scholarly journals A Conewise Linear Elasticity Mixture Model for the Analysis of Tension-Compression Nonlinearity in Articular Cartilage

2000 ◽  
Vol 122 (6) ◽  
pp. 576-586 ◽  
Author(s):  
Michael A. Soltz ◽  
Gerard A. Ateshian
Keyword(s):  
Articular Cartilage ◽  
Mixture Model ◽  
Linear Elasticity ◽  
Solid Phase ◽  
Poisson Ratio ◽  
Fluid Pressure ◽  
Unconfined Compression ◽  
Cubic Symmetry ◽  
Torsional Shear ◽  
Biphasic Mixture

A biphasic mixture model is developed that can account for the observed tension-compression nonlinearity of cartilage by employing the continuum-based Conewise Linear Elasticity (CLE) model of Curnier et al. (J. Elasticity, 37, 1–38, 1995) to describe the solid phase of the mixture. In this first investigation, the orthotropic octantwise linear elasticity model was reduced to the more specialized case of cubic symmetry, to reduce the number of elastic constants from twelve to four. Confined and unconfined compression stress-relaxation, and torsional shear testing were performed on each of nine bovine humeral head articular cartilage cylindrical plugs from 6 month old calves. Using the CLE model with cubic symmetry, the aggregate modulus in compression and axial permeability were obtained from confined compression (H−A=0.64±0.22 MPa, kz=3.62±0.97×10−16 m4/Ns˙s,r2=0.95±0.03), the tensile modulus, compressive Poisson ratio, and radial permeability were obtained from unconfined compression (E+Y=12.75±1.56 MPa, v−=0.03±0.01,kr=6.06±2.10×10−16 m4/Ns˙s,r2=0.99±0.00), and the shear modulus was obtained from torsional shear (μ=0.17±0.06 MPa). The model was also employed to predict the interstitial fluid pressure successfully at the center of the cartilage plug in unconfined compression r2=0.98±0.01. The results of this study demonstrate that the integration of the CLE model with the biphasic mixture theory can provide a model of cartilage that can successfully curve-fit three distinct testing configurations while producing material parameters consistent with previous reports in the literature. [S0148-0731(00)00306-X]

An Analysis of the Unconfined Compression of Articular Cartilage

1984 ◽  
Vol 106 (2) ◽  
pp. 165-173 ◽  
Author(s):  
C. G. Armstrong ◽  
W. M. Lai ◽  
V. C. Mow
Keyword(s):  
Fluid Flow ◽  
Articular Cartilage ◽  
Fluid Pressure ◽  
Constant Strain Rate ◽  
Elastic Solid ◽  
Solid Matrix ◽  
Unconfined Compression ◽  
Creep Stress ◽  
Internal Fluid ◽  
Equilibrium Response

Analytical solutions have been obtained for the internal deformation and fluid-flow fields and the externally observable creep, stress relaxation, and constant strain-rate behaviors which occur during the unconfined compression of a cylindrical specimen of a fluid-filled, porous, elastic solid, such as articular cartilage, between smooth, impermeable plates. Instantaneously, the “biphasic” continuum deforms without change in volume and behaves like an incompressible elastic solid of the same shear modulus. Radial fluid flow then allows the internal fluid pressure to equilibrate with the external environment. The equilibrium response is controlled by the Young’s modulus and Poisson’s ratio of the solid matrix.

