Modeling Inelastic Responses Using Constrained Reactive Mixtures.

This study reviews the progression of our research, from modeling growth theories for cartilage tissue engineering, to the formulation of constrained reactive mixture theories to model inelastic responses in any solid material, such as theories for damage mechanics, viscoelasticity, plasticity, and elasto-plastic damage. In this framework, multiple solid generations α can co-exist at any given time in the mixture. The oldest generation is denoted by α=s and is called the master generation, whose reference configuration Xs is observable. The solid generations α are all constrained to share the same velocity vs, but may have distinct reference configurations Xα. An important element of this formulation is that the time-invariant mapping Fαs=∂Xα/∂Xs between these reference configurations is a function of state, whose mathematical formulation is postulated by constitutive assumption. Thus, reference configurations Xα are not observable (α≠s). This formulation employs only observable state variables, such as the deformation gradient Fs of the master generation and the referential mass concentrations ρrα of each generation, in contrast to classical formulations of inelastic responses which rely on internal state variable theory, requiring evolution equations for those hidden variables. In constrained reactive mixtures, the evolution of the mass concentrations is governed by the axiom of mass balance, using constitutive models for the mass supply densities ρˆrα. Classical and constrained reactive mixture approaches share considerable mathematical analogies, as they both introduce a multiplicative decomposition of the deformation gradient, also requiring evolution equations to track some of the state variables. However, they also differ at a fundamental level, since one adopts only observable state variables while the other introduces hidden state variables. In summary, this review presents an alternative foundational approach to the modeling of inelastic responses in solids, grounded in the classical framework of mixture theory.

Damage Mechanics of Biological Tissues in Relation to Viscoelasticity.

This study examines the theoretical foundations for the damage mechanics of biological tissues in relation to viscoelasticity. Its primary goal is to provide a mechanistic understanding of well-known experimental observations in biomechanics, which show that the ultimate tensile strength of viscoelastic biological tissues typically increases with increasing strain rate. The basic premise of this framework is that tissue damage occurs when strong bonds, such as covalent bonds in the solid matrix of a biological tissue, break in response to loading. This type of failure is described as elastic damage, under the idealizing assumption that strong bonds behave elastically. Viscoelasticity arises from three types of dissipative mechanisms: (1) Friction between molecules of the same species, which is represented by the tissue viscosity. (2) Friction between fluid and solid constituents of a porous medium, which is represented by the tissue hydraulic permeability. (3) Dissipative reactions arising from weak bonds breaking in response to loading, and reforming in a stress-free state, such as hydrogen bonds and other weak electrostatic bonds. When a viscoelastic tissue is subjected to loading, some of that load may be temporarily supported by those frictional and weak bond forces, reducing the amount of load supported by elastic strong bonds and thus, the extent of elastic damage sustained by those bonds. This protective effect depends on the characteristic time response of viscoelastic mechanisms in relation to the loading history. This study formalizes these concepts by presenting general equations that can model the damage mechanics of viscoelastic tissues.

The Strain Response to Intraocular Pressure Decrease in the Lamina Cribrosa of Patients with Glaucoma.

OBJECTIVE: To measure biomechanical strains in the lamina cribrosa (LC) of living human eyes with intraocular pressure (IOP) lowering. DESIGN: Cohort study. PARTICIPANTS: Patients with glaucoma underwent imaging before and after laser suturelysis after trabeculectomy surgery (29 image pairs; 26 persons). INTERVENTION: Noninvasive imaging of the eye. MAIN OUTCOME MEASURES: Strains in optic nerve head tissue and changes in depths of the anterior border of the LC. RESULTS: Intraocular pressure decreases caused the LC to expand in thickness in the anterior-posterior strain (Ezz = 0.94 ± 1.2%; P = 0.00020) and contract in radius in the radial strain (Err = - 0.19 ± 0.33%; P = 0.0043). The mean LC depth did not significantly change with IOP lowering (1.33 ± 6.26 µm; P = 0.26). A larger IOP decrease produced a larger, more tensile Ezz (P < 0.0001), greater maximum principal strain (Emax; P < 0.0001), and greater maximum shear strain (Γmax; P < 0.0001). The average LC depth change was associated with the Γmax and radial-circumferential shear strain (Erθ; P < 0.02) but was not significantly related to tensile or compressive strains. An analysis by clock hour showed that in temporal clock hours 3 to 6, a more anterior LC movement was associated with a more positive Emax, and in clock hours 3, 5, and 6, it was associated with a more positive Γmax. At 10 o'clock, a more posterior LC movement was related to a more positive Emax (P < 0.004). Greater compliance (strain/ΔIOP) of Emax (P = 0.044), Γmax (P = 0.052), and Erθ (P = 0.018) was associated with a thinner retinal nerve fiber layer. Greater compliance of Emax (P = 0.041), Γmax (P = 0.021), Erθ (P = 0.024), and in-plane shear strain (Erz; P = 0.0069) was associated with more negative mean deviations. Greater compliance of Γmax (P = 0.055), Erθ (P = 0.040), and Erz (P = 0.015) was associated with lower visual field indices. CONCLUSIONS: With IOP lowering, the LC moves either into or out of the eye but, on average, expands in thickness and contracts in radius. Shear strains are nearly as substantial as in-plane strains. Biomechanical strains are more compliant in eyes with greater glaucoma damage. This work was registered at ClinicalTrials.gov as NCT03267849.

