Anisotropic strain-dependent material properties of bovine articular cartilage in the transitional range from tension to compression

Cartilage reshaping via in vitro mechanical loading

Shaped cartilage grafts can be used in the restoration of injured joints and the reconstruction of deformities of the head and neck. This study describes a novel method for altering cartilage shape, based on the hypothesis that mechanical loading coupled with in vitro tissue growth and remodeling facilitates tissue reshaping. Static bending deformations were imposed on strips of immature articular cartilage, and retention of the imposed shape and structural and biochemical measures of growth were assessed after 2, 4, and 6 days of incubation. The results show that mechanical reshaping of tissue is feasible, because shape retention was greater than 86% after 6 days of culture. The imposed mechanical deformations had little effect on measures of tissue viability or growth within the 6-day culture period. The addition of cycloheximide to the culture medium only slightly reduced the ability to reshape these tissues, but cycloheximide plus a lower culture temperature of 4 degrees C markedly inhibited the reshaping response. These results suggest a limited role for chondrocyte biosynthesis but a potentially important role for metabolic reactions in the cartilage matrix in the reshaping process. The ability to modulate cartilage shape in vitro may prove useful for tissue engineering of shaped cartilage grafts.

Measuring fixed charge density of goat articular cartilage using indentation methods and biochemical analysis

An important indicator of osteoarthritis (OA) progression is the loss of proteoglycan (PG) aggregates from the cartilage tissue. Using the indentation creep test, two analytical methods, as previously developed by Lu et al. [Lu, X. L., Miller, C., Chen, F. H., Guo, X. E., Mow, V. C., 2007. The generalized triphasic correspondence principle for simultaneous determination of the mechanical properties and proteoglycan content of articular cartilage by indentation. Journal of Biomechanics 40, 2434-2441 (EPub).], for predicting the fixed charge density (FCD) of goat knee articular cartilage in the normal (control) and degenerated states were compared: (1) a "dual-stage" method to calculate FCD from the mechanical properties of the tissue when tested in isotonic and hypertonic solutions; and (2) a "single-stage" method to predict FCD (as in (1)) assuming an intrinsic Poisson's ratio of 0.05 in the hypertonic state. A biochemical analysis using 1,9-dimethylmethylene blue (DMMB) assay was conducted to directly measure PG content, and hence FCD. The association between the FCD and the aggregate modulus of the tissue was also explored. The mean (+/-S.D.) FCD values measured using the dual-stage method were the closest (control: 0.129+/-0.039, degenerated: 0.046+/-029) to the DMMB results (control: 0.125+/-0.034, degenerated: 0.057+/-0.024) as compared to those of the single-stage method (control: 0.147+/-0.035, degenerated: 0.063+/-0.026). The single-stage method was more reliable (r(2)=0.81) when compared to the dual-stage method (r(2)=0.79). A prediction of FCD from the aggregate modulus generated the least reliable FCD prediction (r(2)=0.68). Because both the dual- and single-stage methods provided reliable FCD estimates for normal and degenerated tissue, the less time-consuming single-stage method was concluded to be the ideal technique for predicting FCD and hence PG content of the tissue.

A mechanical composite spheres analysis of engineered cartilage dynamics

In the preparation of bioengineered reparative strategies for damaged or diseased tissues, the processes of biomaterial degradation and neotissue synthesis combine to affect the developing mechanical state of multiphase, composite engineered tissues. Here, cell-polymer constructs for engineered cartilage have been fabricated by seeding chondrocytes within three-dimensional scaffolds of biodegradable polymers. During culture, synthetic scaffolds degraded passively as the cells assembled an extracellular matrix (ECM) composed primarily of glycosaminoglycan and collagen. Biochemical and biomechanical assessment of the composite (cells, ECM, and polymer scaffold) were modeled at a unit-cell level to mathematically solve stress-strain relationships and thus construct elastic properties (n=4 samples per seven time points). This approach employed a composite spheres, micromechanical analysis to determine bulk moduli of: (1) the cellular-ECM inclusion within the supporting scaffold structure; and (2) the cellular inclusion within its ECM. Results indicate a dependence of constituent volume fractions with culture time (p<0.05). Overall mean bulk moduli were variably influenced by culture, as noted for the cell-ECM inclusion (K(c-m)=29.7 kPa, p=0.1439), the cellular inclusion (K(c)=5.5 kPa, p=0.0067), and its surrounding ECM (K(m)=373.9 kPa, p=0.0748), as well as the overall engineered construct (K=165.0 kPa, p=0.6899). This analytical technique provides a framework to describe the time-dependent contribution of cells, accumulating ECM, and a degrading scaffold affecting bioengineered construct mechanical properties.

