Determination of nonlinear fibre-reinforced biphasic poroviscoelastic constitutive parameters of articular cartilage using stress relaxation indentation testing and an optimizing finite element analysis

Spatial and Temporal Mapping of Articular Cartilage Poro-Viscoelastic Material Properties Using Indentation

Biphasic poro-viscoelastic constitutive material model (BPVE) captures both the fluid flow dependent and independent behavior of cartilage under stress relaxation type indentation. A finite element model based on BPVE formulation was developed to explore the sensitivity of the model to Young's modulus, Poisson's ratio, permeability, and viscoelastic constitutive parameters expressed in terms of Prony series coefficients. Then we fit the numerical model to experimental force versus time curves from stress relaxation indents on bovine tibial plateaus to extract the material properties. Measurements were made over the period of two days to capture the material property changes that resulted from trypsin-induced degradation. We measured spatial and temporal changes in mechanical properties in the cartilage. The areas of degradation were characterized by an increase in both permeability and summation of Prony series shear relaxation amplitude constants. These findings suggest that cartilage degradation reduces the intrinsic viscoelastic properties of the solid phase of the tissue in addition to impairing its capacity to offer frictional drag to the interstitial fluid flow (permeability). The changes in material properties are measurable well before structural degradation occurs.

Characterization and comparison of hyper-viscoelastic properties of normal and osteoporotic bone using stress-relaxation experiment

Bone tissue behavior under various loads is nonlinear elastic due to irreversible energy absorption. Also, viscoelasticity is one of the most important properties of bone which is very important in dynamic analyses and helps a lot in making artificial bone. In this study, rat tibia bone specimens were subjected to compression stress-relaxation test for normal (n = 5) and osteoporotic (n = 5) groups in order to characterize their mechanical properties using finite element modeling coupled with an optimization algorithm. Using this method, the structural equation parameters for the Neo-Hookean model and the Prony series coefficients were used to describe the hyper-elastic and the viscoelastic behavior of specimens, respectively; moreover, the properties of materials including the bulk, shear and Young's moduli for both groups were obtained and compared. The shear modulus was also gained as a function of time. In addition, the percentage of stress reduction and its relation to the initial stress were investigated for specimens. Finally, the effect of changes in each of the parameters of the hyper-viscoelastic structural equation on the force response was determined. Results of this study can be used in predicting the transient response and dynamic analysis of the bone.

Obtaining the sGAG distribution profile in articular cartilage color images

The articular cartilage tissue is an essential component of joints as it reduces the friction between the two bones. Its load-bearing properties depend mostly on proteoglycan distribution, which can be analyzed through the study of the presence of sulfated glycosaminoglycan (sGAG). Currently, sGAG distribution in articular cartilage is not completely known; it is calculated by means of laboratory tests that imply the inherent inaccuracy of a manual procedure. This paper presents an easy-to-use desktop software application for obtaining the sGAG distribution profile in tissue. This app uses color images of stained cartilage tissues taken under a microscope, so researchers at the Trinity Centre for Bioengineering (Dublin, Ireland) can understand the qualitative distribution of sGAG with depth in the studied tissues.

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.

Biphasic Finite Element Modeling Reconciles Mechanical Properties of Tissue-Engineered Cartilage Constructs Across Testing Platforms

Cartilage tissue engineering is emerging as a promising treatment for osteoarthritis, and the field has progressed toward utilizing large animal models for proof of concept and preclinical studies. Mechanical testing of the regenerative tissue is an essential outcome for functional evaluation. However, testing modalities and constitutive frameworks used to evaluate in vitro grown samples differ substantially from those used to evaluate in vivo derived samples. To address this, we developed finite element (FE) models (using FEBio) of unconfined compression and indentation testing, modalities commonly used for such samples. We determined the model sensitivity to tissue radius and subchondral bone modulus, as well as its ability to estimate material parameters using the built-in parameter optimization tool in FEBio. We then sequentially tested agarose gels of 4%, 6%, 8%, and 10% weight/weight using a custom indentation platform, followed by unconfined compression. Similarly, we evaluated the ability of the model to generate material parameters for living constructs by evaluating engineered cartilage. Juvenile bovine mesenchymal stem cells were seeded (2 × 10⁷ cells/mL) in 1% weight/volume hyaluronic acid hydrogels and cultured in a chondrogenic medium for 3, 6, and 9 weeks. Samples were planed and tested sequentially in indentation and unconfined compression. The model successfully completed parameter optimization routines for each testing modality for both acellular and cell-based constructs. Traditional outcome measures and the FE-derived outcomes showed significant changes in material properties during the maturation of engineered cartilage tissue, capturing dynamic changes in functional tissue mechanics. These outcomes were significantly correlated with one another, establishing this FE modeling approach as a singular method for the evaluation of functional engineered and native tissue regeneration, both in vitro and in vivo.

