Abstract
A computational study is presented for the transport of liquids and insoluble surfactant through the lung airways, delivered from a source at the distal end of the trachea. Four distinct transport regimes are considered: 1) the instilled bolus may create a liquid plug that occludes the large airways but is forced peripherally during mechanical ventilation;2) the bolus creates a deposited film on the airway walls, either from the liquid plug transport or from direct coating, that drains under the influence of gravity through the first few airway generations; 3) in smaller airways, surfactant species form a surface layer that spreads due to surfacetension gradients, i.e., Marangoni flows; and4) the surfactant finally reaches the alveolar compartment where it is cleared according to firstorder kinetics. The time required for a quasisteadystate transport process to evolve and for the subsequent delivery of the dose is predicted. Following fairly rapid transients, on the order of seconds, steadystate transport develops and is governed by the interaction of Marangoni flow and alveolar kinetics. Total delivery time is ∼24 h for a typical first dose. Numerical solutions show that both transit and delivery times are strongly influenced by the strength of the preexisting surfactant and the geometric properties of the airway network. Delivery times for followup doses can increase significantly as the level of preexisting surfactant rises.
 pulmonary surfactant
 drug delivery
 surfactant replacement therapy
 respiratory distress syndrome
 Marangoni flow
 airway liquid
 surface tension dynamics
 pulmonary fluid mechanics
direct instillation of a liquid bolus into the lung is common to a number of pulmonary events and clinical treatments. For example, partial liquid ventilation, when using perfluorocarbon liquids, has been suggested for treating respiratory distress syndrome (RDS) either in place of, or in conjunction with, surfactantreplacement therapy (SRT) (20, 50, 72, 79). Perfluorocarbon liquids have low surface tension and high oxygen and carbon dioxide solubilities and have been shown to improve lung mechanics and gas exchange. As another example, present investigations of gene therapy for cystic fibrosis and α1 antitrypsin deficiency utilize delivery of the vector (e.g., adenovirus, liposome) onto the airway epithelial cells by liquid bolus (4, 8, 10, 37). Liquid delivery has also been recognized as a potential means to “piggyback” delivery of drugs (e.g., during cardiopulmonary resuscitation) and unwanted environmental toxins (22, 44, 49). Introduction of liquids into the lung also occurs in therapeutic and diagnostic bronchial alveolar lavage. A very prevalent application is SRT.
The delivery of exogenous surfactants into the lung for SRT is now a standard treatment for neonates with RDS (9, 46, 48, 54). In some studies, it has reduced infant mortality by onehalf (54). The delivery method may be a bolus instilled into the trachea or an aerosol mixture (51, 81) and has been studied either as a prophylactic dose at birth or as rescue doses given several hours after delivery (48). At this juncture, the more popular treatment is the intratracheal bolus that spreads by a combination of various physical forces. The initial spreading can be quite rapid (11), reaching substantial amounts of the lung fields in 20 s. The early response of improved oxygenation for the patient appears to be due to an increase in functional residual capacity (25). Exogenous surfactant administration has also been used as a therapy for acute RDS (ARDS) (53, 69), for sepsisinduced ARDS (3) by aerosol, for mitigation of oxygentoxic lung injury (56) and woodsmoke inhalation injury (18), for improvement of lung transplant results (58), and for treatment of meconium aspiration (78).
Strategies for optimizing liquid delivery into the lung depend, necessarily, on the particular application (SRT, liquid ventilation, gene therapy, drug delivery, etc.). In some cases, it may be desirable to transport the liquid primarily to the alveoli, in others, it may be more effective to coat primarily the airways. It may be important for the liquid to spread homogeneously or to be directed preferentially to specific lobes or generations. The residence time could be long or short. It may be advantageous to “blow” the liquid as a plug into the airways or to let it drain slowly into the lung.
In SRT, several parameters involving the physiology and the delivery technique may affect the transport (67): the bolus volume (24); its injection rate (73); gravity and orientation (73); development of airway occlusion by the liquid; ventilation parameters at normal or high frequency (38, 65); the viscosity and surface tension of the fluid injected; the dose strength; the instillation site; and repeatdosing protocols and intervals. There is evidence, for example, that a second dose of SRT tends to distribute to lung regions where the first dose was transported (73), possibly because of the opening of airways and ease of transport for the second dose through them. On the other hand, there may be delays in seconddose transport because the first dose ultimately lowers the surfacetension gradient driving the flow of the second dose (28). It is known, for example, that the second and following doses can be much less effective than the first dose (54), possibly because of the reduced gradient. The clearance of instilled surfactants is also very important in the overall transport, as is discussed below. In clinical studies, the nonresponse rate to instilled surfactants ranges from 15 to 35%, for example, depending on the study and patient group. Could the lack of response be due, in part, to inadequate surfactant transport and delivery? Consider the delivery pathway of a liquid bolus as it makes its way from the trachea to the alveoli. It may start as a liquid plug, progress to a deposited film lining the airways, establish a surface layer, and then reach the alveolar compartment. These four transport regimes are dominated by different physical forces.
The liquidplug transport regime occurs if the liquid volume instilled is large enough and given over a short enough period for it to occlude the airway. Then the plug flow is driven by the pressure drop across the plug during inspiration, and the resulting motion depends on its viscosity, density, surface tension, and gravity. As the plug is blown peripherally, it deposits its liquid onto the airway wall, leaving behind a trailing film the thickness of which depends on the system parameters. Eventually, through the action of subdividing at airway bifurcations and depositing its mass onto the airway wall, the plug will lose enough liquid that it ruptures. This mode is likely to be operative in the trachea and larger airways.
The depositedfilm transport regime occurs after plug rupture or direct coating. The resulting film coating the airways will flow from combinations of gravity, airflow shear effects, and surface tension, and these effects may compete depending on the system parameters. This mode is probably dominant in the largetomediumsized airways.
When the liquid and its constituents (such as surfactant) form a surface layer, then surfacetension gradients (when present) become significant whenever gravity and capillarity are weak, as is the case in thin layers. These gradients cause Marangoni flows that distribute the surfactant. This regime is likely to be present in the mediumtosmall airways. The fundamental fluid mechanics and transport phenomena of surfacelayer surfactant spreading were reviewed in Refs.26 and 27. The available theoretical models of the Marangoni flow on thin, viscous films are based on lubrication theory (5, 16, 22, 28, 31,42, 43, 45, 71), from which coupled evolution equations for the film depth and the surfactant concentration are derived. If the surfactant is localized on an otherwise clean interface (Fig.1 A), the unsteady spreading flow generates a wave that travels in the direction of lower surfactant concentration (higher surface tension) (Fig. 1 B). If surface diffusion of the monolayer and gravity are negligible, the wave behaves like a shock wave, with rapid changes in height and surfacetension gradients over a very short distance. The film thickens to twice its undisturbed height at the traveling shock, and the film thins significantly behind it, so much so that it may rupture there (21, 42). Film rupture causes the spreading to stop, an unwanted result for SRT. The speed of this advancing shock wave depends on the surfacetension difference driving the flow, the film thickness, the surfactant activity, and the fluid viscosity.
