A transversely isotropic, transversely homogeneous microstructural-statistical model of articular cartilage

Articular cartilage is a multi-phasic, composite, fibre-reinforced material. Therefore, its mechanical properties are determined by the tissue microstructure. The presence of cells (chondrocytes) and collagen fibres within the proteoglycan matrix influences, at a local and a global level, the material symmetries. The volumetric concentration and shape of chondrocytes, and the volumetric concentration and spatial arrangement of collagen fibres have been observed to change as a function of depth in articular cartilage. In particular, collagen fibres are perpendicular to the bone-cartilage interface in the deep zone, their orientation is almost random in the middle zone, and they are parallel to the surface in the superficial zone. The aim of this work is to develop a model of elastic properties of articular cartilage based on its microstructure. In previous work, we addressed this problem based on Piola's notation for fourth-order tensors. Here, mathematical tools initially developed for transversely isotropic composite materials comprised of a statistical orientation of spheroidal inclusions are extended to articular cartilage, while taking into account the dependence of the elastic properties on cartilage depth. The resulting model is transversely isotropic and transversely homogeneous (TITH), the transverse plane being parallel to the bone-cartilage interface and the articular surface. Our results demonstrate that the axial elastic modulus decreases from the deep zone to the articular surface, a result that is in good agreement with experimental findings. Finite element simulations were carried out, in order to explore the TITH model's behaviour in articular cartilage compression tests. The force response, fluid flow and displacement fields obtained with the TITH model were compared with the classical linear elastic, isotropic, homogeneous (IH) model, showing that the IH model is unable to predict the non-uniform behaviour of the tissue. Based on considerations that the mechanical stability of the tissue depends on its topological and microstructural properties, our long-term goal is to clearly understand the stability conditions in topological terms, and the relationship with the growth and remodelling mechanisms in the healthy and diseased tissue.     

Convection and Diffusion in Charged Hydrated Soft Tissues: A Mixture Theory Approach

The extracellular matrix of cartilage is a charged porous fibrous material. Transport phenomena in such a medium are very complex. In this study, solute diffusive flux and convective flux in porous fibrous media were investigated using a continuum mixture theory approach. The intrinsic diffusion coefficient of solute in the mixture was defined and its relation to drag coefficients was presented. The effect of mechanical loading on solute diffusion in cartilage under unconfined compression with a frictionless boundary condition was analyzed numerically using the model developed. Both strain-dependent hydraulic permeability and diffusivity were considered. Analyses and results show that (1) In porous media, the convective velocity for each solute phase is different. (2) The solute convection in tissue is governed by the relative convective velocity (i.e., relative to solid velocity). (3) Under the assumption that all the frictional interactions among solutes are negligible, the relative convective velocity for α-solute phase is equal to the relative solvent velocity multiplied by its convective coefficient (H ^α) which is also known as the hindrance factor in the literature. The relationship between the convective coefficient and the relative diffusivity of solute is presented. (4) Solute concentration profile within the cartilage sample depends on the phase of dynamic compression.     

The fate of mechanically induced cartilage in an unloaded environment.

According to mechanobiologic theories, persistent intermittent mechanical stimulation is required to maintain differentiated cartilage. In a rat model for bone repair, we studied the fate of mechanically induced cartilage after unloading. In three groups of rats, regenerating mesenchymal tissue was submitted to different loading conditions in bone chambers. Two groups were immediately killed after loading periods of 3 or 6 weeks (the 3-group and the 6-group). The third group was loaded for 3 weeks and then kept unloaded for another 3 weeks (the (3 + 3)-group). Cartilage was found in all loaded groups. Without loading, cartilage does not appear in this model. In the 3-group there was no clear ongoing endochondral ossification, the 6-group showed ossification in 2 out of 5 cartilage containing specimens, and in the (3 + 3)-group all cartilage was undergoing ossification. These results suggest a tendency of the cartilage to be maintained also under unloaded conditions until it is reached by bone that can replace it through endochondral ossification.Additional measurements showed less amount of new bone in the loaded specimens. In most of the loaded specimens in the 3-group, necrotic bone fragments were seen embedded in the fibrous tissue layer close to the loading piston, indicating that bone tissue had been resorbed due to the hydrostatic compressive load. In some specimens, a continuous cartilage layer covered the end of the specimen and seemed to protect the underlying bone from pressure-induced resorption. We suggest that one of the functions of the cartilage forming in the compressive loaded parts of a bone callus is to protect the surrounding bone callus from pressure-induced fluid flow leading to resorption.     

