Coupled macroscopic and microscopic scale modeling of fibrillar tissues and tissue equivalents

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.

Simultaneous Wide-Field Planar Strain-Fiber Orientation Distribution Measurement Using Polarized Spatial Domain Imaging

In the present study, we demonstrate that soft tissue fiber architectural maps captured using polarized spatial frequency domain imaging (pSFDI) can be utilized as an effective texture source for DIC-based planar surface strain analyses. Experimental planar biaxial mechanical studies were conducted using pericardium as the exemplar tissue, with simultaneous pSFDI measurements taken. From these measurements, the collagen fiber preferred direction [Formula: see text] was determined at the pixel level over the entire strain range using established methods ( https://doi.org/10.1007/s10439-019-02233-0 ). We then utilized these pixel-level [Formula: see text] maps as a texture source to quantify the deformation gradient tensor [Formula: see text] as a function of spatial position [Formula: see text] within the specimen at time t. Results indicted that that the pSFDI approach produced accurate deformation maps, as validated using both physical markers and conventional particle based method derived from the DIC analysis of the same specimens. We then extended the pSFDI technique to extract the fiber orientation distribution [Formula: see text] as a function of [Formula: see text] from the pSFDI intensity signal. This was accomplished by developing a calibration procedure to account for the optical behavior of the constituent fibers for the soft tissue being studied. We then demonstrated that the extracted [Formula: see text] was accurately computed in both the referential (i.e. unloaded) and deformed states. Moreover, we noted that the measured [Formula: see text] agreed well with affine kinematic deformation predictions. We also demonstrated this calibration approach could also be effectively used on electrospun biomaterials, underscoring the general utility of the approach. In a final step, using the ability to simultaneously quantify [Formula: see text] and [Formula: see text], we examined the effect of deformation and collagen structural measurements on the measurement region size. For pericardial tissues, we determined a critical length of [Formula: see text] 8 mm wherein the regional variations sufficiently dissipated. This result has immediate potential in the identification of optimal length scales for meso-scale strain measurement in soft tissues and fibrous biomaterials.

A computational framework for biomaterials containing three-dimensional random fiber networks based on the affine kinematics

Understanding the structure-function relationship of biomaterials can provide insights into different diseases and advance numerous biomedical applications. This paper presents a finite element-based computational framework to model biomaterials containing a three-dimensional fiber network at the microscopic scale. The fiber network is synthetically generated by a random walk algorithm, which uses several random variables to control the fiber network topology such as fiber orientations and tortuosity. The geometric information of the generated fiber network is stored in an array-like data structure and incorporated into the nonlinear finite element formulation. The proposed computational framework adopts the affine fiber kinematics, based on which the fiber deformation can be expressed by the nodal displacement and the finite element interpolation functions using the isoparametric relationship. A variational approach is developed to linearize the total strain energy function and derive the nodal force residual and the stiffness matrix required by the finite element procedure. Four numerical examples are provided to demonstrate the capabilities of the proposed computational framework, including a numerical investigation about the relationship between the proposed method and a class of anisotropic material models, a set of synthetic examples to explore the influence of fiber locations on material local and global responses, a thorough mesh-sensitivity analysis about the impact of mesh size on various numerical results, and a detailed case study about the influence of material structures on the performance of eggshell-membrane-hydrogel composites. The proposed computational framework provides an efficient approach to investigate the structure-function relationship for biomaterials that follow the affine fiber kinematics.

Complex mechanical behavior of human articular cartilage and hydrogels for cartilage repair

The mechanical behavior of cartilage tissue plays a crucial role in physiological mechanotransduction processes of chondrocytes and pathological changes like osteoarthritis. Therefore, intensive research activities focus on the identification of implant substitute materials that mechanically mimic the cartilage extracellular matrix. This, however, requires a thorough understanding of the complex mechanical behavior of both native cartilage and potential substitute materials to treat cartilage lesions. Here, we perform complex multi-modal mechanical analyses of human articular cartilage and two surrogate materials, commercially available ChondroFiller_liquid, and oxidized alginate-gelatin (ADA-GEL) hydrogels. We show that all materials exhibit nonlinearity and compression-tension asymmetry. However, while hyaline cartilage yields higher stresses in tension than in compression, ChondroFiller^liquid and ADA-GEL exhibit the opposite trend. These characteristics can be attributed to the materials' underlying microstructure: Both cartilage and ChondroFiller^liquid contain fibrillar components, but the latter constitutes a bi-phasic structure, where the 60% nonfibrillar hydrogel proportion dominates the mechanical response. Of all materials, ChondroFiller^liquid shows the most pronounced viscous effects. The present study provides important insights into the microstructure-property relationship of cartilage substitute materials, with vital implications for mechanically-driven material design in cartilage engineering. In addition, we provide a data set to create mechanical simulation models in the future.

