Continuum theory of fibrous tissue damage mechanics using bond kinetics: application to cartilage tissue engineering

Investigation of the biodegradation kinetics and associated mechanical properties of 3D-printed polycaprolactone during long-term preclinical testing

Polycaprolactone (PCL) is a bioresorbable polymer increasingly utilized for customized tissue reconstruction as it is readily 3D printed. A critical design requirement for PCL devices is determining the in vivo bioresorption rate and the resulting change in device mechanics suited for target tissue repair applications. The primary challenge with meeting this requirement involves accurate prediction of degradation in the target tissues. PCL undergoes bulk hydrolytic degradation following first order kinetics until an 80-90 % drop in the starting number average molecular weight (Mn) after 2-3 years in vivo. However, initial polymer architecture, composite incorporation, manufacturing modality, device architecture, and target tissue can impact degradation. In vitro models do not fully capture device degradation, and the limited long-term (2-3 year) models primarily utilize subcutaneous implants. In this study, we investigate the degradation rate of 3D-printed airway support devices (ASDs) comprised of PCL and 4 % hydroxyapatite (HA) when implanted on Yucatan porcine tracheas for two years. After one year of degradation, we report a mass loss of less than 1 % and Mn reduction of 25 %. After two years, mass and Mn decreased by 10 % and 50 % respectively. These changes are accompanied by an increase in elastic modulus from 146.7 ± 5.2 MPa for freshly printed ASDs to 291.7 ± 16.0 MPa after one year and 362.5 ± 102.4 MPa after two years. Additionally, there was a decrease in yield strain, and increase in yield stress from implantation to 1-year (p < 0.001). Plastic strain completely diminished by two years, resulting in brittle failure at a yield stress of 12.5 MPa. The significantly lower rate of hydrolysis coupled with hydrolytic embrittlement indicates alternate degradation kinetics compared to subcutaneous models. Fitting a new model for degradation and predicting elastic and damage properties of this new degradation paradigm provide significant advancements for 3D-printed device design in clinical repair applications.

Modeling Fatigue Failure of Cartilage and Fibrous Biological Tissues Using Constrained Reactive Mixture Theory

Fatigue failure in biological soft tissues plays a critical role in the etiology of chronic soft tissue injuries and diseases such as osteoarthritis (OA). Understanding failure mechanisms is hindered by the decades-long timescales over which damage takes place. Analyzing the factors contributing to fatigue failure requires the help of validated computational models developed for soft tissues. This study presents a framework for fatigue failure of fibrous biological tissues based on reaction kinetics, where the composition of intact and fatigued material regions can evolve via degradation and breakage over time, in response to energy-based fatigue and damage criteria. Using reactive constrained mixture theory, material region mass fractions are governed by the axiom of mass balance. Progression of fatigue is controlled by an energy-based reaction rate, with user-selected probability functions defining the damage propensity of intact and fatigued material regions. Verification of this reactive theory, which is implemented in the open-source FEBio finite element software, is provided in this study. Validation is also demonstrated against experimental data, showing that predicted damage can be linked to results from biochemical assays. The framework is also applied to study fatigue failure during frictional contact of cartilage. Simulating previous experiments suggests that frictional effects slightly increase fatigue progression, but the main driver is cyclic compressive contact loading. This study demonstrated the ability of theoretical models to complement and extend experimental findings, advancing our understanding of the time progression of fatigue in biological tissues.

Continuum Growth Mechanics: Reconciling Two Common Frameworks

The objective of this study was to investigate whether the two most common growth mechanics modeling frameworks, the constrained-mixture growth model and the kinematic growth model, could be reconciled mathematically. The purpose of this effort was to provide practical guidelines for potential users of these modeling frameworks. Results showed that the kinematic growth model is mathematically consistent with a special form of the constrained-mixture growth model, where only one generation of a growing solid exists at any given time, overturning its entire solid mass at each instant of growth in order to adopt the reference configuration dictated by the growth deformation. The thermodynamics of the kinematic growth model, along with the specialized constrained-mixture growth model, requires a cellular supply of chemical energy to allow deposition of solid mass under a stressed state. A back-of-the-envelope calculation shows that the amount of chemical energy required to sustain biological growth under these models is negligibly small, when compared to the amount of energy normally consumed daily by the human body. In conclusion, this study successfully reconciled the two most popular growth theories for biological growth and explained the special circumstances under which the constrained-mixture growth model reduces to the kinematic growth model.

