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

Constituent-based quasi-linear viscoelasticity: a revised quasi-linear modelling framework to capture nonlinear viscoelasticity in arteries

Arteries exhibit fully nonlinear viscoelastic behaviours (i.e. both elastically and viscously nonlinear). While elastically nonlinear arterial models are well established, effective mathematical descriptions of nonlinear viscoelasticity are lacking. Quasi-linear viscoelasticity (QLV) offers a convenient way to mathematically describe viscoelasticity, but its viscous linearity assumption is unsuitable for whole-wall vascular applications. Conversely, application of fully nonlinear viscoelastic models, involving deformation-dependent viscous parameters, to experimental data is impractical and often reduces to identifying specific solutions for each tested loading condition. The present study aims to address this limitation: By applying QLV theory at the wall constituent rather than at the whole-wall level, the deformation-dependent relative contribution of the constituents allows to capture nonlinear viscoelasticity with a unique set of deformation-independent model parameters. Five murine common carotid arteries were subjected to a protocol of quasi-static and harmonic, pseudo-physiological biaxial loading conditions to characterise their viscoelastic behaviour. The arterial wall was modelled as a constrained mixture of an isotropic elastin matrix and four families of collagen fibres. Constituent-based QLV was implemented by assigning different relaxation functions to collagen- and elastin-borne parts of the wall stress. Nonlinearity in viscoelasticity was assessed via the pressure dependency of the dynamic-to-quasi-static stiffness ratio. The experimentally measured ratio increased with pressure, from 1.03 [Formula: see text] 0.03 (mean [Formula: see text] standard deviation) at 80-40 mmHg to 1.58 [Formula: see text] 0.22 at 160-120 mmHg. Constituent-based QLV captured well this trend by attributing the wall viscosity predominantly to collagen fibres, whose recruitment starts at physiological pressures. In conclusion, constituent-based QLV offers a practical and effective solution to model arterial viscoelasticity.

Tensile Viscoelastic Properties of the Sclera after Glycosaminoglycan Depletion

PURPOSE

Fibrillar collagen network and glycosaminoglycans (GAGs) are the primary components of extracellular matrix (ECM) of the sclera. The main goal of this study was to investigate the possible structural roles of GAGs in the scleral tensile properties as a function of preconditioning and displacement rate.

METHODS

Four-step uniaxial stress-relaxation tests were used for characterizing the viscoelastic tensile response of the posterior porcine sclera with and without enzymatic GAG removal. The scleral strips were divided into different groups based on the displacement rate and the presence or absence of a preconditioning step in the loading protocol. The groups were (1) displacement rate of 0.2 mm/min without preconditioning, (2) displacement rate of 1 mm/min without preconditioning, (3) displacement rate of 0.2 mm/min with preconditioning, and (4) displacement rate of 1 mm/min with preconditioning. The peak stress, equilibrium stress, and the equilibrium elastic modulus were calculated for all specimens and compared against each other.

RESULTS

Increasing the displacement rate from 0.2 mm/min to 1.0 mm/min was found to cause an insignificant change in the equilibrium stress and equilibrium elastic modulus of porcine scleral strips. Removal of GAGs resulted in an overall stiffer tensile behavior independent of the displacement rate in samples that were not preconditioned (P < .05). The behavior of preconditioned samples with and without GAG removal was not significantly different from each other.

CONCLUSIONS

The experimental measurements of the present study showed that GAGs play an important role in the mechanical properties of the posterior porcine sclera. Furthermore, using a preconditioning step in the uniaxial testing protocol resulted in not being able to identify any significant difference in the tensile behavior of GAG depleted and normal scleral strips.

