A transversely isotropic, transversely homogeneous microstructural-statistical model of articular cartilage

Discovering 3D hidden elasticity in isotropic and transversely isotropic materials with physics-informed UNets

Three-dimensional variation in structural components or fiber alignments results in complex mechanical property distribution in tissues and biomaterials. In this paper, we use a physics-informed UNet-based neural network model (El-UNet) to discover the three-dimensional (3D) internal composition and space-dependent material properties of heterogeneous isotropic and transversely isotropic materials without a priori knowledge of the composition. We then show the capabilities of El-UNet by validating against data obtained from finite-element simulations of two soft tissues, namely, brain tissue and articular cartilage, under various loading conditions. We first simulated compressive loading of 3D brain tissue comprising of distinct white matter and gray matter mechanical properties undergoing small strains with isotropic linear elastic behavior, where El-UNet reached mean absolute relative errors under 1.5 % for elastic modulus and Poisson's ratio estimations across the 3D volume. We showed that the 3D solution achieved by El-UNet was superior to relative stiffness mapping by inverse of axial strain and two-dimensional plane stress/plane strain approximations. Additionally, we simulated a transversely isotropic articular cartilage with known fiber orientations undergoing compressive loading, and accurately estimated the spatial distribution of all five material parameters, with mean absolute relative errors under 5 %. Our work demonstrates the application of the computationally efficient physics-informed El-UNet in 3D elasticity imaging and provides methods for translation to experimental 3D characterization of soft tissues and other materials. The proposed El-UNet offers a powerful tool for both in vitro and ex vivo tissue analysis, with potential extensions to in vivo diagnostics. STATEMENT OF SIGNIFICANCE: Elasticity imaging is a technique that reconstructs mechanical properties of tissue using deformation and force measurements. Given the complexity of this reconstruction, most existing methods have mostly focused on 2D problems. Our work is the first implementation of physics-informed UNets to reconstruct three-dimensional material parameter distributions for isotropic and transversely isotropic linear elastic materials by having deformation and force measurements. We comprehensively validate our model using synthetic data generated using finite element models of biological tissues with high bio-fidelity-the brain and articular cartilage. Our method can be implemented in elasticity imaging scenarios for in vitro and ex vivo mechanical characterization of biomaterials and biological tissues, with potential extensions to in vivo diagnostics.

Human Cartilage Biomechanics: Experimental and Theoretical Approaches towards the Identification of Mechanical Properties in Healthy and Osteoarthritic Conditions

Articular cartilage is a complex connective tissue with the fundamental functions of load bearing, shock absorption and lubrication in joints. However, traumatic events, aging and degenerative pathologies may affect its structural integrity and function, causing pain and long-term disability. Osteoarthritis represents a health issue, which concerns an increasing number of people worldwide. Moreover, it has been observed that this pathology also affects the mechanical behavior of the articular cartilage. To better understand this correlation, the here proposed review analyzes the physiological aspects that influence cartilage microstructure and biomechanics, with a special focus on the pathological changes caused by osteoarthritis. Particularly, the experimental data on human articular cartilage are presented with reference to different techniques adopted for mechanical testing and the related theoretical mechanical models usually applied to articular cartilage are briefly discussed.

Modeling one-handed grip strangulation: Intentionality of the gesture and age influence

Strangulation is a violent act which can be lethal and is often studied in forensic context. The neck includes several anatomical elements that can evolve with aging. We therefore created a numerical human neck model including the main anatomical elements and simulated one-handed grip strangulation cases. In addition, we created 3 models each representing age groups: 20-30 years old, 30-50 years old and over 50 years old. The main changes between the different age groups are the ossification of the cartilages and the muscles mechanical properties. Several initial and boundary conditions have been tested to perform a realistic simulation of one-handed grip strangulation. Stress analysis and fracture observation were compared with the grip strength of an average man, 552 N, to look at the intentionality of the gesture. In each age group, the results show no model fracture for a force of 552 N. It is necessary to reach a minimum of 1406 N before observing a first fracture on the hyoid bone. However, it is possible to get stresses on the hyoid bone and on the thyroid cartilage way before 552 N. It thus appears that the force created by one-handed grip strangulation is not sufficient to cause fractures of the bony elements of the neck, but it remains sufficient to compress the larynx and at least reduce airflow.

