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Tissue Mechanics


Experimental and Computational Investigation of Viscoelasticity of Native and Engineered Ligament and Tendon

The important mechanisms by which soft collagenous tissues such as ligament and tendon respond to mechanical deformation include non-linear elasticity, viscoelasticity and poroelasticity. These contributions to the mechanical response are modulated by the content and morphology of structural proteins such as type I collagen and elastin, other molecules such as glycosaminoglycans, and fluid. Our ligament and tendon constructs, engineered from either primary cells or bone marrow stromal cells and their autogenous matricies, exhibit histological and mechanical characteristics of native tissues of different levels of maturity. In order to establish whether the constructs have optimal mechanical function for implantation and utility for regenerative medicine, constitutive relationships for the constructs and native tissues at different developmental levels must be established. A micromechanical model incorporating viscoelastic collagen and non-linear elastic elastin is used to describe the non-linear viscoelastic response of our homogeneous engineered constructs in vitro. This model is incorporated within a finite element framework to examine the heterogeneity of the mechanical responses of native ligament and tendon.
J. Ma, H. Narayanan, K. Garikipati, K. Grosh, E. M. Arruda

A Comparison of a Nonlinear and Quasilinear Viscoelastic Anisotropic Model for Fibrous Tissues

This paper presents a nonlinear and quasilinear viscoelasticity model for the behavior of soft fibrous tissues that incorporates the effects of matrix and fiber-level viscoelasticity. Both models treat the tissue as a composite of N fiber families in an isotropic matrix. For the nonlinear model, the anisotropic contribution of the fibers to the constitutive behavior of the tissue is developed by first defining the stress response and viscous flow response of the fiber families, then averaging by the fiber orientation. Similarly, the anisotropic contribution of the quasilinear model is developed from a hereditary integral formulation for the stress response of the fiber families. The stress relaxation response of the nonlinear and quasilinear models are compared for different applied strains. As expect, the time-dependent response of the two formulations are nearly identical for small strain, but they exhibit significant differences at large strain.
T. D. Nguyen

Hysteretic Behavior of Ligaments and Tendons: Microstructural Analysis of Damage, Softening and Non-Recoverable Strain

A microstructural analysis of the hysteretic behavior of ligaments and tendons is proposed from the interaction of their extra-cellular matrix (ECM) components. The tensile response of the tissues during cyclic loading is modeled through a viscoelastic strain energy function. A transition-state theory is used to define the cooperative behavior of the temporary fibrillar network. The viscoelastic model incorporates four internal variables, describing the kinetics of two kinds of adaptive junctions in the ECM microstructure. Two softening variables ξ m , ξ f account for the number density of active matter that is actively connected in the rearranging network of temporary junctions. Conversely, two damage variables η m , η f provide the number density of matter that have been damaged and cannot be rearranged. A dissipation energy functionΦ(t) is linked to the internal variables by thermodynamically consistent evolution equations, describing the irreversible energy dissipation in the tensile cycle of loading and unloading. The model demonstrates the fundamental role of the ECM interactions in determining the time-dependent storage and release of elastic strain energy in ligaments and tendons.
P. Ciarletta, M. Ben Amar

On Measuring Stress Distributions in Epithelia

Many biological tissues are organized as epithelia (i.e., thin cell sheets). Herein, we present a technique to estimate the stress distribution and local material properties in an epithelial membrane. Circular holes are perforated through the tissue to determine the principal stretch ratios; experimentally measured changes in hole geometry are used in combination with finite element modeling to evaluate the stresses and constitutive response. The method is demonstratively applied to the embryonic chick blastoderm, since mechanical stresses have been identified as potential regulators of early development. Due to its small scale, other more traditional mechanical tests have proven intractable for this tissue.
V. D. Varner, L. A. Taber

A Viscoelastic Anisotropic Model for Soft Collageneous Tissues Based on Distributed Fiber–Matrix Units

In this contribution, a nonlinear anisotropic model for the viscoelastic behavior of soft biological tissues is presented. The model is based on micromechanical considerations that take into account the interplay between collagen fibers and the surrounding ground substance. To this end, the stretch along a collagen fiber is multiplicatively decomposed into a part relating to the straightening of the crimped fibers and a part describing the stretch in the fiber itself. The current straightening state of the fibers is described by internal variables. Including a nonuniform distribution of the collagen fibers, the anisotropic three-dimensional constitutive equations are obtained by integration over the unit sphere. The model is applicable for large strains, describes both time and rate dependent behavior and allows to account for particular viscoelastic characteristics of various soft tissues. The performance of the model is illustrated by uniaxial tension as well as sinusoidal simple shear tests and compared to recently published experimental data on ligament tissue.
A. E. Ehret, M. Itskov, G. Weinhold

