentration. Effects of surfactant bulk concentration, liquid layer viscosity, initial liquid layer thickness, initial extent of the monolayer, gravity, and shear stress from airflow are presented. Transit times and percentage of surfactant transported to the periphery are reported.

Phase 1: Bolus Dispersal

1

1

View larger version (20K): [in this window] [in a new window]   View larger version (20K): [in this window] [in a new window]   View larger version (14K): [in this window] [in a new window]   View larger version (21K): [in this window] [in a new window]   View larger version (21K): [in this window] [in a new window]   Fig. 4.   Profiles formed after first inspiration. A: h/h0. B: capillary number (Ca). C: surface tension at meniscus (*). D: liquid layer thickness-to-radius ratio (h/R) along airway tree. E: accumulated volume (Vacc) along airway tree for  = 7 ml/s and µ = 0.01, 0.1, 0.2, 0.4, and 0.8 g · s1 · cm1 (where is airflow and µ is viscosity), with h0 = 103 cm. Generation numbers (1-14) are shown along top of plots. x/L0, normalized distance along airway tree.

The liquid layer thickness in the intermediate airways ranges from approximately h/h₀ = 12 for µ = 0.01 g · s¹ · cm¹ to h/h₀ = 95 for µ = 0.8 g · s¹ · cm¹, where h₀ = 10³ cm (Fig. 4A). The layer thickness decreases as one travels distally, primarily because the local velocity of the advancing meniscus slows due to the increase in total airway cross-sectional area. That is, Ca is decreasing (Fig. 4B). Overall, Ca is near maximum in the trachea, ranging from 10² for µ = 0.01 g · s¹ · cm¹ to 1 for µ = 0.8 g · s¹ · cm¹, and diminishes as one travels distally to generation 14.

Surface tension of the meniscus front as it passes through each generation is plotted in Fig. 4C. Little variation exists for the different values of viscosity, with the results for µ = 0.1 g · s¹ · cm¹ only plotted for clarity. The rapid area expansion at the meniscus momentarily decreases the surface concentration of surfactant, raising the surface tension to ~37 dyn/cm through generation 8 and dropping off to 25 dyn/cm in generation 14. However, surfactant is rapidly adsorbed into the interface along the entire network once the bolus has passed (in ~0.02 s, results not shown), returning the surface tension to 22 dyn/cm.

h/R_c increases as flow rate and viscosity increase, reducing the lumen of the airways (Fig. 4D). This is likely to lead to airway obstruction by liquid bridging, which can occur when the volume of liquid lining a cylindrical tube is >5.6R^e_c (31). For an airway, this requires (R²_c  R²_i)L _a > 5.6R³_c, where L_a is the length of an airway and L_a ~ 6R_c. Letting R_i = R_c  h and solving for h/R_c, airway closure can occur for h/R_c > 0.16. Thus, in examining the results for h/R_c, it is useful to keep this value (h/R_c = 0.16) in mind, inasmuch as when it is exceeded, liquid bridging is likely to occur. In Fig. 4D, h/R_c is <0.16 along the entire airway tree only for µ = 0.01 g · s¹ · cm¹. As viscosity increases to µ = 0.1 g · s¹ · cm¹, h/R_c > 0.16 in a region extending from the trachea through generation 6. h/R_c is >0.16 more and more distally as viscosity increases. Hence, the opportunity for airway obstruction in smaller airways increases as flow rate and viscosity increase.

In Fig. 4E, plots of accumulated volume along the airways provide a measure of the volume of liquid left behind in the conducting airways with  = 7 ml/s. For example, the curve for a surfactant with µ = 0.1 g · s¹ · cm¹ reveals that a little more than 0.75 ml of suspension would coat generations 1-14. Conversely, the volume immediately deposited in the periphery (distal to generation 14) for a given bolus volume can be computed using these same results. For example, using the illustration just presented, if one starts with a bolus volume of 1.5 ml at the trachea, a little more than 0.5 ml immediately reaches the periphery ( = 7 ml/s). A family of curves for the total volume of liquid lining the airways (V_A) as a function of ventilation rate is plotted for different viscosities in Fig. 5. Thus, by knowing the viscosity of the surfactant instillation and the ventilation rate, the volume of liquid left in the airways or reaching the alveoli after the initial inspiration can be predicted. Controlling the volume of liquid left in the airways has clinical significance (see DISCUSSION).

