Abstract
Measurements of surface tension in the lung have shown that a time-mean gradient exists with the potential to generate clearance flows toward the mouth in the thin liquid layer that lines the airways. A model is developed to explore this phenomenon in the simple case of a membrane with linear variation in strain along its length, coupled with the unique properties of pulmonary surfactant. The evolution equations are solved numerically for liquid layer thickness and surfactant concentration during a single oscillatory cycle, and the net volume exchanged is computed. The parameters governing the flow are shown to be time scales for viscous effects, {ie913-1}, surface diffusion, {ie913-2}, surfactant adsorption, {ie913-3}, surfactant desorption, {ie913-4}, oscillation, {ie913-5}, and the average membrane strain {ie913-6}. The volume pumped toward the less compliant end on the initial cycle is maximized when {ie913-7} and is relatively insensitive to {ie913-8}. Rapid adsorption generally augments liquid transport for {ie913-9}. Pumping drops precipitously if {ie913-10}. Effects of strain amplitude are reported as well. For parameter values approximating those in the lung, pumping rates are near optimal; the mean surface velocity is ∼0.05 mm/sec, compared with 0.2 mm/sec produced by the action of cilia on the mucus layer. This mechanism might therefore be important in assisting clearance from the lung or maintaining a liquid layer over alveolar facets.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Abbreviations
- C :
-
bulk concentration of surfactant
- Ds :
-
surface diffusivity of surfactant
- h :
-
liquid layer thickness
- ho :
-
initial liquid layer thickness
- k 1 :
-
surfactant adsorption constant
- k 2 :
-
surfactant desorption constant
- L(t) :
-
length of membrane as a function of time
- Lo :
-
initial length of membrane
- ΔL :
-
mean amplitude of membrane distension
- Q :
-
flow rate evaluated at midplane of the membrane, ζ=0.5
- t :
-
time
- {ie925-1}:
-
mean velocity in across liquid layer, relative to laboratory
- us :
-
surface velocity of liquid layer, relative to laboratory
- V net :
-
net volume of liquid transported past midpoint of membrane
- {ie925-2}:
-
mean velocity in across liquid layer, relative to membrane
- Vs :
-
surface velocity of liquid layer, relative to membrane
- x,y :
-
Cartesian coordinates relative to the laboratory frame of reference
- α:
-
coefficient in linear isotherm relating surface concentration to surface tension
- ∈(ζ):
-
local strain along distending membrane
- {ie925-3}:
-
mean strain of distending membrane
- Γ:
-
surface concentration of surfactant
- {ie925-4}:
-
surface concentration of surfactant when surface tension equals 0
- {ie925-5}:
-
maximum equilibrium surface concentration asC→∞
- {ie925-6}:
-
time-mean surface concentration over 1 cycle
- μ:
-
dynamic viscosity of liquid layer
- σ:
-
air-liquid surface tension
- {ie925-7}:
-
surface tension when no surfactant present
- {ie925-8}:
-
adsorption time scale, 1/k 1 C
- {ie925-9}:
-
desorption time scale, 1/k 2
- {ie925-10}:
-
diffusion across liquid layer time scale,ho 2/D
- {ie925-11}:
-
surface diffusion time scale,Lo 2/Ds
- {ie925-12}:
-
oscillation time scale
- {ie925-13}:
-
viscous time scale, {ie925-14}
- ζ:
-
material coordinate attached to the membrane
References
Adamson, A. W. Physical Chemistry of Surfaces. New York: John Wiley and Sons, Inc., 1990, pp. 777.
Anderson, D. A., J. C. Tannhill, and R. H. Pletcher. Computational Fluid Mechanics and Heat Transfer. Washington DC, Hemisphere Publishing Corp., 1984, pp. 599.
Bachofen, H., S. Schürch, M. Urbinelli, and E. R. Weibel. Relations among alveolar surface tension, surface area, volume, and recoil pressure.J. Appl. Physiol. 62:1878–1887, 1987.
Bastacky, J., C. Y. C. Lee, J. Goerke, H. Koushafar, D. Yager, L. Kenaga, T. P. Speed, Y. Chen, and J. A. Clements. Alveolar lining layer is thin and continuous: low-temperature scanning electron microscopy of rat lung.J. Appl. Physiol. 79:1615–1628, 1995.
Borgas, M. S., and J. B. Grotberg. Monolayer flow on a thin film.J. Fluid Mech. 193:151–170, 1988.