Biomechanics of annulus fibrosus: Elastic fiber simplification and degenerative impact on damage initiation and propagation

This study addresses three primary objectives related to lumbar intervertebral disc (IVD) biomechanics under ramping quasi-static loading conditions. First, we explore the conditions justifying the simplification of axisymmetric elastic fiber families into single fiber bundles through discretized strain energy functions. Simulations reveal that a concentration factor exceeding 10 allows for a consistent deviation below 10% between simplified and non-simplified responses. Second, we investigate the impact of elastic fibers on the physiological stiffness in IVDs, revealing minimal influence on biological motions but significant effects on degeneration. Lastly, we examine the initiation and progression of annulus fibrosus (AF) damage. Our findings confirm the validity of simplifying elastic fiber families and underscore the necessity of considering elastic fiber damage in biomechanical studies of AF tissues. Elastic fibers contribute to increased biaxial stretch stiffness, and their damage significantly affects the loading capacity of the inner AF. Additionally, degeneration significantly alters the susceptibility to damage in the AF, with specific regions exhibiting higher vulnerability. Damage tends to extend circumferentially and radially, emphasizing the regional variations in collagen and elastic fiber properties. This study offers useful insights for refining biomechanical models, paving the way for a more comprehensive understanding of IVD responses and potential clinical implications.

Comprehensive modeling of annulus fibrosus: From biphasic refined characterization to damage accumulation under viscous loading

The annulus fibrosus (AF), a permeable, hydrated, and fiber-reinforced soft tissue, exhibits complex responses influenced by fluid pressure, osmotic pressure, and structural mechanics. Existing models struggle to comprehensively represent these intricate interactions and the heterogeneous solid responses within the AF. Additionally, the mechanisms driving differential damage accumulation between non-degenerative and degenerative intervertebral discs remain poorly understood. In this study, we introduce a biphasic-swelling damage model for the AF. We conceptually develop and rigorously validate this model through tissue-level tests employing various loading modes, consistently aligning model predictions with experimental data. Leveraging parametric geometric algorithms and custom Python scripts, we construct models simulating both non-degenerative and degenerative discs. Following calibration, we subject these models to viscous loading protocols. Our findings reveal the posterior AF's susceptibility to damage, contingent upon loading rate and water content. We elucidate the underlying mechanisms by examining the temporal evolution of fluid pressure, osmotic pressure, and the regionally dependent fiber network. This research presents a highly accurate model of the AF, providing valuable insights into disc damage. Future research endeavors should expand this model to incorporate ionic transport and diffusion, enabling a more profound exploration of intervertebral disc mechanobiology. This comprehensive model contributes to a better understanding of AF behavior and may inform therapeutic strategies for disc-related pathologies. STATEMENT OF SIGNIFICANCE: This research presents a comprehensive model of the annulus fibrosus (AF), a crucial component of the intervertebral disc that provides structural support and resists deformation. The study introduces a biphasic-swelling damage model for the AF and validates it through tissue-level tests. The model accounts for fluid pressure, osmotic pressure, and matrix mechanics, providing a more accurate representation of the AF's behavior. The study also investigates the differential damage accumulation between non-degenerative and degenerative discs, shedding light on the mechanisms driving disc degeneration. The findings have significant implications for medical treatments and interventions, as they highlight the posterior AF's susceptibility to damage. This research is of great interest to readers interested in biomechanics, tissue engineering, and medical treatments for disc degeneration.

A novel knee joint model in FEBio with inhomogeneous fibril-reinforced biphasic cartilage simulating tissue mechanical responses during gait: data from the osteoarthritis initiative

We developed a novel knee joint model in FEBio to simulate walking. Knee cartilage was modeled using a fibril-reinforced biphasic (FRB) formulation with depth-wise collagen architecture and split-lines to account for cartilage structure. Under axial compression, the knee model with FRB cartilage yielded contact pressures, similar to reported experimental data. Furthermore, gait analysis with FRB cartilage simulated spatial and temporal trends in cartilage fluid pressures, stresses, and strains, comparable to those of a fibril-reinforced poroviscoelastic (FRPVE) material in Abaqus. This knee joint model in FEBio could be used for further studies of knee disorders using physiologically relevant loading.