A nonlinear finite element model of cartilage growth

The long range objective of this work is to develop a cartilage growth finite element model (CGFEM), based on the theories of growing mixtures that has the capability to depict the evolution of the anisotropic and inhomogeneous mechanical properties, residual stresses, and nonhomogeneities that are attained by native adult cartilage. The CGFEM developed here simulates isotropic in vitro growth of cartilage with and without mechanical stimulation. To accomplish this analysis a commercial finite element code (ABAQUS) is combined with an external program (MATLAB) to solve an incremental equilibrium boundary value problem representing one increment of growth. This procedure is repeated for as many increments as needed to simulate the desired growth protocol. A case study is presented utilizing a growth law dependent on the magnitude of the diffusive fluid velocity to simulate an in vitro dynamic confined compression loading protocol run for 2 weeks. The results include changes in tissue size and shape, nonhomogeneities that develop in the tissue, as well as the variation that occurs in the tissue constitutive behavior from growth.

Equivalence between short-time biphasic and incompressible elastic material responses

Porous-permeable tissues have often been modeled using porous media theories such as the biphasic theory. This study examines the equivalence of the short-time biphasic and incompressible elastic responses for arbitrary deformations and constitutive relations from first principles. This equivalence is illustrated in problems of unconfined compression of a disk, and of articular contact under finite deformation, using two different constitutive relations for the solid matrix of cartilage, one of which accounts for the large disparity observed between the tensile and compressive moduli in this tissue. Demonstrating this equivalence under general conditions provides a rationale for using available finite element codes for incompressible elastic materials as a practical substitute for biphasic analyses, so long as only the short-time biphasic response is sought. In practice, an incompressible elastic analysis is representative of a biphasic analysis over the short-term response deltat<<Delta(2) / //parallelC(4)//K//, where Delta is a characteristic dimension, C(4) is the elasticity tensor, and K is the hydraulic permeability tensor of the solid matrix. Certain notes of caution are provided with regard to implementation issues, particularly when finite element formulations of incompressible elasticity employ an uncoupled strain energy function consisting of additive deviatoric and volumetric components.

Extraction of mechanical properties of articular cartilage from osmotic swelling behavior monitored using high frequency ultrasound

Articular cartilage is a biological weight-bearing tissue covering the bony ends of articulating joints. Negatively charged proteoglycan (PG) in articular cartilage is one of the main factors that govern its compressive mechanical behavior and swelling phenomenon. PG is nonuniformly distributed throughout the depth direction, and its amount or distribution may change in the degenerated articular cartilage such as osteoarthritis. In this paper, we used a 50 MHz ultrasound system to study the depth-dependent strain of articular cartilage under the osmotic loading induced by the decrease of the bathing saline concentration. The swelling-induced strains under the osmotic loading were used to determine the layered material properties of articular cartilage based on a triphasic model of the free-swelling. Fourteen cylindrical cartilage-bone samples prepared from fresh normal bovine patellae were tested in situ in this study. A layered triphasic model was proposed to describe the depth distribution of the swelling strain for the cartilage and to determine its aggregate modulus H(a) at two different layers, within which H(a) was assumed to be linearly dependent on the depth. The results showed that H(a) was 3.0+/-3.2, 7.0+/-7.4, 24.5+/-11.1 MPa at the cartilage surface, layer interface, and deep region, respectively. They are significantly different (p<0.01). The layer interface located at 70%+/-20% of the overall thickness from the uncalcified-calcified cartilage interface. Parametric analysis demonstrated that the depth-dependent distribution of the water fraction had a significant effect on the modeling results but not the fixed charge density. This study showed that high-frequency ultrasound measurement together with triphasic modeling is practical for quantifying the layered mechanical properties of articular cartilage nondestructively and has the potential for providing useful information for the detection of the early signs of osteoarthritis.