A 2.5D approach to the mechanics of electrospun fibre mats

In this paper, a discrete random network modelling approach specific to electrospun networks is presented. Owing to the manufacturing process, fibres in these materials systems have an enormous length, as compared to their diameters, and form sparse networks since fibre contact over thickness is limited to a narrow range. Representative volume elements are generated, in which fibres span the entire domain, and a technique is developed to apply computationally favourable periodic boundary conditions despite the initial non-periodicity of the networks. To capture sparsity, a physically motivated method is proposed to distinguish true fibre cross-links, in which mechanical interaction takes place, from mere fibre intersections. The model is exclusively informed by experimentally accessible parameters, demonstrates excellent agreement with the mechanical response of electrospun fibre mats, captures typical microscopic deformation patterns, and provides information on the kinematics of fibres and pores. This ability to address relevant mechanisms of deformation at both micro- and macroscopic length scales, together with the moderate computational cost, render the proposed modelling approach a highly qualified tool for the computer-based design and optimization of electrospun networks.

A novel micro-to-macro structural approach for mechanical characterization of adipose tissue extracellular matrix

Mechanical characterization of adipose tissue micro-components is important for various biomedical applications such as tissue engineering and predicting adipose tissue response to forces involved in relevant medical intervention procedures (e.g. breast needle biopsy). For this characterization, we introduce a novel structural method for micromechanical modeling of the adipose tissue. The micromechanical model was developed using fluid-structure interaction (FSI) formulation. We utilized this model within an inverse problem framework to estimate the hyperelastic parameters of adipose tissue extracellular matrix (ECM). Using this framework, the ECM hyperelastic parameters were changed in the FSI model systematically using an optimization algorithm such that the mechanical response obtained from the FSI model matches the corresponding experimental response reported in previous studies. To account for adipocyte size variation, the hyperelastic parameters were determined for different adipocyte sizes in the FSI model. Results obtained in this investigation indicate that at various strains under quasi-static conditions, the stiffness of adipose tissue ECM is ~ (2-3) times higher than that of the adipose tissue. The results also indicate a very good fit between the FSI model responses and their experimental counterparts. This indicates the reliability of the proposed FSI model in capturing major elements of the adipose tissue micromechanics. As such, it is potentially useful in applications such as tissue engineering, estimating tissue deformation pertaining to medical intervention and cataloging the mechanical properties of adipose tissue under health and pathological conditions. It can also be utilized as a forward model for developing inversion algorithms designed to determine pathological adipose microstructural alterations.

Numerical investigation of the influence of pattern topology on the mechanical behavior of PEGDA hydrogels

Poly(ethylene glycol) diacrylate (PEGDA) hydrogels can be potentially used as scaffold material for tissue engineered heart valves (TEHVs) due to their good biocompatibility and biomechanical tunability. The photolithographic patterning technique is an effective approach to pattern PEGDA hydrogels to mimic the mechanical behavior of native biological tissues that are intrinsically anisotropic. The material properties of patterned PEGDA hydrogels largely depend on the pattern topology. In this paper, we adopt a newly proposed computational framework for fibrous biomaterials to numerically investigate the influence of pattern topology, including pattern ratio, orientation and waviness, on the mechanical behavior of patterned PEGDA hydrogels. The material parameters for the base hydrogel and the pattern stripes are directly calibrated from published experimental data. Several experimental observations reported in the literature are captured in the simulation, including the nonlinear relationship between pattern ratio and material linear modulus, and the decrease of material anisotropy when pattern ratio increases. We further numerically demonstrate that a three-region (toe-heel-linear) stress-strain relationship typically exhibited by biological tissues can be obtained by tuning the pattern waviness and the relative stiffness between the base hydrogel and pattern stripes. The numerical strategy and simulation results presented here can provide helpful guidance to optimize pattern design of PEGDA hydrogels toward the targeted material mechanical properties, therefore advance the development of TEHVs.