In physiological applications, there is a preexisting or background surfactant already on the interface before exogenous surfactant is added. It may arise from natural (endogenous) sources or from previous SRT treatments. If the surfactant is localized on an interface with preexisting surfactant (Fig. 1, C andD), the leading edge of the new exogenous material (L _{ex}), spreads more slowly because of the background surfactant (28). This is due to the smaller surfacetension gradients. However, a second phenomenon arises: compression of the background surfactant as the exogenous surfactant spreads. This compression wave causes the background surfactant particles to move closer together, i.e., to increase in concentration, and the wave speed is faster than the spreading speed of the exogenous surfactant. Thus the leading edge of the compression wave (L _{D}) travels ahead of L _{ex}(Fig. 1 D). For larger initial background concentrations, the compression speed actually increases because of greater mobility in the interface while the spreading speed decreases. These phenomena were presented and discussed in Ref.28.
Components within the liquid (surfactants, drugs) may reach the alveolar compartment where the transport conditions may include removal and production kinetics. The clearance mechanisms for instilled surfactants, both lipid and protein fractions, are not entirely understood, although several studies have addressed some of the key issues (80). Alveolar type II cells appear to take in the vast majority (7, 64) and can recycle a portion for secretion. Minor amounts appear to be taken up by alveolar macrophages and bronchial Clara cells (7,60). A few percent exits by the proximal airway (61). Treatment doses of surfactant are often as much as ten times the endogenous pool of surfactant. There have been a number of clearance studies examining recovery of radiolabeled surfactant from alveolar wash or lung tissue. Although several early papers have viewed surfactant as being cleared at a fixed percentage (of the initial mass) per hour (19, 60, 62, 68), it has become more clear that the kinetics is first order (63, 64). It has been shown that clearance rates can be modified if there is lung injury. For example, it was found by Novotny et al. (59) that clearance rates for adult rabbit lungs with prolonged 100% oxygen exposure were lowered. Such changes become important in determining dosing regimens for the injured lung, as may occur in ARDS. Also, some acute injuries may not affect clearance. The acute lung injury models shown in Refs.35 and 52 were made with injections ofNnitrosoNmethylurethane. Clearance of instilled surfactant was similar to that in controls (52), there was altered endogenous surfactant metabolism in response to surfactant treatment in the injured animals, and exogenous surfactant was beneficial to the injured animals (35).
In an earlier study (44), we examined surfactant spreading in a lung model based on Marangoni flow alone. That model allowed for the rapid increase in airway surface area due to airway branching, which can quickly dilute the spreading surfactant. This surfacearea dilution reduces the Marangoni mechanism locally and dramatically slows the process: transit times of the order of 2–3 h for an adult and 10–20 min for an infant were predicted by using zero flux end conditions. In the present work, we extend and improve this model to account for the other three transport regimes mentioned above. The model remains one dimensional, so that much of the geometric complexity of the bronchial tree is ignored, although the salient geometric features are retained. We shall estimate transit times and the time required for essentially complete delivery of the surfactant dose to the alveoli. How these transport times depend on the system parameters will be a main focus of the work. Through this modeling, we seek to develop an understanding of the fluid mechanics and transport of liquid delivery into the lung. Such an approach to overall lung transport for instilled liquid delivery, including exogenous surfactants, can provide a rational basis for developing strategies to optimize their delivery.
FORMULATION OF THE MODEL
Glossary
It will be useful and instructive to cast several of our variables in dimensional terms and in their dimensionless counterparts. We shall adopt the convention of using lowercase symbols to denote dimensional variables and uppercase symbols to denote their dimensionless version.
 α
 Fraction of fluid in draining region
 a, A
 Total airway crosssectional area
 a_{e}
 Total airway crosssectional area exposed to air
 a_{n}
 Total crosssectional area at generationn
 a_{0}
 Tracheal crosssectional area
 A_{A}
 Total alveolar surface area
 A_{tr}
 Crosssectional area of endotracheal tube
 b, B
 Total airway perimeter
 B̂
 Scaled perimeter function used in Marangoni flow regime
 b_{n}
 Total airway perimeter at generation n
 b_{0}
 Tracheal perimeter
 β
 Perimeter parameter
 Ca
 Capillary number
 Ca_{tr}
 Tracheal capillary number
 d_{n}
 Mean airway diameter at generation n
 d, D
 Airway diameter
 d_{0}
 Tracheal diameter
 Δ
 Airway taper parameter
 Δς_{n}
 Surfacetension difference over lengthl_{n}
 F
 Ratio of surfactant delivery to the alveolar space to the rate of uptake
 F̂
 Rescaled delivery of surfactanttouptake ratio
 g
 Gravitational acceleration
 G
 Ratio of typical gravity draining speed to Marangoni speed
 γ,Γ, Γ̂
 Surfactant concentration
 γ_{A}, Γ_{A}
 Alveolar surfactant concentration
 Γ_{eq}
 Equilibrium alveolar surfactant concentration
 Γ̂_{eq}
 Equilibrium surfactant concentration in Marangoni flow regime
 h, H, Ĥ
 Film depth
 H_{eq}
 Equilibrium film thickness in the Marangoni regime
 H_{M}
 Critical film thickness for transition from gravity to Marangoni regimes
 h_{n}
 Liquid lining thickness at generationn
 K
 Rate constant for alveolar surfactant uptake
 l_{n}
 Mean airway length of generation n
 l_{0}
 Tracheal length
 L_{D}
 Leading edge of surfacecompression wave
 L_{ex}
 Leading edge of exogenous surfactant
 L_{M}
 Marangoni regime length
 L_{0}
 Total airway path length
 λ
 Leading edge of bolus draining under gravity
 Λ
 Ratio of tracheal length to four times the path length
 m, M
 Mass of exogenous surfactant delivered to the alveoli
 m_{dose}
 Dose of surfactant delivered
 μ
 Fluid viscosity
 n
 Airway generation number
 N_{A}
 Avogadro’s number
 n_{r}
 Critical generation number for plug rupture
 p_{A}, P_{A}
 Alveolar surfactant production rate
 q,Q
 Surfactant flux and fluid volume flux in Marangoni regime
 q_{a,}Q_{A}
 Surfactant flux into the alveolar compartment
 r, R, R̂
 Airway radius
 r_{n}
 Mean airway radius at generation n
 r_{0}
 Tracheal radius
 ρ
 Liquid density
 S_{M}
 Surfacetension difference across the Marangoni regime
 S_{0}
 Surfacetension difference along the surface layer
 ς
 Surface tension
 ς_{n}
 Surface tension at generation n
 t
 Time
 T
 Dimensionless time used for deposited film flow
 T_{α}
 Transit time for gravitydriven flow
 Ti
 Inspiration time
 T_{M}
 Marangoni time scale
 T̂_{s}
 Surfactant transit time at steady state
 T̂
 Dimensionless time for surfacelayer transport
 τ
 Alveolar uptake time variable
 τ_{D}
 Exogenous surfactant delivery time
 θ
 Surfactant activity parameter
 φ
 Film thickness correlation function
 U
 Speed of propagation of liquid plug
 U_{g}
 Typical speed of gravitydriven drainage
 U_{M}
 Marangoni velocity scale
 Û_{s}
 Marangoni velocity at airliquid interface
 U_{tr}
 Tracheal velocity scale
 v, V
 Liquid bolus volume
 V_{b}
 Initial bolus volume
 V˙
 Airflow rate
 V_{p}
 Initial liquid plug volume
 Vt
 Tidal volume
 V_{tr}
 Tracheal volume
 W_{b}
 Ratio of bolus volume to tracheal volume
 W_{p}
 Ratio of initial plug volume to tracheal volume
 W_{r}
 Value of W_{p} that will rupture at generation n
 x, X
 Distance along fluid layer, measured from the tracheal carina
 X̂
 Dimensionless distance along fluid layer starting atgeneration 7
 X̂_{a}
 Size of domain of Marangoni regime
 x_{n} , X_{n}
 Distance along fluid layer to generationn
 ξ_{1}, ξ_{2}
 Initial condition parameters for surfacelayer flow
Lung morphometry.