Articular cartilage mechanical and biochemical property relations before and after in vitro growth

The aim of this study was to design in vitro growth protocols that can comprehensively quantify articular cartilage structure-function relations via measurement of mechanical and biochemical properties. Newborn bovine patellofemoral groove articular cartilage explants were tested sequentially in confined compression (CC), unconfined compression (UCC), and torsional shear before (D0, i.e. day zero) and after (D14, i.e. day 14) unstimulated in vitro growth. The contents of collagen (COL), collagen-specific pyridinoline (PYR) crosslinks, glycosaminoglycan, and DNA significantly decreased during in vitro growth; consequently, a wide range of biochemical properties existed for investigating structure-function relations when pooling the D0 and D14 groups. All D0 mechanical properties were independent of compression strain while only Poisson's ratios were dependent on direction (i.e. anisotropic). Select D0 and D14 group mechanical properties were correlated with biochemical measures; including (but not limited to) results that CC/UCC moduli and UCC Poisson's ratios were correlated with COL and PYR. COL network weakening during in vitro growth due to reduced COL and PYR was accompanied by reduced CC/UCC moduli and increased UCC Poisson's ratios.

A bimodular polyconvex anisotropic strain energy function for articular cartilage

A strain energy function for finite deformations is developed that has the capability to describe the nonlinear, anisotropic, and asymmetric mechanical response that is typical of articular cartilage. In particular, the bimodular feature is employed by including strain energy terms that are only mechanically active when the corresponding fiber directions are in tension. Furthermore, the strain energy function is a polyconvex function of the deformation gradient tensor so that it meets material stability criteria. A novel feature of the model is the use of bimodular and polyconvex "strong interaction terms" for the strain invariants of orthotropic materials. Several regression analyses are performed using a hypothetical experimental dataset that captures the anisotropic and asymmetric behavior of articular cartilage. The results suggest that the main advantage of a model employing the strong interaction terms is to provide the capability for modeling anisotropic and asymmetric Poisson's ratios, as well as axial stress-axial strain responses, in tension and compression for finite deformations.

Compressive biomechanical properties of human nasal septal cartilage

BACKGROUND

Nasal septal cartilage is frequently used in nasal reconstruction and is a common source of chondrocytes for cartilage tissue engineering. The biomechanical properties of septal cartilage have yet to be fully defined and this limits the ability to compare it to the various alternative tissue-implant materials or tissue-engineered neocartilage. Given the unique structure and orientation of the septum within the nose, we sought to investigate anisotropic behaviors of septal cartilage in compression and correlate this to the concentration of glycosaminoglycans (GAG) and collagen within the cartilage.

METHODS

Human nasal septal cartilage specimens were tested in confined compression, with each sample analyzed in a medial orientation and also either a vertical or caudal-cephalic orientation, with the order of tests randomized. The equilibrium confined compression (aggregate) modulus, HAO, and the permeability, kp, at different offset compression levels were obtained for each compression test. After testing, the cartilage samples were solubilized, and the concentrations of GAG and collagen were obtained.

RESULTS

Forty-nine compression tests (24 medial, 12 vertical, 13 caudal-cephalic) were run on cartilage specimens obtained from 21 patients. There was a significant effect of orientation on compression modulus, HAO, with the vertical (0.7 +/- 0.12 MPa) and caudal-cephalic (0.66 +/- 0.01 MPa) orientations being significantly stiffer (p = 0.05) than the medial orientation (0.44 +/- 0.04 MPa). There was a trend of an orientation effect on kp at 15% offset compression (p = 0.12) and a borderline significant effect of orientation on kp at 30% offset compression (p = 0.05), demonstrating the M orientation to be more permeable than both the vertical and caudal-cephalic orientations. Both univariate and multivariate analysis did not demonstrate a significant effect of order of compression, age, gender, thickness, dry/wet weight, GAG, or collagen on either HAO, or kp values (p > 0.05).

CONCLUSION

This study provides new information on the compressive properties of septal cartilage along different axes of compression. The results demonstrate that human septal cartilage is anisotropic; the compressive stiffness is higher in the vertical and caudal-cephalic orientations than in the medial orientation. Additionally, the medial orientation tends to have the greatest permeability. The data obtained in this study provide a reference to which various craniofacial reconstruction materials and tissue-engineered neocartilage can be compared.

