Unconfined creep compression of chondrocytes

The study of single cell mechanics offers a valuable tool for understanding cellular milieus. Specific knowledge of chondrocyte biomechanics could lead to elucidation of disease etiologies and the biomechanical factors most critical to stimulating regenerative processes in articular cartilage. Recent studies in our laboratory have suggested that it may be acceptable to approximate the shape of a single chondrocyte as a disc. This geometry is easily utilized for generating models of unconfined compression. In this study, three continuum mechanics models of increasing complexity were formulated and used to fit unconfined compression creep data. Creep curves were obtained from middle/deep zone chondrocytes (n = 15) and separately fit using the three continuum models. The linear elastic solid model yielded a Young's modulus of 2.55+/-0.85 kPa. The viscoelastic model (adapted from the Kelvin model) generated an instantaneous modulus of 2.47+/-0.85 kPa, a relaxed modulus of 1.48+/-0.35 kPa, and an apparent viscosity of 1.92+/-1.80 kPa-s. Finally, a linear biphasic model produced an aggregate modulus of 2.58+/-0.87 kPa, a permeability of 2.57 x 10(-12)+/-3.09 m(4)/N-s, and a Poisson's ratio of 0.069+/-0.021. The results of this study demonstrate that similar values for the cell modulus can be obtained from three models of increasing complexity. The elastic model provides an easy method for determining the cell modulus, however, the viscoelastic and biphasic models generate additional material properties that are important for characterizing the transient response of compressed chondrocytes.     

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

At mechanical equilibrium, articular cartilage is usually characterized as an isotropic elastic material with no interstitial fluid flow. In this study, the equilibrium properties (Young's modulus, aggregate modulus and Poisson's ratio) of bovine humeral, patellar and femoral cartilage specimens (n=26) were investigated using unconfined compression, confined compression, and indentation tests. Optical measurements of the Poisson's ratio of cartilage were also carried out. Mean values of the Young's modulus (assessed from the unconfined compression test) were 0.80+/-0.33, 0.57+/-0.17 and 0.31+/-0.18MPa and of the Poisson's ratio (assessed from the optical test) 0.15+/-0.06, 0.16+/-0.05 and 0.21+/-0.05 for humeral, patellar, and femoral cartilages, respectively. The indentation tests showed 30-79% (p<0.01) higher Young's modulus values than the unconfined compression tests. In indentation, values of the Young's modulus were independent of the indenter diameter only in the humeral cartilage. The mean values of the Poisson's ratio, obtained indirectly using the mathematical relation between the Young's modulus and the aggregate modulus in isotropic material, were 0.16+/-0.06, 0.21+/-0.05, and 0.26+/-0.08 for humeral, patellar, and femoral cartilages, respectively. We conclude that the values of the elastic parameters of the cartilage are dependent on the measurement technique in use. Based on the similar values of Poisson's ratios, as determined directly or indirectly, the equilibrium response of articular cartilage under unconfined and confined compression is satisfactorily described by the isotropic elastic model. However, values of the isotropic Young's modulus obtained from the in situ indentation tests are higher than those obtained from the in vitro unconfined or confined compression tests and may depend on the indenter size in use.     