The transport models we develop require as input a mathematical description of airway geometry. We have employed the model used in Ref. 76, which assumes that the adult lung is a symmetric, dichotomous branching tree, in which the mean length of an airway is proportional to its diameter and for which the airway volume for each generation is constant. According to this model, the number of airways at generation n is 2^{n}, for 0 ≤n ≤ 23, and the mean airway diameter (radius) is d_{n}
(r_{n}
), the mean airway length isl_{n}
, the total crosssectional area of the airways isa_{n}
, the total airway perimeter isb_{n}
, given respectively by
For example, using the above formulation, the path distance to the beginning of generation 9 isX _{9} =x _{9}/L _{0}= 1 − 2^{−3} = 0.875. At that location, the airway diameter (radius) is 0.125 times the tracheal diameter (radius), the crosssectional area is 8 times the tracheal crosssectional area, and the total airway perimeter is 64 times the tracheal perimeter. The beginning of the alveolated region of the lung can be represented by generation 18, say, which is atX _{18} =x _{18}/L _{0}= 1 − 2^{−6} = 0.984. We use this value as our boundary with the alveoli, so that the singularities inA(X) andB(X) as X → 1 (seeEq. 3 ) are always avoided. The Weibel model describes the mean diameter of the first 10 generations reasonably accurately but underpredicts the diameter for generations beyond n = 10 and overestimates the number of airways at large n (77). The derived formula for distance along the path as a function of airway generation, Eq. 2 , approximates measurements of path length (29, 75) within a few percent, except forgenerations 1 and2 where the error is larger. No allowance is made for asymmetry in airway branching. However, we use this model to keep the analysis relatively simple. Employment of more sophisticated functions of X will be possible in future studies.
In the Weibel model (76), the pulmonary tree is selfsimilar, so that the scaling relationship between adjacent airway generations is independent of generation number. It is, therefore, possible to employ the same functional forms forD(X),R(X),A(X), andB(X) to represent truncated portions of the pulmonary tree, although the reference path lengthL
_{0}, the reference perimeter b
_{0}, and the X range of interest must be redefined appropriately. This is particularly useful for representing infant lung morphometry, necessary for predictions of liquid and surfactant transport in our analyses. We assume for simplicity that the neonatal lung may be modeled by equating the neonatal trachea to the adult generation 7 airway, and then use the distal adult lung section, 7 ≤n ≤ 18, as the remaining neonatal lung. Therefore, the neonatal trachea diameter is equivalent tod
_{7} = 0.36 cm according to Eq. 1
, a value typical for premature infants. Then the reference quantities inEq. 3
would be replaced by thegeneration 7 values for a single airway, i.e., d
_{7}replaces d
_{0},r
_{7} replacesr
_{0}, πd
_{7} replacesb
_{0,} and π
We now present some analyses of the four transport regimes: liquid plug, deposited film, surface layer, and alveolar compartment. For delivery of surfactants, for example, we shall see that the liquid plug flow and the initial drainage of a deposited film due to gravity occur on the order of seconds. The ultimate delivery of the surfactant to the alveoli is governed by a balance of surfactant supply along the surface layer and surfactant uptake in the alveolar compartment, which occurs on the order of hours. The details of the first two regimes are given to demonstrate the relevant transport mechanisms. The initial distribution of surfactant from these relatively rapid events then provides input to the second two regimes.
Liquid plug flow. After a liquid bolus of surfactant is delivered into the trachea, it may be large enough to occlude the airway. If so, it will initially be pushed into the distal regions of the lung by the ventilatory airflow. As this liquid plug propagates through the tracheobronchial tree, driven by a constant airflow rate V˙, it leaves behind a trailing liquid film of thickness h coating the airway (see Fig.2
A). As long as it is not picking up comparable amounts of liquid from the airway wall ahead, the size of the plug will diminish until it ruptures (Fig. 2
B). More complex situations involving airway liquid linings, which are comparable in thickness to the trailing film and in which airway flexibility is important, are not treated here. An estimate of how far a liquid plug travels through the lung before it ruptures is made by using simple massconservation arguments. The change of plug volumev, with respect to distancex, is given by
This type of flow has been studied by previous investigators who examine the motion of long bubbles in tubes (6, 70). Their results indicate that the ratio of deposited film thickness to airway radius,H =h/r, depends on the capillary number,Ca = μV˙ /(ς a
_{e}), which is a dimensionless airflow speed. Here, μ is the fluid’s viscosity, and ς is its surface tension (assumed for the present to be constant). Note that Ca is a decreasing function of x, sincea
_{e} increases withx. It is convenient to relate the variable Ca to its tracheal valueCa
_{tr}, such that
A liquid plug in the airways can only proceed distally if it is inflating the lung region ahead of it. Blowing a plug into the airways, as discussed above, accomplishes this. Gravity, on the other hand, is not likely to provide enough force for the distal motion of an intact liquid plug. However, gravity can disrupt its motion and cause it to drain along the walls.
It is important to consider under what conditions a liquid plug is formed during tracheal instillation. Experimental studies of the criteria for plug formation during instillation have been presented in the work of Espinosa and Kamm (15), for example, in which effects of flow speed and duration, along with fluid properties, are examined. For the purposes intended here, we shall consider an initial liquid bolus instilled into the trachea as immediately coating the tracheal wall uniformly. Then the pertinent issue becomes what liquid bolus volume, when delivered into the trachea, would be large enough to form a liquid plug. From stability studies of liquidlined tubes (17, 32, 33, 47), a uniform film coating the walls will form a plug when the liquidfilm thickness divided by the tube radius His roughly >0.12–0.16. The range depends mainly on the surfactant concentration, its strength or activity, the tube length, and the relative wall flexibility (33). Once the film becomes unstable, it will quickly form a plug over a time interval on the order of μr /ς (34), which is much shorter than 1 s over a wide range of parameter values. From simple volume calculations, the film’s initial thickness in the trachea is
A significant issue in the practical aspects of surfactant and liquid delivery into the lung is the regurgitation, or reflux, of material out of the trachea following instillation. One potential explanation of this phenomenon is related to the criterion for plug formation discussed above. For any airway, not just the trachea, if the liquid lining becomes too thick, i.e., H ≥ 0.2, then it will form a plug, given sufficient time. As the tracheal plug is blown distally during inspiration, the trailing film thickness may exceed this criterion in some airway generations that will be subject to formation of their own plugs. Depending on which airway generation is involved and when this happens in the respiratory cycle, these newly formed plugs may be convected out of the trachea during expiration. The clinician who encounters reflux may respond by trying to blow in the tracheal bolus more forcefully with the intent of quickly pushing it to the alveolar region. Our model indicates that this approach could be counterproductive, since the reflux may be a result of a newly formed plug and not the original plug, which could have reached the distal parts of the lungs. Also, more forceful delivery implies a larger Ca and, hence, a thicker film.
