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.     

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.     

Analysis of the mechanical behavior of chondrocytes in unconfined compression tests for cyclic loading

Experimental evidence indicates that the biosynthetic activity of chondrocytes is associated with the mechanical environment. For example, excessive, repetitive loading has been found to induce cell death, morphological and cellular damage, as seen in degenerative joint disease, while cyclic, physiological-like loading has been found to trigger a partial recovery of morphological and ultrastructural aspects in osteoarthritic human articular chondrocytes. Mechanical stimuli are believed to influence the biosynthetic activity via the deformation of cells. However, the in situ deformation of chondrocytes for cyclic loading conditions has not been investigated experimentally or theoretically. The purpose of the present study was to simulate the mechanical response of chondrocytes to cyclic loading in unconfined compression tests using a finite element model. The material properties of chondrocytes and extracellular matrix were considered to be biphasic. The time-histories of the shape and volume variations of chondrocytes at three locations (i.e., surface, center, and bottom) within the cartilage were predicted for static and cyclic loading conditions at two frequencies (0.02 and 0.1 Hz) and two amplitudes (0.1 and 0.2 MPa). Our results show that cells at different depths within the cartilage deform differently during cyclic loading, and that the depth dependence of cell deformation is influenced by the amplitude of the cyclic loading. Cell deformations under cyclic loading of 0.02 Hz were found to be similar to those at 0.1 Hz. We conclude from the simulation results that, in homogeneous cartilage layers, cell deformations are location-dependent, and further are affected by load magnitude. In physiological conditions, the mechanical environment of cells are even more complex due to the anisotropy, depth-dependent inhomogeneity, and tension-compression non-linearity of the cartilage matrix. Therefore, it is feasible to speculate that biosynthetic responses of chondrocytes to cyclic loading depend on cell location and load magnitude.     

Depth-dependent analysis of the role of collagen fibrils, fixed charges and fluid in the pericellular matrix of articular cartilage on chondrocyte mechanics

The pericellular matrix of articular cartilage has been shown to regulate the mechanical environment of chondrocytes. However, little is known about the mechanical role of collagen fibrils in the pericellular matrix, and how fibrils might help modulate strains acting on chondrocytes when cartilage is loaded. The primary objective was to clarify the effect of pericellular collagen fibrils on cell volume changes and strains during cartilage loading. Secondary objectives were to investigate the effects of pericellular fixed charges and fluid on cell responses. A microstructural model of articular cartilage, in which chondrocytes and pericellular matrices were represented with depth-dependent structural and morphological properties, was created. The extracellular matrix and pericellular matrices were modeled as fibril-reinforced, biphasic materials with swelling capabilities, while chondrocytes were assumed to be isotropic and biphasic with swelling properties. Collagen fibrils in the extracellular matrix were represented with an arcade-like architecture, whereas pericellular fibrils were assumed to run tangential to the cell surface. In the early stages of a stress-relaxation test, pericellular fibrils were found to sensitively affect cell volume changes, even producing a reversal from increasing to decreasing cell volume with increasing fibril stiffness in the superficial zone. Consequently, steady-state volume of the superficial zone cell decreased with increasing pericellular fibril stiffness. Volume changes in the middle and deep zone chondrocytes were smaller and opposite to those observed in the superficial zone chondrocyte. An increase in the pericellular fixed charge density reduced cell volumes substantially in every zone. The sensitivity of cell volume changes to pericellular fibril stiffness suggests that pericellular fibrils play an important, and as of yet largely neglected, role in regulating the mechanical environment of chondrocytes, possibly affecting matrix synthesis during cartilage development and degeneration, and affecting biosynthetic responses associated with articular cartilage loading.     

Simulating the swelling and deformation behaviour in soft tissues using a convective thermal analogy

Background  

It is generally accepted that cartilage adaptation and degeneration are mechanically mediated. Investigating the swelling behaviour of cartilage is important because the stress and strain state of cartilage is associated with the swelling and deformation behaviour. It is well accepted that the swelling of soft tissues is associated with mechanical, chemical, and electrical events.     

Short-term changes in cell and matrix damage following mechanical injury of articular cartilage explants and modelling of microphysical mediators

The short-term responses of articular cartilage to mechanical injury have important implications for prevention and treatment of degenerative disease. Cell and matrix responses were monitored for 11 days following injurious compression of cartilage in osteochondral explants. Injury was applied as a single ramp compression to 14 MPa peak stress at one of three strain rates: 7 x 10(-1), 7 x 10(-3) or 7 x 10(-5) s(-1). Responses were quantified in terms of the appearance of macroscopic matrix cracks, changes in cell viability, and changes in cartilage wet weights. Loading at the highest strain rate resulted in acute cell death near the superficial zone in association with cracks, followed over the 11 days after compression by a gradual increase in cell death and loss of demarcation between matrix zones containing viable versus nonviable cells. In contrast, loading at the lowest strain rate resulted in more severe, nearly full-depth cell death acutely, but with no apparent worsening over the 11 days following compression. Between days 4 and 11, all mechanically injured explants significantly increased in wet weight, suggesting loss of matrix mechanical integrity independent of compression strain rate. Results demonstrate that short-term responses of cartilage depend upon the biomechanical characteristics of injurious loading, and suggest multiple independent pathways of mechanically-induced cell death and matrix degradation. Modifications to an existing fiber-reinforced poroelastic finite element model were introduced and the model was used for data interpretation and identification of microphysical events involved in cell and matrix injury. The model performed reasonably well at the slower strain rates and exhibited some capacity for anticipating the formation of superficial cracks during injurious loading. However, several improvements appear to be necessary before such a model could reliably be used to draw upon in vitro experimental results for prediction of injurious loading situations in vivo.     

