A Constitutive Model of Soft Tissue: From Nanoscale Collagen to Tissue Continuum

Modeling the mechanosensitive collective migration of cells on the surface and the interior of morphing soft tissues

Cellular contractility, migration, and extracellular matrix (ECM) mechanics are critical for a wide range of biological processes including embryonic development, wound healing, tissue morphogenesis, and regeneration. Even though the distinct response of cells near the tissue periphery has been previously observed in cell-laden microtissues, including faster kinetics and more prominent cell-ECM interactions, there are currently no models that can fully combine coupled surface and bulk mechanics and kinetics to recapitulate the morphogenic response of these constructs. Mailand et al. (Biophys J 117(5):975-986, 2019) had shown the importance of active elastocapillarity in cell-laden microtissues, but modeling the distinct mechanosensitive migration of cells on the periphery and the interior of highly deforming tissues has not been possible thus far, especially in the presence of active elastocapillary effects. This paper presents a framework for understanding the interplay between cellular contractility, migration, and ECM mechanics in dynamically morphing soft tissues accounting for distinct cellular responses in the bulk and the surface of tissues. The major novelty of this approach is that it enables modeling the distinct migratory and contractile response of cells residing on the tissue surface and the bulk, where concurrently the morphing soft tissues undergo large deformations driven by cell contractility. Additionally, the simulation results capture the changes in shape and cell concentration for wounded and intact microtissues, enabling the interpretation of experimental data. The numerical procedure that accounts for mechanosensitive stress generation, large deformations, diffusive migration in the bulk and a distinct mechanism for diffusive migration on deforming surfaces is inspired from recent work on bulk and surface poroelasticity of hydrogels involving elastocapillary effects, but in this work, a two-field weak form is proposed and is able to alleviate numerical instabilities that were observed in the original method that utilized a three-field mixed finite element formulation.

Multiscale modeling of unfolding and bond dissociation of rubredoxin metalloprotein

Mechanical properties of proteins that have a crucial effect on their operation. This study used a molecular dynamics simulation package to investigate rubredoxin unfolding on the atomic scale. Different simulation techniques were applied, and due to the dissociation of covalent/hydrogen bonds, this protein demonstrates several intermediate states in force-extension behavior. A conceptual model based on the cohesive finite element method was developed to consider the intermediate damages that occur during unfolding. This model is based on force-displacement curves derived from molecular dynamics results. The proposed conceptual model is designed to accurately identify bond rupture points and determine the associated forces. This is achieved by conducting a thorough comparison between molecular dynamics and cohesive finite element results. The utilization of a viscoelastic cohesive zone model allows for the consideration of loading rate effects. This rate-dependent model can be further developed and integrated into the multiscale modeling of large assemblies of metalloproteins, providing a comprehensive understanding of mechanical behavior while maintaining a reduced computational cost.

Review paper: The importance of consideration of collagen cross-links in computational models of collagen-based tissues

Collagen as the main protein in Extra Cellular Matrix (ECM) is the main load-bearing component of fibrous tissues. Nanostructure and architecture of collagen fibrils play an important role in mechanical behavior of these tissues. Extensive experimental and theoretical studies have so far been performed to capture these properties, but none of the current models realistically represent the complexity of network mechanics because still less is known about the collagen's inner structure and its effect on the mechanical properties of tissues. The goal of this review article is to emphasize the significance of cross-links in computational modeling of different collagen-based tissues, and to reveal the need for continuum models to consider cross-links properties to better reflect the mechanical behavior observed in experiments. In addition, this study outlines the limitations of current investigations and provides potential suggestions for the future work.

A multiscale investigation into the role of collagen-hyaluronan interface shear on the mechanical behaviour of collagen fibers in annulus fibrosus - Molecular dynamics-cohesive finite element-based study

Multi-directional deformation exhibited by annulus fibrosus (AF) is contributed by chemo-mechanical interactions among its biomolecular constituents' collagen type I (COL-I), collagen type II (COL-II), proteoglycans (aggrecan and hyaluronan) and water. However, the nature and role of such interactions on AF mechanics are unclear. This work employs a molecular dynamics-cohesive finite element-based multiscale approach to investigate role of COL-I-COL-II interchanging distribution and water concentration (WC) variations from outer annulus (OA) to inner annulus (IA) on collagen-hyaluronan (COL-HYL) interface shear, and the mechanisms by which interface shear impacts fibril sliding during collagen fiber deformation. At first, COL-HYL interface atomistic models are constructed by interchanging COL-I with COL-II and increasing COL-II and WC from 0 to 75%, and 65%-75% respectively. Thereafter, a multiscale approach is employed to develop representative volume elements (RVEs) of collagen fibers by incorporating COL-HYL shear as traction-separation behaviour at fibril-hyaluronan contact. Results show that increasing COL-II and WC increases interface stiffness from 0.6 GPa/nm to 1.2 GPa/nm and reduces interface strength from 155 MPa to 58 MPa from OA to IA, contributed by local hydration alterations. A stiffer and weaker interface enhances fibril sliding with increased straining at the contact - thereby contributing to reduction in modulus from 298 MPa to 198 MPa from OA to IA. Such reduction further contributes to softer mechanical response towards IA, as reported by earlier studies. Presented multiscale analysis provides deeper understanding of hierarchical structure-mechanics relationships in AF and can further aid in developing better substitutes for AF repair.

Knee joint biomechanics and cartilage damage prediction during landing: A hybrid MD-FE-musculoskeletal modeling

Understanding the mechanics behind knee joint injuries and providing appropriate treatment is crucial for improving physical function, quality of life, and employability. In this study, we used a hybrid molecular dynamics-finite element-musculoskeletal model to determine the level of loads the knee can withstand when landing from different heights (20, 40, 60 cm), including the height at which cartilage damage occurs. The model was driven by kinematics-kinetics data of asymptomatic subjects at the peak loading instance of drop landing. Our analysis revealed that as landing height increased, the forces on the knee joint also increased, particularly in the vastus muscles and medial gastrocnemius. The patellar tendon experienced more stress than other ligaments, and the medial plateau supported most of the tibial cartilage contact forces and stresses. The load was mostly transmitted through cartilage-cartilage interaction and increased with landing height. The critical height of 126 cm, at which cartilage damage was initiated, was determined by extrapolating the collected data using an iterative approach. Damage initiation and propagation were mainly located in the superficial layers of the tibiofemoral and patellofemoral cartilage. Finally, this study provides valuable insights into the mechanisms of landing-associated cartilage damage and could help limit joint injuries and improve training programs.

