Comprehensive Physiology Wiley Online Library

Distribution of Stresses Within the Lung

Full Article on Wiley Online Library



Abstract

The sections in this article are:

1 Why is Stress Distribution Nonuniform?
2 Why is it Important to Know Stress Distribution?
3 Simplifications Introduced to Analysis of Stress Distribution
4 Relationship Between Microscopic Structure and Macroscopic Properties
5 Material Properties of Lung Parenchyma
6 Meaning of Other Elastic Constants
7 Implications of Relative Magnitudes of Bulk and Shear Moduli
8 Macroscopic Lung Stress Distribution
9 Local Stress Distribution
Figure 1. Figure 1.

Linear analogues of pulmonary stress distribution. Three‐dimensional parenchymal deformation is illustrated by a spring. A: uniform expansion of spring stretched between 2 plates. Stress is distributed uniformly throughout body, and strain is uniform, as indicated by uniform spacing of coils. B: vertically oriented spring deformed by its own weight. Superimposed on uniform stress determined by separation of plates, weight of spring causes nonuniform stress and strain manifested by nonuniform spacing of coils. C: increased uniform stress by greater separation of end plates changes strain distribution and apparent magnitude of gravitational deformation because of nonlinear length‐tension relationship of spring. D: either stiff or soft rings can be inserted in body of spring, further changing strain distribution.

Figure 2. Figure 2.

Bulk modulus (K) of parenchyma of excised dog lungs obtained by small perturbations in volume as a function of trans‐pulmonary pressure (P) in left panel and as a function of lung volume in right panel. Inflation (open circles) and deflation (closed circles) volume histories are shown; K ≃ 4P during both inflation and deflation maneuvers.

From Lai‐Fook
Figure 3. Figure 3.

Shear modulus (μ) of excised dog lungs determined by punch‐indentation tests as a function of transpulmonary pressure (P); μ is essentially a linear function of P and is approximated by μ ≃ 0.7P.

From Lai‐Fook
Figure 4. Figure 4.

Gravitational deformation of cylindrical body constrained within rigid container of same size and shape prior to action of gravity. Material properties are those of normal lung parenchyma at functional residual capacity as in Figs. and and density (σ) of 0.13 g/cm3. Dotted bars in both panels, gradients predicted by linear‐elasticity model using values stated above. Left: vertical gradient in radial stress. Crosshatched bar, gradient that would occur if shear modulus (μ) were zero. Right: vertical gradient in regional volume. Finely stippled bar, gradient that would be predicted from vertical gradient in stress (dotted bar) and pressure‐volume curve [bulk modulus (K)]. Vertically hatched bar, gradient from hydrostatic pressure gradient and pressure‐volume curve.

Adapted from Rodarte
Figure 5. Figure 5.

Effect of change in shape without gravity. Cylindrical body with properties as in Fig. but without gravity is subjected to a rotation of sides of angle (α) of 1.5°. Left: vertical gradient in radial stress. Right: dotted bar, vertical gradient in regional volume; finely stippled bar, gradient that would be predicted from regional stress and uniform pressure‐volume curve.

Adapted from Rodarte
Figure 6. Figure 6.

Regional volume gradients caused by shape and gravity. Cylindrical body with properties as in Figs. and is subjected to gravity and then angular deformation required to create vertical gradient of radial stress (Δτrr2). Left: angle α required to produce vertical gradient of pressure of 0.25 cmH2O/cm height in presence of gravity. Right: vertical gradient in regional volume. Undotted portion of left bar, gradient caused by gravity; dotted portion, gradient caused by shape change; finely stippled bar, gradient that would be predicted from pressure gradient and uniform pressure‐volume curve.

Adapted from Rodarte
Figure 7. Figure 7.

Schematic representation of effect of gravity on material with idealized lung shape. εzz, Vertical strain; ɛrr, radial strain; ɛτzz, vertical stress caused by weight of lungs; E, Young's modulus; ν, Poisson's ratio. Subscript p indicates value is function of pressure.

Figure 8. Figure 8.

Fractional radial expansion of cylindrical hole (U/R) and nondimensional displacement of boundary of cylindrical holes in lung parenchyma as function of transpulmonary pressure (P). μ, Shear modulus. Data fall within range predicted by equation U/R = (P/2) μ, as indicated by dotted lines for μ = 0.7P and μ = 0.5P. Because μ is proportional to P, prediction is independent of P.

