A chondral modeling theory revisited

The mechanical environment of limb joints constantly changes during growth due to growth-related changes in muscle and tendon lengths, long bone dimensions, and body mass. The size and shape of limb joint surfaces must therefore also change throughout post-natal development in order to maintain normal joint function. Frost's (1979, 1999) chondral modeling theory proposed that joint congruence is maintained in mammalian limbs throughout postnatal ontogeny because cartilage growth in articular regions is regulated in part by mechanical load. This paper incorporates recent findings concerning the distribution of stress in developing articular units, the response of chondrocytes to mechanically induced deformation, and the development of articular cartilage in order to expand upon Frost's chondral modeling theory. The theory presented here assumes that muscular contraction during post-natal locomotor development produces regional fluctuating, intermittent hydrostatic pressure within the articular cartilage of limb joints. The model also predicts that peak levels of hydrostatic pressure in articular cartilage increase between birth and adulthood. Finally, the chondral modeling theory proposes that the cell-cell and cell-extracellular matrix interactions within immature articular cartilage resulting from mechanically induced changes in hydrostatic pressure regulate the metabolic activity of chondrocytes. Site-specific rates of articular cartilage growth are therefore regulated in part by the magnitude, frequency, and orientation of prevailing loading vectors. The chondral modeling response maintains a normal kinematic pathway as the magnitude and direction of joint loads change throughout ontogeny. The chondral modeling theory also explains ontogenetic scaling patterns of limb joint curvature observed in mammals. The chondral modeling response is therefore an important physiological mechanism that maintains the match between skeletal structure, function, and locomotor performance throughout mammalian ontogeny and phylogeny.     

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.     

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

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

The "instantaneous" deformation of cartilage: effects of collagen fiber orientation and osmotic stress

The present study was undertaken with two objectives in view. The first was to distinguish between the "instantaneous" deformation and creep of articular cartilage when subjected to a step loading in unconfined compression. This was done by observing changes in the specimen's diameter rather than its thickness. The second objective was to investigate experimentally the anisotropic behaviour of cartilage in a compressive loading mode, corresponding to the physiological situation. An apparatus was thus developed and constructed which enabled us to follow the "instantaneous" changes of the surface area of the sample as the latter was being loaded in unconfined compression. Specimens of human articular cartilage from normal femoral heads and condyles were tested. Full thickness specimens were tested with and without the underlying bone, as well as partial thickness specimens, characterizing the different zones of cartilage. Solutions of different ionic strength were used to vary the osmotic stress and specimens covering a considerable range of proteoglycan concentrations were selected. The effects of hydration and proteoglycan removal on the "instantaneous" deformation were also studied. The "instantaneous" deformation was found to be of a strongly anisotropic nature in all zones. The deformation was always smaller along the Indian-ink prick pattern than at 90 degrees to it, and this effect was most pronounced in the superficial zone of cartilage. The results reveal an analogy with the tensile properties of cartilage and indicate that the collagen network is mainly responsible for controlling the "instantaneous" deformation. The proteoglycans play an indirect role by modulating the stiffness of the collagen network through their osmotic pressure.     

Swelling and curling behaviors of articular cartilage.

A new experimental method was developed to quantify parameters of swelling-induced shape change in articular cartilage. Full-thickness strips of cartilage were studied in free-swelling tests and the swelling-induced stretch, curvature, and areal change were measured. In general, swelling-induced stretch and curvature were found to increase in cartilage with decreasing ion concentration, reflecting an increasing tendency to swell and "curl" at higher swelling pressures. An exception was observed at the articular surface, which was inextensible for all ionic conditions. The swelling-induced residual strain at physiological ionic conditions was estimated from the swelling-induced stretch and found to be tensile and from 3-15 percent. Parameters of swelling were found to vary with sample orientation, reflecting a role for matrix anisotropy in controlling the swelling-induced residual strains. In addition, the surface zone was found to be a structurally important element, which greatly limits swelling of the entire cartilage layer. The findings of this study provide the first quantitative measures of swelling-induced residual strain in cartilage ex situ, and may be readily adapted to studies of cartilage swelling in situ.     