On the theory of reactive mixtures for modeling biological growth

Mixture theory, which can combine continuum theories for the motion and deformation of solids and fluids with general principles of chemistry, is well suited for modeling the complex responses of biological tissues, including tissue growth and remodeling, tissue engineering, mechanobiology of cells and a variety of other active processes. A comprehensive presentation of the equations of reactive mixtures of charged solid and fluid constituents is lacking in the biomechanics literature. This study provides the conservation laws and entropy inequality, as well as interface jump conditions, for reactive mixtures consisting of a constrained solid mixture and multiple fluid constituents. The constituents are intrinsically incompressible and may carry an electrical charge. The interface jump condition on the mass flux of individual constituents is shown to define a surface growth equation, which predicts deposition or removal of material points from the solid matrix, complementing the description of volume growth described by the conservation of mass. A formulation is proposed for the reference configuration of a body whose material point set varies with time. State variables are defined which can account for solid matrix volume growth and remodeling. Constitutive constraints are provided on the stresses and momentum supplies of the various constituents, as well as the interface jump conditions for the electrochemical potential of the fluids. Simplifications appropriate for biological tissues are also proposed, which help reduce the governing equations into a more practical format. It is shown that explicit mechanisms of growth-induced residual stresses can be predicted in this framework.

Experimental–numerical analysis of minipig's multi-rooted teeth

The paper pertains to the analysis of the biomechanical behaviour of the periodontal ligament (PDL) by using a combined experimental and numerical approach. Experimental analysis provides information about a two-rooted pig premolar tooth in its socket with regard to morphological configuration and deformational response. The numerical analysis developed for the present investigation adopts a specific anisotropic hyperelastic formulation, accounting for tissue structural arrangement. The parameters to be adopted for the PDL constitutive model are evaluated with reference to data deducted from experimental in vitro tests on different specimens taken from literature. According to morphometric data relieved, solid models are provided as basis for the development of numerical models that adopt the constitutive formulation proposed. A reciprocal validation of experimental and numerical data allows for the evaluation of reliability of results obtained. The work is intended as preliminary investigation to study the correlation between mechanical status of PDL and induction to cellular activity in orthodontic treatments.

A biphasic viscohyperelastic fibril-reinforced model for articular cartilage: Formulation and comparison with experimental data

Experiments in articular cartilage have shown highly nonlinear stress-strain curves under finite deformations, nonlinear tension-compression response as well as intrinsic viscous effects of the proteoglycan matrix and the collagen fibers. A biphasic viscohyperelastic fibril-reinforced model is proposed here, which is able to describe the intrinsic viscoelasticity of the fibrillar and nonfibrillar components of the solid phase, the nonlinear tension-compression response and the nonlinear stress-strain curves under tension and compression. A viscohyperelastic constitutive equation was used for the matrix and the fibers encompassing, respectively, a hyperelastic function used previously for the matrix and a hyperelastic law used before to represent biological connective tissues. This model, implemented in an updated Lagrangian finite element code, displayed good ability to follow experimental stress-strain equilibrium curves under tension and compression for human humeral cartilage. In addition, curve fitting of experimental reaction force and lateral displacement unconfined compression curves showed that the inclusion of viscous effects in the matrix allows the description of experimental data with material properties for the fibers consistent with experimental tensile tests, suggesting that intrinsic viscous effects in the matrix of articular cartilage plays an important role in the mechanical response of the tissue.

Mechanical strains induced in osteoblasts by use of point femtosecond laser targeting

A study demonstrating how ultrafast laser radiation stimulates osteoblasts is presented. The study employed a custom made optical system that allowed for simultaneous confocal cell imaging and targeted femtosecond pulse laser irradiation. When femtosecond laser light was focused onto a single cell, a rise in intracellular Ca²⁺ levels was observed followed by contraction of the targeted cell. This contraction caused deformation of neighbouring cells leading to a heterogeneous strain field throughout the monolayer. Quantification of the strain fields in the monolayer using digital image correlation revealed local strains much higher than threshold values typically reported to stimulate extracellular bone matrix production in vitro. This use of point targeting with femtosecond pulse lasers could provide a new method for stimulating cell activity in orthopaedic tissue engineering.

Mitochondrial dynamics in chondrocytes and their connection to the mechanical properties of the cytoplasm

BACKGROUND

The motion and redistribution of intracellular organelles is a fundamental process in cells. Organelle motion is a complex phenomenon that depends on a large number of variables including the shape of the organelle, the type of motors with which the organelles are associated, and the mechanical properties of the cytoplasm. This paper presents a study that characterizes the diffusive motion of mitochondria in chondrocytes seeded in agarose constructs and what this implies about the mechanical properties of the cytoplasm.