View larger version (20K): [in this window] [in a new window]   Fig. 5.   Total volume of liquid remaining in conducting airways (VA) as a function of flow rate applied at mouth for µ = 0.01, 0.1, 0.2, 0.4, and 0.8 g · s1 · cm1. Generation numbers (1-14) are shown along top of plot.

Phase 2: Recruitment Model

1

1

Simulation parameter values examined and the physical phenomena they represent are listed in Table 1. Dimensional quantities used to compute these parameters are as follows: h₀ = 10³ cm (23, 42), c_ref = 10 mg/cm³, _ref = 3 × 10⁴ mg/cm², k₁ = 10⁵ cm³ · g¹ · min¹, k₂ = 1 min¹ (36), L₀ = 8.7 cm, _max = 70 dyn/cm,  = 1 g/cm³, µ = 0.1 g · s¹ · cm¹, and g = 981 cm/s². Tidal volume and breathing period were set to 6.6 ml and 2 s/breath, respectively.

The temporal character of transport is analyzed by tracking the extent to which 50, 75, 95, and 100% of the exogenous surfactant mass penetrates into the lungs. Trajectories of x/L₀ vs. t/_v are given for each fraction of surfactant tracked, where x/L₀ is the axial location along the airway tree. x/L₀ is determined by integrating along the surface and bulk concentrations of surfactant, starting at the trachea and proceeding distally. Once a particular percentage of exogenous surfactant has been "recovered," the value of x/L₀ is noted. Recovering 100% of the exogenous surfactant mass corresponds to tracking the leading edge of the exogenous surfactant layer. This process is described by

(5)

where the asterisk denotes dimensionless quantities defined in the APPENDIX.  = _ref/c_ref h₀, W * is the total airway circumference along the airway tree, f_exo is the fraction of exogenous surfactant to be tracked, M *_exo is the total exogenous surfactant mass, and M *_fendo is the endogenous surfactant mass associated with a given fraction of exogenous surfactant. A specific fraction of exogenous mass is tracked by evaluating the integral until it equals the expression on the right for given values of f_exo, M *_exo, and M *_fendo (15).

Evolution profiles of h/h₀, /_ref , and c/c_ref vs. x/L₀ illustrate how the liquid layer deforms and how the surfactant distributes in time. These results along with the temporal trajectories are used to characterize and evaluate the recruitment process. Effects of sorption, liquid layer viscosity and thickness, initial exogenous penetration, gravity, and shear stress due to ventilation are now examined.

Sorption. The exogenous surfactant used clinically is highly concentrated [25 mg/ml (40)] and rapidly adsorbs to the air-liquid interface. However, to provide a comparison of the present model to previous models, an insoluble surfactant is initially considered, where the surfactant in the bulk does not adsorb. Plots of h/h₀, /_ref, and surface tension are presented in Fig. 6 for t/_v = 0, 0.1, 0.2, and 0.3, where _v = 142 s. Parameter values are H_bol = 8, n_bol = 4, Bo = 0, and f = 0. The initial condition is shown as a solid line. Qualitatively, an upwelling of liquid is generated at the leading edge of the monolayer as spreading begins (Fig. 6A). The diffuse, rather than sharp, front of this wave is characteristic of transport when an endogenous surfactant is present (9, 24). Behind the crest, the liquid thins rapidly to a local minimum in thickness before rising again in the direction of the trachea. The liquid layer over the entire domain is pulled distally by the action of surface tension gradients, with the thickness in the trachea falling as time progresses. Figure 6B shows the interfacial concentration decreasing as the insoluble surfactant spreads into the lung. Gradients in surface concentration and, correspondingly, surface tension are established over the entire domain and diminish as time proceeds (Fig. 6C). A small increase in the distal surface concentration is observed as surfactant reaches the periphery. These results are similar to others in the literature for an insoluble surfactant (9, 11, 24).