Modelling the inclusion of swelling pressure in a tissue level poroviscoelastic model of cartilage deformation

Swelling pressure in the interstitial fluid within the pores of cartilage tissue is known to have a significant effect on the rheology of cartilage tissue. The swelling pressure varies rapidly within thin regions inside pores known as Debye layers, caused by the presence of fixed charge, as observed in cartilage. Tissue level calculation of cartilage deformation therefore requires resolution of three distinct spatial scales: the Debye lengthscale within individual pores; the lengthscale of an individual pore; and the tissue lengthscale. We use asymptotics to construct a leading order approximation to the swelling pressure within pores, allowing the swelling pressure to be systematically included within a fluid-solid interaction model at the level of pores in cartilage. We then use homogenization to derive tissue level equations for cartilage deformation that are very similar to those governing the finite deformation of a poroviscoelastic body. The equations derived permit the spatial variations in porosity and electric charge that occur in cartilage tissue. Example solutions are then used to confirm the plausibility of the model derived and to consider the impact of fixed charge heterogeneity, illustrating that local fixed charge loss is predicted to increase deformation gradients under confined compression away from, rather than at, the site of loss.

The potential for intercellular mechanical interaction: simulations of single chondrocyte versus anatomically based distribution

Computational studies of chondrocyte mechanics, and cell mechanics in general, have typically been performed using single cell models embedded in an extracellular matrix construct. The assumption of a single cell microstructural model may not capture intercellular interactions or accurately reflect the macroscale mechanics of cartilage when higher cell concentrations are considered, as may be the case in many instances. Hence, the goal of this study was to compare cell-level response of single and eleven cell biphasic finite element models, where the latter provided an anatomically based cellular distribution representative of the actual number of cells for a commonly used [Formula: see text] edge cubic representative volume in the middle zone of cartilage. Single cell representations incorporated a centered single cell model and eleven location-corrected single cell models, the latter to delineate the role of cell placement in the representative volume element. A stress relaxation test at 10% compressive strain was adopted for all simulations. During transient response, volume- averaged chondrocyte mechanics demonstrated marked differences (up to 60% and typically greater than 10%) for the centered single versus the eleven cell models, yet steady-state loading was similar. Cell location played a marked role, due to inhomogeneity of the displacement and fluid pressure fields at the macroscopic scale. When the single cell representation was corrected for cell location, the transient response was consistent, while steady-state differences on the order of 1-4% were realized, which may be attributed to intercellular mechanical interactions. Anatomical representations of the superficial and deep zones, where cells reside in close proximity, may exhibit greater intercellular interactions, but these have yet to be explored.

Stiffness characterization of anisotropic trabecular meshwork

Elevation of intraocular pressure has been correlated to changes in stiffness of trabecular meshwork (TM) in glaucomatous eyes although mechanical properties of the TM remain to be quantitatively determined. Data in the literature suggest that the TM cannot be considered mechanically as a uniform layer of isotropic elastic material, because the value of its Young's modulus depends on the methods of measurements and can vary up to five orders of magnitude. To this end, we proposed a new theoretical framework for mechanical analysis of the TM, in which the inner wall of Schlemm's canal and the juxtacanalicular tissue in the TM were treated as a uniform layer of isotropic elastic material, and the rest of the TM, i.e., the uveal and corneoscleral meshworks, were modeled as a uniform layer of transversely isotropic material. Using the model, we demonstrated that the large discrepancy in the apparent Young's modulus reported in the literature could be caused by the anisotropy of the meshwork that was significantly stiffer in the longitudinal direction than in the transverse direction. The theoretical framework could be used to integrate existing data of the stiffness, investigate anisotropic behaviors of the tissues, and develop new methods to measure mechanical properties of the TM.

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.

Articular Contact Mechanics from an Asymptotic Modeling Perspective: A Review

In the present paper, we review the current state-of-the-art in asymptotic modeling of articular contact. Particular attention has been given to the knee joint contact mechanics with a special emphasis on implications drawn from the asymptotic models, including average characteristics for articular cartilage layer. By listing a number of complicating effects such as transverse anisotropy, non-homogeneity, variable thickness, nonlinear deformations, shear loading, and bone deformation, which may be accounted for by asymptotic modeling, some unsolved problems and directions for future research are also discussed.

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.