STATEMENT OF SIGNIFICANCE

Poly(ethylene glycol) diacrylate (PEGDA) hydrogels can be used as scaffold material for tissue engineered heart values (TEHVs) providing a promising alternative to generate suitable heart valve replacement method. The patterning of PEGDA hydrogels using photolithographic techniques creates materials that mimic the mechanical behavior of native heart valve tissues. However, targeted material properties are obtained via a trial-and-error process. Depending on experiments alone to explore the influence of pattern topology is expensive and time-consuming. We combine a newly proposed computational framework with published experimental data to numerically investigate the influence of pattern geometry on the mechanical behavior of patterned PEGDA hydrogels. The numerical strategy and simulation results presented here can provide guidance to optimize the design of PEGDA hydrogels with targeted material properties, therefore advance the development of TEHVs.

Finite element implementation of a multiscale model of the human lens capsule

An axisymmetric finite element implementation of a previously described structural constitutive model for the human lens capsule (Burd in Biomech Model Mechanobiol 8(3):217-231, 2009) is presented. This constitutive model is based on a hyperelastic approach in which the network of collagen IV within the capsule is represented by an irregular hexagonal planar network of hyperelastic bars, embedded in a hyperelastic matrix. The paper gives a detailed specification of the model and the periodic boundary conditions adopted for the network component. Momentum balance equations for the network are derived in variational form. These balance equations are used to develop a nonlinear solution scheme to enable the equilibrium configuration of the network to be computed. The constitutive model is implemented within a macroscopic finite element framework to give a multiscale model of the lens capsule. The possibility of capsule wrinkling is included in the formulation. To achieve this implementation, values of the first and second derivatives of the strain energy density with respect to the in-plane stretch ratios need to be computed at the local, constitutive model, level. Procedures to determine these strain energy derivatives at equilibrium configurations of the network are described. The multiscale model is calibrated against previously published experimental data on isolated inflation and uniaxial stretching of ex vivo human capsule samples. Two independent example lens capsule inflation analyses are presented.

Geometric characterization and simulation of planar layered elastomeric fibrous biomaterials

Many important biomaterials are composed of multiple layers of networked fibers. While there is a growing interest in modeling and simulation of the mechanical response of these biomaterials, a theoretical foundation for such simulations has yet to be firmly established. Moreover, correctly identifying and matching key geometric features is a critically important first step for performing reliable mechanical simulations. The present work addresses these issues in two ways. First, using methods of geometric probability, we develop theoretical estimates for the mean linear and areal fiber intersection densities for 2-D fibrous networks. These densities are expressed in terms of the fiber density and the orientation distribution function, both of which are relatively easy-to-measure properties. Secondly, we develop a random walk algorithm for geometric simulation of 2-D fibrous networks which can accurately reproduce the prescribed fiber density and orientation distribution function. Furthermore, the linear and areal fiber intersection densities obtained with the algorithm are in agreement with the theoretical estimates. Both theoretical and computational results are compared with those obtained by post-processing of scanning electron microscope images of actual scaffolds. These comparisons reveal difficulties inherent to resolving fine details of multilayered fibrous networks. The methods provided herein can provide a rational means to define and generate key geometric features from experimentally measured or prescribed scaffold structural data.

Effects of fabrication on the mechanics, microstructure and micromechanical environment of small intestinal submucosa scaffolds for vascular tissue engineering

In small intestinal submucosa scaffolds for functional tissue engineering, the impact of scaffold fabrication parameters on success rate may be related to the mechanotransductory properties of the final microstructural organization of collagen fibers. We hypothesized that two fabrication parameters, 1) preservation (P) or removal (R) of a dense collagen layer present in SIS and 2) SIS in a final dehydrated (D) or hydrated (H) state, have an effect on scaffold void area, microstructural anisotropy (fiber alignment) and mechanical anisotropy (global mechanical compliance). We further integrated our experimental measurements in a constitutive model to explore final effects on the micromechanical environment inside the scaffold volume. Our results indicated that PH scaffolds might exhibit recurrent and large force fluctuations between layers (up to 195 pN), while fluctuations in RH scaffolds might be larger (up to 256 pN) but not as recurrent. In contrast, both PD and RD groups were estimated to produce scarcer and smaller fluctuations (not larger than 50 pN). We concluded that the hydration parameter strongly affects the micromechanics of SIS and that an adequate choice of fabrication parameters, assisted by the herein developed method, might leverage the use of SIS for functional tissue engineering applications, where forces at the cellular level are of concern in the guidance of new tissue formation.

