INTRODUCTION

Top Abstract Introduction Results Discussion Appendix References

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, surfactant-replacement 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 -1 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 one-half (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 sepsis-induced ARDS (3) by aerosol, for mitigation of oxygen-toxic lung injury (56) and wood-smoke 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 repeat-dosing 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 second-dose transport because the first dose ultimately lowers the surface-tension 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 liquid-plug 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 deposited-film 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 large-to-medium-sized airways.

When the liquid and its constituents (such as surfactant) form a surface layer, then surface-tension 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 medium-to-small airways. The fundamental fluid mechanics and transport phenomena of surface-layer 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. 1A), the unsteady spreading flow generates a wave that travels in the direction of lower surfactant concentration (higher surface tension) (Fig. 1B). If surface diffusion of the monolayer and gravity are negligible, the wave behaves like a shock wave, with rapid changes in height and surface-tension 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 surface-tension difference driving the flow, the film thickness, the surfactant activity, and the fluid viscosity.

View larger version (17K): [in this window] [in a new window]   View larger version (18K): [in this window] [in a new window]   Fig. 1.   A: surfactant monolayer on a uniform clean film of height h0 before spreading, at time t = 0. Lex denotes position of leading edge of exogenous surfactant. B: surfactant spreading on a deforming interface, where h(x,t) is the film height and (x,t) is surfactant concentration at a distance x from trachea and at t > 0. C: surfactant monolayer on a contaminated film before spreading begins, where exogenous surfactant is depicted with  and endogenous surfactant with . D: effect of preexisting endogenous surfactant on film deformation, surfactant distribution, and position of Lex front with respect to leading edge of compression wave LD. See Glossary for other definitions.

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 and D), 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 surface-tension 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. 1D). 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 of N-nitroso-N-methylurethane. 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 surface-area 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

	Fraction of fluid in draining region
a, A	Total airway cross-sectional area
ae	Total airway cross-sectional area exposed to air
an	Total cross-sectional area at generation n
a0	Tracheal cross-sectional area
AA	Total alveolar surface area
Atr	Cross-sectional area of endotracheal tube
b, B	Total airway perimeter
	Scaled perimeter function used in Marangoni flow regime
bn	Total airway perimeter at generation n
b0	Tracheal perimeter
	Perimeter parameter
Ca	Capillary number
Catr	Tracheal capillary number
dn	Mean airway diameter at generation n
d, D	Airway diameter
d0	Tracheal diameter
	Airway taper parameter
n	Surface-tension difference over length ln
F	Ratio of surfactant delivery to the alveolar space to the rate of uptake

	Rescaled delivery of surfactant-to-uptake ratio
g	Gravitational acceleration
G	Ratio of typical gravity draining speed to Marangoni speed
,,	Surfactant concentration
A, A	Alveolar surfactant concentration
eq	Equilibrium alveolar surfactant concentration
eq	Equilibrium surfactant concentration in Marangoni flow regime
h, H,	Film depth
Heq	Equilibrium film thickness in the Marangoni regime
HM	Critical film thickness for transition from gravity to Marangoni regimes
hn	Liquid lining thickness at generation n
K	Rate constant for alveolar surfactant uptake
ln	Mean airway length of generation n
l0	Tracheal length
LD	Leading edge of surface-compression wave
Lex	Leading edge of exogenous surfactant
LM	Marangoni regime length
L0	Total airway path length
	Leading edge of bolus draining under gravity
	Ratio of tracheal length to four times the path length
m, M	Mass of exogenous surfactant delivered to the alveoli
mdose	Dose of surfactant delivered
µ	Fluid viscosity
n	Airway generation number
NA	Avogadro's number
nr	Critical generation number for plug rupture
pA, PA	Alveolar surfactant production rate
q, Q	Surfactant flux and fluid volume flux in Marangoni regime
qa, QA	Surfactant flux into the alveolar compartment
r, R,	Airway radius
rn	Mean airway radius at generation n
r0	Tracheal radius
	Liquid density
SM	Surface-tension difference across the Marangoni regime
S0	Surface-tension difference along the surface layer
	Surface tension
n	Surface tension at generation n
t	Time
T	Dimensionless time used for deposited film flow
T	Transit time for gravity-driven flow
TI	Inspiration time
TM	Marangoni time scale
s	Surfactant transit time at steady state
	Dimensionless time for surface-layer transport
	Alveolar uptake time variable
D	Exogenous surfactant delivery time
	Surfactant activity parameter
	Film thickness correlation function
U	Speed of propagation of liquid plug
Ug	Typical speed of gravity-driven drainage
UM	Marangoni velocity scale
Ûs	Marangoni velocity at air-liquid interface
Utr	Tracheal velocity scale
v, V	Liquid bolus volume
Vb	Initial bolus volume
	Airflow rate
Vp	Initial liquid plug volume
VT	Tidal volume
Vtr	Tracheal volume
Wb	Ratio of bolus volume to tracheal volume
Wp	Ratio of initial plug volume to tracheal volume
Wr	Value of Wp that will rupture at generation n
x, X	Distance along fluid layer, measured from the tracheal carina
	Dimensionless distance along fluid layer starting at generation 7
a	Size of domain of Marangoni regime
xn, Xn	Distance along fluid layer to generation n
1, 2	Initial condition parameters for surface-layer 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 2ⁿ, for 0  n  23, and the mean airway diameter (radius) is d_n (r_n), the mean airway length is l_n, the total cross-sectional area of the airways is a_n, the total airway perimeter is b_n, given respectively by