An overview of multiphase cartilage mechanical modelling and its role in understanding function and pathology

There is a long history of mathematical and computational modelling with the objective of understanding the mechanisms governing cartilage׳s remarkable mechanical performance. Nonetheless, despite sophisticated modelling development, simulations of cartilage have consistently lagged behind structural knowledge and thus the relationship between structure and function in cartilage is not fully understood. However, in the most recent generation of studies, there is an emerging confluence between our structural knowledge and the structure represented in cartilage modelling. This raises the prospect of further refinement in our understanding of cartilage function and also the initiation of an engineering-level understanding for how structural degradation and ageing relates to cartilage dysfunction and pathology, as well as informing the potential design of prospective interventions. Aimed at researchers entering the field of cartilage modelling, we thus review the basic principles of cartilage models, discussing the underlying physics and assumptions in relatively simple settings, whilst presenting the derivation of relatively parsimonious multiphase cartilage models consistent with our discussions. We proceed to consider modern developments that start aligning the structure captured in the models with observed complexities. This emphasises the challenges associated with constitutive relations, boundary conditions, parameter estimation and validation in cartilage modelling programmes. Consequently, we further detail how both experimental interrogations and modelling developments can be utilised to investigate and reduce such difficulties before summarising how cartilage modelling initiatives may improve our understanding of cartilage ageing, pathology and intervention.

Finite element modeling of finite deformable, biphasic biological tissues with transversely isotropic statistically distributed fibers: toward a practical solution

The distribution of collagen fibers across articular cartilage layers is statistical in nature. Based on the concepts proposed in previous models, we developed a methodology to include the statistically distributed fibers across the cartilage thickness in the commercial FE software COMSOL which avoids extensive routine programming. The model includes many properties that are observed in real cartilage: finite hyperelastic deformation, depth-dependent collagen fiber concentration, depth- and deformation-dependent permeability, and statistically distributed collagen fiber orientation distribution across the cartilage thickness. Numerical tests were performed using confined and unconfined compressions. The model predictions on the depth-dependent strain distributions across the cartilage layer are consistent with the experimental data in the literature.

In Vivo Dynamic Deformation of Articular Cartilage in Intact Joints Loaded by Controlled Muscular Contractions

When synovial joints are loaded, the articular cartilage and the cells residing in it deform. Cartilage deformation has been related to structural tissue damage, and cell deformation has been associated with cell signalling and corresponding anabolic and catabolic responses. Despite the acknowledged importance of cartilage and cell deformation, there are no dynamic data on these measures from joints of live animals using muscular load application. Research in this area has typically been done using confined and unconfined loading configurations and indentation testing. These loading conditions can be well controlled and allow for accurate measurements of cartilage and cell deformations, but they have little to do with the contact mechanics occurring in a joint where non-congruent cartilage surfaces with different material and functional properties are pressed against each other by muscular forces. The aim of this study was to measure in vivo, real time articular cartilage deformations for precisely controlled static and dynamic muscular loading conditions in the knees of mice. Fifty and 80% of the maximal knee extensor muscular force (equivalent to approximately 0.4N and 0.6N) produced average peak articular cartilage strains of 10.5±1.0% and 18.3±1.3% (Mean ± SD), respectively, during 8s contractions. A sequence of 15 repeat, isometric muscular contractions (0.5s on, 3.5s off) of 50% and 80% of maximal muscular force produced cartilage strains of 3.0±1.1% and 9.6±1.5% (Mean ± SD) on the femoral condyles of the mouse knee. Cartilage thickness recovery following mechanical compression was highly viscoelastic and took almost 50s following force removal in the static tests.

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.

Measuring microscale strain fields in articular cartilage during rapid impact reveals thresholds for chondrocyte death and a protective role for the superficial layer

Articular cartilage is a heterogeneous soft tissue that dissipates and distributes loads in mammalian joints. Though robust, cartilage is susceptible to damage from loading at high rates or magnitudes. Such injurious loads have been implicated in degenerative changes, including chronic osteoarthritis (OA), which remains a leading cause of disability in developed nations. Despite decades of research, mechanisms of OA initiation after trauma remain poorly understood. Indeed, although bulk cartilage mechanics are measurable during impact, current techniques cannot access microscale mechanics at those rapid time scales. We aimed to address this knowledge gap by imaging the microscale mechanics and corresponding acute biological changes of cartilage in response to rapid loading. In this study, we utilized fast-camera and confocal microscopy to achieve roughly 85 µm spatial resolution of both the cartilage deformation during a rapid (~3 ms), localized impact and the chondrocyte death following impact. Our results showed that, at these high rates, strain and chondrocyte death were highly correlated (p<0.001) with a threshold of 8% microscale strain norm before any cell death occurred. Additionally, chondrocyte death had developed by two hours after impact, suggesting a time frame for clinical therapeutics. Moreover, when the superficial layer was removed, strain - and subsequently chondrocyte death - penetrated deeper into the samples (p<0.001), suggesting a protective role for the superficial layer of articular cartilage. Combined, these results provide insight regarding the detailed biomechanics that drive early chondrocyte damage after trauma and emphasize the importance of understanding cartilage and its mechanics on the microscale.