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.     

力学环境对软骨基质代谢的影响

正常关节软骨所受压力是由动态压力与静态压力交替完成。压力引起软骨一系列生理变化包括细胞及细胞外基质成分变形、组织内液体流动、水流电位和生理生化变化。这些变化直接调控细胞外基质代谢。体外构建有良好功能的组织工程化软骨是目前软骨病变、缺损理想的修复方法。研究力学环境对软骨基质代谢的影响,对构建组织工程化软骨有深远意义。     

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.     

Functional tissue engineering of chondral and osteochondral constructs

Due to the prevalence of osteoarthritis (OA) and damage to articular cartilage, coupled with the poor intrinsic healing capacity of this avascular connective tissue, there is a great demand for an articular cartilage substitute. As the bearing material of diarthrodial joints, articular cartilage has remarkable functional properties that have been difficult to reproduce in tissue-engineered constructs. We have previously demonstrated that by using a functional tissue engineering approach that incorporates mechanical loading into the long-term culture environment, one can enhance the development of mechanical properties in chondrocyte-seeded agarose constructs. As these gel constructs begin to achieve material properties similar to that of the native tissue, however, new challenges arise, including integration of the construct with the underlying native bone. To address this issue, we have developed a technique for producing gel constructs integrated into an underlying bony substrate. These osteochondral constructs develop cartilage-like extracellular matrix and material properties over time in free swelling culture. In this study, as a preliminary to loading such osteochondral constructs, finite element modeling (FEM) was used to predict the spatial and temporal stress, strain, and fluid flow fields within constructs subjected to dynamic deformational loading. The results of these models suggest that while chondral ("gel alone") constructs see a largely homogenous field of mechanical signals, osteochondral ("gel bone") constructs see a largely inhomogeneous distribution of mechanical signals. Such inhomogeneity in the mechanical environment may aid in the development of inhomogeneity in the engineered osteochondral constructs. Together with experimental observations, we anticipate that such modeling efforts will provide direction for our efforts aimed at the optimization of applied physical forces for the functional tissue engineering of an osteochondral articular cartilage substitute.     

Effects of damage in the articular surface on the cartilage response to injurious compression in vitro

Macroscopic structural damage to the cartilage articular surface can occur due to slicing in surgery, cracking in mechanical trauma, or fibrillation in early stage osteoarthrosis. These alterations may render cartilage matrix and chondrocytes susceptible to subsequent mechanical injury and contribute to progression of degenerative disease. To examine this hypothesis, single 300 microm deep vertical slices were introduced across a diameter of the articular surface of osteochondral explant disks on day 6 after dissection. Then a single uniaxial unconfined ramp compression at 7 x 10(-5) or 7 x 10(-2) s(-1) strain rate to a peak stress of 3.5 or 14 MPa was applied on day 13 during which mechanical behavior was monitored. Effects of slices alone and together with compression were measured in terms of explant swelling and cell viability on days 10 and 17. Slicing alone induced tissue swelling without significant cell death, while compression alone induced cell death without significant tissue swelling. Under low strain rate loading, no differences in the response to injurious compression were found between sliced and unsliced explants. Under high strain rate loading, slicing rendered cartilage more easily compressible and appeared to slightly reduce compression-induced cell and matrix injury. Findings highlight microphysical factors important to cartilage mechanical injury, and suggest ways that macroscopic structural damage may accelerate or, in certain cases, possibly slow the progression of cartilage degeneration.     

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.     

The deformation behavior and viscoelastic properties of chondrocytes in articular cartilage

Chondrocytes in articular cartilage utilize mechanical signals in conjunction with other environmental factors to regulate their metabolic activity. However, the sequence of biomechanical and biochemical events involved in the process of mechanical signal transduction has not been fully deciphered. A fundamental step in determining the role of various factors in regulating chondrocyte activity is to characterize accurately the biophysical environment within the tissue under physiological conditions of mechanical loading. Microscopic imaging studies have revealed that chondrocytes as well as their nuclei undergo shape and volume changes in a coordinated manner with deformation of the tissue matrix. Through micromechanical experiments, it has been shown that the chondrocyte behaves as a viscoelastic solid material with a mechanical stiffness that is several orders of magnitude lower than that of the cartilage extracellular matrix. These properties seem to be due to the structure of the chondrocyte cytoskeleton, and in part, the viscoelastic properties of the cell nucleus. The mechanical properties of the pericellular matrix that immediately surrounds the chondrocyte significantly differ from those of the chondrocyte and the extracellular matrix, suggesting that the pericellular matrix plays an important role in defining the mechanical environment of the chondrocyte. These experimentally measured values for chondrocyte and cartilage mechanical properties have been used in combination with theoretical constitutive modeling of the chondrocyte within articular cartilage to predict the non-uniform and time-varying stress-strain and fluid flow environment of the cell. The ultimate goal of these studies has been to elucidate the sequence of biomechanical and biochemical events through which mechanical stress influences chondrocyte activity in both health and in disease.     

