Department of Mechanical Engineering and Center for Biomedical
Engineering, Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139
A model is presented of
surfactant replacement therapy. An instilled bolus is pushed into the
lungs on the first inspiration, coating the airways with a layer of
surfactant and depositing some in the alveoli. Layer thickness depends
on the capillary number (µU/
, where µ, U, and
are bolus viscosity, advancing meniscus velocity, and surface
tension, respectively). Larger capillary number leads to thicker
layers, reducing alveolar delivery. Subsequently, surface tension
gradients sweep surfactant into alveoli not receiving surfactant during
the first inspiration. The effects on spreading of sorption kinetics,
bolus viscosity, initial layer thickness, initial penetration of
surfactant, gravity, and shear stress are examined. Sorption nearly
eliminates surface tension gradients in central airways but produces a
sharp transition at the leading edge of the exogenous layer. Local
thinning of the liquid layer results, trapping 95% of the surfactant
in the airways. Gravity and ventilation augment transport somewhat.
Transport to the periphery takes 4-170 s for the leading edge but
considerably longer for the bulk of the surfactant. The model
demonstrates how the various physical parameters governing surfactant
distribution might alter the response to surfactant replacement therapy.
 |
INTRODUCTION |
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: t1/4
for a droplet (11), t1/3 for a strip (9, 24), and
t1/2 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
· s
1 · cm
1, 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
After bolus instillation (Fig.
1A), surfactant is distributed
through the airway network as the bolus is advanced distally by the
ensuing inspiration. As the bolus advances, it coats the airway walls
with a thin layer of surfactant suspension, the thickness depending on
the bolus properties (viscosity and surface tension) and the rate at
which it is convected distally. Because of the heterogeneity of the
lung in RDS, the initial distribution of surfactant will likely be
nonuniform: the regions with higher compliance will receive a greater
fraction of surfactant during the initial inspiration (Fig.
1B).

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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.
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Phase 2
The liquid layer left behind in the airways by the advancing bolus
during phase 1 provides the "reservoir" of surfactant
that can potentially be delivered to terminal units not initially
reached. Regions of the lung that lack endogenous surfactant (and,
therefore, are less compliant) have a lower probability of being
reached during phase 1. It is assumed that, although the
expansion of these regions is impaired as a result of, e.g., the lack
of surfactant, they are nonetheless patent with a continuous layer of
liquid on the walls. Along those pathways, surface tension gradients exist that will draw the exogenous surfactant distally to locations where the surface tension remains high, thereby recruiting additional alveoli (Fig. 1C).
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 simulation is based on the assumption that the injected surfactant
bolus forms a plug in the central airways when first introduced into
the lungs. The conditions under which this occurs are examined in a
separate study (7). This section describes the model developed for how
the bolus of surfactant initially filling the trachea and major bronchi
is distributed along the airways on inspiration (phase 1).
Published experimental results revealing the factors that control this
phase of the dispersal process are employed.
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
Rc. The radius of the air-liquid interface for the
liquid left behind is Ri. 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).

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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.
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As a brief review, the experimental results of Fairbrother and Stubbs
(10) indicate that m ~ Ca1/2 for 10
4 < Ca < 10
2. Taylor (41) later confirmed and
extended this finding to Ca ~ 10
1. For larger Ca, m
tapers off more rapidly than Ca1/2, 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
Ca2/3, indicating that less liquid should remain in the
tube than was found experimentally. Betherton's experimental results
were in good agreement with m
Ca2/3 for Ca > 10
4 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
10
5 < Ca < 10
3. Ratulowski
and Chang (37) showed that surfactants can generate surface tension
gradients that increase m by a maximum of 42/3 over the
result of Bretherton at small Ca (Ca ~ 10
6). This
difference diminishes as Ca increases, with their theory giving m
Ca2/3 for larger Ca (Ca > 10
4).
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,
R2i2n,
where n is the airway generation number. Letting
R2i = R2c(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 · s
1 · cm
1,
= 37 dyn/cm) to 10
4 in generation 14 (
= 2 ml/s, µ = 0.01
g · s
1 · cm
1,
= 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, Ri 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
Regions in the lung not receiving surfactant during the initial passage
of the bolus (phase 1) must rely on surface tension gradients to deliver exogenous surfactant (phase 2).
Phase 2 can be viewed as transporting surfactant to
surfactant-deficient, low-compliant regions of the lung. The
action of phase 1 leaves exogenous surfactant coating the
airways through generation n with a nonzero endogenous
surfactant concentration distal to this location. This initial
condition is shown in Fig. 3.

