Anisotropy of fibrous tissues in relation to the distribution of tensed and buckled fibers

Fibrous tissues are characterized by a much higher stiffness in tension than compression. This study uses microstructural modeling to analyze the material symmetry of fibrous tissues undergoing tension and compression, to better understand how material symmetry relates to the distribution of tensed and buckled fibers. The analysis is also used to determine whether the behavior predicted from a microstructural model can be identically described by phenomenological continuum models. The analysis confirms that in the case when all the fibers are in tension in the current configuration, the material symmetry of a fibrous tissue in the corresponding reference configuration is dictated by the symmetry of its fiber angular distribution in that configuration. However, if the strain field exhibits a mix of tensile and compressive principal normal strains, the fibrous tissue is represented by a material body which consists only of those fibers which are in tension; the material symmetry of this body may be deduced from the superposition of the planes of symmetry of the strain and the planes of symmetry of the angular fiber distribution. Thus the material symmetry is dictated by the symmetry of the angular distribution of only those fibers which are in tension. Examples are provided for various fiber angular distribution symmetries. In particular, it is found that a fibrous tissue with isotropic fiber angular distribution exhibits orthotropic symmetry when subjected to a mix of tensile and compressive principal normal strains, with the planes of symmetry normal to the principal directions of the strain. This anisotropy occurs even under infinitesimal strains and is distinct from the anisotropy induced from the finite rotation of fibers. It is also noted that fibrous materials are not stable under all strain states due to the inability of fibers to sustain compression along their axis; this instability can be overcome by the incorporation of a ground matrix. It is concluded that the material response predicted using a microstructural model of the fibers cannot be described exactly by phenomenological continuum models. These results are also applicable to nonbiological fiber-composite materials.     

On the anisotropy of the canine diaphragmatic central tendon

We studied the mechanical and anatomical anisotropy of the canine diaphragmatic central tendon (CT). Dumb-bell-shaped strips with effective dimensions of 10 x 2 mm (length x width) were cut from different regions of the canine diaphragmatic CT in two different orientations relative to the direction of neighboring muscle fibers. Specimens sampled with their long axial dimension oriented parallel to the neighboring muscle fibers were named Group-1 and those sampled with an orientation perpendicular to the neighboring muscle fibers were named Group-2. Results from one-dimensional stress-strain and tensile failure strength tests revealed that the CT is a nonlinear, inelastic, and anisotropic material. Group-1 specimens were found to have a higher stiffness, higher failure strength and higher strain energy density at failure than Group-2 specimens. Polarized microscopy showed that multiple sheets of collagen fiber bundles formed an orthogonal network in the tendon. Collagen fiber bundles along Group-1 direction formed parallel trajectory lines connecting the neighboring costal and crural muscles; bundles along Group-2 direction were observed to orient 90 degrees away. At the central apex region of the CT, collagen bundles of Group-1 formed a fan-like trajectory pattern. This collagen network architecture was compared favorably to the trajectories of an approximated principal stress field in the CT due to simulated contractile forces from its adjacent costal and crural muscles. These combined results suggest a structure-function relationship for the anatomical and mechanical anisotropy in the canine diaphragmatic CT.     

From single muscle fiber to whole muscle mechanics: a finite element model of a muscle bundle with fast and slow fibers

Muscles exhibit highly complex, multi-scale architecture with thousands of muscle fibers, each with different properties, interacting with each other and surrounding connective structures. Consequently, the results of single-fiber experiments are scarcely linked to the macroscopic or whole muscle behavior. This is especially true for human muscles where it would be important to understand of how skeletal muscles disorders affect patients’ life. In this work, we developed a mathematical model to study how fast and slow muscle fibers, well characterized in single-fiber experiments, work and generate together force and displacement in muscle bundles. We characterized the parameters of a Hill-type model, using experimental data on fast and slow single human muscle fibers, and comparing experimental data with numerical simulations obtained from finite element (FE) models of single fibers. Then, we developed a FE model of a bundle of 19 fibers, based on an immunohistochemically stained cross section of human diaphragm and including the corresponding properties of each slow or fast fiber. Simulations of isotonic contractions of the bundle model allowed the generation of its apparent force–velocity relationship. Although close to the average of the force–velocity curves of fast and slow fibers, the bundle curve deviates substantially toward the fast fibers at low loads. We believe that the present model and the characterization of the force–velocity curve of a fiber bundle represents the starting point to link the single-fiber properties to those of whole muscle with FE application in phenomenological models of human muscles.     