(1)

Here, d₀ (r₀), l₀, a₀, and b₀ represent the tracheal diameter (radius), length, cross-sectional area, and perimeter, respectively. The tracheal values from Ref. 76 are d₀ = 1.8 cm and l₀ = 12 cm, from which r₀, a₀, and b₀ may be computed. The distance from the tracheal carina to the end of generation n is denoted by the discrete variable, x_n. It may be expressed as the sum of the intervening airway lengths l_n, as given in Eq. 1. This geometric sum yields a simpler form

(2)

where the total path length is L₀ = l₀/(2^1/3  1)  15 cm in the adult. Using Eq. 2, we eliminate n from Eq. 1 and replace the discrete variable x_n with the continuous variable x. Then the functions in Eq. 1 become continuous functions of x. These are further simplified by representing them in dimensionless form as follows

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where the dimensionless pathway distance is now X = x/L₀, and we note that 0  X < 1. Upper (lower) case variables are used to indicate nondimensional (dimensional) variables. The introduction of continuous variables and functions will allow us to apply conservation equations for mass and momentum as they arise in the analyses.

3

6

7/3

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View larger version (16K): [in this window] [in a new window]   Fig. 2.   A: liquid plug of volume (V) propagating down an airway of radius (r) due to airflow rate (), leaving behind a trailing film of thickness (h). B: progression of liquid plug through a symmetric airway network illustrating loss of volume as it travels from trachea to generation 3. See Glossary for other definitions.

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1

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View larger version (11K): [in this window] [in a new window]   Fig. 3.   Liquid bolus of thickness h draining due to gravity on a vertical wall. Leading edge of bolus is denoted by X = . See Glossary for other definitions.

,

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View larger version (21K): [in this window] [in a new window]   Fig. 4.   (,) (A) and (,) (B) vs. at  = 0, 0.1, 0.25, 0.5, 1, and 1.5 for spreading with no surface-area expansion or curvature effects (total airway perimeter B = 1,  = 0) and an initially flat film with A = 0.2.  in B show leading edge of exogenous surfactant distribution at times 0, 0.1, 0.25, 0.5. See Glossary for other definitions.

View larger version (20K): [in this window] [in a new window]   Fig. 5.   (,) vs. on a logarithmic scale (A) and (,) vs. generation number n at  = 0, 0.1, 1, 2, 4, and 8 (B, C), incorporating effects of surface-area expansion, obtained by solving Eq. 1A with A = 0.2 and  = 0. Solid curves with symbols show ultimate steady state;  in B show leading edge of exogenous surfactant distribution at times 0, 0.1, 1, 2, and 3.07. See Glossary for other definitions.

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d

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View larger version (10K): [in this window] [in a new window]   Fig. 6.   First-order alveolar compartmental model, indicating that alveolar surfactant concentration A increases due to alveolar production pA and airway surfactant flux qA but decreases due to constant rate of uptake K.

