A nonlinear biphasic viscohyperelastic model for articular cartilage

Experiments on articular cartilage have shown nonlinear stress-strain curves under finite deformations as well as intrinsic viscous effects of the solid phase. The aim of this study was to propose a nonlinear biphasic viscohyperelastic model that combines the intrinsic viscous effects of the proteoglycan matrix with a nonlinear hyperelastic constitutive equation. The proposed equation satisfies objectivity and reduces for uniaxial loading to a solid type viscous model in which the actions of the springs are represented by the hyperelastic function proposed by Holmes and Mow [1990. J. Biomechanics 23, 1145-1156.]. Results of the model, that were efficiently implemented in an updated Lagrangian algorithm, were compared with experimental infinitesimal data reported by DiSilverstro and Suh [2001. J. Biomechanics 34, 519-525.] and showed acceptable fitting for the axial force (R(2)=0.991) and lateral displacement (R(2)=0.914) curves in unconfined compression as well as a good fitting of the axial indentation force curve (R(2)=0.982). In addition, the model showed an excellent fitting of finite-deformation confined compression stress relaxation data reported by Ateshian et al. [1997. J. Biomechanics 30, 1157-1164.] and Huang et al. [2005. J. Biomechanics 38, 799-809.] (R(2)=0.993 and R(2)=0.995, respectively). The constitutive equation may be used to represent the mechanical behavior of the proteoglycan matrix in a fiber reinforced model of articular cartilage.     

Finite element formulation of biphasic poroviscoelastic model for articular cartilage.

The purpose of the present study was to develop a computationally efficient finite element model that could be useful for parametric analysis of the biphasic poroviscoelastic (BPVE) behavior of articular cartilage under various loading conditions. The articular cartilage was modeled as the BPVE mixture of a porous, linear viscoelastic, and incompressible solid and an inviscid and incompressible fluid. A finite element (FE) formulation of the BPVE model was developed using two different algorithms, the continuous and discrete spectrum relaxation functions for the viscoelasticity of the solid matrix. These algorithms were applied to the creep and stress relaxation responses to the confined compression of articular cartilage, and a comparison of their performances was made. It was found that the discrete spectrum algorithm significantly saved CPU time and memory, as compared to the continuous spectrum algorithm. The consistency analysis for the present FE formulation was performed in comparison with the IMSL, a commercially available numerical software package. It was found that the present FE formulation yielded consistent results in predicting model behavior, whereas the IMSL subroutine produced inconsistent results in the velocity field, and thereby in the strain calculation.     

Tensile fatigue behaviour of articular cartilage

Although fatigue has been implicated in cartilage failure there are only two studies by the same author, and in both of which cartilage was tested in the direction parallel to the collagen orientation in the surface layer. In the present work articular cartilage was tested also along the perpendicular direction, being the direction in which cartilage possesses lower tensile strength.Specimens were tested under cyclic tensile load. Number of cycles at failure was recorded as well as elongation of the specimen. To date 72 specimens have been tested all from one knee joint.The number of cycles to failure ranged between two and 1.5 million. The surface and deep layers have better fatigue properties whether tested in the parallel or the perpendicular direction, while the middle layer was far weaker. Better fatigue behaviour was observed with specimens tested in parallel than in perpendicular direction to the fibres.     

Frictional behaviour of bovine articular cartilage

The experimentally measured indentation displacement and friction of normal and degraded (treated with chondroitinase AC) bovine articular cartilage plugs against a smooth steel plate were compared with the predictions based on the biphasic theory using the finite element method. It was found that the measured indentation displacement of both cartilage specimens could be predicted from the biphasic theory and the permeability for the degraded cartilage specimen was increased approximately three times. However, the measured friction coefficient was much lower for short period of loading, and the difference in the finite element prediction of friction coefficient between the normal and degraded cartilage specimens was not observed in the experiment. Therefore, it was concluded that both biphasic and other mechanisms were important in controlling the frictional and lubricating characteristics of articular cartilage in mixed and boundary lubrication regimes.     

A numerical method for the continuous spectrum biphasic poroviscoelastic model of articular cartilage

A method for numerical solution of the continuous spectrum linear biphasic poroviscoelastic (BPVE) model of articular cartilage is presented. The method is based on an alternate formulation of the continuous spectrum stress-strain law that is implemented using Gaussian quadrature integration combined with quadratic interpolation of the strain history. For N time steps, the cost of the method is O(N). The method is applied to a finite difference solution of the one-dimensional confined compression BPVE stress-relaxation problem. For a range of relaxation times that are representative of articular cartilage, accuracy of the method is demonstrated by direct comparison to a theoretical Laplace transform solution.     