In situ cell-matrix mechanics in tendon fascicles and seeded collagen gels: implications for the multiscale design of biomaterials

Designing biomaterials to mimic and function within the complex mechanobiological conditions of connective tissues requires a detailed understanding of the micromechanical environment of the cell. The objective of our study was to measure the in situ cell-matrix strains from applied tension in both tendon fascicles and cell-seeded type I collagen scaffolds using laser scanning confocal microscopy techniques. Tendon fascicles and collagen gels were fluorescently labelled to simultaneously visualise the extracellular matrix and cell nuclei under applied tensile strains of 5%. There were significant differences observed in the micromechanics at the cell-matrix scale suggesting that the type I collagen scaffold did not replicate the pattern of native tendon strains. In particular, although the overall in situ tensile strains in the matrix were quite similar (∼2.5%) between the tendon fascicles and the collagen scaffolds, there were significant differences at the cell-matrix boundary with visible shear across cell nuclei of >1 μm measured in native tendon which was not observed at all in the collagen scaffolds. Similarly, there was significant non-uniformity of intercellular strains with relative sliding observed between cell rows in tendon which again was not observed in the collagen scaffolds where the strain environment was much more uniform. If the native micromechanical environment is not replicated in biomaterial scaffolds, then the cells may receive incorrect or mixed mechanical signals which could affect their biosynthetic response to mechanical load in tissue engineering applications. This study highlights the importance of considering the microscale mechanics in the design of biomaterial scaffolds and the need to incorporate such features in computational models of connective tissues.

Microstructural manipulation of electrospun scaffolds for specific bending stiffness for heart valve tissue engineering

Biodegradable thermoplastic elastomers are attractive for application in cardiovascular tissue construct development due to their amenability to a wide range of physical property tuning. For heart valve leaflets, while low flexural stiffness is a key design feature, control of this parameter has been largely neglected in the scaffold literature where electrospinning is being utilized. This study evaluated the effect of processing variables and secondary fiber populations on the microstructure, tensile and bending mechanics of electrospun biodegradable polyurethane scaffolds for heart valve tissue engineering. Scaffolds were fabricated from poly(ester urethane) urea (PEUU) and the deposition mandrel was translated at varying rates in order to modify fiber intersection density. Scaffolds were also fabricated in conjunction with secondary fiber populations designed either for mechanical reinforcement or to be selectively removed following fabrication. It was determined that increasing fiber intersection densities within PEUU scaffolds was associated with lower bending moduli. Further, constructs fabricated with stiff secondary fiber populations had higher bending moduli whereas constructs with secondary fiber populations which were selectively removed had noticeably lower bending moduli. Insights gained from this work will be directly applicable to the fabrication of soft tissue constructs, specifically in the development of cardiac valve tissue constructs.

The mechanical stress-strain properties of single electrospun collagen type I nanofibers

Knowledge of the mechanical properties of electrospun fibers is important for their successful application in tissue engineering, material composites, filtration and drug delivery. In particular, electrospun collagen has great potential for biomedical applications due to its biocompatibility and promotion of cell growth and adhesion. Using a combined atomic force microscopy (AFM)/optical microscopy technique, the single fiber mechanical properties of dry, electrospun collagen type I were determined. The fibers were electrospun from a 80 mg ml(-1) collagen solution in 1,1,1,3,3,3-hexafluro-2-propanol and collected on a striated surface suitable for lateral force manipulation by AFM. The small strain modulus, calculated from three-point bending analysis, was 2.82 GPa. The modulus showed significant softening as the strain increased. The average extensibility of the fibers was 33% of their initial length, and the average maximum stress (rupture stress) was 25 MPa. The fibers displayed significant energy loss and permanent deformations above 2% strain.

On the biomechanical function of scaffolds for engineering load-bearing soft tissues

Replacement or regeneration of load-bearing soft tissues has long been the impetus for the development of bioactive materials. While maturing, current efforts continue to be confounded by our lack of understanding of the intricate multi-scale hierarchical arrangements and interactions typically found in native tissues. The current state of the art in biomaterial processing enables a degree of controllable microstructure that can be used for the development of model systems to deduce fundamental biological implications of matrix morphologies on cell function. Furthermore, the development of computational frameworks which allow for the simulation of experimentally derived observations represents a positive departure from what has mostly been an empirically driven field, enabling a deeper understanding of the highly complex biological mechanisms we wish to ultimately emulate. Ongoing research is actively pursuing new materials and processing methods to control material structure down to the micro-scale to sustain or improve cell viability, guide tissue growth, and provide mechanical integrity, all while exhibiting the capacity to degrade in a controlled manner. The purpose of this review is not to focus solely on material processing but to assess the ability of these techniques to produce mechanically sound tissue surrogates, highlight the unique structural characteristics produced in these materials, and discuss how this translates to distinct macroscopic biomechanical behaviors.

