Cartilage electromechanics--I. Electrokinetic transduction and the effects of electrolyte pH and ionic strength

Articular cartilage contains a high fixed charge density under physiological conditions associated primarily with the ionized proteoglycan molecules of the extracellular matrix. Oscillatory compression of cartilage using physiological loads produces electrical potentials that have been shown previously to be the result of an electrokinetic (streaming) transduction mechanism. We have now observed two additional electromechanical phenomena not previously seen in cartilage or other soft tissues: 'streaming current' and 'current-generated stress'. Sinusoidal mechanical compression induced a sinusoidal streaming current density through cartilage disks when the Ag/AgCl electrodes that were used to compress the cartilage were shorted together externally. Conversely, a sinusoidal current density applied to the tissue generated a sinusoidal mechanical stress within the tissue. Both these phenomena were found to be consistent with the same electrokinetic transduction mechanism responsible for the streaming potential. Changes in the measured streaming potential response that resulted from modification of bath ionic strength and pH have provided additional insights into the molecular origins of these transduction processes. Finally, we have now observed streaming potentials in living cartilage maintained in organ culture, as well as in previously frozen tissue.     

Evidence for increased de novo synthesis of NAD in immune-activated RAW264.7 macrophages: a self-protective mechanism?

Chondrogenesis in cartilage development and repair and cartilage degeneration in arthritis can be regulated by mechanical-load-induced physical factors such as tissue deformation, interstitial fluid flow and pressure, and electrical fields or streaming potentials. Previous animal and tissue explant studies have shown that time-varying dynamic tissue loading can increase the synthesis and deposition of matrix molecules in an amplitude-, frequency-, and spatially dependent manner. To provide information on the cell-level physical factors which may stimulate chondrocytes to increase production and export of aggrecan, the main proteoglycan component of the cartilage matrix, we characterized local changes in aggrecan synthesis within cyclically loaded tissue explant disks and compared those changes to values of predicted local physical factors. Aggrecan synthesis following a 23-h compression/radiolabel protocol was measured with a spatial resolution of approximately 0.1 mm across the 1.5-mm radius of explanted disks using a quantitative autoradiography method. A uniform stimulation of aggrecan synthesis was observed at an intermediate frequency of 0.01 Hz, while, at a higher frequency of 0.1 Hz, stimulation was only seen at peripheral radial positions. Profiles of radial solid matrix deformation and interstitial fluid pressure and velocity predicted to be occurring across the radius of the disk during sinusoidal loading were estimated using a composite poroelastic model. Tissue regions experiencing high interstitial fluid velocities corresponded to those displaying increased aggrecan synthesis. These results reinforce the role of load-induced flow of interstitial fluid in the stimulation of aggrecan production during dynamic loading of cartilage.     

Hydrostatic pressurization and depletion of trapped lubricant pool during creep contact of a rippled indenter against a biphasic articular cartilage layer

This study presents an analysis of the contact of a rippled rigid impermeable indenter against a cartilage layer, which represents a first simulation of the contact of rough cartilage surfaces with lubricant entrapment. Cartilage was modeled with the biphasic theory for hydrated soft tissues, to account for fluid flow into or out of the lubricant pool. The findings of this study demonstrate that under contact creep, the trapped lubricant pool gets depleted within a time period on the order of seconds or minutes as a result of lubricant flow into the articular cartilage. Prior to depletion, hydrostatic fluid load support across the contact interface may be enhanced by the presence of the trapped lubricant pool, depending on the initial geometry of the lubricant pool. According to friction models based on the biphasic nature of the tissue, this enhancement in fluid load support produces a smaller minimum friction coefficient than would otherwise be predicted without a lubricant pool. The results of this study support the hypothesis that trapped lubricant decreases the initial friction coefficient following load application, independently of squeeze-film lubrication effects.     

