Abstract
A continuum model for lung parenchyma is constructed. The model describes the thermomechanical response over a range of loading rates—from static to dynamic to shock waves—and a range of stress states, including isotropic expansion, triaxial extension, simple shear, and plane wave compression. Nonlinear elasticity, viscoelasticity, and damage are included, with the latter associated with changes of biological function as well as mechanical stiffness. A Gram–Schmidt decomposition of the deformation gradient leads to strain attributes that enter the thermodynamic potentials as state variables. A free energy function is designed for loading at low to moderate rates and tensile pressures, whereby the tissue response, with surface tension, is preeminent. An internal energy function is designed for wave propagation analysis, including shock waves, whereby compressibility of the air inside the alveoli is addressed via a composite stiffness based on a closed-cell assumption. The model accurately represents the response to triaxial loading, pressure relaxation, and dynamic torsion with relatively few parameters. Longitudinal wave speeds are reasonable for ranges of internal airway pressure and transpulmonary pressure. Airway pressure strongly affects the response to plane wave compression. Criteria for local injury and damage progression depend on a normalized energy density and its gradient, where the latter is paramount for impact problems involving fast pressure rises. Results suggest that local damage associated with edema is induced at load intensities much lower than those that would cause significant stiffness changes due to rupture, major tearing, or local collapse.
Similar content being viewed by others
References
Fung, Y. -C.: Biomechanics: Motion, Flow, Stress, and Growth. Springer, New York (1990)
Fung, Y. -C.: Biomechanics: Mechanical Properties of Living Tissues, 2nd edn. Springer, New York (1993)
Humphrey, J. D.: Continuum biomechanics of soft biological tissues. Proc. R. Soc. Lond. A 459, 3–46 (2003)
Freed, A. D.: Soft Solids. Birkhauser, Cham (2016)
Clayton, J. D., Freed, A.D.: Viscoelastic-damage theory based on a QR decomposition of deformation gradient. Technical Report ARL-8840, Army Research Laboratory, Aberdeen Proving Ground, MD (2019)
Clayton, J.D., Banton, R.J., Freed, A.D.: A nonlinear thermoelastic-viscoelastic continuum model of lung mechanics for shock wave analysis. AIP Conference Proceedings, in press (2019)
Clayton, J. D., Freed, A. D.: A constitutive framework for finite viscoelasticity and damage based on the Gram-Schmidt decomposition. Acta Mechanica submitted (2019)
Grimal, Q., Gama, B. A., Naili, S., Watzky, A. l., Gillespie, J. W.: Finite element study of high-speed blunt impact on thorax: linear elastic considerations. International Journal of Impact Engineering 30, 665–683 (2004)
Grimal, Q., Naili, S., Watzky, A.: A high-frequency lung injury mechanism in blunt thoracic impact. J. Biomech. 38, 1247–1254 (2005)
Shen, W., Niu, Y., Mattrey, R. F., Fournier, A., Corbeil, J., Kono, Y., Stuhmiller, J. H.: Development and validation of subject-specific finite element models for blunt trauma study. J. Biomech. Eng. 130, 021022 (2008)
Bowen, I. G., Fletcher, E. R., Richmond, D.R.: Estimate of man’s tolerance to the direct effects of air blast. Technical report, Lovelace Foundation for Medical Education and Research, Albuquerque NM (1968)
