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

(3)

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

(4)

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.

(5)

(6)

(7)

(8)

1

(9)

(10)

(11)

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.

,

(12)

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.

(13)

d

(14)

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.

(15)

	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