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Research Papers

Geometric Hysteresis of Alveolated Ductal Architecture

[+] Author and Article Information
M. Kojic

Molecular and Integrative Physiological Sciences, Harvard School of Public Health, Boston, MA 02115; Research and Development Center for Bioengineering, Kragujevac, Serbia 34000;  The Methodist Hospital Research Institute, Houston, TX 77030

J. P. Butler

Molecular and Integrative Physiological Sciences, Harvard School of Public Health, Boston, MA 02115

I. Vlastelica

 Metropolitan University, Belgrade, Serbia 11000

B. Stojanovic

Faculty of Science,  University of Kragujevac, Serbia, Kragujevac, Serbia 34000

V. Rankovic

Faculty of Economics,  University of Kragujevac, Serbia, Kragujevac, Serbia 34000

A. Tsuda1

Molecular and Integrative Physiological Sciences, Harvard School of Public Health, Boston, MA 02115atsuda@hsph.harvard.edu

1

Corresponding author.

J Biomech Eng 133(11), 111005 (Nov 29, 2011) (11 pages) doi:10.1115/1.4005380 History: Received September 11, 2011; Accepted October 19, 2011; Published November 29, 2011; Online November 29, 2011

Low Reynolds number airflow in the pulmonary acinus and aerosol particle kinetics therein are significantly conditioned by the nature of the tidal motion of alveolar duct geometry. At least two components of the ductal structure are known to exhibit stress-strain hysteresis: smooth muscle within the alveolar entrance rings, and surfactant at the air-tissue interface. We hypothesize that the geometric hysteresis of the alveolar duct is largely determined by the interaction of the amount of smooth muscle and connective tissue in ductal rings, septal tissue properties, and surface tension-surface area characteristics of surfactant. To test this hypothesis, we have extended the well-known structural model of the alveolar duct by Wilson and Bachofen (1982, “A Model for Mechanical Structure of the Alveolar Duct,” J. Appl. Physiol. 52 (4), pp. 1064–1070) by adding realistic elastic and hysteretic properties of (1) the alveolar entrance ring, (2) septal tissue, and (3) surfactant. With realistic values for tissue and surface properties, we conclude that: (1) there is a significant, and underappreciated, amount of geometric hysteresis in alveolar ductal architecture; and (2) the contribution of smooth muscle and surfactant to geometric hysteresis are of opposite senses, tending toward cancellation. Quantitatively, the geometric hysteresis found experimentally by Miki (1993, “Geometric Hysteresis in Pulmonary Surface-to-Volume Ratio during Tidal Breathing,” J. Appl. Physiol. 75 (4), pp. 1630–1636) is consistent with little or no smooth muscle tone in anesthetized rabbits in control conditions, and with substantial smooth muscle activation following methacholine challenge. The observed local hysteretic boundary motion of the acinar duct would result in irreversible acinar flow fields, which might be important mechanistic contributors to aerosol mixing and deposition deep in the lung.

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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

(a) Schematic of the model of alveolar duct: left panel – cross-section in the plane orthogonal to the duct longitudinal axis, right panel – axial cross-section of duct; (b) representative geometric location of the model in the acinus, with morphology of alveolar space in lung [10]

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Figure 2

Material models used in the analysis. (a) Uniaxial model for connective tissue of alveolar ring [29]; (b) Biaxial model for septum tissue [(29),30]; (c) Hysteretic model of muscle constituent of ring material [21]; (d) Hysteretic characteristic of surfactant [35].

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Figure 3

Finite element model of alveolar duct. Septum is modeled by membrane (shell) finite elements and ring is modeled by line finite elements. Deformations of ring and septum are calculated for prescribed radial displacement of the outer (acinar) boundary ΔRa  = ΔRa , max sin t. Time is a parameter controlling volume history; no dynamics are associated with it. Solution is obtained incrementally using 250 steps over the breathing cycle.

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Figure 4

Geometric hysteresis within the alveolar duct. (a) Case without surfactant. Geometric hystersesis is due to hysteresis of muscle constituent within the alveolar entrance ring and it increases with increasing m. The S-V hysteresis loop has a clockwise direction for one breathing cycle. (b) Case with surfactant. The S-V hysteresis loop due to surfactant has counterclockwise direction (condition with no muscle constituent, m = 0). Since S-V hysteresis loops corresponding to muscle and surfactant have the opposite senses, the resulting hysteresis is smaller when m = 0.5 with respect to m = 0 and the loop changes the direction for m = 1.

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Figure 5

Coefficient of geometric hysteresis ηgeom plotted against m; its geometric construction is shown in the cartoon. In the absence of muscle contribution, the hysteresis loop is entirely controlled by surfactant, and the loop is counterclockwise with positive ηgeom . Its value progressively decreases with muscle activation, crossing zero into a regime dominated by muscle, where the loops are clockwise and ηgeom is negative. Also shown is the hysteretic behavior in the absence of surfactant, showing the uniformly negative ηgeom .

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Figure 6

Radial displacements of internal rim (radius Rd ), outer boundary (radius Ra ), and points at initial radius R0  = 190 μm during one cycle; for two extreme cases: condition with no muscle constituent (m = 0), and with only muscle constituent within the ring (m = 1). Due to hysteretic characteristic of surfactant, and muscle radial displacement curves are not symmetric with respect to the middle line (end of inspiration), except for the outer boundary, with asymmetry more pronounced for the domain closer to internal rim. Displacements and asymmetry are smaller for m = 1.

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Figure 7

Particle trajectories within radial plane (x-y) of material points at outer boundary, internal rim and at two membrane radii. Axial maximum displacement is (ΔL)max  = 0.5 (ΔRd )max following from geometrical similarity during alveolar deformation (see equation 1). (a) Case with no muscle constituent (m = 0): the trajectories display clockwise hysteretic loops due to dominant surfactant action during inspiration (see Fig. 2), with loops diminishing from the internal to the external membrane boundary; (b) Case m = 1: The muscle hysteresis acts in the opposite sense from the surfactant effects (see Fig. 2), and hysteretic loops become counterclockwise in the regime where muscle hysteretic effect is dominant (expiration regime); the loops diminish when approaching to external boundary.

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Figure 8

Distribution of displacement hysteresis coefficient Shyster . The ratio R(t)/R(0) displays a hysteretic character over a breathing cycle, and a measure of this hysteresis is defined as the ratio Shyster  = Ahyster /Atotal . (a) Case without muscle content within the ring (m = 0): the hysteretic loops are clockwise (considered positive) since displacements are larger during inspiration, and diminish toward the external boundary; (b) Case m = 1: the overall hysteretic muscle characteristic is dominant and the surface Ahyster is negative (see Fig. 7).

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Figure 9

Sketch of stress calculation. Stresses correspond to the previous iteration (indexed by i−1 in equation 1). (a) At each material point of the radial septum the stress is represented as a sum of the stresses due to material deformation (radial stress σrad and circumferential stress σcircular ) and due to surfactant σγ . These stresses are evaluated from constitutive laws for material (biaxial model) and surfactant (hysteretic model). The stresses produce the finite element nodal forces Fint  = Fmat  + Fγ entering the equilibrium equations for the finite element assemblage. (b) and (c) stresses within the ring evaluated for nonlinear elastic and hysteretic material models.

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Figure 10

Validation of methodology: Comparison of analytic and finite element solutions for distribution of radial displacements along the radius

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