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.     

Strain-rate dependent stiffness of articular cartilage in unconfined compression

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 reinfored 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.     

Anisotropic hydraulic permeability in compressed articular cartilage

The extent to which articular cartilage hydraulic permeability is anisotropic is largely unknown, despite its importance for understanding mechanisms of joint lubrication, load bearing, transport phenomena, and mechanotransduction. We developed and applied new techniques for the direct measurement of hydraulic permeability within statically compressed adult bovine cartilage explant disks, dissected such that disk axes were perpendicular to the articular surface. Applied pressure gradients were kept small to minimize flow-induced matrix compaction, and fluid outflows were measured by observation of a meniscus in a glass capillary under a microscope. Explant disk geometry under radially unconfined axial compression was measured by direct microscopic observation. Pressure, flow, and geometry data were input to a finite element model where hydraulic permeabilities in the disk axial and radial directions were determined. At less than 10% static compression, near free-swelling conditions, hydraulic permeability was nearly isotropic, with values corresponding to those of previous studies. With increasing static compression, hydraulic permeability decreased, but the radially directed permeability decreased more dramatically than the axially directed permeability such that strong anisotropy (a 10-fold difference between axial and radial directions) in the hydraulic permeability tensor was evident for static compression of 20-40%. Results correspond well with predictions of a previous microstructurally-based model for effects of tissue mechanical deformations on glycosaminoglycan architecture and cartilage hydraulic permeability. Findings inform understanding of structure-function relationships in cartilage matrix, and suggest several biomechanical roles for compression-induced anisotropic hydraulic permeability in articular cartilage.     

A finite deformation theory for cartilage and other soft hydrated connective tissues--I. Equilibrium results

The determination of valid stress-strain relations for articular cartilage under finite deformation conditions is a prerequisite for constructing models for synovial joint lubrication. Under physiological conditions of high strain rates and/or high stresses in the joint, large strains occur in cartilage. A finite deformation theory valid for describing cartilage, as well as other soft hydrated connective tissues under large loads, has been developed. This theory is based on the choice of a specific Helmholtz energy function which satisfies the generalized Coleman-Noll (GCN0) condition and the Baker-Ericksen (B-E) inequalities established in finite elasticity theory. In addition, the finite deformation biphasic theory includes the effects of strain-dependent porosity and permeability. These nonlinear effects are essential for properly describing the biomechanical behavior of articular cartilage, even when strain rates are low and strains are infinitesimal. The finite deformation theory describes the large strain behavior of cartilage observed in one-dimensional confined compression experiments at equilibrium, and it reduces to the linear biphasic theory under infinitesimal strain and slow strain rate conditions. Using this theory, we have determined the material coefficients of both human and bovine articular cartilages under large strain conditions at equilibrium. The theory compares very well with experimental results.     

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.     

Comparison of nonlinear mechanical properties of bovine articular cartilage and meniscus

Nonlinear, linear and failure properties of articular cartilage and meniscus in opposing contact surfaces are poorly known in tension. Relationships between the tensile properties of articular cartilage and meniscus in contact with each other within knee joints are also not known. In the present study, rectangular samples were prepared from the superficial lateral femoral condyle cartilage and lateral meniscus of bovine knee joints. Tensile tests were carried out with a loading rate of 5 mm/min until the tissue rupture. Nonlinear properties of the toe region, linear properties in larger strains, and failure properties of both tissues were analysed. The strain-dependent tensile modulus of the toe region, Young's modulus of the linear region, ultimate tensile stress and toughness were on average 98.2, 8.3, 4.0 and 1.9 times greater (p<0.05) for meniscus than for articular cartilage. In contrast, the toe region strain, yield strain and failure strain were on average 9.4, 3.1 and 2.3 times greater (p<0.05) for cartilage than for meniscus. There was a significant negative correlation between the strain-dependent tensile moduli of meniscus and articular cartilage samples within the same joints (r=−0.690, p=0.014). In conclusion, the meniscus possesses higher nonlinear and linear elastic stiffness and energy absorption capability before rupture than contacting articular cartilage, while cartilage has longer nonlinear region and can withstand greater strains before failure. These findings point out different load carrying demands that both articular cartilage and meniscus have to fulfil during normal physiological loading activities of knee joints.     

Load sharing between solid and fluid phases in articular cartilage: II--Comparison of experimental results and u-p finite element predictions.