Strain-rate Dependent Stiffness of Articular Cartilage in Unconfined Compression

2003 ◽  
Vol 125 (2) ◽  
pp. 161-168 ◽  
Author(s):  
L. P. Li ◽  
M. D. Buschmann ◽  
A. Shirazi-Adl
Keyword(s):  
Articular Cartilage ◽  
Strain Rate ◽  
Compressive Strain ◽  
Fluid Pressure ◽  
Nonlinear Function ◽  
Protective Mechanism ◽  
Unconfined Compression ◽  
Rate Dependent ◽  
Dependent Behavior ◽  
Elastic Loading

The stiffness of articular cartilage is a nonlinear function of the strain amplitude and strain rate as well as the loading history, as a consequence of the flow of interstitial water and the stiffening of the collagen fibril network. This paper presents a full investigation of the interplay between the fluid kinetics and fibril stiffening of unconfined cartilage disks by analyzing over 200 cases with diverse material properties. The lower and upper elastic limits of the stress (under a given strain) are uniquely established by the instantaneous and equilibrium stiffness (obtained numerically for finite deformations and analytically for small deformations). These limits could be used to determine safe loading protocols in order that the stress in each solid constituent remains within its own elastic limit. For a given compressive strain applied at a low rate, the loading is close to the lower limit and is mostly borne directly by the solid constituents (with little contribution from the fluid). In contrast, however in case of faster compression, the extra loading is predominantly transported to the fibrillar matrix via rising fluid pressure with little increase of stress in the nonfibrillar matrix. The fibrillar matrix absorbs the loading increment by self-stiffening: the quicker the loading the faster the fibril stiffening until the upper elastic loading limit is reached. This self-protective mechanism prevents cartilage from damage since the fibrils are strong in tension. The present work demonstrates the ability of the fibril reinforced poroelastic models to describe the strain rate dependent behavior of articular cartilage in unconfined compression using a mechanism of fibril stiffening mainly induced by the fluid flow.

Effects of Friction on the Unconfined Compressive Response of Articular Cartilage: A Finite Element Analysis

1990 ◽  
Vol 112 (2) ◽  
pp. 138-146 ◽  
Author(s):  
Robert L. Spilker ◽  
Jun-Kyo Suh ◽  
Van C. Mow
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Articular Cartilage ◽  
Soft Tissues ◽  
Solid Phase ◽  
Mechanical Test ◽  
Fluid Phase ◽  
Element Formulation ◽  
Unconfined Compression ◽  
Element Analysis

A finite element analysis is used to study a previously unresolved issue of the effects of platen-specimen friction on the response of the unconfined compression test; effects of platen permeability are also determined. The finite element formulation is based on the linear KLM biphasic model for articular cartilage and other hydrated soft tissues. A Galerkin weighted residual method is applied to both the solid phase and the fluid phase, and the continuity equation for the intrinsically incompressible binary mixture is introduced via a penalty method. The solid phase displacements and fluid phase velocities are interpolated for each element in terms of unknown nodal values, producing a system of first order differential equations which are solved using a standard numerical finite difference technique. An axisymmetric element of quadrilateral cross-section is developed and applied to the mechanical test problem of a cylindrical specimen of soft tissue in unconfined compression. These studies show that interfacial friction plays a major role in the unconfined compression response of articular cartilage specimens with small thickness to diameter ratios.

The Influence of the Fixed Negative Charges on Mechanical and Electrical Behaviors of Articular Cartilage Under Unconfined Compression

2004 ◽  
Vol 126 (1) ◽  
pp. 6-16 ◽  
Author(s):  
D. D. Sun ◽  
X. E. Guo ◽  
M. Likhitpanichkul ◽  
W. M. Lai ◽  
V. C. Mow
Keyword(s):  
Articular Cartilage ◽  
Osmotic Pressure ◽  
Young’S Modulus ◽  
Poisson’S Ratio ◽  
Young's Modulus ◽  
Axial Strain ◽  
Fluid Pressure ◽  
Poisson's Ratio ◽  
Unconfined Compression ◽  
Fixed Charges