An investigation of biphasic failure criteria for impact-induced fissuring of articular cartilage.

Articular cartilage consists of both solid and fluid phases with fissures observed on the surface occurring in the solid portion. In order to determine which of the solid phase stresses provides the best predictor for the initiation of a fissure, elastic stresses from a series of in vitro impact experiments were used to derive stresses in the solid phase of the cartilage. This stress information was then analyzed using a logistic regression to identify the best predictor of fissuring. The mechanical analysis indicated that low-magnitude tensile solid hoop stress develops in the solid phase within the contact zone in impacts involving the two smaller radius interfaces. The logistic regression, however, indicated that maximum shear stress in the solid (which is equal to the shear stress from the elastic analysis) was the best predictor of the occurrence of a fissure. This study helps support the suggestion that in stress fields dominated by compression, the maximum shear stress from an elastic analysis may be used to predict fissure initiation in cartilage.     

A fast quadrature-based numerical method for the continuous spectrum biphasic poroviscoelastic model of articular cartilage

A new and efficient method for numerical solution of the continuous spectrum biphasic poroviscoelastic (BPVE) model of articular cartilage is presented. Development of the method is based on a composite Gauss–Legendre quadrature approximation of the continuous spectrum relaxation function that leads to an exponential series representation. The separability property of the exponential terms in the series is exploited to develop a numerical scheme that can be reduced to an update rule requiring retention of the strain history at only the previous time step. The cost of the resulting temporal discretization scheme is O(N) for N time steps. Application and calibration of the method is illustrated in the context of a finite difference solution of the one-dimensional confined compression BPVE stress-relaxation problem. Accuracy of the numerical method is demonstrated by comparison to a theoretical Laplace transform solution for a range of viscoelastic relaxation times that are representative of articular cartilage.     

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

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

Variations in the composition of bovine hip articular cartilage with distance from the articular surface.

Punch biopsies of bovine hip articular cartilage was sectioned according to depth and the proteoglycans were isolated. The mid-sections of the cartilage contained more proteoglycans than did either the superficial or the deepest portions of the cartilage proteoglycans than did either the superficial or the deepest portions of the cartilage. The most superficial 40 micrometer of the cartilage contained relatively more glucosaminoglycans compared with the remainder of the cartilage. The proteoglycans recovered from the surface 200 micrometer layer contained less chondroitin sulphate, were smaller and almost all of these molecules were able to interact with hyaluronic acid to form aggregates. From about 200 micrometer and down to 1040 micrometer from the surface, the proteoglycans became gradually somewhat smaller, probably owing to decreasing size of the chondroitin sulphate-rich region. The proportion of molecules that were able to interact with the hyaluronic acid was about 90% and remained constant with depth. The proteoglycans from the deepest layer near the cartilage-bone junction contained a large proportion of non-aggregating molecules, and the average size of the proteoglycans was somewhat larger. The alterations of proteoglycan structure observed with increasing depth of the articular cartilage beneath the surface layer (to 200 micrometer) are of the same nature as those observed with increasing age in full-thickness articular cartilage. The articular-cartilage proteoglycans were smaller and had much higher keratan sulphate and protein contents that did molecules isolated from bovine nasal or tracheal cartilage.     

The role of flow-independent viscoelasticity in the biphasic tensile and compressive responses of articular cartilage

