Bulletin of Mathematical Biology

, Volume 57, Issue 1, pp 77–98 | Cite as

Ion induced deformation of soft tissue

  • T. G. Myers
  • G. K. Aldis
  • S. Naili
Article

Abstract

In this paper the effects of changing the ion concentration in and around a sample of soft tissue are investigated. The triphasic theory developed by Laiet al. (1990,Biomechanics of Diarthrodial Joints, Vol. 1, Berlin, Springer-Verlag) is reduced to two coupled partial differential equations involving fluid ion concentration and tissue solid deformation. These equations are given in general form for Cartesian, cylindrical and spherical geometries. After solving the two equations quantities such as fluid velocity, fluid pressure, chemical potentials and chemical expansion stress may be easily calculated. In the Cartesian geometry comparison is made with the experimental and theoretical work of Myerset al. (1984,ASME J. biomech. Engng,106, 151–158). This dealt with changing the ion concentration of a salt shower on a strip of bovine articular cartilage. Results were obtained in both free swelling and isometric tension states, using an empirical formula to acount for ion induced deformation. The present theory predicts lower ion concentrations inside the tissue than this earlier work. A spherical sample of tissue subjected to a change in salt bath ion concentration is also considered. Numerical results are obtained for both hypertonic and hypotonic bathing solutions. Of particular interest is the finding that tissue may contract internally before reaching a final swollen equilibrium state or swell internally before finally contracting. By considering the relative magnitude, and also variation throughout the time course of terms in the governing equations, an even simpler system is deduced. As well as being linear the concentration equation in the new system is uncoupled. Results obtained from the linear system compare well with those from the spherical section. Thus, biological swelling situations may be modelled by a simple system of equations with the possibility, of approximate analytic solutions in certain cases.

Keywords

Articular Cartilage Fluid Pressure Spherical Geometry Initial Displacement Displacement Equation 

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Copyright information

© Society for Mathematical Biology 1994

Authors and Affiliations

  • T. G. Myers
    • 1
  • G. K. Aldis
    • 2
  • S. Naili
    • 3
  1. 1.Mathematical InstituteOxfordUK
  2. 2.Department of Mathematics, University College, University of New South WalesAustralian Defence Force AcademyCanberraAustralia
  3. 3.Laboratoire de Mecanique PhysiqueUniversite de ParisCreteil CedexFrance

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