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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 = 10 3 cm.
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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, cref,
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/k1cref, a ratio
of surfactant adsorption to desorption time, and
v/
D, a ratio of viscous spreading to
desorption time. Here
A = 1/k1cref,
D = 1/k2, and
v = µL20/h0
max, 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). Hbol = h/h0 is the maximum liquid layer thickness
in the trachea resulting from the initial dispersal process, and
nbol 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, Hbol,
nbol, nbol, 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/Rc), and accumulated volume
(Vacc) 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
Profiles of h/h0, Ca,
*,
h/Rc, and Vacc along the airway
tree (Fig. 4) were calculated for
= 7 ml/s with µ = 0.01, 0.1, 0.2, 0.4, and 0.8 g · s
1 · cm
1. The
accumulated volume is the cumulative amount of liquid left in the
airways as the bolus is pushed from the trachea completely through the
last conducting airway (generation 14). In general, increasing
viscosity or flow rate (all else being equal) increases Ca (Eq. 2), which leads to a larger fraction of liquid being left on the
airway walls.
The liquid layer thickness in the intermediate airways ranges from
approximately h/h0 = 12 for µ = 0.01 g · s
1 · cm
1 to
h/h0 = 95 for µ = 0.8 g · s
1 · cm
1,
where h0 = 10
3 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 10
2 for µ = 0.01 g · s
1 · cm
1 to 1 for µ = 0.8 g · s
1 · cm
1, 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 · s
1 · cm
1 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/Rc 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
>5.6Rec (31). For an
airway, this requires
(R2c
R2i)L
a > 5.6R3c,
where La is the length of an airway and
La ~ 6Rc. Letting
Ri = Rc
h and
solving for h/Rc, airway closure can occur
for h/Rc > 0.16. Thus, in examining the
results for h/Rc, it is useful to keep this
value (h/Rc = 0.16) in mind, inasmuch as
when it is exceeded, liquid bridging is likely to occur. In Fig.
4D, h/Rc is <0.16 along the
entire airway tree only for µ = 0.01 g · s
1 · cm
1. As
viscosity increases to µ = 0.1 g · s
1 · cm
1,
h/Rc > 0.16 in a region extending from
the trachea through generation 6. h/Rc 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 · s
1 · cm
1
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
(VA) 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).

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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 · s 1 · cm 1.
Generation numbers (1-14) are shown along top of
plot.
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Phase 2: Recruitment Model
As mentioned earlier, when h/Rc > 0.16, airway obstruction by liquid bridging is likely to occur (31). This
instability has been studied by several research groups (19, 20, 29, 30, 34) and is certain to play a role in transport (see
DISCUSSION), but it is not the subject of this analysis.
Hence, to study transport by surface tension gradients, stable liquid
layers that would not lead to liquid bridging
(h/Rc < 0.16) were chosen. Therefore, the liquid layer resulting when
= 0.28 ml/s and µ = 0.1 g · s
1 · cm
1 from
phase 1 was chosen as the initial condition for the recruitment model. Although this flow rate is quite small (an order of magnitude smaller than those clinically used), it is only the initial inspiration rate for delivering the surfactant and not the ventilatory rate used in
the course of treatment. Once phase 1 is completed, standard ventilatory support can be resumed. Alternatively, a stable liquid layer similar to this one could result under current clinical conditions where ventilation rate is much higher (
= 7
ml/s). In this case, occlusion by liquid bridging and reopening of the airways occurs over several given breaths. Each time occlusion occurs,
the applied ventilation drives the mensci into the lungs. In this
manner, liquid is carried distally, and the liquid layer thins until a
stable liquid lining results. The physics of this process are not
addressed here; rather, we have chosen conditions that provide a stable
liquid layer to systematically study transport after a stable layer is established.
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 = 10
3 cm (23, 42),
cref = 10 mg/cm3,
ref = 3 × 10
4 mg/cm2, k1 = 105
cm3 · g
1 · min
1,
k2 = 1 min
1 (36),
L0 = 8.7 cm,
max = 70 dyn/cm,
= 1 g/cm3, µ = 0.1 g · s
1 · cm
1, 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/cref h0,
W * is the total airway circumference along the airway tree,
fexo 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 fexo, M *exo, and
M *fendo (15).
Evolution profiles of h/h0,
/
ref , and c/cref 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 Hbol = 8, nbol = 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).