Direct measurement of the Poisson's ratio of human patella cartilage in tension

Articular cartilage has been shown to exhibit large transverse contractions when loaded in tension, suggesting the existence of large values for the Poisson's ratio. Previous studies have suggested that this effect is dependent on amplitude of applied strain, so that a single Poisson's ratio may not be sufficient to describe cartilage behavior. In this study, the Poisson's ratio (v), toe region modulus (Eo), and linear region modulus (E) of human patellar articular cartilage were calculated in simple tension tests from optical analysis of the two-dimensional strain fields at equilibrium. The Poisson's ratio was found to be independent of strain due to the absence of viscoelastic effects during testing. The Poisson's ratio was found to be significantly higher in the surface zone (1.87 +/- 1.11, p<0.01) than in the middle zone (0.62 +/- 0.23), with no significant correlation of v with age of the cartilage. In general, values for Poisson's ratio were greater than 0.5, suggesting cartilage behavior in tension deviates from isotropy. Reported values for the Poisson's ratio of cartilage in compression have been much lower than values measured here in tension, reflecting a mechanical contribution of the collagen fibers to anisotropy in tension but not compression. The toe-region modulus (Eo) was significantly higher in the surface zone (4.51 +/- 2.78 MPa, n=8) compared to the middle zone (2.51 +/- 1.93 MPa, n=10). In addition, the linear-region modulus (E) in the surface zone, but not middle zone (3.42 +/- 2.17 MPa, n=10), was found to correlate with age (R=0.97, p<0.02) with values of surface zone E equal to 23.92 +/- 12.29 MPa (n=5) for subjects under 70 yr of age, and 4.27 +/- 2.89 MPa (n=3) for subjects over 70 yr. Moduli values and trends with depth were consistent with previous studies of human and animal cartilage. From direct measures of two independent material properties, v and E, we calculated a shear modulus, G, which had not been previously reported for cartilage from tensile testing. Calculated values for surface zone G were 3.64 +/- 1.80 MPa for subjects under 70 yr old and 0.96 +/- 0.69 MPa for subjects over 70 yr old, and were significantly higher in the surface zone than in the middle zone (1.10 +/- 0.78 MPa). This study provides an intrinsic measure for the Poisson's ratio of articular cartilage and its dependence on depth which will be important in understanding the nonlinear tension-compression and anisotropic behaviors of articular cartilage.     

Alterations in the mechanical properties of the human chondrocyte pericellular matrix with osteoarthritis

In articular cartilage, chondrocytes are surrounded by a pericellular matrix (PCM), which together with the chondrocyte have been termed the "chondron." While the precise function of the PCM is not know there has been considerable speculation that it plays a role in regulating the biomechanical environment of the chondrocyte. In this study, we measured the Young's modulus of the PCM from normal and osteoarthritic cartilage using the micropipette aspiration technique, coupled with a newly developed axisymmetric elastic layered half-space model of the experimental configuration. Viable, intact chondrons were extracted from human articular cartilage using a new microaspiration-based isolation technique. In normal cartilage, the Young's modulus of the PCM was similar in chondrons isolated from the surface zone (68.9 +/- 18.9 kPa) as compared to the middle and deep layers (62.0 +/- 30.5 kPa). However, the mean Young's modulus of the PCM (pooled for the two zones) was significantly decreased in osteoarthritic cartilage (66.5 +/- 23.3 kPa versus 41.3 +/- 21.1 kPa, p < 0.001). In combination with previous theoretical models of cell-matrix interactions in cartilage, these findings suggest that the PCM has an important influence on the stress-strain environment of the chondrocyte that potentially varies with depth from the cartilage surface. Furthermore, the significant loss of PCM stiffness that was observed in osteoarthritic cartilage may affect the magnitude and distribution of biomechanical signals perceived by the chondrocytes.     

The Mechanical Behaviour of Chondrocytes Predicted with a Micro-structural Model of Articular Cartilage

The integrity of articular cartilage depends on the proper functioning and mechanical stimulation of chondrocytes, the cells that synthesize extracellular matrix and maintain tissue health. The biosynthetic activity of chondrocytes is influenced by genetic factors, environmental influences, extracellular matrix composition, and mechanical factors. The mechanical environment of chondrocytes is believed to be an important determinant for joint health, and chondrocyte deformation in response to mechanical loading is speculated to be an important regulator of metabolic activity. In previous studies of chondrocyte deformation, articular cartilage was described as a biphasic material consisting of a homogeneous, isotropic, linearly elastic solid phase, and an inviscid fluid phase. However, articular cartilage is known to be anisotropic and inhomogeneous across its depth. Therefore, isotropic and homogeneous models cannot make appropriate predictions for tissue and cell stresses and strains. Here, we modelled articular cartilage as a transversely isotropic, inhomogeneous (TI) material in which the anisotropy and inhomogeneity arose naturally from the microstructure of the depth-dependent collagen fibril orientation and volumetric fraction, as well as the chondrocyte shape and volumetric fraction. The purpose of this study was to analyse the deformation behaviour of chondrocytes using the TI model of articular cartilage. In order to evaluate our model against experimental results, we simulated indentation and unconfined compression tests for nominal compressions of 15%. Chondrocyte deformations were analysed as a function of location within the tissue. The TI model predicted a non-uniform behaviour across tissue depth: in indentation testing, cell height decreased by 43% in the superficial zone and between 11 and 29% in the deep zone. In unconfined compression testing, cell height decreased by 32% in the superficial zone, 25% in the middle, and 18% in the deep zones. This predicted non-uniformity is in agreement with experimental studies. The novelty of this study is the use of a cartilage material model accounting for the intrinsic inhomogeneity and anisotropy of cartilage caused by its microstructure.     