Modeling and design of heterogeneous hierarchical bioinspired spider web structures using deep learning and additive manufacturing

Spider webs are incredible biological structures, comprising thin but strong silk filament and arranged into complex hierarchical architectures with striking mechanical properties (e.g., lightweight but high strength, achieving diverse mechanical responses). While simple 2D orb webs can easily be mimicked, the modeling and synthesis of 3D-based web structures remain challenging, partly due to the rich set of design features. Here, we provide a detailed analysis of the heterogeneous graph structures of spider webs and use deep learning as a way to model and then synthesize artificial, bioinspired 3D web structures. The generative models are conditioned based on key geometric parameters (including average edge length, number of nodes, average node degree, and others). To identify graph construction principles, we use inductive representation sampling of large experimentally determined spider web graphs, to yield a dataset that is used to train three conditional generative models: 1) an analog diffusion model inspired by nonequilibrium thermodynamics, with sparse neighbor representation; 2) a discrete diffusion model with full neighbor representation; and 3) an autoregressive transformer architecture with full neighbor representation. All three models are scalable, produce complex, de novo bioinspired spider web mimics, and successfully construct graphs that meet the design objectives. We further propose an algorithm that assembles web samples produced by the generative models into larger-scale structures based on a series of geometric design targets, including helical and parametric shapes, mimicking, and extending natural design principles toward integration with diverging engineering objectives. Several webs are manufactured using 3D printing and tested to assess mechanical properties.

Effect of Surgical Design Variations on the Knee Contact Behavior during Anterior Cruciate Ligament Reconstruction

In this study, we aimed to develop an in-silico synthesis of the effect of critical surgical design parameters on articular contact behavior for a bone-patellar-tendon-bone anterior cruciate ligament reconstruction (ACL-R) surgery. A previously developed finite element model of the knee joint consisting of all relevant soft tissues was employed. The knee model was further updated with additional features to develop the parametric FE model of the biomechanical experiments that depicted the ACL-R surgery. The parametricity was created involving femoral tunnel architecture (orientations and locations) and graft fixation characteristics (pretension and angle of fixation). A global sensitivity analysis based on variance decomposition was used to investigate the contribution of the surgical parameters to the uncertainty in response to the ACL-R joint. Our examinations indicated that the total contact force was primarily influenced by either combined or individual action of the graft pretension and fixation angle, with a modest contribution of the graft insertion sites. The joint contact center and area were affected mainly by the angle of fixation and the tunnel placements. Graft pretension played the dominant role in the maximum contact pressure variability, an observation that has been well-documented in the literature. Interestingly, the joint contact behavior was almost insensitive to the tunnel's coronal and sagittal orientations. Our data provide an evaluation of how the surgical parameters affect the knee joint's contact behavior after ACL-R and may provide additional information to better explain the occurrence of osteoarthritis as an aftermath of such surgery.

Surrogate modeling of articular cartilage degradation to understand the synergistic role of MMP-1 and MMP-9: a case study

A characteristic feature of arthritic diseases is cartilage extracellular matrix (ECM) degradation, often orchestrated by the overexpression of matrix metalloproteinases (MMPs) and other proteases. The interplay between fibril level degradation and the tissue-level aggregate response to biomechanical loading was explored in this work by a computational multiscale cartilaginous model. We considered the relative abundance of collagenases (MMP-1) and gelatinases (MMP-9) in surrogate models, where the diffusion (spatial distribution) of these enzymes and the subsequent, co-localized fibrillar damage were spatially randomized with Latin Hypercube Sampling. The computational model was constructed by incorporating the results from prior molecular dynamics simulations (tensile test) of microfibril degradation into a hyper-elastoplastic fibril-reinforced cartilage model. Including MMPs-mediated collagen fibril-level degradation in computational models may help understand the ECM pathomechanics at the tissue level. The mechanics of cartilage tissue and fibril show variations in mechanical integrity depending on the different combinations of MMPs-1 and 9 with a concentration ratio of 1:1, 3:1, and 1:3 in simulated indentation tests. The fibril yield (local failure) was initiated at 20.2 ± 3.0 (%) and at 23.0 ± 2.8 (%) of bulk strain for col 1:gel 3 and col 3: gel 1, respectively. The reduction in failure stress (global response) was 39.8% for col 1:gel 3, 37.5% for col 1:gel 1, and 36.7% for col 3:gel 1 compared with the failure stress of the degradation free tissue. These findings indicate that cartilage's global and local mechanisms of failure largely depend on the relative abundance of the two key enzymes-collagenase (MMP-1) and gelatinase (MMP-9) and the spatial characteristics of diffusion across the layers of the cartilage ECM.

Multiscale modeling of skin mechanical Behavior: Effect of dehydrating agent on Collagen's mechanical properties

The dermis, second layer of human skin, is mainly responsible for mechanical response of the skin. The unique viscoelastic nature of this layer arises from the characteristic hierarchical structure of collagen at various length scales. The effect of topical formulation on skin's mechanical properties of great importance for several personal-care applications. Understanding the transport of an active ingredient across skin layer and its effects on the structure of collagen assembly is crucial for successful design of these applications. In this study, we report a multiscale modelling framework mimicking the skin's mechanical behavior. The framework captures the details from the nanoscale (tropocollagen) to microscale (fibers). At first, atomistic molecular dynamics simulations (MDS) of tropocollagen (TC) molecules of various lengths (∼100 nm) were performed to obtain the molecular modulus of TC. The stress-strain response data obtained from these simulations, were utilized in macroscopic models of fibrils and fibers. The modulus obtained from the mentioned framework was in good agreement with earlier reported experimental data. Further, we have utilized this framework to show the effect of dehydrating agent on skin's mechanical response. The hydration effect is utilized in many anti-ageing strategies to improve the overall mechanical property of skin. We showed that on incorporation of hydrating agent, the collagen structure changes significantly at molecular scale which effects the overall response of the skin at macroscopic scale. The reported multiscale framework can further be explored to gain insights into interlinked properties of collagen at much larger scales without extensive molecular simulations and detailed experiments.