A long-standing challenge in the biomechanics of connective tissues (e.g., articular cartilage, ligament, tendon) has been the reported disparities between their tensile and compressive properties. In general, the intrinsic tensile properties of the solid matrices of these tissues are dictated by the collagen content and microstructural architecture, and the intrinsic compressive properties are dictated by their proteoglycan content and molecular organization as well as water content. These distinct materials give rise to a pronounced and experimentally well-documented nonlinear tension-compression stress-strain responses, as well as biphasic or intrinsic extracellular matrix viscoelastic responses. While many constitutive models of articular cartilage have captured one or more of these experimental responses, no single constitutive law has successfully described the uniaxial tensile and compressive responses of cartilage within the same framework. The objective of this study was to combine two previously proposed extensions of the biphasic theory of Mow et al. [1980, ASME J. Biomech. Eng., 102, pp. 73-84] to incorporate tension-compression nonlinearity as well as intrinsic viscoelasticity of the solid matrix of cartilage. The biphasic-conewise linear elastic model proposed by Soltz and Ateshian [2000, ASME J. Biomech. Eng., 122, pp. 576-586] and based on the bimodular stress-strain constitutive law introduced by Curnier et al. [1995, J. Elasticity, 37, pp. 1-38], as well as the biphasic poroviscoelastic model of Mak [1986, ASME J. Biomech. Eng., 108, pp. 123-130], which employs the quasi-linear viscoelastic model of Fung [1981, Biomechanics: Mechanical Properties of Living Tissues, Springer-Verlag, New York], were combined in a single model to analyze the response of cartilage to standard testing configurations. Results were compared to experimental data from the literature and it was found that a simultaneous prediction of compression and tension experiments of articular cartilage, under stress-relaxation and dynamic loading, can be achieved when properly taking into account both flow-dependent and flow-independent viscoelasticity effects, as well as tension-compression nonlinearity.     

Altered mechano-chemical environment in hip articular cartilage: effect of obesity

The production of extracellular matrix (ECM) components of articular cartilage is regulated, among other factors, by an intercellular signaling mechanism mediated by the interaction of cell surface receptors (CSR) with insulin-like growth factor-1 (IGF-1). In ECM, the presence of binding proteins (IGFBP) hinders IGF-1 delivery to CSR. It has been reported that levels of IGF-1 and IGFBP in obese population are, respectively, lower and higher than those found in normal population. In this study, an experimental–numerical approach was adopted to quantify the effect of this metabolic alteration found in obese population on the homeostasis of femoral hip cartilage. A new computational model, based on the mechano-electrochemical mixture theory, was developed to describe competitive binding kinetics of IGF-1 with IGFBP and CSR, and associated glycosaminoglycan (GAG) biosynthesis. Moreover, a gait analysis was carried out on obese and normal subjects to experimentally characterize mechanical loads on hip cartilage during walking. This information was deployed into the model to account for effects of physiologically relevant tissue deformation on GAG production in ECM. Numerical simulations were performed to compare GAG biosynthesis in femoral hip cartilage of normal and obese subjects. Results indicated that the lower ratio of IGF-1 to IGFBP found in obese population reduces cartilage GAG concentration up to 18 % when compared to normal population. Moreover, moderate physical activity, such as walking, has a modest beneficial effect on GAG production. The findings of this study suggest that IGF-1/IGFBP metabolic unbalance should be accounted for when considering the association of obesity with hip osteoarthritis.     

A system for modelling forces on the hip joint in one-legged stance

We describe a method of testing hip prostheses. The prosthesis is implanted in a proximal section of femur, which is loaded to model the forces applied through the acetabulum via the greater trochanter.     

A finite element analysis of the indentation stress-relaxation response of linear biphasic articular cartilage.

The indentation problem of a thin layer of hydrated soft tissue such as cartilage or meniscus by a circular plane-ended indenter is investigated. The tissue is represented by a biphasic continuum model consisting of a solid phase (collagen and proteoglycan) and a fluid phase (interstitial water). A finite element formulation of the linear biphasic continuum equations is used to solve an axisymmetric approximation of the indentation problem. We consider stress-relaxation problems for which analytic solution is intractable; where the indenter is impermeable (solid) and/or when the interface between the indenter and tissue is perfectly adhesive. Thicknesses corresponding to a thin and thick specimen are considered to examine the effects of tissue thickness. The different flow, pressure, stress and strain fields which are predicted within the tissue, over time periods typically used in the mechanical testing of soft tissues, will be presented. Results are compared with the case of a porous free-draining indenter with a perfectly lubricated tissue-indenter interface, for which an analytic solution is available, to show the effects of friction at the tissue-indenter interface, and the effects of an impermeable indenter. While these effects are present for both thin and thick tissues, they are shown to be more significant for the thin tissue. We also examine the effects of the stiffness of the subchondral bone on the response of the soft tissue and demonstrate that the subchondral bone substrate can be modeled as a rigid, impermeable boundary. The effects of a curved tissue-subchondral bone interface, and the early time response are also studied. For physiologically reasonable levels of curvature, we will show that the curved tissue-subchrondal bone interface has negligible influence on the tissue response away from the interface. In addition, the short-time stress-relaxation responses of the tissue (e.g., at times less than 1s) demonstrate the essential role of the fluid phase in supporting the load applied to the tissue, and by extrapolation to shorter times characteristics of normal joint motion, suggest the essential role of a biphasic model in representing soft tissue behavior in joint response.     