From Lai‐Fook, Rodarte, et al.
Figure 9. Figure 9.

Graphic representation of relationships among uniform hole diameter (Du), intact bronchial diameter (Di), excised bronchial diameter (De), difference between peribronchial and pleural pressure (ΔPx), and parenchymal shear modulus (μ), when Du > De. Du and Di are plotted against transpulmonary pressure (PL) and De is plotted against transmural pressure (Ptm). Nonuniform behavior of parenchymal hole at constant transpulmonary pressure (PL′) is given by line ab. With specific values for Du and Di (a and c) at PL′, d on De‐Ptm curve is determined. Alternatively, given values for Du and De, value for Di (c on line af) is determined.

From Lai‐Fook, Hyatt, and Rodarte
Figure 10. Figure 10.

Comparison of intact bronchial pressure (PL)‐diameter (Di) behavior for deflation pressure‐volume maneuvers in which lung volume changes with PL (dashed curve) and the isovolume case in which lung volume is held at initial PL and intrabronchial pressure is reduced (dasheddotted curve). Diagonal line from uniform hole diameter (Du) represents pressure change (ΔPx) required to reduce Du. Solid curve (De‐Ptm) represents excised bronchial behavior. Reduction of intact bronchial diameter (Di) at constant lung volume requires total reduction in intrabronchial pressure that equals sum of ΔPx required to reduce diameter of lung parenchyma and change in transmural pressure (ΔPtm) required to reduce excised bronchial diameter (De).

From Lai‐Fook, Hyatt, and Rodarte
Figure 11. Figure 11.

Representative pressure‐diameter behavior for major veins in excised dog lobe at various static values of transpulmonary pressure (PL) indicated on individual curves. Symbols, measurements obtained by roentgenograms; solid curves, predicted relationships computed from pressure‐diameter behavior of uniform hole (Du vs. PL, dashed curve) and computed excised pressure‐diameter behavior (De, dotted curve). Du varies with cube root of lung volume and was chosen so that De computed from Du, and measured Dv at one PL value, when used with Du predicted the measured Dv at the other PL values.

From Lai‐Fook
Figure 12. Figure 12.

Difference between perivascular pressure and pleural pressure (Px′) vs, transpulmonary pressure (PL) for constant arterial pressures of 10, 25, and 35 cmH2O and venous pressures of 0, 10, and 25 cmH2O. Vascular pressures (Pv) are measured relative to pleural pressure. A: data for arteries in 10 lobes. B: data for veins in 11 lobes. Vertical bars, 1 SE.

From Lai‐Fook
Figure 13. Figure 13.

Interdependence between bronchus and arteries. A: distribution of 2 principal stresses (open and closed circles) for points on walls of deformed bronchus and artery corresponding to circles in B. Note stress concentration at junction of airway and vessel. B: dashed curves, luminal surface of bronchus and artery when parenchyma is uniformly expanded at a transpulmonary pressure of 4 cmH2O; solid curves, distortion of shape of luminal surface when arterial pressure is reduced to −40 cmH2O.

From Lai‐Fook and Kallok


Figure 1.

Linear analogues of pulmonary stress distribution. Three‐dimensional parenchymal deformation is illustrated by a spring. A: uniform expansion of spring stretched between 2 plates. Stress is distributed uniformly throughout body, and strain is uniform, as indicated by uniform spacing of coils. B: vertically oriented spring deformed by its own weight. Superimposed on uniform stress determined by separation of plates, weight of spring causes nonuniform stress and strain manifested by nonuniform spacing of coils. C: increased uniform stress by greater separation of end plates changes strain distribution and apparent magnitude of gravitational deformation because of nonlinear length‐tension relationship of spring. D: either stiff or soft rings can be inserted in body of spring, further changing strain distribution.



Figure 2.

Bulk modulus (K) of parenchyma of excised dog lungs obtained by small perturbations in volume as a function of trans‐pulmonary pressure (P) in left panel and as a function of lung volume in right panel. Inflation (open circles) and deflation (closed circles) volume histories are shown; K ≃ 4P during both inflation and deflation maneuvers.

From Lai‐Fook


Figure 3.

Shear modulus (μ) of excised dog lungs determined by punch‐indentation tests as a function of transpulmonary pressure (P); μ is essentially a linear function of P and is approximated by μ ≃ 0.7P.

From Lai‐Fook


Figure 4.