Nondestructive Techniques to Evaluate the Characteristics and Development of Engineered Cartilage

In this review, methods for evaluating the properties of tissue engineered (TE) cartilage are described. Many of these have been developed for evaluating properties of native and osteoarthritic articular cartilage. However, with the increasing interest in engineering cartilage, specialized methods are needed for nondestructive evaluation of tissue while it is developing and after it is implanted. Such methods are needed, in part, due to the large inter- and intra-donor variability in the performance of the cellular component of the tissue, which remains a barrier to delivering reliable TE cartilage for implantation. Using conventional destructive tests, such variability makes it near-impossible to predict the timing and outcome of the tissue engineering process at the level of a specific piece of engineered tissue and also makes it difficult to assess the impact of changing tissue engineering regimens. While it is clear that the true test of engineered cartilage is its performance after it is implanted, correlation of pre and post implantation properties determined non-destructively in vitro and/or in vivo with performance should lead to predictive methods to improve quality-control and to minimize the chances of implanting inferior tissue.

Strain-rate-dependent non-linear tensile properties of the superficial zone of articular cartilage

AIM OF THE STUDY

The tensile properties of articular cartilage play an important role in the compressive behavior and integrity of the tissue. The stress-strain relationship of cartilage in compression was observed previously to depend on the strain-rate. This strain-rate dependence has been thought to originate mainly from fluid pressurization. However, it was not clear to what extent the tensile properties of cartilage contribute to the strain-rate dependence in compressive behavior of cartilage. The aim of the present study was to quantify the strain-rate dependent stress-strain relationship and hysteresis of articular cartilage in tension.

METHODS

Uniaxial tensile tests were performed to examine the strain-rate dependent non-linear tensile properties of the superficial zone of bovine knee cartilage. Tensile specimens were oriented in the fiber direction indicated by the India ink method. Seven strain-rates were used in the measurement ranging from 0.1 to 80%/s, which corresponded to nearly static to impact joint loadings.

RESULTS

The experimental data showed substantial strain-rate and strain-magnitude dependent load response: for a given strain-magnitude, the tensile stress could vary by a factor of 1.95 while the modulus by a factor of 1.58 with strain-rate; for a given strain-rate, the modulus at 15% strain could be over four times the initial modulus at no strain. The energy loss in cartilage tension upon unloading exhibited a complex variation with the strain-rate.

CONCLUSION

The strain-rate dependence of cartilage in tension observed from the present study is relatively weaker than that in compression observed previously, but is considerable to contribute to the strain-rate dependent load response in compression.

An Equilibrium Constitutive Model of Anisotropic Cartilage Damage to Elucidate Mechanisms of Damage Initiation and Progression

Traumatic injuries and gradual wear-and-tear of articular cartilage (AC) that can lead to osteoarthritis (OA) have been hypothesized to result from tissue damage to AC. In this study, a previous equilibrium constitutive model of AC was extended to a constitutive damage articular cartilage (CDAC) model. In particular, anisotropic collagen (COL) fibril damage and isotropic glycosaminoglycan (GAG) damage were considered in a 3D formulation. In the CDAC model, time-dependent effects, such as viscoelasticity and poroelasticity, were neglected, and thus all results represent the equilibrium response after all time-dependent effects have dissipated. The resulting CDAC model was implemented in two different finite-element models. The first simulated uniaxial tensile loading to failure, while the second simulated spherical indentation with a rigid indenter displaced into a bilayer AC sample. Uniaxial tension to failure simulations were performed for three COL fibril Lagrangian failure strain (i.e., the maximum elastic COL fibril strain) values of 15%, 30%, and 45%, while spherical indentation simulations were performed with a COL fibril Lagrangian failure strain of 15%. GAG damage parameters were held constant for all simulations. Our results indicated that the equilibrium postyield tensile response of AC and the macroscopic tissue failure strain are highly dependent on COL fibril Lagrangian failure strain. The uniaxial tensile response consisted of an initial nonlinear ramp region due to the recruitment of intact fibrils followed by a rapid decrease in tissue stress at initial COL fibril failure, as a result of COL fibril damage which continued until ultimate tissue failure. In the spherical indentation simulation, damage to both the COL fibril and GAG constituents was located only in the superficial zone (SZ) and near the articular surface with tissue thickening following unloading. Spherical indentation simulation results are in agreement with published experimental observations. Our results indicate that the proposed CDAC model is capable of simulating both initial small magnitude damage as well as complete failure of AC tissue. The results of this study may help to elucidate the mechanisms of AC tissue damage, which initiate and propagate OA.

