# Mathematical modelling and numerical solution of swelling of cartilaginous tissues. Part 1: Modelling of incompressible charged porous media

8 Citations (Scopus)

### 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.
Original language English 661-678 ESAIM : Mathematical Modelling and Numerical Analysis 41 4 https://doi.org/10.1051/m2an:2007036 Published - 2007

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Swelling
Mathematical Modeling
Porous Media
Porous materials
Numerical Solution
Tissue
Bearings (structural)
Modeling
Skeleton
Ions
Constitutive equations
Flow of fluids
Mixture Theory
Differential equations
Negative ions
Positive ions
Parabolic Differential Equations
Balance Laws
Second Law of Thermodynamics
Biological Tissue

### Cite this

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title = "Mathematical modelling and numerical solution of swelling of cartilaginous tissues. Part 1: Modelling of incompressible charged porous media",
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 = "K. Malakpoor and E.F. Kaasschieter and J.M.R.J. Huyghe",
year = "2007",
doi = "10.1051/m2an:2007036",
language = "English",
volume = "41",
pages = "661--678",
journal = "ESAIM : Mathematical Modelling and Numerical Analysis",
issn = "0764-583X",
publisher = "EDP Sciences",
number = "4",

}

In: ESAIM : Mathematical Modelling and Numerical Analysis, Vol. 41, No. 4, 2007, p. 661-678.

TY - JOUR

T1 - Mathematical modelling and numerical solution of swelling of cartilaginous tissues. Part 1: Modelling of incompressible charged porous media

AU - Malakpoor, K.

AU - Kaasschieter, E.F.

AU - Huyghe, J.M.R.J.

PY - 2007

Y1 - 2007

N2 - 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.

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.

U2 - 10.1051/m2an:2007036

DO - 10.1051/m2an:2007036

M3 - Article

VL - 41

SP - 661

EP - 678

JO - ESAIM : Mathematical Modelling and Numerical Analysis

JF - ESAIM : Mathematical Modelling and Numerical Analysis

SN - 0764-583X

IS - 4

ER -