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.

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.

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

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.

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]

Biphasic Poroviscoelastic Simulation of the Unconfined Compression of Articular Cartilage: II—Effect of Variable Strain Rates

2000 ◽  
Vol 123 (2) ◽  
pp. 198-200 ◽  
Author(s):  
Mark R. DiSilvestro, ◽  
Qiliang Zhu, ◽  
Jun-Kyo Francis Suh
Keyword(s):  
Fluid Flow ◽  
Articular Cartilage ◽  
Strain Rate ◽  
Viscoelastic Behavior ◽  
Viscoelastic Response ◽  
Strain Rates ◽  
Unconfined Compression ◽  
Rate Dependent ◽  
Linear Viscoelastic

This study investigated the abilities of the linear biphasic poroviscoelastic (BPVE) model and the linear biphasic poroelastic (BPE) model to simulate the effect of variable ramp strain rates on the unconfined compression stress relaxation response of articular cartilage. Curve fitting of experimental data showed that the BPVE model was able to successfully account for the ramp strain rate-dependent viscoelastic behavior of articular cartilage under unconfined compression, while the BPE model was able to account for the complete viscoelastic response at a slow strain rate, but only the long-term viscoelastic response at faster strain rates. We concluded that the short-term viscoelastic behavior of articular cartilage, when subjected to a fast ramp strain rate, is primarily governed by a fluid flow-independent (intrinsic) viscoelastic mechanism, whereas the long-term viscoelastic behavior is governed by a fluid flow-dependent (biphasic) viscoelastic mechanism. Furthermore, a linear viscoelastic representation of the solid stress was found to be a valid model assumption for the simulation of ramp strain rate-dependent relaxation behaviors of articular cartilage within the range of ramp strain rates investigated.

Experimental Verification of the Roles of Intrinsic Matrix Viscoelasticity and Tension-Compression Nonlinearity in the Biphasic Response of Cartilage

2003 ◽  
Vol 125 (1) ◽  
pp. 84-93 ◽  
Author(s):  
Chun-Yuh Huang ◽  
Michael A. Soltz ◽  
Monika Kopacz ◽  
Van C. Mow ◽  
Gerard A. Ateshian
Keyword(s):  
Articular Cartilage ◽  
Stress Relaxation ◽  
Strain Rate ◽  
Dynamic Loading ◽  
Dynamic Compression ◽  
Solid Matrix ◽  
Supporting Evidence ◽  
Compression Stress ◽  
Unconfined Compression ◽  
Support Mechanism

A biphasic-CLE-QLV model proposed in our recent study [2001, J. Biomech. Eng., 123, pp. 410–417] extended the biphasic theory of Mow et al. [1980, J. Biomech. Eng., 102, pp. 73–84] to include both tension-compression nonlinearity and intrinsic viscoelasticity of the cartilage solid matrix by incorporating it with the conewise linear elasticity (CLE) model [1995, J. Elasticity, 37, pp. 1–38] and the quasi-linear viscoelasticity (QLV) model [Biomechanics: Its foundations and objectives, Prentice Hall, Englewood Cliffs, 1972]. This model demonstrates that a simultaneous prediction of compression and tension experiments of articular cartilage, under stress-relaxation and dynamic loading, can be achieved when properly taking into account both flow-dependent and flow-independent viscoelastic effects, as well as tension-compression nonlinearity. The objective of this study is to directly test this biphasic-CLE-QLV model against experimental data from unconfined compression stress-relaxation tests at slow and fast strain rates as well as dynamic loading. Twelve full-thickness cartilage cylindrical plugs were harvested from six bovine glenohumeral joints and multiple confined and unconfined compression stress-relaxation tests were performed on each specimen. The material properties of specimens were determined by curve-fitting the experimental results from the confined and unconfined compression stress relaxation tests. The findings of this study demonstrate that the biphasic-CLE-QLV model is able to describe the strain-rate-dependent mechanical behaviors of articular cartilage in unconfined compression as attested by good agreements between experimental and theoretical curvefits (r2=0.966±0.032 for testing at slow strain rate; r2=0.998±0.002 for testing at fast strain rate) and predictions of the dynamic response r2=0.91±0.06. This experimental study also provides supporting evidence for the hypothesis that both tension-compression nonlinearity and intrinsic viscoelasticity of the solid matrix of cartilage are necessary for modeling the transient and equilibrium responses of this tissue in tension and compression. Furthermore, the biphasic-CLE-QLV model can produce better predictions of the dynamic modulus of cartilage in unconfined dynamic compression than the biphasic-CLE and biphasic poroviscoelastic models, indicating that intrinsic viscoelasticity and tension-compression nonlinearity of articular cartilage may play important roles in the load-support mechanism of cartilage under physiologic loading.