METHOD OF APPROACH

Images showing mitochondrial motion in individual cells at 30 s intervals for 15 min were captured with a confocal microscope. Digital image correlation was used to quantify the motion of the mitochondria, and the mean square displacement (MSD) was calculated. Statistical tools for testing whether the characteristic motion of mitochondria varied throughout the cell were developed. Calculations based on statistical mechanics were used to establish connections between the measured MSDs and the mechanical nature of the cytoplasm.

RESULTS

The average MSD of the mitochondria varied with time according to a power law with the power term greater than 1, indicating that mitochondrial motion can be viewed as a combination of diffusion and directional motion. Statistical analysis revealed that the motion of the mitochondria was not uniform throughout the cell, and that the diffusion coefficient may vary by over 50%, indicating intracellular heterogeneity. High correlations were found between movements of mitochondria when they were less than 2 microm apart. The correlation is probably due to viscoelastic properties of the cytoplasm. Theoretical analysis based on statistical mechanics suggests that directed diffusion can only occur in a material that behaves like a fluid on large time scales.

CONCLUSIONS

The study shows that mitochondria in different regions of the cell experience different characteristic motions. This suggests that the cytoplasm is a heterogeneous viscoelastic material. The study provides new insight into the motion of mitochondria in chondrocytes and its connection with the mechanical properties of the cytoplasm.

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.

Constitutive Modeling of Cartilaginous Tissues: A Review

An important and longstanding field of research in orthopedic biomechanics is the elucidation and mathematical modeling of the mechanical response of cartilaginous tissues. Traditional approaches have treated such tissues as continua and have described their mechanical response in terms of macroscopic models borrowed from solid mechanics. The most important of such models are the biphasic and single-phase viscoelastic models, and the many variations thereof. These models have reached a high level of maturity and have been successful in describing a wide range of phenomena. An alternative approach that has received considerable recent interest, both in orthopedic biomechanics and in other fields, is the description of mechanical response based on consideration of a tissue's structure—so-called microstructural modeling. Examples of microstructurally based approaches include fibril-reinforced biphasic models and homogenization approaches. A review of both macroscopic and microstructural constitutive models is given in the present work.

The effect of finite compressive strain on chondrocyte viability in statically loaded bovine articular cartilage

Recent studies have reported that certain regimes of compressive loading of articular cartilage result in increased cell death in the superficial tangential zone (STZ). The objectives of this study were (1) to test the prevalent hypothesis that preferential cell death in the STZ results from excessive compressive strain in that zone, relative to the middle and deep zones, by determining whether cell death correlates with the magnitude of compressive strain and (2) to test the corollary hypothesis that the viability response of cells is uniform through the thickness of the articular layer when exposed to the same loading environment. Live cartilage explants were statically compressed by approximately 65% of their original thickness, either normal to the articular surface (axial loading) or parallel to it (transverse loading). Cell viability after 12 h was compared to the local strain distribution measured by digital image correlation. Results showed that the strain distribution in the axially loaded samples was highest in the STZ (77%) and lowest in the deep zone (55%), whereas the strain was uniformly distributed in the transversely loaded samples (64%). In contrast, axially and transversely loaded samples exhibited very similar profiles of cell death through the depth, with a preferential distribution in the STZ. Unloaded control samples showed negligible cell death. Thus, under prolonged static loading, depth-dependent variations in chondrocyte death did not correlate with the local depth-dependent compressive strain, and the prevalent hypothesis must be rejected. An alternative hypothesis, suggested by these results, is that superficial zone chondrocytes are more vulnerable to prolonged static loading than chondrocytes in the middle and deep zones.

Spatially varying material properties of the rat caudal intervertebral disc

STUDY DESIGN

The use of a microscopy based material testing technique to assess the local material properties of rat caudal intervertebral discs under uniaxial compression.

OBJECTIVES

To better understand the cell environment of rat caudal intervertebral discs during mechanical loading and elucidate better the role of the nucleus pulposus to the overall disc material properties.