Cooper, G. J., Pearce, B. P., Sedman, A. J., Bush, I. S., Oakley, C.W.: Experimental evaluation of a rig to simulate the response of the thorax to blast loading. Journal of Trauma and Acute Care Surgery 40, 38S–41S (1996)
Cooper, G.J., Jonsson, A.: Protection against blast injury. In: Cooper, G.J., Dudley, H.A.F., Gann, D.S., Little, R.A., Maynard, R.L. (eds.) Scientific Foundations of Trauma, pp 258–283. Butterworth Heinemann, Oxford (1997)
Rafaels, K. A., Cameron, R., Panzer, M. B., Salzar, R. S.: Pulmonary injury risk assessment for long-duration blasts: a meta-analysis. Journal of Trauma and Acute Care Surgery 69, 368–374 (2010)
Gibbons, M. M., Dang, X., Adkins, M., Powell, B., Chan, P.: Finite element modeling of blast lung injury in sheep. J. Biomech. Eng. 137, 041002 (2015)
Stitzel, J. D., Gayzik, F. S., Hoth, J. J., Mercier, J., Gage, H. D., Morton, K. A., Duma, S. M., Payne, R. M.: Development of a finite element-based injury metric for pulmonary contusion part I: model development and validation. Stapp Car Crash J. 49, 271–289 (2005)
Stuhmiller, J. H., Chuong, C. J., Phillips, Y. Y., Dodd, K. T.: Computer modeling of thoracic response to blast. Journal of Trauma 28, S132–S139 (1988)
Vlessis, A.A., Trunkey, D.D.: Non-penetrating injury of the thorax. In: Cooper, G.J., Dudley, H.A.F., Gann, D.S., Little, R.A., Maynard, R.L. (eds.) Scientific Foundations of Trauma, pp 127–143. Butterworth Heinemann, Oxford (1997)
Fung, Y. -C.: Stress, deformation, and atelectasis of the lung. Circ. Res. 37, 481–496 (1975)
Fung, Y. -C., Patitucci, P., Tong, P.: Stress and strain in the lung. ASCE Journal of Engineering Mechanics 104, 201–223 (1978)
Vawter, D. L., Fung, Y. -C., West, J. B.: Constitutive equation of lung tissue elasticity. J. Biomech. Eng. 101, 38–45 (1979)
Vawter, D. L.: A finite element model for macroscopic deformation of the lung. J. Biomech. Eng. 102, 1–7 (1980)
Fung, Y. -C.: Elasticity of soft tissues in simple elongation. Am J Physiol 213, 1532–1544 (1967)
Bachofen, H., Hildebrandt, J., Bachofen, M.: Pressure-volume curves of air-and liquid-filled excised lungs-surface tension in situ. J. Appl. Physiol. 29, 422–431 (1970)
Suki, B., Bates, J. H.: A nonlinear viscoelastic model of lung tissue mechanics. J. Appl. Physiol. 71, 826–833 (1991)
Gayzik, F. S., Hoth, J. J., Daly, M., Meredith, J. W., Stitzel, J. D.: A finite element-based injury metric for pulmonary contusion: investigation of candidate metrics through correlation with computed tomography. Stapp Car Crash J. 51, 189–209 (2007)
Gayzik, F. S., Hoth, J. J., Stitzel, J. D.: Finite element–based injury metrics for pulmonary contusion via concurrent model optimization. Biomech. Model. Mechanobiol. 10, 505–520 (2011)
Hallquist, J. O.: LS-DYNA Theory Manual. Livermore Software Technology Corporation (2006)
Cronin, D. S.: Model for pulmonary response resulting from high deformation rate loading. In: Proceedings of the 2011 International Research Council on Biomechanics of Injury (IRCOBI) Conference, pp 181–192 (2011)
Rice, D. A.: Sound speed in pulmonary parenchyma. J. Appl. Physiol. 54, 304–308 (1983)
Butler, J. P., Lehr, J. L., Drazen, J. M.: Longitudinal elastic wave propagation in pulmonary parenchyma. J. Appl. Physiol. 62, 1349–1355 (1987)
Freed, A. D., Einstein, D. R.: An implicit elastic theory for lung parenchyma. Int. J. Eng. Sci. 62, 31–47 (2013)
Stamenovic, D.: Micromechanical foundations of pulmonary elasticity. Physiol. Rev. 70, 1117–1134 (1990)