Depositedfilm flow. The advancing plug leaves behind itself a film of thicknessh. Once the plug ruptures, this trailing film contains the liquids or surfactants that may need to reach the alveoli. The transport of this film then becomes an important issue. Here, we want to determine whether this film flow is dominated by gravity or by Marangoni forces. Although airways are oriented in many different directions, clinically, the patient may be positioned at several angles during the delivery process, so that the majority of airways may experience appreciable gravitational forces directed distally.
The speed of gravitydriven drainage of the liquid lining in a single airway generation n is approximatelyU
_{g} = ρ g h
We can estimate the magnitude ofH _{M} for delivery of a liquid bolus to an adult lung by taking ρ ≈ 1 g/cm^{3};g ≈ 10^{3}cm/s^{2}, andS _{0} ≈ 50 dyn/cm. This surface tension difference is initially distributed across the path length L _{0}≈ 15 cm, yieldingH _{M} = 50 μm. As shown later, for typical ventilation rates and tracheal plug volumes in an adult, the trailing film reaches this value ofH _{M} neargeneration 7. For a neonate, the initial surfacetension gradient is distributed over a length of only 3.75 cm, so H _{M} = 200 μm. This value ofH _{M} is 11% of the tracheal radius, a value that may lead to plug formation. Then transport will, again, be dominated by airflow.
We first consider the gravitydriven drainage regime in an adult. For simplicity, we are considering that the deposition by the liquid bolus occurs first, followed by drainage. It is helpful to consider two extreme cases. One case is when the instilled bolus forms a plug in the trachea and it ruptures in the trachea. Equivalently, this starting condition could be achieved by direct deposit of the initial liquid bolus on the tracheal walls. Either way, this would leave the entire bolus volume to drain from the trachea to the distal airways. The other case is when the plug ruptures or persists in the alveolus, leaving a coating over all airways.
If the bolus ruptures in the trachea, then the initial bolus volumeV
_{b} drains unsteadily and nonuniformly down the airway walls (see Fig.3). We model this process, by extending existing theories (40, 57) of flow down a vertical surface to include the increase of surface perimeterB(X) (see Eq. 3
), along the draining axis, as occurs in the lung. Applying conservation equations for mass and momentum for lubrication flow, the resulting evolution equation for the dimensionless film thicknessH (X,T) is found to be
The other extreme case is when the plug ruptures or persists at the alveolar level. Now there is liquid deposited betweengenerations 0 and18. BecauseH (X,T) in Eq. 11 is a similarity solution, eventually, the deposited liquid is likely to evolve to this distribution. Now the appropriate range oft is for times greater than the time to rupture. So we see that the two extremes yield similar features: front transit on the order of seconds, and further drainage of the liquid lining on the order of hours.
Surfacelayer transport. Surfacetension effects dominate spreading once the film becomes sufficiently thin, as shown in the previous section. We examine two types of surfacetensiondriven flows: Marangoni flow driven by surfacetension gradients (Fig. 1) and flows driven by axially varying pressure gradients associated with nonuniform curvature of tapering airways. The analysis (given in ) shows that Marangoni flows are eventually much stronger than those due to nonuniform curvature. Both the initial rapid transient behavior of these flows and their subsequent steady states are considered.
The unsteady, transient flow created by the surfacetension gradients, although shortlived, is important to understand, since it may cause certain undesirable events to occur. For example, as the flow is initiated, the airway liquidlining thickness,Ĥ(X̂,T̂) =h (x,t)/h
_{7}and the surfactant concentration,Γ̂(X̂,T̂) = γ(x,t)/γ_{7}, change as functions of X̂ andT̂, where the hat over the variables indicates new scalings that better represent the Marangoni regime (see
). Theh
_{7} is the reference film thickness at generation 7 in the adult. The surfacetension difference betweengenerations 7 and18 isS
_{M} = ς_{18} − ς_{7} = −θ(γ_{18} − γ_{7}), where the surfactant activity θ represents the surfacetensionreducing capacity of the monolayer and is taken to be constant. This is equivalent to assuming a linear equation of state for the surface tensionsurface concentration relationship. The scaling for the axial variable is the Marangoni regime length,L
_{M} =x
_{18} −x
_{7}, and the scaling for time isT
_{M} = μ
Conservation of mass and momentum lead to the governing equations forĤ and Γ̂, Eq.EA1 in . These include the parameter Δ, representing the effects of surfacetensiondriven flows due to airway taper. The equations are solved numerically in the domain 0 ≤ X̂≤ X̂ _{a}, which corresponds to the pathway segment from the beginning ofgeneration 7 to the beginning ofgeneration 18, where the alveolar boundary is located. If the lining becomes too thin at a particular value of X̂, it may rupture there because of destabilizing van der Waals forces, i.e., Ĥ = 0, which may lead to a cessation of spreading (42). However, ifĤ becomes too large, then there may be plug formation, as discussed above, which will also stop spreading. Before examining the effects of surfacearea expansion, we consider an unsteady solution of Eq. EA1 forĤ and Γ̂ for a single, uniform tube.
Figure 4 shows the time evolution ofĤ and Γ̂ for the case of a single, uniform tube with the upstream and downstream surfactant concentrations fixed at Γ̂ (X̂ = 0,T̂) = 1 and Γ̂(X̂ =X̂
_{a},T̂) = Γ_{A} = 0.2, respectively, where Γ_{A} = γ_{A}/γ_{7}. The initial conditions for Ĥ and Γ̂ are discussed in the
. Shear stresses from the initially large negative gradient in surfactant concentration drive a flow in the X̂ direction, causing the fluid layer to well up behind the leading edge of the advancing disturbance (e.g., at T̂ = 0.1). As the monolayer advances, the disturbance first grows and then diminishes in size. At T̂ ≈ 2, the leading edge of the disturbance in Γ̂ reachesX̂ =X̂
_{a}, and a nonzero surfactant gradient is established there. This surfactant gradient increases and induces film thinning at the downstream end until T̂ ≈ 4, when the fluxes of surfactant at X̂ = 0 andX̂ = X̂
_{a}equalize, and Γ̂ has essentially reached a steady state. The Ĥ distribution evolves for a slightly longer time. These steady solutions are obtained by settingΓ̂
_{T̂} =Ĥ
_{T̂} = 0 in Eq. EA1
, which can be integrated to yield
The leading edge of the exogenous surfactant distribution, shown by the black markers in Fig. 4 B, takes approximately half a time unit to reach the downstream end, significantly longer than the time taken for the disturbance first to reach X̂ =X̂ _{a}. As was shown in Ref. 28, an increase inΓ _{A} causes the transit time of exogenous surfactant to increase, since the surfacetension gradient driving the flow is reduced.