SUMMARY OF BACKGROUND DATA

Rat tail models of disc degeneration have been widely used for their similarity with the degeneration phenomena in human beings. Degenerative patterns in the disc are often inhomogeneous, however, only average material properties of rodent discs have been studied. Knowledge of the spatially varying properties within the disc is necessary to understand the disc cell milieu during tissue loading.

METHODS

Rat caudal motion segments were tested intact, sectioned, and with alterations of nucleus pulposus using microscopy based techniques. Local displacements and strains were obtained using digital image correlation. Strains and load measurements were used to get the average apparent Young's modulus, peak stress, local Young's modulus, and local Poisson's ratio.

RESULTS

There was no difference observed in the average apparent Young's modulus among experimental groups. Peak stresses decreased significantly when the nucleus pulposus was replaced with extremely fluid-like materials. The axial displacement field showed 3 distinct linear distributions in samples which were sectioned. The center region in all groups had significantly smaller axial strain and showed a higher local Young's modulus.

CONCLUSIONS

The average equilibrium Young's modulus may be dependent on short-range ultrastructural organization. Spatially varying material properties within the intervertebral disc may be caused by orientation of fiber bundles in the different regions of the anulus fibrosus. The fiber bundles are better able to resist compressive loads when oriented parallel rather than perpendicular to the loading direction. At equilibrium, the anulus fibrosus also appears to have a shielding effect independent of the material filling up the nucleus pulposus space.

Analysis of the mechanical behavior of chondrocytes in unconfined compression tests for cyclic loading

Experimental evidence indicates that the biosynthetic activity of chondrocytes is associated with the mechanical environment. For example, excessive, repetitive loading has been found to induce cell death, morphological and cellular damage, as seen in degenerative joint disease, while cyclic, physiological-like loading has been found to trigger a partial recovery of morphological and ultrastructural aspects in osteoarthritic human articular chondrocytes. Mechanical stimuli are believed to influence the biosynthetic activity via the deformation of cells. However, the in situ deformation of chondrocytes for cyclic loading conditions has not been investigated experimentally or theoretically. The purpose of the present study was to simulate the mechanical response of chondrocytes to cyclic loading in unconfined compression tests using a finite element model. The material properties of chondrocytes and extracellular matrix were considered to be biphasic. The time-histories of the shape and volume variations of chondrocytes at three locations (i.e., surface, center, and bottom) within the cartilage were predicted for static and cyclic loading conditions at two frequencies (0.02 and 0.1 Hz) and two amplitudes (0.1 and 0.2 MPa). Our results show that cells at different depths within the cartilage deform differently during cyclic loading, and that the depth dependence of cell deformation is influenced by the amplitude of the cyclic loading. Cell deformations under cyclic loading of 0.02 Hz were found to be similar to those at 0.1 Hz. We conclude from the simulation results that, in homogeneous cartilage layers, cell deformations are location-dependent, and further are affected by load magnitude. In physiological conditions, the mechanical environment of cells are even more complex due to the anisotropy, depth-dependent inhomogeneity, and tension-compression non-linearity of the cartilage matrix. Therefore, it is feasible to speculate that biosynthetic responses of chondrocytes to cyclic loading depend on cell location and load magnitude.

Depth-dependent Compressive Equilibrium Properties of Articular Cartilage Explained by its Composition

For this study, we hypothesized that the depth-dependent compressive equilibrium properties of articular cartilage are the inherent consequence of its depth-dependent composition, and not the result of depth-dependent material properties. To test this hypothesis, our recently developed fibril-reinforced poroviscoelastic swelling model was expanded to include the influence of intra- and extra-fibrillar water content, and the influence of the solid fraction on the compressive properties of the tissue. With this model, the depth-dependent compressive equilibrium properties of articular cartilage were determined, and compared with experimental data from the literature. The typical depth-dependent behavior of articular cartilage was predicted by this model. The effective aggregate modulus was highly strain-dependent. It decreased with increasing strain for low strains, and increases with increasing strain for high strains. This effect was more pronounced with increasing distance from the articular surface. The main insight from this study is that the depth-dependent material behavior of articular cartilage can be obtained from its depth-dependent composition only. This eliminates the need for the assumption that the material properties of the different constituents themselves vary with depth. Such insights are important for understanding cartilage mechanical behavior, cartilage damage mechanisms and tissue engineering studies.