The mechanical environment of the chondrocyte: a biphasic finite element model of cell-matrix interactions in articular cartilage

Mechanical compression of the cartilage extracellular matrix has a significant effect on the metabolic activity of the chondrocytes. However, the relationship between the stress–strain and fluid-flow fields at the macroscopic “tissue” level and those at the microscopic “cellular” level are not fully understood. Based on the existing experimental data on the deformation behavior and biomechanical properties of articular cartilage and chondrocytes, a multi-scale biphasic finite element model was developed of the chondrocyte as a spheroidal inclusion embedded within the extracellular matrix of a cartilage explant. The mechanical environment at the cellular level was found to be time-varying and inhomogeneous, and the large difference (3 orders of magnitude) in the elastic properties of the chondrocyte and those of the extracellular matrix results in stress concentrations at the cell–matrix border and a nearly two-fold increase in strain and dilatation (volume change) at the cellular level, as compared to the macroscopic level. The presence of a narrow “pericellular matrix” with different properties than that of the chondrocyte or extracellular matrix significantly altered the principal stress and strain magnitudes within the chondrocyte, suggesting a functional biomechanical role for the pericellular matrix. These findings suggest that even under simple compressive loading conditions, chondrocytes are subjected to a complex local mechanical environment consisting of tension, compression, shear, and fluid pressure. Knowledge of the local stress and strain fields in the extracellular matrix is an important step in the interpretation of studies of mechanical signal transduction in cartilage explant culture models.     

Mechanoregulation of chondrocyte proliferation, maturation, and hypertrophy: ion-channel dependent transduction of matrix deformation signals

Mechanical stress-induced matrix deformation plays a fundamental role in regulating cellular activities; however, little is known about its underlying mechanisms. To understand the effects of matrix deformation on chondrocytes, we characterized primary chondrocytes cultured on three-dimensional collagen scaffoldings, which can be loaded mechanically with a computer-controlled "Bio-Stretch" device. Cyclic matrix deformation greatly stimulated proliferation of immature chondrocytes, but not that of hypertrophic chondrocytes. This indicates that mechanical stimulation of chondrocyte proliferation is developmental stage specific. Synthesis of cartilage matrix protein (CMP/matrilin-1), a mature chondrocyte marker, and type X collagen, a hypertrophic chondrocyte marker, was up-regulated by stretch-induced matrix deformation. Therefore, genes of CMP and type X collagen are responsive to mechanical stress. Mechanical stimulation of the mRNA levels of CMP and type X collagen occurred exactly at the same time points when these markers were synthesized by nonloading cells. This indicates that cyclic matrix deformation does not alter the speed of differentiation, but affects the extent of differentiation. The addition of the stretch-activated channel blocker gadolinium during loading abolished mechanical stimulation of chondrocyte proliferation, but did not affect the up-regulation of CMP mRNA by mechanical stretch. In contrast, the calcium channel blocker nifedipine inhibited both the stretch-induced proliferation and the increase of CMP mRNA. This suggests that stretch-induced matrix deformation regulates chondrocyte proliferation and differentiation via two signal transduction pathways, with stretch-activated channels involved in transducing the proliferative signals and calcium channels involved in transducing the signals for both proliferation and differentiation.     

Modelling of location- and time-dependent deformation of chondrocytes during cartilage loading.

Experimental evidence suggests that the biosynthetic activity of chondrocytes is regulated primarily by the mechanical environment. In order to study the mechanisms underlying remodeling, adaptation, and degeneration of articular cartilage in a joint subjected to changing loads, it is important to know the time-dependent fluid pressure and stress-strain state in chondrocytes. The purpose of the present study was to develop a theoretical model to simulate the mechanical behaviour of articular cartilage and to describe the time-dependent stress-strain state and fluid pressure distribution in chondrocytes during cartilage deformation. It was assumed that the volume occupied by the chondrocytes is small and that cartilage can be treated as a macroscopically homogenized material with effective material properties which depend on the material properties of the cells and matrix and the volumetric fraction of the cells. Model predictions on the time-dependent distribution of fluid pressure and stress and on the time-dependent cell deformation during confined and unconfined compression tests agree with previous theoretical predictions and experimental observations. The proposed model supplies the tools to study the mechanisms of degeneration, adaptation and remodelling of cartilage associated with cell loading and deformation.