An analytical model of joint contact

The stress distribution in the region of contact between a layered elastic sphere and a layered elastic cavity is determined using an analytical model to stimulate contact of articulating joints. The purpose is to use the solution to analyze the effects of cartilage thickness and stiffness, bone stiffness and joint curvature on the resulting stress field, and investigate the possibility of cracking of the material due to tensile and shear stresses. Vertical cracking of cartilage as well as horizontal splitting at the cartilage-calcified cartilage interface has been observed in osteoarthritic joints. The current results indicate that for a given system (material properties mu and nu constant), the stress distribution is a function of the ratio of contact radius to layer thickness (a/h), and while tensile stresses are seen to occur only when a/h is small, tensile strain is observed for all a/h values. Significant shear stresses are observed at the cartilage-bone interface. Softening of cartilage results in an increase in a/h, and a decrease in maximum normal stress. Cartilage thinning increases a/h and the maximum contact stress, while thickening has the opposite effect. A reduction in the indenting radius reduces a/h and increases the maximum normal stress. Bone softening is seen to have negligible effect on the resulting contact parameters and stress distribution.     

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.     

Residual periosteum tension is insufficient to directly modulate bone growth

Periosteal incision is one of the less severe interventions used to correct mild long bone growth pathologies. The mechanism responsible for this growth modulation is still unclear. A generally adopted hypothesis is that incision releases compressive force created by tensioned periosteum. We set out to evaluate the feasibility of this hypothesis by quantifying the stress level imposed on cartilage by periosteum tension in the rapid growth phase of chick embryos and evaluating if tension release could be responsible for modulating growth.Residual force in embryonic periosteum was measured in a tensile tester. A finite element model was developed, based on geometry determined using optical projection tomography in combination with histology. This model was then used to calculate the stress-distribution throughout the cartilage imposed by the periosteum force and to evaluate its possible contribution in modulating growth.Residual periosteal force in e17 chick tibiotarsi resulted in compressive stresses of 6 kPa in the proliferative zone and tensile stresses up to 9 kPa in the epiphyseal cartilage. Based on the literature, these compressive stresses are estimated to reduce growth rates by 1.1% and calculated tensile stresses increase growth rates by 1.7%. However, growth rate modulations between 8% and 28% are reported in the literature upon periosteum release.We therefore conclude that the increased growth, initiated by periosteal incision, is unlikely to be predominantly the result of mechanical release of cartilage compression by periosteum tension. However, increased epiphyseal growth rates due to periosteal tension, may contribute to bone morphogenesis by widening the epiphysis.     

Load transmission in the nasofrontal suture of the pig, Sus scrofa

The nasofrontal suture links the nasal complex with the braincase and is subject to compressive strain during mastication and (theoretically) tensile strain during growth of nasal soft tissues. The suture's ability to transmit compressive and tensile loads therefore affects both cranioskeletal stress distribution and growth. This study investigated the in vitro viscoelastic and failure properties of the nasofrontal suture in the pig, Sus scrofa. Suture specimens from two ages were tested in compression and tension and at fast and slow rates. In additional specimens, strain gauges were applied to the suture and nasal bone for strain measurement during testing. Relaxation testing demonstrated higher elastic moduli in tension than compression, regardless of test rate or pig age. In contrast, maximum elastic moduli from failure tests, as well as peak stresses, were significantly higher in compression than in tension. Sutures from older pigs tended to have higher elastic moduli and peak stresses, significantly so for tensile relaxation moduli. Strain gauge results showed that deformation at the suture was much greater than that of the nasal bone. These data demonstrate the viscoelasticity and deformability of the nasofrontal sutural ligament. The suture achieved maximal resistance to tensile deformation at low loads, corresponding with the low tensile loads likely to occur during growth of nasal soft tissues. In contrast, the maximal stiffness in compression at high loads indicates that the suture functions with a substantial safety factor during mastication.     