Understanding the inelastic response of collagen fibrils: A viscoelastic-plastic constitutive model

Collagen fibrils, which are the lowest level fibrillar unit of organization of collagen, are thus of primary interest towards understanding the mechanical behavior of load-bearing soft tissues. The deformation of collagen fibrils shows unique mechanical features; namely, their high energy dissipation is even superior compared to most engineering materials. Additionally, there are indications that cyclic loading can further improve the toughness of collagen fibrils. Recent experiments from Liu at al. (2018) focused on the response of type I collagen fibrils to uniaxial cyclic loading, revealing some interesting results regarding their rate-dependent and inelastic response. In this work, we aim to develop a model that allows interpreting the complex nonlinear and inelastic response of collagen fibrils under cyclic loading. We propose a constitutive model that accounts for viscoelastic deformations through a decoupled strain-energy density function (into an elastic and a viscous parts), and for plastic deformations through plastic evolution laws. The stress-stretch response results obtained using this constitutive law showed good agreement with experimental data over complex loading paths. Ultimately we use the model to gain more insights on how cyclic loading and rate effects control the interplay between viscoelastic and plastic deformation in collagen fibrils, and to extrapolate the results from experimental data, analyzing how complex cyclic load influences energy dissipation and deformation mechanisms. STATEMENT OF SIGNIFICANCE: In this work, we develop a viscoelastic-plastic constitutive model for collagen fibrils with the aim of analyzing the effects of inelasticity and energy dissipation in this material, and more specifically the competition between viscoelasticity and plasticity in the context of cyclic loading and overload. Experimental and theoretical approaches so far have not fully clarified the interplay between viscous and plastic deformations during cyclic loading of collagen fibrils. Here, we aim to interpret the complex nonlinear response of collagen fibrils and, ultimately, suggest predictive capabilities that can inform tissue-level response and injury. To validate our model, we compare our results against the stress-stretch data obtained from experiments of cyclic loaded single fibrils performed by Liu et al. (2018).

A model for 3D deformation and reconstruction of contractile microtissues

Tissue morphogenesis and regeneration are essentially mechanical processes that involve coordination of cellular forces, production and structural remodeling of extracellular matrix (ECM), and cell migration. Discovering the principles of cell-ECM interactions and tissue-scale deformation in mechanically-loaded tissues is instrumental to the development of novel regenerative therapies. The combination of high-throughput three-dimensional (3D) culture systems and experimentally-validated computational models accelerate the study of these principles. In our previous work [E. Mailand, et al., Biophys. J., 2019, 117, 975-986], we showed that prominent surface stresses emerge in constrained fibroblast-populated collagen gels, driving the morphogenesis of fibrous microtissues. Here, we introduce an active material model that allows the embodiment of surface and bulk contractile stresses while maintaining the passive elasticity of the ECM in a 3D setting. Unlike existing models, the stresses are driven by mechanosensing and not by an externally applied signal. The mechanosensing component is incorporated in the model through a direct coupling of the local deformation state with the associated contractile force generation. Further, we propose a finite element implementation to account for large deformations, nonlinear active material response, and surface effects. Simulation results quantitatively capture complex shape changes during tissue formation and as a response to surgical disruption of tissue boundaries, allowing precise calibration of the parameters of the 3D model. The results of this study imply that the organization of the extracellular matrix in the bulk of the tissue may not be a major factor behind the morphogenesis of fibrous tissues at sub-millimeter length scales.

The Effect of Enzymatic Crosslink Degradation on the Mechanics of the Anterior Cruciate Ligament: A Hybrid Multi-Domain Model

The anterior cruciate ligament’s (ACL) mechanics is an important factor governing the ligament’s integrity and, hence, the knee joint’s response. Despite many investigations in this area, the cause and effect of injuries remain unclear or unknown. This may be due to the complexity of the direct link between macro- and micro-scale damage mechanisms. In the first part of this investigation, a three-dimensional coarse-grained model of collagen fibril (type I) was developed using a bottom-up approach to investigate deformation mechanisms under tensile testing. The output of this molecular level was used later to calibrate the parameters of a hierarchical multi-scale fibril-reinforced hyper-elastoplastic model of the ACL. Our model enabled us to determine the mechanical behavior of the ACL as a function of the basic response of the collagen molecules. Modeled elastic response and damage distribution were in good agreement with the reported measurements and computational investigations. Our results suggest that degradation of crosslink content dictates the loss of the stiffness of the fibrils and, hence, damage to the ACL. Therefore, the proposed computational frame is a promising tool that will allow new insights into the biomechanics of the ACL.

Constitutive modeling using structural information on collagen fiber direction and dispersion in human superficial femoral artery specimens of different ages

Arterial mechanics plays an important role in vascular pathophysiology and repair, and advanced imaging can inform constitutive models of vascular behavior. We have measured the mechanical properties of 14 human superficial femoral arteries (SFAs) (age 12-70, mean 48±19 years) using planar biaxial extension, and determined the preferred collagen fiber direction and dispersion using multiphoton microscopy. The collagen fiber direction and dispersion were evaluated using second-harmonic generation imaging and modeled using bivariate von Mises distributions. The microstructures of elastin and collagen were assessed using two-photon fluorescence imaging and conventional bidirectional histology. The mechanical and structural data were used to describe the SFA mechanical behavior using two- and four-fiber family invariant-based constitutive models. Older SFAs were stiffer and mechanically more nonlinear than younger specimens. In the adventitia, collagen fibers were undulated and diagonally-oriented, while in the media, they were straight and circumferentially-oriented. The media was rich in collagen that surrounded the circumferentially-oriented smooth muscle cells, and the elastin was present primarily in the internal and external elastic laminae. Older SFAs had a more circumferential collagen fiber alignment, a decreased circumferential-radial fiber dispersion, but the same circumferential-longitudinal fiber dispersion as younger specimens. Both the two- and the four-fiber family constitutive models were able to capture the experimental data, and the fits were better for the four-fiber family formulation. Our data provide additional details on the SFA intramural structure and inform structurally-based constitutive models.