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 Model Simultaneously Predicts Axial Reaction Force and Lateral Displacement of Articular Cartilage During an Unconfined Compression-Stress Relaxation Experiment

1999 ◽  
Author(s):  
Mark R. DiSilvestro ◽  
Qiliang Zhu ◽  
Marcy Wong ◽  
Jukka Jurvelin ◽  
Jun-Kyo Suh
Keyword(s):  
Fluid Flow ◽  
Articular Cartilage ◽  
Stress Relaxation ◽  
Mechanical Behavior ◽  
Lateral Displacement ◽  
Reaction Force ◽  
Simple Extension ◽  
Unconfined Compression ◽  
Relaxation Experiment ◽  
Diarthrodial Joints

Abstract Articular cartilage lining the articulating surfaces in diarthrodial joints is composed of an extracellular matrix and interstitial fluid. The complex mechanical behavior of this tissue has been successfully modeled by the linear biphasic poroviscoelastic (BPVE) model first introduced by Mak (1986). This model, a simple extension of the well-known biphasic theory first proposed by Mow et al. (1980), accounts for both fluid flow-dependent and fluid flow-independent viscoelastic mechanisms which contribute to the overall mechanical behavior exhibited by the tissue. Despite the success of the linear BPVE model for indentation (Suh and Bai, 1997), as well as that described for unconfined compression (Suh and DiSilvestro, 1997, 1998), the model’s ability to account for more than one measurable variable with a single parameter set has not been established. Therefore, the objective, of this study was to assess the ability of the linear BPVE model to account for both the axial reaction force and lateral deformation of a cylindrical plug of articular cartilage subjected to unconfined compression under a stress relaxation protocol.

Application of the u-p Finite Element Method to the Study of Articular Cartilage

1991 ◽  
Vol 113 (4) ◽  
pp. 397-403 ◽  
Author(s):  
Jennifer S. Wayne ◽  
Savio L.-Y. Woo ◽  
Michael K. Kwan
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Articular Cartilage ◽  
Numerical Solutions ◽  
Virtual Work ◽  
Fluid Pressure ◽  
Numerical Procedure ◽  
Solid Matrix ◽  
Confined Compression ◽  
Element Method

The finite element method using the principle of virtual work was applied to the biphasic theory to establish a numerical routine for analyses of articular cartilage behavior. The matrix equations that resulted contained displacements of the solid matrix (u) and true fluid pressure (p) as the unknown variables at the element nodes. Both small and large strain conditions were considered. The algorithms and computer code for the analysis of two-dimensional plane strain, plane stress, and axially symmetric cases were developed. The u-p finite element numerical procedure demonstrated excellent agreement with available closed-form and numerical solutions for the configurations of confined compression and unconfined compression under small strains, and for confined compression under large strains. The model was also used to examine the behavior of a repaired articular surface. The differences in material properties between the repair tissue and normal cartilage resulted in significant deformation gradients across the repair interface as well as increased fluid efflux from the tissue.

A Biphasic Transversely Isotropic Poroviscoelastic Model for the Unconfined Compression of Hydrated Soft Tissue

2016 ◽  
Vol 138 (3) ◽  
Author(s):  
H. Hatami-Marbini ◽  
R. Maulik
Keyword(s):  
Articular Cartilage ◽  
Soft Tissue ◽  
Stress Relaxation ◽  
Soft Tissues ◽  
Transverse Isotropy ◽  
Viscoelastic Model ◽  
Transversely Isotropic ◽  
Solid Matrix ◽  
Unconfined Compression ◽  
Material Parameters

The unconfined compression experiments are commonly used for characterizing the mechanical behavior of hydrated soft tissues such as articular cartilage. Several analytical constitutive models have been proposed over the years to analyze the unconfined compression experimental data and subsequently estimate the material parameters. Nevertheless, new mathematical models are still required to obtain more accurate numerical estimates. The present study aims at developing a linear transversely isotropic poroviscoelastic theory by combining a viscoelastic material law with the transversely isotropic biphasic model. In particular, an integral type viscoelastic model is used to describe the intrinsic viscoelastic properties of a transversely isotropic solid matrix. The proposed constitutive theory incorporates viscoelastic contributions from both the fluid flow and the intrinsic viscoelasticity to the overall stress-relaxation behavior. Moreover, this new material model allows investigating the biomechanical properties of tissues whose extracellular matrix exhibits transverse isotropy. In the present work, a comprehensive parametric study was conducted to determine the influence of various material parameters on the stress–relaxation history. Furthermore, the efficacy of the proposed theory in representing the unconfined compression experiments was assessed by comparing its theoretical predictions with those obtained from other versions of the biphasic theory such as the isotropic, transversely isotropic, and viscoelastic models. The unconfined compression behavior of articular cartilage as well as corneal stroma was used for this purpose. It is concluded that while the proposed model is capable of accurately representing the viscoelastic behavior of any hydrated soft tissue in unconfined compression, it is particularly useful in modeling the behavior of those with a transversely isotropic skeleton.

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