Experimental measurements in conjunction with theoretical predictions were used to determine the extent of load supported by the fluid phase of cartilage at the articular surface. The u-p finite element model was used to simulate the loading of six separate porcine knee joints and to predict surface deformations of the cartilage layer on the lateral femoral condyle. Representative geometry for the condyle, contact pressures, and intrinsic material properties of the cartilage layer were supplied from experimental measures (see Part I). The u-p finite element predictions for surface deformations of the cartilage layer were obtained for several load partitioning states between the solid and fluid phases of cartilage at the articular surface. These were then compared to actual surface deformations obtained experimentally. It appeared from the comparison that approximately 75 percent of the applied load was borne by the fluid phase at the articular surface under this loading regime. This was qualitatively in agreement with the hypothesis that an applied load to articular joints is partitioned at the surface to the two phases according to the surface area ratios of the solid and fluid phases. It appeared that the solid phase was shielded from the total applied stress on the articular surface by the fluid and could be a reason for the excellent durability of the tissue under the demanding conditions in a diarthrodial joint.     

A transversely isotropic hyperelastic constitutive model of the PDL. Analytical and computational aspects

This study describes the development of a constitutive law for the modelling of the periodontal ligament (PDL) and its practical implementation into a commercial finite element code. The constitutive equations encompass the essential mechanical features of this biological soft tissue: non-linear behaviour, large deformations, anisotropy, distinct behaviour in tension and compression and the fibrous characteristics. The approach is based on the theory of continuum fibre-reinforced composites at finite strain where a compressible transversely isotropic hyperelastic strain energy function is defined. This strain energy density function is further split into volumetric and deviatoric contributions separating the bulk and shear responses of the material. Explicit expressions of the stress tensors in the material and spatial configurations are first established followed by original expressions of the elasticity tensors in the material and spatial configurations. As a simple application of the constitutive model, two finite element analyses simulating the mechanical behaviour of the PDL are performed. The results highlight the significance of integrating the fibrous architecture of the PDL as this feature is shown to be responsible for the complex strain distribution observed.     

A Transversely Isotropic Hyperelastic Constitutive Model of the PDL. Analytical and Computational Aspects

This study describes the development of a constitutive law for the modelling of the periodontal ligament (PDL) and its practical implementation into a commercial finite element code. The constitutive equations encompass the essential mechanical features of this biological soft tissue: non-linear behaviour, large deformations, anisotropy, distinct behaviour in tension and compression and the fibrous characteristics. The approach is based on the theory of continuum fibre-reinforced composites at finite strain where a compressible transversely isotropic hyperelastic strain energy function is defined. This strain energy density function is further split into volumetric and deviatoric contributions separating the bulk and shear responses of the material. Explicit expressions of the stress tensors in the material and spatial configurations are first established followed by original expressions of the elasticity tensors in the material and spatial configurations. As a simple application of the constitutive model, two finite element analyses simulating the mechanical behaviour of the PDL are performed. The results highlight the significance of integrating the fibrous architecture of the PDL as this feature is shown to be responsible for the complex strain distribution observed.     

Mechanical characterisation of in vivo human skin using a 3D force-sensitive micro-robot and finite element analysis

The complex mechanical properties of skin have been the subject of much study in recent years. Several experimental methods developed to measure the mechanical properties of skin in vivo, such as suction or torsion, are unable to measure skin’s anisotropic characteristics. An experiment characterising the mechanical properties of in vivo human skin using a novel force-sensitive micro-robot is presented. The micro-robot applied in-plane deformations to the anterior forearm and the posterior upper arm. The behaviour of the skin in each area is highly nonlinear, anisotropic, and viscoelastic. The response of the upper arm skin is very dependent on the orientation of the arm. A finite element model consisting of an Ogden strain energy function and quasi-linear viscoelasticity was developed to simulate the experiments. An orthogonal initial stress field, representing the in vivo skin tension, was used as an additional model parameter. The model simulated the experiments accurately with an error-of-fit of 17.5% for the anterior lower forearm area, 6.5% for the anterior upper forearm and 9.3% for the posterior upper arm. The maximum in vivo tension in each area determined by the model was 6.2 Nm⁻¹ in the anterior lower forearm, 11.4 Nm⁻¹ in anterior upper forearm and 5.6 Nm⁻¹ in the posterior upper arm. The results also show that a finite element model with a neo-Hookean strain energy function cannot simulate the experiments with the same accuracy.     