On mechanical mechanism of damage evolution in articular cartilage

Superficial lesions of cartilage are the direct indication of osteoarthritis. To investigate the mechanical mechanism of cartilage with micro-defect under external loading, a new plain strain numerical model with micro-defect was proposed and damage evolution progression in cartilage over time has been simulated, the parameter were studied including load style, velocity of load and degree of damage. The new model consists of the hierarchical structure of cartilage and depth-dependent arched fibers. The numerical results have shown that not only damage of the cartilage altered the distribution of the stress but also matrix and fiber had distinct roles in affecting cartilage damage, and damage in either matrix or fiber could promote each other. It has been found that the superficial cracks in cartilage spread preferentially along the tangent direction of the fibers. It is the arched distribution form of fibers that affects the crack spread of cartilage, which has been verified by experiment. During the process of damage evolution, its extension direction and velocity varied constantly with the damage degree. The rolling load could cause larger stress and strain than sliding load. Strain values of the matrix initially increased and then decreased gradually with the increase of velocity, and velocity had a greater effect on matrix than fibers. Damage increased steadily before reaching 50%, sharply within 50 to 85%, and smoothly and slowly after 85%. The finding of the paper may help to understand the mechanical mechanism why the cracks in cartilage spread preferentially along the tangent direction of the fibers.

Properties of the cartilage layer from the cam-type hip impingement deformity

Femoro-acetabular impingement (FAI) is associated with significant acetabular cartilage damage and degenerative arthritis. To understand the contact stress and thus biomechanical mechanisms that may contribute to degeneration, the material behaviour of the cartilage layer is required. The objective of this study is to determine the fibril-reinforced poroelastic properties and composition of cartilage from cam deformities and to compare to those of normal cartilage. Patients undergoing surgical treatment of a symptomatic cam FAI deformity were recruited from the clinical practice of one of the authors. Osteochondral specimens were retrieved from the deformity during surgery using a trephine. Control specimens were retrieved from the anterior femoral head bearing surface during autopsy procedures. Indentation stress-relaxation tests were performed to determine the modulus (E_S), Poisson's ratio (ν) and permeability (k₀) of the poroelastic component, and the strain-independent (E₀) and -dependent (E_ε) moduli of the fibril-reinforcement using finite element analysis and optimization. Safranin-O staining was used to quantify proteoglycan content. E_S and ν were 71% and 37% lower, respectively, in Cam specimens compared to controls, and k₀ was approximately triple that of Control specimens (p<0.05). No significant differences were seen in the fibrillar components, E₀ and E_ε. Proteoglycan content was substantially depleted in Cam specimens, and was correlated with E_S, ν and k₀. This study showed that cartilage from the cam deformity exhibits severe degeneration in terms of the mechanical behaviour and composition changes, and is consistent with osteoarthritis. This further supports the hypothesis that FAI is a cause of hip osteoarthritis.

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.

An optimized transversely isotropic, hyper-poro-viscoelastic finite element model of the meniscus to evaluate mechanical degradation following traumatic loading

Inverse finite element (FE) analysis is an effective method to predict material behavior, evaluate mechanical properties, and study differences in biological tissue function. The meniscus plays a key role in load distribution within the knee joint and meniscal degradation is commonly associated with the onset of osteoarthritis. In the current study, a novel transversely isotropic hyper-poro-viscoelastic constitutive formulation was incorporated in a FE model to evaluate changes in meniscal material properties following tibiofemoral joint impact. A non-linear optimization scheme was used to fit the model output to indentation relaxation experimental data. This study is the first to investigate rate of relaxation in healthy versus impacted menisci. Stiffness was found to be decreased (p=0.003), while the rate of tissue relaxation increased (p=0.010) at twelve weeks post impact. Total amount of relaxation, however, did not change in the impacted tissue (p=0.513).