An approach for the stress analysis of transversely isotropic biphasic cartilage under impact load.

Stress analysis of contact models for isotropic articular cartilage under impacting loads shows high shear stresses at the interface with the subchondral bone and normal compressive stresses near the surface of the cartilage. These stress distributions are not consistent, with lesions observed on the cartilage surface of rabbit patellae from blunt impact, for example, to the patello-femoral joint. The purpose of the present study was to analyze, using the elastic capabilities of a finite element code, the stress distribution in more morphologically realistic transversely isotropic biphasic contact models of cartilage. The elastic properties of an incompressible material, equivalent to those of the transversely isotropic biphasic material at time zero, were derived algebraically using stress-strain relations. Results of the stress analysis showed the highest shear stresses on the surface of the solid skeleton of the cartilage and tensile stresses in the zone of contact. These results can help explain the mechanisms responsible for surface injuries observed during blunt insult experiments.     

The effects of damage and microcracking on the impact strength of bone

Microcracking has been shown to occur when bone is 'damaged' as shown by a loss of stiffness. The effect on bone's toughness of the types of damage produced at low losses of stiffness are not known. We loaded bovine bone specimens in bending and tension to stiffness losses of up to 27%, and examined the microcracking produced. The tensile specimens had diffuse arrays of microcracks of 2-20 microm in length, characteristic of tensile loading, on all surfaces. The bending specimens showed tensile microcracking on the tensile surface and characteristic long, straight, cross-hatched compression cracks on the compressive surface. Specimens were then broken in impact. Those that had been damaged in bending were divided into two groups, in one group the part of the specimen which had undergone compression damage was placed in tension, and in the other group the tensile damage was placed in tension. Tensile damage loaded in tension did not reduce the bone's energy-absorbing ability in impact until a modulus reduction of over 20%. However compression damage loaded in tension did severely reduce the bone's energy absorption capabilities (by an average of about 40%).     

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.     

Nonlinear tensile properties of bovine articular cartilage and their variation with age and depth

Tensile stiffness of articular cartilage is much greater than its compressive stiffness and plays an essential role even in compressive properties by increasing transient fluid pressures during physiological loading. Recent studies of nonlinear properties of articular cartilage in compression revealed several physiologically pertinent nonlinear behaviors, all of which required that cartilage tensile stiffness increase significantly with stretch. We therefore performed sequences of uniaxial tension tests on fresh bovine articular cartilage slices using a protocol that allowed several hours to attain equilibrium and measured longitudinal and transverse tissue strain. By testing bovine cartilage from different ages (6 months to 6 years) we found that equilibrium and transient tensile modulus increased significantly with maturation and age, from 0 to 15 MPa at equilibrium and from 10 to 28 MPa transiently. Our results indicate that cartilage stiffens with age in a manner similar to other highly hydrated connective tissues, possibly due to age-dependent content of enzymatic and nonenzymatic collagen cross links. The long relaxation period used in our tests (5-10 hours) was necessary in order to attain equilibrium and avoid a very significant overestimation of equilibrium modulus that occurs when much shorter times are used (15-30 minutes). We also found that equilibrium and transient tensile modulus increased nonlinearly when cartilage is stretched from 0 to 10% strain without any previous tare load. Although our results estimate a nonlinear increase in tensile stiffness with stretch that is an order of magnitude lower than that required to predict nonlinear properties in compression, they are in agreement with previous results from other uniaxial tension tests of collagenous materials. We therefore speculate that biaxial tensile moduli may be much higher and thereby more compatible with observed nonlinear compressive properties.     