Multiscale modeling of knee ligament biomechanics

Knee connective tissues are mainly responsible for joint stability and play a crucial role in restraining excessive motion during regular activities. The damage mechanism of these tissues is directly linked to the microscale collagen level. However, this mechanical connection is still unclear. During this investigation, a multiscale fibril-reinforced hyper-elastoplastic model was developed and statistically calibrated. The model is accounting for the structural architecture of the soft tissue, starting from the tropocollagen molecule that forms fibrils to the whole soft tissue. Model predictions are in agreement with the results of experimental and numerical studies. Further, damage initiation and propagation in the collagen fiber were computed at knee ligaments and located mainly in the superficial layers. Results indicated higher crosslink density required higher tensile stress to elicit fibril damage. This approach is aligned with a realistic simulation of a damaging process and repair attempt. To the best of our knowledge, this is the first model published in which the connective tissue stiffness is simultaneously predicted by encompassing the mesoscopic scales between the molecular and macroscopic levels.

Computational frame of ligament in situ strain in a full knee model

The biomechanical function of connective tissues in a knee joint is to stabilize the kinematics-kinetics of the joint by augmenting its stiffness and limiting excessive coupled motion. The connective tissues are characterized by an in vivo reference configuration (in situ strain) that would significantly contribute to the mechanical response of the knee joint. In this work, a novel iterative method for computing the in situ strain at reference configuration was presented. The framework used an in situ strain gradient approach (deformed reference configuration) and a detailed finite element (FE) model of the knee joint. The effect of the predicted initial configuration on the mechanical response of the joint was then investigated under joint axial compression, passive flexion, and coupled rotations (adduction and internal), and during the stance phase of gait. The inclusion of the reference configuration has a minimal effect on the knee joint mechanics under axial compression, passive flexion, and at two instances (0% and 50%) of the stance phase of gait. However, the presence of the ligaments in situ strains significantly increased the joint stiffness under passive adduction and internal rotations, as well as during the other simulated instances (25%, 75% and 100%) of the stance phase of gait. Also, these parameters substantially altered the local loading state of the ligaments and resulted in better agreement with the literature during joint flexion. Therefore, the proposed computational framework of ligament in situ strain will help to overcome the challenges in considering this crucial biological aspect during knee joint modeling. Besides, the current construct is advantageous for a better understanding of the mechanical behavior of knee ligaments under physiological and pathological states and provide relevant information in the design of reconstructive treatments and artificial grafts.

Implementing a micromechanical model into a finite element code to simulate the mechanical and microstructural response of arteries

A computational strategy based on the finite element method for simulating the mechanical response of arterial tissues is herein proposed. The adopted constitutive formulation accounts for rotations of the adventitial collagen fibers and introduces parameters which are directly measurable or well established. Moreover, the refined constitutive model is readily utilized in finite element analyses, enabling the simulation of mechanical tests to reveal the influence of microstructural and histological features on macroscopic material behavior. Employing constitutive parameters supported by histological examinations, the results herein validate the model's ability to predict the micro- and macroscopic mechanical behavior, closely matching previously observed experimental findings. Finally, the capabilities of the adopted constitutive description are shown investigating the influence of some collagen disorders on the macroscopic mechanical response of the arterial tissues.

Microstructural modeling of Achilles Tendon biomechanics focusing on bone insertion site

The interface between the Achilles Tendon (AT) and calcaneus comprises soft and hard connective tissues. Such interfaces are vulnerable to mechanical damage. Tendon to Bone Insertion Region (TBIR) has unique microstructural characteristics for reinforcement. This region constitutes almost 10% of the AT's distal end. The rest of the tendon (tendon proper) has longitudinal fiber orientation with no mineral content. Although, the TBIR lacks longitudinally organized fibers and at the same time, incorporates mineral molecules. In this study, a 3D computational model of the TBIR proposed to underline several reinforcement mechanisms. The obtained results showed that off-axis alignment of fibers, when coupled with the mineral deposition, shifts the stress concentration region to the tendon proper. In the case of altering each parameter individually, probable failure observed in the bone interface, which causes complications in surgical procedure or during healing. A gradual increase of mineral compensates for the stiffness mismatch between the AT and calcaneus. The proposed computational framework illustrated the complementary roles of fiber orientation and mineral molecules: nearly transverse orientation of fibers alleviated the stress concentration locally, while mineral deposition directly enhanced the TBIR stiffness.

Molecular-level collagen damage explains softening and failure of arterial tissues: A quantitative interpretation of CHP data with a novel elasto-damage model

The present experimental-modelling study provides a quantitative interpretation of mechanical data and damage measurements obtained from collagen hybridizing peptide (CHP) techniques on overstretched sheep cerebral arterial tissues. To this aim, a structurally-motivated constitutive model is developed in the framework of continuum damage mechanics. The model includes two internal variables for describing the effects of collagen triple-helical unfolding via interstrand delamination: one governs plastic mechanisms in collagen fibers, leading to a stress softening response of the tissue at the macroscale; the other one describes the loss of fiber structural integrity, leading to tissue final failure. The proposed model is calibrated using the obtained mechanical experimental data, showing excellent fitting capabilities. The predicted evolution of internal variables agree well with independent measurements of molecular-level CHP-based damage data, obtaining an independent a posteriori validation of damage predictions. Moreover, available data on inelastic tissue elongation following supraphysiological loads are successfully reproduced. These outcomes further the hypothesis that the accumulation of interstrand delamination is a primary cause for the evolution of inelastic mechanisms in tissues, and in particular of stress softening up to failure.

Multiscale mechanical effects of native collagen cross-linking in tendon

The hierarchical structure of tendon allows for attenuation of mechanical strain down decreasing length scales. While reorganization of collagen fibers accounts for microscale strain attenuation, cross-linking between collagen molecules contributes to deformation mechanisms at the fibrillar and molecular scales. Divalent and trivalent enzymatic cross-links form during the development of collagen fibrils through the enzymatic activity of lysyl oxidase (LOX). By establishing connections between telopeptidyl and triple-helical domains of adjacent molecules within collagen fibrils, these cross-links stiffen the fibrils by resisting intermolecular sliding. Ultimately, greater enzymatic cross-linking leads to less compliant and stronger tendon as a result of stiffer fibrils. In contrast, nonenzymatic cross-links such as glucosepane and pentosidine are not produced during development but slowly accumulate through glycation of collagen. Therefore, these cross-links are only expected to be present in significant quantities in advanced age, where there has been sufficient time for glycation to occur, and in diabetes, where the presence of more free sugar in the extracellular matrix increases the rate of glycation. Unlike enzymatic cross-links, current evidence suggests that nonenzymatic cross-links are at least partially isolated to the surface of collagen fibers. As a result, glycation has been proposed to primarily impact tendon mechanics by altering molecular interactions at the fiber interface, thereby diminishing sliding between fibers. Thus, increased nonenzymatic cross-linking decreases microscale strain attenuation and the viscous response of tendon. In conclusion, enzymatic and nonenzymatic collagen cross-links have demonstrable and distinct effects on the mechanical properties of tendon across different length scales.