A Finite Element Formulation and Program to Study Transient Swelling and Load-carriage in Healthy and Degenerate Articular Cartilage

The theory of poroelasticity is extended to include physico-chemical swelling and used to predict the transient responses of normal and degenerate articular cartilage to both chemical and mechanical loading; with emphasis on isolating the influence of the major parameters which govern its deformation. Using a new hybrid element, our mathematical relationships were implemented in a purpose-built poroelastic finite element analysis algorithm (u–π–c fortran program) which was used to resolve the nature of the coupling between the mechanical and chemical responses of cartilage when subjected to ionic transport across its membranous skeleton. Our results demonstrate that one of the roles of the strain-dependent matrix permeability is to limit the rate of transmission of stresses from the fluid to the collagen-proteoglycan solid skeleton in the incipient stages of loading, and that the major contribution of the swelling pressure is that of preventing any excessive deformation of the matrix.     

A comparison between mechano-electrochemical and biphasic swelling theories for soft hydrated tissues

Biological tissues like intervertebral discs and articular cartilage primarily consist of interstitial fluid, collagen fibrils and negatively charged proteoglycans. Due to the fixed charges of the proteoglycans, the total ion concentration inside the tissue is higher than in the surrounding synovial fluid (cation concentration is higher and the anion concentration is lower). This excess of ion particles leads to an osmotic pressure difference, which causes swelling of the tissue. In the last decade several mechano-electrochemical models, which include this mechanism, have been developed. As these models are complex and computationally expensive, it is only possible to analyze geometrically relatively small problems. Furthermore, there is still no commercial finite element tool that includes such a mechano-electrochemical theory. Lanir (Biorheology, 24, pp. 173-187, 1987) hypothesized that electrolyte flux in articular cartilage can be neglected in mechanical studies. Lanir's hypothesis implies that the swelling behavior of cartilage is only determined by deformation of the solid and by fluid flow. Hence, the response could be described by adding a deformation-dependent pressure term to the standard biphasic equations. Based on this theory we developed a biphasic swelling model. The goal of the study was to test Lanir's hypothesis for a range of material properties. We compared the deformation behavior predicted by the biphasic swelling model and a full mechano-electrochemical model for confined compression and 1D swelling. It was shown that, depending on the material properties, the biphasic swelling model behaves largely the same as the mechano-electrochemical model, with regard to stresses and strains in the tissue following either mechanical or chemical perturbations. Hence, the biphasic swelling model could be an alternative for the more complex mechano-electrochemical model, in those cases where the ion flux itself is not the subject of the study. We propose thumbrules to estimate the correlation between the two models for specific problems.     

How changes in interconnectivity affect the bulk properties of articular cartilage: a fibre network study

The remarkable compressive strength of articular cartilage arises from the mechanical interactions between the tension-resisting collagen fibrils and swelling proteoglycan proteins within the tissue. These interactions are facilitated by a significant level of interconnectivity between neighbouring collagen fibrils within the extracellular matrix. A reduction in interconnectivity is suspected to occur during the early stages of osteoarthritic degeneration. However, the relative contribution of these interconnections towards the bulk mechanical properties of articular cartilage has remained an open question. In this study, we present a simple 2D fibre network model which explicitly represents the microstructure of articular cartilage as collection of discrete nodes and linear springs. The transverse stiffness and swelling properties of this fibre network are studied, and a semi-analytic relationship which relates these two macroscopic properties via microscopic interconnectivity is derived. By comparing this derived expression to previously published experimental data, we show that although a reduction in network interconnectivity accounts for some of the observed changes in the mechanical properties of articular cartilage as degeneration occurs, a decrease in matrix interconnectivity alone do not provide a full account of this process.     

