A Linear Viscoelastic Biphasic Model for Soft Tissues Based on the Theory of Porous Media

2001 ◽  
Vol 123 (5) ◽  
pp. 418-424 ◽  
Author(s):  
Wolfgang Ehlers ◽  
Bernd Markert
Keyword(s):  
Porous Media ◽  
Articular Cartilage ◽  
Soft Tissues ◽  
Differential Algebraic Equations ◽  
Solid Matrix ◽  
Numerical Treatment ◽  
Algebraic Equations ◽  
Theory Of Porous Media ◽  
Biphasic Model ◽  
Linear Viscoelastic

Based on the Theory of Porous Media (mixture theories extended by the concept of volume fractions), a model describing the mechanical behavior of hydrated soft tissues such as articular cartilage is presented. As usual, the tissue will be modeled as a materially incompressible binary medium of one linear viscoelastic porous solid skeleton saturated by a single viscous pore-fluid. The contribution of this paper is to combine a descriptive representation of the linear viscoelasticity law for the organic solid matrix with an efficient numerical treatment of the strongly coupled solid-fluid problem. Furthermore, deformation-dependent permeability effects are considered. Within the finite element method (FEM), the weak forms of the governing model equations are set up in a system of differential algebraic equations (DAE) in time. Thus, appropriate embedded error-controlled time integration methods can be applied that allow for a reliable and efficient numerical treatment of complex initial boundary-value problems. The applicability and the efficiency of the presented model are demonstrated within canonical, numerical examples, which reveal the influence of the intrinsic dissipation on the general behavior of hydrated soft tissues, exemplarily on articular cartilage.

Biphasic Creep and Stress Relaxation of Articular Cartilage in Compression: Theory and Experiments

1980 ◽  
Vol 102 (1) ◽  
pp. 73-84 ◽  
Author(s):  
V. C. Mow ◽  
S. C. Kuei ◽  
W. M. Lai ◽  
C. G. Armstrong
Keyword(s):  
Articular Cartilage ◽  
Stress Relaxation ◽  
Relative Motion ◽  
Interstitial Fluid ◽  
Fluid Phase ◽  
Solid Content ◽  
Solid Matrix ◽  
Frictional Drag ◽  
Tissue Mass ◽  
Biphasic Model

Articular cartilage is a biphasic material composed of a solid matrix phase (∼ 20 percent of the total tissue mass by weight) and an interstitial fluid phase (∼ 80 percent). The intrinsic mechanical properties of each phase as well as the mechanical interaction between these two phases afford the tissue its interesting rheological behavior. In this investigation, the solid matrix was assumed to be intrinsically incompressible, linearly elastic and nondissipative while the interstitial fluid was assumed to be intrinsically incompressible and nondissipative. Further, it was assumed that the only dissipation comes from the frictional drag of relative motion between the phases. However, more general constitutive equations, including a viscoelastic dissipation of the solid matrix as well as a viscous dissipation of interstitial fluid were also developed. A constant “average” permeability of the tissue was assumed, i.e., independent of deformation, and a solid content function Vs/Vf (the ratio of the volume of each of the phases) was assumed to vary with depth in accordance with the experimentally determined weight ratios. This linear, nonhomogeneous theory was applied to describe the experimentally obtained biphasic creep and biphasic stress relaxation data via a nonlinear regression technique. The determined intrinsic “aggregate” elastic modulus, from ten creep experiments, is 0.70 ± 0.09 MN/m2 and, from six stress relaxation experiments, is 0.76 ± 0.03 MN/m2. The “average” permeability of the tissue is (0.76 ± 0.42) × 10−14 m4 /N•s. We concluded that the large spread in the permeability coefficients is due to the assumption of a constant deformation independent permeability. We also concluded that 1) a nonlinearly permeable biphasic model, where the permeability function is given by an experimentally determined empirical law: k = A(p) exp [α(p)e], can be used to describe more accurately the rheological properties of articular cartilage, and 2) the frictional drag of relative motion is the most important factor governing the fluid/solid viscoelastic properties of the tissue in compression.

Numerical Treatment of Differential-Algebraic Equations with Index 2 by Reducing the Index

2008 ◽  
Vol 6 (1) ◽  
pp. 49-60 ◽  
Author(s):  
Basem Attili ◽  
Alaa Abu-Hatab ◽  
Aseel Ayoub ◽  
Ayat Ismail ◽  
Shefa Abu-Ghneim
Keyword(s):  
Differential Algebraic Equations ◽  
Numerical Treatment ◽  
Algebraic Equations ◽  
Index 2

Articular Cartilage Biomechanics Modeled via an Intrinsically Compressible Biphasic Model: Implications and Deviations From an Incompressible Biphasic Approach

2013 ◽  
Author(s):  
Francesco Travascio ◽  
Roberto Serpieri ◽  
Shihab Asfour
Keyword(s):  
Articular Cartilage ◽  
Experimental Testing ◽  
Solid Matrix ◽  
Hydraulic Permeability ◽  
Theoretical Frameworks ◽  
Theoretical Formulation ◽  
New Model ◽  
Biphasic Model ◽  
Testing Procedures ◽  
Cartilage Biomechanics

Biphasic continuum models have been extensively deployed for modeling macroscopic articular cartilage biomechanics [1,2]. This consolidated theoretical approach schematizes tissue as a mixture of an elastic solid matrix embedded in a fluid phase. In physiological conditions, intrinsic compressibility of each phase is very limited when compared to the whole tissue macroscopic counterpart. Based on such experimental evidence, intrinsic phase compressibility is generally reasonably neglected [3]. Hence, traditionally, cartilage biomechanics models have been developed on the basis of incompressible biphasic formulations [3–5], often referred to as Incompressible Theories of Mixtures (ITM). Alternatively, a more general biphasic model for cartilage biomechanics, accounting for full intrinsic compressibility of phases, may be considered. A consistent theoretical formulation of this type has been recently made available [6,7], hereby referred to as Theory of Microscopically Compressible Porous Media (TMCPM). In the present contribution, a new model for articular cartilage biomechanics, based on TMCPM, was developed. Predictions of this new model, and its deviations from a traditional ITM approach were studied. In particular, deviations between compressible and incompressible theoretical frameworks were investigated with a specific focus on the repercussions on models’ capability of characterizing fundamental tissue properties, such as hydraulic permeability, via established experimental testing procedures.

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