A nonlinear biphasic viscohyperelastic model for articular cartilage

Cartilage biomechanics: From the basic facts to the challenges of tissue engineering

Articular cartilage (AC) is the thin tissue that covers the long bone ends in the joints and that ensures the transmission of forces between adjacent bones while allowing nearly frictionless movements between them. AC repair is a technologic and scientific challenge that has been addressed with numerous approaches. A major deadlock is the capacity to take in account its complex mechanical properties in repair strategies. In this review, we first describe the major mechanical behaviors of AC for the non-specialists. Then, we show how researchers have progressively identified specific mechanical parameters using mathematical models. There are still gaps in our understanding of some of the observations concerning AC biomechanical properties, particularly the differences in extracellular matrix stiffness measured at the microscale and at the millimetric scale. Nevertheless, for bioengineering applications, AC repair strategies must take into account what are commonly considered the main mechanical features of cartilage: its ability to withstand high stresses through three main behaviors (elasticity, poroelasticity and swelling). Finally, we emphasize that future studies need to investigate AC mechanical properties at different scales, particularly the gradient of mechanical properties around cells and across the cartilage depth, and the differences in mechanical properties at different scales. This multi-scale approach could greatly enhance the success of AC restorative approaches.

Biphasic Properties of PVAH (Polyvinyl Alcohol Hydrogel) Reflecting Biomechanical Behavior of the Nucleus Pulposus of the Human Intervertebral Disc

PVAH is a mixture of solid and fluid, but its mechanical behavior has usually been described using solid material models. The purpose of this study was to obtain material properties that can reflect the mechanical behavior of polyvinyl alcohol hydrogel (PVAH) using finite element analysis, a biphasic continuum model, and to optimize the composition ratio of PVAH to replace the nucleus pulposus (NP) of the human intervertebral disc. Six types of PVAH specimens (3, 5, 7, 10, 15, 20 wt%) were prepared, then unconfined compression experiments were performed to acquire their material properties using the Holmes–Mow biphasic model. With an increasing weight percentage of PVA in PVAH, the Young’s modulus increased while the permeability parameter decreased. The Young’s modulus and permeability parameter were similar to those of the NP at 15 wt% and 20 wt%. The range of motion, facet joint force, and NP pressures measured from dynamic motional analysis of the lumbar segments with the NP model also exhibited similar values to those with 15~20 wt% PVAH models. Considering the structural stability and pain of the lumbar segments, it appears that 20 wt% PVAH is most suitable for replacing the NP.

Comparison Between the Hyperelastic Behavior of Fresh and Frozen Equine Articular Cartilage in Various Joints

Fresh and frozen cartilage samples of the fetlock, carpus, and stifle were collected from 12 deceased horses. Half were measured immediately following extraction, and half were frozen for seven days and then measured. Seven indentations (various normalized displacements) were implemented with an indention rate of 0.1 mm/s. Solid phase aggregate modulus (Es), hyperelastic material constant (α), and fluid load fraction (F') of equine articular cartilage were assessed using the Ogden hyperelastic model. The properties were statistically compared in various joints (fetlock, carpus, and stifle), and between fresh and frozen samples using various statistical models. There was no statistical difference between the fetlock and carpus in the aggregate modulus (p = 0.5084), while both were significantly different from the stifle (fetlock: p = 0.0017 and carpus: p = 0.0406). For the hyperelastic material constant, no statistical differences between joints were observed (p = 0.3310). For the fluid load fraction, the fetlock and stifle comparison showed a difference (p = 0.0333), while the carpus was not different from the fetlock (p = 0.1563) or stifle (p = 0.3862). Comparison between the fresh and frozen articular cartilage demonstrated no significant difference among the joints in the three material properties: p = 0.9418, p = 0.7031, and p = 0.9313 for the aggregate modulus, the hyperelastic material constant, and the fluid load fraction, respectively.

An elasto-viscoplastic model to describe the ratcheting behavior of articular cartilage

In the present work, a constitutive model for articular cartilage is proposed in finite elasto-viscoplasticity. For simplification, articular cartilage is supposed to be a typical composite composed of a soft basis and a fiber assembly. The stress tensor and free energy function are hence accordingly divided into two components. The high nonlinear stress-strain response is assumed to be mainly related to the fiber assembly and described by an exponential-type hypoelastic relation. Ratcheting is considered according to the viscoplasticity, the evolution rule of which is deduced from the dissipative inequality by the co-directionality hypotheses. Then, the capability of the proposed model is validated by comparing its predictions with related experimental observations. Results show that the ratcheting behavior and stress-strain hysteresis loops are reasonably captured by the proposed model.