Gravitational deformation of cylindrical body constrained within rigid container of same size and shape prior to action of gravity. Material properties are those of normal lung parenchyma at functional residual capacity as in Figs. and and density (σ) of 0.13 g/cm3. Dotted bars in both panels, gradients predicted by linear‐elasticity model using values stated above. Left: vertical gradient in radial stress. Crosshatched bar, gradient that would occur if shear modulus (μ) were zero. Right: vertical gradient in regional volume. Finely stippled bar, gradient that would be predicted from vertical gradient in stress (dotted bar) and pressure‐volume curve [bulk modulus (K)]. Vertically hatched bar, gradient from hydrostatic pressure gradient and pressure‐volume curve.

Adapted from Rodarte


Figure 5.

Effect of change in shape without gravity. Cylindrical body with properties as in Fig. but without gravity is subjected to a rotation of sides of angle (α) of 1.5°. Left: vertical gradient in radial stress. Right: dotted bar, vertical gradient in regional volume; finely stippled bar, gradient that would be predicted from regional stress and uniform pressure‐volume curve.

Adapted from Rodarte


Figure 6.

Regional volume gradients caused by shape and gravity. Cylindrical body with properties as in Figs. and is subjected to gravity and then angular deformation required to create vertical gradient of radial stress (Δτrr2). Left: angle α required to produce vertical gradient of pressure of 0.25 cmH2O/cm height in presence of gravity. Right: vertical gradient in regional volume. Undotted portion of left bar, gradient caused by gravity; dotted portion, gradient caused by shape change; finely stippled bar, gradient that would be predicted from pressure gradient and uniform pressure‐volume curve.

Adapted from Rodarte


Figure 7.

Schematic representation of effect of gravity on material with idealized lung shape. εzz, Vertical strain; ɛrr, radial strain; ɛτzz, vertical stress caused by weight of lungs; E, Young's modulus; ν, Poisson's ratio. Subscript p indicates value is function of pressure.



Figure 8.

Fractional radial expansion of cylindrical hole (U/R) and nondimensional displacement of boundary of cylindrical holes in lung parenchyma as function of transpulmonary pressure (P). μ, Shear modulus. Data fall within range predicted by equation U/R = (P/2) μ, as indicated by dotted lines for μ = 0.7P and μ = 0.5P. Because μ is proportional to P, prediction is independent of P.

From Lai‐Fook, Rodarte, et al.


Figure 9.

Graphic representation of relationships among uniform hole diameter (Du), intact bronchial diameter (Di), excised bronchial diameter (De), difference between peribronchial and pleural pressure (ΔPx), and parenchymal shear modulus (μ), when Du > De. Du and Di are plotted against transpulmonary pressure (PL) and De is plotted against transmural pressure (Ptm). Nonuniform behavior of parenchymal hole at constant transpulmonary pressure (PL′) is given by line ab. With specific values for Du and Di (a and c) at PL′, d on De‐Ptm curve is determined. Alternatively, given values for Du and De, value for Di (c on line af) is determined.

From Lai‐Fook, Hyatt, and Rodarte


Figure 10.

Comparison of intact bronchial pressure (PL)‐diameter (Di) behavior for deflation pressure‐volume maneuvers in which lung volume changes with PL (dashed curve) and the isovolume case in which lung volume is held at initial PL and intrabronchial pressure is reduced (dasheddotted curve). Diagonal line from uniform hole diameter (Du) represents pressure change (ΔPx) required to reduce Du. Solid curve (De‐Ptm) represents excised bronchial behavior. Reduction of intact bronchial diameter (Di) at constant lung volume requires total reduction in intrabronchial pressure that equals sum of ΔPx required to reduce diameter of lung parenchyma and change in transmural pressure (ΔPtm) required to reduce excised bronchial diameter (De).

From Lai‐Fook, Hyatt, and Rodarte


Figure 11.

Representative pressure‐diameter behavior for major veins in excised dog lobe at various static values of transpulmonary pressure (PL) indicated on individual curves. Symbols, measurements obtained by roentgenograms; solid curves, predicted relationships computed from pressure‐diameter behavior of uniform hole (Du vs. PL, dashed curve) and computed excised pressure‐diameter behavior (De, dotted curve). Du varies with cube root of lung volume and was chosen so that De computed from Du, and measured Dv at one PL value, when used with Du predicted the measured Dv at the other PL values.