Cell-substrate Interactions


Chemical and Mechanical Micro-Diversity of the Extracellular Matrix

Interaction with the extracellular matrix (ECM) triggers multiple physiological responses in living cells, affecting their structure, function and fate. Recent studies have demonstrated that cells can sense a wide variety of chemical and physical features of the ECM, and differentially respond to them. Thus, cells cultured on flat surfaces coated with two different integrin-reactive adhesive proteins, fibronectin and vitronectin, display varying degrees of spreading on these matrices, and form morphologically distinct types of matrix adhesions, with variable prominence and spatial distribution of both focal and fibrillar adhesions. It was further shown, using labeling with different antibodies which bind to distinct sites on the fibronectin molecule, that even a “molecularly homogeneous” matrix displays spatial micro-heterogeneity, exposing distinct epitopes at different locations. Diversification of the adhesive surface can be induced by the application of mechanical force to the elastic fibronectin matrix, resulting in the formation of different patterns of fibrillar ECM arrays. Time-lapse monitoring of matrix fibrillogenesis by cells expressing fluorescently tagged fibronectin demonstrated that the assembly of fibrils in such cell cultures occurs when the leading lamella of the cell advances, attaches to the substrate-bound fibronectin, and then retracts backwards, thus applying tensile forces to the attached fibronectin. These results indicate that the ECM is a highly complex cellular environment, whose chemical and physical properties are directly regulated by the attached cells.
T. Volberg, J. Ulmer, J. Spatz, B. Geiger

Tissue-to-Cellular Deformation Coupling in Cell-Microintegrated Elastomeric Scaffolds

Long term efficacy of tissue replacements or regenerative therapies rely on the critical processes of cell proliferation and differentiation, the production of organized matrix, and concurrent tissue remodeling or growth. Recent studies have shown that mechanical and chemical factors modulate cell function which has profound implications on tissue growth and remodeling. As such, creating engineered tissue replacement options requires a detailed command of the complex, dynamic, and reciprocal interactions which occur at the cell-ECM interface. To gain a better understanding of the coupled tissue-cellular deformation response, we propose the use of cell-microintegrated elastomeric scaffolds which provide a unique platform to investigate cellular deformations within a three dimensional fibrous scaffold. Scaffold specimens micro-integrated with vascular smooth muscle cells (VSMC) were subjected to controlled biaxial stretch with 3D cellular deformations and local fiber micro-architecture simultaneously quantified. Interestingly, local cellular deformations exhibited a non-linear deformation response with scaffold strain which was attributed to unique microarchitectural morphologies. Local scaffold microstructural changes induced by macro-level applied strain dominated cellular deformations, so that monotonic increases in scaffold strain do not necessitate similar levels of cellular deformation. This result has fundamental implications when attempting to elucidate the events of de-novo tissue development and remodeling in engineeredtissues.
J. A. Stella, J. Liao, Y. Hong, W. D. Merryman, W. R. Wagner, M. S. Sacks

Orientational Polarizability and Stress Response of Biological Cells

We present a theoretical treatment of the orientational response to external stress of active, contractile cells embedded in a gel-like elastic medium. The theory includes random forces as well as forces that arise from the deformation of the matrix and those due to the internal regulation of the stress fibers and focal adhesions of the cell. We calculate both the static and high frequency limits of the orientational response in terms of the cellular polarizability. For systems in which the forces due to regulation and activity dominate the mechanical forces, we show that there is a non-linear dynamical response which, in the high frequency limit, causes the cell to orient nearly perpendicular to the direction of the applied stress.
S. A. Safran, R. De, A. Zemel

Universal Temporal Response of Fibroblasts Adhering on Cyclically Stretched Substrates