Price A, A. Hulley P, S. Gill H, Doblaré M, Hamdy Doweidar M. Inhomogeneous Response of Articular Cartilage: A Three-Dimensional Multiphasic Heterogeneous Study

Articular cartilage exhibits complex mechano-electrochemical behaviour due to its anisotropy, inhomogeneity and material non-linearity. In this work, the thickness and radial dependence of cartilage properties are incorporated into a 3D mechano-electrochemical model to explore the relevance of heterogeneity in the behaviour of the tissue. The model considers four essential phenomena: (i) osmotic pressure, (ii) convective and diffusive processes, (iii) chemical expansion and (iv) three-dimensional through-the-thickness heterogeneity of the tissue. The need to consider heterogeneity in computational simulations of cartilage behaviour and in manufacturing biomaterials mimicking this tissue is discussed. To this end, healthy tibial plateaus from pigs were mechanically and biochemically tested in-vitro. Heterogeneous properties were included in the mechano-electrochemical computational model to simulate tissue swelling. The simulation results demonstrated that swelling of the heterogeneous samples was significantly lower than swelling under homogeneous and isotropic conditions. Furthermore, there was a significant reduction in the flux of water and ions in the former samples. In conclusion, the computational model presented here can be considered as a valuable tool for predicting how the variation of cartilage properties affects its behaviour, opening up possibilities for exploring the requirements of cartilage-mimicking biomaterials for tissue engineering. Besides, the model also allows the establishment of behavioural patterns of swelling and of water and ion fluxes in articular cartilage.

Micromechanical model of biphasic biomaterials with internal adhesion: Application to nanocellulose hydrogel composites

The mechanical properties of hydrated biomaterials are non-recoverable upon unconfined compression if adhesion occurs between the structural components in the material upon fluid loss and apparent plastic behaviour. We explore these micromechanical phenomena by introducing an aggregation force and a critical yield pressure into the constitutive biphasic formulation for transversely isotropic tissues. The underlying hypothesis is that continual fluid pressure build-up during compression temporarily supresses aggregation. Once compression stops and the pressure falls below some critical value, internal aggregation occurs over a time scale comparable to the poroelastic time. We demonstrate this model by predicting the mechanical response of bacterial nanocellulose hydrogel composites, which are promising biomaterials and a structural mimetic for the plant cell wall. Cross-linking of cellulose by xyloglucan creates an extensional resistance and substantially increases the compressive modulus under large compression and densification. In comparison, incorporating non-crosslinking arabinoxylan into the hydrogel has little effect on its mechanics at the strain rates investigated. These results assist in elucidating the mechanical role of these polysaccharides in the complex plant cell wall structure. They also suggest xyloglucan is a suitable candidate to tailor the stiffness of nanocellulose hydrogels in biomaterial design, which includes modulating cell-adhesion in tissue engineering applications. The model and overall approach may be utilised to characterise and design a myriad of biomaterials and mammalian tissues, particularly those with a fibrillar structure.

STATEMENT OF SIGNIFICANCE

The mechanical properties of hydrated biomaterials can be non-recoverable upon compression due to increased adhesion occurring between the structural components in the material. Cellulose-hemicellulose composite hydrogels constitute a classical example of this phenomenon, since fibres can freely re-orient and adhere upon fluid loss to produce significant variations in the mechanical response to compression. Here, we model their micromechanics by introducing an aggregation force and a critical yield pressure into the constitutive formulation for transversely isotropic biphasic materials. The resulting model is easy to implement for routine characterization of this type of hydrated biomaterials through unconfined compression testing and produces physically meaningful and reproducible mechanical parameters.

Parameter-dependent behavior of articular cartilage: 3D mechano-electrochemical computational model

BACKGROUND AND OBJECTIVE

Changes in mechano-electrochemical properties of articular cartilage play an essential role in the majority of cartilage diseases. Despite of this importance, the specific effect of each parameter into tissue behavior remains still obscure. Parametric computational modeling of cartilage can provide some insights into this matter, specifically the study of mechano-electrochemical properties variation and their correlation with tissue swelling, water and ion fluxes. Thus, the aim of this study is to evaluate the influence of the main mechanical and electrochemical parameters on the determination of articular cartilage behavior by a parametric analysis through a 3D finite element model.

