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Distribution of Stresses Within the Lung

Joseph R. Rodarte, Y. C. Fung

10.1002/cphy.cp030315

Source: Supplement 12: Handbook of Physiology, The Respiratory System, Mechanics of Breathing

Originally published: 1986

Published online: January 2011

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 18
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 17
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. 2 and 3 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. Cross‐hatched 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 31
Figure 5. Figure 5.

Effect of change in shape without gravity. Cylindrical body with properties as in Fig. 4 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 31
Figure 6. Figure 6.

Regional volume gradients caused by shape and gravity. Cylindrical body with properties as in Figs. 4 and 5 is subjected to gravity and then angular deformation required to create vertical gradient of radial stress (Δτrr/Δ2). 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 31
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. 20
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 a–b. 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 a–f) is determined.

From Lai‐Fook, Hyatt, and Rodarte 19
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 (dashed‐dotted 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 19
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 17
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 17
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 21


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 18


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 17


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. 2 and 3 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. Cross‐hatched 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 31


Figure 5.

Effect of change in shape without gravity. Cylindrical body with properties as in Fig. 4 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 31


Figure 6.

Regional volume gradients caused by shape and gravity. Cylindrical body with properties as in Figs. 4 and 5 is subjected to gravity and then angular deformation required to create vertical gradient of radial stress (Δτrr/Δ2). 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 31


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. 20


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 a–b. 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 a–f) is determined.

From Lai‐Fook, Hyatt, and Rodarte 19


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 (dashed‐dotted 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 19


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 17


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 17


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 21
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Article Title: Distribution of Stresses Within the Lung
 

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