Zonal and directional variations in tensile properties of bovine articular cartilage with special reference to strain rate variation

THE AIMS of this study were: (i) to investigate the variation in the tensile properties of articular cartilage with depth through cartilage thickness and fibre orientation; (ii) to determine the effect of strain rate on tensile properties of articular cartilage. MATERIALS AND METHOD: All experimental work was performed on cartilage specimens taken from two bovine knee joints. Osteochondral plugs 12 mm in diameter were harvested with a special reamer from the femur and the tibial plateaux of each knee. Slices (0.2 mm thick), of articular cartilage were cut from the plug with a microtome. The predominant orientation of the collagen fibres on the cartilage surface was determined using the pinpricking technique. Each specimen used for the tensile test was cut, so as to produce a dumbbell shape, with a gauge length of 6 mm. Uniaxial tensile tests were performed on each specimen in order to determine the tensile Young's modulus, and ultimate tensile strength (UTS). In this investigation, these tensile tests were carried out at different strain rate: 1, 20, 50 and 70%/sec. RESULTS: As regards the zonal properties, it was found that tensile stiffness was greater in the superficial layer than in deep layer. However, a few specimens from the deep layer displayed similar or greater stiffness compared to the superficial layer. With respect to the directional properties, the specimens oriented parallel to the predominant alignment of collagen, were stiffer than those, which were perpendicular to it in each layer. However, only the results regarding the deep layer can be considered statistically significant. In regard to the variation of modulus with the strain-rate, the results showed that there is no significant increase of the modulus with increasing strain rate from 20 to 50% per second. However, at 70% per second, articular cartilage stiffness considerably increased by up to one order of magnitude greater than that determined at lower strain rates in both the superficial and deep layer. Moreover, the UTS of cartilage specimens tested at 70% per second showed a significant rise, reaching values of four to five times that of those measured at 1, 20 or 50% per second. CONCLUSION: The steep increases in both the stiffness and ultimate tensile strength of cartilage at high strain rates point to the existence in cartilage of a mechanism for its protection from damage by stresses arising in trauma, which are usually applied at high rates. This mechanism needs to be elucidated. The reduced anisotropy found in the present study pointed out that collagen is likely to be less organized in bovine cartilage than in the human and therefore, a study of its ultra-structure would be appropriate.     

Anisotropy, inhomogeneity, and tension-compression nonlinearity of human glenohumeral cartilage in finite deformation

The tensile and compressive properties of human glenohumeral cartilage were determined by testing 120 rectangular strips in uniaxial tension and 70 cylindrical plugs in confined compression, obtained from five human glenohumeral joints. Specimens were harvested from five regions across the articular surface of the humeral head and two regions on the glenoid. Tensile strips were obtained along two orientations, parallel and perpendicular to the split-line directions. Two serial slices through the thickness, corresponding to the superficial and middle zones of the cartilage layers, were prepared from each tensile strip and each compressive plug. The equilibrium tensile modulus and compressive aggregate modulus of cartilage were determined from the uniaxial tensile and confined compression tests, respectively. Significant differences in the tensile moduli were found with depth and orientation relative to the local split-line direction. Articular cartilage of the humeral head was significantly stiffer in tension than that of the glenoid. There were significant differences in the aggregate compressive moduli of articular cartilage between superficial and middle zones in the humeral head. Furthermore, tensile and compressive stress-strain responses exhibited nonlinearity under finite strain, while the tensile modulus differed by up to two orders of magnitude from the compressive aggregate modulus at 0% strain, demonstrating a high degree of tension-compression nonlinearity. The complexity of the mechanical properties of human glenohumeral cartilage was exposed in this study, showing anisotropy, inhomogeneity, and tension-compression nonlinearity within the same joint. The observed differences in the tensile properties of human glenohumeral cartilage suggest that the glenoid may be more susceptible to cartilage degeneration than the humeral head.     