View larger version (18K): [in this window] [in a new window]   View larger version (18K): [in this window] [in a new window]   View larger version (19K): [in this window] [in a new window]   Fig. 6.   Insoluble surfactant. Evolution profiles of liquid layer thickness (h/h0), /ref, and * vs. x/L0.  = 0.03, A/D = 1, v/D = 0, Hbol = 8, nbol = 4, Bo = 0, f = 0; h0 = 103 cm, and v = 142 s. Spreading occurs from left to right. Generation numbers (1-14) are shown along top of plots. Profiles for t/v = 0, 0.1, 0.2, and 0.3. v, Viscous spreading; A/D, ratio of surfactant adsorption to desorption time; v/D, ratio of viscous spreading to desorption time; Bo, Bond number; f ratio of time for inspiration to time for a complete breath; t, time.

View larger version (21K): [in this window] [in a new window]   Fig. 7.   Trajectories for different percentages of exogenous surfactant mass along airway tree (x/L0) vs. time (t/v) for an insoluble surfactant. Generation numbers (1-14) are shown along right axis. v = 142 s.

3

1

1

View larger version (18K): [in this window] [in a new window]   View larger version (18K): [in this window] [in a new window]   Fig. 8.   Effect of sorption. Evolution profiles of liquid layer thickness (A) and details at leading edge (B) vs. x/L0.  = 0.03, A/D = 103, v/D = 2.38, Hbol = 8, nbol = 4, Bo = 0, f = 0, h0 = 103 cm, and v = 142 s. Generation numbers (1-14) are shown along top of plots. Profiles for t/v = 0, 0.1, 0.2, and 0.3.

View larger version (17K): [in this window] [in a new window]   View larger version (18K): [in this window] [in a new window]   View larger version (17K): [in this window] [in a new window]   Fig. 9.   Effect of sorption. Evolution profiles of /ref, *, and bulk concentration (c/cref) vs. x/L0.  = 0.03, A/D = 103, v/D = 2.38, Hbol = 8, nbol = 4, Bo = 0, f = 0, h0 = 103 cm, and v = 142 s. Generation numbers (1-14) are shown along top of plots. Profiles for t/v = 0, 0.1, 0.2, and 0.3.

2

View larger version (24K): [in this window] [in a new window]   Fig. 10.   Trajectories for different percentages of exogenous surfactant mass along airway tree (x/L0) vs. time (t/v). Effect of sorption: solid lines, A/D = 103; dashed lines, A/D = 102 (c = 1 mg/ml,  = 0.3). Generation numbers (1-14) are shown along right axis. v = 142 s.

Viscosity. The dynamic viscosity of an exogenous surfactant bolus is controlled by many factors such as lipid concentration and temperature. To examine the effect of viscosity, transport is examined for a higher viscosity (µ = 0.4 g · s¹ · cm¹). The simulation parameters are _A/_D = 10³, _v/_D = 9.53, H_bol = 8, n_bol = 4, Bo = 0, f = 0, and _v = 572 s. As is evident from Fig. 11A, liquid layer thinning is even more accentuated and extends further up the airway tree (generation 4). The gradient in surface tension becomes situated over the thinned region and continues to slowly pump fluid from this location (Fig. 11B). The net effect is a dramatic reduction in the rate at which surfactant is transported toward the periphery. Profiles in /_ref are essentially stationary (Fig. 11B), with 95% of the surfactant "trapped" behind generation 3 (Fig. 12). For further comparison, trajectories for a lower viscosity are also plotted in Fig. 12 (µ = 0.01 g · s¹ · cm¹, _v/_D = 0.238, _v = 14.2). In this case the leading edge propagates to the periphery in <2 s, with 5% residing between generations 5 and 14.

View larger version (18K): [in this window] [in a new window]   View larger version (17K): [in this window] [in a new window]   Fig. 11.   Effect of viscosity. Evolution profiles of h/h0 and /ref vs. x/L0.  = 0.03, A/D = 103, v/D = 9.53, Hbol = 8, nbol = 4, Bo = 0, f = 0, h0 = 103 cm, and v = 572 s. Generation numbers (1-14) are shown along top of plots. Profiles for t/v = 0, 0.1, 0.2, and 0.3.

View larger version (24K): [in this window] [in a new window]   Fig. 12.   Trajectories for different percentages of exogenous surfactant mass along airway tree (x/L0) vs. time (t/v). Effect of viscosity: solid lines, v/D = 9.53 (v = 572 s); dashed lines, v/D = 0.238 (v = 14.2 s). Generation numbers (1-14) are shown along right axis.