Characterization of the complete fiber network topology of planar fibrous tissues and scaffolds

Understanding how engineered tissue scaffold architecture affects cell morphology, metabolism, phenotypic expression, as well as predicting material mechanical behavior has recently received increased attention. In the present study, an image-based analysis approach that provides an automated tool to characterize engineered tissue fiber network topology is presented. Micro-architectural features that fully defined fiber network topology were detected and quantified, which include fiber orientation, connectivity, intersection spatial density, and diameter. Algorithm performance was tested using scanning electron microscopy (SEM) images of electrospun poly(ester urethane)urea (ES-PEUU) scaffolds. SEM images of rabbit mesenchymal stem cell (MSC) seeded collagen gel scaffolds and decellularized rat carotid arteries were also analyzed to further evaluate the ability of the algorithm to capture fiber network morphology regardless of scaffold type and the evaluated size scale. The image analysis procedure was validated qualitatively and quantitatively, comparing fiber network topology manually detected by human operators (n = 5) with that automatically detected by the algorithm. Correlation values between manual detected and algorithm detected results for the fiber angle distribution and for the fiber connectivity distribution were 0.86 and 0.93 respectively. Algorithm detected fiber intersections and fiber diameter values were comparable (within the mean +/- standard deviation) with those detected by human operators. This automated approach identifies and quantifies fiber network morphology as demonstrated for three relevant scaffold types and provides a means to: (1) guarantee objectivity, (2) significantly reduce analysis time, and (3) potentiate broader analysis of scaffold architecture effects on cell behavior and tissue development both in vitro and in vivo.

Computational predictions of the tensile properties of electrospun fibre meshes: effect of fibre diameter and fibre orientation

The mechanical properties of biomaterial scaffolds are crucial for their efficacy in tissue engineering and regenerative medicine. At the microscopic scale, the scaffold must be sufficiently rigid to support cell adhesion, spreading, and normal extracellular matrix deposition. Concurrently, at the macroscopic scale the scaffold must have mechanical properties that closely match those of the target tissue. The achievement of both goals may be possible by careful control of the scaffold architecture. Recently, electrospinning has emerged as an attractive means to form fused fibre scaffolds for tissue engineering. The diameter and relative orientation of fibres affect cell behaviour, but their impact on the tensile properties of the scaffolds has not been rigorously characterized. To examine the structure-property relationship, electrospun meshes were made from a polyurethane elastomer with different fibre diameters and orientations and mechanically tested to determine the dependence of the elastic modulus on the mesh architecture. Concurrently, a multiscale modelling strategy developed for type I collagen networks was employed to predict the mechanical behaviour of the polyurethane meshes. Experimentally, the measured elastic modulus of the meshes varied from 0.56 to 3.0 MPa depending on fibre diameter and the degree of fibre alignment. Model predictions for tensile loading parallel to fibre orientation agreed well with experimental measurements for a wide range of conditions when a fitted fibre modulus of 18 MPa was used. Although the model predictions were less accurate in transverse loading of anisotropic samples, these results indicate that computational modelling can assist in design of electrospun artificial tissue scaffolds.

Multiscale, structure-based modeling for the elastic mechanical behavior of arterial walls

Passive elastic behavior of arterial wall remains difficult to model. Although phenomenological and structural models exist, the question of how the three-dimensional network structure of the collagen in the artery determines its mechanical properties is still open. A model is presented that incorporates a collagen network as well as the noncollagenous material that comprise the artery. The collagen architecture is represented as a network of interconnected fibers, and a neo-Hookean constitutive equation is used to describe the contribution of the noncollagenous matrix. The model is multiscale in that volume-averaging theory is applied to the collagen network, and it is structural in that parameters of the microstructure of the collagen network were considered instead of a macroscopic constitutive law. The computational results provided a good fit to published experimental data for decellularized porcine carotid arteries. The model predicted increased circumferential compliance for increased axial stretch, consistent with previously published reports, and a relatively small sensitivity to open angle. Even at large extensions, the model predicted that the noncollagenous matrix would be in compression, preventing collapse of the collagen network. The incorporation of fiber-fiber interactions led to an accurate model of artery wall behavior with relatively few parameters. The counterintuitive result that the noncollagenous component is in compression during extension and inflation of the tissue suggests that the collagen is important even at small strains, with the noncollagenous components supporting the network, but not resisting the load directly. More accurate representation of the microstructure of the artery wall is needed to explore this issue further.

