Mathematical modelling and numerical solution of swelling of cartilaginous tissues. Part I: Modelling of incompressible charged porous media

Kamyar Malakpoor; Enrique F. Kaasschieter; Jacques M. Huyghe

ESAIM: Mathematical Modelling and Numerical Analysis (2007)

  • Volume: 41, Issue: 4, page 661-678
  • ISSN: 0764-583X

Abstract

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The swelling and shrinkage of biological tissues are modelled by a four-component mixture theory in which a deformable and charged porous medium is saturated with a fluid with dissolved ions. Four components are defined: solid, liquid, cations and anions. The aim of this paper is the construction of the Lagrangian model of the four-component system. It is shown that, with the choice of Lagrangian description of the solid skeleton, the motion of the other components can be described in terms of Lagrangian initial system of the solid skeleton as well. Such an approach has a particularly important bearing on computer-aided calculations. Balance laws are derived for each component and for the whole mixture. In cooperation of the second law of thermodynamics, the constitutive equations are given. This theory results in a coupled system of nonlinear parabolic differential equations together with an algebraic constraint for electroneutrality. In this model, it is desirable to obtain an accurate approximation of the fluid flow and ions flow. Such an accurate approximation can be determined by the mixed finite element method. Part II is devoted to this task.

How to cite

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Malakpoor, Kamyar, Kaasschieter, Enrique F., and Huyghe, Jacques M.. "Mathematical modelling and numerical solution of swelling of cartilaginous tissues. Part I: Modelling of incompressible charged porous media." ESAIM: Mathematical Modelling and Numerical Analysis 41.4 (2007): 661-678. <http://eudml.org/doc/250022>.

@article{Malakpoor2007,
abstract = { The swelling and shrinkage of biological tissues are modelled by a four-component mixture theory in which a deformable and charged porous medium is saturated with a fluid with dissolved ions. Four components are defined: solid, liquid, cations and anions. The aim of this paper is the construction of the Lagrangian model of the four-component system. It is shown that, with the choice of Lagrangian description of the solid skeleton, the motion of the other components can be described in terms of Lagrangian initial system of the solid skeleton as well. Such an approach has a particularly important bearing on computer-aided calculations. Balance laws are derived for each component and for the whole mixture. In cooperation of the second law of thermodynamics, the constitutive equations are given. This theory results in a coupled system of nonlinear parabolic differential equations together with an algebraic constraint for electroneutrality. In this model, it is desirable to obtain an accurate approximation of the fluid flow and ions flow. Such an accurate approximation can be determined by the mixed finite element method. Part II is devoted to this task. },
author = {Malakpoor, Kamyar, Kaasschieter, Enrique F., Huyghe, Jacques M.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Mixture theory; porous media; hydrated soft tissue.},
language = {eng},
month = {10},
number = {4},
pages = {661-678},
publisher = {EDP Sciences},
title = {Mathematical modelling and numerical solution of swelling of cartilaginous tissues. Part I: Modelling of incompressible charged porous media},
url = {http://eudml.org/doc/250022},
volume = {41},
year = {2007},
}

TY - JOUR
AU - Malakpoor, Kamyar
AU - Kaasschieter, Enrique F.
AU - Huyghe, Jacques M.
TI - Mathematical modelling and numerical solution of swelling of cartilaginous tissues. Part I: Modelling of incompressible charged porous media
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2007/10//
PB - EDP Sciences
VL - 41
IS - 4
SP - 661
EP - 678
AB - The swelling and shrinkage of biological tissues are modelled by a four-component mixture theory in which a deformable and charged porous medium is saturated with a fluid with dissolved ions. Four components are defined: solid, liquid, cations and anions. The aim of this paper is the construction of the Lagrangian model of the four-component system. It is shown that, with the choice of Lagrangian description of the solid skeleton, the motion of the other components can be described in terms of Lagrangian initial system of the solid skeleton as well. Such an approach has a particularly important bearing on computer-aided calculations. Balance laws are derived for each component and for the whole mixture. In cooperation of the second law of thermodynamics, the constitutive equations are given. This theory results in a coupled system of nonlinear parabolic differential equations together with an algebraic constraint for electroneutrality. In this model, it is desirable to obtain an accurate approximation of the fluid flow and ions flow. Such an accurate approximation can be determined by the mixed finite element method. Part II is devoted to this task.
LA - eng
KW - Mixture theory; porous media; hydrated soft tissue.
UR - http://eudml.org/doc/250022
ER -

References

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  7. A.J.H. Frijns, A four-component mixture theory applied to cartilaginous tissues. Ph.D. thesis, Eindhoven University of Technology (2001).  Zbl0966.92002
  8. W.Y. Gu, W.M. Lai and V.C. Mow, A triphasic analysis of negative osmotic flows through charged hydrated soft tissues J. Biomechanics30 (1997) 71–78.  
  9. W.Y. Gu, W.M. Lai and V.C. Mow, Transport of multi-electrolytes in charged hydrated biological soft tissues. Transport Porous Med.34 (1999) 143–157.  
  10. F. Helfferich, Ion exchange. McGraw-Hill, New York (1962).  
  11. G.A. Holzapfel, Nonlinear Solid Mechanics. Wiley (2000).  
  12. J.M. Huyghe and J.D. Janssen, Quadriphasic mechanics of swelling incompressible porous media. Int. J. Engng. Sci.35 (1997) 793–802.  Zbl0903.73004
  13. J.M. Huyghe, M.M. Molenaar and F.P.T. Baaijens, Poromechanics of compressible charged porous media using the theory of mixtures. J. Biomech. Eng. (2007) in press.  
  14. W.M. Lai, J.S. Houa and V.C. Mow, A triphasic theory for the swelling and deformation behaviours of articular cartilage. ASME J. Biomech. Eng.113 (1991) 245–258.  
  15. E.G. Richards, An introduction to physical properties of large molecules in solute. Cambridge University Press, Cambridge (1980).  
  16. C. Truesdell and R.A. Toupin, The classical field theories, in Handbuch der Physik, Vol. III/1, S. Flügge Ed., Springer-Verlag, Berlin (1960) 226–902.  

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