Modeling fatigue failure in soft tissue using a visco-hyperelastic model with discontinuous damage

Soft tissue is susceptible to injury from single high-magnitude static loads and from repetitive low-magnitude fatigue loads. While many constitutive formulations have been developed and validated to model static failure in soft tissue, a modeling framework is not well-established for fatigue failure. Here we determined the feasibility of using a visco-hyperelastic damage model with discontinuous damage (strain energy-based damage criterion) to simulate low- and high-cycle fatigue failure in soft fibrous tissue. Cyclic creep data from six uniaxial tensile fatigue experiments of human medial meniscus were used to calibrate the specimen-specific material parameters. The model was able to successfully simulate all three characteristic stages of cyclic creep, and predict the number of cycles until tissue rupture. Mathematically, damage propagated under constant cyclic stress due to time-dependent viscoelastic increases in tensile stretch that in turn increased strain energy. Our results implicate solid viscoelasticity as a fundamental regulator of fatigue failure in soft tissue, where tissue with slow stress relaxation times will be more resistant to fatigue injury. In a validation study, the visco-hyperelastic damage model was able to simulate characteristic stress-strain curves of pull to failure experiments (static failure) when using material parameters curve fit to the fatigue experiments. For the first time, we've shown that a visco-hyperelastic discontinuous damage framework can model cyclic creep and predict material rupture in soft tissue, and may enable the reliable simulation of both fatigue and static failure behavior from a single constitutive formulation.

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 X s is observable. The solid generations α are all constrained to share the same velocity v s , but may have distinct reference configurations X α . An important element of this formulation is that the time-invariant mapping F α s = ∂ X α / ∂ X s between these reference configurations is a function of state, whose mathematical formulation is postulated by constitutive assumption. Thus, reference configurations X α are not observable ( α ≠ s ) . This formulation employs only observable state variables, such as the deformation gradient F s of the master generation and the referential mass concentrations ρ r α 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 ρ ˆ r α . 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.

Finite element modeling of meniscal tears using continuum damage mechanics and digital image correlation

Meniscal tears are a common, painful, and debilitating knee injury with limited treatment options. Computational models that predict meniscal tears may help advance injury prevention and repair, but first these models must be validated using experimental data. Here we simulated meniscal tears with finite element analysis using continuum damage mechanics (CDM) in a transversely isotropic hyperelastic material. Finite element models were built to recreate the coupon geometry and loading conditions of forty uniaxial tensile experiments of human meniscus that were pulled to failure either parallel or perpendicular to the preferred fiber orientation. Two damage criteria were evaluated for all experiments: von Mises stress and maximum normal Lagrange strain. After we successfully fit all models to experimental force-displacement curves (grip-to-grip), we compared model predicted strains in the tear region at ultimate tensile strength to the strains measured experimentally with digital image correlation (DIC). In general, the damage models underpredicted the strains measured in the tear region, but models using von Mises stress damage criterion had better overall predictions and more accurately simulated experimental tear patterns. For the first time, this study has used DIC to expose strengths and weaknesses of using CDM to model failure behavior in soft fibrous tissue.

Dots-on-Plots: A Web Application to Analyze Stress-Strain Curves From Tensile Tests of Soft Tissue

The calculation of tensile mechanical properties from stress-strain curves is a fundamental step in characterizing material behavior, yet no standardized method exists to perform these calculations for soft tissue. To address this deficiency, we developed a free web application called Dots-on-Plots2 that fully automates the calculation of tensile mechanical properties from stress-strain curves. The analyzed mechanical properties include the strength, strain, and energy at four points of interest (transition, yield, ultimate, and rupture), and the linear modulus. Users of Dots-on-Plots can upload multiple files, view and download results, and adjust threshold settings. This study determined a threshold setting that minimized error when calculating the transition point, where the stress-strain curve "transitions" from a nonlinear "toe" region to a linear region. Using the optimal threshold (2% stress deviation from a linear region fit), Dots-on-Plots calculated the transition strains from twenty tensile experiments of human meniscus to be 0.049 ± 0.007, which nearly matched the known transition strain values of 0.050 ± 0.006 (determined using finite element parameter optimization). The sensitivity of the calculated transition strain to the shape of various stress-strain curves was analyzed using sets of model-generated synthetic data. This free web application offers a convenient and reliable tool to systematically enhance the speed, transparency, and consistency of mechanical analysis across biomedical research groups.