Unconfined compression test has been frequently used to study the mechanical behaviors of articular cartilage, both theoretically and experimentally. It has also been used in explant and gel-cell-complex studies in tissue engineering. In biphasic and poroelastic theories, the effect of charges fixed on the proteoglycan macromolecules in articular cartilage is embodied in the apparent compressive Young’s modulus and the apparent Poisson’s ratio of the tissue, and the fluid pressure is considered to be the portion above the osmotic pressure. In order to understand how proteoglycan fixed charges might affect the mechanical behaviors of articular cartilage, and in order to predict the osmotic pressure and electric fields inside the tissue in this experimental configuration, it is necessary to use a model that explicitly takes into account the charged nature of the tissue and the flow of ions within its porous interstices. In this paper, we used a finite element model based on the triphasic theory to study how fixed charges in the porous-permeable soft tissue can modulate its mechanical and electrochemical responses under a step displacement in unconfined compression. The results from finite element calculations showed that: 1) A charged tissue always supports a larger load than an uncharged tissue of the same intrinsic elastic moduli. 2) The apparent Young’s modulus (the ratio of the equilibrium axial stress to the axial strain) is always greater than the intrinsic Young’s modulus of an uncharged tissue. 3) The apparent Poisson’s ratio (the negative ratio of the lateral strain to the axial strain) is always larger than the intrinsic Poisson’s ratio of an uncharged tissue. 4) Load support derives from three sources: intrinsic matrix stiffness, hydraulic pressure and osmotic pressure. Under the unconfined compression, the Donnan osmotic pressure can constitute between 13%–22% of the total load support at equilibrium. 5) During the stress-relaxation process following the initial instant of loading, the diffusion potential (due to the gradient of the fixed charge density and the associated gradient of ion concentrations) and the streaming potential (due to fluid convection) compete against each other. Within the physiological range of material parameters, the polarity of the electric potential depends on both the mechanical properties and the fixed charge density (FCD) of the tissue. For softer tissues, the diffusion effects dominate the electromechanical response, while for stiffer tissues, the streaming potential dominates this response. 6) Fixed charges do not affect the instantaneous strain field relative to the initial equilibrium state. However, there is a sudden increase in the fluid pressure above the initial equilibrium osmotic pressure. These new findings are relevant and necessary for the understanding of cartilage mechanics, cartilage biosynthesis, electromechanical signal transduction by chondrocytes, and tissue engineering.

Biphasic Poroviscoelastic Simulation of the Unconfined Compression of Articular Cartilage: I—Simultaneous Prediction of Reaction Force and Lateral Displacement

2000 ◽  
Vol 123 (2) ◽  
pp. 191-197 ◽  
Author(s):  
Mark R. DiSilvestro ◽  
Qiliang Zhu ◽  
Marcy Wong ◽  
Jukka S. Jurvelin ◽  
Jun-Kyo Francis Suh
Keyword(s):  
Fluid Flow ◽  
Articular Cartilage ◽  
Solid Phase ◽  
Lateral Displacement ◽  
Transverse Isotropy ◽  
Reaction Force ◽  
Solid Matrix ◽  
Inviscid Fluid ◽  
Unconfined Compression ◽  
Interstitial Fluid Flow

This study investigated the ability of the linear biphasic poroelastic (BPE) model and the linear biphasic poroviscoelastic (BPVE) model to simultaneously predict the reaction force and lateral displacement exhibited by articular cartilage during stress relaxation in unconfined compression. Both models consider articular cartilage as a binary mixture of a porous incompressible solid phase and an incompressible inviscid fluid phase. The BPE model assumes the solid phase is elastic, while the BPVE model assumes the solid phase is viscoelastic. In addition, the efficacy of two additional models was also examined, i.e., the transversely isotropic BPE (TIBPE) model, which considers transverse isotropy of the solid matrix within the framework of the linear BPE model assumptions, and a linear viscoelastic solid (LVE) model, which assumes that the viscoelastic behavior of articular cartilage is solely governed by the intrinsic viscoelastic nature of the solid matrix, independent of the interstitial fluid flow. It was found that the BPE model was able to accurately account for the lateral displacement, but unable to fit the short-term reaction force data of all specimens tested. The TIBPE model was able to account for either the lateral displacement or the reaction force, but not both simultaneously. The LVE model was able to account for the complete reaction force, but unable to fit the lateral displacement measured experimentally. The BPVE model was able to completely account for both lateral displacement and reaction force for all specimens tested. These results suggest that both the fluid flow-dependent and fluid flow-independent viscoelastic mechanisms are essential for a complete simulation of the viscoelastic phenomena of articular cartilage.