Tensorial electrokinetics in articular cartilage

Electrokinetic phenomena contribute to biomechanical functions of articular cartilage and underlie promising methods for early detection of osteoarthritic lesions. Although some transport properties, such as hydraulic permeability, are known to become anisotropic with compression, the direction-dependence of cartilage electrokinetic properties remains unknown. Electroosmosis experiments were therefore performed on adult bovine articular cartilage samples, whereby fluid flows were driven by electric currents in directions parallel and perpendicular to the articular surface of statically compressed explants. Magnitudes of electrokinetic coefficients decreased slightly with compression (from approximately -7.5 microL/As in the range of 0-20% compression to -6.0 microL/As in the 35-50% range) consistent with predictions of microstructure-based models of cartilage material properties. However, no significant dependence on direction of the electrokinetic coupling coefficient was detected, even for conditions where the hydraulic permeability tensor is known to be anisotropic. This contrast may also be interpreted using microstructure-based models, and provides insights into structure-function relationships in cartilage extracellular matrix and physical mediators of cell responses to tissue compression. Findings support the use of relatively simple isotropic modeling approaches for electrokinetic phenomena in cartilage and related materials, and indicate that measurement of electrokinetic properties may provide particularly robust means for clinical evaluation of cartilage matrix integrity.     

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.     

Force-mediated dissociation of proteoglycan aggregate in articular cartilage

Proteoglycan aggregate is the primary component in articular cartilage responsible for resisting compressive loading. It consists of a core molecule of hyaluronan and a number of side chains of aggrecan bound to hyaluronan non-covalently. The loss of aggrecan from articular cartilage is considered to be a major factor in the development of osteoarthritis. Though enzymatic digestion of aggrecan is believed to be responsible for the release of aggrecan from osteoarthritic cartilage, other mechanisms, such as direct force-mediated detachment of aggrecan from hyaluronan may also be involved. In this study, the rupture force of the single bond between hyaluronan and aggrecan in articular cartilage was directly quantified using experimental measurement and Monte Carlo simulation. Low rupture force of this bond, as determined in this study suggested a possible direct force-mediated detachment of aggrecan from proteoglycan aggregate in osteoarthritic cartilage.     

Cyclic compression of articular cartilage explants is associated with progressive consolidation and altered expression pattern of extracellular matrix proteins.

In this study, we investigated the biosynthetic response of full thickness, adult bovine articular cartilage explants to 45 h of static and cyclic unconfined compression. The cyclic compression of articular cartilage resulted in a progressive consolidation of the cartilage matrix. The oscillatory loading increased protein synthesis ([35S]methionine incorporation) by as much as 50% above free swelling control values, but had an inhibitory influence on proteoglycan synthesis ([35SO4] incorporation). As expected, static compression was associated with a dose-dependent decrease in biosynthetic activity. ECM oligomeric proteins which were most affected by mechanical loading were fibronectin and cartilage oligomeric matrix protein (COMP). Static compression at all amplitudes caused a significant increase in fibronectin synthesis over free swelling control levels. Cyclic compression of articular cartilage at 0.1 Hz and higher was consistently associated with a dramatic increase in the synthesis of COMP as well as fibronectin. The biosynthetic activity of chondrocytes appears to be sensitive to both the frequency and amplitude of the applied load. The results of this study support the hypothesis that cartilage tissue can remodel its extracellular matrix in response to alterations in functional demand.     

Quantitative analysis of collagen gel contractile forces generated by dermal fibroblasts and the relationship to cell morphology

The force generated in granulation tissue during wound contraction is thought to be cell mediated; however, it is unclear whether contractile forces are generated by fibroblast locomotion or contraction of myofibroblasts. To help clarify this question the force of this contraction can now be determined accurately in a human dermal fibroblast collagen lattice system using a novel instrument known as a Culture Force Monitor. Three distinct phases of contraction of such collagen gels could be identified over the first 24 hours. Most of the force generated by human dermal fibroblasts was produced during the first stage in parallel with cell attachment and associated changes in cell shape, and the appearance of cell processes. During this initial 24 hours no evidence could be found for the presence of myofibroblasts, but stereoscopic and electron microscopic analysis at a range of time points indicated that migratory fibroblasts were present in the system. Comparison of the contraction profiles of cells extracted from other tissues (tendon and articular cartilage), and extracted by different means from the same tissue specimen, indicated that different populations of fibroblasts can be distinguished on the basis of their pattern of contractions. It would seem that most of the force generated in this model is a result of fibroblast attachment and movement within the collagen lattice. Furthermore, different groups of fibroblasts, even within the same tissue, may vary in their contraction (hence locomotory) activity. © 1996 Wiley-Liss, Inc.     