Superficial zone chondrocytes can get compacted under physiological loading: A multiscale finite element analysis.

Recent in vivo and in vitro studies have demonstrated that superficial zone (SZ) chondrocytes within articular layers of diarthrodial joints die under normal physiologic loading conditions. In order to further explore the implications of this observation in future investigations, we first needed to understand the mechanical environment of SZ chondrocytes that might cause them to die under physiological sliding contact conditions. In this study we performed a multiscale finite element analysis of articular contact to track the temporal evolution of a SZ chondrocyte's interstitial fluid pressure, hydraulic permeability, and volume under physiologic loading conditions. The effect of the pericellular matrix modulus and permeability was parametrically investigated. Results showed that SZ chondrocytes can lose ninety percent of their intracellular fluid after several hours of intermittent or continuous contact loading, resulting in a reduction of intracellular hydraulic permeability by more than three orders of magnitude. These findings are consistent with loss of cell viability due to the impediment of cellular metabolic pathways induced by the loss of fluid. They suggest that there is a simple mechanical explanation for the vulnerability of SZ chondrocytes to sustained physiological loading conditions. Future studies will focus on validating these specific findings experimentally. STATEMENT OF SIGNIFICANCE: As with any mechanical system, normal 'wear and tear' of cartilage tissue lining joints is expected. Yet incidences of osteoarthritis are uncommon in individuals younger than 45. This counter-intuitive observation suggests there must be an intrinsic repair mechanism compensating for this wear and tear over many decades of life. Recent experimental studies have shown superficial zone chondrocytes die under physiologic loading conditions, suggesting that this repair mechanism may involve cell replenishment. To better understand the mechanical environment of these cells, we performed a multiscale computational analysis of articular contact under loading. Results indicated that normal activities like walking or standing can induce significant loss of intracellular fluid volume, potentially hindering metabolic activity and fluid transport properties, and causing cell death.

A Finite Element Algorithm for Large Deformation Biphasic Frictional Contact Between Porous-Permeable Hydrated Soft Tissues.

The frictional response of porous and permeable hydrated biological tissues such as articular cartilage is significantly dependent on interstitial fluid pressurization. To model this response, it is common to represent such tissues as biphasic materials, consisting of a binary mixture of a porous solid matrix and an interstitial fluid. However, no computational algorithms currently exist in either commercial or open-source software that can model frictional contact between such materials. Therefore, this study formulates and implements a finite element algorithm for large deformation biphasic frictional contact in the open-source finite element software FEBio. This algorithm relies on a local form of a biphasic friction model that has been previously validated against experiments, and implements the model into our recently-developed surface-to-surface (STS) contact algorithm. Contact constraints, including those specific to pressurized porous media, are enforced with the penalty method regularized with an active-passive augmented Lagrangian scheme. Numerical difficulties specific to challenging finite deformation biphasic contact problems are overcome with novel smoothing schemes for fluid pressures and Lagrange multipliers. Implementation accuracy is verified against semi-analytical solutions for biphasic frictional contact, with extensive validation performed using canonical cartilage friction experiments from prior literature. Essential details of the formulation are provided in this paper, and the source code of this biphasic frictional contact algorithm is made available to the general public.

Continuum Thermodynamics of Constrained Reactive Mixtures.

Mixture theory models continua consisting of multiple constituents with independent motions. In constrained mixtures, all constituents share the same velocity but they may have different reference configurations. The theory of constrained reactive mixtures was formulated to analyze growth and remodeling in living biological tissues. It can also reproduce and extend classical frameworks of damage mechanics and viscoelasticity under isothermal conditions, when modeling bonds that can break and reform. This study focuses on establishing the thermodynamic foundations of constrained reactive mixtures under more general conditions, for arbitrary reactive processes where temperature varies in time and space. By incorporating general expressions for reaction kinetics, it is shown that the residual dissipation statement of the Clausius-Duhem inequality must include a reactive power density, while the axiom of energy balance must include a reactive heat supply density. Both of these functions are proportional to the molar production rate of a reaction, and they depend on the chemical potentials of the mixture constituents. We present novel formulas for the classical thermodynamic concepts of energy of formation and heat of reaction, making it possible to evaluate the heat supply generated by reactive processes from the knowledge of the specific free energy of mixture constituents as well as the reaction rate. We illustrate these novel concepts with mixtures of ideal gases, and isothermal reactive damage mechanics and viscoelasticity, as well as reactive thermoelasticity. This framework facilitates the analysis of reactive tissue biomechanics and physiological and biomedical engineering processes where temperature variations cannot be neglected.