Unconfined compression of articular cartilage: nonlinear behavior and comparison with a fibril-reinforced biphasic model

Mechanical behavior of articular cartilage was characterized in unconfined compression to delineate regimes of linear and nonlinear behavior, to investigate the ability of a fibril-reinforced biphasic model to describe measurements, and to test the prediction of biphasic and poroelastic models that tissue dimensions alter tissue stiffness through a specific scaling law for time and frequency. Disks of full-thickness adult articular cartilage from bovine humeral heads were subjected to successive applications of small-amplitude ramp compressions cumulating to a 10 percent compression offset where a series of sinusoidal and ramp compression and ramp release displacements were superposed. We found all equilibrium behavior (up to 10 percent axial compression offset) to be linear, while most nonequilibrium behavior was nonlinear, with the exception of small-amplitude ramp compressions applied from the same compression offset. Observed nonlinear behavior included compression-offset-dependent stiffening of the transient response to ramp compression, nonlinear maintenance of compressive stress during release from a prescribed offset, and a nonlinear reduction in dynamic stiffness with increasing amplitudes of sinusoidal compression. The fibril-reinforced biphasic model was able to describe stress relaxation response to ramp compression, including the high ratio of peak to equilibrium load. However, compression offset-dependent stiffening appeared to suggest strain-dependent parameters involving strain-dependent fibril network stiffness and strain-dependent hydraulic permeability. Finally, testing of disks of different diameters and rescaling of the frequency according to the rule prescribed by current biphasic and poroelastic models (rescaling with respect to the sample's radius squared) reasonably confirmed the validity of that scaling rule. The overall results of this study support several aspects of current theoretical models of articular cartilage mechanical behavior, motivate further experimental characterization, and suggest the inclusion of specific nonlinear behaviors to models.     

Cartilage thickness distribution affects computational model predictions of cervical spine facet contact parameters

With motion-sparing disk replacement implants gaining popularity as an alternative to anterior cervical discectomy and fusion (ACDF) for the treatment of certain spinal degenerative disorders, recent laboratory investigations have studied the effects of disk replacement and implant design on spinal kinematics and kinetics. Particularly relevant to cervical disk replacement implant design are any postoperative changes in solid stresses or contact conditions in the articular cartilage of the posterior facets, which are hypothesized to lead to adjacent-level degeneration. Such changes are commonly investigated using finite element methods, but significant simplification of the articular geometry is generally employed. The impact of such geometric representations has not been thoroughly investigated. In order to assess the effects of different models of cartilage geometry on load transfer and contact pressures in the lower cervical spine, a finite element model was generated using cadaver-based computed tomography imagery. Mesh resolution was varied in order to establish model convergence, and cadaveric testing was undertaken to validate model predictions. The validated model was altered to include four different geometric representations of the articular cartilage. Model predictions indicate that the two most common representations of articular cartilage geometry result in significant reductions in the predictive accuracy of the models. The two anatomically based geometric models exhibited less computational artifact, and relatively minor differences between them indicate that contact condition predictions of spatially varying thickness models are robust to anatomic variations in cartilage thickness and articular curvature. The results of this work indicate that finite element modeling efforts in the lower cervical spine should include anatomically based and spatially varying articular cartilage thickness models. Failure to do so may result in loss of fidelity of model predictions relevant to investigations of physiological import.     

Hemiarthroplasty of hip joint: An experimental validation using porcine acetabulum

Biphasic properties of articular cartilage allow it to be an excellent bearing material and have been studied through several simplified experiments as well as finite element modelling. However, three-dimensional biphasic finite element (FE) models of the whole joint are rare. The current study was carried out to experimentally validate FE methodology for modelling hemiarthroplasty. Material properties such as equilibrium elastic modulus and permeability of porcine acetabular cartilage were initially derived by curve-fitting an experimental deformation curve with that obtained using FE. These properties were then used in the hemiarthroplasty hip joint modelling. Each porcine acetabular cup was loaded with 400N using a 34mm diameter CoCr femoral head. A specimen-specific FE model of each acetabular cup was created using μCT and a series of software processes. Each model was analysed under conditions similar to those tested experimentally. Contact stresses and contact areas predicted by the model, immediately after loading, were then compared with the corresponding experimentally measured values. Very high peak contact stresses (maximum experimental: 14.09MPa) were recorded. A maximum difference of 12.42% was found in peak contact stresses. The corresponding error for contact area was 20.69%. Due to a fairly good agreement in predicted and measured values of contact stresses and contact areas, the integrated methodology developed in this study can be used as a basis for future work. In addition, FE predicted total fluid load support was around 80% immediately after loading. This was lower than that observed in conforming contact problems involving biphasic cartilage and was due to a smaller local contact area and variable clearance making fluid exudation easier.     