METHODS

For this purpose, a previous 3D mechano-electrochemical model, developed by the same authors, of articular cartilage behavior has been used. Young's modulus, Poisson coefficient, ion diffusivities and ion activity coefficients variations have been analyzed and quantified through monitoring tissue simulated response.

RESULTS

Simulation results show how Young's modulus and Poisson coefficient control tissue behavior rather than electrochemical properties. Meanwhile, ion diffusivity and ion activity coefficients appear to be vital in controlling velocity of incoming and outgoing fluxes.

CONCLUSIONS

This parametric study establishes a basic guide when defining the main properties that are essential to be included into computational modeling of articular cartilage providing a helpful tool in tissue simulations.

Altered swelling and ion fluxes in articular cartilage as a biomarker in osteoarthritis and joint immobilization: a computational analysis

In healthy cartilage, mechano-electrochemical phenomena act together to maintain tissue homeostasis. Osteoarthritis (OA) and degenerative diseases disrupt this biological equilibrium by causing structural deterioration and subsequent dysfunction of the tissue. Swelling and ion flux alteration as well as abnormal ion distribution are proposed as primary indicators of tissue degradation. In this paper, we present an extension of a previous three-dimensional computational model of the cartilage behaviour developed by the authors to simulate the contribution of the main tissue components in its behaviour. The model considers the mechano-electrochemical events as concurrent phenomena in a three-dimensional environment. This model has been extended here to include the effect of repulsion of negative charges attached to proteoglycans. Moreover, we have studied the fluctuation of these charges owning to proteoglycan variations in healthy and pathological articular cartilage. In this sense, standard patterns of healthy and degraded tissue behaviour can be obtained which could be a helpful diagnostic tool. By introducing measured properties of unhealthy cartilage into the computational model, the severity of tissue degeneration can be predicted avoiding complex tissue extraction and subsequent in vitro analysis. In this work, the model has been applied to monitor and analyse cartilage behaviour at different stages of OA and in both short (four, six and eight weeks) and long-term (11 weeks) fully immobilized joints. Simulation results showed marked differences in the corresponding swelling phenomena, in outgoing cation fluxes and in cation distributions. Furthermore, long-term immobilized patients display similar swelling as well as fluxes and distribution of cations to patients in the early stages of OA, thus, preventive treatments are highly recommended to avoid tissue deterioration.

Biomimetic Model of Articular Cartilage Based on In Vitro Experiments

Articular cartilage is a complicated material to model for a variety of reasons: its biphasic/triphasic properties, heterogeneous structure, compressibility, unique geometry, and variance between samples. However, the applications for a biomimetic, cartilage-like material are numerous and include: porous bearings, viscous dampers, robotic linkages, artificial joints, etc. This work reports experimental results on the stress-relaxation of equine articular cartilage in unconfined compression. The response is consistent with simple spring and damper systems, and gives a storage and loss moduli. This model is proposed for use in evaluating biomimetic materials, and can be incorporated into large-scale dynamic analyses to account for motion or impact. The proposed characterization is suited for high-level analysis of multi-phase materials, where separating the contribution of each phase is not desired.

Validation and application of an intervertebral disc finite element model utilizing independently constructed tissue-level constitutive formulations that are nonlinear, anisotropic, and time-dependent

Finite element (FE) models are advantageous in the study of intervertebral disc mechanics as the stress-strain distributions can be determined throughout the tissue and the applied loading and material properties can be controlled and modified. However, the complicated nature of the disc presents a challenge in developing an accurate and predictive disc model, which has led to limitations in FE geometry, material constitutive models and properties, and model validation. The objective of this study was to develop a new FE model of the intervertebral disc, to validate the model's nonlinear and time-dependent responses without tuning or calibration, and to evaluate the effect of changes in nucleus pulposus (NP), cartilaginous endplate (CEP), and annulus fibrosus (AF) material properties on the disc mechanical response. The new FE disc model utilized an analytically-based geometry. The model was created from the mean shape of human L4/L5 discs, measured from high-resolution 3D MR images and averaged using signed distance functions. Structural hyperelastic constitutive models were used in conjunction with biphasic-swelling theory to obtain material properties from recent tissue tests in confined compression and uniaxial tension. The FE disc model predictions fit within the experimental range (mean ± 95% confidence interval) of the disc's nonlinear response for compressive slow loading ramp, creep, and stress-relaxation simulations. Changes in NP and CEP properties affected the neutral-zone displacement but had little effect on the final stiffness during slow-ramp compression loading. These results highlight the need to validate FE models using the disc's full nonlinear response in multiple loading scenarios.