Indentation properties and glycosaminoglycan content of human menisci in the deep zone

Menisci are two crescent shaped fibrocartilaginous structures that provide fundamental load distribution and support within the knee joint. Their unique shape transmits axial stresses (i.e. "body force") into hoop or radial stresses. The menisci are primarily an inhomogeneous aggregate of glycosaminoglycans (GAGs) supporting bulk compression and type I collagen fibrils sustaining tension. It has been shown that the superficial meniscal layers are functionally homogeneous throughout the three distinct regions (anterior, central and posterior) using a 300 μm diameter spherical indenter tip, but the deep zone of the meniscus has yet to be mechanically characterized at this scale. Furthermore, the distribution and content of GAG throughout the human meniscal cross-section have not been examined. This study investigated the mechanical properties, via indentation, of the human deep zone meniscus among three regions of the lateral and medial menisci. The distribution of GAGs through the cross-section was also documented. Results for the deep zone of the meniscus showed the medial posterior region to have a significantly greater instantaneous elastic modulus than the central region. No significant differences in the equilibrium modulus were seen when comparing regions or the hemijoint. Histological results revealed that GAGs are not present until at least ~600 μm from the meniscal surface. Understanding the role and distribution of GAG within the human meniscus in conjunction with the material properties of the meniscus will aid in the design of tissue engineered meniscal replacements.

A mathematical model for analyzing the elasticity, viscosity, and failure of soft tissue: comparison of native and decellularized porcine cardiac extracellular matrix for tissue engineering

The clinical success of tissue-engineered constructs commonly requires mechanical properties that closely mimic those of the human tissue. Determining the viscoelastic properties of such biomaterials and the factors governing their failure profiles, however, has proven challenging, although collecting extensive data regarding their tensile behavior is straightforward. The easily calculated Young's modulus remains the most reported mechanical measure, regardless of its limitations, even though single-relaxation-time (SRT) models can provide much more information, which remain scarce due to a lack of manageable tools for implementing these models. We developed an easy-to-use algorithm for applying the Zener SRT model and determining the elastic moduli, viscosity, and failure profiles of materials under different mechanical tests in a user-independent manner. The algorithm was validated on the data resulting from tensile tests on native and decellularized porcine cardiac tissue, previously suggested as a promising scaffold material for cardiac tissue engineering. This analysis yields new and more accurate measurements such as the elastic moduli and viscosity, the model's relaxation time, and information on the factors governing the materials' failure profiles. These measurements indicate that the viscoelasticity and strength of the decellularized acellular extracellular matrix (ECM) are similar to those of native tissue, although its elasticity and apparent viscosity are higher. Nonetheless, reseeding and culturing the ECM with mesenchymal stem cells was shown to partially restore the mechanical properties lost after decellularization. We propose this algorithm as a platform for soft-tissue analysis that can provide comparable and unbiased measures for characterizing viscoelastic biomaterials commonly used in tissue engineering.