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	RESULTS

Top Abstract Introduction Results Discussion Appendix References

Liquid plug flow. As was shown in FORMULATION 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 volume W_p. The adult tracheal volume is V_tr = r²₀l₀ = 30 cm³, whereas a premature neonatal value is V_tr = 2⁷r²₀l₀ = 0.25 cm³, roughly equivalent to the volume of a single adult generation 7 airway. For neonates, a typical dose of surfactant liquid is two half-doses of 2.5 ml/kg. The first half-dose is instilled in small portions in time with each mechanical inspiration. Normally, the drug is administered over a 1- to 2-min period, corresponding to 30-50 mechanical breaths. Therefore, plug volumes may be 0.25 cm³ or lower, so that W_p  1. The range W_p  1 is also reasonable for adults who experience instillation of surfactants, resuscitative drugs, or perfluorocarbons. Our analysis is not limited to W_p  1; however, this is the range we will consider.

The airflow rate, , can be expressed in terms of the tidal volume VT and an inspiration time TI as  = VT/TI. Defining a tracheal velocity scale to be U_tr = VT/(TI A_tr), where A_tr is the cross-sectional area of the endotracheal tube, the Ca_tr is then

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For a neonate, let the inner diameter of the endotracheal tube be 0.3 cm, which is slightly smaller than d₇, so that A_tr = 0.07 cm². A typical value for inspiratory time, under conditions of assisted ventilation, would be TI = 0.7 s, whereas a typical VT could be 6 cm³; then, U_tr = 122 cm/s. For an adult, A_tr  r²₀ = 2.5 cm², TI  2 s, VT  500 cm³, yielding U_tr = 100 cm/s, approximately the same as the neonatal value. For either adult or neonate, the range of µ/ depends only on the instilled substance. If that substance has a low surface tension, i.e.,  = 5 dyn/cm, and a high viscosity, i.e., µ = 0.6 g · cm¹ · s, or a high surface tension, i.e.,  = 50 dyn/cm, and a lower viscosity, i.e., µ = 0.01 g · cm¹ · s, then the range of µ/ is roughly 2.0 × 10⁴ s/cm  µ/  0.12 s/cm. For our representative value of U_tr = 100 cm/s, the range for the Ca_tr is ~0.02  Ca_tr  12.

In Fig. 7A, the plug volume given by Eq. 7 is plotted vs. generation number for various values of W_p and Ca_tr that fall in the range calculated above. The plots show that the plug will rupture proximally for higher values of Ca_tr because the thickness of the deposited film is larger (see Eq. 6), and for smaller values of W_p, since there is less volume to distribute. Therefore, the generation number at which the plug ruptures decreases with decreasing W_p and increasing Ca_tr. From Eq. 7 it is possible to compute the critical value of W_p (call it W_r) that will rupture at a specified generation number n = n_r. Figure 7B shows plots of W_r as a function of Ca_tr for selected values of n_r where clearly W_r increases with Ca_tr and airway generation. As shown in Fig. 7A, 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. If W_p is sufficiently large or Ca_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.

View larger version (16K): [in this window] [in a new window]   Fig. 7.   A: liquid plug volume vs. airway generation n for a range of Catr and Wp; B: critical plug volume that ruptures at generation nr, Wr, vs. Catr for nr = 7, 12, and 18. See Glossary for definitions.

Direct instillation into the alveoli may be a desirable goal. Figure 8 is a plot of the alveolar plug volume, i.e., V(X₁₈) from Eq. 7, as a function of Ca_tr for several values of W_p. This figure shows that direct instillation into the alveoli is enhanced by larger volumes and smaller Ca_tr. As inferred from previous figures, the plug volume at generation 18 decreases with increasing Ca_tr and decreasing W_p. For the range of parameters considered, rupture does not occur at generation 18 (and earlier) if Ca_tr falls below 0.0035 and if W_p  0.5. Also, the Ca_tr needed for rupture increases significantly if W_p is doubled, for example from 0.5 to 1. 

View larger version (13K): [in this window] [in a new window]   Fig. 8.   Alveolar plug volume V(X18) vs. tracheal capillary number Catr, for various initial plug volumes Wp.