The effect of the lung’s surfacearea expansion on the unsteady spreading of surfactant is shown in Fig. 5.Ĥ and Γ̂ are plotted as functions of X̂ (on a logarithmic scale) and the equivalent generation number n. Pressuredriven flows due to changes in airway radius are neglected for the time being. AsĤ (X̂,T̂) in Fig. 5 A evolves from the initial conditions (given in ), a kinematic wave propagates from left to right (as in Fig.4 A), with thinning occurring at the upstream end of the domain. After the wave reaches the downstream end, the film begins to thicken, and the wave is damped. Compared with Fig.4 B, the leading edge of the surfactant front in Fig. 5 B progresses to the distal airways more slowly because surfactant has to distribute itself over an expanding surface area and also because the initial film thickness (given by Eq. EA4 ) is thinner than that used in Fig. 4. At T̂ = 0.6, a nonzero surfactant flux at the downstream end is established, which is weaker than the uniformtube case. Whereas the surfactant concentration increases monotonically with time at fixed X̂ in the uniform tube (Fig. 4 B), this is not the case (Fig. 5, B andC) for an expanding surface area. When the disturbance first reaches the downstream end of the domain, the fluid layer is relatively thin, so that large shear stresses are needed to drive the flow (T̂ = 1, 2), and, hence,Γ̂ rises to relatively high values ingenerations 13–17 (Fig.5 C). Later, as fluid is driven distally and the liquid layer thickens in this region (Fig.5 A), the viscous resistance to flow falls and the surfactant gradients fall also, causingΓ̂ to fall (T̂ ≥ 4).
A steady state is reached once T̂ ≈ 7. Analytical steadystate solutions can be obtained from the governing equations (Eq. EA1
) and are given by
Alveolar compartment transport. When liquid and surfactant from the instilled bolus finally reach the alveolar region, alveolar surfactant kinetics begin to play a central role. As surfactant accumulates in the alveoli, the average concentration there, γ_{A}, will slowly rise and weaken the Marangoni flow. To determine the delivery time more accurately, a timedependent model of alveolar surfactant uptake is developed. Treating the alveolar space as well mixed, the average surfactant concentration there can be modeled by using a simple model for the kinetics of alveolar surfactant (Fig.6)
It appears that a halflife in the range of 5–15 h occurs for many of the surfactants used in alveolarwash kinetics studies (60, 63, 64). Therefore, a reasonable estimate of the rate constant range is 0.046/h ≤ K ≤ 0.138/h. It is convenient and instructive to recast Eq.14
in nondimensional terms. We define the nondimensional variables as follows: the time τ =Kt, which is scaled on the uptake rate; the alveolar surfactant concentration Γ_{A} = γ_{A}/γ_{7}; the flux Q
_{A} =q
_{A}/q
_{7,}such that q
_{7} = γ_{7}
b
_{7}
L
_{M}/T
_{M}and P
_{A} =p
_{A}/(γ_{7}
K) is a parameter representing the ratio of natural surfactant supply to its uptake. Expressing Eq. 14
in these nondimensional variables, we have
RESULTS
Liquid plug flow. As was shown informulation of the model,Liquid plug flow, the volume of a liquid plug (Eq. 7
) depends critically on the tracheal capillary number,Ca
_{tr}, and on the ratio of initial plug volume to tracheal volumeW
_{p}. The adult tracheal volume isV
_{tr} = π
The airflow rate, V˙, can be expressed in terms of the tidal volumeVt
and an inspiration timeTi
asV˙ =Vt
/Ti
. Defining a tracheal velocity scale to beU
_{tr} =Vt
/(Ti A
_{tr}), where A
_{tr} is the crosssectional area of the endotracheal tube, theCa
_{tr} is then
In Fig.7 A, the plug volume given by Eq. 7 is plotted vs. generation number for various values ofW _{p} andCa _{tr} that fall in the range calculated above. The plots show that the plug will rupture proximally for higher values ofCa _{tr} because the thickness of the deposited film is larger (see Eq.6 ), and for smaller values ofW _{p}, since there is less volume to distribute. Therefore, the generation number at which the plug ruptures decreases with decreasingW _{p} and increasingCa _{tr}. FromEq. 7 it is possible to compute the critical value ofW _{p} (call itW _{r}) that will rupture at a specified generation numbern =n _{r}. Figure7 B shows plots ofW _{r} as a function of Ca _{tr} for selected values ofn _{r} where clearlyW _{r} increases withCa _{tr} and airway generation. As shown in Fig. 7 A, an increase in Ca _{tr}by a factor of 10, say from 1.2 to 12, does not have a significant effect on the deposited liquid distribution because of the asymptotic behavior of Eq. 6 . IfW _{p} is sufficiently large orCa _{tr} is sufficiently small, then the liquid plug may not rupture in the airway domain. This could represent direct convection of instilled liquid into the alveolus. However, this is unlikely to occur, since there is always some gas trapped between the plug and the alveoli. The pressure in the gas would rise until the motion of the plug stopped somewhere proximal to the alveoli.
Direct instillation into the alveoli may be a desirable goal. Figure8 is a plot of the alveolar plug volume, i.e.,V(X _{18}) from Eq. 7 , as a function ofCa _{tr} for several values of W _{p}. This figure shows that direct instillation into the alveoli is enhanced by larger volumes and smallerCa _{tr}. As inferred from previous figures, the plug volume at generation 18 decreases with increasingCa _{tr} and decreasing W _{p}. For the range of parameters considered, rupture does not occur atgeneration 18 (and earlier) ifCa _{tr} falls below 0.0035 and if W _{p}≥ 0.5. Also, theCa _{tr} needed for rupture increases significantly ifW _{p} is doubled, for example from 0.5 to 1.
From Eq. 5 , asCa increases, so does the thickness of the trailing film, H (seeEq. 6 ). This is illustrated in Fig.9, where His plotted as a function of airwaygeneration numbern for several values ofCa _{tr} andW _{p}. It shows that for fixed Ca _{tr}and W _{p},H is a decreasing function ofn, sinceCa decreases as the total crosssectional area increases with n.H does not exceed 0.14, which is within the range necessary for closure, at any airway generation, provided Ca _{tr} is <0.05.
Depositedfilm flow. From the similarity solution, Eq. 11
,H(X) is plotted for several values of timeT in Fig.10. The smallest dimensionless value ofT, T= 200, was chosen so that H does not exceed the critical value for closure. Also, in dimensional terms, this value is quite small, implying that we can apply this similarity solution to accurately predict the position of the leading edge of the bolus with time. Figure 10 shows thatH =h/rincreases with distance for fixed T, but the dimensional film thickness hdiminishes with X. Because surfacetension effects are being neglected, there is a sharp leading edge where H drops to 0. With time, the leading edge of the bolus advances, causing the film to thin. It takes ∼6,000 time units for the leading edge to reachgeneration 7, whereH is of the same order asH
_{M}. In Fig.11, the front location λ(T) is plotted for several values of the productW
_{b}Λ. It shows that initially the leading edge advances quite rapidly, but, as λ approaches the more distal airways, the rate of increase slows down as the liquid bolus is shared between an increasing number of airways. From Eq. 11
, the dimensionless time taken for the front of the deposited film to move a distance λ by gravity is found by inverting the definition for λ(T) in Eq.11
For a liquid bolus withV _{b} = 3 cm^{3}, having the viscosity of water, the leadingedge film thickness falls beneath the critical film thickness of 50 μm (at which gravitational and Marangoni forces are of equal magnitude in the adult) after ∼3 s (Liquivent) or 181 s (Survanta) at X = 0.832, very close togeneration 7. [For largerV _{b}, the bolus travels quicker and deeper into the lung before reaching the critical height: for example, ifV _{b} = 1 (or 5) cm^{3}, the critical height is reached at generation 4 (or9) after 6.3 (or 2.02) s.]
