The role of viscoelasticity of collagen fibers in articular cartilage: theory and numerical formulation

The relative importance of fluid-dependent and fluid-independent transient mechanical behavior in articular cartilage was examined for tensile and unconfined compression testing using a fibril reinforced model. The collagen matrix of articular cartilage was modeled as viscoelastic using a quasi-linear viscoelastic formulation with strain-dependent elastic modulus, while the proteoglycan matrix was considered as linearly elastic. The collagen viscoelastic properties were obtained by fitting experimental data from a tensile test. These properties were used to investigate unconfined compression testing, and the sensitivity of the properties was also explored. It was predicted that the stress relaxation observed in tensile tests was not caused by fluid pressurization at the macroscopic level. A multi-step tensile stress relaxation test could be approximated using a hereditary integral in which the elastic fibrillar modulus was taken to be a linear function of the fibrillar strain. Applying the same formulation to the radial fibers in unconfined compression, stress relaxation could not be simulated if fluid pressurization were absent. Collagen viscoelasticity was found to slightly weaken fluid pressurization in unconfined compression, and this effect was relatively more significant at moderate strain rates. Therefore, collagen viscoelasticity appears to play an import role in articular cartilage in tensile testing, while fluid pressurization dominates the transient mechanical behavior in compression. Collagen viscoelasticity plays a minor role in the mechanical response of cartilage in unconfined compression if significant fluid flow is present.     

Experimental verification of the roles of intrinsic matrix viscoelasticity and tension-compression nonlinearity in the biphasic response of cartilage

A biphasic-CLE-QLV model proposed in our recent study [2001, J. Biomech. Eng., 123, pp. 410-417] extended the biphasic theory of Mow et al. [1980, J. Biomech. Eng., 102, pp. 73-84] to include both tension-compression nonlinearity and intrinsic viscoelasticity of the cartilage solid matrix by incorporating it with the conewise linear elasticity (CLE) model [1995, J. Elasticity, 37, pp. 1-38] and the quasi-linear viscoelasticity (QLV) model [Biomechanics: Its foundations and objectives, Prentice Hall, Englewood Cliffs, 1972]. This model demonstrates that a simultaneous prediction of compression and tension experiments of articular cartilage, under stress-relaxation and dynamic loading, can be achieved when properly taking into account both flow-dependent and flow-independent viscoelastic effects, as well as tension-compression nonlinearity. The objective of this study is to directly test this biphasic-CLE-QLV model against experimental data from unconfined compression stress-relaxation tests at slow and fast strain rates as well as dynamic loading. Twelve full-thickness cartilage cylindrical plugs were harvested from six bovine glenohumeral joints and multiple confined and unconfined compression stress-relaxation tests were performed on each specimen. The material properties of specimens were determined by curve-fitting the experimental results from the confined and unconfined compression stress relaxation tests. The findings of this study demonstrate that the biphasic-CLE-QLV model is able to describe the strain-rate-dependent mechanical behaviors of articular cartilage in unconfined compression as attested by good agreements between experimental and theoretical curvefits (r2 = 0.966 +/- 0.032 for testing at slow strain rate; r2 = 0.998 +/- 0.002 for testing at fast strain rate) and predictions of the dynamic response (r2 = 0.91 +/- 0.06). This experimental study also provides supporting evidence for the hypothesis that both tension-compression nonlinearity and intrinsic viscoelasticity of the solid matrix of cartilage are necessary for modeling the transient and equilibrium responses of this tissue in tension and compression. Furthermore, the biphasic-CLE-QLV model can produce better predictions of the dynamic modulus of cartilage in unconfined dynamic compression than the biphasic-CLE and biphasic poroviscoelastic models, indicating that intrinsic viscoelasticity and tension-compression nonlinearity of articular cartilage may play important roles in the load-support mechanism of cartilage under physiologic loading.     