Determination of poroelastic properties of cartilage using constrained optimization coupled with finite element analysis

The feasibility of determining biphasic material properties using a finite element model of stress relaxation coupled with two types of constrained optimization to match measured data was investigated. Comparison of these two approaches, a zero-order method and a gradient-based algorithm, validated the predicted material properties. Optimizations were started from multiple different initial guesses of material properties (design variables) to establish the robustness of the optimization. Overall, the optimal values are close to those found by Cohen et al. (1998) but these small differences produced a marked improvement in the fit to the measured stress relaxation. Despite the greater deviation in the optimized values obtained from the zero-order method, both optimization procedures produced material properties that gave equally good overall fits to the measured data. Furthermore, optimized values were all within the expected range of material properties. Modeling stress relaxation using the optimized material properties showed an excellent fit to the entire time history of the measured data.

Computational analysis of cartilage implants based on an interpenetrated polymer network for tissue repairing

Interpenetrated polymer networks (IPNs), composed by two independent polymeric networks that spatially interpenetrate, are considered as valuable systems to control permeability and mechanical properties of hydrogels for biomedical applications. Specifically, poly(ethyl acrylate) (PEA)-poly(2-hydroxyethyl acrylate) (PHEA) IPNs have been explored as good hydrogels for mimicking articular cartilage. These lattices are proposed as matrix implants in cartilage damaged areas to avoid the discontinuity in flow uptake preventing its deterioration. The permeability of these implants is a key parameter that influences their success, by affecting oxygen and nutrient transport and removing cellular waste products to healthy cartilage. Experimental try-and-error approaches are mostly used to optimize the composition of such structures. However, computational simulation may offer a more exhaustive tool to test and screen out biomaterials mimicking cartilage, avoiding expensive and time-consuming experimental tests. An accurate and efficient prediction of material's permeability and internal directionality and magnitude of the fluid flow could be highly useful when optimizing biomaterials design processes. Here we present a 3D computational model based on Sussman-Bathe hyperelastic material behaviour. A fluid structure analysis is performed with ADINA software, considering these materials as two phases composites where the solid part is saturated by the fluid. The model is able to simulate the behaviour of three non-biodegradable hydrogel compositions, where percentages of PEA and PHEA are varied. Specifically, the aim of this study is (i) to verify the validity of the Sussman-Bathe material model to simulate the response of the PEA-PHEA biomaterials; (ii) to predict the fluid flux and the permeability of the proposed IPN hydrogels and (iii) to study the material domains where the passage of nutrients and cellular waste products is reduced leading to an inadequate flux distribution in healthy cartilage tissue. The obtained results show how the model predicts the permeability of the PEA-PHEA hydrogels and simulates the internal behaviour of the samples and shows the distribution and quantification of fluid flux.

Experiment and hydro-mechanical coupling simulation study on the human periodontal ligament

In this paper, a new method involving an experiment in vivo and hydro-mechanical coupling simulations was proposed to investigate the biomechanical property of human periodontal ligament (PDL). Teeth were loaded and their displacements were measured in vivo. The finite element model of the experiment was built and hydro-mechanical coupling simulations were conducted to test some PDL's constitutive models. In the simulations, the linear elastic model, the hyperfoam model, and the Ogden model were assumed for the solid phase of the PDL coupled with a model of the fluid phase of the PDL. The displacements of the teeth derived from the simulations were compared with the experimental data to validate these constitutive models. The study shows that a proposed constitutive model of the PDL can be reliably tested by this method. Furthermore, the influence of species, areas, and the fluid volume ratio on PDL's mechanical property should be considered in the modeling and simulation of the mechanical property of the PDL.

The effect of collagen crosslinking on the biphasic poroviscoelastic cartilage properties determined from a semi-automated microindentation protocol for stress relaxation

Given the important role of the collagenous structure in cartilage mechanics, there is considerable interest in the relationship between collagen crosslinking and the mechanical behavior of the cartilage matrix. While crosslink-induced alterations to the elastic modulus of cartilage have been described, changes to time-dependent behavior have not yet been determined. The objective of the study was to quantify changes to cartilage material properties, including viscoelastic coefficients, with crosslinking via indentation. To accomplish this, a semi-autonomous microindentation stress relaxation protocol was first developed, validated and then applied to cartilage specimens before and after crosslinking. The change in mechanical properties with crosslinking was analyzed both in the unloading portions of the test via the Oliver-Pharr method and in the holding portion with an inverse iterative finite element model that represented cartilage as a biphasic poroviscoelastic material. Although both techniques suggested a similar increase in equilibrium modulus in the crosslinked specimens as compared to the controls, distinct differences in the control specimens were apparent, suggesting that the two different techniques may be capturing different aspects of the material behavior. No differences in time-dependent properties were observed between the crosslinked and the control specimens. These results give further insight into the effects of crosslinking in cartilage mechanical behavior. Additionally, the microindentation stress relaxation protocol may enable increased automation for high-throughput testing.