Topographical variation of the elastic properties of articular cartilage in the canine knee

Equilibrium response of articular cartilage to indentation loading is controlled by the thickness (h) and elastic properties (shear modulus, mu, and Poisson's ratio, nu) of the tissue. In this study, we characterized topographical variation of Poisson's ratio of the articular cartilage in the canine knee joint (N=6). Poisson's ratio was measured using a microscopic technique. In this technique, the shape change of the cartilage disk was visualized while the cartilage was immersed in physiological solution and compressed in unconfined geometry. After a constant 5% axial strain, the lateral strain was measured during stress relaxation. At equilibrium, the lateral-to-axial strain ratio indicates the Poisson's ratio of the tissue. Indentation (equilibrium) data from our prior study (Arokoski et al., 1994. International Journal of Sports Medicine 15, 254-260) was re-analyzed using the Poisson's ratio results at the test site to derive values for shear and aggregate moduli. The lowest Poisson's ratio (0.070+/-0.016) located at the patellar surface of femur (FPI) and the highest (0.236+/-0.026) at the medial tibial plateau (TMI). The stiffest cartilage was found at the patellar groove of femur (micro=0.964+/-0.189MPa, H(a)=2.084+/-0. 409MPa) and the softest at the tibial plateaus (micro=0.385+/-0. 062MPa, H(a)=1.113+/-0.141MPa). Comparison of the mechanical results and the biochemical composition of the tissue (Jurvelin et al., 1988. Engineering in Medicine 17, 157-162) at the matched sites of the canine knee joint indicated a negative correlation between the Poisson's ratio and collagen-to-PG content ratio. This is in harmony with our previous findings which suggested that, in unconfined compression, the degree of lateral expansion in different tissue zones is related to collagen-to-PG ratio of the zone.     

Importance of collagen orientation and depth-dependent fixed charge densities of cartilage on mechanical behavior of chondrocytes

The collagen network and proteoglycan matrix of articular cartilage are thought to play an important role in controlling the stresses and strains in and around chondrocytes, in regulating the biosynthesis of the solid matrix, and consequently in maintaining the health of diarthrodial joints. Understanding the detailed effects of the mechanical environment of chondrocytes on cell behavior is therefore essential for the study of the development, adaptation, and degeneration of articular cartilage. Recent progress in macroscopic models has improved our understanding of depth-dependent properties of cartilage. However, none of the previous works considered the effect of realistic collagen orientation or depth-dependent negative charges in microscopic models of chondrocyte mechanics. The aim of this study was to investigate the effects of the collagen network and fixed charge densities of cartilage on the mechanical environment of the chondrocytes in a depth-dependent manner. We developed an anisotropic, inhomogeneous, microstructural fibril-reinforced finite element model of articular cartilage for application in unconfined compression. The model consisted of the extracellular matrix and chondrocytes located in the superficial, middle, and deep zones. Chondrocytes were surrounded by a pericellular matrix and were assumed spherical prior to tissue swelling and load application. Material properties of the chondrocytes, pericellular matrix, and extracellular matrix were obtained from the literature. The loading protocol included a free swelling step followed by a stress-relaxation step. Results from traditional isotropic and transversely isotropic biphasic models were used for comparison with predictions from the current model. In the superficial zone, cell shapes changed from rounded to elliptic after free swelling. The stresses and strains as well as fluid flow in cells were greatly affected by the modulus of the collagen network. The fixed charge density of the chondrocytes, pericellular matrix, and extracellular matrix primarily affected the aspect ratios (height/width) and the solid matrix stresses of cells. The mechanical responses of the cells were strongly location and time dependent. The current model highlights that the collagen orientation and the depth-dependent negative fixed charge densities of articular cartilage have a great effect in modulating the mechanical environment in the vicinity of chondrocytes, and it provides an important improvement over earlier models in describing the possible pathways from loading of articular cartilage to the mechanical and biological responses of chondrocytes.     