The forward problem of electroarthrography: modeling load-induced electrical potentials at the surface of the knee

Electroarthrography (EAG) is a novel technology recently proposed to detect cartilage degradation. EAG consists of recording electrical potentials on the knee surface while the joint is undergoing compressive loading. Previous results show that these signals originating from streaming potentials in the cartilage reflect joint cartilage health. The aim of this study is to contribute to the understanding of the generation of the EAG signals and to the development of interpretation criteria using computer models of the human knee. The knee is modeled as a volume conductor composed of different regions characterized by specific electrical conductivities. The source of the EAG signal is the load-induced interstitial fluid flow that transports ions within the compressed cartilage. It is modeled as an impressed current density in different sections of the articular cartilage. The finite-element method is used to compute the potential distribution in two knee models with a realistic geometry. The simulated potential distributions correlate very well with previously measured potential values, which further supports the hypothesis that the EAG signals originate from compressed cartilage. Also, different localized cartilage defects simulated as a reduced impressed current density produce specific potential distributions that may be used to detect and localize cartilage degradation. In conclusion, given the structural and electrophysiological complexity of the knee, computer modeling constitutes an important tool to improve our understanding of the generation of EAG signals and of the various factors that affect the EAG signals so as to help develop the EAG technology as a useful clinical tool.

Age-related changes in aortic valve hemostatic protein regulation

OBJECTIVE

Although valvular endothelial cells have unique responses compared with vascular endothelial cells, valvular regulation of hemostasis is not well-understood. Heart valves remodel throughout a person's lifetime, resulting in changes in extracellular matrix composition and tissue mechanical properties that may affect valvular endothelial cell hemostatic function. This work assessed valvular endothelial cell regulation of hemostasis in situ and in vitro as a function of specimen age.

APPROACH AND RESULTS

Porcine aortic valves were assigned to 1 of 3 age groups: Young (YNG) (6 weeks); Adult (ADT) (6 months); or Elderly (OLD) (2 years). Histological examination of valves showed that secreted thrombotic/antithrombotic proteins localize at the valve endothelium and tissue interior. Gene expression and immunostains for von Willebrand factor (VWF), tissue factor pathway inhibitor, and tissue plasminogen activator in YNG porcine aortic valve endothelial cells were higher than they were for OLD, whereas plasminogen activator inhibitor 1 levels in OLD were higher than those for YNG and ADT. Histamine-stimulated YNG porcine aortic valve endothelial cells released higher concentrations of VWF proteins than OLD, and the fractions of VWF-140 fragments was not different between age groups. A calcific aortic valve disease in vitro model using valvular interstitial cells confirmed that VWF in culture significantly increased valvular interstitial cell nodule formation and calcification.

CONCLUSIONS

Hemostatic protein regulation in aortic valve tissues and in valvular endothelial cells changes with age. The presence of VWF and other potential hemostatic proteins increase valvular interstitial cell calcification in vitro. Therefore, the increased capacity of elderly valves to sequester the hemostatic proteins, together with age-associated loss of extracellular matrix organization, warrants investigation into potential role of these proteins in the formation of calcific nodules.

Comparison between FEBio and Abaqus for biphasic contact problems

Articular cartilage plays an important role in the function of diarthrodial joints. Computational methods have been used to study the biphasic mechanics of cartilage, and Abaqus has been one of the most widely used commercial software packages for this purpose. A newly developed open-source finite element solver, FEBio, has been developed specifically for biomechanical applications. The aim of this study was to undertake a direct comparison between FEBio and Abaqus for some practical contact problems involving cartilage. Three model types, representing a porous flat-ended indentation test, a spherical-ended indentation test, and a conceptual natural joint contact model, were compared. In addition, a parameter sensitivity study was also performed for the spherical-ended indentation test to investigate the effects of changes in the input material properties on the model outputs, using both FEBio and Abaqus. Excellent agreement was found between FEBio and Abaqus for all of the model types and across the range of material properties that were investigated.