Freed, A. D., Einstein, D. R., Carson, J. P., Jacob, R. E.: Viscoelastic model for lung parenchyma for multi-scale modeling of respiratory system, phase II: Dodecahedral micro-model. Technical report, Pacific Northwest National Laboratory (PNNL), Richland, WA (US) (2012)
D’yachenko, A. I., Manyuhina, O. V.: Modeling of weak blast wave propagation in the lung. J. Biomech. 39, 2113–2122 (2006)
Regueiro, R. A., Zhang, B., Wozniak, S. L.: Large deformation dynamic three-dimensional coupled finite element analysis of soft biological tissues treated as biphasic porous media. Computer Modeling in Engineering and Sciences (CMES) 98, 1–39 (2014)
Fankell, D. P., Regueiro, R. A., Kramer, E. A., Ferguson, V. L., Rentschler, M. E.: A small deformation thermoporomechanics finite element model and its application to arterial tissue fusion. J. Biomech. Eng. 140, 031007 (2018)
Freed, A. D., Zamani, S.: On the use of convected coordinate systems in the mechanics of continuous media derived from a QR factorization of F. Int. J. Eng. Sci. 127, 145–161 (2018)
Freed, A. D., Graverend, J. B., Rajagopal, K. R.: A decomposition of Laplace stretch with applications in inelasticity. Acta Mech. 230, 3423–3429 (2019)
Freed, A. D., Zamani, S.: Elastic Kelvin-Poisson-Poynting solids described through scalar conjugate stress/strain pairs derived from a QR factorization of F. Journal of the Mechanics and Physics of Solids 129, 278–293 (2019)
Freed, A. D.: A note on stress/strain conjugate pairs: explicit and implicit theories of thermoelasticity for anisotropic materials. Int. J. Eng. Sci. 120, 155–171 (2017)
Srinivasa, A. R.: On the use of the upper triangular (or QR) decomposition for developing constitutive equations for Green-elastic materials. Int. J. Eng. Sci. 60, 1–12 (2012)
McLellan, A. G.: The Classical Thermodynamics of Deformable Materials. Cambridge University Press, Cambridge (1980)
Clayton, J. D.: Nonlinear Mechanics of Crystals. Springer, Dordrecht (2011)
Holzapfel, G. A., Simo, J. C.: A new viscoelastic constitutive model for continuous media at finite thermomechanical changes. Int. J. Solids Struct. 33, 3019–3034 (1996)
Holzapfel, G. A.: On large strain viscoelasticity: continuum formulation and finite element applications to elastomeric structures. Int. J. Numer. Methods Eng. 39, 3903–3926 (1996)
Simo, J. C.: On a fully three-dimensional finite-strain viscoelastic damage model: formulation and computational aspects. Comput. Methods Appl. Mech. Eng. 60, 153–173 (1987)
Balzani, D., Brinkhues, S., Holzapfel, G. A.: Constitutive framework for the modeling of damage in collagenous soft tissues with application to arterial walls. Comput. Methods Appl. Mech. Eng. 213, 139–151 (2012)
Krajcinovic, D.: Damage Mechanics. North-Holland, Amsterdam (1996)
Clayton, J. D., Tonge, A.: A nonlinear anisotropic elastic-inelastic constitutive model for polycrystalline ceramics and minerals with application to boron carbide. Int J Solids Struct 64–65, 191–207 (2015)
Hoppin, F. G., Lee, G. C., Dawson, S. V.: Properties of lung parenchyma in distortion. J. Appl. Physiol. 39, 742–751 (1975)
Lee, G. C., Frankus, A.: Elasticity properties of lung parenchyma derived from experimental distortion data. Biophys. J. 15, 481–493 (1975)
Lai-Fook, S. J., Wilson, T. A., Hyatt, R. E., Rodarte, J. R.: Elastic constants of inflated lobes of dog lungs. J. Appl. Physiol. 40, 508–513 (1976)