A biphasic multiscale study of the mechanical microenvironment of chondrocytes within articular cartilage under unconfined compression

Computational analyses have been used to study the biomechanical microenvironment of the chondrocyte that cannot be assessed by in vitro experimental studies; yet all computational studies thus far have focused on the effect of zonal location (superficial, middle, and deep) on the mechanical microenvironment of chondrocytes. The aim of this paper was to study the effect of both zonal and radial locations on the biomechanical microenvironment of chondrocytes in inhomogeneous cartilage under unconfined stress relaxation. A biphasic multiscale approach was employed and nine chondrocytes in different locations were studied. Hyperelastic biphasic theory and depth-dependent aggregate modulus and permeability of articular cartilage were included in the models. It was found that both zonal and radial locations affected the biomechanical stresses and strains of the chondrocytes. Chondrocytes in the mid-radial location had increased volume during the early stage of the loading process. Maximum principal shear stress at the interface between the chondrocyte and the extracellular matrix (ECM) increased with depth, yet that at the ECM–pericellular matrix (PCM) interface had an inverse trend. Fluid pressure decreased with depth, while the fluid pressure difference between the top and bottom boundaries of the microscale model increased with depth. Regardless of location, fluid was exchanged between the chondrocyte, PCM, and ECM. These findings suggested that even under simple compressive loading conditions, the biomechanical microenvironment of the chondrocytes, PCM and ECM was spatially dependent. The current study provides new insight on chondrocyte biomechanics.     

Inhomogeneous cartilage properties enhance superficial interstitial fluid support and frictional properties, but do not provide a homogeneous state of stress

It has been well established that articular cartilage is compositionally and mechanically inhomogenous through its depth. To what extent this structural inhomogeneity is a prerequisite for appropriate cartilage function and integrity is not well understood. The first hypothesis to be tested in this study was that the depth-dependent inhomogeneity of the cartilage acts to maximize the interstitial fluid load support at the articular surface, to provide efficient frictional and wear properties. The second hypothesis was that the inhomogeneity produces a more homogeneous state of elastic stress in the matrix than would be achieved with uniform properties. We have, for the first time, simultaneously determined depth-dependent tensile and compressive properties of human patellofemoral cartilage from unconfined compression stress relaxation tests. The results show that the tensile modulus increases significantly from 4.1 +/- 1.9 MPa in the deep zone to 8.3 +/- 3.7 MPa at the superficial zone, while the compressive modulus decreases from 0.73 +/- 0.26 MPa to 0.28 +/- 0.16 MPa. The experimental measurements were then implemented with the finite-element method to compute the response of an inhomogeneous and homogeneous cartilage layer to loading. The finite-element models demonstrate that structural inhomogeneity acts to increase the interstitial fluid load support at the articular surface. However, the state of stress, strain, or strain energy density in the solid matrix remained inhomogeneous through the depth of the articular layer, whether or not inhomogeneous material properties were employed. We suggest that increased fluid load support at the articular surface enhances the frictional and wear properties of articular cartilage, but that the tissue is not functionally adapted to produce homogeneous stress, strain, or strain energy density distributions. Interstitial fluid pressurization, but not a homogeneous elastic stress distribution, appears thus to be a prerequisite for the functional and morphological integrity of the cartilage.     