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

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

Dynamic unconfined compression of articular cartilage under a cyclic compressive load

To study the effect of dynamic mechanical force on cartilage metabolism, many investigators have applied a cyclic compressive load to cartilage disc explants in vitro. The most frequently used in vitro testing protocol has been the cyclic unconfined compression of articular cartilage in a bath of culture medium. Cyclic compression has been achieved by applying either a prescribed cyclic displacement or a prescribed cyclic force on a loading platen placed on the top surface of a cylindrical cartilage disc. It was found that the separation of the loading platen from the tissue surface was likely when a prescribed cyclic displacement was applied at a high frequency.The purpose of the present study was to simulate mathematically the dynamic behavior of a cylindrical cartilage disc subjected to cyclic unconfined compression under a dynamic force boundary condition protocol, and to provide a parametric analysis of mechanical deformations within the extracellular matrix. The frequency-dependent dynamic characteristics of dilatation, hydrostatic pressure and interstitial fluid velocity were analyzed over a wide range of loading frequencies without the separation of the loading platen. The result predicted that a cyclic compressive force created an oscillating positive-negative hydrostatic pressure together with a forced circulation of interstitial fluid within the tissue matrix. It was also found that the load partitioning mechanism between the solid and fluid phases was a function of loading frequency. At a relatively high loading frequency, a localized dynamic zone was developed near the peripheral free surface of the cartilage disc, where a large dynamic pressure gradient exists, causing vigorous interstitial fluid flow.     

Effect of sulfated glycosaminoglycan digestion on the transverse permeability of medial collateral ligament

Dermatan and chondroitin sulfate glycosaminoglycans (GAGs) comprise over 90% of the GAG content in ligament. Studies of their mechanical contribution to soft tissues have reported conflicting results. Measuring the transient compressive response and biphasic material parameters of the tissue may elucidate the contributions of GAGs to the viscoelastic response to deformation. The hypotheses of the current study were that digestion of sulfated GAGs would decrease compressive stress and aggregate modulus while increasing the permeability of porcine medial collateral ligament (MCL). Confined compression stress relaxation experiments were carried out on porcine MCL and tissue treated with chondroitinase ABC (ChABC). Results were fit to a biphasic constitutive model to derive permeability and aggregate modulus. Bovine articular cartilage was used as a benchmark tissue to verify that the apparatus provided reliable results. GAG digestion removed up to 88% of sulfated GAGs from the ligament. Removal of sulfated GAGs increased the permeability of porcine MCL nearly 6-fold versus control tissues. Peak stress decreased significantly. Bovine articular cartilage exhibited the typical reduction of GAG content and resultant decreases in stress and modulus and increases in permeability with ChABC digestion. Given the relatively small amount of GAG in ligament (<1% of tissue dry weight) and the significant change in peak stress and permeability upon removal of GAGs, sulfated GAGs may play a significant role in maintaining the apposition of collagen fibrils in the transverse direction, thus supporting dynamic compressive loads experienced by the ligament during complex joint motion.     

The Mechanical Behaviour of Chondrocytes Predicted with a Micro-structural Model of Articular Cartilage