A Numerical Scheme for Anisotropic Reactive Nonlinear Viscoelasticity

Reactive viscoelasticity is a theoretical framework based on the theory of reactive constrained mixtures that encompasses nonlinear viscoelastic responses. It models a viscoelastic solid as a mixture of strong and weak bonds that maintain the cohesiveness of the molecular constituents of the solid matter. Strong bonds impart the elastic response while weak bonds break and reform into a stress-free state in response to loading. The process of bonds breaking and reforming is modeled as a reaction where loaded bonds are the reactants and bonds reformed into a stress-free state are the products of a reaction. The reaction is triggered by the evolving state of loading. The state of stress in strong bonds is a function of the total strain in the material, whereas the state of stress in weak bonds is based on the state of strain relative to the time that these bonds were reformed. This study introduces two important practical contributions to the reactive nonlinear viscoelasticity framework: (1) normally, the evaluation of the stress tensor involves taking a summation over a continually increasing number of weak bond generations, which is poorly suited for a computational scheme. Therefore, this study presents an effective numerical scheme for evaluating the strain energy density, the Cauchy stress, and spatial elasticity tensors of reactive viscoelastic materials. (2) We provide the conditions for satisfying frame indifference for anisotropic nonlinear viscoelasticity, including for tension-bearing fiber models. Code verifications and model validations against experimental data provide evidence in support of this updated formulation.

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.

Finite Element Implementation of Biphasic-Fluid Structure Interactions in febio

In biomechanics, solid-fluid mixtures have commonly been used to model the response of hydrated biological tissues. In cartilage mechanics, this type of mixture, where the fluid and solid constituents are both assumed to be intrinsically incompressible, is often called a biphasic material. Various physiological processes involve the interaction of a viscous fluid with a porous-hydrated tissue, as encountered in synovial joint lubrication, cardiovascular mechanics, and respiratory mechanics. The objective of this study was to implement a finite element solver in the open-source software febio that models dynamic interactions between a viscous fluid and a biphasic domain, accommodating finite deformations of both domains as well as fluid exchanges between them. For compatibility with our recent implementation of solvers for computational fluid dynamics (CFD) and fluid-structure interactions (FSI), where the fluid is slightly compressible, this study employs a novel hybrid biphasic formulation where the porous skeleton is intrinsically incompressible but the fluid is also slightly compressible. The resulting biphasic-FSI (BFSI) implementation is verified against published analytical and numerical benchmark problems, as well as novel analytical solutions derived for the purposes of this study. An illustration of this BFSI solver is presented for two-dimensional (2D) airflow through a simulated face mask under five cycles of breathing, showing that masks significantly reduce air dispersion compared to the no-mask control analysis. In addition, we model three-dimensional (3D) blood flow in a bifurcated carotid artery assuming porous arterial walls and verify that mass is conserved across all fluid-permeable boundaries. The successful formulation and implementation of this BFSI solver offers enhanced multiphysics modeling capabilities that are accessible via an open-source software platform.

A synthetic mechanogenetic gene circuit for autonomous drug delivery in engineered tissues

Mechanobiologic signals regulate cellular responses under physiologic and pathologic conditions. Using synthetic biology and tissue engineering, we developed a mechanically responsive bioartificial tissue that responds to mechanical loading to produce a preprogrammed therapeutic biologic drug. By deconstructing the signaling networks induced by activation of the mechanically sensitive ion channel transient receptor potential vanilloid 4 (TRPV4), we created synthetic TRPV4-responsive genetic circuits in chondrocytes. We engineered these cells into living tissues that respond to mechanical loading by producing the anti-inflammatory biologic drug interleukin-1 receptor antagonist. Chondrocyte TRPV4 is activated by osmotic loading and not by direct cellular deformation, suggesting that tissue loading is transduced into an osmotic signal that activates TRPV4. Either osmotic or mechanical loading of tissues transduced with TRPV4-responsive circuits protected constructs from inflammatory degradation by interleukin-1α. This synthetic mechanobiology approach was used to develop a mechanogenetic system to enable long-term, autonomously regulated drug delivery driven by physiologically relevant loading.

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.

Historical Review of Combined Experimental and Computational Approaches for Investigating Annulus Fibrosus Mechanics

Intervertebral disc research has sought to develop a deeper understanding of spine biomechanics, the complex relationship between disc health and back pain, and the mechanisms of spinal injury and repair. To do so, many researchers have focused on characterizing tissue-level properties of the disc, where the roles of tissue subcomponents can be more systematically investigated. Unfortunately, experimental challenges often limit the ability to measure important disc tissue- and subtissue-level behaviors, including fiber-matrix interactions, transient nutrient and electrolyte transport, and damage propagation. Numerous theoretical and numerical modeling frameworks have been introduced to explain, complement, guide, and optimize experimental research efforts. The synergy of experimental and computational work has significantly advanced the field, and these two aspects have continued to develop independently and jointly. Meanwhile, the relationship between experimental and computational work has become increasingly complex and interdependent. This has made it difficult to interpret and compare results between experimental and computational studies, as well as between solely computational studies. This paper seeks to explore issues of model translatability, robustness, and efficient study design, and to propose and motivate potential future directions for experimental, computational, and combined tissue-level investigations of the intervertebral disc.

