Abstract
The behaviour of a cavity during an injection of fluid into biological tissue is considered. High cavity pressure drives fluid into the neighbouring tissue where it is absorbed by capillaries and lymphatics. The tissue is modelled as a nonlinear deformable porous medium with the injected fluid absorbed by the tissue at a rate proportional to the local pressure. A model with a spherical cavity in an infinite medium is used to find the pressure and displacement of the tissue as a function of time and radial distance. Analytical and numerical solutions for a step change in cavity pressure show that the flow induces a radial compression in the medium together with an annular expansion, the net result being an overall expansion of the medium. Thus any flow induced deformation of the material will aid in the absorption of fluid.
Similar content being viewed by others
Literature
Abramowitz, M. and I. A. Stegun. 1972.Handbook of Mathematical Functions. New York: Dover.
Barry, S. I. 1990. Flow in a deformable porous medium. Ph.D. Thesis, University College, University of New South Wales, Australia.
Barry, S. I. and G. K. Aldis. 1990. Comparison of models for flow induced deformation of soft biological tissue.J. Biomech. 23, 647–654.
Barry, S. I. and G. K. Aldis. 1991a. Unsteady flow induced deformation of porous materials.Int. J. Non-Linear Mech. 26, 687–699.
Barry, S. I. and G. K. Aldis. 1991b. Fluid Flow over a thin deformable porous layer.Z. A. M. P. 42, 633–648.
Biot, M. A. 1941. General theory of three-dimensional consolidation.J. appl. Phys. 12, 155–164.
Biot, M. A. 1955. Theory of elasticity and consolidation for a porous anisotropic solid.J. appl. Phys. 26, 182–185.
Biot, M. A.. 1956. Mechanics of deformation and acoustic propagation in porous media.J. appl. Phys. 27, 1482–1498.
Bowen, R. M. 1980. Incompressible porous media models by the theory of mixtures.Int. J. Engng. Sci. 18, 1129–1148.
Ford, T. R., J. S. Sachs, J. B. Grotberg and M. R. Glucksberg. 1990.Mechanics of the perialveolar interstitium of the lung. First World Congress of Biomechanics, La Jolla, Vol. 1, p. 31.
Friedman, M. H. 1971. General theory of tissue swelling with application to the corneal stroma.J. theor. Biol. 30, 93–109.
Holmes, M. H. 1983. A nonlinear diffusion equation arising in the study of soft tissue.Quart. App. Math. 61, 209–220.
Holmes, M. H. 1984. Comparison theorems and similarity solution approximation for a nonlinear diffusion equation arising in the study of soft tissue.SIAM J. appl. Math. 44, 545–556.
Holmes, M. H. 1985. A theoretical analysis for determining the nonlinear hydraulic permeability of a soft tissue from a permeation experiment.Bull. math. Biol. 47, 669–683.
Holmes, M. H. 1986. Finite deformation of soft tissue: analysis of a mixture model in uni-axial compression.J. biomech. Engng 108, 372–381.
Holmes, M. H. and V. C. Mow. 1990. The nonlinear characteristics of soft gels and hydrated connective tissue in ultrafiltration.J. Biomech. 23, 1145–1156.
Hou, J. S., M. H. Holmes, W. M. Lai and V. C. Mow. 1989. Boundary conditions at the cartilage-synovial fluid interface for joint lubrication and theoretical verfications.J. biomech. Engng. 111, 78–87.
Jain, R. and G. Jayaraman. 1987. A theoretical model for water flux through the arterial wall.J. biomech. Engng. 109, 311–317.
Jayaraman, G. 1983. Water transport in the arterial wall—a theoretical study.J. Biomech. 16, 833–840.
Keele C. A. and E. Neil, 1971.Samson Wright's Applied Physiology, 12th Edn. London: Oxford University Press.
Kenyon, D. E. 1976a. The theory of an incompressible solid-fluid mixture.Arch. Rat. Mech. Anal. 62, 131–147.
Kenyon, D. E. 1976b. Transient filtration in a porous elastic cylinder.J. appl. Mech. 98, 594–598.
Kenyon, D. E. 1978. Consolidation in compressible mixtures.J. appl. Mech. 45, 727–732.
Kenyon, D. E. 1979. A mathematical model of water flux through aortic tissue.Bull. math. Biol. 41, 79–90.
Klanchar, M. and J. M. Tarbell. 1987. Modeling water flow through arterial tissue.Bull. math. Biol. 49, 651–669.
Lai, W. M. and V. C. Mow. 1980. Drag induced compression of articular cartilage during a permeation experiment.Biorheology 17, 111–123.
Lanir, Y. 1987. Biorheology and fluid flux in swelling tissues. I. Bicomponent theory for small deformations, including concentration effects.Biorheology 24, 173–187.
Lanir, Y., S. Dikstein, A. Hartzshtark and V. Manny. 1990.In-vivo indentation of human skin.J. biomech. Engng. 112, 63–69.
Middleman, S. 1972.Transport Phenomena in the Cardiovascular System, New York: Wiley-Interscience.
Mow, V. C. and W. M. Lai. 1979. Mechanics of animal joints.Ann. Rev. Fluid Mech. 11, 247–288.
Mow, V. C., S. C. Kuei, W. M. Lai and C. G. Armstrong. 1980. Biphasic creep and stress relaxation of articular cartilage in compression: Theory and experiment.J. biomech. Engng. 102, 73–84.
Mow, V. C., M. H. Holmes and W. M. Lai. 1984. Fluid transport and mechanical properties of articular cartilage: a review.J. Biomech. 17, 377–394.
Mow, V. C., M. K. Kwan, W. M. Lai and M. H. Holmes. 1985. A finite deformation theory for nonlinearly permeable soft hydrated biological tissues. InFrontiers in Biomechanics, G. Schmid-Schoenbein, S. L-Y. Woo and B. W. Zweifach (Eds), pp. 153–179. New York: Springer-Verlag.
Nicholson, C. 1985. Diffusion from an injected volume of a substance in brain tissue with arbitrary volume fraction and tortuosity.Brain Res. 333, 325–329.
Oomens, C. W. J., D. H. Van Campen and H. J. Grootenboer. 1987. A mixture approach to the mechanics of skin.J. Biomech. 20, 877–885.
Oomens, C. W. J. 1987.In vitro compression of a soft tissue layer on the a rigid foundation.J. Biomech. 20, 923–935.
Simon, B. R. and M. Gaballa. 1988. Poroelastic finite element models for the spinal motion segement including ionic swelling. InComputational Methods in Bioengineering, ASME R. L. Spilker and B. R. Simon (Eds).
Terzaghi, K. 1925.Erdbaumechanik auf Bodenphysikalischen Grundlagen. Wien: Deuticke.
Yang, M. and L. A. Taber. 1990.A nonlinear poroelastic model for the developing chick heart. First World Congress of Biomechanics, La Jolla, Vol. 1, p. 152.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Barry, S.I., Aldis, G.K. Flow-induced deformation from pressurized cavities in absorbing porous tissues. Bltn Mathcal Biology 54, 977–997 (1992). https://doi.org/10.1007/BF02460662
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02460662