A new microrheometric approach reveals individual and cooperative roles for TGF-beta1 and IL-1beta in fibroblast-mediated stiffening of collagen gels

The stiffness of the extracellular matrix can profoundly influence cell and tissue behaviors. Thus there is an emerging emphasis on understanding how matrix mechanical environments are established, regulated, and modified. Here we develop a microrheometric assay to measure the mechanical properties of a model extracellular matrix (type I collagen gel) and use it to explore cytokine-induced, cell-mediated changes in matrix mechanical properties. The microrheometric assay uses micron-scale ferrimagnetic beads embedded within collagen gels during fibrillogenesis. The beads are magnetized, then subjected to a twisting field, with the aggregate rotation of the beads measured by a magnetometer. The degree of bead rotation reflects the stiffness of the surrounding matrix. We show that the microscale assay provides stiffness measures for collagen gels comparable to those obtained with standard macroscale rheometry. To demonstrate the utility of the assay for biological discovery, we measure stiffness changes in fibroblast-populated collagen gels exposed to three concentrations of six cytokines over 2 to 14 days. Among the cytokines tested, transforming growth factor-beta1 and interleukin-1beta enhanced matrix stiffness, and together exerted cooperative effects on cellular modulation of matrix mechanics. The microrheometry approach developed here should accelerate the discovery of biological pathways orchestrating cellular modulation of matrix mechanics.

Fiber kinematics of small intestinal submucosa under biaxial and uniaxial stretch

Improving our understanding of the design requirements of biologically derived collagenous scaffolds is necessary for their effective use in tissue reconstruction. In the present study, the collagen fiber kinematics of small intestinal submucosa (SIS) was quantified using small angle light scattering (SALS) while the specimen was subjected to prescribed uniaxial or biaxial strain paths. A modified biaxial stretching device based on Billiar and Sacks (J. Biomech., 30, pp. 753-7, 1997) was used, with a real-time analysis of the fiber kinematics made possible due to the natural translucency of SIS. Results indicated that the angular distribution of collagen fibers in specimens subjected to 10% equibiaxial strain was not significantly different from the initial unloaded condition, regardless of the loading path (p=0.31). Both 10% strip biaxial stretch and uniaxial stretches of greater than 5% in the preferred fiber direction led to an increase in the collagen fiber alignment along the same direction, while 10% strip biaxial stretch in the cross preferred fiber direction led to a broadening of the distribution. While an affine deformation model accurately predicted the experimental findings for a biaxial strain state, uniaxial stretch paths were not accurately predicted. Nonaffine structural models will be necessary to fully predict the fiber kinematics under large uniaxial strains in SIS.

Computational modeling of morphogenesis regulated by mechanical feedback

Mechanical forces cause changes in form during embryogenesis and likely play a role in regulating these changes. This paper explores the idea that changes in homeostatic tissue stress (target stress), possibly modulated by genes, drive some morphogenetic processes. Computational models are presented to illustrate how regional variations in target stress can cause a range of complex behaviors involving the bending of epithelia. These models include growth and cytoskeletal contraction regulated by stress-based mechanical feedback. All simulations were carried out using the commercial finite element code ABAQUS, with growth and contraction included by modifying the zero-stress state in the material constitutive relations. Results presented for bending of bilayered beams and invagination of cylindrical and spherical shells provide insight into some of the mechanical aspects that must be considered in studying morphogenetic mechanisms.

Effects of extracellular fiber architecture on cell membrane shear stress in a 3D fibrous matrix

Interstitial fluid flow has been shown to affect the organization and behavior of cells in 3D environments in vivo and in vitro, yet the forces driving such responses are not clear. Due to the complex architecture of the extracellular matrix (ECM) and the difficulty of measuring fluid flow near cells embedded in it, the levels of shear stress experienced by cells in this environment are typically estimated using bulk-averaged matrix parameters such as hydraulic permeability. While this is useful for estimating average stresses, it cannot yield insight into how local matrix fiber architecture-which is cell-controlled in the immediate pericellular environment-affects the local stresses imposed on the cell surface. To address this, we used computational fluid dynamics to study flow through an idealized mesh constructed of a cubic lattice of fibers simulating a typical in vitro collagen gel. We found that, in such high porosity matrices, the fibers strongly affect the flow fields near the cell, with peak shear stresses up to five times higher than those predicted by the Brinkman equation. We also found that minor remodeling of the fibers near the cell surface had major effects on the shear stress profile on the cell. These findings demonstrate the importance of fiber architecture to the fluid forces on a cell embedded in a 3D matrix, and also show how small modifications in the local ECM can lead to large changes in the mechanical environment of the cell.