THE DIFFERENTIAL CONSTITUTIVE EQUATION AND MODEL OF ABALONE NACRE BY NANOINDENTER

Nacre has a complex hierarchical microarchitecture that spans over multiple length scales from nanoscale to macroscale. Its structures are optimized leading to extraordinary mechanical performance and energy absorption. Nacre's special characteristics of the self-assembly method have attracted the interest of material scientists to develop laminated composite materials, molecular scale self-assembly and biomineralization. Nanoindentation testing can determine a material's anisotropic properties through a single indentation. In the present study, nanoindentation stress–strain curves were used to characterize the complete mechanical behavior of nacre of abalone shell. A differential constitutive equation was developed with time-dependent spring constants k and viscosities η. Furthermore, to describe the complex viscoelastic behavior of abalone nacre, a descriptive representation of the linear viscoelasticity law for the multilayer matrix was formulated. A qualitative model for the relationship between nacre structure and mechanical properties of nacre may help develop bionic composite materials for micro-aircraft, bionic tribology, bionic medical apparatus and bionic organs (tissue engineering).

Mechanical properties of the extra-fibrillar matrix of human annulus fibrosus are location and age dependent

The mechanical behavior of the annulus fibrosus (AF) of the intervertebral disc can be modeled as a mixture of fibers, extra-fibrillar matrix (EFM), ions, and fluid. However, the properties of the EFM have not been measured directly. We measured mechanical properties of the human EFM at several locations, determined the effect of age and degeneration, and evaluated whether changes in EFM properties correspond to AF compositional changes. EFM mechanical properties were measured using a method that combines osmotic loading and confined compression. AF samples were dissected from several locations, and mechanical properties were correlated with age, degeneration, and composition. EFM modulus was found to range between 10 and 50 kPa, increasing nonlinearly with compression magnitude and being highest in the AF outer-anterior region. EFM properties were not correlated with composition or degeneration. However, the EFM modulus, its relative contribution to tissue modulus, and model parameters were correlated with age. These measurements will result in more accurate predictions of deformations in the intervertebral disc. Additionally, parameters such as permeability and diffusivity used for biotransport analysis of glucose and other solutes depend on EFM deformation. Consequently, the accuracy of biotransport simulations will be greatly improved.

Biaxial tension of fibrous tissue: using finite element methods to address experimental challenges arising from boundary conditions and anisotropy

Planar biaxial tension remains a critical loading modality for fibrous soft tissue and is widely used to characterize tissue mechanical response, evaluate treatments, develop constitutive formulas, and obtain material properties for use in finite element studies. Although the application of tension on all edges of the test specimen represents the in situ environment, there remains a need to address the interpretation of experimental results. Unlike uniaxial tension, in biaxial tension the applied forces at the loading clamps do not transmit fully to the region of interest (ROI), which may lead to improper material characterization if not accounted for. In this study, we reviewed the tensile biaxial literature over the last ten years, noting experimental and analysis challenges. In response to these challenges, we used finite element simulations to quantify load transmission from the clamps to the ROI in biaxial tension and to formulate a correction factor that can be used to determine ROI stresses. Additionally, the impact of sample geometry, material anisotropy, and tissue orientation on the correction factor were determined. Large stress concentrations were evident in both square and cruciform geometries and for all levels of anisotropy. In general, stress concentrations were greater for the square geometry than the cruciform geometry. For both square and cruciform geometries, materials with fibers aligned parallel to the loading axes reduced stress concentrations compared to the isotropic tissue, resulting in more of the applied load being transferred to the ROI. In contrast, fiber-reinforced specimens oriented such that the fibers aligned at an angle to the loading axes produced very large stress concentrations across the clamps and shielding in the ROI. A correction factor technique was introduced that can be used to calculate the stresses in the ROI from the measured experimental loads at the clamps. Application of a correction factor to experimental biaxial results may lead to more accurate representation of the mechanical response of fibrous soft tissue.