Subject-specific analysis of joint contact mechanics: application to the study of osteoarthritis and surgical planning

Advances in computational mechanics, constitutive modeling, and techniques for subject-specific modeling have opened the door to patient-specific simulation of the relationships between joint mechanics and osteoarthritis (OA), as well as patient-specific preoperative planning. This article reviews the application of computational biomechanics to the simulation of joint contact mechanics as relevant to the study of OA. This review begins with background regarding OA and the mechanical causes of OA in the context of simulations of joint mechanics. The broad range of technical considerations in creating validated subject-specific whole joint models is discussed. The types of computational models available for the study of joint mechanics are reviewed. The types of constitutive models that are available for articular cartilage are reviewed, with special attention to choosing an appropriate constitutive model for the application at hand. Issues related to model generation are discussed, including acquisition of model geometry from volumetric image data and specific considerations for acquisition of computed tomography and magnetic resonance imaging data. Approaches to model validation are reviewed. The areas of parametric analysis, factorial design, and probabilistic analysis are reviewed in the context of simulations of joint contact mechanics. Following the review of technical considerations, the article details insights that have been obtained from computational models of joint mechanics for normal joints; patient populations; the study of specific aspects of joint mechanics relevant to OA, such as congruency and instability; and preoperative planning. Finally, future directions for research and application are summarized.

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.

Contribution of proteoglycan osmotic swelling pressure to the compressive properties of articular cartilage

The negatively charged proteoglycans (PG) provide compressive resistance to articular cartilage by means of their fixed charge density (FCD) and high osmotic pressure (π(PG)), and the collagen network (CN) provides the restraining forces to counterbalance π(PG). Our objectives in this work were to: 1), account for collagen intrafibrillar water when transforming biochemical measurements into a FCD-π(PG) relationship; 2), compute π(PG) and CN contributions to the compressive behavior of full-thickness cartilage during bovine growth (fetal, calf, and adult) and human adult aging (young and old); and 3), predict the effect of depth from the articular surface on π(PG) in human aging. Extrafibrillar FCD (FCD(EF)) and π(PG) increased with bovine growth due to an increase in CN concentration, whereas PG concentration was steady. This maturation-related increase was amplified by compression. With normal human aging, FCD(EF) and π(PG) decreased. The π(PG)-values were close to equilibrium stress (σ(EQ)) in all bovine and young human cartilage, but were only approximately half of σ(EQ) in old human cartilage. Depth-related variations in the strain, FCD(EF), π(PG), and CN stress profiles in human cartilage suggested a functional deterioration of the superficial layer with aging. These results suggest the utility of the FCD-π(PG) relationship for elucidating the contribution of matrix macromolecules to the biomechanical properties of cartilage.

Cartilage Thickness Distribution Affects Computational Model Predictions of Cervical Spine Facet Contact Parameters

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

In vitro modulation of cartilage shape plasticity by biochemical regulation of matrix remodeling

With consideration of the need for cartilage grafts of specific sizes and shapes in orthopedics and other fields, immature cartilage explants and grafts have recently been molded in vitro and in vivo. Nonsurgical correction of cartilage deformities and malformations often uses mechanical stimuli and further demonstrates the plasticity of cartilage shape. Cartilage shape plasticity appears to diminish with maturation, coincident with changes in matrix composition. This study's objectives were to characterize shape plasticity of articular cartilage from immature and mature bovines and test whether altering proteoglycan and collagen (COL) remodeling modulates shape plasticity in vitro. Cartilage explants were analyzed fresh on day 0 or after 14 days of culture in the presence of β-D-xyloside to suppress glycosaminoglycan accumulation or β-aminopropionitrile (BAPN) to inhibit lysyl oxidase-mediated COL crosslinking. Culture with β-d-xyloside and BAPN differentially regulated cartilage size, composition, and shape plasticity, with an inverse association between shape plasticity and the ratio of tissue COL to glycosaminoglycan. Retention of a mechanically imposed contour was increased by culture with BAPN compared to day 0 calf cartilage (90% vs. 69%), and BAPN-treated samples had higher shape retention than β-D-xyloside-treated samples for both calf (90% vs. 74%) and adult cartilage (54% vs. 31%). The findings provide quantitative measures of cartilage shape plasticity at immature and mature stages and are consistent with the concept of diminishing shape plasticity with maturation. The ability to modulate cartilage shape plasticity by varying in vitro biochemical conditions may be a useful tool for the formation of contoured chondral grafts.