Finite Element Framework for Computational Fluid Dynamics in FEBio

The mechanics of biological fluids is an important topic in biomechanics, often requiring the use of computational tools to analyze problems with realistic geometries and material properties. This study describes the formulation and implementation of a finite element framework for computational fluid dynamics (CFD) in FEBio, a free software designed to meet the computational needs of the biomechanics and biophysics communities. This formulation models nearly incompressible flow with a compressible isothermal formulation that uses a physically realistic value for the fluid bulk modulus. It employs fluid velocity and dilatation as essential variables: The virtual work integral enforces the balance of linear momentum and the kinematic constraint between fluid velocity and dilatation, while fluid density varies with dilatation as prescribed by the axiom of mass balance. Using this approach, equal-order interpolations may be used for both essential variables over each element, contrary to traditional mixed formulations that must explicitly satisfy the inf-sup condition. The formulation accommodates Newtonian and non-Newtonian viscous responses as well as inviscid fluids. The efficiency of numerical solutions is enhanced using Broyden's quasi-Newton method. The results of finite element simulations were verified using well-documented benchmark problems as well as comparisons with other free and commercial codes. These analyses demonstrated that the novel formulation introduced in FEBio could successfully reproduce the results of other codes. The analogy between this CFD formulation and standard finite element formulations for solid mechanics makes it suitable for future extension to fluid-structure interactions (FSIs).

Evaluating Plastic Deformation and Damage as Potential Mechanisms for Tendon Inelasticity using a Reactive Modeling Framework

Inelastic behaviors, such as softening, a progressive decrease in modulus before failure, occur in tendon and are important aspect in degeneration and tendinopathy. These inelastic behaviors are generally attributed to two potential mechanisms: plastic deformation and damage. However, it is not clear which is primarily responsible. In this study, we evaluated these potential mechanisms of tendon inelasticity by using a recently developed reactive inelasticity model (RIE), which is a structurally-inspired continuum mechanics framework that models tissue inelasticity based on the molecular bond kinetics. Using RIE, we formulated two material models, one specific to plastic deformation and the other to damage. The models were independently fit to published experimental tensile tests of rat tail tendons. We quantified the inelastic effects and compared the performance of the two models in fitting the mechanical response during loading, relaxation, unloading, and reloading phases. Additionally, we validated the models by using the resulting fit parameters to predict an independent set of experimental stress-strain curves from ramp-to-failure tests. Overall, the models were both successful in fitting the experiments and predicting the validation data. However, the results did not strongly favor one mechanism over the other. As a result, to distinguish between plastic deformation and damage, different experimental protocols will be needed. Nevertheless, these findings suggest the potential of RIE as a comprehensive framework for studying tendon inelastic behaviors.

A Formulation for Fluid Structure-Interactions in FEBio Using Mixture Theory

Many physiological systems involve strong interactions between fluids and solids, posing a signicant challenge when modeling biomechanics. The objective of this study was to implement a fluid-structure interaction (FSI) solver in the free, open-source finite element code FEBio (febio.org), that combined the existing solid mechanics and rigid body dynamics solver with a recently-developed computational fluid dynamics (CFD) solver. A novel Galerkin-based finite element FSI formulation was introduced based on mixture theory, where the FSI domain was described as a mixture of fluid and solid constituents that have distinct motions. The mesh was defined on the solid domain, specialized to have zero mass, negligible stiffness and zero frictional interactions with the fluid, whereas the fluid was modeled as isothermal and compressible. The mixture framework provided the foundation for evaluating material time derivatives in a material frame for the solid and in a spatial frame for the fluid. Similar to our recently reported CFD solver, our FSI formulation did not require stabilization methods to achieve good convergence, producing a compact set of equations and code implementation. The code was successfully verified against benchmark problems and an analytical solution for squeeze-film lubrication. It was validated against experimental measurements of the flow rate in a peristaltic pump, and illustrated using non-Newtonian blood flow through a bifurcated carotid artery with a thick arterial wall. The successful formulation and implementation of this FSI solver enhances the multiphysics modeling capabilities in FEBio relevant to the biomechanics and biophysics communities.