An analytical model to predict interstitial lubrication of cartilage in migrating contact areas

For nearly a century, articular cartilage has been known for its exceptional tribological properties. For nearly as long, there have been research efforts to elucidate the responsible mechanisms for application toward biomimetic bearing applications. It is now widely accepted that interstitial fluid pressurization is the primary mechanism responsible for the unusual lubrication and load bearing properties of cartilage. Although the biomechanics community has developed elegant mathematical theories describing the coupling of solid and fluid (biphasic) mechanics and its role in interstitial lubrication, quantitative gaps in our understanding of cartilage tribology have inhibited our ability to predict how tribological conditions and material properties impact tissue function. This paper presents an analytical model of the interstitial lubrication of biphasic materials under migrating contact conditions. Although finite element and other numerical models of cartilage mechanics exist, they typically neglect the important role of the collagen network and are limited to a specific set of input conditions, which limits general applicability. The simplified approach taken in this work aims to capture the broader underlying physics as a starting point for further model development. In agreement with existing literature, the model indicates that a large Peclet number, Pe, is necessary for effective interstitial lubrication. It also predicts that the tensile modulus must be large relative to the compressive modulus. This explains why hydrogels and other biphasic materials do not provide significant interstitial pressure under high Pe conditions. The model quantitatively agrees with in-situ measurements of interstitial load support and the results have interesting implications for tissue engineering and osteoarthritis problems. This paper suggests that a low tensile modulus (from chondromalacia or local collagen rupture after impact, for example) may disrupt interstitial pressurization, increase shear stresses, and activate a condition of progressive surface damage as a potential precursor of osteoarthritis.     

Biphasic poroviscoelastic simulation of the unconfined compression of articular cartilage: I--Simultaneous prediction of reaction force and lateral displacement

This study investigated the ability of the linear biphasic poroelastic (BPE) model and the linear biphasic poroviscoelastic (BPVE) model to simultaneously predict the reaction force and lateral displacement exhibited by articular cartilage during stress relaxation in unconfined compression. Both models consider articular cartilage as a binary mixture of a porous incompressible solid phase and an incompressible inviscid fluid phase. The BPE model assumes the solid phase is elastic, while the BPVE model assumes the solid phase is viscoelastic. In addition, the efficacy of two additional models was also examined, i.e., the transversely isotropic BPE (TIBPE) model, which considers transverse isotropy of the solid matrix within the framework of the linear BPE model assumptions, and a linear viscoelastic solid (LVE) model, which assumes that the viscoelastic behavior of articular cartilage is solely governed by the intrinsic viscoelastic nature of the solid matrix, independent of the interstitial fluid flow. It was found that the BPE model was able to accurately account for the lateral displacement, but unable to fit the short-term reaction force data of all specimens tested. The TIBPE model was able to account for either the lateral displacement or the reaction force, but not both simultaneously. The LVE model was able to account for the complete reaction force, but unable to fit the lateral displacement measured experimentally. The BPVE model was able to completely account for both lateral displacement and reaction force for all specimens tested. These results suggest that both the fluid flow-dependent and fluid flow-independent viscoelastic mechanisms are essential for a complete simulation of the viscoelastic phenomena of articular cartilage.     

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.     

Cartilage electromechanics--II. A continuum model of cartilage electrokinetics and correlation with experiments

We have formulated a continuum model for linear electrokinetic transduction in cartilage. Expressions are derived for the streaming potential and streaming current induced by oscillatory, uniaxial confined compression of the tissue, as well as the mechanical stress generated by a current density or potential difference applied to the tissue. The experimentally observed streaming potential and current-generated stress response, measured on the same specimens, are compared with the predictions of the theory over a wide frequency range. The theory compares well with the data for reasonable values of cartilage intrinsic mechanical parameters and electrokinetic coupling coefficients. Experiments also show a linear relationship between the stimulus amplitude and the transduction response amplitude, within the range of stimulus amplitudes of interest. This observation is shown to be consistent with the predictions of the linear theory.     

