A mixed finite element formulation of triphasic mechano-electrochemical theory for charged, hydrated biological soft tissues

Abstract

SUMMARYAn equivalent new expression of the triphasic mechano-electrochemical theory [9] is presented and a mixed "nite element formulation is developed using the standard Galerkin weighted residual method. Solid displacement u s , modi,ed electrochemical/chemical potentials , >and \ (with dimensions of concentration) for water, cation and anion are chosen as the four primary degrees of freedom (DOFs) and are independently interpolated. The modi"ed Newton}Raphson iterative procedure is employed to handle the non-linear terms. The resulting "rst-order Ordinary Di!erential Equations (ODEs) with respect to time are solved using the implicit Euler backward scheme which is unconditionally stable. One-dimensional (1-D) linear isoparametric element is developed. The "nal algebraic equations form a non-symmetric but sparse matrix system. With the current choice of primary DOFs, the formulation has the advantage of small amount of storage, and the jump conditions between elements and across the interface boundary are satis"ed automatically. The "nite element formulation has been used to investigate a 1-D triphasic stress relaxation problem in the con"ned compression con"guration and a 1-D triphasic free swelling problem. The formulation accuracy and convergence for 1-D cases are examined with independent "nite di!erence methods. The FEM results are in excellent agreement with those obtained from the other methods.

Altered swelling and ion fluxes in articular cartilage as a biomarker in osteoarthritis and joint immobilization: a computational analysis

“…R is the universal gas constant and T is the absolute temperature [42]. Considering F a as the volume fraction of each component, thus we have the following saturation condition for the tissue …”

“…u ¼ divu s is the solid matrix dilatation related to the infinitesimal strain tensor of the solid matrix [42]. Small strains are considered in this model as well as linear-elastic isotropic and homogeneous material.…”

“…In our previous work [38], we developed a threedimensional computational model of cartilage behaviour, based on the triphasic theory for hydrated soft tissue [10,42], which was able to accurately simulate the contribution of the main tissue properties to its behaviour. The model considers the mechano-electrochemical events as a concurrent phenomenon.…”

Tribology of Hemiarthroplasty

“…Swelling of articular cartilage depends on both its fixed charge density and distribution, the stiffness of cartilage collagen-proteoglycan matrix, and the ion concentrations in the interstitial fluid (Lai et al, 1991) determined to playa small role compared to the solid and fluid phases of cartilage (Gu et al, 1997;Gu et al, 1998;Narmoneva et al, 1999;Sun et al, 1999;Ateshian et al, 2004).…”

Parameter-dependent behavior of articular cartilage: 3D mechano-electrochemical computational model

“…To our knowledge, this is the first parametric study that determines the influence of these properties in an isolate manner in cartilage behavior, resulting in a basic guide to select the main parameters required for articular cartilage simulation. Note that the model includes essential biological phenomena, previously described in literature, that affect cartilage behaviour (diffusive-convective events and mechano-electrochemical effects) (Mow and Guo, 2002;Sun et al, 1999;Lai et al, 1991). However, it excludes those that experimentally show less influence as the viscosity of the solid matrix.…”