A Reactive Inelasticity Theoretical Framework for Modeling Viscoelasticity, Plastic Deformation, and Damage in Fibrous Soft Tissue

Fibrous soft tissues are biopolymeric materials that are made of extracellular proteins, such as different types of collagen and proteoglycans, and have a high water content. These tissues have nonlinear, anisotropic, and inelastic mechanical behaviors that are often categorized into viscoelastic behavior, plastic deformation, and damage. While tissue's elastic and viscoelastic mechanical properties have been measured for decades, there is no comprehensive theoretical framework for modeling inelastic behaviors of these tissues that is based on their structure. To model the three major inelastic mechanical behaviors of tissue's fibrous matrix, we formulated a structurally inspired continuum mechanics framework based on the energy of molecular bonds that break and reform in response to external loading (reactive bonds). In this framework, we employed the theory of internal state variables (ISV) and kinetics of molecular bonds. The number fraction of bonds, their reference deformation gradient, and damage parameter were used as state variables that allowed for consistent modeling of all three of the inelastic behaviors of tissue by using the same sets of constitutive relations. Several numerical examples are provided that address practical problems in tissue mechanics, including the difference between plastic deformation and damage. This model can be used to identify relationships between tissue's mechanical response to external loading and its biopolymeric structure.

FEBio: History and Advances

The principal goal of the FEBio project is to provide an advanced finite element tool for the biomechanics and biophysics communities that allows researchers to model mechanics, transport, and electrokinetic phenomena for biological systems accurately and efficiently. In addition, because FEBio is geared toward the research community, the code is designed such that new features can be added easily, thus making it an ideal tool for testing novel computational methods. Finally, because the success of a code is determined by its user base, integral goals of the FEBio project have been to offer support and outreach to our community; to provide mechanisms for dissemination of results, models, and data; and to encourage interaction between users. This review presents the history of the FEBio project, from its initial developments through its current funding period. We also present a glimpse into the future of FEBio.

Reactive Constrained Mixtures for Modeling the Solid Matrix of Biological Tissues

This article illustrates our approach for modeling the solid matrix of biological tissues using reactive constrained mixtures. Several examples are presented to highlight the potential benefits of this approach, showing that seemingly disparate fields of mechanics and chemical kinetics are actually closely interrelated and may be elegantly expressed in a unified framework. Thus, constrained mixture models recover classical theories for fibrous materials with bundles oriented in different directions or having different reference configurations, that produce characteristic fiber recruitment patterns under loading. Reactions that exchange mass among various constituents of a mixture may be used to describe tissue growth and remodeling, which may also alter the material's anisotropy. Similarly, reactions that describe the breaking and reforming of bonds may be used to model free energy dissipation in a viscoelastic material. Therefore, this framework is particularly well suited for modeling biological tissues.

An anisotropic multiphysics damage model with application to annulus fibrosus

An anisotropic multiphysics damage model is developed to characterize the couplings among multiple physical fields within soft tissues and the tissue damage based on thermodynamic principles. This anisotropic multiphysics damage model integrates the continuum mixture theory and a continuum damage model, and the anisotropic damage is considered by evolution of internal damage variables governing the anisotropic mechanical behaviors of tissues. The energy dissipation associated with the transport of fluid and ions is generally related to tissue damage. The anisotropic multiphysics damage model is applied to simulate a case of annulus fibrosus (AF) damage in an isolated intervertebral disc under compression, to understand the damage initiation and propagation. It is found that, for this case (with 1000N/s of compression rate and neglected ground matrix damage), the damage initiated in the outer and middle posterior regions of AF at about 700N of axial compression. The region-dependent yield stretch ratio predicted by this model is consistent with experimental findings. A sensitive study on the damage parameters is also presented. This study provides an additional insight into AF damage in the isolated disc under mechanical compression.

* Constrained Cage Culture Improves Engineered Cartilage Functional Properties by Enhancing Collagen Network Stability