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.

Computational aspects in mechanical modeling of the articular cartilage tissue

This review focuses on the modeling of articular cartilage (at the tissue level), chondrocyte mechanobiology (at the cell level) and a combination of both in a multiscale computation scheme. The primary objective is to evaluate the advantages and disadvantages of conventional models implemented to study the mechanics of the articular cartilage tissue and chondrocytes. From monophasic material models as the simplest form to more complicated multiscale theories, these approaches have been frequently used to model articular cartilage and have contributed significantly to modeling joint mechanics, addressing and resolving numerous issues regarding cartilage mechanics and function. It should be noted that attentiveness is important when using different modeling approaches, as the choice of the model limits the applications available. In this review, we discuss the conventional models applicable to some of the mechanical aspects of articular cartilage such as lubrication, swelling pressure and chondrocyte mechanics and address some of the issues associated with the current modeling approaches. We then suggest future pathways for a more realistic modeling strategy as applied for the simulation of the mechanics of the cartilage tissue using multiscale and parallelized finite element method.

Multiscale mechanics of articular cartilage: potentials and challenges of coupling musculoskeletal, joint, and microscale computational models

Articular cartilage experiences significant mechanical loads during daily activities. Healthy cartilage provides the capacity for load bearing and regulates the mechanobiological processes for tissue development, maintenance, and repair. Experimental studies at multiple scales have provided a fundamental understanding of macroscopic mechanical function, evaluation of the micromechanical environment of chondrocytes, and the foundations for mechanobiological response. In addition, computational models of cartilage have offered a concise description of experimental data at many spatial levels under healthy and diseased conditions, and have served to generate hypotheses for the mechanical and biological function. Further, modeling and simulation provides a platform for predictive risk assessment, management of dysfunction, as well as a means to relate multiple spatial scales. Simulation-based investigation of cartilage comes with many challenges including both the computational burden and often insufficient availability of data for model development and validation. This review outlines recent modeling and simulation approaches to understand cartilage function from a mechanical systems perspective, and illustrates pathways to associate mechanics with biological function. Computational representations at single scales are provided from the body down to the microstructure, along with attempts to explore multiscale mechanisms of load sharing that dictate the mechanical environment of the cartilage and chondrocytes.

A depth-dependent model of the pericellular microenvironment of chondrocytes in articular cartilage

Experimental studies suggest that the magnitude of chondrocyte deformation is much smaller than expected based on the material properties of extracellular matrix (ECM) and cells, and that this result could be explained by a structural unit, the chondron, that is thought to protect chondrocytes from large deformations in situ. We extended an existing numerical model of chondrocyte, ECM and pericellular matrix (PCM) to include depth-dependent structural information. Our results suggest that superficial zone chondrocytes, which lack a pericellular capsule (PC), are relatively stiff, and therefore are protected from excessive deformations, whereas middle and deep zone chondrocytes are softer but are protected by the PC that limits cell deformations in these regions. We conclude that cell deformations sensitively depend on the immediate structural environment of the PCM in a depth-dependent manner, and that the functional stiffness of chondrocytes in situ is much larger than experiments on isolated cells would suggest.

Temperature effects in articular cartilage biomechanics

Articular cartilage is the soft tissue that covers contacting surfaces of bones in synovial joints. Cartilage is composed of chondrocytes and an extracellular matrix containing numerous biopolymers, cations and water. Healthy cartilage functions biomechanically to provide smooth and stable joint movement. Degenerative joint diseases such as osteoarthritis involve cartilage deterioration, resulting in painful and cumbersome joint motion. Temperature is a fundamental quantity in mechanics, yet the effects of temperature on cartilage mechanical behavior are unknown. This study addressed the questions of whether cartilage stiffness and stress relaxation change with temperature. Samples of middle-zone bovine calf patellofemoral cartilage were tested in unconfined compression first at 24°C and then again after heating to 60°C. The data reveal that when temperature increases: (1) both peak and equilibrium stiffness increase by 150 and 8%, respectively, and (2) stress relaxation is faster at higher temperature, as shown by a 60% decrease in the time constant. The increases in temperature-dependent stiffness are consistent with polymeric mechanisms of matrix viscoelasticity but not with interstitial fluid flow. The changes in the time constant are consistent with a combination of both fluid flow and matrix viscoelasticity. Furthermore, we discovered a novel phenomenon: at stress-relaxation equilibrium, compressive stress increased with temperature. These data demonstrate a rich area of cartilage mechanics that has previously been unexplored and emphasize the role of polymer dynamics in cartilage viscoelasticity. Further studies of cartilage polymer dynamics may yield additional insight into mechanisms of cartilage material behavior that could improve treatments for cartilage degeneration.