Thereafter, fluid drains past this location, providing a continuous flux of material to the distal generations. By integrating the volume flux at generation 7, for example, with respect to time, it may be shown by using Eq.11
that the time taken for a fraction (1 − α)V
_{b} of the fluid bolus to pass through generation 7 is
Equations 10, 11, 17
, and
18
can be applied pastgeneration 7 if a constantsurfacetension bolus of perfluorocarbon is used. For example, the leading edge of a 3cm^{3} liquid bolus reaches generation 17 of an adult lung at T = [(1 −X
_{17})^{−1/2}− 1]^{2}, or 54 s, and 95% of the initial bolus arrives afterT
_{α} ≈ 400/[(1 −X
_{17})
For a neonate, the initial surfacetension gradient is distributed over a length of only 3.75 cm, so the representative critical film thickness is 200 μm. Taking the infant tracheal radius to be 0.18 cm, this thickness represents ∼11% of the tracheal radius. With a bolus volume V _{b} = 0.5 cm^{3}, for example, we find that the leading edge reaches generation 7 in ∼0.05 s, and 95% passes through generation 7 in 64 min if Liquivent is used or in 2.94 s and 64 h, respectively, if Survanta is used. If such a large bolus volume is delivered at a single instant, the bolus fluid would occupy a substantial proportion of each airway, with the potential for temporarily occluding some airways. This could happen, for example, in liquid ventilation, where a bolus of fluorocarbon is forced along the trachea. However, if a liquid bolus is delivered slowly, a thin film forms on the airway walls. Under such conditions, gravitational effects would be dominated everywhere by surfacetension gradients, and our Marangoniflow model would be appropriate.
Surfacelayer transport. The following parameter values are chosen for the surfactantspreading model derived in formulation of the model,Surfacelayer transport. The bolus viscosity, and that of the liquid lining in the generations of interest, is taken to be that of water, so μ = 0.01 g ⋅ cm^{−1} ⋅ s. The reference film thickness for an adult is chosen to beh
_{7} = 20 μm, which corresponds to 1% of the diameter of airwaygeneration 7. The path length from trachea to alveoli in the adult is ∼15 cm, so the path length fromgeneration 7 to the acinar region gives a length scaleL
_{M} = 2.74 cm. The computational domain is taken to be 11 generations, so the downstream boundary condition is imposed at generation 18, and we set the domain lengthX̂
_{a} = 0.921. The scale for the surfacetension difference driving the spreading is taken to be S
_{7} = 50 dyn/cm. The Marangoni time scaleT
_{M} = μ
The effect of increasing preexisting surfactant concentration, Γ_{A}, is shown in Fig.12 where all parameter values are as in Fig. 5, except that Γ_{A} = 0.5 and the computations are carried out for a longer time until a steadystate solution is obtained. As predicted by Refs. 16, 28, and 42, the effect of increasing Γ_{A} is to forceΓ̂ and Ĥ to have smoother distributions near the disturbance’s leading edge at early times [compare, for example,Γ̂(X̂,0.1) in Figs.5 B and12 B]. However, once the disturbance reaches the downstream end of the domain, the leading edge of Ĥ becomes steeper and the film thicker behind the leading edge. This is because surfacetension gradients are very small in the distal generations, and, therefore, a greater volume of fluid is entering these airways than is leaving. The film thickness at the downstream end,Ĥ(n = 18,T̂), increases with T̂until there is a balance of fluxes atn = 7 andn = 18. When this occurs, the liquidlining thickness is approximately six times its initial value. However, airway closure is not likely to occur because it takes a very long time to reach steady state, and we are assuming that the liquid lining is initially very thin.
Because the effects of surfacearea expansion cause the surfacetension gradient to be small at the distal end, other effects such as a pressuregradientdriven flow due to the decrease in airway radius with distance from the trachea may become important. In Fig.13, we have plottedĤ and Γ̂ for Γ_{A} = 0.5 and Δ = 0.01, obtained by solving Eq. EA1 numerically. Note that the preexisting surfactant is assumed to have an initially nonuniform distributionΓ̂ _{eq}(X̂) (given by Eq. EA3 , and visible forn > 9 in Fig. 13,B andC) in which Γ̂increases slowly with respect to X̂: this generates a weak, proximally directed shear stress at the free surface that opposes the shear stress generated by the curvaturedriven flow, ensuring that the free surface is initially immobile. Once the exogenous surfactant starts to spread, the disturbances at the upstream end of the domain are very similar to those shown in Fig. 12, since the flow induced by surfactant gradients is much larger than the curvaturedriven flow. Similarly, at the downstream end, where at first the surfactant concentration is relatively uniform, the distally directed flow causes preexisting surfactant to be swept towardX̂ =X̂ _{a}, so thatΓ̂ increases rapidly above the levelΓ̂ _{eq}(X̂ _{a}) for 0 < T̂ < 1 (see Fig.13 C). However, as the surfactantdriven flow approaches equilibrium, it weakens relative to the curvaturedriven flow, and Γ̂ again falls beneathΓ̂ _{eq}(X̂ _{a}) (e.g., T̂ = 12 in Fig.13 C), until ultimately a steady state is achieved. An important effect of the weak curvaturedriven flow is to prevent the film thickness at the downstream end from becoming too big, reducing the likelihood of airway closure (Fig.13 A).
The leading edges of the exogenous surfactant distributionL _{ex} and of the surfacecompression waveL _{D} are plotted vs. time in Fig. 14. As in Ref. 28,L _{ex} lags significantly behindL _{D}, since Γ_{A} > 0. However, whereasL _{D} is relatively insensitive to Γ_{A} and the effects of surfacearea expansion, reaching generation 18 at T̂ ≈ 0.55, or 0.41 s (Fig.14 B), this is not so forL _{ex}: for Γ_{A} = 0.2, it takes approximately three time units, or 2.25 s, for exogenous surfactant to reach the downstream end, an order of magnitude longer than if there were no surfacearea expansion. The rate of advance ofL _{ex} falls in the distal generations due to the effects of surfacearea expansion and the increased flow resistance due to a decrease in liquidlining thickness. An increase in Γ_{A} to 0.5 causes the arrival time of exogenous surfactant to more than double. The effect of curvaturedriven flow on the time taken for exogenous surfactant to reach the respiratory bronchioles is not significant (Fig. 14 A). It causes exogenous surfactant to reach generation 18marginally later than the Δ = 0 case and has its biggest impact in the distal generations where surfacetension gradients are weakest.