Collagenous Extracellular Matrix Biomaterials for Tissue Engineering: Lessons from the Common Sea Urchin Tissue

Scaffolds for tissue engineering application may be made from a collagenous extracellular matrix (ECM) of connective tissues because the ECM can mimic the functions of the target tissue. The primary sources of collagenous ECM material are calf skin and bone. However, these sources are associated with the risk of having bovine spongiform encephalopathy or transmissible spongiform encephalopathy. Alternative sources for collagenous ECM materials may be derived from livestock, e.g., pigs, and from marine animals, e.g., sea urchins. Collagenous ECM of the sea urchin possesses structural features and mechanical properties that are similar to those of mammalian ones. However, even more intriguing is that some tissues such as the ligamentous catch apparatus can exhibit mutability, namely rapid reversible changes in the tissue mechanical properties. These tissues are known as mutable collagenous tissues (MCTs). The mutability of these tissues has been the subject of on-going investigations, covering the biochemistry, structural biology and mechanical properties of the collagenous components. Recent studies point to a nerve-control system for regulating the ECM macromolecules that are involved in the sliding action of collagen fibrils in the MCT. This review discusses the key attributes of the structure and function of the ECM of the sea urchin ligaments that are related to the fibril-fibril sliding action-the focus is on the respective components within the hierarchical architecture of the tissue. In this context, structure refers to size, shape and separation distance of the ECM components while function is associated with mechanical properties e.g., strength and stiffness. For simplicity, the components that address the different length scale from the largest to the smallest are as follows: collagen fibres, collagen fibrils, interfibrillar matrix and collagen molecules. Application of recent theories of stress transfer and fracture mechanisms in fibre reinforced composites to a wide variety of collagen reinforcing (non-mutable) connective tissue, has allowed us to draw general conclusions concerning the mechanical response of the MCT at specific mechanical states, namely the stiff and complaint states. The intent of this review is to provide the latest insights, as well as identify technical challenges and opportunities, that may be useful for developing methods for effective mechanical support when adapting decellularised connective tissues from the sea urchin for tissue engineering or for the design of a synthetic analogue.

Functional adaptation of crustacean exoskeletal elements through structural and compositional diversity: a combined experimental and theoretical study

The crustacean cuticle is a composite material that covers the whole animal and forms the continuous exoskeleton. Nano-fibers composed of chitin and protein molecules form most of the organic matrix of the cuticle that, at the macroscale, is organized in up to eight hierarchical levels. At least two of them, the exo- and endocuticle, contain a mineral phase of mainly Mg-calcite, amorphous calcium carbonate and phosphate. The high number of hierarchical levels and the compositional diversity provide a high degree of freedom for varying the physical, in particular mechanical, properties of the material. This makes the cuticle a versatile material ideally suited to form a variety of skeletal elements that are adapted to different functions and the eco-physiological strains of individual species. This review presents our recent analytical, experimental and theoretical studies on the cuticle, summarising at which hierarchical levels structure and composition are modified to achieve the required physical properties. We describe our multi-scale hierarchical modeling approach based on the results from these studies, aiming at systematically predicting the structure-composition-property relations of cuticle composites from the molecular level to the macro-scale. This modeling approach provides a tool to facilitate the development of optimized biomimetic materials within a knowledge-based design approach.

Microstructure-based biomechanics of coronary arteries in health and disease

Coronary atherosclerosis is the major cause of mortality and disability in developed nations. A deeper understanding of mechanical properties of coronary arteries and hence their mechanical response to stress is significant for clinical prevention and treatment. Microstructure-based models of blood vessels can provide predictions of arterial mechanical response at the macro- and micro-mechanical level for each constituent structure. Such models must be based on quantitative data of structural parameters (constituent content, orientation angle and dimension) and mechanical properties of individual adventitia and media layers of normal arteries as well as change of structural and mechanical properties of atherosclerotic arteries. The microstructural constitutive models of healthy coronary arteries consist of three major mechanical components: collagen, elastin, and smooth muscle cells, while the models of atherosclerotic arteries should account for additional constituents including intima, fibrous plaque, lipid, calcification, etc. This review surveys the literature on morphology, mechanical properties, and microstructural constitutive models of normal and atherosclerotic coronary arteries. It also provides an overview of current gaps in knowledge that must be filed in order to advance this important area of research for understanding initiation, progression and clinical treatment of vascular disease. Patient-specific structural models are highlighted to provide diagnosis, virtual planning of therapy and prognosis when realistic patient-specific geometries and material properties of diseased vessels can be acquired by advanced imaging techniques.

A multi-scale elasto-plastic model of articular cartilage

Collagen damage is one of the earliest signs of cartilage degeneration and the onset of osteoarthritis (OA), but the connection between the microscale damage and macroscale tissue function is unclear. We argue that a multiscale model can help elucidate the biochemical and mechanical underpinnings of OA by connecting the microscale defects in collagen fibrils to the macroscopic cartilage mechanics. We investigated this connection using a multiscale fibril reinforced hyperelastoplastic (MFRHEP) model that accounts for the structural architecture of the soft tissue, starting from tropocollagen molecules that form fibrils, and moving to the complete soft tissue. This model was driven by reported experimental data from unconfined compression testing of cartilage. The model successfully described the observed transient response of the articular cartilage in unconfined and indentation tests with low and high loading rates. We used this model to understand damage initiation and propagation as a function of the cross-link density between tropocollagen molecules. This approach appeared to provide a realistic simulation of damage when compared with certain published studies. The current construct presents the first attempt to express the aggregate cartilage damage in terms of the cross-link density at the microfibril level.