Anisotropic dynamic changes in the pore network structure, fluid diffusion and fluid flow in articular cartilage under compression

A compression cell designed to fit inside an NMR spectrometer was used to investigate the in situ mechanical strain response, structural changes to the internal pore structure, and the diffusion and flow of interstitial water in full-thickness cartilage samples as it was deforming dynamically under a constant compressive load (pressure). We distinguish between the hydrostatic pressure acting on the interstitial fluid and the pore pressure acting on the cartilage fibril network. Our results show that properties related to the pore matrix microstructure such as diffusion and hydraulic conductivity are strongly influenced by the hydrostatic pressure in the interstitial fluid of the dynamically deforming cartilage which differ significantly from the properties measured under static i.e. equilibrium loading conditions (when the hydrostatic pressure has relaxed back to zero). The magnitude of the hydrostatic fluid pressure also appears to affect the way cartilage's pore matrix changes during deformation with implications for the diffusion and flow-driven fluid transport through the deforming pore matrix. We also show strong evidence for a highly anisotropic pore structure and deformational dynamics that allows cartilage to deform without significantly altering the axial porosity of the matrix even at very large strains. The insensitivity of the axial porosity to compressive strain may be playing a critical function in directing the flow of pressurized interstitial fluid in the compressed cartilage to the surface, to support the load, and provide a protective interfacial fluid film that 'weeps' out from the deforming tissue and thereby enhances the (elasto)hydrodynamic efficacy of sliding joints. Our results appear to show a close synergy between the structure of cartilage and both the hydrodynamic and boundary lubrication mechanisms.

Liposomes act as effective biolubricants for friction reduction in human synovial joints

Phospholipids (PL) form the matrix of biological membranes and of the lipoprotein envelope monolayer, and are responsible for many of the unique physicochemical, biochemical, and biological properties of these supermolecular bioassemblies. It was suggested that phospholipids present in the synovial fluid (SF) and on the surface of articular cartilage have major involvement in the low friction of cartilage, which is essential for proper mobility of synovial joints. In pathologies, such as impaired biolubrication (leading to common joint disorders such as osteoarthritis), the level of phospholipids in the SF is reduced. Using a human-sourced cartilage-on-cartilage setup, we studied to what extent and how phospholipids act as highly effective cartilage biolubricants. We found that large multilamellar vesicles (MLV), >800 nm in diameter, composed of 1,2-dimyristoyl-sn-glycero-3-phosphocholine (DMPC) or of a mixture of DMPC and 1,2-dipalmitoyl-sn-glycero-3-phosphocholine (DPPC) are superior lubricants in comparison to MLV composed of other phosphatidylcholines. Introducing cholesterol into liposomes resulted in less effective lubricants. DMPC-MLV was also superior to small unilamellar vesicles (SUV), <100 nm in diameter, composed of DMPC. MLV are superior to SUV due to MLV retention at and near (<200 microm below) the cartilage surface, while SUV penetrate deeper into the cartilage (450-730 microm). Superiority of specific PL compositions is explained by the thermotropic behavior (including compressibility) of the lipid bilayer. Correlating physicochemical properties of the MLV with the friction results suggests that MLV having lipid bilayers in the liquid-disordered phase and having a solid-ordered to liquid-disordered phase transition temperature slightly below physiological temperature are optimal for lubrication. High phospholipid headgroup hydration, high compressibility, and softness are the common denominators of all efficient PL compositions. The high efficiency of DMPC-MLV and DMPC/DPPC-MLV as cartilage lubricants combined with their resistance to degradation at 37 degrees C supports further evaluation of these MLV for treatment of joint impairments related to poor lubrication. This work also demonstrates the relevance of basic physicochemical properties of phospholipids to their activities in biological systems.