From Eq. 5, as Ca increases, so does the thickness of the trailing film, H (see Eq. 6). This is illustrated in Fig. 9, where H is plotted as a function of airway-generation number n for several values of Ca_tr and W_p. It shows that for fixed Ca_tr and W_p, H is a decreasing function of n, since Ca decreases as the total cross-sectional 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.

View larger version (17K): [in this window] [in a new window]   Fig. 9.   Effect of Catr on film thickness-to-airway radius ratio H vs. generation number n. Symbols denote where plug ruptures were for given Wp.

Deposited-film flow. From the similarity solution, Eq. 11, H(X) is plotted for several values of time T in Fig. 10. The smallest dimensionless value of T, 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 that H = h/r increases with distance for fixed T, but the dimensional film thickness h diminishes with X. Because surface-tension 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 reach generation 7, where H is of the same order as H_M. In Fig. 11, the front location (T) is plotted for several values of the product W_b. It shows that initially the leading edge advances quite rapidly, but, as  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  by gravity is found by inverting the definition for (T) in Eq. 11

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For example, the time required for the film front to reach generation 7 in an adult, where X = 0.8, is T  1.5/(W_b)². For an adult lung, this corresponds to 0.000833/(W_b)² s, and for a premature neonate the time taken is 0.0052/(W_b)² s, when using µ = 0.01 g · cm¹ · s. Approximately the same values are obtained when a perfluorocarbon liquid such as Liquivent is used, since it has approximately the same kinematic viscosity as water (µ = 0.0184 g · cm¹ · s; see Ref. 74), but the dimensional time can be considerably larger if Survanta is used, which is 60 times more viscous than water (F. F. Espinosa, personal communication).

View larger version (12K): [in this window] [in a new window]   Fig. 10.   Gravitational spreading: similarity solution, H (Eq. 11), is plotted against airway generation number n at times indicated. Dotted line shows height of bolus at its leading edge as a function of position. Parameter values are Wp = 0.098 and  = 0.2.

View larger version (13K): [in this window] [in a new window]   Fig. 11.   Leading edge of bolus  vs. time T for different values of  Wb. See Glossary for definitions.

For a liquid bolus with V_b = 3 cm³, having the viscosity of water, the leading-edge 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 to generation 7. [For larger V_b, the bolus travels quicker and deeper into the lung before reaching the critical height: for example, if V_b = 1 (or 5) cm³, the critical height is reached at generation 4 (or 9) 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  )V_b of the fluid bolus to pass through generation 7 is

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For  = 0.05, 95% of the fluid drains past generation 7 in time T = 2 × 10³/(W²_b²). For an adult, by using the values of W_b shown in Fig. 11, the time taken for 95% of the bolus to reach generation 7 ranges from 1.85 to 185 min, whereas for a neonate it ranges from 11.57 min to 19.29 h. With the parameter values used above, the leading edge of a bolus with V_b = 3 cm³ reaches generation 7 in 3 s (181 s for Survanta), and then 95% of the fluid volume drains past this location in the next 48 min (48 h for Survanta). We note that these transport times are linearly dependent on 1/g, from the dimensional versions of Eqs. 17 and 18. Airways that, on the average, are not vertically oriented will have a smaller gravity component, and transit time will be lengthened accordingly in those regions.

Equations 10, 11, 17, and 18 can be applied past generation 7 if a constant-surface-tension bolus of perfluorocarbon is used. For example, the leading edge of a 3-cm³ liquid bolus reaches generation 17 of an adult lung at T = [(1  X₁₇)^1/2  1]², or 54 s, and 95% of the initial bolus arrives after T  400/[(1  X₁₇)W²_b²], or ~8 h. This order of magnitude increase in time between generations 7 and 17 is primarily due to the rapid increase in surface area with path length.

For a neonate, the initial surface-tension 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 cm³, 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 surface-tension gradients, and our Marangoni-flow model would be appropriate.