Hajji, M. A., Wilson, T. A., Lai-Fook, S. J.: Improved measurements of shear modulus and pleural membrane tension of the lung. J. Appl. Physiol. 47, 175–181 (1979)
Jahed, M., Lai-Fook, S. J., Bhagat, P. K., Kraman, S. S.: Propagation of stress waves in inflated sheep lungs. J. Appl. Physiol. 66, 2675–2680 (1989)
Jahed, M., Lai-Fook, S. J., Bhagat, P. K.: Effect of vascular volume and edema on wave propagation in canine lungs. J. Appl. Physiol. 68, 2171–2176 (1990)
Jahed, M., Lai-Fook, S. J.: Stress wave velocity measured in intact pig lungs with cross-spectral analysis. J. Appl. Physiol. 76, 565–571 (1994)
Yen, R. T., Fung, Y. C., Ho, H. H., Butterman, G.: Speed of stress wave propagation in lung. J. Appl. Physiol. 61, 701–705 (1986)
Zeng, Y. J., Yager, D., Fung, Y. C.: Measurement of the mechanical properties of the human lung tissue. J. Biomech. Eng. 109, 169–174 (1987)
Lai-Fook, S. J.: The elastic constants of lung parenchyma: the effect of pressure-volume hysteresis on the behavior of blood vessels. J. Biomech. 12, 757–764 (1979)
Denny, E., Schroter, R. C.: A model of non-uniform lung parenchyma distortion. J. Biomech. 39, 652–663 (2006)
Hoppin, F. G., Stothert, J. C., Greaves, I. A., Lai, Y. -L., Hildebrandt, J.: Lung recoil: elastic and rheological properties. In: Handook of Physiology. The Respiratory System. Mechanics of Breathing, pp 195–216. American Physiological Society, Bethesda (1986)
McGee, K. P., Mariappan, Y. K., Hubmayr, R. D., Carter, R. E., Bao, Z., Levin, D. L., Manduca, A., Ehman, R. L.: Magnetic resonance assessment of parenchymal elasticity in normal and edematous, ventilator-injured lung. J. Appl. Physiol. 113, 666–676 (2012)
Carney, D., DiRocco, J., Nieman, G.: Dynamic alveolar mechanics and ventilator-induced lung injury. Crit. Care Med. 33, S122–S128 (2005)
Perlman, C. E., Lederer, D. J., Bhattacharya, J.: Micromechanics of alveolar edema. Am. J. Respir. Cell Mol. Biol. 44, 34–39 (2011)
Ingenito, E. P., Mark, L., Davison, B.: Effects of acute lung injury on dynamic tissue properties. J. Appl. Physiol. 77, 2689–2697 (1994)
Fung, Y. -C., Yen, R. T., Tao, Z. L., Liu, S. Q.: A hypothesis on the mechanism of trauma of lung tissue subjected to impact load. J. Biomech. Eng. 110, 50–56 (1988)
Tao, Z. L., Fung, Y. C.: Lungs under cyclic compression and expansion. J. Biomech. Eng. 109, 160–162 (1987)
Hughes, R., May, A. J., Widdicombe, J. G.: Stress relaxation in rabbits’ lungs. J Physiol 146, 85–97 (1959)
Fredberg, J. J., Stamenovic, D.: On the imperfect elasticity of lung tissue. J. Appl. Physiol. 67, 2408–2419 (1989)
Suki, B., Barabasi, A. L., Lutchen, K. R.: Lung tissue viscoelasticity: a mathematical framework and its molecular basis. J. Appl. Physiol. 76, 2749–2759 (1994)
Navajas, D., Maksym, G. N., Bates, J. H.: Dynamic viscoelastic nonlinearity of lung parenchymal tissue. J. Appl. Physiol. 79, 348–356 (1995)
Dai, Z., Peng, Y., Mansy, H. A., Sandler, R. H., Royston, T. J.: A model of lung parenchyma stress relaxation using fractional viscoelasticity. Medical Engineering and Physics 37, 752–758 (2015)
Sanborn, B., Nie, X., Chen, W., Weerasooriya, T.: High strain rate pure shear and axial compressive response of porcine lung tissue. J. Appl. Mech. 80, 011029 (2013)
Saraf, H., Ramesh, K. T., Lennon, A. M., Merkle, A. C., Roberts, J. C.: Mechanical properties of soft human tissues under dynamic loading. J. Biomech. 40, 1960–1967 (2007)
Davison, L.: Fundamentals of Shock Wave Propagation in Solids. Springer, Berlin (2008)