From Lai‐Fook


Figure 12.

Difference between perivascular pressure and pleural pressure (Px′) vs, transpulmonary pressure (PL) for constant arterial pressures of 10, 25, and 35 cmH2O and venous pressures of 0, 10, and 25 cmH2O. Vascular pressures (Pv) are measured relative to pleural pressure. A: data for arteries in 10 lobes. B: data for veins in 11 lobes. Vertical bars, 1 SE.

From Lai‐Fook


Figure 13.

Interdependence between bronchus and arteries. A: distribution of 2 principal stresses (open and closed circles) for points on walls of deformed bronchus and artery corresponding to circles in B. Note stress concentration at junction of airway and vessel. B: dashed curves, luminal surface of bronchus and artery when parenchyma is uniformly expanded at a transpulmonary pressure of 4 cmH2O; solid curves, distortion of shape of luminal surface when arterial pressure is reduced to −40 cmH2O.

From Lai‐Fook and Kallok
References
 1. Agostoni, E., and G. Miserocchi. Vertical gradient of trans‐pulmonary pressure with active and artificial lung expansion. J. Appl. Physiol. 29: 705–712, 1970.
 2. Chevalier, P. A., J. F. Greenleaf, R. A. Robb, and E. H. Wood. Biplane videoroentgenographic analysis of dynamic regional lung strains in dogs. J. Appl. Physiol. 40: 118–122, 1976.
 3. Chevalier, P. A., J. R. Rodarte, and L. D. Harris. Regional lung expansion at total lung capacity in intact vs. excised canine lungs. J. Appl. Physiol.: Respirat. Environ. Exercise Physiol. 45: 363–369, 1978.
 4. Crystal, R. G. Lung collagen: definition, diversity and development. Federation Proc. 33: 2248–2255, 1974.
 5. Fukaya, H., C. J. Martin, A. C. Young, and S. Katsura. Mechanical properties of alveolar walls. J. Appl. Physiol. 25: 689–695, 1968.
 6. Fung, Y. C. Foundations of Solid Mechanics. Englewood Cliffs, NJ: Prentice‐Hall, 1965.
 7. Fung, Y. C. A theory of elasticity of the lung. J. Appl. Mech. 41: 8–14, 1974.
 8. Fung, Y. C. Stress, deformation, and atelectasis of the lung. Circ. Res. 37: 481–496, 1975.
 9. Fung, Y. C., P. Tong, and P. Patitucci. Stress and strain in the lung. J. Eng. Mech. Div. 104: 201–223, 1978.
 10. Goshy, M., S. J. Lai‐Fook, and R. E. Hyatt. Perivascular pressure measurements by wick‐catheter technique in isolated dog lobes. J. Appl. Physiol: Respirat. Environ. Exercise Physiol. 46: 950–955, 1979.
 11. Hajji, M. A., T. A. Wilson, and S. J. Lai‐Fook. Improved measurements of shear modulus and pleural membrane tension of the lung. J. Appl. Physiol.: Respirat. Environ. Exercise Physiol. 47: 175–181, 1979.
 12. Hoppin, F. G., Jr., G. C. Lee, and S. V. Dawson. Properties of lung parenchyma in distortion. J. Appl. Physiol. 39: 742–751, 1975.
 13. Hubmayr, R. D., B. J. Walters, P. A. Chevalier, J. R. Rodarte, and L. E. Olson. Topographical distribution of regional lung volume in anesthetized dogs. J. Appl. Physiol.: Respirat. Environ. Exercise Physiol. 54: 1048–1056, 1983.
 14. Inoue, H., C. Inoue, and J. Hildebrandt. Vascular and airway pressures, and interstitial edema, affect peribronchial fluid pressure. J. Appl. Physiol.: Respirat. Environ. Exercise Physiol. 48: 177–185, 1980.
 15. Kallok, M. J., T. A. Wilson, J. R. Rodarte, S. J. Lai‐Fook, P. A. Chevalier, and L. D. Harris. Distribution of regional volumes and ventilation in excised canine lobes. J. Appl. Physiol.: Respirat. Environ. Exercise Physiol. 47: 182–191, 1979.
 16. Karakaplan, A. D., M. P. Bieniek, and R. Skalak. A mathematical model of lung parenchyma. J. Biomech. Eng. 102: 124–136, 1980.