Deterministic Material-Based Averaging Theory Model of Collagen Gel Micromechanics

Mechanics of collagen gels, like that of many tissues, is governed by events occurring on a length scale much smaller than the functional scale of the material. To deal with the challenge of incorporating deterministic micromechanics into a continuous macroscopic model, we have developed an averaging-theory-based modeling framework for collagen gels. The averaging volume, which is constructed around each integration point in a macroscopic finite-element model, is assumed to experience boundary deformations homogeneous with the macroscopic deformation field, and a micromechanical problem is solved to determine the average stress at the integration point. A two-dimensional version was implemented with the microstructure modeled as a network of nonlinear springs, and 500 segments were found to be sufficient to achieve statistical homogeneity. The method was then used to simulate the experiments of Tower et al. (Ann. Biomed. Eng., 30, pp. 1221–1233) who performed uniaxial extension of prealigned collagen gels. The simulation captured many qualitative features of the experiments, including a toe region and the realignment of the fibril network during extension. Finally, the method was applied to an idealized wound model based on the characterization measurements of Bowes et al. (Wound Repair Regen., 7, pp. 179–186). The model consisted of a strongly aligned “wound” region surrounded by a less strongly aligned “healthy” region. The alignment of the fibrils in the wound region led to reduced axial strains, and the alignment of the fibrils in the healthy region, combined with the greater effective stiffness of the wound region, caused rotation of the wound region during uniaxial stretch. Although the microscopic model in this study was relatively crude, the multiscale framework is general and could be employed in conjunction with any microstructural model.

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.

Affine versus non-affine fibril kinematics in collagen networks: theoretical studies of network behavior

The microstructure of tissues and tissue equivalents (TEs) plays a critical role in determining the mechanical properties thereof. One of the key challenges in constitutive modeling of TEs is incorporating the kinematics at both the macroscopic and the microscopic scale. Models of fibrous microstructure commonly assume fibrils to move homogeneously, that is affine with the macroscopic deformation. While intuitive for situations of fibril-matrix load transfer, the relevance of the affine assumption is less clear when primary load transfer is from fibril to fibril. The microstructure of TEs is a hydrated network of collagen fibrils, making its microstructural kinematics an open question. Numerical simulation of uniaxial extensile behavior in planar TE networks was performed with fibril kinematics dictated by the network model and by the affine model. The average fibril orientation evolved similarly with strain for both models. The individual fibril kinematics, however, were markedly different. There was no correlation between fibril strain and orientation in the network model, and fibril strains were contained by extensive reorientation. As a result, the macroscopic stress given by the network model was roughly threefold lower than the affine model. Also, the network model showed a toe region, where fibril reorientation precluded the development of significant fibril strain. We conclude that network fibril kinematics are not governed by affine principles, an important consideration in the understanding of tissue and TE mechanics, especially when load bearing is primarily by an interconnected fibril network.

Modeling cell and matrix anisotropy in fibroblast populated collagen vessels

Microstructurally based models for bio-artificial tissues are needed to predict in vivo mechanical behavior and to validate assumptions for models of biologic tissues. We develop a microstructural model, based on on Zahalak et al. (2000) [Biophys 79(5):2369-2381], to describe matrix and tissue anisotropy observed in recent biaxial tests of fibroblast populated collagen vessels (FPCVs) with different cell orientations (Wagenseil et al. in Ann Biomed Eng 32(5):720-731 2004). The model includes pseudo-elastic cell behavior and pseudo-elastic, non-linear matrix behavior with recruitment of initially buckled collagen fibers. We obtained estimates of collagen matrix parameters from measurements of FPCVs treated with 2x 10(-6) M Cytochalasin D and used these estimates to determine cell parameters in FPCVs activated with 5% fetal calf serum. The estimated stiffness of individual fibroblasts was 41-1,165 kPa. Parameter estimates for both cell and matrix were influenced by the non-linearity of the biaxial test data, making it difficult to obtain unique parameter values for some experiments. Additional microstructural measurements of the collagen matrix may help to more precisely determine the relative contributions of cells and matrix.