Liquid layer thickness. To mitigate the barrier produced by localized thinning, it might be desirable to increase the initial liquid layer thickness as, for example, by pushing the initial bolus into the lung more rapidly. Changing overall liquid layer thickness is examined for H_bol = 26, with _A/_D = 10³, _v/_D = 2.38, n_bol = 4, Bo = 0, f = 0, and _v = 142 s. This corresponds to a flow rate during inspiration of approximately  = 2 ml/s and a viscosity of µ = 0.1 g · s¹ · cm¹. Profiles in h/h₀ (Fig. 13) are similar to previous cases, except the steep leading edge of the wave is more pronounced and more liquid in the trachea is carried distally. Surfactant is transported toward the periphery more rapidly (cf. Fig. 14). The leading edge now carries to the periphery in 14 s.

View larger version (18K): [in this window] [in a new window]   Fig. 13.   Effect of liquid layer thickness. Evolution profiles of h/h0 vs. x/L0.  = 0.03, A/D = 103, v/D = 2.38, Hbol = 26, nbol = 4, Bo = 0, f = 0, h0 = 103 cm, and v = 142 s. Generation numbers (1-14) are shown along top of plot. Profiles for t/v = 0, 0.1, 0.2, and 0.3.

View larger version (22K): [in this window] [in a new window]   Fig. 14.   Trajectories for different percentages of exogenous surfactant mass along airway tree (x/L0) vs. time (t/v). Effect of liquid layer thickness: solid lines, Hbol = 26; dashed lines, Hbol = 8. Generation numbers (1-14) are shown along right axis. v = 142 s.

Extent of initial monolayer. One means of assessing how heterogeneity in the lungs affects transit time is to vary the initial extent to which the exogenous surfactant penetrates; depth of penetration will generally be greatest in more compliant regions. This is examined by extending the exogenous surfactant to generation 8, with _A/_D = 10³, _v/_D = 2.38, H_bol = 8, n_bol = 8, Bo = 0, f = 0, and _v = 142 s. The major outcome of this simulation is that the leading edge of the instilled surfactant, and therefore the gradient in surface tension, is placed over an increasingly thinner liquid layer as n_bol increases (not shown). This results in a diminished upwelling of the liquid at the leading edge and more significant thinning in the distal airways. With a shorter distance to travel, transit time of the leading edge to the periphery decreases (Fig. 15), but the amount of surfactant trapped in the intermediate airways increases (compare with reference curves).

View larger version (22K): [in this window] [in a new window]   Fig. 15.   Trajectories for different percentages of exogenous surfactant mass along airway tree (x/L0) vs. time (t/v). Effect of initial extent of monolayer: solid lines, nbol = 8; dashed lines, nbol = 4. Generation numbers (1-14) are shown along right axis. v = 142 s.

Gravity. Orientation of the infant is thought to influence the distribution of surfactant, favoring the dependent region of the lungs with respect to gravity. The effect of gravity is examined here in the maximized situation in which gravity acts along the axis of each airway. The Bo for this simulation is 0.15, with _A/_D = 10³, _V/_D = 2.38, H_bol = 8, n_bol = 4, f = 0, and _v = 142 s. Figure 16 illustrates the gravitational effects on the liquid layer, whereas Fig. 17 presents concentration data. As shown in Fig. 16, a steep front appears at the leading edge, consistent with earlier results. However, rather than remain stagnant, the liquid in the trachea and intermediate airways flows toward the periphery, propelled by gravity. The slope of the progressing film sharpens into a "kinematic shock." As time proceeds (Fig. 16), a second wave structure forms further upstream. Flow continues down into the distal airways, with the liquid layer in the proximal airways stabilizing, changing little between t/_v = 0.15 and 0.2 (Fig. 16B). Gravity has significantly augmented the surfactant transport in the airways, with one-half of the mass residing between generations 6 and 14 (Fig. 18); however, the effect of gravity is not as pronounced at the leading edge, where the liquid layer is much thinner.

View larger version (19K): [in this window] [in a new window]   View larger version (18K): [in this window]