Changes in pore morphology and fluid transport in compressed articular cartilage and the implications for joint lubrication

Cartilage sections were cut from the middle zone of pig knee articular cartilage and attached to substrates in two different kinds of newly designed 'pressure cells', one for fluorescence the other for NMR measurements. The fluorescence cell was filled with buffer solution containing fluorescently marked 70 kDa dextran which was allowed to diffuse into the cartilage pores. A second glass surface was then pressed down onto the thin cartilage sample under different loads (pressures), and the resulting compression (strain) and change in pore volume were measured as a function of time, simultaneously with measurements of the lateral diffusion and flow pattern of the dextran molecules using Fluorescence Recovery After Photobleaching (FRAP). Complementary experiments were made on the normal diffusion coefficients of pure electrolyte solutions (no dextran) in thicker cartilage sections with pulse-gradient NMR using a new pressure cell suitable for such measurements. Taken together our results show that the highly anisotropic structure of cartilage has a strong effect on the way fluid diffuses laterally and normally at different stages of compression. Our results also show how geometric constraints on a cartilage network and trapped high MW polymer such as HA during normal compressions are likely to affect both the normal and the lateral mobilities of polyelectrolytes and water.

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.

On the anisotropy and inhomogeneity of permeability in articular cartilage

Articular cartilage is known to be anisotropic and inhomogeneous because of its microstructure. In particular, its elastic properties are influenced by the arrangement of the collagen fibres, which are orthogonal to the bone-cartilage interface in the deep zone, randomly oriented in the middle zone, and parallel to the surface in the superficial zone. In past studies, cartilage permeability has been related directly to the orientation of the glycosaminoglycan chains attached to the proteoglycans which constitute the tissue matrix. These studies predicted permeability to be isotropic in the undeformed configuration, and anisotropic under compression. They neglected tissue anisotropy caused by the collagen network. However, magnetic resonance studies suggest that fluid flow is "directed" by collagen fibres in biological tissues. Therefore, the aim of this study was to express the permeability of cartilage accounting for the microstructural anisotropy and inhomogeneity caused by the collagen fibres. Permeability is predicted to be anisotropic and inhomogeneous, independent of the state of strain, which is consistent with the morphology of the tissue. Looking at the local anisotropy of permeability, we may infer that the arrangement of the collagen fibre network plays an important role in directing fluid flow to optimise tissue functioning.

A finite element exploration of cartilage stress near an articular incongruity during unstable motion

Both instability and residual articular incongruity are implicated in the development of post-traumatic osteoarthritis (OA) following intra-articular fracture, but currently no information exists regarding cartilage stresses for unstable residual incongruities. In this study, a transversely isotropic poroelastic cartilage finite element model was implemented and validated within physiologically relevant loading ranges. This material model was then used to simulate the loading of cartilage during stable and unstable motion accompanying a step-off incongruity residual from intra-articular fracture, using load data from previous cadaver tests of ankle instability. Peak solid-phase stresses and fluid pressure were found to increase markedly in the presence of instability. Solid-phase transients of normal stress increased from 2.00 to 13.8 MPa/s for stable compared to unstable motion, and tangential stress transients increased from 17.1 to 118.1 MPa/s. Corresponding fluid pressure transients increased from 15.1 to 117.9 MPa/s for unstable motion. In the most rapidly loaded sections of cartilage, the fluid was found to carry nearly all of the normal load, with the pressurization of the fluid resulting in high solid matrix tangential stresses.

Noninvasive determination of ligament strain with deformable image registration

Ligament function and propensity for injury are directly related to regional stresses and strains. However, noninvasive techniques for measurement of strain are currently limited. This study validated the use of Hyperelastic Warping, a deformable image registration technique, for noninvasive strain measurement in the human medial collateral ligament using direct comparisons with optical measurements. Hyperelastic Warping determines the deformation map that aligns consecutive images of a deforming material, allowing calculation of strain. Diffeomorphic deformations are ensured by representing the deformable image as a hyperelastic material. Ten cadaveric knees were subjected to six loading scenarios each. Tissue deformation was documented with magnetic resonance imaging (MRI) and video-based experimental measurements. MRI datasets were analyzed using Hyperelastic Warping, representing the medial collateral ligament (MCL) with a hexahedral finite element (FE) model projected to a manually segmented ligament surface. The material behavior was transversely isotropic hyperelastic. Warping predictions of fiber stretch were strongly correlated with experimentally measured strains (R (2) = 0.81). Both sets of measurements were in agreement with previous ex vivo studies. Warping predictions of fiber stretch were insensitive to bulk:shear modulus ratio, fiber stiffness, and shear modulus in the range of +2.5SD to -1.0SD. Correlations degraded when the shear modulus was decreased to 2.5SD below the mean (R (2) = 0.56), and when an isotropic constitutive model was substituted for the transversely isotropic model (R (2) = 0.65). MCL strains in the transitional region near the joint line, where the material behavior and material symmetry are more complex, showed the most sensitivity to changes in shear modulus. These results demonstrate that Hyperelastic Warping requires the use of a constitutive model that reflects the material symmetry, but not subject-specific material properties for accurate strain predictions for this application. Hyperelastic Warping represents a powerful technique for noninvasive strain measurement of musculoskeletal tissues and has many advantages over other image-based strain measurement techniques.