Mechanobiology in the third dimension

Cells are mechanically coupled to their extracellular environments, which play critical roles in both communicating the state of the mechanical environment to the cell as well as in mediating cellular response to a variety of stimuli. Along with the molecular composition and mechanical properties of the extracellular matrix (ECM), recent work has demonstrated the importance of dimensionality in cell-ECM associations for controlling the sensitive communication between cells and the ECM. Matrix forces are generally transmitted to cells differently when the cells are on two-dimensional (2D) vs. within three-dimensional (3D) matrices, and cells in 3D environments may experience mechanical signaling that is unique vis-à-vis cells in 2D environments, such as the recently described 3D-matrix adhesion assemblies. This review examines how the dimensionality of the extracellular environment can affect in vitro cell mechanobiology, focusing on collagen and fibrin systems.

Degeneration affects the fiber reorientation of human annulus fibrosus under tensile load

The angled, lamellar structure of the annulus fibrosus is integral to its load-bearing function. Reorientation of this fiber structure with applied load may contribute to nonlinear mechanical behavior and to large increases in tensile modulus. Fiber reorientation has not yet been quantified for loaded non-degenerated and degenerated annulus fibrosus tissue. The objective of this study was to measure fiber reorientation and mechanical properties (toe- and linear-region modulus, transition strain, and Poisson's ratio) of loaded outer annulus fibrosus tissue using a new application of FFT image processing techniques. This method was validated for quantification of annulus fiber reorientation during loading in this study. We hypothesized that annulus fibrosus fibers would reorient under circumferential tensile load, and that fiber reorientation would be affine. Additionally, we hypothesized that degeneration would affect fiber reorientation, toe-region modulus and Poisson's ratio. Annulus fibrosus fibers were found to reorient toward the loading direction, and degeneration significantly decreased fiber reorientation (the fiber reorientation parameter, m(FFT)=-1.70 degrees /% strain for non-degenerated and -0.95 degrees /% strain for degenerated tissue). Toe-region modulus was significantly correlated with age (r=0.6). Paired t-tests showed no significant difference in the fiber reorientation parameter calculated experimentally with that calculated using an affine prediction. Thus, an affine prediction is a good approximation of fiber reorientation. The findings of this study add to the understanding of overall disc mechanical behavior and degeneration.

Local, Three-Dimensional Strain Measurements Within Largely Deformed Extracellular Matrix Constructs

The ability to create extracellular matrix (ECM) constructs that are mechanically and biochemically similar to those found in vivo and to understand how their properties affect cellular responses will drive the next generation of tissue engineering strategies. To date, many mechanisms by which cells biochemically communicate with the ECM are known. However, the mechanisms by which mechanical information is transmitted between cells and their ECM remain to be elucidated. “Self-assembled” collagen matrices provide an in vitro-model system to study the mechanical behavior of ECM. To begin to understand how the ECM and the cells interact mechanically, the three-dimensional (3D) mechanical properties of the ECM must be quantified at the micro-(local) level in addition to information measured at the macro-(global) level. Here we describe an incremental digital volume correlation (IDVC) algorithm to quantify large (>0.05) 3D mechanical strains in the microstructure of 3D collagen matrices in response to applied mechanical loads. Strain measurements from the IDVC algorithm rely on 3D confocal images acquired from collagen matrices under applied mechanical loads. The accuracy and the precision of the IDVC algorithm was verified by comparing both image volumes collected in succession when no deformation was applied to the ECM (zero strain) and image volumes to which simulated deformations were applied in both 1D and 3D (simulated strains). Results indicate that the IDVC algorithm can accurately and precisely determine the 3D strain state inside largely deformed collagen ECMs. Finally, the usefulness of the algorithm was demonstrated by measuring the microlevel 3D strain response of a collagen ECM loaded in tension.

Microstructural mechanics of collagen gels in confined compression: poroelasticity, viscoelasticity, and collapse

BACKGROUND

Collagen gels are important as platforms for in vitro study of cell behavior and as prototypical bioartificial tissues, but their mechanical behavior, particularly on the microscopic scale, is still poorly understood.

METHOD OF APPROACH

Collagen gels were studied in step (10% strain in 0.05 s) and ramp (0.1%/s strain rate for 100 s) confined compression. Real-time birefringence mapping gave the local collagen concentration and orientation along with piston stress. Variations in the retardation allowed material-point tracking and qualitative determination of the strain distribution.

RESULTS

Ramp tests showed classical poroelastic behavior: compression near the piston and relaxation to a uniform state. Step tests, however, showed an irreversibly collapsed region near the piston.

CONCLUSIONS

Our results suggest that interstitial flow and fibril bending at crosslinks are the dominant mechanical processes during compression, and that fibril bending is reversible before collapse.