The integrity of articular cartilage depends on the proper functioning and mechanical stimulation of chondrocytes, the cells that synthesize extracellular matrix and maintain tissue health. The biosynthetic activity of chondrocytes is influenced by genetic factors, environmental influences, extracellular matrix composition, and mechanical factors. The mechanical environment of chondrocytes is believed to be an important determinant for joint health, and chondrocyte deformation in response to mechanical loading is speculated to be an important regulator of metabolic activity. In previous studies of chondrocyte deformation, articular cartilage was described as a biphasic material consisting of a homogeneous, isotropic, linearly elastic solid phase, and an inviscid fluid phase. However, articular cartilage is known to be anisotropic and inhomogeneous across its depth. Therefore, isotropic and homogeneous models cannot make appropriate predictions for tissue and cell stresses and strains. Here, we modelled articular cartilage as a transversely isotropic, inhomogeneous (TI) material in which the anisotropy and inhomogeneity arose naturally from the microstructure of the depth-dependent collagen fibril orientation and volumetric fraction, as well as the chondrocyte shape and volumetric fraction. The purpose of this study was to analyse the deformation behaviour of chondrocytes using the TI model of articular cartilage. In order to evaluate our model against experimental results, we simulated indentation and unconfined compression tests for nominal compressions of 15%. Chondrocyte deformations were analysed as a function of location within the tissue. The TI model predicted a non-uniform behaviour across tissue depth: in indentation testing, cell height decreased by 43% in the superficial zone and between 11 and 29% in the deep zone. In unconfined compression testing, cell height decreased by 32% in the superficial zone, 25% in the middle, and 18% in the deep zones. This predicted non-uniformity is in agreement with experimental studies. The novelty of this study is the use of a cartilage material model accounting for the intrinsic inhomogeneity and anisotropy of cartilage caused by its microstructure.     

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.     

Fluid transport and mechanical properties of articular cartilage: a review

This review is aimed at unifying our understanding of cartilage viscoelastic properties in compression, in particular the role of compression-dependent permeability in controlling interstitial fluid flow and its contribution to the observed viscoelastic effects. During the previous decade, it was shown that compression causes the permeability of cartilage to drop in a functional manner described by k = ko exp (epsilon M) where ko and M were defined as intrinsic permeability parameters and epsilon is the dilatation of the solid matrix (epsilon = tr delta u). Since permeability is inversely related to the diffusive drag coefficient of relative fluid motion with respect to the porous solid matrix, the measured load-deformation response of the tissue must therefore also depend on the non-linearly permeable nature of the tissue. We have summarized in this review our understanding of this non-linear phenomenon. This understanding of these flow-dependent viscoelastic effects are put into the historical perspective of a comprehensive literature review of earlier attempts to model the compressive viscoelastic properties of articular cartilage.     

Transport of neutral solute in articular cartilage: effect of microstructure anisotropy

Due to the avascular nature of articular cartilage, solute transport through its extracellular matrix is critical for the maintenance and the functioning of the tissue. What is more, diffusion of macromolecules may be affected by the microstructure of the extracellular matrix in both undeformed and deformed cartilage and experiments demonstrate diffusion anisotropy in the case of large solute. However, these phenomena have not received sufficient theoretical attention to date. We hypothesize here that the diffusion anisotropy of macromolecules is brought about by the particular microstructure of the cartilage network. Based on this hypothesis, we then propose a mathematical model that correlates the diffusion coefficient tensor with the structural orientation tensor of the network. This model is shown to be successful in describing anisotropic diffusion of macromolecules in undeformed tissue and is capable of clarifying the effects of network reorientation as the tissue deforms under mechanical load. Additionally, our model explains the anomaly that at large strain, in a cylindrical plug under unconfined compression, solute diffusion in the radial direction increases with strain. Our results indicate that in cartilage the degree of diffusion anisotropy is site specific, but depends also on the size of the diffusing molecule. Mechanical loading initiates and/or further exacerbates this anisotropy. At small deformation, solute diffusion is near isotropic in a tissue that is isotropic in its unstressed state, becoming anisotropic as loading progresses. Mechanical loading leads to an attenuation of solute diffusion in all directions when deformation is small. However, loading, if it is high enough, enhances solute transport in the direction perpendicular to the load line, instead of inhibiting it.     

A case for poroelasticity in skeletal muscle finite element analysis: experiment and modeling

Finite element models of skeletal muscle typically ignore the biphasic nature of the tissue, associating any time dependence with a viscoelastic formulation. In this study, direct experimental measurement of permeability was conducted as a function of specimen orientation and strain. A finite element model was developed to identify how various permeability formulations affect compressive response of the tissue. Experimental and modeling results suggest the assumption of a constant, isotropic permeability is appropriate. A viscoelastic only model differed considerably from a visco-poroelastic model, suggesting the latter is more appropriate for compressive studies.     