Depth-dependent biomechanical and biochemical properties of fetal, newborn, and tissue-engineered articular cartilage

Adult articular cartilage has depth-dependent mechanical and biochemical properties which contribute to zone-specific functions. The compressive moduli of immature cartilage and tissue-engineered cartilage are known to be lower than those of adult cartilage. The objective of this study was to determine if such tissues exhibit depth-dependent compressive properties, and how these depth-varying properties were correlated with cell and matrix composition of the tissue. The compressive moduli of fetal and newborn bovine articular cartilage increased with depth (p<0.05) by a factor of 4-5 from the top 0.1 mm (28+/-13 kPa, 141+/-10 kPa, respectively) to 1 mm deep into the tissue. Likewise, the glycosaminoglycan and collagen content increased with depth (both p<0.001), and correlated with the modulus (both p<0.01). In contrast, tissue-engineered cartilage formed by either layering or mixing cells from the superficial and middle zone of articular cartilage exhibited similarly soft regions at both construct surfaces, as exemplified by large equilibrium strains. The properties of immature cartilage may provide a template for developing tissue-engineered cartilage which aims to repair cartilage defects by recapitulating the natural development and growth processes. These results suggest that while depth-dependent properties may be important to engineer into cartilage constructs, issues other than cell heterogeneity must be addressed to generate such tissues.     

Biomechanical properties of human articular cartilage under compressive loads

The function of articular cartilage is to support and distribute loads and to provide lubrication in the diarthrodial joints. Cartilage function is described by proper mechanical and rheological properties, strain and depth-dependent, which are not completely assessed. Unconfined and confined compression are commonly used to evaluate the Young's modulus (E) and the aggregate modulus (H(A)), respectively. The Poisson's ratio (nu) can be calculated indirectly from the equilibrium compression data, or using the biphasic indentation technique; it has recently been optically evaluated by using video microscopy during unconfined compression. The transient response of articular cartilage during confined compression depends on its permeability k; a constant value of k can be easily identified by a simple analytical model of confined compression tests, whereas more complex models or direct measurements (permeation tests) are needed to study the permeability dependence on deformation. A poroelastic finite element model of articular cartilage was developed for this purpose. The elastic parameters (E,nu) of the model were evaluated performing unconfined compression creep tests on human articular cartilage disks, whereas k was identified from the confined test response. Our combined experimental and computational method can be used to identify the parameters that define the permeability dependence on deformation, as a function of depth from articular surface.     

Zonal changes in the three-dimensional morphology of the chondron under compression: the relationship among cellular, pericellular, and extracellular deformation in articular cartilage

The pericellular matrix (PCM) is a narrow region of tissue that completely surrounds chondrocytes in articular cartilage. Previous theoretical models of the "chondron" (the PCM with enclosed cells) suggest that the structure and properties of the PCM may significantly influence the mechanical environment of the chondrocyte. The objective of this study was to quantify changes in the three-dimensional (3D) morphology of the chondron in situ at different magnitudes of compression applied to the cartilage extracellular matrix. Fluorescence immunolabeling for type-VI collagen was used to identify the boundaries of the cell and PCM, and confocal microscopy was used to form 3D images of chondrons from superficial, middle, and deep zone cartilage in explants compressed to 0%, 10%, 30%, and 50% surface-to-surface strain. Lagrangian tissue strain, determined locally using texture correlation, was highly inhomogeneous and revealed depth-dependent compressive stiffness and Poisson's ratio of the extracellular matrix. Compression significantly decreased cell and chondron height and volume, depending on the zone and magnitude of compression. In the superficial zone, cellular-level strains were always lower than tissue-level strains. In the middle and deep zones, however, tissue strains below 25% were amplified at the cellular level, while tissue strains above 25% were decreased at the cellular level. These findings are consistent with previous theoretical models of the chondron, suggesting that the PCM can serve as either a protective layer for the chondrocyte or a transducer that amplifies strain, such that cellular-level strains are more homogenous throughout the tissue depth despite large inhomogeneities in local ECM strains.     