Two different correlations between nanoindentation modulus and mineral content in the bone-cartilage interface

The biomechanical properties of the zone of calcified cartilage (ZCC) in articulating joints are of clinical relevance due to the role ZCC plays in load transfer from cartilage to bone. To determine the micron-level mechanical properties and their correlation to mineral concentration in the ZCC, we combined nanoindentation (for micrometer level stiffness E(r) and hardness H) and quantitative back-scattered electron imaging or qBEI (for micrometer level mean calcium concentration Ca(Mean)) to study the ZCC-subchondral bone junction in 3 embedded human patellae. Nanoindentation line scans were correlated to qBEI analysis in the ZCC. The correlation between local stiffness and local mineral content was different in calcified cartilage compared to bone. The stiffness and hardness of calcified cartilage was typically lower than subchondral bone for the same mineral content. ZCC showed a wider range of variation in calcium content (1-28 wt %) compared to subchondral bone (16-26 wt %). 2D material property maps of the ZCC were generated from the mechanical-mineral correlation, showing that bands of high and low stiffness were found between the bone and tidemark, and between the ZCC and the unmineralized cartilage.     

The tensile properties of the cartilage of human femoral condyles related to the content of collagen and glycosaminoglycans

A method is described of measuring the tensile stiffness and fracture stress of human femoral condylar cartilage in planes parallel to and at increasing depth below the articular surface. The axis of tension was either parallel or perpendicular to the predominant collagen fibre direction in the superficial zone. Specimens were analysed for their collagen and glycosaminoglycan contents and partial correlation coefficients were determined between the tensile properties and each of the chemical constituents.The correlations between the tensile properties and the collagen content of specimens oriented parallel to the collagen fibre direction was statistically significant in the superficial zone but the significance level decreased with increasing depth. In specimens which were oriented perpendicularly to the collagen fibre direction the correlations between the above variables were less significant.There was no significant correlation between the tensile properties and the glycosaminoglycans in cartilage.Visibly normal specimens from the superficial layer which were situated adjacent to visibly degenerate cartilage were weaker and less stiff than specimens situated on normal joints or remote from visibly degenerate cartilage. Such differences decreased with depth below articular surface and were greater in parallel-oriented specimens.     

The effects of proteolytic enzymes on the mechanical properties of adult human articular cartilage.

The effects of the lysosomal proteinase cathepsin D on the mechanical properties of adult human articular cartilage were examined in detail in 7 joints within the age range 21 to 72 years. The results of a preliminary study on the effects of the lysosomal proteinase cathepsin B1 and clostridial collagenase on the mechanical properties of cartilage are also presented. Cartilage which had been incubated with either cathepsin D or cathepsin B1 showed increased deformation in uniaxial compression perpendicular to the articular surface. The enzyme-treated cartilage also showed decreased tensile stiffness at low values of stress. This effect was more pronounced in specimens from the deeper zone of cartilage than in specimens from the superficial zone. It was also more pronounced in specimens which were aligned perpendicular to the predominant alignment of the collagen fibres in the superficial zone than in specimens which were parallel to the collagen fibres. At higher stresses the tensile stiffness of the treated cartilage was not significantly different from that of the untreated tissue. The tensile fracture stress of the cartilage was also not significantly reduced by the action of cathepsin D. In contrast to the effects observed with the cathepsins, the preliminary results obtained by incubating cartilage for 24 h with clostridial collagenase showed that both the tensile stiffness and the fracture stress were considerably lower than the corresponding values for the untreated tissue. Biochemical analysis of the incubation media, and the specimens, revealed that a large proportion of the proteoglycans was released from the cartilage by each of the three enzymes. The proportion of the total collagen which was released from the cartilage was different for each enzyme: cathepsin D released between 0 and 1.5 per cent, cathepsin B1 released between 2.3 and 4.3 per cent and collagenase released between 5.3 and 27.8 per cent of the collagen after 24 h.     