Integrating qPLM and biomechanical test data with an anisotropic fiber distribution model and predictions of TGF-β1 and IGF-1 regulation of articular cartilage fiber modulus

A continuum mixture model with distinct collagen (COL) and glycosaminoglycan elastic constituents was developed for the solid matrix of immature bovine articular cartilage. A continuous COL fiber volume fraction distribution function and a true COL fiber elastic modulus ([Formula: see text] were used. Quantitative polarized light microscopy (qPLM) methods were developed to account for the relatively high cell density of immature articular cartilage and used with a novel algorithm that constructs a 3D distribution function from 2D qPLM data. For specimens untreated and cultured in vitro, most model parameters were specified from qPLM analysis and biochemical assay results; consequently, [Formula: see text] was predicted using an optimization to measured mechanical properties in uniaxial tension and unconfined compression. Analysis of qPLM data revealed a highly anisotropic fiber distribution, with principal fiber orientation parallel to the surface layer. For untreated samples, predicted [Formula: see text] values were 175 and 422 MPa for superficial (S) and middle (M) zone layers, respectively. TGF-[Formula: see text]1 treatment was predicted to increase and decrease [Formula: see text] values for the S and M layers to 281 and 309 MPa, respectively. IGF-1 treatment was predicted to decrease [Formula: see text] values for the S and M layers to 22 and 26 MPa, respectively. A novel finding was that distinct native depth-dependent fiber modulus properties were modulated to nearly homogeneous values by TGF-[Formula: see text]1 and IGF-1 treatments, with modulated values strongly dependent on treatment.

A detailed and validated three dimensional dynamic model of the patellofemoral joint

A detailed 3D anatomical model of the patellofemoral joint was developed to study the tracking, force, contact and stability characteristics of the joint. The quadriceps was considered to include six components represented by 15 force vectors. The patellar tendon was modeled using four bundles of viscoelastic tensile elements. Each of the lateral and medial retinaculum was modeled by a three-bundle nonlinear spring. The femur and patella were considered as rigid bodies with their articular cartilage layers represented by an isotropic viscoelastic material. The geometrical and tracking data needed for model simulation, as well as validation of its results, were obtained from an in vivo experiment, involving MR imaging of a normal knee while performing isometric leg press against a constant 140 N force. The model was formulated within the framework of a rigid body spring model and solved using forth-order Runge-Kutta, for knee flexion angles between zero and 50 degrees. Results indicated a good agreement between the model predictions for patellar tracking and the experimental results with RMS deviations of about 2 mm for translations (less than 0.7 mm for patellar mediolateral shift), and 4 degrees for rotations (less than 3 degrees for patellar tilt). The contact pattern predicted by the model was also consistent with the results of the experiment and the literature. The joint contact force increased linearly with progressive knee flexion from 80 N to 210 N. The medial retinaculum experienced a peak force of 18 N at full extension that decreased with knee flexion and disappeared entirely at 20 degrees flexion. Analysis of the patellar time response to the quadriceps contraction suggested that the muscle activation most affected the patellar shift and tilt. These results are consistent with the recent observations in the literature concerning the significance of retinaculum and quadriceps in the patellar stability.

[Development of finite element models for cartilage replacement material]

In the development of cartilage replacement materials, mechanical models are necessary to quantify elasticity and damping properties of the artificial tissue. The aim is to identify parameters from material tests leading to an objective assessment of elasticity and damping for tissue replacement. This is especially important as the evolution of material properties is investigated during a cultivation period of several weeks. For this reason, in the present study a method is proposed to identify all necessary material parameters by means of a finite element model. The numerical simulations are based on a phenomenological material model exhibiting as few parameters as possible for covering elastic material and damping properties. This allows a practical identification of material parameters. Thus, deformation dependent damping properties of the replacement material are covered by parameters identified from material tests without extensive determination of pore, solid or fluid fractions.