A Bimodular Theory for Finite Deformations: Comparison of Orthotropic Second-order and Exponential Stress Constitutive Equations for Articular Cartilage

Cartilaginous tissues, such as articular cartilage and the annulus fibrosus, exhibit orthotropic behavior with highly asymmetric tensile-compressive responses. Due to this complex behavior, it is difficult to develop accurate stress constitutive equations that are valid for finite deformations. Therefore, we have developed a bimodular theory for finite deformations of elastic materials that allows the mechanical properties of the tissue to differ in tension and compression. In this paper, we derive an orthotropic stress constitutive equation that is second-order in terms of the Biot strain tensor as an alternative to traditional exponential type equations. Several reduced forms of the bimodular second-order equation, with six to nine parameters, and a bimodular exponential equation, with seven parameters, were fit to an experimental dataset that captures the highly asymmetric and orthotropic mechanical response of cartilage. The results suggest that the bimodular second-order models may be appealing for some applications with cartilaginous tissues.

Chondrocyte deformation induces mitochondrial distortion and heterogeneous intracellular strain fields

Chondrocyte mechanotransduction is poorly understood but may involve cell deformation and associated distortion of intracellular structures and organelles. This study quantifies the intracellular displacement and strain fields associated with chondrocyte deformation and in particular the distortion of the mitochondria network, which may have a role in mechanotransduction. Isolated articular chondrocytes were compressed in agarose constructs and simultaneously visualised using confocal microscopy. An optimised digital image correlation technique was developed to calculate the local intracellular displacement and strain fields using confocal images of fluorescently labelled mitochondria. The mitochondria formed a dynamic fibrous network or reticulum, which co-localised with microtubules and vimentin intermediate filaments. Cell deformation induced distortion of the mitochondria, which collapsed in the axis of compression with a resulting loss of volume. Compression generated heterogeneous intracellular strain fields indicating mechanical heterogeneity within the cytoplasm. The study provides evidence supporting the potential involvement of mitochondrial deformation in chondrocyte mechanotransduction, possibly involving strain-mediated release of reactive oxygen species. Furthermore the heterogeneous strain fields, which appear to be influenced by intracellular structure and organisation, may generate significant heterogeneity in mechanotransduction behaviour for cells subjected to identical levels of deformation.

Effect of Compressive Strain on Cell Viability in Statically Loaded Articular Cartilage

Physiological loading of articulating joints is necessary for normal cartilage function. However, conditions of excessive overloading or trauma can cause cartilage injury resulting in matrix damage and cell death. The objective of this study was to evaluate chondrocyte viability within mechanically compressed articular cartilage removed from immature and mature bovine knees. Twenty-three mature and 68 immature cartilage specimens were subjected to static uniaxial confined-creep compressions of 0-70% and the extent of cell death was measured using fluorescent microscopic imaging. In both age groups, cell death was always initiated at the articular surface and increased linearly in depth with increasing strain magnitude. However, most of the cell death was localized within the superficial zone (SZ) of the cartilage matrix with the depth never greater than approximately 500 microm or 25% of the thickness of the test specimen. The immature cartilage was found to have a significantly greater (> 2 times) amount (depth) of cell death compared to the mature cartilage, especially at the higher strains. This finding was attributed to the lower compressive modulus of the immature cartilage in the SZ compared to that of the mature cartilage, resulting in a greater local matrix strain and concomitant cell surface membrane strain in this zone when the matrix was compressed. These results provide further insight into the capacity of articular cartilage in different age groups to resist the severity of traumatic injury from compressive loads.

Analysis of articular cartilage as a composite using nonlinear membrane elements for collagen fibrils

To develop a composite fibre-reinforced model of the cartilage, membrane shell elements were introduced to represent collagen fibrils reinforcing the isotropic porous solid matrix filled with fluid. Nonlinear stress-strain curve of pure collagen fibres and collagen volume fraction were explicitly presented in the formulation of these membrane elements. In this composite model, in accordance with tissue structure, the matrix and fibril membrane network experienced dissimilar stresses despite identical strains in the fibre directions. Different unconfined compression and indentation case studies were performed to determine the distinct role of membrane collagen fibrils in nonlinear poroelastic mechanics of articular cartilage. The importance of nonlinear fibril membrane elements in the tissue relaxation response as well as in temporal and spatial variations of pore pressure and solid matrix stresses was demonstrated. By individual adjustments of the collagen volume fraction and collagen mechanical properties, the model allows for the simulation of alterations in the fibril network structure of the tissue towards modelling damage processes or repair attempts. The current model, which is based on a physiological description of the tissue structure, is promising in improvement of our understanding of the cartilage pathomechanics.