Surface-layer transport. The following parameter values are chosen for the surfactant-spreading model derived in FORMULATION OF THE MODEL, Surface-layer 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 · cm¹ · s. The reference film thickness for an adult is chosen to be h₇ = 20 µm, which corresponds to 1% of the diameter of airway generation 7. The path length from trachea to alveoli in the adult is ~15 cm, so the path length from generation 7 to the acinar region gives a length scale L_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 length _a = 0.921. The scale for the surface-tension difference driving the spreading is taken to be S₇ = 50 dyn/cm. The Marangoni time scale T_M = µL²_M/(S₇h₇) is, therefore, ~T_M = 0.75 s for an adult. For a neonate, where the bolus is delivered slowly so that the fluid lining remains thin, the same time scale applies. A key aim of our calculations is to determine how surface-area expansion, and the effects of preexisting surfactant, leads to transit and delivery times for exogenous surfactant that may be significantly in excess of T_M.

The effect of increasing preexisting surfactant concentration, _A, is shown in Fig. 12 where all parameter values are as in Fig. 5, except that _A = 0.5 and the computations are carried out for a longer time until a steady-state solution is obtained. As predicted by Refs. 16, 28, and 42, the effect of increasing _A is to force and to have smoother distributions near the disturbance's leading edge at early times [compare, for example, (,0.1) in Figs. 5B and 12B]. However, once the disturbance reaches the downstream end of the domain, the leading edge of becomes steeper and the film thicker behind the leading edge. This is because surface-tension 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, (n = 18,), increases with until there is a balance of fluxes at n = 7 and n = 18. When this occurs, the liquid-lining 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.

View larger version (20K): [in this window] [in a new window]   Fig. 12.   (,) (A) and (,) (B, C) vs. and n at  = 0, 0.1, 1, 3, 6, 9, and 12, incorporating effects of surface-area expansion, with A = 0.5 and  = 0. Solid curves with symbols show the ultimate steady state;  in B show leading edge of exogenous surfactant distribution at times 0, 0.1, 1, 3, 6, and 6.95. See Glossary for definitions.

Because the effects of surface-area expansion cause the surface-tension gradient to be small at the distal end, other effects such as a pressure-gradient-driven flow due to the decrease in airway radius with distance from the trachea may become important. In Fig. 13, we have plotted and for _A = 0.5 and  = 0.01, obtained by solving Eq. A1 numerically. Note that the preexisting surfactant is assumed to have an initially nonuniform distribution _eq() (given by Eq. A3, and visible for n > 9 in Fig. 13, B and C) in which increases slowly with respect to : this generates a weak, proximally directed shear stress at the free surface that opposes the shear stress generated by the curvature-driven 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 curvature-driven 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 toward  = _a, so that increases rapidly above the level _eq(_a) for 0 <  < 1 (see Fig. 13C). However, as the surfactant-driven flow approaches equilibrium, it weakens relative to the curvature-driven flow, and again falls beneath _eq(_a) (e.g.,  = 12 in Fig. 13C), until ultimately a steady state is achieved. An important effect of the weak curvature-driven flow is to prevent the film thickness at the downstream end from becoming too big, reducing the likelihood of airway closure (Fig. 13A).

View larger version (20K): [in this window] [in a new window]   Fig. 13.   (,) (A) and (,) (B, C) vs. and n at  = 0, 0.1, 1, 3, 6, and 12, including a weak curvature-driven flow, with A = 0.5 and  = 0.01.  in B show leading edge of the exogenous surfactant distribution at times 0, 0.1, 1, 3, 6, and 7.02. See Glossary for definitions.

The leading edges of the exogenous surfactant distribution L_ex and of the surface-compression wave L_D are plotted vs. time in Fig. 14. As in Ref. 28, L_ex lags significantly behind L_D, since _A > 0. However, whereas L_D is relatively insensitive to _A and the effects of surface-area expansion, reaching generation 18 at   0.55, or 0.41 s (Fig. 14B), this is not so for L_ex: for _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 surface-area expansion. The rate of advance of L_ex falls in the distal generations due to the effects of surface-area expansion and the increased flow resistance due to a decrease in liquid-lining thickness. An increase in _A to 0.5 causes the arrival time of exogenous surfactant to more than double. The effect of curvature-driven flow on the time taken for exogenous surfactant to reach the respiratory bronchioles is not significant (Fig. 14A). It causes exogenous surfactant to reach generation 18 marginally later than the  = 0 case and has its biggest impact in the distal generations where surface-tension gradients are weakest.