Use of a Poroelastic Model to Predict Intramuscular Pressure

Measurement of individual muscle tension in a clinical setting has yet to be achieved. Previous investigators have suggested that the tension in skeletal muscle, comprised of approximately 70% fluid, could be determined using interstitial muscle fluid pressure (IMP). A computational model is needed to aid in understanding IMP distribution in muscles of varying geometry and contractile states without exhaustive testing. The first aim of this study was to determine a set of transversely isotropic material properties (i.e., permeability, relaxed modulus, and drained Poisson's ratio) for excised skeletal muscle using inverse finite element analysis with a poroelastic constitutive formulation on tension data from either longitudinal or transverse uniaxial load-relaxation tests of skeletal muscle tissue. The second aim was to compare pore pressure estimated from a model to experimental pressure measurements to assess its ability to accurately predict IMP. Results of this study indicated that skeletal muscle was transversely isotropic under load-relaxation as demonstrated by significant differences in the drained Poisson's ratio. It was also noted that the drained Poisson's ratios under both longitudinal and transverse loading were negative in these tests of excised muscle tissue. Pore pressure calculated with this model provided a good prediction of the development of IMP. These results point to the benefit of using a poroelastic model of skeletal muscle to predict IMP.

A review of the combination of experimental measurements and fibril-reinforced modeling for investigation of articular cartilage and chondrocyte response to loading

The function of articular cartilage depends on its structure and composition, sensitively impaired in disease (e.g. osteoarthritis, OA). Responses of chondrocytes to tissue loading are modulated by the structure. Altered cell responses as an effect of OA may regulate cartilage mechanotransduction and cell biosynthesis. To be able to evaluate cell responses and factors affecting the onset and progression of OA, local tissue and cell stresses and strains in cartilage need to be characterized. This is extremely challenging with the presently available experimental techniques and therefore computational modeling is required. Modern models of articular cartilage are inhomogeneous and anisotropic, and they include many aspects of the real tissue structure and composition. In this paper, we provide an overview of the computational applications that have been developed for modeling the mechanics of articular cartilage at the tissue and cellular level. We concentrate on the use of fibril-reinforced models of cartilage. Furthermore, we introduce practical considerations for modeling applications, including also experimental tests that can be combined with the modeling approach. At the end, we discuss the prospects for patient-specific models when aiming to use finite element modeling analysis and evaluation of articular cartilage function, cellular responses, failure points, OA progression, and rehabilitation.

How changes in fibril-level organization correlate with the macrolevel behavior of articular cartilage

The primary structural components of articular cartilage are the zonally differentiated interconnected network of collagen fibrils and proteoglycans, the latter having the potential to bind large amounts of water. Both components exist in a coupled relationship that gives rise to its remarkable mechanical properties. The response of cartilage to compression is governed both by the degree to which the hydrated proteoglycans are constrained within this fibrillar network and the ease with which the matrix fluid can be displaced. The functional properties of cartilage are therefore closely linked to the integrity of the fibrillar network. Our current understanding of this network has been derived via studies conducted at the macro, micro, and ultrastructural levels. Of particular interest to joint researchers and clinicians are issues relating to how the network structure varies both directionally and with zonal depth, how its integrity is maintained via mechanisms of fibril interconnectivity, and how it is modified by ageing, degeneration, and trauma. Physical models have been developed to explore modes of interconnectivity. Combined micromechanical and structural studies confirm the critical role that this interconnectivity must play but detailed descriptions at the molecular level remain elusive. Current computationally based models of cartilage have in some cases implemented the fibrillar component, albeit simplistically, as a separate structure. Considering how important a role fibril network interconnectivity plays in actual tissue structure and mechanical behavior, and especially how it changes with degeneration, a major challenge facing joint tissue modellers is how to incorporate such a feature in their models.