The role of viscoelasticity of collagen fibers in articular cartilage: theory and numerical formulation

The relative importance of fluid-dependent and fluid-independent transient mechanical behavior in articular cartilage was examined for tensile and unconfined compression testing using a fibril reinforced model. The collagen matrix of articular cartilage was modeled as viscoelastic using a quasi-linear viscoelastic formulation with strain-dependent elastic modulus, while the proteoglycan matrix was considered as linearly elastic. The collagen viscoelastic properties were obtained by fitting experimental data from a tensile test. These properties were used to investigate unconfined compression testing, and the sensitivity of the properties was also explored. It was predicted that the stress relaxation observed in tensile tests was not caused by fluid pressurization at the macroscopic level. A multi-step tensile stress relaxation test could be approximated using a hereditary integral in which the elastic fibrillar modulus was taken to be a linear function of the fibrillar strain. Applying the same formulation to the radial fibers in unconfined compression, stress relaxation could not be simulated if fluid pressurization were absent. Collagen viscoelasticity was found to slightly weaken fluid pressurization in unconfined compression, and this effect was relatively more significant at moderate strain rates. Therefore, collagen viscoelasticity appears to play an import role in articular cartilage in tensile testing, while fluid pressurization dominates the transient mechanical behavior in compression. Collagen viscoelasticity plays a minor role in the mechanical response of cartilage in unconfined compression if significant fluid flow is present.     

Determination of the Poisson's ratio of the cell: recovery properties of chondrocytes after release from complete micropipette aspiration

Chondrocytes in articular cartilage are regularly subjected to compression and recovery due to dynamic loading of the joint. Previous studies have investigated the elastic and viscoelastic properties of chondrocytes using micropipette aspiration techniques, but in order to calculate cell properties, these studies have generally assumed that cells are incompressible with a Poisson's ratio of 0.5. The goal of this study was to measure the Poisson's ratio and recovery properties of the chondrocyte by combining theoretical modeling with experimental measures of complete cellular aspiration and release from a micropipette. Chondrocytes isolated from non-osteoarthritic and osteoarthritic cartilage were fully aspirated into a micropipette and allowed to reach mechanical equilibrium. Cells were then extruded from the micropipette and cell volume and morphology were measured throughout the experiment. This experimental procedure was simulated with finite element analysis, modeling the chondrocyte as either a compressible two-mode viscoelastic solid, or as a biphasic viscoelastic material. By fitting the experimental data to the theoretically predicted cell response, the Poisson's ratio and the viscoelastic recovery properties of the cell were determined. The Poisson's ratio of chondrocytes was found to be 0.38 for non-osteoarthritic cartilage and 0.36 for osteoarthritic chondrocytes (no significant difference). Osteoarthritic chondrocytes showed an increased recovery time following full aspiration. In contrast to previous assumptions, these findings suggest that chondrocytes are compressible, consistent with previous studies showing cell volume changes with compression of the extracellular matrix.     

Del1: a new protein in the superficial layer of articular cartilage

Articular cartilage contains four distinct zones, extending from the surface to the subchondral bone. Freshly isolated chondrocytes from the superficial zone of articular cartilage retain a collagenase-P-resistant cell-associated matrix. In the studies described here, the protein Del1 was identified as a component of the cell-associated matrix of superficial zone chondrocytes from adult bovine articular cartilage. Very little Del1 was associated with freshly isolated deep zone chondrocytes. Western blot analysis of articular cartilage cell and tissue extracts using polyclonal antibodies specific for Del1 showed Del1 was present in an insoluble cell-associated fraction. Extracts of the superficial zone of articular cartilage were found to be enriched in Del1 compared to the deeper layers of the tissue. Immunohistochemical staining of full-thickness articular cartilage with anti-Del1 antibodies also showed an enrichment of Del1 in the superficial zone. These observations are the first to describe the protein Del1 in a nonendothelial, nonfetal tissue.     