View larger version (15K): [in this window] [in a new window]   Fig. 14.   A: Lex vs. for A = 0.2 and 0.5 and  = 0, 0.01, and 0.1. B: LD vs. for A = 0.2,  = 0.01 and A = 0.5,  = 0.01. See Glossary for definitions.

Surface-layer transit time in the steady state. The effect of the initial preexisting surfactant concentration _A on surface-area expansion can be further investigated by determining the transit time, _s, for exogenous surfactant to cross the whole domain once steady state is achieved (neglecting curvature-driven flow). _s is computed from the equation that governs the Lagrangian motion of a particle on the air-liquid interface (28)

(19)

where Û_s(,) is the surface velocity (assuming for simplicity that  = 0). At steady state, with the boundary conditions used above,  = . Hence

(20)

In the absence of surface-area expansion, the right-hand side of Eq. 20 can be integrated analytically, using Eq. 12, to yield

(21)

Figure 15 shows that in this case _s is only very weakly dependent on _A for 0 < _A  0.5 but increases by a factor of approximately five as _A is increased from 0.1 to 0.9; the transit time becomes infinite as _A  1. The dependence of _s on _A is influenced by the choice of boundary conditions: Jensen et al. (44), who used no-flux conditions on and at  = 0 rather than assuming a steady source of exogenous surfactant there, so that their flow was always unsteady, found a much stronger dependence of _s on _A. However, when the effects of surface-area expansion are included in Eq. 19, _s increases by more than a factor of 10 as _A is increased from 0.01 to 0.95 (Fig. 15).

View larger version (13K): [in this window] [in a new window]   Fig. 15.   Transit time (s) for exogenous surfactant to cross domain at steady state as a function of A. Solid curve includes surface-area expansion effects, and dashed curve does not.

Dose delivery times: steady-state alveolar surface concentration. After the initial transients have decayed (within a few seconds of the initial delivery of the first surfactant dose), the steady-state flow in generations 7-18 delivers a steady flux of surfactant to the alveolar space. This (nondimensional) flux, B, evaluated at  = _a, where and are given by Eq. 13, is

(22)

where _A is the surface concentration of surfactant in the alveoli, which we treat as a well-mixed compartment. We assume that the alveolar compartment comprises the entire transport regime distal to generation 18.

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 half-doses, 12 h apart (14). Therefore, an infant weighing 500 g receives two doses of 25 mg. So we take m_dose = 25 × 10³ 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 70-kg adult. We take the micelle concentration to be the reference concentration at generation 7 in an adult, so ₇ = 2 × 10¹⁴ molecules/cm² (66). Because dipalmitoylphosphatidylcholine has molecular weight of 734 and N_A = 6.02 × 10²³, ₇ = 2.44 × 10⁷ g/cm². For an infant, L_M = 3.75 cm, b₇ = 1.13 cm, so ₇ L_M b₇ = 1.03 × 10⁶ g, and T_M = 0.75 s (where this time scale is that defined for surfactant-driven flow in FORMULATION OF THE MODEL, Deposited-film flow). Thus the scale of the exogenous surfactant flux is q₇ = ₇ L_M b₇/T_M = 1.37 × 10⁶ 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 (with q_A = q₇) yields a delivery time = 25 × 10³ g/(1.37 × 10⁶ g/s) = ~5 h.

Dose delivery times: quasi-steady 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 half-life, i.e., 5-15 h. This time scale, used to define  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 flux Q_A depends only on _A when Eq. 22 is used. Then, using Eq. 22, Eq. 15 becomes

(23)

where  = F/[1  (1  _a)³]. If  = 0, so that there is no supply of exogenous surfactant, the alveolar surfactant concentration has an equilibrium value in which natural supply and uptake balance, i.e., _A = P_A. This value will be smaller than the level of exogenous concentrations, so that 0 < P_A < 1. We now consider the evolution of the system for  > 0.