On the use of constrained reactive mixtures of solids to model finite deformation isothermal elastoplasticity and elastoplastic damage mechanics.

This study presents a framework for plasticity and elastoplastic damage mechanics by treating materials as reactive solids whose internal composition evolves in response to applied loading. Using the framework of constrained reactive mixtures, plastic deformation is accounted for by allowing loaded bonds within the material to break and reform in a stressed state. Bonds which break and reform represent a new generation with a new reference configuration, which is time-invariant and provided by constitutive assumption. The constitutive relation for the reference configuration of each generation may depend on the selection of a suitable yield measure. The choice of this measure and the resulting plastic flow conditions are constrained by the Clausius-Duhem inequality. We show that this framework remains consistent with classical plasticity approaches and principles. Verification of this reactive plasticity framework, which is implemented in the open source FEBio finite element software (febio.org), is performed against standard 2D and 3D benchmark problems. Damage is incorporated into this reactive framework by allowing loaded bonds to break permanently according to a suitable damage measure, where broken bonds can no longer store free energy. Validation is also demonstrated against experimental data for problems involving plasticity and plastic damage. This study demonstrates that it is possible to formulate simple elastoplasticity and elastoplastic damage models within a consistent framework which uses measures of material mass composition as theoretically observable state variables. This theoretical frame can be expanded in scope to account for more complex behaviors.

Direct Osmotic Pressure Measurements in Articular Cartilage Demonstrate Nonideal and Concentration-Dependent Phenomena.

The osmotic pressure in articular cartilage serves an important mechanical function in healthy tissue. Its magnitude is thought to play a role in advancing osteoarthritis. The aims of this study were to: (1) isolate and quantify the magnitude of cartilage swelling pressure in situ; and (2) identify the effect of salt concentration on material parameters. Confined compression stress-relaxation testing was performed on 18 immature bovine and six mature human cartilage samples in solutions of varying osmolarities. Direct measurements of osmotic pressure revealed nonideal and concentration-dependent osmotic behavior, with magnitudes approximately 1/3 those predicted by ideal Donnan law. A modified Donnan constitutive behavior was able to capture the aggregate behavior of all samples with a single adjustable parameter. Results of curve-fitting transient stress-relaxation data with triphasic theory in febio demonstrated concentration-dependent material properties. The aggregate modulus HA increased threefold as the external concentration decreased from hypertonic 2 M to hypotonic 0.001 M NaCl (bovine: HA=0.420±0.109 MPa to 1.266±0.438 MPa; human: HA=0.499±0.208 MPa to 1.597±0.455 MPa), within a triphasic theory inclusive of osmotic effects. This study provides a novel and simple analytical model for cartilage osmotic pressure which may be used in computational simulations, validated with direct in situ measurements. A key finding is the simultaneous existence of Donnan osmotic and Poisson-Boltzmann electrostatic interactions within cartilage.

Tissue engineered autologous cartilage-bone grafts for temporomandibular joint regeneration.

Joint disorders can be detrimental to quality of life. There is an unmet need for precise functional reconstruction of native-like cartilage and bone tissues in the craniofacial space and particularly for the temporomandibular joint (TMJ). Current surgical methods suffer from lack of precision and comorbidities and frequently involve multiple operations. Studies have sought to improve craniofacial bone grafts without addressing the cartilage, which is essential to TMJ function. For the human-sized TMJ in the Yucatan minipig model, we engineered autologous, biologically, and anatomically matched cartilage-bone grafts for repairing the ramus-condyle unit (RCU), a geometrically intricate structure subjected to complex loading forces. Using image-guided micromilling, anatomically precise scaffolds were created from decellularized bone matrix and infused with autologous adipose-derived chondrogenic and osteogenic progenitor cells. The resulting constructs were cultured in a dual perfusion bioreactor for 5 weeks before implantation. Six months after implantation, the bioengineered RCUs maintained their predefined anatomical structure and regenerated full-thickness, stratified, and mechanically robust cartilage over the underlying bone, to a greater extent than either autologous bone-only engineered grafts or acellular scaffolds. Tracking of implanted cells and parallel bioreactor studies enabled additional insights into the progression of cartilage and bone regeneration. This study demonstrates the feasibility of TMJ regeneration using anatomically precise, autologous, living cartilage-bone grafts for functional, personalized total joint replacement. Inclusion of the adjacent tissues such as soft connective tissues and the TMJ disc could further extend the functional integration of engineered RCUs with the host.