Volumetric changes of articular cartilage during stress relaxation in unconfined compression

The time-dependent lateral expansion and load relaxation of cartilage cylinders subjected to unconfined compression were simultaneously recorded. These measurements were used to (1) test the assumption of incompressibility for articular cartilage, (2) measure the Poisson's ratio of articular cartilage in compression and (3) investigate the relationship between stress relaxation and volumetric change. Mechanical tests were performed on fetal, calf, and adult humeral head articular cartilage. The instantaneous Poisson's ratio of adult cartilage was 0.49+/-0.08 (mean+S.D.), thus confirming the assumption of incompressibility for this tissue. The instantaneous Poisson's ratio was significantly lower for calf (0. 38+/-0.04) and fetal cartilage (0.36+/-0.04). The equilibrium Poisson's ratio, i.e. true Poisson's ratio of the solid matrix, was significantly higher for the adult tissue (0.26+/-0.11) compared to both the fetal (0.09+/-0.02) and calf (0.11+/-0.03) cartilage. A linear relationship between time-matched load and lateral expansion after the first minute of stress relaxation was observed.     

Osteoarthritic changes in the biphasic mechanical properties of the chondrocyte pericellular matrix in articular cartilage

The pericellular matrix (PCM) is a narrow region of cartilaginous tissue that surrounds chondrocytes in articular cartilage. Previous modeling studies indicate that the mechanical properties of the PCM relative to those of the extracellular matrix (ECM) can significantly affect the stress-strain, fluid flow, and physicochemical environments of the chondrocyte, suggesting that the PCM plays a biomechanical role in articular cartilage. The goals of this study were to measure the mechanical properties of the PCM using micropipette aspiration coupled with a linear biphasic finite element model, and to determine the alterations in the mechanical properties of the PCM with osteoarthritis (OA). Using a recently developed isolation technique, chondrons (the chondrocyte and its PCM) were mechanically extracted from non-degenerate and osteoarthritic human cartilage. The transient mechanical behavior of the PCM was well-described by a biphasic model, suggesting that the viscoelastic response of the PCM is attributable to flow-dependent effects, similar to that of the ECM. With OA, the mean Young's modulus of the PCM was significantly decreased (38.7+/-16.2 kPa vs. 23.5+/-12.9 kPa, p < 0.001), and the permeability was significantly elevated (4.19+/-3.78 x10(-17) m(4)/Ns vs. 10.2+/-9.38 x 10(-17) m(4)/Ns, p < 0.01). The Poisson's ratio was similar for both non-degenerate and OA PCM (0.044+/-0.063 vs. 0.030+/-0.068, p > 0.6). These findings suggest that the PCM may undergo degenerative processes with OA, similar to those occurring in the ECM. In combination with previous theoretical models of cell-matrix interactions in cartilage, our findings suggest that changes in the properties of the PCM with OA may have an important influence on the biomechanical environment of the chondrocyte.     

Dynamic response of immature bovine articular cartilage in tension and compression, and nonlinear viscoelastic modeling of the tensile response

Very limited information is currently available on the constitutive modeling of the tensile response of articular cartilage and its dynamic modulus at various loading frequencies. The objectives of this study were to (1) formulate and experimentally validate a constitutive model for the intrinsic viscoelasticity of cartilage in tension, (2) confirm the hypothesis that energy dissipation in tension is less than in compression at various loading frequencies, and (3) test the hypothesis that the dynamic modulus of cartilage in unconfined compression is dependent upon the dynamic tensile modulus. Experiment 1: Immature bovine articular cartilage samples were tested in tensile stress relaxation and cyclical loading. A proposed reduced relaxation function was fitted to the stress-relaxation response and the resulting material coefficients were used to predict the response to cyclical loading. Adjoining tissue samples were tested in unconfined compression stress relaxation and cyclical loading. Experiment 2: Tensile stress relaxation experiments were performed at varying strains to explore the strain-dependence of the viscoelastic response. The proposed relaxation function successfully fit the experimental tensile stress-relaxation response, with R2 = 0.970+/-0.019 at 1% strain and R2 = 0.992+/-0.007 at 2% strain. The predicted cyclical response agreed well with experimental measurements, particularly for the dynamic modulus at various frequencies. The relaxation function, measured from 2% to 10% strain, was found to be strain dependent, indicating that cartilage is nonlinearly viscoelastic in tension. Under dynamic loading, the tensile modulus at 10 Hz was approximately 2.3 times the value of the equilibrium modulus. In contrast, the dynamic stiffening ratio in unconfined compression was approximately 24. The energy dissipation in tension was found to be significantly smaller than in compression (dynamic phase angle of 16.7+/-7.4 deg versus 53.5+/-12.8 deg at 10(-3) Hz). A very strong linear correlation was observed between the dynamic tensile and dynamic compressive moduli at various frequencies (R2 = 0.908+/-0.100). The tensile response of cartilage is nonlinearly viscoelastic, with the relaxation response varying with strain. A proposed constitutive relation for the tensile response was successfully validated. The frequency response of the tensile modulus of cartilage was reported for the first time. Results emphasize that fluid-flow dependent viscoelasticity dominates the compressive response of cartilage, whereas intrinsic solid matrix viscoelasticity dominates the tensile response. Yet the dynamic compressive modulus of cartilage is critically dependent upon elevated values of the dynamic tensile modulus.     