A biphasic multiscale study of the mechanical microenvironment of chondrocytes within articular cartilage under unconfined compression

Computational analyses have been used to study the biomechanical microenvironment of the chondrocyte that cannot be assessed by in vitro experimental studies; yet all computational studies thus far have focused on the effect of zonal location (superficial, middle, and deep) on the mechanical microenvironment of chondrocytes. The aim of this paper was to study the effect of both zonal and radial locations on the biomechanical microenvironment of chondrocytes in inhomogeneous cartilage under unconfined stress relaxation. A biphasic multiscale approach was employed and nine chondrocytes in different locations were studied. Hyperelastic biphasic theory and depth-dependent aggregate modulus and permeability of articular cartilage were included in the models. It was found that both zonal and radial locations affected the biomechanical stresses and strains of the chondrocytes. Chondrocytes in the mid-radial location had increased volume during the early stage of the loading process. Maximum principal shear stress at the interface between the chondrocyte and the extracellular matrix (ECM) increased with depth, yet that at the ECM–pericellular matrix (PCM) interface had an inverse trend. Fluid pressure decreased with depth, while the fluid pressure difference between the top and bottom boundaries of the microscale model increased with depth. Regardless of location, fluid was exchanged between the chondrocyte, PCM, and ECM. These findings suggested that even under simple compressive loading conditions, the biomechanical microenvironment of the chondrocytes, PCM and ECM was spatially dependent. The current study provides new insight on chondrocyte biomechanics.     

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.     

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.     

Fluid load support during localized indentation of cartilage with a spherical probe

Interstitial fluid pressurization, a consequence of a biphasic tissue structure, is essential to the load bearing and lubrication properties of articular cartilage. Focal tissue degradation may interfere with this protective mechanism, eventually leading to gross degeneration and osteoarthritis. Our long-term goal is to determine whether local contacts can be used as a means to probe local tissue integrity and functionality. In the present work, Hertzian rate-controlled microindentation was used as a model of the more complicated sliding system to directly determine the effects of contact radius and deformation rate on interstitial load support. During localized contact between a steel spherical probe and bovine articular cartilage, the equilibrium and non-equilibrium responses were well-fit by the Hertz model (R(2)>0.998) with a mean equilibrium contact modulus of 0.93 MPa. The effective contact modulus and fluid load fraction were independent of indentation depth, contact radius, and normal force; both increased monotonically with indentation rate. At 21 μm/s indentation rate, the cartilage was effectively stiffened by 6-fold with the fluid pressure supporting 85% of the contact force. The results motivated a simple analytical model that directly links the tribomechanical response (including fluid load support) and the Peclet number to measurable material properties and controllable experimental variables. This paper demonstrates that tribological contacts can be used to probe local functional properties. Such measurements can add important insights into the roles of focal tissue damage and impaired local functionality in the pathogenesis of osteoarthritis.     

The influence of the fixed negative charges on mechanical and electrical behaviors of articular cartilage under unconfined compression

Unconfined compression test has been frequently used to study the mechanical behaviors of articular cartilage, both theoretically and experimentally. It has also been used in explant and gel-cell-complex studies in tissue engineering. In biphasic and poroelastic theories, the effect of charges fixed on the proteoglycan macromolecules in articular cartilage is embodied in the apparent compressive Young's modulus and the apparent Poisson's ratio of the tissue, and the fluid pressure is considered to be the portion above the osmotic pressure. In order to understand how proteoglycan fixed charges might affect the mechanical behaviors of articular cartilage, and in order to predict the osmotic pressure and electric fields inside the tissue in this experimental configuration, it is necessary to use a model that explicitly takes into account the charged nature of the tissue and the flow of ions within its porous interstices. In this paper, we used a finite element model based on the triphasic theory to study how fixed charges in the porous-permeable soft tissue can modulate its mechanical and electrochemical responses under a step displacement in unconfined compression. The results from finite element calculations showed that: 1) A charged tissue always supports a larger load than an uncharged tissue of the same intrinsic elastic moduli. 2) The apparent Young's modulus (the ratio of the equilibrium axial stress to the axial strain) is always greater than the intrinsic Young's modulus of an uncharged tissue. 3) The apparent Poisson's ratio (the negative ratio of the lateral strain to the axial strain) is always larger than the intrinsic Poisson's ratio of an uncharged tissue. 4) Load support derives from three sources: intrinsic matrix stiffness, hydraulic pressure and osmotic pressure. Under the unconfined compression, the Donnan osmotic pressure can constitute between 13%-22% of the total load support at equilibrium. 5) During the stress-relaxation process following the initial instant of loading, the diffusion potential (due to the gradient of the fixed charge density and the associated gradient of ion concentrations) and the streaming potential (due to fluid convection) compete against each other. Within the physiological range of material parameters, the polarity of the electric potential depends on both the mechanical properties and the fixed charge density (FCD) of the tissue. For softer tissues, the diffusion effects dominate the electromechanical response, while for stiffer tissues, the streaming potential dominates this response. 6) Fixed charges do not affect the instantaneous strain field relative to the initial equilibrium state. However, there is a sudden increase in the fluid pressure above the initial equilibrium osmotic pressure. These new findings are relevant and necessary for the understanding of cartilage mechanics, cartilage biosynthesis, electromechanical signal transduction by chondrocytes, and tissue engineering.     