Clayton, J.D.: Nonlinear Elastic and Inelastic Models for Shock Compression of Crystalline Solids. Springer, Cham (2019)
Clayton, J. D., Knap, J.: A geometrically nonlinear phase field theory of brittle fracture. Int. J. Fract. 189, 139–148 (2014)
Clayton, J. D., Knap, J.: Phase field modeling of coupled fracture and twinning in single crystals and polycrystals. Comput. Methods Appl. Mech. Eng. 312, 447–467 (2016)
Clayton, J. D.: Finsler geometry of nonlinear elastic solids with internal structure. J. Geom. Phys. 112, 118–146 (2017)
Clayton, J. D.: Generalized finsler geometric continuum physics with applications in fracture and phase transformations. Zeitschrift fur Angewandte Mathematik und Physik (ZAMP) 68, 9 (2017)
Malvern, L. E.: Introduction to the Mechanics of a Continuous Medium. Prentice-Hall, Englewood Cliffs (1969)
Graff, K. F.: Wave Motion in Elastic Solids. Oxford University Press London, Oxford (1975)
McLellan, A. G.: Finite strain coordinates and the stability of solid phases. Journal of Physics C: Solid State Physics 9, 4083–4094 (1976)
Thurston, R.N.: Waves in solids. In: Truesdell, C. (ed.) Handbuch der Physik, vol. VI, pp 109–308. Springer, Berlin (1974)
Clayton, J. D., Lloyd, J. T.: Analysis of nonlinear elastic aspects of precursor attenuation in shock-compressed metallic crystals. Journal of Physics Communications 2, 045032 (2018)
Weir, C. E.: Effect of temperature on the volume of leather and collagen in water. J. Res. Natl. Bur. Stand. 41, 279–285 (1948)
Kanagy, J. R.: Specific heats of collagen and leather. J. Res. Natl. Bur. Stand. 55, 191–195 (1955)
Kakivaya, S. R., Hoeve, C. A.: The glass point of elastin. Proc. Natl. Acad. Sci. 72, 3505–3507 (1975)
Lillie, M. A., Gosline, J. M.: Unusual swelling of elastin. Biopolymers: Original Research on Biomolecules 64, 115–126 (2002)
McQueen, R.G., Marsh, S.P., Taylor, J.W., Fritz, J.N., Carter, W.J.: The equation of state of solids from shock wave studies. In: Kinslow, R. (ed.) High-Velocity Impact Phenomena, pp 294–417. Academic Press, New York (1970)
Clayton, J. D.: Analysis of shock compression of strong single crystals with logarithmic thermoelastic-plastic theory. Int. J. Eng. Sci. 79, 1–20 (2014)
Yen, R. T., Fung, Y. C., Liu, S. Q.: Trauma of lung due to impact load. J. Biomech. 21, 745–753 (1988)
Clayton, J. D.: Finsler-geometric continuum dynamics and shock compression. Int. J. Fract. 208, 53–78 (2017)
Stuhmiller, J. H., Ho, K., Vander Vorst, M. J., Dodd, K. T., Fitzpatrick, T., Mayorga, M.: A model of blast overpressure injury to the lung. J. Biomech. 29, 227–234 (1996)
Grimal, Q., Watzky, A., Naili, S.: A one-dimensional model for the propagation of transient pressure waves through the lung. J. Biomech. 35, 1081–1089 (2002)
Tsokos, M., Paulsen, F., Petri, S., Madea, B., Puschel, K., Turk, E. E.: Histologic, immunohistochemical, and ultrastructural findings in human blast lung injury. Am. J. Respir. Crit. Care Med. 168, 549–555 (2003)
Acknowledgments
J.D.C. acknowledges support of the CCDC Army Research Laboratory. A.D.F. acknowledges support of a Joint Faculty Appointment with the CCDC Army Research Laboratory.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Clayton, J.D., Freed, A.D. A constitutive model for lung mechanics and injury applicable to static, dynamic, and shock loading. Mech Soft Mater 2, 3 (2020). https://doi.org/10.1007/s42558-020-0018-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s42558-020-0018-9