Three-dimensional arrangement of muscle bundles in the outer layers of rodentia vasa deferentia

Three-dimensional arrangement of the smooth muscle bundles of the outer layer of the vas deferens musculature in mammals (guinea-pigs, rats and mice) was examined under the scanning electron microscope (SEM) after removal of fibrous connective tissue elements. Muscle fibers of all examined animals formed bundles. In the guinea-pig, similar sized bundles extended longitudinally along the tubular vas deferens and branched to anastomose with branches of neighboring bundles to create a net which was regular in form. In the rat, longitudinal muscle bundles constituted an outer layer in the form of a net, which was roughly enmeshed with variously-sized, transverse or oblique bundles in anastomosis with underlying longitudinal bundles. In the mouse, longitudinal bundles of irregular thickness branched into many small bundles and anastomosed not only with neighboring bundles to create an irregular net. In both the rat and the mouse there were bundles extending over many other bundles to anastomose with them at a far point. Junctional structures were well developed between neighboring fibers. Myofibrils were represented as thin streaks on muscle fiber surfaces. Varicosed nerve fibers existed between muscle fibers and in narrow cytoplasmic grooves in all the examined animal species. The findings are discussed in correlation with electrophysiological data.     

Micromechanical models of helical superstructures in ligament and tendon fibers predict large Poisson's ratios

Experimental measurements of the Poisson's ratio in tendon and ligament tissue greatly exceed the isotropic limit of 0.5. This is indicative of volume loss during tensile loading. The microstructural origin of the large Poisson's ratios is unknown. It was hypothesized that a helical organization of fibrils within a fiber would result in a large Poisson's ratio in ligaments and tendons, and that this helical organization would be compatible with the crimped nature of these tissues, thus modeling their classic nonlinear stress–strain behavior. Micromechanical finite element models were constructed to represent crimped fibers with a super-helical organization, composed of fibrils embedded within a matrix material. A homogenization procedure was performed to determine both the effective Poisson's ratio and the Poisson function. The results showed that helical fibril organization within a crimped fiber was capable of simultaneously predicting large Poisson's ratios and the nonlinear stress–strain behavior seen experimentally. Parametric studies revealed that the predicted Poisson's ratio was strongly dependent on the helical pitch, crimp angle and the material coefficients. The results indicated that, for physiologically relevant parameters, the models were capable of predicting the large Poisson's ratios seen experimentally. It was concluded that helical organization within a crimped fiber can produce both the characteristic nonlinear stress–strain behavior and large Poisson's ratios, while fiber crimp alone could only account for the nonlinear stress–strain behavior.     

A multi-modular tensegrity model of an actin stress fiber

Stress fibers are contractile bundles in the cytoskeleton that stabilize cell structure by exerting traction forces on the extracellular matrix. Individual stress fibers are molecular bundles composed of parallel actin and myosin filaments linked by various actin-binding proteins, which are organized end-on-end in a sarcomere-like pattern within an elongated three-dimensional network. While measurements of single stress fibers in living cells show that they behave like tensed viscoelastic fibers, precisely how this mechanical behavior arises from this complex supramolecular arrangement of protein components remains unclear. Here we show that computationally modeling a stress fiber as a multi-modular tensegrity network can predict several key behaviors of stress fibers measured in living cells, including viscoelastic retraction, fiber splaying after severing, non-uniform contraction, and elliptical strain of a puncture wound within the fiber. The tensegrity model can also explain how they simultaneously experience passive tension and generate active contraction forces; in contrast, a tensed cable net model predicts some, but not all, of these properties. Thus, tensegrity models may provide a useful link between molecular and cellular scale mechanical behaviors and represent a new handle on multi-scale modeling of living materials.     

Force heterogeneity in a two-dimensional network model of lung tissue elasticity

We have developed a model of forces developed inlung tissue in which the stress-bearing units are heterogeneous. Eachelement of the fiber network is composed of an idealized elastin andcollagen element in parallel. Elastin is represented by linear springs and collagen by stiff strings that extend without resistance until taut. The model can quantitatively account for the nonlinear shape ofthe length-tension curve of lung tissue strips when the knee lengths ofthe collagen fibers are distributed according to an inverse power law.The novel feature of this model is that as macroscopic strain increasesthe load is carried by progressively fewer elements with progressivelyhigher forces, and preferential pathways of force transmission emergewithin the matrix. The topology of these self-organizing pathways offorce transmission takes the rough appearance of cracks, but, unlikereal cracks, they represent the locus of force concentration ratherthan force release.