A validated 3D microstructure-based constitutive model of coronary artery adventitia

A structure-based model that accurately predicts micro- or macromechanical behavior of blood vessels is necessary to understand vascular physiology. Based on recently measured microstructural data, we propose a three-dimensional microstructural model of coronary adventitia that incorporates the elastin and collagen distributions throughout the wall. The role of ground substance was found to be negligible under physiological axial stretch λz = 1.3, based on enzyme degradation of glycosaminoglycans in swine coronary adventitia (n = 5). The thick collagen bundles of outer adventitia (n = 4) were found to be undulated and unengaged at physiological loads, whereas the inner adventitia consisted of multiple sublayers of entangled fibers that bear the majority of load at higher pressures. The microstructural model was validated against biaxial (inflation and extension) experiments of coronary adventitia (n = 5). The model accurately predicted the nonlinear responses of the adventitia, even at high axial force (axial stretch ratio λz = 1.5). The model also enabled a reliable estimation of material parameters of individual fibers that were physically reasonable. A sensitivity analysis was performed to assess the effect of using mean values of the distributions for fiber orientation and waviness as opposed to the full distributions. The simplified mean analysis affects the fiber stress-strain relation, resulting in incorrect estimation of mechanical parameters, which underscores the need for measurements of fiber distribution for a rigorous analysis of fiber mechanics. The validated structure-based model of coronary adventitia provides a deeper understanding of vascular mechanics in health and can be extended to disease conditions.

Investigation of ethanol infiltration into demineralized dentin collagen fibrils using molecular dynamics simulations

The purpose of this study is to investigate the interaction of neat ethanol with bound and non-bound water in completely demineralized dentin that is fully hydrated, using molecular dynamics (MD) simulation method. The key to creating ideal resin-dentin bonds is the removal of residual free water layers and its replacement by ethanol solvent in which resin monomers are soluble, using the ethanol wet-bonding technique. The test null hypotheses were that ethanol cannot remove any collagen-bound water, and that ethanol cannot infiltrate into the spacing between collagen triple helix due to narrow interlayer spacing. Collagen fibrillar structures of overlap and gap regions were constructed by aligning the collagen triple helix of infinite length in hexagonal packing. Three layers of the water molecules were specified as the layers of 0.15-0.22nm, 0.22-0.43nm and 0.43-0.63nm from collagen atoms by investigating the water distribution surrounding collagen molecules. Our simulation results show that ethanol molecules infiltrated into the intermolecular spacing in the gap region, which increased due to the lateral shrinkage of the collagen structures in contact with ethanol solution, while there was no ethanol infiltration observed in the overlap region. Infiltrated ethanol molecules in the gap region removed residual water molecules via modifying mostly the third water layer (50% decrease), which would be considered as a loosely-bound water layer. The first and second hydration layers, which would be considered as tightly bound water layers, were not removed by the ethanol molecules, thus maintaining the helical structures of the collagen molecules.

Modelling approaches for evaluating multiscale tendon mechanics

Tendon exhibits anisotropic, inhomogeneous and viscoelastic mechanical properties that are determined by its complicated hierarchical structure and varying amounts/organization of different tissue constituents. Although extensive research has been conducted to use modelling approaches to interpret tendon structure-function relationships in combination with experimental data, many issues remain unclear (i.e. the role of minor components such as decorin, aggrecan and elastin), and the integration of mechanical analysis across different length scales has not been well applied to explore stress or strain transfer from macro- to microscale. This review outlines mathematical and computational models that have been used to understand tendon mechanics at different scales of the hierarchical organization. Model representations at the molecular, fibril and tissue levels are discussed, including formulations that follow phenomenological and microstructural approaches (which include evaluations of crimp, helical structure and the interaction between collagen fibrils and proteoglycans). Multiscale modelling approaches incorporating tendon features are suggested to be an advantageous methodology to understand further the physiological mechanical response of tendon and corresponding adaptation of properties owing to unique in vivo loading environments.

Crack Propagation Versus Fiber Alignment in Collagen Gels: Experiments and Multiscale Simulation

It is well known that the organization of the fibers constituting a collagenous tissue can affect its failure behavior. Less clear is how that effect can be described computationally so as to predict the failure of a native or engineered tissue under the complex loading conditions that can occur in vivo. Toward the goal of a general predictive strategy, we applied our multiscale model of collagen gel mechanics to the failure of a double-notched gel under tension, comparing the results for aligned and isotropic samples. In both computational and laboratory experiments, we found that the aligned gels were more likely to fail by connecting the two notches than the isotropic gels. For example, when the initial notches were 30% of the sample width (normalized tip-to-edge distance = 0.7), the normalized tip-to-tip distance at which the transition occurred from between-notch failure to across-sample failure shifted from 0.6 to 1.0. When the model predictions for the type of failure event (between the two notches versus across the sample width) were compared to the experimental results, the two were found to be strongly covariant by Fisher's exact test (p < 0.05) for both the aligned and isotropic gels with no fitting parameters. Although the double-notch system is idealized, and the collagen gel system is simpler than a true tissue, it presents a simple model system for studying failure of anisotropic tissues in a controlled setting. The success of the computational model suggests that the multiscale approach, in which the structural complexity is incorporated via changes in the model networks rather than via changes to a constitutive equation, has the potential to predict tissue failure under a wide range of conditions.

Evaluation of Biaxial Mechanical Properties of Aortic Media Based on the Lamellar Microstructure

Evaluation of the mechanical properties of arterial wall components is necessary for establishing a precise mechanical model applicable in various physiological and pathological conditions, such as remodeling. In this contribution, a new approach for the evaluation of the mechanical properties of aortic media accounting for the lamellar structure is proposed. We assumed aortic media to be composed of two sets of concentric layers, namely sheets of elastin (Layer I) and interstitial layers composed of mostly collagen bundles, fine elastic fibers and smooth muscle cells (Layer II). Biaxial mechanical tests were carried out on human thoracic aortic samples, and histological staining was performed to distinguish wall lamellae for determining the dimensions of the layers. A neo-Hookean strain energy function (SEF) for Layer I and a four-parameter exponential SEF for Layer II were allocated. Nonlinear regression was used to find the material parameters of the proposed microstructural model based on experimental data. The non-linear behavior of media layers confirmed the higher contribution of elastic tissue in lower strains and the gradual engagement of collagen fibers. The resulting model determines the nonlinear anisotropic behavior of aortic media through the lamellar microstructure and can be assistive in the study of wall remodeling due to alterations in lamellar structure during pathological conditions and aging.