Mechano-acoustic determination of Young's modulus of articular cartilage

The compressive stiffness of an elastic material is traditionally characterized by its Young's modulus. Young's modulus of articular cartilage can be directly measured using unconfined compression geometry by assuming the cartilage to be homogeneous and isotropic. In isotropic materials, Young's modulus can also be determined acoustically by the measurement of sound speed and density of the material. In the present study, acoustic and mechanical techniques, feasible for in vivo measurements, were investigated to quantify the static and dynamic compressive stiffness of bovine articular cartilage in situ. Ultrasound reflection from the cartilage surface, as well as the dynamic modulus were determined with the recently developed ultrasound indentation instrument and compared with the reference mechanical and ultrasound speed measurements in unconfined compression (n=72). In addition, the applicability of manual creep measurements with the ultrasound indentation instrument was evaluated both experimentally and numerically. Our experimental results indicated that the sound speed could predict 47% and 53% of the variation in the Young's modulus and dynamic modulus of cartilage, respectively. The dynamic modulus, as determined manually with the ultrasound indentation instrument, showed significant linear correlations with the reference Young's modulus (r(2)=0.445, p<0.01, n=70) and dynamic modulus (r(2)=0.779, p<0.01, n=70) of the cartilage. Numerical analyses indicated that the creep measurements, conducted manually with the ultrasound indentation instrument, were sensitive to changes in Young's modulus and permeability of the tissue, and were significantly influenced by the tissue thickness. We conclude that acoustic parameters, i.e. ultrasound speed and reflection, are indicative to the intrinsic mechanical properties of the articular cartilage. Ultrasound indentation instrument, when further developed, provides an applicable tool for the in vivo detection of cartilage mechano-acoustic properties. These techniques could promote the diagnostics of osteoarthrosis.     

An instrumented tissue tester for measuring soft tissue property under the metatarsal heads in relation to metatarsophalangeal joint angle

Identification of the localized mechanical response of the plantar soft tissue pads underneath the metatarsal heads (i.e., sub-MTH pad) to external loading is key to understand and predict how it functions in a gait cycle. The mechanical response depends on various parameters, such as the external load (direction and rate), the sub-MTH tissue properties (anisotropy and viscoelasticity), and the configuration of the metatarsophalangeal (MTP) joint overlying the tissue. In this study, an instrument-driven tissue tester that incorporates a portable motorized indentor within a special foot positioning apparatus was developed for realistic in vivo mechanical characterization (i.e. tissue stiffness and force relaxation behavior) of the local sub-MTH pad with the MTP joint configured at various dorsiflexion angles associated with gait. The tester yields consistent results for tests on the 2nd sub-MTH pad. Measurement errors for the initial stiffness (for indentation depths ≤ 1 mm), end-point stiffness, and percentage force relaxation were less than 0.084 N/mm, 0.133 N/mm, and 0.127%, respectively, across all test configurations. The end-point tissue stiffness, which increased by 104.2% due to a 50° MTP joint dorsiflexion, also agreed with a previous investigation. In vivo tissue's force relaxation was shown to be pronounced (avg. = 8.1%), even for a short holding-time interval. The proposed technique to facilitate study of the dependence of the local sub-MTH pad and tissue response on the MTP joint angle might be preferable to methods that focus solely on measurement of tissue property because under physiologic conditions the sub-MTH pad elasticity may vary in gait, to adapt to drastically changing mechanical demands in the sub-MTH region of the terminal stance-phase, where MTP joint dorsiflexion occurs.     