Suppose that at  = 0, _A = P_A, and  > 0. Then _A will rise with time, approaching a new equilibrium value _A = _eq, say, where _eq = _eq(,P_e) satisfies

(24)

Figure 16 shows how _eq depends on and P_A. As increases for fixed P_A, _eq rises from P_A (when  = 0) toward 1. For small , the delivered flux is weak, so that alveolar concentrations are close to their undisturbed values. For large , the rate of uptake is relatively weak compared with the flux of exogenous material, so that over sufficiently long times the alveolar surfactant concentration can rise close to the level of the exogenous surfactant supply. As P_A increases, the equilibrium levels all rise: an increase in natural surfactant production may be stimulated by the delivery of exogenous material. Figure 17 shows how the alveolar concentration _A() rises slowly to its equilibrium value, driven by the flux of exogenous material. This equilibrium is reached in principle after an infinite time. In practice, however, the dose delivered to generation 7 is of finite size, so that after a finite time the source dries up. We must therefore compute the time taken for the full dose to be delivered to the alveoli.

View larger version (12K): [in this window] [in a new window]   Fig. 16.   Steady-state alveolar surfactant concentration eq vs. rate of supply of exogenous surfactant for several rates of supply of endogenous surfactant, PA.

View larger version (12K): [in this window] [in a new window]   Fig. 17.   Alveolar equilibrium surfactant concentration A() vs.  for several values of PA and .

Therefore, we compute the integral of the flux of surfactant delivered to the alveoli. The mass of exogenous surfactant delivered in time _D is

(25)

In terms of the nondimensional variables used in Eq. 23

(26)

where _D = t_DK. Using parameter values given in Dose delivery times: steady-state alveolar surface concentration, with A_A = 3.6 m² for an infant and 90 m² for an adult, delivery times for M = m_dose/(A_{A 7}) = 3 (for an infant) and M = 16 (for an adult), computed by using Eqs. 23 and 26, are plotted for various values of P_A and in Fig. 18. _A will rise from P_A at  = 0 toward its equilibrium value _eq(P_A,) < 1 as  increases, until a dose M is delivered at  = _D. Figure 18 shows that _D decreases if the rate of delivery of exogenous surfactant is increased () or if the supply of endogenous surfactant is increased. When the dose is fully used up, the alveolar surfactant concentration will fall again to its equilibrium value P_A, which may now be larger if P_A has increased as _A increased. For a neonate, 2.5  _D  12, given 2.6    7.2. Therefore, for a rate constant K = 0.138/h, the total delivery time ranges from 18 to 87 h. Usually a second dose is delivered 12-24 h after the first dose. This correlates well with our findings, provided P_A is sufficiently large. For an adult, given K = 0.138/h,   1 and, from Fig. 18B, 25  _D  104. This corresponds to delivery times ranging from 1 wk to 1 mo and may provide an explanation for the lack of long-term success of SRT with ARDS patients (36).

View larger version (14K): [in this window] [in a new window]   Fig. 18.   Delivery time D vs. for different pA. A: M = 3 (infant); B: M = 16 (adult). See Glossary for definitions.

	DISCUSSION

Top Abstract Introduction Results Discussion Appendix References

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 cm³, 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 surface-area expansion effects that gravitational forces weaken, relative to surface-tension gradients, so that Marangoni flows drive the surfactant deeper into the lung. Distally directed curvature-driven 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 surface-area expansion that the monolayer experiences as it advances further, the model predicts that surface-tension reduction is initially experienced at the terminal bronchioles very rapidly, within ~0.5 s (Fig. 14B) of the bolus reaching generation 7. This is so because the advancing exogenous surfactant compresses preexisting surfactant: the higher the concentration of preexisting material, the quicker an initial surface-tension 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 at generation 18. Even so, the level of preexisting surfactant, _A, remains an important parameter. Exogenous surfactant was predicted to arrive at generation 18 within ~2.25 s of leaving generation 7 (Fig. 14A) when _A = 0.2; this transit time rose to ~4.5 s with _A = 0.5. A steady balance of fluxes between material provided by gravity-driven 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 _A  0 to a few minutes as _A approaches its tracheal concentration (_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 quasi-steady-state 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 2.5-cm³ 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 thin-film 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 surface-tension 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 at generation 7. The calculations appropriate for the adult lung then carry over directly to this case: surface-tension 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 function B(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 gravity-driven and Marangoni-driven 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 gravity-driven 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 gravity-driven 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 < _A < 0.7 (approximately) the dependence of spreading rates on _A is greater if the effects of surface-area expansion are included in the model than if they are absent. Rapid spreading of fresh exogenous material occurs if _A is low; once _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 slow-moving 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.