NUMERICAL STUDY OF BIDIRECTIONAL GLENN WITH UNILATERAL PULMONARY ARTERY STENOSIS

Purpose: Hypoplastic left heart syndrome (HLHS) is a congenital heart disease and is usually associated with pulmonary artery stenosis. The superior vena cava-to-pulmonary artery (bidirectional Glenn) shunt is used primarily as a staging procedure to the total cava-to-pulmonary connection for single-ventricle complex. When HLHS coexists with pulmonary artery stenosis, the surgeons then face a multiple problem. This leads to high demand of optimized structure of Glenn surgery. The objective of this article is to investigate the influence of various anastomotic structures and the direction of superior vena cava (SVC) in Glenn on hemodynamics under pulse inflow conditions and try to find an optimal structure of SVC in Glenn surgery with unilateral pulmonary artery stenosis.

Method: First, 3D patient-specific models were constructed from medical images of a HLHS patient before any surgery by using the commercial software Mimics, and another software Free-form was used to deform the reconstructed models in the computer. Four 3D patient-specific Glenn models were constructed: model-1 (normal Glenn), model-2 (lean the SVC back to the stenotic pulmonary artery), model-3 (lean the SVC towards the stenotic pulmonary artery), model-4 (add patch at junction of the SVC toward stenosis at pulmonary artery). Second, a lumped parameter model (LPM) was established to predict boundary conditions for computational fluid dynamics (CFD). In addition, numerical simulations were conducted using CFD through the finite volume method. Finally, hemodynamic parameters were obtained and evaluated.

Results: It was showed that model-4 have relatively balanced vena cava blood perfusion into the left pulmonary artery (LPA) and right pulmonary artery (RPA), this may be due to less helical flow and the patch at junction of the SVC. Near stenosis of pulmonary artery, model-4 performed with the higher wall shear stress (WSS), which would benefit endothelial cell function and gene expression. In addition, results showed that model-4 performed with the lower oscillatory shear index (OSI) and wall shear stress gradient (WSSG), which would decrease the opportunity of vascular intimal hyperplasia.

Conclusion: It is benefited that surgeons adds patch at junction of the SVC towards stenosis at pulmonary artery. These results can impact the surgical design and planning of the Glenn surgery with unilateral pulmonary artery stenosis.

A coupled fiber-matrix model demonstrates highly inhomogeneous microstructural interactions in soft tissues under tensile load

A soft tissue's macroscopic behavior is largely determined by its microstructural components (often a collagen fiber network surrounded by a nonfibrillar matrix (NFM)). In the present study, a coupled fiber-matrix model was developed to fully quantify the internal stress field within such a tissue and to explore interactions between the collagen fiber network and nonfibrillar matrix (NFM). Voronoi tessellations (representing collagen networks) were embedded in a continuous three-dimensional NFM. Fibers were represented as one-dimensional nonlinear springs and the NFM, meshed via tetrahedra, was modeled as a compressible neo-Hookean solid. Multidimensional finite element modeling was employed in order to couple the two tissue components and uniaxial tension was applied to the composite representative volume element (RVE). In terms of the overall RVE response (average stress, fiber orientation, and Poisson's ratio), the coupled fiber-matrix model yielded results consistent with those obtained using a previously developed parallel model based upon superposition. The detailed stress field in the composite RVE demonstrated the high degree of inhomogeneity in NFM mechanics, which cannot be addressed by a parallel model. Distributions of maximum/minimum principal stresses in the NFM showed a transition from fiber-dominated to matrix-dominated behavior as the matrix shear modulus increased. The matrix-dominated behavior also included a shift in the fiber kinematics toward the affine limit. We conclude that if only gross averaged parameters are of interest, parallel-type models are suitable. If, however, one is concerned with phenomena, such as individual cell-fiber interactions or tissue failure that could be altered by local variations in the stress field, then the detailed model is necessary in spite of its higher computational cost.

Microablation of collagen-based substrates for soft tissue engineering

Noting the abundance and importance of collagen as a biomaterial, we have developed a facile method for the production of a dense fibrillar extracellular matrix mimicking collagen-elastin hybrids with tunable mechanical properties. Through the use of excimer-laser technology, we have optimized conditions for the ablation of collagen lamellae without denaturation of protein, maintenance of fibrillar ultrastructure and preservation of native D-periodicity. Strengths of collagen-elastin hybrids ranged from 0.6 to 13 MPa, elongation at break from 9 to 70% and stiffness from 2.9 to 94 MPa, allowing for the design of a wide variety of tissue specific scaffolds. Further, large (centimeter scale) lamellae can be fabricated and embedded with recombinant elastin to generate collagen-elastin hybrids. Exposed collagen in hybrids act as cell adhesive sites for rat mesenchymal stem cells that conform to ablate waveforms. The ability to modulate these features allows for the generation of a class of biopolymers that can architecturally and physiologically replicate native tissue.

Stress-whitening occurs in demineralized bone

INTRODUCTION

The incidence of age-related bone fracture is increasing with average population age. Bone scatters more light (stress-whitens) during loading, immediately prior to failure, in a manner visually similar to polymer crazing. We wish to understand the stress-whitening process because of its possible effect on bone toughness. The goals of this investigation were a) to establish that stress-whitening is a property of the demineralized organic matrix of bone rather than only a property of mineralized tissue and that stress whitening within the demineralized bone is dependent upon both b) hydrogen bonding and, c) the orientation of loading.

METHODS

Demineralized cortical bone specimens were loaded in tension to failure (0.08 strain/s). The effect of hydrogen bonding on mechanical properties and the stress-whitening process was probed by altering the Hansen's hydrogen bonding parameter (δh) of the immersing solution.

RESULTS

Stress-whitening occurred in the demineralized bone. Stress-whitening was negatively correlated with δh (R(2)=0.81, p<0.0001). Stress-whitening was significantly lower (p<0.0001) in specimens loaded orthogonally compared to those loaded parallel to the long (strong) axis.

CONCLUSION

The stress-whitening observed was consistent with increased Mie scattering. We suggest that the change in Mie scattering was due to collagen fibril dehydration driven by the externally applied stress. The presence of stress-whitening in demineralized bone suggests that this process may be a property of the collagenous matrix and hence may be present in other collagenous tissues rather than an emergent property of the bone composite.