It is well known that many cells adherent on a cyclically stretched substrate reorient nearly perpendicular to the applied stretching direction. Such periodic mechanical signals are characterized by the stretching amplitude and frequency ad many studies focus on the influence of the amplitude on the orientation behavior. However, little is know about the temporal characteristic and dynamics of this cellular response. Consequently, we developed an experimental stretch system for live cell imaging. Using this setup, we observed the dynamic reorientation of different human fibroblast types over a frequency range 0.01–10 s−1 and a constant stretching amplitude of 8%. We demonstrate an increasing mean cell orientation with an exponentially time characteristics. The characteristic time τ for the reorientation is frequency-dependent and is in a range from 1 to 5 h. This characteristic time is a function of frequency and follows a power law for frequencies below 1s−1,τ decreases with a power law as the frequency increases. For frequencies above 1s−1,τ is nearly constant and the kinetics of cell reorientation is in saturation. In addition, a threshold frequency is found below which no significant cell reorientation occurs. Our results are consistent for the two different human fibroblast types and indicate a saturation of molecular mechanisms of mechanotransduction or response machinery for subconfluent cells within the frequency regime under investigation.
S. Jungbauer, B. Aragües, J. P. Spatz, R. Kemkemer

Mechanics of DNA


Elastic and Electrostatic Model for DNA in Rotation–Extension Experiments

We present a self-contained theory for the mechanical response of DNA in extension–rotation single molecule experiments. The theory is based on the elasticity of the double-helix and the electrostatic repulsion between two DNA duplex. The configuration of the molecule at large imposed rotation is assumed to comprise two phases, linear and superhelical DNA. Thermal fluctuations are accounted for in the linear phase and electrostatic repulsion is treated in the superhelical phase. This analytical model enables the computation of the supercoiling radius and angle of DNA during experiments. The torsional stress in the molecule and the slope of the linear region of the experimental curve are also predicted and compared successfully with experimental data.
S. Neukirch, N. Clauvelin, B. Audoly

Shape and Energetics of DNA Plectonemes

The mechanics of DNA at length scales of few hundred nanometers is described by a fluctuating elastic rod model. In this paper we couple the fluctuating rod model with a variational method for describing plectonemes to unravel the mechanics of some recent single molecule experiments. We are able to reproduce some features seen in these experiments which analyze plectoneme formation in DNA under tension by continuously twisting it and tracking its end-to-end distance, torque, etc., as a function of the added link. Our model accounts for configurational entropy and electrostatics in the plectoneme. We find that configurational entropy makes a significant contribution to the mechanics of the plectoneme while electrostatics (in the presence of monovalent counterions) plays a relatively minor role.
P. K. Purohit

Mechanics of Biopolymer Networks


Constitutive Models for the Force-Extension Behavior of Biological Filaments

Biopolymer filaments form the molecular backbone of biological structures throughout the body. The biomechanical response of single filaments yields insight into their individual behavior at the molecular level as well as their concerted networked behavior at the cellular and tissue scales. This paper focuses on modeling approaches for axial force vs. extension behavior of single biopolymer filaments within three stiffness regimes: flexible, semiflexible, and stiff. The end-to-end force-extension behaviors of flexible and semiflexible filaments arise as a result of a reduction in configurational space as the filament is straightened and are captured with entropic models including the freely jointed chain model and the worm-like chain model. As the filament is straightened and the end-to-end distance approaches the filament contour length, the contour length is directly axially extended and an internal energy contribution governs the force-extension behavior in this limiting extension regime. On the other hand, for stiff filaments in originally crimped or kinked configurations, the end-to-end force vs. extension behavior results from the unbending (straightening) of the crimped configuration as governed by an internal energy based elastica approximation which is also complemented by an axial stretching contribution once the end-to-end distance approaches the contour length of the filament. Simplified, analytical force-extension relationships are developed for the worm-like chain model for semiflexible filaments, and for the Euler elastica model for stiffer, wavy fibers. For the case of flexible molecules containing modular folded domains, the influence of force-induced unfolding on the force-extension behavior of single molecules and assemblies of multiple molecules is also presented.
J. S. Palmer, C. E. Castro, M. Arslan, M. C. Boyce

Small Strain Topological Effects of Biopolymer Networks with Rigid Cross-Links

Networks of cross-linked filamentous biopolymers form topological structures characterized by L, T and X cross-link types of connectivity 2, 3 and 4, respectively. The distribution of cross-links over these three types proofs to be very important for the initial elastic shear stiffness of isotropic rigidly cross-linked biopolymer networks. After proper scaling of this stiffness, we identify the topological function \(f(\mathcal{T} ) = ({l}_{\mathrm{c}}/\xi ){({n}_{\mathrm{X}}/{n}_{\mathrm{cl}})}^{2.5}\) that describes these effects in terms of the network parameters: mean section length l c, mesh size ξ and relative number of X cross-links n Xn cl.
G. Žagar, P. R. Onck, E. Van der Giessen