Surfacelayer transit time in the steady state. The effect of the initial preexisting surfactant concentration Γ_{A} on surfacearea expansion can be further investigated by determining the transit time,T̂
_{s}, for exogenous surfactant to cross the whole domain once steady state is achieved (neglecting curvaturedriven flow).T̂
_{s} is computed from the equation that governs the Lagrangian motion of a particle on the airliquid interface (28)
Dose delivery times: steadystate alveolar surface concentration. After the initial transients have decayed (within a few seconds of the initial delivery of the first surfactant dose), the steadystate flow in generations 7–18 delivers a steady flux of surfactant to the alveolar space. This (nondimensional) flux, −BΓ̂ĤΓ̂
_{X̂}, evaluated at X̂ =X̂
_{a}, whereΓ̂ and Ĥ are given byEq. 13
, is
An estimate for the delivery time of the entire exogenous surfactant bolus can be made by using Eq. 22 . Typical neonatal dose sizes are 100 mg/kg, given in two halfdoses, 12 h apart (14). Therefore, an infant weighing 500 g receives two doses of 25 mg. So we takem _{dose} = 25 × 10^{−3} g. Adult dose sizes are also taken to be 100 mg/kg (13, 39, 69). This corresponds to an m _{dose} = 3.5 g for a 70kg adult. We take the micelle concentration to be the reference concentration at generation 7 in an adult, so γ_{7} = 2 × 10^{14}molecules/cm^{2} (66). Because dipalmitoylphosphatidylcholine has molecular weight of 734 andN _{A} = 6.02 × 10^{23}, γ_{7} = 2.44 × 10^{−7}g/cm^{2}. For an infant,L _{M} = 3.75 cm,b _{7} = 1.13 cm, so γ_{7} L _{M} b _{7}= 1.03 × 10^{−6} g, andT _{M} = 0.75 s (where this time scale is that defined for surfactantdriven flow informulation of the model,Depositedfilm flow). Thus the scale of the exogenous surfactant flux isq _{7} = γ_{7} L _{M} b _{7}/T _{M}= 1.37 × 10^{−6} g/s approximately, which is an upper bound on the actual flux. We can therefore estimate the time for one dose to be delivered, assuming the conditions are steady state. Dividing the mass delivered by the mass flux (withq _{A} =q _{7}) yields a delivery time = 25 × 10^{−3} g/(1.37 × 10^{−6} g/s) = ∼5 h.
Dose delivery times: quasisteady alveolar surface concentration. The exogenous surfactant that arrives at the alveolar compartment is not taken up immediately. It accumulates and is removed by processes with a long halflife, i.e., 5–15 h. This time scale, used to define τ in Eq.15
, is long compared with the time required to reach steady state in the delivery process. Therefore, we may treat the alveolar concentration, at generation 18 in our model, as time dependent on the slow time scale and suppose the fluxQ
_{A} depends only on Γ_{A} when Eq.22
is used. Then, using Eq.22
, Eq. 15
becomes
Suppose that at τ = 0, Γ_{A} =P
_{A}, andF̂ > 0. Then Γ_{A} will rise with time, approaching a new equilibrium value Γ_{A} = Γ_{eq}, say, where Γ_{eq} = Γ_{eq}(F̂,P
_{e}) satisfies
Therefore, we compute the integral of the flux of surfactant delivered to the alveoli. The mass of exogenous surfactant delivered in time τ_{D} is
DISCUSSION
This model, although a greatly simplified picture of the true spreading process, captures some important elements of the transport of a bolus of exogenous surfactant to the lung periphery. A liquid plug of surfactant is initially driven by the inspiratory airflow in the upper airways. It loses volume as it propagates to the distal airways, leaving behind a deposited liquid layer that coats the airways. A simple theory shows that rupture of the plug, when its volume decreases to zero, depends on the magnitude of the airflow rate and the ratio of initial plug volume to tracheal volume. Our model indicates that plugs with initial volumes <1 cm^{3}, approximately, will rupture in the first seven generations, even at high flow rates. Larger plugs may never rupture in one breath but they may stop once they reach a collapsed region of the lung. The thickness of the deposited liquid lining may become sufficiently thick that the liquid coating will start to drain under the effect of gravity.
The effects of gravity are dominant in the upper airways of the adult lung: a simple similarity solution, Eq.11 , suggests that, for typical bolus volumes, gravity carries the surfactant to generations 5–9 in a few seconds (Fig. 10) and provides thereafter a continuous flux of surfactant to the distal portions of the lung for many minutes. The bolus fluid, which is distributed around the airway walls, becomes so thin near these generations because of surfacearea expansion effects that gravitational forces weaken, relative to surfacetension gradients, so that Marangoni flows drive the surfactant deeper into the lung. Distally directed curvaturedriven flow was shown to have only a very weak effect on transit times (Fig.13), although it had a much more pronounced effect on film distributions. Despite the significant surfacearea expansion that the monolayer experiences as it advances further, the model predicts that surfacetension reduction is initially experienced at the terminal bronchioles very rapidly, within ∼0.5 s (Fig.14 B) of the bolus reachinggeneration 7. This is so because the advancing exogenous surfactant compresses preexisting surfactant: the higher the concentration of preexisting material, the quicker an initial surfacetension reduction is experienced. Hence, surface compression of a preexisting surfactant layer, possibly occurring well before the exogenous surfactant actually arrives at a target airway or alveolus, may be a therapeutically valuable effect.
We model surfactant uptake by alveolar type II cells (7) as a sink by fixing the surfactant concentration to be constant atgeneration 18. Even so, the level of preexisting surfactant, Γ_{A}, remains an important parameter. Exogenous surfactant was predicted to arrive at generation 18 within ∼2.25 s of leaving generation 7 (Fig.14 A) when Γ_{A} = 0.2; this transit time rose to ∼4.5 s with Γ_{A} = 0.5. A steady balance of fluxes between material provided by gravitydriven flow at generation 7 and being taken up at generation 18 was established in most cases within ∼10 s; the time taken for exogenous surfactant to be advected to the periphery from generation 7 by this steady flow is strongly dependent on the level of preexisting surfactant, ranging from a few seconds for Γ_{A} → 0 to a few minutes as Γ_{A} approaches its tracheal concentration (Γ_{A} → 1, Fig. 15). This is consistent with the fact that spreading rates are reduced as the surfactant level builds up in the lung (28) and correlates with observations that neonatal patients may respond to the first intratracheal bolus of surfactant, but that subsequent instillations, e.g., beyond two, are not as effective (54). They may follow the same opened pathways as the first dose (73) and, therefore, may not be going to untreated regions of the lung.
The surfactant concentration at the distal end of the Marangoni flow regime is held constant because the time scale for alveolar surfactant uptake is much longer than the time scale for the Marangoni flow. The alveolar surfactant concentration actually increases slowly as a result of the flux of exogenous surfactant generated by the Marangoni flow. A surfactant kinetics model is employed by using available data for the time constant of uptake. The quasisteadystate balance of Marangoni delivery with alveolar uptake determines the delivery time of a given dose of surfactant, with the alveolar surfactant production rate as a parameter. For neonates, it takes ∼24 h for a 2.5cm^{3} bolus to be delivered. Delivery times are reduced by increasing either the rate of delivery of exogenous surfactant or the alveolar surfactant production rate with respect to the rate constant for surfactant uptake.