Structural and mechanical multi-scale characterization of white New-Zealand rabbit Achilles tendon

Multi-scale characterization of structures and mechanical behavior of biological tissues are of huge importance in order to evaluate the quality of a biological tissue and/or to provide bio-inspired scaffold for functional tissue engineering. Indeed, the more information on main biological tissue structures we get, the more relevant we will be to design new functional prostheses for regenerative medicine or to accurately evaluate tissues. From this perspective, we have investigated the structures and their mechanical properties from nanoscopic to macroscopic scale of fresh ex-vivo white New-Zealand rabbit Achilles tendon using second harmonic generation (SHG) microscopy, atomic force microscopy (AFM) and tensile tests to provide a "simple" model whose parameters are relevant of its micro or nano structure. Thus, collagen fiber's crimping was identified then measured from SHG images as a plane sine wave with 28.4 ± 5.8 μm of amplitude and 141 ± 41 μm of wavelength. Young's moduli of fibrils (3.0 GPa) and amorphous phases (223 MPa) were obtained using TH-AFM. From these investigations, a non-linear Zener model linking a statistical Weibull's distribution of taut fibers under traction to crimp fibers were developed. This model showed that for small strain (<0.1), the amorphous inter-fibrils phase in collagen fibers is more solicited than collagen fibrils themselves. The results open the way to modeled macroscopic mechanical behavior of aligned-crimped collagen soft tissues using multi-scale tendon observations under static or dynamic solicitations.

A new multiscale model for the mechanical behavior of vein walls

The purpose of the present work is to propose a new multiscale model for the prediction of the mechanical behavior of vein walls. This model is based on one of our previous works which considered scale transitions applied to undulated collagen fibers. In the present work, the scale below was added to take the anisotropy of collagen fibrils into account. One scale above was also added, modeling the global reorientation of collagen fibers inside the vessel wall. The model was verified on experimental data from the literature, leading to a satisfactory agreement. The proposed multiscale approach also allows the extraction of local stresses and strains at each scale. This approach is presented here in the case of vein walls, but can easily be extended to other tissues which contain similar constituents.

Finite element modeling of the viscoelastic responses of the eye during microvolumetric changes

A linear viscoelastic finite element model was built to investigate factors that influenced the intraocular pressure (IOP) elevations due to micro-volumetric changes in the eye at three different rates. The viscoelastic properties of the cornea and the sclera, including the instantaneous modulus, equilibrium modulus, and relaxation time constants, parametrically varied to examine their effects on IOP elevations at different rates of volumetric changes. The simulated responses were in good agreement with the previously reported experimental results obtained from porcine globes, showing the general trend of higher IOP elevations at faster rates. The simulations showed that all viscoelastic properties influenced the profile of the dynamic IOP due to volumetric changes, and the relative significance of a specific parameter was highly dependent on the rate of change.

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.

Nanomaterials design and tests for neural tissue engineering

Nanostructured scaffolds recently showed great promise in tissue engineering: nanomaterials can be tailored at the molecular level and scaffold morphology may more closely resemble features of extracellular matrix components in terms of porosity, framing and biofunctionalities. As a consequence, both biomechanical properties of scaffold microenvironments and biomaterial-protein interactions can be tuned, allowing for improved transplanted cell engraftment and better controlled diffusion of drugs. Easier said than done, a nanotech-based regenerative approach encompasses different fields of know-how, ranging from in silico simulations, nanomaterial synthesis and characterization at the nano-, micro- and mesoscales to random library screening methods (e.g. phage display), in vitro cellular-based experiments and validation in animal models of the target injury. All of these steps of the "assembly line" of nanostructured scaffolds are tightly interconnected both in their standard analysis techniques and in their most recent breakthroughs: indeed their efforts have to jointly provide the deepest possible analyses of the diverse facets of the challenging field of neural tissue engineering. The purpose of this review is therefore to provide a critical overview of the recent advances in and drawbacks and potential of each mentioned field, contributing to the realization of effective nanotech-based therapies for the regeneration of peripheral nerve transections, spinal cord injuries and brain traumatic injuries. Far from being the ultimate overview of such a number of topics, the reader will acknowledge the intrinsic complexity of the goal of nanotech tissue engineering for a conscious approach to the development of a regenerative therapy and, by deciphering the thread connecting all steps of the research, will gain the necessary view of its tremendous potential if each piece of stone is correctly placed to work synergically in this impressive mosaic.

Mechanics of a fiber network within a non-fibrillar matrix: model and comparison with collagen-agarose co-gels

While collagen is recognized as the predominant mechanical component of soft connective tissues, the role of the non-fibrillar matrix (NFM) is less well understood. Even model systems, such as the collagen-agarose co-gel, can exhibit complex behavior, making it difficult to identify relative contributions of specific tissue constituents. In the present study, we developed a two-component microscale model of collagen-agarose tissue analogs and used it to elucidate the interaction between collagen and NFM in uniaxial tension. Collagen fibers were represented with Voronoi networks, and the NFM was modeled as a neo-Hookean solid. Model predictions of total normal stress and Poisson's ratio matched experimental observations well (including high Poisson's values of ~3), and the addition of NFM led to composition-dependent decreases in volume change and increases in fiber stretch. Because the NFM was more resistant to volume change than the fiber network, extension of the composite led to pressurization of the NFM. Within a specific range of parameter values (low shear modulus and moderate Poisson's ratio), the magnitude of the reaction force decreased relative to this pressurization component resulting in a negative (compressive) NFM stress in the loading direction, even though the composite tissue was in tension.

Stress and strain localization in stretched collagenous tissues via a multiscale modelling approach

Mechanobiology of cells in soft collagenous tissues is highly affected by both tissue response at the macroscale and stress/strain localization mechanisms due to features at lower scales. In this paper, the macroscale mechanical behaviour of soft collagenous tissues is modelled by a three-level multiscale approach, based on a multi-step homogenisation technique from nanoscale up to the macroscale. Nanoscale effects, related to both intermolecular cross-links and collagen mechanics, are accounted for, together with geometric nonlinearities at the microscale. Moreover, an effective submodelling procedure is conceived in order to evaluate the local stress and strain fields at the microscale, which is around and within cells. Numerical results, obtained by using an incremental finite element formulation and addressing stretched tendinous tissues, prove consistency and accuracy of the model at both macroscale and microscale, confirming also the effectiveness of the multiscale modelling concept for successfully analysing physiopathological processes in biological tissues.