Stresses and Strains in the Medial Meniscus of an ACL Deficient Knee under Anterior Loading: A Finite Element Analysis with Image-Based Experimental Validation

The menisci are believed to play a stabilizing role in the ACL-deficient knee, and are known to be at risk for degradation in the chronically unstable knee. Much of our understanding of this behavior is based on ex vivo experiments or clinical studies in which we must infer the function of the menisci from external measures of knee motion. More recently, studies using magnetic resonance (MR) imaging have provided more clear visualization of the motion and deformation of the menisci within the tibio-femoral articulation. In this study, we used such images to generate a finite element model of the medial compartment of an ACL-deficient knee to reproduce the meniscal position under anterior loads of 45, 76, and 107N. Comparisons of the model predictions to boundaries digitized from images acquired in the loaded states demonstrated general agreement, with errors localized to the anterior and posterior regions of the meniscus, areas in which large shear stresses were present. Our model results suggest that further attention is needed to characterize material properties of the peripheral and horn attachments. Although overall translation of the meniscus was predicted well, the changes in curvature and distortion of the meniscus in the posterior region were not captured by the model, suggesting the need for refinement of meniscal tissue properties.

Direct measurement of osmotic pressure of glycosaminoglycan solutions by membrane osmometry at room temperature

Articular cartilage is a hydrated soft tissue composed of negatively charged proteoglycans fixed within a collagen matrix. This charge gradient causes the tissue to imbibe water and swell, creating a net osmotic pressure that enhances the tissue's ability to bear load. In this study we designed and utilized an apparatus for directly measuring the osmotic pressure of chondroitin sulfate, the primary glycosaminoglycan found in articular cartilage, in solution with varying bathing ionic strength (0.015 M, 0.15 M, 0.5 M, 1 M, and 2 M NaCl) at room temperature. The osmotic pressure (pi) was found to increase nonlinearly with increasing chondroitin sulfate concentration and decreasing NaCl ionic bath environment. Above 1 M NaCl, pi changes negligibly with further increases in salt concentration, suggesting that Donnan osmotic pressure is negligible above this threshold, and the resulting pressure is attributed to configurational entropy. Results of the current study were also used to estimate the contribution of osmotic pressure to the stiffness of cartilage based on theoretical and experimental considerations. Our findings indicate that the osmotic pressure resulting from configurational entropy is much smaller in cartilage (based on an earlier study on bovine articular cartilage) than in free solution. The rate of change of osmotic pressure with compressive strain is found to contribute approximately one-third of the compressive modulus (H(A)(eff)) of cartilage (Pi approximately H(A)(eff)/3), with the balance contributed by the intrinsic structural modulus of the solid matrix (i.e., H(A) approximately 2H(A)(eff)/3). A strong dependence of this intrinsic modulus on salt concentration was found; therefore, it appears that proteoglycans contribute structurally to the magnitude of H(A), in a manner independent of osmotic pressure.

Indentation testing of human articular cartilage: effects of probe tip geometry and indentation depth on intra-tissue strain

Experimental determination of intra-tissue deformation during clinically applicable rapid indentation testing would be useful for understanding indentation biomechanics and for designing safe indentation probes and protocols. The objectives of this study were to perform two-dimensional (2-D) indentation tests, using indenters and protocols that are analogous to those in clinically oriented probes, of normal adult-human articular cartilage in order to determine: (1) intra-tissue strain maps and regions of high strain magnitude, and (2) the effects on strain of indenter geometry (rectangular prismatic and cylindrical) and indentation depth (40-190 microm). Epifluorescence microscopy of samples undergoing indentation and subsequent video image correlation analysis allowed determination of strain maps. Regions of peak strain were near the "edges" of indenter contact with the cartilage surface, and the strain magnitude in these regions ranged from approximately 0.05 to approximately 0.30 in compression and shear, a range with known biological consequences. With increasing indentation displacement, strain magnitudes generally increased in all regions of the tissue. Compared to indentation using a rectangular prismatic tip, indentation with a cylindrical tip resulted in slightly higher peak strain magnitudes while influencing a smaller region of cartilage. These results may be used to refine clinical indenters and indentation protocols.