When cultured with sufficient nutrient supply, engineered cartilage synthesizes proteoglycans rapidly, producing an osmotic swelling pressure that destabilizes immature collagen and prevents the development of a robust collagen framework, a hallmark of native cartilage. We hypothesized that mechanically constraining the proteoglycan-induced tissue swelling would enhance construct functional properties through the development of a more stable collagen framework. To test this hypothesis, we developed a novel "cage" growth system to mechanically prevent tissue constructs from swelling while ensuring adequate nutrient supply to the growing construct. The effectiveness of constrained culture was examined by testing constructs embedded within two different scaffolds: agarose and cartilage-derived matrix hydrogel (CDMH). Constructs were seeded with immature bovine chondrocytes and cultured under free swelling (FS) conditions for 14 days with transforming growth factor-β before being placed into a constraining cage for the remainder of culture. Controls were cultured under FS conditions throughout. Agarose constructs cultured in cages did not expand after the day 14 caging while FS constructs expanded to 8 × their day 0 weight after 112 days of culture. In addition to the physical differences in growth, by day 56, caged constructs had higher equilibrium (agarose: 639 ± 179 kPa and CDMH: 608 ± 257 kPa) and dynamic compressive moduli (agarose: 3.4 ± 1.0 MPa and CDMH 2.8 ± 1.0 MPa) than FS constructs (agarose: 193 ± 74 kPa and 1.1 ± 0.5 MPa and CDMH: 317 ± 93 kPa and 1.8 ± 1.0 MPa for equilibrium and dynamic properties, respectively). Interestingly, when normalized to final day wet weight, cage and FS constructs did not exhibit differences in proteoglycan or collagen content. However, caged culture enhanced collagen maturation through the increased formation of pyridinoline crosslinks and improved collagen matrix stability as measured by α-chymotrypsin solubility. These findings demonstrate that physically constrained culture of engineered cartilage constructs improves functional properties through improved collagen network maturity and stability. We anticipate that constrained culture may benefit other reported engineered cartilage systems that exhibit a mismatch in proteoglycan and collagen synthesis.

Nutrient Channels Aid the Growth of Articular Surface-Sized Engineered Cartilage Constructs

Symptomatic osteoarthritic lesions span large regions of joint surfaces and the ability to engineer cartilage constructs at clinically relevant sizes would be highly desirable. We previously demonstrated that nutrient transport limitations can be mitigated by the introduction of channels in 10 mm diameter cartilage constructs. In this study, we scaled up our previous system to cast and cultivate 40 mm diameter constructs (2.3 mm overall thickness); 4 mm diameter and channeled 10 mm diameter constructs were studied for comparison. Furthermore, to assess whether prior results using primary bovine cells are applicable for passaged cells-a more clinically realistic scenario-we cast constructs of each size with primary or twice-passaged cells. Constructs were assessed mechanically for equilibrium compressive Young's modulus (EY), dynamic modulus at 0.01 Hz (G*), and friction coefficient (μ); they were also assessed biochemically, histologically, and immunohistochemically for glycosaminoglycan (GAG) and collagen contents. By maintaining open channels, we successfully cultured robust constructs the size of entire human articular cartilage layers (growing to ∼52 mm in diameter, 4 mm thick, mass of 8 g by day 56), representing a 100-fold increase in scale over our 4 mm diameter constructs, without compromising their functional properties. Large constructs reached EY of up to 623 kPa and GAG contents up to 8.9%/ww (% of wet weight), both within native cartilage ranges, had G* >2 MPa, and up to 3.5%/ww collagen. Constructs also exhibited some of the lowest μ reported for engineered cartilage (0.06-0.11). Passaged cells produced tissue of lower quality, but still exhibited native EY and GAG content, similar to their smaller controls. The constructs produced in this study are, to our knowledge, the largest engineered cartilage constructs to date which possess native EY and GAG, and are a testament to the effectiveness of nutrient channels in overcoming transport limitations in cartilage tissue engineering.

Optimizing nutrient channel spacing and revisiting TGF-beta in large engineered cartilage constructs

Cartilage tissue engineering is a promising approach to treat osteoarthritis. However, current techniques produce tissues too small for clinical relevance. Increasingly close-packed channels have helped overcome nutrient transport limitations in centimeter-sized chondrocyte-agarose constructs, yet optimal channel spacings to recapitulate native cartilage compositional and mechanical properties in constructs this large have not been identified. Transient active TGF-β treatment consistently reproduces native compressive Young׳s modulus (EY) and glycosaminoglycan (GAG) content in constructs, but standard dosages of 10ng/mL exacerbate matrix heterogeneity. To ultimately produce articular layer-sized constructs, we must first optimize channel spacing and investigate the role of TGF-β in the utility of channels. We cultured ∅10mm constructs with 0, 12, 19, or 27 nutrient channels (∅1mm) for 6-8 weeks with 0, 1, or 10ng/mL TGF-β; subsequently we analyzed them mechanically, biochemically, and histologically. Constructs with 12 or 19 channels grew the most favorably, reaching EY=344±113kPa and GAG and collagen contents of 10.8±1.2% and 2.2±0.2% of construct wet weight, respectively. Constructs with 27 channels had significantly less deposited GAG than other groups. Channeled constructs given 1 or 10ng/mL TGF-β developed similar properties. Without TGF-β, constructs with 0 or 12 channels exhibited properties that were indistinguishable, and lower than TGF-β-supplemented constructs. Taken together, these results emphasize that nutrient channels are effective only in the presence of TGF-β, and indicate that spacings equivalent to 12 channels in ∅10mm constructs can be employed in articular-layer-sized constructs with reduced dosages of TGF-β.