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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 = 10 3 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.
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Different percentages of exogenous surfactant are tracked in Fig.
7. The leading edge of the exogenous mass
(100%) rapidly travels into the lungs, reaching a plateau between
generations 10 and 12 by t/
v = 0.3 (43 s). Twenty-five percent lies between generations
4 and 12, with 50% remaining in the trachea. The
trajectories for 50, 75, and 95% have not reached a plateau,
signifying that surface tension gradients (surfactant gradients) are
still present (Fig. 6C).

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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.
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Effects of sorption are now considered for a concentration similar to
that used in SRT [c = 10 mg/ml (40)] with
= 0.03,
A/
D = 10
3, and
v/
D = 2.38 (µ = 0.1
g · s
1 · cm
1).
The overall character of the liquid layer changes significantly (Fig.
8). The steep leading edge is steeper, and
dramatic thinning extends up the airway tree over several generations.
A detail of the leading edge highlights this thinning of the liquid
layer (Fig. 8B). Transport of liquid from the trachea is also
noticeably reduced. This can be explained on the basis of Fig.
9A, showing that, rather than
decrease, the surface concentration remains nearly constant behind the
leading edge. This results in a near-zero surface tension gradient
through the intermediate airways (Fig. 9B). Profiles of
/
ref become almost stationary. Because concentration gradients (and, therefore, surface tension gradients) remain fixed, liquid is continually pumped distally from the same airways. The absence of gradients in the more central airways means that the liquid
pumped from the small airways is not replenished and the liquid layer
continues to thin. In Fig. 9C, the bulk concentration is shown
with similar stationary profiles.

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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 = 10 3,
v/ D = 2.38, Hbol = 8, nbol = 4, Bo = 0, f = 0, h0 = 10 3 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.
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Fig. 9.
Effect of sorption. Evolution profiles of / ref, *,
and bulk concentration (c/cref) vs.
x/L0. = 0.03, A/ D = 10 3,
v/ D = 2.38, Hbol = 8, nbol = 4, Bo = 0, f = 0, h0 = 10 3 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.
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Because gradients are reduced over a large extent of the domain
compared with the case of an insoluble surfactant, 95% of the
surfactant mass travels no further than generation 2 during the
first 40 s (Fig. 10). The bulk of the
surfactant is therefore essentially trapped in the central airways and
prevented from reaching the periphery by the thinned liquid layer in
generations 6-10. However, the leading edge preceding
the crest in Fig. 8 has nearly propagated to the periphery in the same
time. For comparison, trajectories for a more dilute bulk
concentration (c = 1 mg/ml,
A/
D = 10
2, and
= 0.3) are also plotted in Fig. 10. Here, all portions are
carried more deeply into the lungs, with the leading edge reaching the
periphery. Although a lower concentration spreads more rapidly, the
total amount of surfactant transported is less.

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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 = 10 3; dashed lines,
A/ D = 10 2 (c = 1 mg/ml,
= 0.3). Generation numbers (1-14) are shown along
right axis. v = 142 s.
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In summary, sorption brings surfactant into the air-liquid interface,
keeping surface tension constant and low behind the leading edge. A
steep, linear gradient connects this surfactant-rich region with the
surfactant-deficient liquid layer ahead of the leading edge, resulting
in strong pumping and severe thinning over these airways.
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 · s
1 · cm
1).
The simulation parameters are
A/
D = 10
3,
v/
D = 9.53, Hbol = 8, nbol = 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 · s
1 · cm
1,
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.

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Fig. 11.
Effect of viscosity. Evolution profiles of
h/h0 and / ref vs.
x/L0. = 0.03, A/ D = 10 3,
v/ D = 9.53, Hbol = 8, nbol = 4, Bo = 0, f = 0, h0 = 10 3 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.
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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.
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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 Hbol = 26, with
A/
D = 10
3,
v/
D = 2.38, nbol = 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 · s
1 · cm
1.
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.

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Fig. 13.
Effect of liquid layer thickness. Evolution profiles of
h/h0 vs. x/L0.
= 0.03, A/ D = 10 3,
v/ D = 2.38, Hbol = 26, nbol = 4, Bo = 0, f = 0, h0 = 10 3 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.
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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.
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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 = 10
3,
v/
D = 2.38, Hbol = 8, nbol = 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 nbol 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).

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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.
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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 = 10
3,
V/
D = 2.38, Hbol = 8, nbol = 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.