Cell adhesion


An Observation on Bell’s Model for Molecular Bond Separation Under Force

Within the context of reaction rate theory, Bell proposed a particular dependence of the reaction off-rate on applied force to describe molecular bond separation under force. Here, the issue is re-examined from the point of view of the diffusive transport of bond states over a landscape of interaction energy for the bond pair in the presence of a time-dependent applied force. We are led to an expression for the off-rate which is perhaps more soundly based and which reduces to Bell’s result for bond separation at force levels that are small in a particular sense. For a given molecular bond, it appears that the condition of small force can be assured only under conditions of relatively slow loading of the bond.
L. B. Freund

A Theoretical Study of the Thermodynamics and Kinetics of Focal Adhesion Dynamics

A theoretical treatment of growth and disassembly of focal adhesions is developed in the framework of rate processes driven by thermodynamics. For this purpose, the structural unit of focal adhesions is a complex consisting of a ligand such as fibronectin, an integrin molecule, and associated plaque proteins. The free energy that drives binding and dissociation of the complexes includes mechanical, chemical and statistical (mixing entropy) contributions. The binding and dissociation of complexes manifests as growth and disassembly, respectively, of focal adhesions. We have complemented fracture mechanics by bond formation to explain this reversible chemo-mechanical process. The reaction-limited case is considered. We have identified a competition between four mechanisms: (i) mechanical work done by actin-transmitted force, (ii) a chemical instability inherent to focal adhesions, (iii) an elastic instability, and (iv) a molecular conformational change, that control focal adhesion dynamics. Our central finding is that for a focal adhesion to slide requires symmetry breaking between its two ends. This happens only with the first of these four mechanisms. The molecular conformational change can contribute symmetric growth modes, while the remaining two mechanisms cause disassembly.
J. E. Olberding, M. D. Thouless, E. M. Arruda, K. Garikipati

Tension-Induced Growth of Focal Adhesions at Cell–Substrate Interface

This paper presents a stochastic-elasticity model on the tension-induced growth of focal adhesions (FAs) at cell–substrate interface. The model is based on a Monte Carlo scheme incorporating applied tension, cell/substrate elasticity, receptor–ligand binding/unbinding and receptor diffusion in the same framework that fully couples elasticity and probabilistic rate processes in the system. We investigate the clustering of receptor molecules and growth of FAs under different levels of applied tension. While overly simplified in a number of aspects, our model seems to give predictions that are consistent with relevant experimental observations on the mechanosensitivity of FAs.
J. Qian, J. Wang, H. Gao

Pattern Formation and Force Generation by Cell Ensembles in a Filamentous Matrix

Adhesion-dependent soft tissue cells both create and sense tension in the extracellular matrix. Therefore cells can actively interact through the mechanics of the surrounding matrix. An intracellular positive feedback loop upregulates cellular contractility in stiff or tensed environments. Here we theoretically address the resulting pattern formation and force generation for the case of a filamentous matrix, which we model as a two-dimensional cable network. Cells are modeled as anisotropic contraction dipoles which move in favor of tensed directions in the matrix. Our Monte Carlo simulations suggest that at small densities, cells align in strings, while at high densities, they form interconnected meshworks. Cellular activation both by biochemical factors and by tension leads to a hyperbolic increase in tissue tension. We also discuss the effect of cell density on tissue tension and shape.
R. Paul, U. S. Schwarz

Mechano-Chemical Coupling in Shell Adhesion

The coupling between cell deformation and chemical segregation during the early stages of cell adhesion is investigated by studying the equilibrium of thin shells adhered to rigid substrates that are either flat or have topography. A finite-range adhesion law is taken to depend on the local shell-substrate separation and on the local concentrations of segregating chemical species. Nonlinear shell kinematics accounting for finite rotations of both closed spherical shells and open spherical caps are coupled with the equilibrium equations for axisymmetric deformations and linearly elastic material response. Representative solutions demonstrate the thermodynamic coupling that results in nonuniform mechanical and chemical fields, effects of substrate topography, and the influence of finite-range adhesive interactions. Strong coupling is predicted between shell deformation and the level of chemical activation which is measured by the total adhesive energy at equilibrium.
R. M. Springman, J. L. Bassani

Catch-to-Slip Bond Transition in Biological Bonds by Entropic and Energetic Elasticity