Davis, S. H., A. K. Liu, and G. R. Sealy. Motion driven by surface tension gradients in a tube lining.J. Fluid Mech. 62:737–751, 1974.
Espinosa, F. F. Exogenous Surfactant Transport through the Pulmonary Airways: Improving Surfactant Replacement Therapy for Neonatal Respiratory Distress Syndrome. Cambridge, MA: Massachusetts Institute of Technology, Doctoral Thesis, 1996.
Espinosa, F. F., A. H. Shapiro, J. J. Fredberg, and R. D. Kamm. Spreading of exogenous surfactant in an airway.J. Appl. Physiol. 75:2028–2039, 1993.
Faridy, E. E. Effect of ventilation on movement of surfactant in airways.Respir. Physiol. 27:323–334, 1976.
Gaver, D. P. III, and J. B. Grotberg. The dynamics of a localized surfactant on a thin film.J. Fluid Mech. 213:127–148, 1990.
Gaver, D. P. III, and J. B. Grotberg. Droplet spreading on a thin viscous film.J. Fluid Mech. 235:399–414, 1992.
Gehr, P., S. Schürch, Y. Berthiaume, V. Im Hof, and M. Geiser. Particle retention in airways by surfactant.J. Aerospace Med. 3:27–43, 1990.
Goerke, J., and J. A. Clements. Alveolar surface tension and lung surfactant. In: Handbook of Physiology. The Respiratory System. Mechanics of Breathing, sect. 3, vol. III, part 1, chap. 16. Bethesda, MD: American Physiological Society, 1986.
Halpern, D., and J. B. Grotberg. Dynamics and transport of a localized soluble surfactant on a thin film.J. Fluid Mech. 237:1–11, 1992.
Jensen, O. E., and J. B. Grotberg. Insoluble surfactant spreading on a thin viscous film: shock evolution and film rupture.J. Fluid Mech. 240:259–288, 1992.
Jensen, O. E., and J. B. Grotberg. The spreading of heat or soluble surfactant along a thin liquid film.Phys. Fluids A 5:58–68, 1993.
Jensen, O. E. Self-similar, surfactant-driven flows.Phys. Fluids 6:1084–1094, 1994.
Kharasch, V. S., T. D. Sweeney, J. J. Fredberg, J. Lehr, A. I. Damokosh, M. E. Avery, and J. D. Brain. Pulmonary surfactants as a vehicle for intratracheal delivery of technetium sulfur colloid and pentamidine in hamster lungs.Am. Rev. Respir. Dis. 144:909–913, 1991.
Martin, H. B., and D. F. Proctor. Pressure-volume measurements of dog bronchi.J. Appl. Physiol. 13:337–343, 1958.
Mason, R. J., and M. C. Williams. Alveolar type II cells. In: The lung: scientific foundations, chap. 3.1.9 edited by R. G. Crystal, J. B. West,et al. New York: Raven Press Ltd., 1991.
Mendenhall, R. M. Surface spreading of lung alveolar surfactant.Respir. Physiol. 16:175–178, 1972.
Otis, D. R. Jr. E. P. Ingenito, R. D. Kamm, and M. Johnson. Dynamic surface tension of surfactant TA: experiments and theory.J. Appl. Physiol. 77:2681–2688, 1994.
Podgorski, A., and L. Gradon. An improved mathematical model of hydrodynamical self cleansing of pulmonary alveoli.Ann. Occup. Hyg. 37:347–365, 1993.
Rensch, H., H. von Seefeld, K. F. Gebhardt, D. Renzow, and P. J. Sell. Stop and go particle transport in the peripheral airways? A model study.Respiration 44:346–350, 1983.
Sakata, E. K., and J. C. Berg. Surface diffusion in monolayers.Ind. Eng. Chem. Fundam. 8:570–575, 1969.
Sleigh, M. A., J. R. Blake, and N. Liron. The propulsion of mucus by cilia.Am. Rev. Respir. Dis. 137:726–741, 1988.
Toremalm, N. G. The daily amount of tracheo-bronchial secretions in man.Acta Oto-laryng. (Suppl.) 158:43–53, 1960.
Weibel, E. R. Morphometry of the Human Lung. New York: Academic Press, Inc., 1963.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Espinosa, F.F., Kamm, R.D. Thin layer flows due to surface tension gradients over a membrane undergoing nonuniform, periodic strain. Ann Biomed Eng 25, 913–925 (1997). https://doi.org/10.1007/BF02684128
Received:
Revised:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02684128