Immature bovine cartilage wear by fatigue failure and delamination.

This study investigated wear damage of immature bovine articular cartilage using reciprocal sliding of tibial cartilage strips against glass or cartilage. Experiments were conducted in physiological buffered saline (PBS) or mature bovine synovial fluid (SF). A total of 63 samples were tested, of which 47 exhibited wear damage due to delamination of the cartilage surface initiated in the middle zone, with no evidence of abrasive wear. There was no difference between the friction coefficient of damaged and undamaged samples, showing that delamination wear occurs even when friction remains low under a migrating contact area configuration. No difference was observed in the onset of damage or in the friction coefficient between samples tested in PBS or SF. The onset of damage occurred earlier when testing cartilage against glass versus cartilage against cartilage, supporting the hypothesis that delamination occurs due to fatigue failure of the collagen in the middle zone, since stiffer glass produces higher strains and tensile stresses under comparable loads. The findings of this study are novel because they establish that delamination of the articular surface, starting in the middle zone, may represent a primary mechanism of failure. Based on preliminary data, it is reasonable to hypothesize that delamination wear via subsurface fatigue failure is similarly the primary mechanism of human cartilage wear under normal loading conditions, albeit requiring far more cycles of loading than in immature bovine cartilage.

A Surface-to-Surface Finite Element Algorithm for Large Deformation Frictional Contact in febio.

This study formulates a finite element algorithm for frictional contact of solid materials, accommodating finite deformation and sliding. The algorithm uses a penalty method regularized with an augmented Lagrangian scheme to enforce contact constraints in a nonmortar surface-to-surface approach. Use of a novel kinematical approach to contact detection and enforcement of frictional constraints allows solution of complex problems previously requiring mortar methods or contact smoothing algorithms. Patch tests are satisfied to a high degree of accuracy with a single-pass penalty method, ensuring formulation errors do not affect the solution. The accuracy of the implementation is verified with Hertzian contact, and illustrations demonstrating the ability to handle large deformations and sliding are presented and validated against prior literature. A biomechanically relevant example addressing finger friction during grasping demonstrates the utility of the proposed algorithm. The algorithm is implemented in the open source software febio, and the source code is made available to the general public.

Roles of the Fibrous Superficial Zone in the Mechanical Behavior of TMJ Condylar Cartilage.

In temporomandibular joints (TMJs), the cartilage on the condylar head displays a unique ultrastructure with a dense layer of type I collagen in the superficial zone, different from hyaline cartilage in other joints. This study aims to elucidate the roles of this fibrous zone in the mechanical behaviors, particularly lubrication, of TMJ under physiological loading regimes. Mechanical tests on porcine condylar cartilage demonstrated that the superficial and middle-deep zones exhibit tension-compression nonlinearity. The tensile and compressive moduli of the superficial zone are 30.73 ± 12.97 and 0.028 ± 0.016 MPa, respectively, while those for the middle-deep zone are 2.43 ± 1.75 and 0.14 ± 0.09 MPa. A nonlinear finite element model of condylar cartilage was built to simulate sliding of a spherical probe over the articular surface. The presence of the superficial zone significantly promoted interstitial fluid pressurization (IFP) inside the loaded cartilage and reduced the friction force on the surface, compared to the case without the superficial zone. Finite element simulations showed that IFP depends on sliding speed but not normal load, which matches the experimental results. This study revealed the presence of the fibrous zone can significantly reduce the deformation of condylar cartilage under compression and the friction force on its surface during sliding.

Determine the equilibrium mechanical properties of articular cartilage from the short-term indentation response.

Indentation testing is widely used to evaluate the mechanical properties of articular cartilage. However, most curve-fitting solutions for indentation analysis require the deformation data of cartilage at the equilibrium state, which often takes the tissue hours to reach. The lengthy testing time reduces the efficiency of indentation, increases the chance of tissue deterioration, and prevents in vivo applications. To overcome these limitations, a novel technique based on principal component analysis (PCA) was developed in this study, which can predict the full indentation creep curve based on the first few minutes' deformation history and the principal components. The accuracy of this technique was confirmed using the indentation data from 40 temporomandibular joint condylar cartilage samples and 17 bovine knee joint samples. The mechanical properties determined by biphasic curve-fitting using predicted and experimental data are in good agreement, with the difference between the two less than 5%. For TMJ and knee cartilages, it is found that any number of full tests beyond eight will not lead to any increase larger than 1% in the accuracy, indicating a low sample number required for prediction. In addition, the principal components of indentation creep curves are consistent for the same type of cartilage tested with identical protocols, but significantly different between two distinct cartilages. Therefore PCA may also represent a new method to compare the mechanical behaviors of different cartilages, as it avoids the assumptions associated with mechanical constitutive models and relies purely on the experimental data.