Biphasic poroviscoelastic simulation of the unconfined compression of articular cartilage: II--Effect of variable strain rates

This study investigated the abilities of the linear biphasic poroviscoelastic (BPVE) model and the linear biphasic poroelastic (BPE) model to simulate the effect of variable ramp strain rates on the unconfined compression stress relaxation response of articular cartilage. Curve fitting of experimental data showed that the BPVE model was able to successfully account for the ramp strain rate-dependent viscoelastic behavior of articular cartilage under unconfined compression, while the BPE model was able to account for the complete viscoelastic response at a slow strain rate, but only the long-term viscoelastic response at faster strain rates. We concluded that the short-term viscoelastic behavior of articular cartilage, when subjected to a fast ramp strain rate, is primarily governed by a fluid flow-independent (intrinsic) viscoelastic mechanism, whereas the long-term viscoelastic behavior is governed by a fluid flow-dependent (biphasic) viscoelastic mechanism. Furthermore, a linear viscoelastic representation of the solid stress was found to be a valid model assumption for the simulation of ramp strain rate-dependent relaxation behaviors of articular cartilage within the range of ramp strain rates investigated.     

An approach to determine pressure profile generated by compression bandage using quasi-linear viscoelastic model

Understanding the stress relaxation behavior of the compression bandage could be very useful in determining the behavior of the interface pressure exerted by the bandage on a limb during the course of the compression treatment. There has been no comprehensive study in the literature to investigate the pressure profile (interface pressure with time) generated by a compression bandage when applied at different levels of strain. The present study attempts to describe the pressure profile, with the use of a quasi-linear viscoelastic model, generated by a compression bandage during compression therapy. The quasi-linear viscoelastic (QLV) theory proposed by Fung (Fung, 1972, "Stress Strain History Relations of Soft Tissues in Simple Elongation," Biomechanics: Its Foundations and Objectives, Y. C. Fung, N. Perrone, and M. Anliker, eds., Prentice-Hall, Englewood Cliffs, NJ, pp. 181-207). was used to model the nonlinear time- and history-dependent relaxation behavior of the bandage using the ramp strain approach. The regression analysis was done to find the correlation between the pressure profile and the relaxation behavior of the bandage. The parameters of the QLV model, describing the relaxation behavior of the bandage, were used to determine the pressure profile generated by the bandage at different levels of strain. The relaxation behaviors of the bandage at different levels of strain were well described by the QLV model parameters. A high correlation coefficient (nearly 0.98) shows a good correlation of the pressure profile with the stress relaxation behavior of the bandage.The prediction of the pressure profile using the QLV model parameters were in agreement with the experimental data. The pressure profile generated by a compression bandage could be predicted using the QLV model describing the nonlinear relaxation behavior of the bandage. This new application of the QLV theory helps in evaluating the bandage performance during compression therapy as scientific wound care management.     

The prediction of stress-relaxation of ligaments and tendons using the quasi-linear viscoelastic model

Recent studies have questioned the ability of the quasi-linear viscoelastic (QLV) model to predict stresses and strains in response to loading conditions other than those used to fit the model. The objective of this study was to evaluate the ability of several models in the literature to predict the elastic stress response of ligament and tendon at strain levels higher than the levels used to fit the model. The constitutive models were then used to evaluate the ability of the QLV model to predict the overall stress response during stress relaxation. The models expressing stress as an exponential function of strain significantly overestimated stress when used at higher strain levels. The polynomial formulation of the Mooney–Rivlin model more accurately predicted the stress–strain behavior of ligament and tendon. The results demonstrate that the ability of the QLV model to accurately predict the stress-relaxation response is dependent in part on the accuracy of the function used to model the elastic response of the soft tissue.     

Dynamic response of immature bovine articular cartilage in tension and compression, and nonlinear viscoelastic modeling of the tensile response