Effect of the anisotropic permeability in the frequency dependent properties of the superficial layer of articular cartilage

Abstract

Articular cartilage is a tissue of fundamental importance for the mechanics of joints, since it provides a smooth and lubricated surface for the proper transfer of loads. From a mechanical point of view, this tissue is an anisotropic poroviscoelastic material: its characteristics at the macroscopic level depend on the complex microscopic architecture. With the ability to probe the local microscopic features, dynamic nanoindentation test is a powerful tool to investigate cartilage mechanics. In this work we focus on a length scale where the time dependent behaviour is regulated by poroelasticity more than viscoelasticity and we aim to understand the effect of the anisotropic permeability on the mechanics of the superficial layer of the articular cartilage. In a previous work, a finite element model for the dynamic nanoindentation test has been presented. In this work, we improve the model by considering the presence of an anisotropic permeability tensor that depends on the collagen fibers distribution. Our sensitivity analysis highlights that the permeability decreases with increasing indentation, thus making the tissue stiffer than the case of isotropic permeability, when solicited at the same frequency. With this improved model, a revised identification of the mechanical and physical parameters for articular cartilage is provided. To this purpose the model was used to simulate experimental data from tests performed on bovine tissue, giving a better estimation of the anisotropy in the elastic properties. A relation between the identified macroscopic anisotropic permeability properties and the microscopic rearrangement of the fiber/matrix structure during indentation is also provided.     

Transport phenomena in articular cartilage cryopreservation as predicted by the modified triphasic model and the effect of natural inhomogeneities

Knowledge of the spatial and temporal distribution of cryoprotective agent (CPA) is necessary for the cryopreservation of articular cartilage. Cartilage dehydration and shrinkage, as well as the change in extracellular osmolality, may have a significant impact on chondrocyte survival during and after CPA loading, freezing, and thawing, and during CPA unloading. In the literature, Fick's law of diffusion is commonly used to predict the spatial distribution and overall concentration of the CPA in the cartilage matrix, and the shrinkage and stress-strain in the cartilage matrix during CPA loading are neglected. In this study, we used a previously described biomechanical model to predict the spatial and temporal distributions of CPA during loading. We measured the intrinsic inhomogeneities in initial water and fixed charge densities in the cartilage using magnetic resonance imaging and introduced them into the model as initial conditions. We then compared the prediction results with the results obtained using uniform initial conditions. The simulation results in this study demonstrate the presence of a significant mechanical strain in the matrix of the cartilage, within all layers, during CPA loading. The osmotic response of the chondrocytes to the cartilage dehydration during CPA loading was also simulated. The results reveal that a transient shrinking occurs to different levels, and the chondrocytes experience a significant decrease in volume, particularly in the middle and deep zones of articular cartilage, during CPA loading.     

Biomechanical properties of human articular cartilage under compressive loads

The function of articular cartilage is to support and distribute loads and to provide lubrication in the diarthrodial joints. Cartilage function is described by proper mechanical and rheological properties, strain and depth-dependent, which are not completely assessed. Unconfined and confined compression are commonly used to evaluate the Young's modulus (E) and the aggregate modulus (H(A)), respectively. The Poisson's ratio (nu) can be calculated indirectly from the equilibrium compression data, or using the biphasic indentation technique; it has recently been optically evaluated by using video microscopy during unconfined compression. The transient response of articular cartilage during confined compression depends on its permeability k; a constant value of k can be easily identified by a simple analytical model of confined compression tests, whereas more complex models or direct measurements (permeation tests) are needed to study the permeability dependence on deformation. A poroelastic finite element model of articular cartilage was developed for this purpose. The elastic parameters (E,nu) of the model were evaluated performing unconfined compression creep tests on human articular cartilage disks, whereas k was identified from the confined test response. Our combined experimental and computational method can be used to identify the parameters that define the permeability dependence on deformation, as a function of depth from articular surface.     

Compressive fatigue and endurance of juvenile bovine articular cartilage explants