Influence of aspect ratios on the creep behaviour of articular cartilage in indentation

Indentation tests are commonly used to determine the mechanical behaviour of articular cartilage with varying properties, thickness, and geometry. This investigation evaluated the effect of changing geometric parameters on the properties determined from creep indentation tests. Finite element analyses simulated the indentation behaviour of two models, an excised cylindrical specimen of cartilage with either normal and repair qualities and an osteochondral defect represented as a cylindrical region of repair cartilage integrated with a surrounding layer of normal tissue. For each model, the ratios of indenter radius to cartilage height (a/h=0.5,1.5) and cartilage radius to indenter radius (r/a=2,5) were varied. The vertical displacement of the cartilage under the indenter obtained through finite element analysis was fitted to a numerical algorithm to determine the aggregate modulus, permeability, and Poisson's ratio. Indentation behaviours of cartilage specimens for either model with a/h=1.5 were not affected by r/a for values of 2 and 5. Aggregate modulus was not greatly affected by the geometric changes studied. Permeability was affected by changes in the ratio of specimen to indenter radii for a/h=0.5. These findings suggest that experimental configurations of excised cylindrical specimens, also representing osteochondral defects with no or unknown degree of integration, where the cartilage layer has a/h=0.5 should not have r/a values on the order of 2 for confidence in the mechanical properties determined. Indentation of osteochondral defects where the repair cartilage is fully integrated to the surrounding cartilage can be performed with confidence for all cases tested.     

Comparison of MRI-based estimates of articular cartilage contact area in the tibiofemoral joint

Knee osteoarthritis (OA) detrimentally impacts the lives of millions of older Americans through pain and decreased functional ability. Unfortunately, the pathomechanics and associated deviations from joint homeostasis that OA patients experience are not well understood. Alterations in mechanical stress in the knee joint may play an essential role in OA; however, existing literature in this area is limited. The purpose of this study was to evaluate the ability of an existing magnetic resonance imaging (MRI)-based modeling method to estimate articular cartilage contact area in vivo. Imaging data of both knees were collected on a single subject with no history of knee pathology at three knee flexion angles. Intra-observer reliability and sensitivity studies were also performed to determine the role of operator-influenced elements of the data processing on the results. The method's articular cartilage contact area estimates were compared with existing contact area estimates in the literature. The method demonstrated an intra-observer reliability of 0.95 when assessed using Pearson's correlation coefficient and was found to be most sensitive to changes in the cartilage tracings on the peripheries of the compartment. The articular cartilage contact area estimates at full extension were similar to those reported in the literature. The relationships between tibiofemoral articular cartilage contact area and knee flexion were also qualitatively and quantitatively similar to those previously reported. The MRI-based knee modeling method was found to have high intra-observer reliability, sensitivity to peripheral articular cartilage tracings, and agreeability with previous investigations when using data from a single healthy adult. Future studies will implement this modeling method to investigate the role that mechanical stress may play in progression of knee OA through estimation of articular cartilage contact area.     

The correspondence between equilibrium biphasic and triphasic material properties in mixture models of articular cartilage

Mixture models have been successfully used to describe the response of articular cartilage to various loading conditions. Mow et al. (J. Biomech. Eng. 102 (1980) 73) formulated a biphasic mixture model of articular cartilage where the collagen-proteoglycan matrix is modeled as an intrinsically incompressible porous-permeable solid matrix, and the interstitial fluid is modeled as an incompressible fluid. Lai et al. (J. Biomech. Eng. 113 (1991) 245) proposed a triphasic model of articular cartilage as an extension of their biphasic theory, where negatively charged proteoglycans are modeled to be fixed to the solid matrix, and monovalent ions in the interstitial fluid are modeled as additional fluid phases. Since both models co-exist in the cartilage literature, it is useful to show how the measured properties of articular cartilage (the confined and unconfined compressive and tensile moduli, the compressive and tensile Poisson's ratios, and the shear modulus) relate to both theories. In this study, closed-form expressions are presented that relate biphasic and triphasic material properties in tension, compression and shear. These expressions are then compared to experimental findings in the literature to provide greater insight into the measured properties of articular cartilage as a function of bathing solutions salt concentrations and proteoglycan fixed-charge density.