On the electric potentials inside a charged soft hydrated biological tissue: streaming potential versus diffusion potential

The main objective of this study is to determine the nature of electric fields inside articular cartilage while accounting for the effects of both streaming potential and diffusion potential. Specifically, we solve two tissue mechano-electrochemical problems using the triphasic theories developed by Lai et al. (1991, ASME J. Biomech Eng., 113, pp. 245-258) and Gu et al. (1998, ASME J. Biomech. Eng., 120, pp. 169-180) (1) the steady one-dimensional permeation problem; and (2) the transient one-dimensional ramped-displacement, confined-compression, stress-relaxation problem (both in an open circuit condition) so as to be able to calculate the compressive strain, the electric potential, and the fixed charged density (FCD) inside cartilage. Our calculations show that in these two technically important problems, the diffusion potential effects compete against the flow-induced kinetic effects (streaming potential) for dominance of the electric potential inside the tissue. For softer tissues of similar FCD (i.e., lower aggregate modulus), the diffusion potential effects are enhanced when the tissue is being compressed (i.e., increasing its FCD in a nonuniform manner) either by direct compression or by drag-induced compaction; indeed, the diffusion potential effect may dominate over the streaming potential effect. The polarity of the electric potential field is in the same direction of interstitial fluid flow when streaming potential dominates, and in the opposite direction of fluid flow when diffusion potential dominates. For physiologically realistic articular cartilage material parameters, the polarity of electric potential across the tissue on the outside (surface to surface) may be opposite to the polarity across the tissue on the inside (surface to surface). Since the electromechanical signals that chondrocytes perceive in situ are the stresses, strains, pressures and the electric field generated inside the extracellular matrix when the tissue is deformed, the results from this study offer new challenges for the understanding of possible mechanisms that control chondrocyte biosyntheses.     

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.     

Three-dimensional finite element modeling of pericellular matrix and cell mechanics in the nucleus pulposus of the intervertebral disk based on in situ morphology

Nucleus pulposus (NP) cells of the intervertebral disk (IVD) have unique morphological characteristics and biologic responses to mechanical stimuli that may regulate maintenance and health of the IVD. NP cells reside as single cell, paired or multiple cells in a contiguous pericellular matrix (PCM), whose structure and properties may significantly influence cell and extracellular matrix mechanics. In this study, a computational model was developed to predict the stress–strain, fluid pressure and flow fields for cells and their surrounding PCM in the NP using three-dimensional (3D) finite element models based on the in situ morphology of cell–PCM regions of the mature rat NP, measured using confocal microscopy. Three-dimensional geometries of the extracellular matrix and representative cell–matrix units were used to construct 3D finite element models of the structures as isotropic and biphasic materials. In response to compressive strain of the extracellular matrix, NP cells and PCM regions were predicted to experience volumetric strains that were 1.9–3.7 and 1.4–2.1 times greater than the extracellular matrix, respectively. Volumetric and deviatoric strain concentrations were generally found at the cell/PCM interface, while von Mises stress concentrations were associated with the PCM/extracellular matrix interface. Cell–matrix units containing greater cell numbers were associated with higher peak cell strains and lower rates of fluid pressurization upon loading. These studies provide new model predictions for micromechanics of NP cells that can contribute to an understanding of mechanotransduction in the IVD and its changes with aging and degeneration.     

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.