Very limited information is currently available on the constitutive modeling of the tensile response of articular cartilage and its dynamic modulus at various loading frequencies. The objectives of this study were to (1) formulate and experimentally validate a constitutive model for the intrinsic viscoelasticity of cartilage in tension, (2) confirm the hypothesis that energy dissipation in tension is less than in compression at various loading frequencies, and (3) test the hypothesis that the dynamic modulus of cartilage in unconfined compression is dependent upon the dynamic tensile modulus. Experiment 1: Immature bovine articular cartilage samples were tested in tensile stress relaxation and cyclical loading. A proposed reduced relaxation function was fitted to the stress-relaxation response and the resulting material coefficients were used to predict the response to cyclical loading. Adjoining tissue samples were tested in unconfined compression stress relaxation and cyclical loading. Experiment 2: Tensile stress relaxation experiments were performed at varying strains to explore the strain-dependence of the viscoelastic response. The proposed relaxation function successfully fit the experimental tensile stress-relaxation response, with R2 = 0.970+/-0.019 at 1% strain and R2 = 0.992+/-0.007 at 2% strain. The predicted cyclical response agreed well with experimental measurements, particularly for the dynamic modulus at various frequencies. The relaxation function, measured from 2% to 10% strain, was found to be strain dependent, indicating that cartilage is nonlinearly viscoelastic in tension. Under dynamic loading, the tensile modulus at 10 Hz was approximately 2.3 times the value of the equilibrium modulus. In contrast, the dynamic stiffening ratio in unconfined compression was approximately 24. The energy dissipation in tension was found to be significantly smaller than in compression (dynamic phase angle of 16.7+/-7.4 deg versus 53.5+/-12.8 deg at 10(-3) Hz). A very strong linear correlation was observed between the dynamic tensile and dynamic compressive moduli at various frequencies (R2 = 0.908+/-0.100). The tensile response of cartilage is nonlinearly viscoelastic, with the relaxation response varying with strain. A proposed constitutive relation for the tensile response was successfully validated. The frequency response of the tensile modulus of cartilage was reported for the first time. Results emphasize that fluid-flow dependent viscoelasticity dominates the compressive response of cartilage, whereas intrinsic solid matrix viscoelasticity dominates the tensile response. Yet the dynamic compressive modulus of cartilage is critically dependent upon elevated values of the dynamic tensile modulus.     

Viscous elements have little impact on measured passive length-tension properties of human gastrocnemius muscle-tendon units in vivo

Several studies have measured the elastic properties of a single human muscle-tendon unit in vivo. However the viscoelastic behavior of single human muscles has not been characterized. In this study, we adapted QLV theory to model the viscoelastic behavior of human gastrocnemius muscle-tendon units in vivo. We also determined the influence of viscoelasticity on passive length-tension properties of human gastrocnemius muscle-tendon units. Eight subjects participated in the experiment, which consisted of two parts. First, the stress relaxation response of human gastrocnemius muscle-tendon units was determined at a range of knee and ankle angles. Subsequently, passive ankle torque and ankle angle were collected during cyclic dorsiflexion and plantarflexion at a range of knee angles. Viscous parameters were determined by fitting the stress relaxation experiment data with a two-term exponential function, and elastic parameters were estimated by fitting the QLV model and viscous parameters to the cyclic experiment data. The model fitted the experimental data well at slow speeds (RMSE: 1.7 ± 0.5N) and at fast speeds (RMSE: 1.9 ± 0.2N). Muscle-tendon units demonstrated a large amount of stress relaxation. Nonetheless, viscoelastic passive length-tension curves estimated with the QLV model were similar to elastic passive length-tension curves obtained using a model that ignored viscosity. There was little difference in the elastic passive length-tension curves at different loading rates. We conclude that (a) the QLV model can be used to quantify viscoelastic behaviors of relaxed human gastrocnemius muscle-tendon units in vivo, and (b) over the range of velocities we examined, the velocity of loading has little effect on the passive length-tension properties of human gastrocnemius muscle-tendon units.     

Confined and unconfined stress relaxation of cartilage: appropriateness of a transversely isotropic analysis.

Previous studies have shown that stress relaxation behavior of calf ulnar growth plate and chondroepiphysis cartilage can be described by a linear transverse isotropic biphasic model. The model provides a good fit to the observed unconfined compression transients when the out-of-plane Poisson's ratio is set to zero. This assumption is based on the observation that the equilibrium stress in the axial direction (deltaz) is the same in confined and unconfined compression, which implies that the radial stress deltar = 0 in confined compression. In our study, we further investigated the ability of the transversely isotropic model to describe confined and unconfined stress relaxation behavior of calf cartilage. A series of confined and unconfined stress relaxation tests were performed on calf articular cartilage (4.5 mm diameter, approximately 3.3 mm height) in a displacement-controlled compression apparatus capable of measuring delta(z) and delta(r). In equilibrium, delta(r) > 0 and delta(z) in confined compression was greater than in unconfined compression. Transient data at each strain were fitted by the linear transversely isotropic biphasic model and the material parameters were estimated. Although the model could provide good fits to the unconfined transients, the estimated parameters overpredicted the measured delta(r). Conversely, if the model was constrained to match equilibrium delta(r), the fits were poor. These findings suggest that the linear transversely isotropic biphasic model could not simultaneously describe the observed stress relaxation and equilibrium behavior of calf cartilage.     