It is revealed recently that the life time of some biological bonds increases in response to small and moderate external tensile forces, decreases with further increasing of tensile forces. Such biological bonds are termed ‘catch bonds’. This work aims to explain the dependence of bond life time on entropic and energetic factors which are controlled by external tensile forces. We count debonding events of a biological bond in a sphere surrounding the bonding complex. For simplicity, the surface is divided into two regions. Region (a) has a surface normal nearly parallel to a tensile force, and region (b) is the rest of the surface. The influence of a tensile force to dissociation in region (a) is by lowering the energy barrier to escape, and that to region (b) is by modifying accessible microstates for dissociation. The lifetime of the biological bond, due to the superimposition of two concurrent dissociation rates in each region, may grow with increasing tensile force to moderate amount and decrease with further increasing load. It is hypothesized that a catch-to-slip bond transition is a generic feature in biological bonds. The model also predicts that catch bonds in compliant molecular structure have longer lifetimes and lower bond strength. Here bond strength is defined as the critical force where the bond lifetime is maximized.
Y. Wei



Dilation and Hypertrophy: A Cell-Based Continuum Mechanics Approach Towards Ventricular Growth and Remodeling

This manuscript presents a continuum approach towards cardiac growth and remodeling that is capable to predict chronic maladaptation of the heart in response to changes in mechanical loading. It is based on the multiplicative decomposition of the deformation gradient into and elastic and a growth part. Motivated by morphological changes in cardiomyocyte geometry, we introduce an anisotropic growth tensor that can capture both hypertrophic wall thickening and ventricular dilation within one generic concept. In agreement with clinical observations, we propose wall thickening to be a stress-driven phenomenon whereas dilation is introduced as a strain-driven process. The features of the proposed approach are illustrated in terms of the adaptation of thin heart slices and in terms overload-induced dilation in a generic bi-ventricular heart model.
J. Ulerich, S. Göktepe, E. Kuhl

A Morpho-Elastic Model of Hyphal Tip Growth in Filamentous Organisms

The growth of filamentous cells is modeled through the use of exact, nonlinear, elasticity theory for shells and membranes. The biomechanical model is able to capture the generic features of growth of a broad array of cells including actinomycetes, fungi, and root hairs. It also provides the means of studying the effects of external surface stresses. The growth mechanism is modeled by a process of incremental elastic growth in which the cell wall responds elastically to the continuous addition of new material.
A. Goriely, M. Tabor, A. Tongen

Extracellular Control of Limb Regeneration

Adult newts possess the ability to completely regenerate organs and appendages. Immediately after limb loss, the extracellular matrix (ECM) undergoes dramatic changes that may provide mechanical and biochemical cues to guide the formation of the blastema, which is comprised of uncommitted stem-like cells that proliferate to replace the lost structure. Skeletal muscle is a known reservoir for blastema cells but the mechanism by which it contributes progenitor cells is still unclear. To create physiologically relevant culture conditions for the testing of primary newt muscle cells in vitro, the spatio-temporal distribution of ECM components and the mechanical properties of newt muscle were analyzed. Tenascin-C and hyaluronic acid (HA) were found to be dramatically upregulated in the amputated limb and were co-expressed around regenerating skeletal muscle. The transverse stiffness of muscle measured in situ was used as a guide to generate silicone-based substrates of physiological stiffness. Culturing newt muscle cells under different conditions revealed that the cells are sensitive to both matrix coating and substrate stiffness: Myoblasts on HA-coated soft substrates display a rounded morphology and become more elongated as the stiffness of the substrate increases. Coating of soft substrates with matrigel or fibronectin enhanced cell spreading and eventual cell fusion.
S. Calve, H.-G. Simon

Poroelasticity of Bone


Bone Composite Mechanics Related to Collagen Hydration State

Bone is considered in two different composite mechanics frameworks: first as an organic–inorganic two phase composite, and second as a fluid-saturated porous solid. Experimental data from previous studies, in which the mechanical responses of bone or collagen were examined following immersion in a range of polar solvents, were used as inputs for both models. The changes in bone elastic modulus with polar solvents could not be predicted by two-phase organic–inorganic composites models. A spherical indentation finite element model is generated within the poroelastic framework with the objective of identifying the permeability coefficient. In particular the effect of the ramp rise-time on the identification results is compared with results that assumed a step-load creep experiment. The results confirm that immersion of bone in polar solvents with decreasing polarity results in decreased hydraulic permeability. The developed identification approach based on the normalization of the indentation time-displacement response results shows potential for the efficient analysis of high throughput indentation tests. Further extension of composites models to include all three phases – water, collagen and mineral – is needed to fully explore the mechanical behavior of bone.
M. L. Oyen, M. Galli


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