High seeding density of human chondrocytes in agarose produces tissue-engineered cartilage approaching native mechanical and biochemical properties

Animal cells have served as highly controllable model systems for furthering cartilage tissue engineering practices in pursuit of treating osteoarthritis. Although successful strategies for animal cells must ultimately be adapted to human cells to be clinically relevant, human chondrocytes are rarely employed in such studies. In this study, we evaluated the applicability of culture techniques established for juvenile bovine and adult canine chondrocytes to human chondrocytes obtained from fresh or expired osteochondral allografts. Human chondrocytes were expanded and encapsulated in 2% agarose scaffolds measuring ∅3-4mm×2.3mm, with cell seeding densities ranging from 15 to 90×10(6)cells/mL. Subsets of constructs were subjected to transient or sustained TGF-β treatment, or provided channels to enhance nutrient transport. Human cartilaginous constructs physically resembled native human cartilage, and reached compressive Young's moduli of up to ~250kPa (corresponding to the low end of ranges reported for native knee cartilage), dynamic moduli of ~950kPa (0.01Hz), and contained 5.7% wet weight (%/ww) of glycosaminoglycans (≥ native levels) and 1.5%/ww collagen. We found that the initial seeding density had pronounced effects on tissue outcomes, with high cell seeding densities significantly increasing nearly all measured properties. Transient TGF-β treatment was ineffective for adult human cells, and tissue construct properties plateaued or declined beyond 28 days of culture. Finally, nutrient channels improved construct mechanical properties, presumably due to enhanced rates of mass transport. These results demonstrate that our previously established culture system can be successfully translated to human chondrocytes.

Integrated multiscale biomaterials experiment and modelling: a perspective

Advances in multiscale models and computational power have enabled a broad toolset to predict how molecules, cells, tissues and organs behave and develop. A key theme in biological systems is the emergence of macroscale behaviour from collective behaviours across a range of length and timescales, and a key element of these models is therefore hierarchical simulation. However, this predictive capacity has far outstripped our ability to validate predictions experimentally, particularly when multiple hierarchical levels are involved. The state of the art represents careful integration of multiscale experiment and modelling, and yields not only validation, but also insights into deformation and relaxation mechanisms across scales. We present here a sampling of key results that highlight both challenges and opportunities for integrated multiscale experiment and modelling in biological systems.

Heterogeneous engineered cartilage growth results from gradients of media-supplemented active TGF-β and is ameliorated by the alternative supplementation of latent TGF-β

Transforming growth factor beta (TGF-β) has become one of the most widely utilized mediators of engineered cartilage growth. It is typically exogenously supplemented in the culture medium in its active form, with the expectation that it will readily transport into tissue constructs through passive diffusion and influence cellular biosynthesis uniformly. The results of this investigation advance three novel concepts regarding the role of TGF-β in cartilage tissue engineering that have important implications for tissue development. First, through the experimental and computational analysis of TGF-β concentration distributions, we demonstrate that, contrary to conventional expectations, media-supplemented exogenous active TGF-β exhibits a pronounced concentration gradient in tissue constructs, resulting from a combination of high-affinity binding interactions and a high cellular internalization rate. These gradients are sustained throughout the entire culture duration, leading to highly heterogeneous tissue growth; biochemical and histological measurements support that while biochemical content is enhanced up to 4-fold at the construct periphery, enhancements are entirely absent beyond 1 mm from the construct surface. Second, construct-encapsulated chondrocytes continuously secrete large amounts of endogenous TGF-β in its latent form, a portion of which undergoes cell-mediated activation and enhances biosynthesis uniformly throughout the tissue. Finally, motivated by these prior insights, we demonstrate that the alternative supplementation of additional exogenous latent TGF-β enhances biosynthesis uniformly throughout tissue constructs, leading to enhanced but homogeneous tissue growth. This novel demonstration suggests that latent TGF-β supplementation may be utilized as an important tool for the translational engineering of large cartilage constructs that will be required to repair the large osteoarthritic defects observed clinically.