Microporous calcium phosphate ceramics as tissue engineering scaffolds for the repair of osteochondral defects: biomechanical results

This work investigated the suitability of microporous β-tricalcium phosphate (TCP) scaffolds pre-seeded with autologous chondrocytes for treatment of osteochondral defects in a large animal model. Microporous β-TCP cylinders (Ø 7 mm; length 25 mm) were seeded with autologous chondrocytes and cultured for 4 weeks in vitro. Only the upper end of the cylinder was seeded with chondrocytes. Chondrocytes formed a multilayer on the top. The implants were then implanted in defects (diameter 7 mm) created in the left medial femoral condyle of ovine knees. The implants were covered with synovial membrane from the superior recess of the same joint. For the right knees, an empty defect with the same dimensions served as control. Twenty-eight sheep were split into 6-, 12-, 26- and 52 week groups of seven animals. Indentation tests with a spherical (Ø 3mm) indenter were used to determine the biomechanical properties of regenerated tissue. A software-based limit switch was implemented to ensure a maximal penetration depth of 200 μm and maximal load of 1.5 N. The achieved load, the absorbed energy and the contact stiffness were measured. Newly formed cartilage was assessed with the International Cartilage Repair Society Visual Assessment Scale (ICRS score) and histomorphometric analysis. Results were analysed statistically using the t-test, Mann-Whitney U-test and Wilcoxon test. Statistical significance was set at p<0.05. After 6 weeks of implantation, the transplanted area tolerated an indentation load of 0.05±0.20 N. This value increased to 0.10±0.06 N after 12 weeks, to 0.27±0.18 N after 26 weeks, and 0.27±0.11 N after 52 weeks. The increase in the tolerated load was highly significant (p<0.0001), but the final value was not significantly different from that of intact cartilage (0.30±0.12 N). Similarly, the increase in contact stiffness from 0.87±0.29 N mm-(1) after 6 weeks to 3.14±0.86 N mm(-1) after 52 weeks was highly significant (p<0.0001). The absorbed energy increased significantly (p=0.02) from 0.74×10(-6)±0.38×10(-6) Nm after 6 weeks to 2.83×10(-6)±1.35×10(-6) Nm after 52 weeks. At 52 weeks, the International Cartilage Repair Society (ICRS) scores for the central area of the transplanted area and untreated defects were comparable. In contrast, the score for the area from the edge to the centre of the transplanted area was significantly higher (p=0.001) than the score for the unfilled defects. A biomechanically stable cartilage was built outside the centre of defect. After 52 weeks, all but one empty control defect were covered by bone and a very thin layer of cartilage (ICRS 7 points). The empty hole could still be demonstrated beneath the bone. The histomorphometric evaluation revealed that 81.0±10.6% of TCP was resorbed after 52 weeks. The increase in TCP resorption and replacement by spongy bone during the observation period was highly significant (p<0.0001). In this sheep trial, the mechanical properties of microporous TCP scaffolds seeded with transplanted autologous chondrocytes were similar to those of natural cartilage after 52 weeks of implantation. However, the central area of the implants had a lower ICRS score than healthy cartilage. Microporous TCP was almost fully resorbed at 52 weeks and replaced by bone.

A nonlinear biphasic fiber-reinforced porohyperviscoelastic model of articular cartilage incorporating fiber reorientation and dispersion

A nonlinear biphasic fiber-reinforced porohyperviscoelastic (BFPHVE) model of articular cartilage incorporating fiber reorientation effects during applied load was used to predict the response of ovine articular cartilage at relatively high strains (20%). The constitutive material parameters were determined using a coupled finite element-optimization algorithm that utilized stress relaxation indentation tests at relatively high strains. The proposed model incorporates the strain-hardening, tension-compression, permeability, and finite deformation nonlinearities that inherently exist in cartilage, and accounts for effects associated with fiber dispersion and reorientation and intrinsic viscoelasticity at relatively high strains. A new optimization cost function was used to overcome problems associated with large peak-to-peak differences between the predicted finite element and experimental loads that were due to the large strain levels utilized in the experiments. The optimized material parameters were found to be insensitive to the initial guesses. Using experimental data from the literature, the model was also able to predict both the lateral displacement and reaction force in unconfined compression, and the reaction force in an indentation test with a single set of material parameters. Finally, it was demonstrated that neglecting the effects of fiber reorientation and dispersion resulted in poorer agreement with experiments than when they were considered. There was an indication that the proposed BFPHVE model, which includes the intrinsic viscoelasticity of the nonfibrillar matrix (proteoglycan), might be used to model the behavior of cartilage up to relatively high strains (20%). The maximum percentage error between the indentation force predicted by the FE model using the optimized material parameters and that measured experimentally was 3%.