Articular cartilage is an enduring tissue. For most individuals, articular cartilage facilitates a lifetime of pain-free ambulation, supporting millions of loading cycles from activities of daily living. Although early studies into osteoarthritis focused on the role of mechanical fatigue in articular cartilage degeneration, much is still unknown regarding its strength and endurance characteristics. The compressive strength of juvenile, bovine articular cartilage explants was determined to be loading rate-dependent, reaching a maximum strength of 29.5 ± 4.8 MPa at a strain rate of 0.10 %/sec. The fatigue and endurance properties of articular cartilage were characterized utilizing a material testing system, as well as a custom, validated instrument termed the two degrees-of-freedom endurance meter (endurometer). These instruments characterized fatigue in articular cartilage explants at loading levels ranging from 10 to 80 % strength (%S), up to 100,000 cycles. Cartilage explants displayed characteristics of fatigue – fatigue life increased as the loading magnitude decreased. All explants failed within 14,000 cycles at loading levels between 50 and 80 %S. At 10 and 20 %S, all explants endured 100,000 loading cycles. There was no significant difference in equilibrium compressive modulus between run-out explants and unloaded controls, although the pooled modulus increased in response to testing. Histological staining and biochemical assays revealed no material changes in collagen, sulfated glycosaminoglycan, or hydration content between unloaded controls and explants cyclically loaded to run-out. These results suggest articular cartilage may have a putative endurance limit of 20 %S (5.86 MPa), with implications for articular cartilage biomechanics and mechanobiology.     

Chondrocyte deformations as a function of tibiofemoral joint loading predicted by a generalized high-throughput pipeline of multi-scale simulations

Cells of the musculoskeletal system are known to respond to mechanical loading and chondrocytes within the cartilage are not an exception. However, understanding how joint level loads relate to cell level deformations, e.g. in the cartilage, is not a straightforward task. In this study, a multi-scale analysis pipeline was implemented to post-process the results of a macro-scale finite element (FE) tibiofemoral joint model to provide joint mechanics based displacement boundary conditions to micro-scale cellular FE models of the cartilage, for the purpose of characterizing chondrocyte deformations in relation to tibiofemoral joint loading. It was possible to identify the load distribution within the knee among its tissue structures and ultimately within the cartilage among its extracellular matrix, pericellular environment and resident chondrocytes. Various cellular deformation metrics (aspect ratio change, volumetric strain, cellular effective strain and maximum shear strain) were calculated. To illustrate further utility of this multi-scale modeling pipeline, two micro-scale cartilage constructs were considered: an idealized single cell at the centroid of a 100×100×100 μm block commonly used in past research studies, and an anatomically based (11 cell model of the same volume) representation of the middle zone of tibiofemoral cartilage. In both cases, chondrocytes experienced amplified deformations compared to those at the macro-scale, predicted by simulating one body weight compressive loading on the tibiofemoral joint. In the 11 cell case, all cells experienced less deformation than the single cell case, and also exhibited a larger variance in deformation compared to other cells residing in the same block. The coupling method proved to be highly scalable due to micro-scale model independence that allowed for exploitation of distributed memory computing architecture. The method's generalized nature also allows for substitution of any macro-scale and/or micro-scale model providing application for other multi-scale continuum mechanics problems.     

Glycosaminoglycan network geometry may contribute to anisotropic hydraulic permeability in cartilage under compression.

Resistance to fluid flow within cartilage extracellular matrix is provided primarily by a dense network of rod-like glycosaminoglycans (GAGs). If the geometrical organization of this network is random, the hydraulic permeability tensor of cartilage is expected to be isotropic. However, experimental data have suggested that hydraulic permeability may become anisotropic when the matrix is mechanically compressed, contributing to cartilage biomechanical functions such as lubrication. We hypothesized that this may be due to preferred GAG rod orientations and directionally-dependent reduction of inter-GAG spacings which reflect molecular responses to tissue deformations. To examine this hypothesis, we developed a model for effects of compression which allows the GAG rod network to deform consistently with tissue-scale deformations but while still respecting limitations imposed by molecular structure. This network deformation model was combined with a perturbation analysis of a classical analytical model for hydraulic permeability based on molecular structure. Finite element analyses were undertaken to ensure that this approach exhibited results similar to those emerging from more exact calculations. Model predictions for effects of uniaxial confined compression on the hydraulic permeability tensor were consistent with previous experimental results. Permeability decreased more rapidly in the direction perpendicular to compression than in the parallel direction, for matrix solid volume fractions associated with fluid transport in articular cartilage. GAG network deformations may therefore introduce anisotropy to the permeability (and other GAG-associated matrix properties) as physiological compression is applied, and play an important role in cartilage lubrication and other biomechanical functions.