 17. Lai‐Fook, S. J. A continuum mechanics analysis of pulmonary vascular interdependence in isolated dog lobes. J. Appl. Physiol: Respirat. Environ. Exercise Physiol. 46: 419–429, 1979.
 18. Lai‐Fook, S. J. Elastic properties of lung parenchyma: the effect of pressure‐volume hysteresis on the behavior of large blood vessels. J. Biomech. 12: 757–764, 1979.
 19. Lai‐Fook, S. J., R. E. Hyatt, and J. R. Rodarte. Effect of parenchymal shear modulus and lung volume on bronchial pressure‐diameter behavior. J. Appl. Physiol.: Respirat. Environ. Exercise Physiol. 44: 859–868, 1978.
 20. Lai‐Fook, S. J., R. E. Hyatt, J. R. Rodarte, and T. A. Wilson. Behavior of artificially produced holes in lung parenchyma. J. Appl. Physiol: Respirat. Environ. Exericse Physiol. 43: 648–655, 1977.
 21. Lai‐Fook, S. J., and M. J. Kallok. Bronchial‐arterial interdependence in isolated dog lung. J. Appl. Physiol.: Respirat. Environ. Exercise Physiol. 52: 1000–1007, 1982.
 22. Lai‐Fook, S. J., and B. Toporoff. Pressure‐volume behavior of perivascular interstitium measured in isolated dog lung. J. Appl Physiol.: Respirat. Environ. Exercise Physiol. 48: 939–946, 1980.
 23. Lai‐Fook, S. J., T. A. Wilson, R. E. Hyatt, and J. R. Rodarte. Elastic constants of inflated lobes of dog lungs. J. Appl. Physiol. 40: 508–513, 1976.
 24. Lambert, R. K., and T. A. Wilson. A model for the elastic properties of the lung and their effect on expiratory flow. J. Appl. Physiol. 34: 34–48, 1973.
 25. Lee, G. C., and A. Frankus. Elasticity properties of lung parenchyma derived from experimental distortion data. Biophys. J. 15: 481–493, 1975.
 26. Lee, G. C., A. Frankus, and P. D. Chen. Small distortion properties of lung parenchyma as a compressible continuum. J. Biomech. 9: 641–648, 1976.
 27. Liu, J.‐T., and G. C. Lee. Static finite deformation analysis of the lung. J. Eng. Mech. Div. 104: 225–238, 1978.
 28. Mead, J., T. Takishima, and D. Leith. Stress distribution in lungs: a model of pulmonary elasticity. J. Appl. Physiol. 28: 596–608, 1970.
 29. Milic‐Emili, J., J. A. M. Henderson, M. B. Dolovich, D. Trop, and K. Kaneko. Regional distribution of inspired gas in the lung. J. Appl. Physiol. 21: 749–759, 1966.
 30. Miserocchi, G., and E. Agostoni. Longitudinal forces acting on the trachea. Respir. Physiol. 17: 62–71, 1973.
 31. Rodarte, J. R. Importance of lung material properties in respiratory system mechanics. Physiologist 20 (5): 21–25, 1977.
 32. Smith, H. C., J. F. Greenleaf, E. H. Wood, D. J. Sass, and A. A. Bove. Measurement of regional pulmonary parenchymal movement in dogs. J. Appl. Physiol. 34: 544–547, 1973.
 33. Vawter, D. L. Poisson's ratio and incompressibility. Biorheology 18: 170–171, 1981.
 34. Vawter, D. L., Y. C. Fung, and J. B. West. Elasticity of excised dog lung parenchyma. J. Appl. Physiol.: Respirat. Environ. Exercise Physiol. 45: 261–269, 1978.
 35. Vawter, D. L., F. L. Matthews, and J. B. West. Effect of shape and size of lung and chest wall on stresses in the lung. J. Appl. Physiol. 39: 9–17, 1975.
 36. West, J. B., and F. L. Matthews. Stresses, strains, and surface pressures in the lung caused by its weight. J. Appl. Physiol. 32: 332–345, 1972.

Contact Editor

Submit a note to the editor about this article by filling in the form below.

* Required Field

How to Cite

Joseph R. Rodarte, Y. C. Fung. Distribution of Stresses Within the Lung. Compr Physiol 2011, Supplement 12: Handbook of Physiology, The Respiratory System, Mechanics of Breathing: 233-245. First published in print 1986. doi: 10.1002/cphy.cp030315