Cartilage interstitial fluid load support in unconfined compression

Under physiological conditions of loading, articular cartilage is subjected to both compressive strains, normal to the articular surface, and tensile strains, tangential to the articular surface. Previous studies have shown that articular cartilage exhibits a much higher modulus in tension than in compression, and theoretical analyses have suggested that this tension–compression nonlinearity enhances the magnitude of interstitial fluid pressurization during loading in unconfined compression, above a theoretical threshold of 33% of the average applied stress. The first hypothesis of this experimental study is that the peak fluid load support in unconfined compression is significantly greater than the 33% theoretical limit predicted for porous permeable tissues modeled with equal moduli in tension and compression. The second hypothesis is that the peak fluid load support is higher at the articular surface side of the tissue samples than near the deep zone, because the disparity between the tensile and compressive moduli is greater at the surface zone. Ten human cartilage samples from six patellofemoral joints, and 10 bovine cartilage specimens from three calf patellofemoral joints were tested in unconfined compression. The peak fluid load support was measured at 79±11% and 69±15% at the articular surface and deep zone of human cartilage, respectively, and at 94±4% and 71±8% at the articular surface and deep zone of bovine calf cartilage, respectively. Statistical analyses confirmed both hypotheses of this study. These experimental results suggest that the tension–compression nonlinearity of cartilage is an essential functional property of the tissue which makes interstitial fluid pressurization the dominant mechanism of load support in articular cartilage.     

Determination of Poisson's ratio of articular cartilage by indentation using different-sized indenters

Articular cartilage is often characterized as an isotropic elastic material with no interstitial fluid flow during instantaneous and equilibrium conditions, and indentation testing commonly used to deduce material properties of Young's modulus and Poisson's ratio. Since only one elastic parameter can be deduced from a single indentation test, some other test method is often used to allow separate measurement of both parameters. In this study, a new method is introduced by which the two material parameters can be obtained using indentation tests alone, without requiring a secondary different type of test. This feature makes the method more suitable for testing small samples in situ. The method takes advantages of the finite layer effect. By indenting the sample twice with different-sized indenters, a nonlinear equation with the Poisson's ratio as the only unknown can be formed and Poisson's ratio obtained by solving the nonlinear equation. The method was validated by comparing the predicted Poisson's ratio for urethane rubber with the manufacturer's supplied value, and comparing the predicted Young's modulus for urethane rubber and an elastic foam material with modulii measured by unconfined compression. Anisotropic and nonhomogeneous finite-element (FE) models of the indentation were developed to aid in data interpretation. Applying the method to bovine patellar cartilage, the tissue Young's modulus was found to be 1.79 +/- 0.59 MPa in instantaneous response and 0.45 +/- 0.26 MPa in equilibrium, and the Poisson's ratio 0.503 +/- 0.028 and 0.463 +/- 0.073 in instantaneous and equilibrium, respectively. The equilibrium Poisson's ratio obtained in our work was substantially higher than those derived from biphasic indentation theory and those optically measured in an unconfined compression test. The finite element model results and examination of viscoelastic-biphasic models suggest this could be due to viscoelastic, inhomogeneity, and anisotropy effects.