The effects of proteolytic enzymes on the mechanical properties of adult human articular cartilage

The effects of the lysosomal proteinase cathepsin D on the mechanical properties of adult human articulage were examined in detail in 7 joints within the age rangee 21 to 72 years. The results of preliminary study on the effects of the lysosomal proteinase cathepsin B1 and clostridial collagenase on the mechanical properties of cartilage are also presented.Cartilage which had been incubated with either cathepsin D or cathepsin B1 showed increased deformation in unixial compression perpendicular to the articular surface.The enzyme-treated cartilage also showed decreased tensile stiffness at low values of stress. This effect was more pronounced in specimens from the deeper zone of cartilage than in specimens from the superficial zone. It was also more pronounced in specimens which were aligned perpendicular to the predominant alignment of the collagen fibres in the superficial zone than in specimens which were parallel to the collagen fibres.At higher stresses the tensile stiffness of the treated cartilage was not significantly different from that of the untreated tissue. The tensile fracture stress of the cartilage was not significantly reduced by the action of cathepsin D.In contrast to the effects observed with the cathepsins, the preliminary results obtained by incubating cartilage for 24 h with clostridial collagenase showed that both the tensile stiffness and the fracture stress were considerably lower than the corresponding values for the untreated tissue.Biochemical analysis of the incubation media, and the specimens, reveled that a large proportion of the proteoglycans was released from the cartilage by each of the freeze enzymes. The proportion of the total collagen which was released from the cartilage was different for each enzyme: cathepsin D released between 0 and 1.5 per cent, cathepsin B1 released between 2.3 and 4.3 per cent and collagenase relesed between 5.3 and 27.8 per cent of the collagen after 24 h.     

Evaluation of a subject-specific finite-element model of the equine metacarpophalangeal joint under physiological load

The equine metacarpophalangeal (MCP) joint is frequently injured, especially by racehorses in training. Most injuries result from repetitive loading of the subchondral bone and articular cartilage rather than from acute events. The likelihood of injury is multi-factorial but the magnitude of mechanical loading and the number of loading cycles are believed to play an important role. Therefore, an important step in understanding injury is to determine the distribution of load across the articular surface during normal locomotion. A subject-specific finite-element model of the MCP joint was developed (including deformable cartilage, elastic ligaments, muscle forces and rigid representations of bone), evaluated against measurements obtained from cadaver experiments, and then loaded using data from gait experiments. The sensitivity of the model to force inputs, cartilage stiffness, and cartilage geometry was studied. The FE model predicted MCP joint torque and sesamoid bone flexion angles within 5% of experimental measurements. Muscle–tendon forces, joint loads and cartilage stresses all increased as locomotion speed increased from walking to trotting and finally cantering. Perturbations to muscle–tendon forces resulted in small changes in articular cartilage stresses, whereas variations in joint torque, cartilage geometry and stiffness produced much larger effects. Non-subject-specific cartilage geometry changed the magnitude and distribution of pressure and the von Mises stress markedly. The mean and peak cartilage stresses generally increased with an increase in cartilage stiffness. Areas of peak stress correlated qualitatively with sites of common injury, suggesting that further modelling work may elucidate the types of loading that precede joint injury and may assist in the development of techniques for injury mitigation.     

Mechanobiology of cartilage: how do internal and external stresses affect mechanochemical transduction and elastic energy storage?

 Articular cartilage is a multilayered structure that lines the surfaces of all articulating joints. It contains cells, collagen fibrils, and proteoglycans with compositions that vary from the surface layer to the layer in contact with bone. It is composed of several zones that vary in structure, composition, and mechanical properties. In this paper we analyze the structure of the extracellular matrix found in articular cartilage in an effort to relate it to the ability of cartilage to store, transmit, and dissipate mechanical energy during locomotion. Energy storage and dissipation is related to possible mechanisms of mechanochemical transduction and to changes in cartilage structure and function that occur in osteoarthritis. In addition, we analyze how passive and active internal stresses affect mechanochemical transduction in cartilage, and how this may affect cartilage behavior in health and disease. Received: 8 February 2002 / Accepted: 9 July 2002     

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.     

