The Mechanics of Lung Tissue under High-Frequency Ventilation

High-frequency ventilation is a radical departure from conventional lung ventilation, with frequencies greater than 2Hz, and volumes per breath much smaller than the anatomical dead-space. Its use has been shown to benefit premature infants and patients with severe respiratory distress, but a vital question concerns ventilator-induced damage to the lung tissue, and a clear protocol for the most effective treatment has not been identified. Mathematical modeling can help in understanding the mechanical effects of lung ventilation, and hence in establishing such a protocol.

In this paper we describe the use of homogenization theory to predict the macroscopic behavior of lung tissue based upon the three dimensional microstructure of respiratory regions, making the simplifying assumption that the microstructure is periodic. This approach yields equations for macroscopic air flow, pressure, and tissue deformation, with parameters which can be determined from a specification of the tissue microstructure and its material properties. We are able to include an alternative hypothesis as to the dependence of lung tissue shear viscosity on the frequency of forcing, known as the structural damping hypothesis.

We then show how, if we consider isotropic tissue, the parameters determining the macroscopic response of the tissue can be estimated from bulk measurements. Finally, we consider the solutions of the macroscopic system when we consider variations in just one spatial dimension. In particular, we demonstrate that the structural damping hypothesis leads to markedly different solution behavior.

  • [1]  M. A. Biot, Generalized theory of acoustic propagation in porous dissipative media, J. Acoust. Soc. Am., 34 (1962), pp. 1254–1264. jas JASMAN 0001-4966 J. Acoust. Soc. Am. CrossrefISIGoogle Scholar

  • [2]  R. Burridge and  and J. B. Keller, Poroelasticity equations derived from microstructure, J. Acoust. Soc. Am., 70 (1981), pp. 1140–1146. jas JASMAN 0001-4966 J. Acoust. Soc. Am. CrossrefISIGoogle Scholar

  • [3]  H. K. Chang, Mechanisms of gas transport during ventilation by high‐frequency oscillation, J. Appl. Physiol., 56 (1984), pp. 553–563. CrossrefISIGoogle Scholar

  • [4]  Alexandre Chorin and , Jerrold Marsden, A mathematical introduction to fluid mechanics, Texts in Applied Mathematics, Vol. 4, Springer‐Verlag, 1993xii+169 94c:76002 CrossrefGoogle Scholar

  • [5]  R. H. Clark, D. R. Gerstmann, D. M. Null, B. A. Yoder, J. D. Cornish, C. M. Glasier, N. B. Ackerman, R. E. Bell and , and R. A. Delemos, Pulmonary interstitial emphysema treated by high‐frequency oscillatory ventilation, Crit. Care Med., 14 (1986), pp. 926–930. 4b8 ZZZZZZ 0090-3493 Crit. Care Med. CrossrefISIGoogle Scholar

  • [6]  C. Darquenne and  and M. Paiva, Two‐ and three‐dimensional simulations of aerosol transport and deposition in alveolar zone of human lung, J. Appl. Physiol., 80 (1996), pp. 1401–1414. hev JAPHEV 8750-7587 J. Appl. Physiol. CrossrefISIGoogle Scholar

  • [7]  B. L. K. Davey and  and J. H. T. Bates, Regional lung impedance from forced oscillations through alveolar capsules, Respir. Physiol., 91 (1993), pp. 165–182. rsp RSPYAK 0034-5687 Respir. Physiol. CrossrefGoogle Scholar

  • [8]  E. Denny and  and R. C. Schroter, The mechanical behavior of a mammalian lung alveolar duct model, J. Biomech. Eng., 117 (1995), pp. 254–261. jby JBENDY 0148-0731 J. Biomech. Eng. CrossrefISIGoogle Scholar

  • [9]  Google Scholar

  • [10]  Aaron Fogelson, A mathematical model and numerical method for studying platelet adhesion and aggregation during blood clotting, J. Comput. Phys., 56 (1984), 111–134 86a:92013 CrossrefISIGoogle Scholar

  • [11]  J. J. Fredberg and  and A. Hoenig, Mechanical response of the lungs at high frequencies, J. Biomech. Eng., 100 (1977), pp. 57–66. jby JBENDY 0148-0731 J. Biomech. Eng. CrossrefGoogle Scholar

  • [12]  J. J. Fredberg and  and J. A. Moore, The distributed response of complex branching duct networks, J. Acoust. Soc. Am., 63 (1978), pp. 954–961. jas JASMAN 0001-4966 J. Acoust. Soc. Am. CrossrefISIGoogle Scholar

  • [13]  J. J. Fredberg and  and D. Stamenovic, On the imperfect elasticity of lung tissue, J. Appl. Physiol., 67 (1989), pp. 2408–2419. hev JAPHEV 8750-7587 J. Appl. Physiol. CrossrefISIGoogle Scholar

  • [14]  H. Fukaya, C. Martin, A. Young and , and S. Katsura, Mechanical properties of alveolar walls, J. Appl. Physiol., 25 (1968), pp. 689–695. CrossrefISIGoogle Scholar

  • [15]  Y. C. Fung, Microrheology and constitutive equation of soft tissue, Biorheology, 25 (1988), pp. 261–270. bry BRHLAU 0006-355X Biorheology CrossrefISIGoogle Scholar

  • [16]  Y. C. Fung, Connecting incremental shear modulus and Poisson’s ratio of lung tissue with morphology and rheology of microstructure, Biorheology, 26 (1989), pp. 279–289. bry BRHLAU 0006-355X Biorheology CrossrefISIGoogle Scholar

  • [17]  K. C. High, J. S. Ultman and , and S. R. Karl, Mechanically induced pendelluft flow in a model airway bifurcation during high frequency oscillation, J. Biomech. Eng., 113 (1991), pp. 342–347. jby JBENDY 0148-0731 J. Biomech. Eng. CrossrefISIGoogle Scholar

  • [18]  K. Horsfield and  and G. Cummings, Morphology of the bronchial tree in men, J. Appl. Physiol., 24 (1968), pp. 373–383. CrossrefISIGoogle Scholar

  • [19]  E. Kimmel, R. D. Kamm and , and A. H. Shapiro, A cellular model of lung elasticity, J. Biomech. Eng., 109 (1987), pp. 126–131. jby JBENDY 0148-0731 J. Biomech. Eng. CrossrefISIGoogle Scholar

  • [20]  Thérèse Lévy, Propagation of waves in a fluid‐saturated porous elastic solid, Internat. J. Engrg. Sci., 17 (1979), 1005–1014 10.1016/0020-7225(79)90022-3 83g:76093 CrossrefISIGoogle Scholar

  • [21]  N. Borodachev, On a form of general solution of a three‐dimensional problem of the theory of elasticity in stresses, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki, (1999), 58–61 2000j:74025 Google Scholar

  • [22]  A. B. Otis, C. B. McKerrow, R. A. Bartlett, J. Mead, M. B. McIlroy, N. J. Selverstone and , and E. P. Radford, Mechanical factors in distribution of pulmonary ventilation, J. Appl. Physiol., 8 (1956), pp. 427–443. CrossrefISIGoogle Scholar

  • [23]  Charles Peskin and , David McQueen, Modeling prosthetic heart valves for numerical analysis of blood flow in the heart, J. Comput. Phys., 37 (1980), 113–132 81g:92011 CrossrefISIGoogle Scholar

  • [24]  Enrique Sánchez‐Palencia, Nonhomogeneous media and vibration theory, Lecture Notes in Physics, Vol. 127, Springer‐Verlag, 1980ix+398 82j:35010 Google Scholar