A viscoelastic constitutive model can accurately represent entire creep indentation tests of human patella cartilage

Cartilage material properties provide important insights into joint health, and cartilage material models are used in whole-joint finite element models. Although the biphasic model representing experimental creep indentation tests is commonly used to characterize cartilage, cartilage short-term response to loading is generally not characterized using the biphasic model. The purpose of this study was to determine the short-term and equilibrium material properties of human patella cartilage using a viscoelastic model representation of creep indentation tests. We performed 24 experimental creep indentation tests from 14 human patellar specimens ranging in age from 20 to 90 years (median age 61 years). We used a finite element model to reproduce the experimental tests and determined cartilage material properties from viscoelastic and biphasic representations of cartilage. The viscoelastic model consistently provided excellent representation of the short-term and equilibrium creep displacements. We determined initial elastic modulus, equilibrium elastic modulus, and equilibrium Poisson's ratio using the viscoelastic model. The viscoelastic model can represent the short-term and equilibrium response of cartilage and may easily be implemented in whole-joint finite element models.

Anisotropic hydraulic permeability under finite deformation

The structural organization of biological tissues and cells often produces anisotropic transport properties. These tissues may also undergo large deformations under normal function, potentially inducing further anisotropy. A general framework for formulating constitutive relations for anisotropic transport properties under finite deformation is lacking in the literature. This study presents an approach based on representation theorems for symmetric tensor-valued functions and provides conditions to enforce positive semidefiniteness of the permeability or diffusivity tensor. Formulations are presented, which describe materials that are orthotropic, transversely isotropic, or isotropic in the reference state, and where large strains induce greater anisotropy. Strain-induced anisotropy of the permeability of a solid-fluid mixture is illustrated for finite torsion of a cylinder subjected to axial permeation. It is shown that, in general, torsion can produce a helical flow pattern, rather than the rectilinear pattern observed when adopting a more specialized, unconditionally isotropic spatial permeability tensor commonly used in biomechanics. The general formulation presented in this study can produce both affine and nonaffine reorientations of the preferred directions of material symmetry with strain, depending on the choice of material functions. This study addresses a need in the biomechanics literature by providing guidelines and formulations for anisotropic strain-dependent transport properties in porous-deformable media undergoing large deformations.

A nonlinear constituent based viscoelastic model for articular cartilage and analysis of tissue remodeling due to altered glycosaminoglycan-collagen interactions

A constituent based nonlinear viscoelastic (VE) model was modified from a previous study (Vena, et al., 2006, "A Constituent-Based Model for the Nonlinear Viscoelastic Behavior of Ligaments," J. Biomech. Eng., 128, pp. 449-457) to incorporate a glycosaminoglycan (GAG)-collagen (COL) stress balance using compressible elastic stress constitutive equations specific to articular cartilage (AC). For uniaxial loading of a mixture of quasilinear VE constituents, time constant and relaxation ratio equations are derived to highlight how a mixture of constituents with distinct quasilinear VE properties is one mechanism that produces a nonlinear VE tissue. Uniaxial tension experiments were performed with newborn bovine AC specimens before and after approximately 55% and approximately 85% GAG depletion treatment with guanidine. Experimental tissue VE parameters were calculated directly from stress relaxation data, while intrinsic COL VE parameters were calculated by curve fitting the data with the nonlinear VE model with intrinsic GAG viscoelasticity neglected. Select tissue and intrinsic COL VE parameters were significantly different from control and experimental groups and correlated with GAG content, suggesting that GAG-COL interactions exist to modulate tissue and COL mechanical properties. Comparison of the results from this and other studies that subjected more mature AC tissue to GAG depletion treatment suggests that the GAGs interact with the COL network in a manner that may be beneficial for rapid volumetric expansion during developmental growth while protecting cells from excessive matrix strains. Furthermore, the underlying GAG-COL interactions appear to diminish as the tissue matures, indicating a distinctive remodeling response during developmental growth.

Mechanical properties of human fetal talus

Mechanical characterization of human cartilage anlagen is required to effectively model congenital musculoskeletal deformities. Such modeling can effectively explore the effect of treatment procedures and potentially suggest enhanced treatment methods. Using serial MRI, we have noted shape changes of the cartilaginous hindfoot anlagen in patients with clubfoot, suggesting they are soft and deformable. We therefore determined the stress relaxation behavior of cartilage plugs obtained from third-trimester stillborn fetuses in unconfined and confined compression geometries. The material parameters determined were the aggregate modulus H(A) = 0.15 +/- 0.07 MPa, Poisson's ratio nu = 0.4 +/- 0.06, Young's modulus E(s) = 0.06 +/- 0.03 MPa, and permeability coefficients k(0) = 2.01 +/- 0.8 x 10(-14) m(4) N(-1) s(-1) and M = 4.6 +/- 1.0. As compared with adult articular cartilage, stiffness was an order of magnitude lower than the values reported in the literature, suggesting the relative softness of the tissue, and the permeability was an order of magnitude higher, indicating relative ease of flow in the tissue. Poisson's ratio also was close to the higher end of the range reported in previous studies. Such material is expected to deform and relax to larger extents. These findings are consistent with the deformability of the cartilage anlagen during manipulation and casting for treatment of clubfoot.