The influence of cyclic tension amplitude on chondrocyte matrix synthesis: experimental and finite element analyses

While not generally viewed as physiologically significant in articular cartilage, substantial tension can develop in fibrocartilage structures and in articular cartilage injuries. This study examined how different amplitudes of cyclic tension influence chondrocyte matrix synthesis. Bovine articular chondrocytes seeded in fibrin gels were loaded continuously for 48 hours at 1.0 Hz with displacements of 5%, 10%, or 20%. Protein and proteoglycan synthesis were measured by (3)H-proline and (35)S-sulfate incorporation, respectively. A poroelastic finite element model of the fibrin gel was developed to determine the strain distributions, hydrostatic pressures, and fluid velocities within the constructs at the various levels of displacement. Compared to unloaded controls, 10% and 20% displacements inhibited proteoglycan synthesis to the same extent, while 5% displacement had no effect. Tensile loading did not significantly affect protein synthesis. The finite element model predicted a wide range of strains and fluid velocities within the region of the gel analyzed for matrix synthesis, and the ranges overlapped for the different levels of displacement. These results indicate that the cyclic tension amplitude influences chondrocyte proteoglycan synthesis and that there may be a threshold in the response.     

An elasto-viscoplastic model to describe the ratcheting behavior of articular cartilage

In the present work, a constitutive model for articular cartilage is proposed in finite elasto-viscoplasticity. For simplification, articular cartilage is supposed to be a typical composite composed of a soft basis and a fiber assembly. The stress tensor and free energy function are hence accordingly divided into two components. The high nonlinear stress-strain response is assumed to be mainly related to the fiber assembly and described by an exponential-type hypoelastic relation. Ratcheting is considered according to the viscoplasticity, the evolution rule of which is deduced from the dissipative inequality by the co-directionality hypotheses. Then, the capability of the proposed model is validated by comparing its predictions with related experimental observations. Results show that the ratcheting behavior and stress-strain hysteresis loops are reasonably captured by the proposed model.     

Do changes in the mechanical properties of articular cartilage promote catabolic destruction of cartilage and osteoarthritis?

Osteoarthritis (OA) is a joint disease characterized by cartilage degeneration, a thickening of subchondral bone, and formation of marginal osteophytes. Previous mechanical characterization of cartilage in our laboratory suggests that energy storage and dissipation is reduced in osteoarthritis as the extent of fibrillation and fissure formation increases. It is not clear whether the loss of energy storage and dissipation characteristics is a result of biochemical and/or biophysical changes that occur to hyaline cartilage in joints. The purpose of this study is to present data, on the strain rate dependence of the elastic and viscous behaviors of cartilage, in order to further characterize changes that occur in the mechanical properties that are associated with OA. We have previously hypothesized that the changes seen in the mechanical properties of cartilage may be due to altered mechanochemical transduction by chondrocytes. Results of incremental tensile stress-strain tests at strain rates between 100%/min and 10,000%/min conducted on OA cartilage indicate that the slope of the elastic stress-strain curve increases with increasing strain rate, unlike the reported behavior of skin and self-assembled collagen fibers. It is suggested that the strain-rate dependence of the elastic stress-strain curve is due to the presence of large quantities of proteoglycans (PGs), which protect articular cartilage by increasing the apparent stiffness. The increased apparent stiffness of articular cartilage at high strain rates may limit the stresses borne and prolong the onset of OA. It is further hypothesized that increased compressive loading of chondrocytes in the intermediate zone of articular cartilage occurs as a result of normal wear to the superficial zone or from excessive impact loading. Once the superficial zone of articular cartilage is worn away, the tension is decreased throughout all cartilage zones leading to increased chondrocyte compressive loading and up-regulation of mechanochemical transduction processes that elaborate catabolic enzymes.     

A continuum theory and an experiment for the ion-induced swelling behavior of articular cartilage

Swelling of normal bovine articular cartilage equilibrated in NaCl solutions was dimensionally measured in thin strips of tissue. The ion-induced strains show that free swelling of articular cartilage is anisotropic and inhomogeneous. For the molar concentrations used, contraction increased linearly with concentration, defining a "coefficient of chemical contraction" (alpha c). Isometrically constrained specimens registered a rise in tensile force followed by stress relaxation. An extension of the biphasic theory incorporating this ion-induced strain is proposed. This theory can describe the equilibrium anisotropic swelling behavior of cartilage and explain the transient force history observed in the isometric experiment.     

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相似文献   

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.     

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.