Constitutive Modeling of Cartilaginous Tissues: A Review

An important and longstanding field of research in orthopedic biomechanics is the elucidation and mathematical modeling of the mechanical response of cartilaginous tissues. Traditional approaches have treated such tissues as continua and have described their mechanical response in terms of macroscopic models borrowed from solid mechanics. The most important of such models are the biphasic and single-phase viscoelastic models, and the many variations thereof. These models have reached a high level of maturity and have been successful in describing a wide range of phenomena. An alternative approach that has received considerable recent interest, both in orthopedic biomechanics and in other fields, is the description of mechanical response based on consideration of a tissue's structure—so-called microstructural modeling. Examples of microstructurally based approaches include fibril-reinforced biphasic models and homogenization approaches. A review of both macroscopic and microstructural constitutive models is given in the present work.

Considerations on Joint and Articular Cartilage Mechanics

When studying joint degeneration leading to osteoarthritis (OA), it seems imperative that local joint tissue loading is known during normal everyday movement and that the adaptive/degenerative effects of this loading are quantified systematically. Philosophically, we believe the best way to approach this problem is by studying joint degeneration and osteoarthritis in long-term experimental models and by representing diarthrodial joints and the associated tissues with accurate, geometric and structural, theoretical models. Here, we present selected examples of our work representing this approach. Experimentally, we demonstrate that the local loading of joints changes continuously in experimental models of OA, not only because of the changing external and internal loading, but also because of the continuous alterations in joint contact geometry and tissue mechanical properties. Furthermore, we show that single bouts of joint loading affect gene expression, and that gene expression, as well as subsequent joint degeneration is site-specific. In fact, opposing articular surfaces that are exposed to the same loading may degenerate at completely different rates. Finally, we propose a series of theoretical models of articular cartilage and contact mechanics, demonstrating that many of the anisotropic and inhomogeneous properties can be explained by structural elements and their orientation and volumetric concentration across the tissue.

The Mechanical Behaviour of Chondrocytes Predicted with a Micro-structural Model of Articular Cartilage

The integrity of articular cartilage depends on the proper functioning and mechanical stimulation of chondrocytes, the cells that synthesize extracellular matrix and maintain tissue health. The biosynthetic activity of chondrocytes is influenced by genetic factors, environmental influences, extracellular matrix composition, and mechanical factors. The mechanical environment of chondrocytes is believed to be an important determinant for joint health, and chondrocyte deformation in response to mechanical loading is speculated to be an important regulator of metabolic activity. In previous studies of chondrocyte deformation, articular cartilage was described as a biphasic material consisting of a homogeneous, isotropic, linearly elastic solid phase, and an inviscid fluid phase. However, articular cartilage is known to be anisotropic and inhomogeneous across its depth. Therefore, isotropic and homogeneous models cannot make appropriate predictions for tissue and cell stresses and strains. Here, we modelled articular cartilage as a transversely isotropic, inhomogeneous (TI) material in which the anisotropy and inhomogeneity arose naturally from the microstructure of the depth-dependent collagen fibril orientation and volumetric fraction, as well as the chondrocyte shape and volumetric fraction. The purpose of this study was to analyse the deformation behaviour of chondrocytes using the TI model of articular cartilage. In order to evaluate our model against experimental results, we simulated indentation and unconfined compression tests for nominal compressions of 15%. Chondrocyte deformations were analysed as a function of location within the tissue. The TI model predicted a non-uniform behaviour across tissue depth: in indentation testing, cell height decreased by 43% in the superficial zone and between 11 and 29% in the deep zone. In unconfined compression testing, cell height decreased by 32% in the superficial zone, 25% in the middle, and 18% in the deep zones. This predicted non-uniformity is in agreement with experimental studies. The novelty of this study is the use of a cartilage material model accounting for the intrinsic inhomogeneity and anisotropy of cartilage caused by its microstructure.