A fluid-saturated poroelastic model of the vocal folds with hydrated tissue

The purpose of this study is to develop a continuous model to describe the vibration of the vocal fold with hydrated tissue. This model is unique because it is based on the fluid-saturated porous solid theory. Therefore, this new model can be used to study some vocal fold characteristics that would be difficult to predict using previous models. Numerical simulations show that this model can generate self-oscillation and that its phonation threshold pressure (PTP) is 0.555 kPa. The basic outputs of this model, including fundamental frequency, maximum lateral displacement, surface dynamics, and empirical eigenfunctions, agree with previous models and experimental studies, which validates this new model. The ability to simulate the flow of liquid through the tissue is one of the important advantages of this new model. It was found that the liquid in the vocal fold tissue could be accumulated at the anterior-posterior midpoint during phonation, which could cause a pressure increase in the liquid. The liquid pressure increased from 0.033 to 0.150 kPa when the subglottal pressure increased from 0.555 kPa (PTP) to 0.7 kPa. It was believed that the liquid dynamics in the tissue during phonation could be related to the development of some vocal diseases, such as vocal nodules, edema, and so on. Therefore, we expect that this model might not only provide a more appropriate description of the vocal fold vibration, but that it could also have clinical value in investigating certain vocal fold diseases.

A zipper network model of the failure mechanics of extracellular matrices

Mechanical failure of soft tissues is characteristic of life-threatening diseases, including capillary stress failure, pulmonary emphysema, and vessel wall aneurysms. Failure occurs when mechanical forces are sufficiently high to rupture the enzymatically weakened extracellular matrix (ECM). Elastin, an important structural ECM protein, is known to stretch beyond 200% strain before failing. However, ECM constructs and native vessel walls composed primarily of elastin and proteoglycans (PGs) have been found to fail at much lower strains. In this study, we hypothesized that PGs significantly contribute to tissue failure. To test this, we developed a zipper network model (ZNM), in which springs representing elastin are organized into long wavy fibers in a zipper-like formation and placed within a network of springs mimicking PGs. Elastin and PG springs possessed distinct mechanical and failure properties. Simulations using the ZNM showed that the failure of PGs alone reduces the global failure strain of the ECM well below that of elastin, and hence, digestion of elastin does not influence the failure strain. Network analysis suggested that whereas PGs drive the failure process and define the failure strain, elastin determines the peak and failure stresses. Predictions of the ZNM were experimentally confirmed by measuring the failure properties of engineered elastin-rich ECM constructs before and after digestion with trypsin, which cleaves the core protein of PGs without affecting elastin. This study reveals a role for PGs in the failure properties of engineered and native ECM with implications for the design of engineered tissues.

Meniscectomy alters the dynamic deformational behavior and cumulative strain of tibial articular cartilage in knee joints subjected to cyclic loads

OBJECTIVE

Meniscectomy-induced osteoarthritis may be mechanically based. We asked how meniscectomy alters time-dependent deformation of physiologically loaded articular cartilage. We hypothesized that meniscectomy alters nominal strain in tibial articular cartilage, and that meniscectomy affects cartilage thickness recovery following cessation of loading.

METHODS

A cyclic load simulating normal gait was applied to four sheep knees. A custom device was used to obtain MR images of cartilage at 4.7T during cyclic loading. Articular cartilage thickness and nominal strain were measured every 2.5 min during 1h of cyclic loading, and during 2.5h after cessation of loading.

RESULTS

Following meniscectomy the loaded joints rapidly developed high strain centrally and minimal strain peripherally. Maximum nominal strains after 1h of loading were about 55% in the intact knees and 72% in the meniscectomized knees. Nominal strains in the peripheral tibial cartilage were significantly reduced in the meniscectomized knees. Strain recovery was markedly prolonged in the meniscectomized knees.