The importance of superficial collagen fibrils for the function of articular cartilage

It has been proposed that the superficial tangential zone (STZ) of articular cartilage is essential to the tissue’s load-distributing function. However, the exact mechanism by which the STZ fulfills this function has not yet been revealed. Using a channel-indentation experiment, it was recently shown that compared to intact tissue, cartilage without STZ behaves slightly stiffer and deforms significantly different in regions adjacent to mechanically compressed areas (Bevill et al. in Osteoarthr Cartil 18:1310–1318, 2010). We aim to further explore the role of STZ in the load-transfer mechanism of AC by thorough biomechanical analysis of these experiments. Using our previously validated fibril-reinforced swelling model of articular cartilage, which accounts for the depth-dependent collagen structure and biochemical composition of articular cartilage, we simulated the above-mentioned channel-indenter compression experiments for both intact and STZ-removed cartilage. First, we show that the composition of the deep zone in cartilage is most effective in carrying cartilage compression, which explains the apparent tissue stiffening after STZ removal. Second, we show that tangential fibrils in the STZ are responsible for transferring compressive loads from directly loaded regions to adjacent tissue. Cartilage with an intact STZ has superior load-bearing properties compared to cartilage in which the STZ is compromised, because the STZ is able to recruit a larger area of deep zone cartilage to carry compressive loads.     

A finite element formulation and program to study transient swelling and load-carriage in healthy and degenerate articular cartilage

The theory of poroelasticity is extended to include physico-chemical swelling and used to predict the transient responses of normal and degenerate articular cartilage to both chemical and mechanical loading; with emphasis on isolating the influence of the major parameters which govern its deformation. Using a new hybrid element, our mathematical relationships were implemented in a purpose-built poroelastic finite element analysis algorithm (u-pi-c program) which was used to resolve the nature of the coupling between the mechanical and chemical responses of cartilage when subjected to ionic transport across its membranous skeleton. Our results demonstrate that one of the roles of the strain-dependent matrix permeability is to limit the rate of transmission of stresses from the fluid to the collagen-proteoglycan solid skeleton in the incipient stages of loading, and that the major contribution of the swelling pressure is that of preventing any excessive deformation of the matrix.     

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.     

Nonuniform swelling-induced residual strains in articular cartilage

Swelling effects in cartilage originate from an interstitial osmotic pressure generated by the presence of negatively charged proteoglycans in the tissue. This swelling pressure gives rise to a non-zero residual strain in the cartilage solid matrix in the absence of externally applied loads. Previous studies have quantified swelling effects in cartilage as volumetric or dimensional change of excised samples in varying osmotically active solutions. This study presents a new optical technique for measuring two-dimensional swelling-induced residual strain fields in planar samples of articular cartilage attached to the bone (i.e., in situ). Osmotic loading was applied to canine cartilage bone samples by equilibration in external baths of varying NaCl concentration. Non-zero swelling-induced strains were measured in physiological saline, giving evidence of the existence of residual strains in articular cartilage. Only one component of planar strain (i.e., in thickness direction) was found to be non-zero. This strain was found to be highly non-uniform in the thickness direction, with evidence of compressive strain in the deep zone of cartilage and tensile strain in the middle and surface zones. The obtained results can be used to characterize the material properties of the articular cartilage solid matrix, with estimated values of 26 M Pa for the tensile modulus for middle zone cartilage. The method provides the basis to obtain material properties of the cartilage solid matrix from a simple, free-swelling test and may be useful for quantifying changes in cartilage properties with injury, degeneration and repair.     

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.