Finite element prediction of fatigue damage growth in cancellous bone

Cyclic stresses applied to bones generate fatigue damage that affects the bone stiffness and its elastic modulus. This paper proposes a finite element model for the prediction of fatigue damage accumulation and failure in cancellous bone at continuum scale. The model is based on continuum damage mechanics and incorporates crack closure effects in compression. The propagation of the cracks is completely simulated throughout the damaged area. In this case, the stiffness of the broken element is reduced by 98% to ensure no stress-carrying capacities of completely damaged elements. Once a crack is initiated, the propagation direction is simulated by the propagation of the broken elements of the mesh. The proposed model suggests that damage evolves over a real physical time variable (cycles). In order to reduce the computation time, the integration of the damage growth rate is based on the cycle blocks approach. In this approach, the real number of cycles is reduced (divided) into equivalent blocks of cycles. Damage accumulation is computed over the cycle blocks and then extrapolated over the corresponding real cycles. The results show a clear difference between local tensile and compressive stresses on damage accumulation. Incorporating stiffness reduction also produces a redistribution of the peak stresses in the damaged region, which results in a delay in damage fracture.     

On foundations of discrete element analysis of contact in diarthrodial joints

Information about the stress distribution on contact surfaces of adjacent bones is indispensable for analysis of arthritis, bone fracture and remodeling. Numerical solution of the contact problem based on the classical approaches of solid mechanics is sophisticated and time-consuming. However, the solution can be essentially simplified on the following physical grounds. The bone contact surfaces are covered with a layer of articular cartilage, which is a soft tissue as compared to the hard bone. The latter allows ignoring the bone compliance in analysis of the contact problem, i.e. rigid bones are considered to interact through a compliant cartilage. Moreover, cartilage shear stresses and strains can be ignored because of the negligible friction between contacting cartilage layers. Thus, the cartilage can be approximated by a set of unilateral compressive springs normal to the bone surface. The forces in the springs can be computed from the equilibrium equations iteratively accounting for the changing contact area. This is the essence of the discrete element analysis (DEA). Despite the success in applications of DEA to various bone contact problems, its classical formulation required experimental validation because the springs approximating the cartilage were assumed linear while the real articular cartilage exhibited non-linear mechanical response in reported tests. Recent experimental results of Ateshian and his co-workers allow for revisiting the classical DEA formulation and establishing the limits of its applicability. In the present work, it is shown that the linear spring model is remarkably valid within a wide range of large deformations of the cartilage. It is also shown how to extend the classical DEA to the case of strong nonlinearity if necessary.     

Stress distribution and consolidation in cartilage constituents is influenced by cyclic loading and osteoarthritic degeneration

The understanding of load support mechanisms in cartilage has evolved with computational models that better mimic the tissue ultrastructure. Fibril-reinforced poroelastic models can reproduce cartilage behaviour in a variety of test conditions and can be used to model tissue anisotropy as well as assess stress and pressure partitioning to the tissue constituents. The goal of this study was to examine the stress distribution in the fibrillar and non-fibrillar solid phase and pressure in the fluid phase of cartilage in axisymmetric models of a healthy and osteoarthritic hip joint. Material properties, based on values from the literature, were assigned to the fibrillar and poroelastic components of cartilage and cancellous and subchondral compact bone regions. A cyclic load representing walking was applied for 25 cycles. Contact stresses in the fibrillar and non-fibrillar solid phase supported less than 1% of the contact force and increased only minimally with load cycles. Simulated proteoglycan depletion increased stresses in the radial and tangential collagen fibrils, whereas fibrillation of the tangential fibrils resulted in increased compressive stress in the non-fibrillar component and tensile stress in the radial fibrils. However neither had an effect on fluid pressure. Subchondral sclerosis was found to have the largest effect, resulting in increased fluid pressure, non-fibrillar compressive stress, tangential fibril stress and greater cartilage consolidation. Subchondral bone stiffening may play an important role in the degenerative cascade and may adversely affect tissue repair and regeneration treatments.