CONCLUSIONS

With meniscectomy, tibial articular cartilage in the central load bearing region remains chronically deformed and dehydrated, even after cessation of loading. Post-meniscectomy osteoarthritis may be initiated in this region by direct damage to the cartilage matrix, or by altering the hydration of the tissue. In peripheral regions, reduced loading and strain may facilitate subchondral vascular invasion, and endochondral ossification. This is consistent with the central fibrillation and peripheral osteophyte formation seen in post-meniscectomy osteoarthritis.

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.

Simulation of high tensile Poisson's ratios of articular cartilage with a finite element fibril-reinforced hyperelastic model

Analyses with a finite element fibril-reinforced hyperelastic model were undertaken in this study to simulate high tensile Poisson's ratios that have been consistently documented in experimental studies of articular cartilage. The solid phase was represented by an isotropic matrix reinforced with four sets of fibrils, two of them aligned in orthogonal directions and two oblique fibrils in a symmetric configuration respect to the orthogonal axes. Two distinct hyperelastic functions were used to represent the matrix and the fibrils. Results of the analyses showed that only by considering non-orthogonal fibrils was it possible to represent Poisson's ratios higher than one. Constrains in the grips and finite deformations played a minor role in the calculated Poisson's ratio. This study also showed that the model with oblique fibrils at 45 degrees was able to represent significant differences in Poisson's ratios near 1 documented in experimental studies. However, even considering constrains in the grips, this model was not capable to simulate Poisson's ratios near 2 that have been reported in other studies. The study also confirmed that only with a high relation between the stiffness of the fibers and that of the matrix was it possible to obtain high Poisson's ratios for the tissue. Results suggest that analytical models with a finite number of fibrils are appropriate to represent main mechanical effects of articular cartilage.

Fluid pressure driven fibril reinforcement in creep and relaxation tests of articular cartilage

Biological tissues exhibit diverse mechanical behaviors because of complex material properties. As has been shown for ligaments and intervertebral discs, mathematical models often appear to well predict load responses individually by adjusting model parameters, but likely fail to describe several different load responses simultaneously using the same model parameters. In the present study, we attempted to describe and explain both creep and relaxation responses of articular cartilage using a fibril-reinforced model, which has been successfully used to account for the load response of the relaxation tests of articular cartilage. Experiments were performed on bovine articular cartilage disks (n=8) using multi-step loading protocols, involving both creep and relaxation in each protocol. The experimental results indicated that mechanical changes, such as fiber recruitment in collagen network during stretch, recovered fully upon unloading. Creep loading did not affect relaxation properties, and vice versa. Relaxation proceeded much faster than creep, because of different fluid pressure profiles. The load sharing among the proteoglycan matrix, collagen network and fluid pressurization was predicted to differ for the creep and relaxation testing. The experimentally observed strong creep and relaxation responses in unconfined compression could not be predicted if either fibril reinforcement or fluid pressurization were neglected. It was essential to consider the interplay between nonlinear fibril reinforcement and fluid pressurization for the transient response (this interplay may be best termed as fluid pressure driven fibril reinforcement). Fibril reinforcement played a relatively insignificant role in the compressive load response at equilibrium, in agreement with previous findings for cartilage stress relaxation testing.

A biphasic viscohyperelastic fibril-reinforced model for articular cartilage: Formulation and comparison with experimental data

Experiments in articular cartilage have shown highly nonlinear stress-strain curves under finite deformations, nonlinear tension-compression response as well as intrinsic viscous effects of the proteoglycan matrix and the collagen fibers. A biphasic viscohyperelastic fibril-reinforced model is proposed here, which is able to describe the intrinsic viscoelasticity of the fibrillar and nonfibrillar components of the solid phase, the nonlinear tension-compression response and the nonlinear stress-strain curves under tension and compression. A viscohyperelastic constitutive equation was used for the matrix and the fibers encompassing, respectively, a hyperelastic function used previously for the matrix and a hyperelastic law used before to represent biological connective tissues. This model, implemented in an updated Lagrangian finite element code, displayed good ability to follow experimental stress-strain equilibrium curves under tension and compression for human humeral cartilage. In addition, curve fitting of experimental reaction force and lateral displacement unconfined compression curves showed that the inclusion of viscous effects in the matrix allows the description of experimental data with material properties for the fibers consistent with experimental tensile tests, suggesting that intrinsic viscous effects in the matrix of articular cartilage plays an important role in the mechanical response of the tissue.