Stress-relaxation of human patellar articular cartilage in unconfined compression: prediction of mechanical response by tissue composition and structure

Mechanical properties of articular cartilage are controlled by tissue composition and structure. Cartilage function is sensitively altered during tissue degeneration, in osteoarthritis (OA). However, mechanical properties of the tissue cannot be determined non-invasively. In the present study, we evaluate the feasibility to predict, without mechanical testing, the stress-relaxation response of human articular cartilage under unconfined compression. This is carried out by combining microscopic and biochemical analyses with composition-based mathematical modeling. Cartilage samples from five cadaver patellae were mechanically tested under unconfined compression. Depth-dependent collagen content and fibril orientation, as well as proteoglycan and water content were derived by combining Fourier transform infrared imaging, biochemical analyses and polarized light microscopy. Finite element models were constructed for each sample in unconfined compression geometry. First, composition-based fibril-reinforced poroviscoelastic swelling models, including composition and structure obtained from microscopical and biochemical analyses were fitted to experimental stress-relaxation responses of three samples. Subsequently, optimized values of model constants, as well as compositional and structural parameters were implemented in the models of two additional samples to validate the optimization. Theoretical stress-relaxation curves agreed with the experimental tests (R=0.95-0.99). Using the optimized values of mechanical parameters, as well as composition and structure of additional samples, we were able to predict their mechanical behavior in unconfined compression, without mechanical testing (R=0.98). Our results suggest that specific information on tissue composition and structure might enable assessment of cartilage mechanics without mechanical testing.     

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.     

A cross-validation of the biphasic poroviscoelastic model of articular cartilage in unconfined compression, indentation, and confined compression

The biphasic poroviscoelastic (BPVE) model was curve fit to the simultaneous relaxation of reaction force and lateral displacement exhibited by articular cartilage in unconfined compression (n=18). Model predictions were also made for the relaxation observed in reaction force during indentation with a porous plane-ended metal indenter (n=4), indentation with a nonporous plane ended metal indenter (n=4), and during confined compression (n=4). Each prediction was made using material parameters resulting from curve fits of the unconfined compression response of the same tissue. The BPVE model was able to account for both the reaction force and the lateral displacement during unconfined compression very well. Furthermore, model predictions for both indentation and confined compression also followed the experimental data well. These results provide substantial evidence for the efficacy of the biphasic poroviscoelastic model for articular cartilage, as no successful cross-validation of a model simulation has been demonstrated using other mathematical models.     

A visco-hyperelastic constitutive approach for modeling polyvinyl alcohol sponge

This study proposes the quasi-linear viscoelastic (QLV) model to characterize the time dependent mechanical behavior of poly(vinyl alcohol) (PVA) sponges. The PVA sponges have implications in many viscoelastic soft tissues, including cartilage, liver, and kidney as an implant. However, a critical barrier to the use of the PVA sponge as tissue replacement material is a lack of sufficient study on its viscoelastic mechanical properties. In this study, the nonlinear mechanical behavior of a fabricated PVA sponge is investigated experimentally and computationally using relaxation and stress failure tests as well as finite element (FE) modeling. Hyperelastic strain energy density functions, such as Yeoh and Neo-Hookean, are used to capture the mechanical behavior of PVA sponge at ramp part, and viscoelastic model is used to describe the viscose behavior at hold part. Hyperelastic material constants are obtained and their general prediction ability is verified using FE simulations of PVA tensile experiments. The results of relaxation and stress failure tests revealed that Yeoh material model can define the mechanical behavior of PVA sponge properly compared with Neo-Hookean one. FE modeling results are also affirmed the appropriateness of Yeoh model to characterize the mechanical behavior of PVA sponge. Thus, the Yeoh model can be used in future biomechanical simulations of the spongy biomaterials. These results can be utilized to understand the viscoelastic behavior of PVA sponges and has implications for tissue engineering as scaffold.     