scholarly journals Study of the Mechanical Environment of Chondrocytes in Articular Cartilage Defects Repaired Area under Cyclic Compressive Loading

2017 ◽  
Vol 2017 ◽  
pp. 1-10
Author(s):  
Hai-Ying Liu ◽  
Hang-Tian Duan ◽  
Chun-Qiu Zhang ◽  
Wei Wang
Keyword(s):  
Articular Cartilage ◽  
Cyclic Loading ◽  
Response Curve ◽  
Fluid Pressure ◽  
Cartilage Defects ◽  
Loading Frequency ◽  
Liquid Pressure ◽  
Mechanical Environment ◽  
Finite Element Software ◽  
Maximum Value

COMSOL finite element software was used to establish a solid-liquid coupling biphasic model of articular cartilage and a microscopic model of chondrocytes, using modeling to take into account the shape and number of chondrocytes in cartilage lacuna in each layer. The effects of cyclic loading at different frequencies on the micromechanical environment of chondrocytes in different regions of the cartilage were studied. The results showed that low frequency loading can cause stress concentration of superficial chondrocytes. Moreover, along with increased frequency, the maximum value of stress response curve of chondrocytes decreased, while the minimum value increased. When the frequency was greater than 0.2 Hz, the extreme value stress of response curve tended to be constant. Cyclic loading had a large influence on the distribution of liquid pressure in chondrocytes in the middle and deep layers. The concentration of fluid pressure changed alternately from intracellular to peripheral in the middle layer. Both the range of liquid pressure in the upper chondrocytes and the maximum value of liquid pressure in the lower chondrocytes in the same lacunae varied greatly in the deep layer. At the same loading frequency, the elastic modulus of artificial cartilage had little effect on the mechanical environment of chondrocytes.

On the Natural Lubrication of Synovial Joints: Normal and Degenerate

1977 ◽  
Vol 99 (2) ◽  
pp. 163-172 ◽  
Author(s):  
Joseph M. Mansour ◽  
Van C. Mow
Keyword(s):  
Articular Cartilage ◽  
Solid Phase ◽  
Articular Surface ◽  
Moving Load ◽  
Frictional Resistance ◽  
Elastic Solid ◽  
Surface Load ◽  
Two Phase ◽  
Tissue Properties ◽  
Synovial Joints

Fluid flow and mass transport mechanisms associated with articular cartilage function are important biomechanical processes of normal and pathological synovial joints. A three-layer permeable, two-phase medium of an incompressible fluid and a linear elastic solid are used to model the flow and deformational behavior of articular cartilage. The frictional resistance of the relative motion of the fluid phase with respect to the solid phase is given by a linear diffusive dissipation term. The subchondral bony substrate is represented by an elastic solid. The three-layer model of articular cartilage is chosen because of the known histological, ultrastructural, and biomechanical variations of the tissue properties. The calculated flow field shows that for material properties of normal healthy articular cartilage the tissue creates a naturally lubricated surface. The movement of the interstitial fluid at the surface is circulatory in manner, being exuded in front and near the leading half of the moving surface load and imbibed behind and near the trailing half of the moving load. The flow fields of healthy tissues are capable of sustaining a film of fluid at the articular surface whereas pathological tissues cannot.

Comparison of the equilibrium response of articular cartilage in unconfined compression, confined compression and indentation

2002 ◽  
Vol 35 (7) ◽  
pp. 903-909 ◽  
Author(s):  
R.K Korhonen ◽  
M.S Laasanen ◽  
J Töyräs ◽  
J Rieppo ◽  
J Hirvonen ◽  
...  
Keyword(s):  
Articular Cartilage ◽  
Unconfined Compression ◽  
Confined Compression ◽  
Equilibrium Response
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