To apply the thinfilm model to the infant lung, it was necessary to assume that the surfactant bolus was delivered to the trachea sufficiently slowly that the bolus fluid always formed a thin layer around the lung airways. If this is the case, then the liquid layer is everywhere so thin that surfacetension gradients dominate over gravitational forces. To model the morphometry of the immature infant lung, we made the crude assumption that it can be represented by that portion of an adult lung distal to a single airway atgeneration 7. The calculations appropriate for the adult lung then carry over directly to this case: surfacetension reduction at the periphery is experienced within ∼0.5 s; fresh material reaches the alveoli within 3–6 s. These times will obviously be reduced if a larger bolus is introduced rapidly into the trachea, because gravitational spreading will enhance the rate of delivery, but the larger volume of fluid may possibly occlude the airways temporarily and cause choking.
There are many aspects of surfactant transport that have been neglected, which should be considered in future models. The lining fluid has been taken to be a Newtonian fluid with viscosity similar to that of water, which is unreasonable in the upper airways. In addition, we have omitted clearance by ciliary transport. The surfactant has been supposed to have a linear equation of state (i.e., constant surface activity), whereas, in practice, the relationship between surface tension and surfactant concentration is typically nonlinear, particularly at high concentrations. We have assumed that the exogenous and preexisting surfactants have the same surface activity. It is, instead, possible that there could be physicochemical interactions between the chemically distinct exogenous and preexisting materials, which would contribute to these nonlinearities and, hence, influence spreading rates.
The model of the lung is necessarily simple, to make calculations tractable. The dichotomous branching model used to develop a formula for the airway circumference functionB(X) (Eq. 3 ) is highly idealized, particularly since asymmetries in the lung are neglected; there is a need for more reliable functions to be developed by using realistic morphometric data, especially for developing neonatal lungs. A particularly important aspect of SRT, which this model is presently unable to address, is inhomogeneous spreading, which arises both in gravitydriven and Marangonidriven spreading. In the former case, the surfactant bolus will drain rapidly vertically downward, and, even if the patient is rolled around it, it is likely that the spreading may be nonuniform; there is scope for improving the model for gravitydriven flow (see formulation of the model,Surface layer transport) to take this and other factors into account. In the latter case, a portion of the lung that is well ventilated will have patent airways down which surfactant can progress relatively easily, whereas the monolayer is unlikely to advance as quickly into atelectatic regions. The mechanism by which airways reopen, and the role played by surfactants in this process, is being investigated elsewhere (23). The effects of inhomogeneity are felt particularly strongly when the lung is being ventilated, since the patent airways tend to overdistend.
The interaction between surfactant transport and ventilation, either spontaneous or forced, deserves attention: the unsteady expansion and contraction of the airway walls, the importance of dynamic surface tension, and the role of air shear stresses should be investigated. At smaller scales, inhomogeneities in the advance of the monolayer are likely to develop in the neighborhood of an airway bifurcation. The nonuniform curvature of the airway wall may cause the airway liquid lining to assume a configuration in which it is of nonuniform thickness (41) (by draining away from the vertex of a bifurcation, for example). An advancing monolayer will travel quicker over thicker fluid layers, so that by neglecting details of airway structure we may be neglecting important features of the spreading process.
Finally, with use of our results, it is interesting to consider clinical delivery strategies. Making the instilled volume larger enhances rapid gravitydriven spreading but risks both ventilatory obstruction and strongly inhomogeneous delivery. Also, the larger the plug the further it penetrates into the lung before rupturing. For delivery of liquids that may carry drugs or genetic material intended for airway walls, the smaller plug will rupture before reaching the alveoli, so that the contents coat the targeted airways. Surfactant transport to the periphery by the Marangoni mechanism is controlled predominantly by the level of preexisting surfactant: Fig. 15 shows that for 0.1 < Γ_{A} < 0.7 (approximately) the dependence of spreading rates on Γ_{A} is greater if the effects of surfacearea expansion are included in the model than if they are absent. Rapid spreading of fresh exogenous material occurs if Γ_{A} is low; once Γ_{A} increases beyond, say, 0.9 (as it might after repeated small doses), there may be no significant gain in delivering additional exogenous surfactant. However, the film deformations arising in this case are potentially severe: although those in Fig. 5, for example, are not particularly dramatic, under different boundary conditions (e.g., Ref. 43), we found that severe thickening or thinning of the film can occur, increasing the risks of either airway closure or rupture of the lining film. (In some cases, it was even possible to cause the film to overturn.) These findings support the study of Alvarez et al. (2), who found that four fractional surfactant doses given to rats with damaged lungs gave no enhancement over the same total dose delivered as a single bolus. Although combining the bolus and aerosol techniques may be effective (28), the bolus to open airways and the aerosol to bypass any slowmoving Marangoni regions, there remains the difficulty that aerosolized material may go where it is least needed, i.e., to the best ventilated portions of the lungs.
Acknowledgments
This study was supported by North Atlantic Treaty Organization Grants CRG930189 and CRG950725; by the Whitaker Foundation; by National Science Foundation (NSF) Grant CTS9412523; by National Heart, Lung, and Blood Institute Grant HL41126; and by NSF Experimental Program to Stimulate Competitive Research in Alabama.
Footnotes

Address for reprint requests: D. Halpern, Dept. of Mathematics, Univ. of Alabama, Tuscaloosa, AL 35487.

Present address of J. B. Grotberg: Biomedical Engineering, Univ. of Michigan, 3304 G.G. Brown, 2350 Hayward, Ann Arbor, MI 48109.
 Copyright © 1998 the American Physiological Society
Appendix
Evolution Equations for SurfaceArea Expansion Models
We suppose that as the surfactant advances because of surfacetension forces, its concentration and the film thickness of the airways it traverses are dependent only on distance along the pulmonary tree and on time. We are not considering variations in film thickness or surfactant concentration among airways of the same generation, so no allowance is made for inhomogeneous spreading. The liquid lining is assumed to be sufficiently thin everywhere so that it can be treated as locally planar. By considering conservation of fluid mass and conservation of surfactant, we obtain the dimensionless governing equations for the film thickness,Ĥ (X̂,T̂) =h (x,t)/h
_{7}, and the surfactant concentration,Γ̂(X̂,T̂) = γ(x,t)/γ_{7}, which are derived in a manner similar to our previous studies (22, 28,42, 44)
The boundary conditions on the governing Eq.EA1
are as follows. The surfactant concentration is fixed at the proximal boundary of the Marangoni domain,X̂ = 0, to be unity and, at the distal end of the domain, X̂ = X̂
_{a}(generation 18) to be the alveolar value, Γ_{A}, that is
Steadystate solutions to Eq. EA1
may be derived by setting the time derivatives to zero. An interesting subset of such solutions is an equilibrium solution that balances Marangoni and capillarity effects, i.e., no surfactant flux. This state could represent the preexisting surfactant distribution before any new surfactant is added to the system. The equilibrium solution is obtained by setting q = 0 andQ = constant inEq. EA1
, and is chosen to satisfy the distal boundary condition for Γ̂ and the proximal boundary condition for Ĥ. The result is
These equations (A1, A2, andA4) are solved numerically by using the method of lines. We assume that the liquid lining is Newtonian with uniform viscosity, although the model can readily be modified to allow for viscosity variation. The terms involvingΔ are generally very small compared with the Marangoni terms. The two mechanisms may become comparable only in the most distal airways where R̂ becomes very small. For the case of a uniform tube, B̂ = 1, the initial conditions are given by Eq. EA4 withĤ _{eq}(X̂) = 1 and Γ̂_{eq}(X̂) = Γ_{A}.