Biomechanical properties of human articular cartilage under compressive loads

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

Viscoelastic properties of passive skeletal muscle in compression—Cyclic behaviour

Skeletal muscle relaxation behaviour in compression has been previously reported, but the anisotropic behaviour at higher loading rates remains poorly understood. In this paper, uniaxial unconfined cyclic compression tests were performed on fresh porcine muscle samples at various fibre orientations to determine muscle viscoelastic behaviour. Mean compression level of 25% was applied and cycles of 2% and 10% amplitude were performed at 0.2–80 Hz. Under cycles of low frequency and amplitude, linear viscoelastic cyclic relaxation was observed. Fibre/cross-fibre results were qualitatively similar, but the cross-fibre direction was stiffer (ratio of 1.2). In higher amplitude tests nonlinear viscoelastic behaviour with a frequency dependent increase in the stress cycles amplitude was found (factor of 4.1 from 0.2 to 80 Hz).The predictive capability of an anisotropic quasi-linear viscoelastic model previously fitted to stress-relaxation data from similar tissue samples was investigated. Good qualitative results were obtained for low amplitude cycles but differences were observed in the stress cycle amplitudes (errors of 7.5% and 31.8%, respectively, in the fibre/cross-fibre directions). At higher amplitudes significant qualitative and quantitative differences were evident. A nonlinear model formulation was therefore developed which provided a good fit and predictions to high amplitude low frequency cyclic tests performed in the fibre/cross-fibre directions. However, this model gave a poorer fit to high frequency cyclic tests and to relaxation tests. Neither model adequately predicts the stiffness increase observed at frequencies above 5 Hz.Together with data previously presented, the experimental data presented here provide a unique dataset for validation of future constitutive models for skeletal muscle in compression.     

Polymer dynamics as a mechanistic model for the flow-independent viscoelasticity of cartilage

The initial, rapid, flow independent, apparent stress relaxation of articular cartilage disks deformed by unconfined compressive displacement is shown to be consistent with the theory of polymer dynamics. A relaxation function for polymers based upon a mechanistic model of molecular interaction (reptation) appropriately approximated early, flow independent relaxation of stress. It is argued that the theory of polymer dynamics, with its reliance on mechanistic models of molecular interaction, is an appropriate technique for application to and the understanding of rapid, flow independent, stress relaxation in cartilage.     

Unconfined compression of hydrated viscoelastic tissues: a biphasic poroviscoelastic analysis

A biphasic poroviscoelastic theory was used to analyze the unconfined compression creep and stress relaxation of a hydrated viscoelastic tissue. The intrinsic shear properties of the tissue matrix was described by an integral-type viscoelastic constitutive law while the intrinsic bulk property of the matrix was assumed to be linearly elastic. Parametric data were presented to show how the two major energy dissipative mechanisms, namely the interstitial fluid flow and the intrinsic matrix viscoelasticity, may each contribute to the apparent viscoelastic behavior of the whole tissue under unconfined compression. The hydraulic permeability of the tissue was found to enter in only as a scaling factor for time.     

A simplified approach to quasi-linear viscoelastic modeling

The fitting of quasi-linear viscoelastic (QLV) constitutive models to material data often involves somewhat cumbersome numerical convolution. A new approach to treating quasi-linearity in 1-D is described and applied to characterize the behavior of reconstituted collagen. This approach is based on a new principle for including nonlinearity and requires considerably less computation than other comparable models for both model calibration and response prediction, especially for smoothly applied stretching. Additionally, the approach allows relaxation to adapt with the strain history. The modeling approach is demonstrated through tests on pure reconstituted collagen. Sequences of "ramp-and-hold" stretching tests were applied to rectangular collagen specimens. The relaxation force data from the "hold" was used to calibrate a new "adaptive QLV model" and several models from literature, and the force data from the "ramp" was used to check the accuracy of model predictions. Additionally, the ability of the models to predict the force response on a reloading of the specimen was assessed. The "adaptive QLV model" based on this new approach predicts collagen behavior comparably to or better than existing models, with much less computation.     

Singular perturbation analysis of the nonlinear, flow-dependent compressive stress relaxation behavior of articular cartilage

The dominant mechanism giving rise to the viscoelastic response of articular cartilage during compression is the nonlinear diffusive interaction of the fluid and solid phases of the tissue as they flow relative to one another. The present study is concerned with the role of this interaction under uniaxial stress relaxation in compression. The model is a biphasic mixture of fluid and solid which incorporates the strain-dependent permeability found earlier from permeation experiments. When a ramp-displacement is imposed on the articular surface, simple, but accurate, asymptotic approximations are derived for the deformation and stress fields in the tissue for slow and moderately fast rates of compression. They are shown to agree very well with experiment and they provide a simple means for determining the material parameters. Moreover, they lead to important insights into the role of the flow-dependent viscoelastic nature of articular cartilage and other hydrated biological tissues.