     

Biaxial tensile testing and constitutive modeling of human supraspinatus tendon

The heterogeneous composition and mechanical properties of the supraspinatus tendon offer an opportunity for studying the structure-function relationships of fibrous musculoskeletal connective tissues. Previous uniaxial testing has demonstrated a correlation between the collagen fiber angle distribution and tendon mechanics in response to tensile loading both parallel and transverse to the tendon longitudinal axis. However, the planar mechanics of the supraspinatus tendon may be more appropriately characterized through biaxial tensile testing, which avoids the limitation of nonphysiologic traction-free boundary conditions present during uniaxial testing. Combined with a structural constitutive model, biaxial testing can help identify the specific structural mechanisms underlying the tendon's two-dimensional mechanical behavior. Therefore, the objective of this study was to evaluate the contribution of collagen fiber organization to the planar tensile mechanics of the human supraspinatus tendon by fitting biaxial tensile data with a structural constitutive model that incorporates a sample-specific angular distribution of nonlinear fibers. Regional samples were tested under several biaxial boundary conditions while simultaneously measuring the collagen fiber orientations via polarized light imaging. The histograms of fiber angles were fit with a von Mises probability distribution and input into a hyperelastic constitutive model incorporating the contributions of the uncrimped fibers. Samples with a wide fiber angle distribution produced greater transverse stresses than more highly aligned samples. The structural model fit the longitudinal stresses well (median R(2) ≥ 0.96) and was validated by successfully predicting the stress response to a mechanical protocol not used for parameter estimation. The transverse stresses were fit less well with greater errors observed for less aligned samples. Sensitivity analyses and relatively affine fiber kinematics suggest that these errors are not due to inaccuracies in measuring the collagen fiber organization. More likely, additional strain energy terms representing fiber-fiber interactions are necessary to provide a closer approximation of the transverse stresses. Nevertheless, this approach demonstrated that the longitudinal tensile mechanics of the supraspinatus tendon are primarily dependent on the moduli, crimp, and angular distribution of its collagen fibers. These results add to the existing knowledge of structure-function relationships in fibrous musculoskeletal tissue, which is valuable for understanding the etiology of degenerative disease, developing effective tissue engineering design strategies, and predicting outcomes of tissue repair.     

Structural Characteristics of the Short-Tail Fibers of T4 Bacteriophage

The characteristics of pure preparations of short-tail fibers of bacteriophage T4 have been studied in the optical and electron microscope. Three main structures were observed: 1) spheres of 8.1 nm diameter; 2) fibers 43 nm long and 3.8 nm thick; and 3) fibers 54 nm long and 3.2 nm thick. Both types of fibers exhibited a regular beaded appearance. The 43-nm fibers were the most abundant structure. During the process of purification of the short-tail fibers, the formation of aggregates was observed each time the material containing the short-tail fibers was dialyzed against saline solutions. These aggregates became increasingly fibrous (as observed in the optical microscope) as the material used was increasingly enriched in short-tail fibers. Finally, most of the aggregates were of the fibrous type when they were formed from a purified preparation of short-tail fibers. In the electron microscope, it was found that the filamentous aggregates were organized in well-defined bundles. The amino acid composition of the highly purified short-tail fibers was also determined. Among the known fibrous proteins, the ones that most resemble the amino acid composition of the short-tail fibers are actin and fibrinogen. These observations are discussed in relation to the T4 short-tail fiber structure and their localization on the hexagonal baseplate of the T4 tail structure.     

Constitutive modeling of an electrospun tubular scaffold used for vascular tissue engineering

In this study, we sought to model the mechanical behavior of an electrospun tubular scaffold previously reported for vascular tissue engineering with hyperelastic constitutive equations. Specifically, the scaffolds were made by wrapping electrospun polycaprolactone membranes that contain aligned fibers around a mandrel in such a way that they have microstructure similar to the native arterial media. The biaxial stress-stretch data of the scaffolds made of moderately or highly aligned fibers with three different off-axis fiber angles \(\alpha \) (30\(^\circ \), 45\(^\circ \), and 60\(^\circ \)) were fit by a phenomenological Fung model and a series of structurally motivated models considering fiber directions and fiber angle distributions. In particular, two forms of fiber strain energy in the structurally motivated model for a linear and a nonlinear fiber stress–strain relation, respectively, were tested. An isotropic neo-Hookean strain energy function was also added to the structurally motivated models to examine its contribution. The two forms of fiber strain energy did not result in significantly different goodness of fit for most groups of the scaffolds. The absence of the neo-Hookean term in the structurally motivated model led to obvious nonlinear stress-stretch fits at a greater axial stretch, especially when fitting data from the scaffolds with a small \(\alpha \). Of the models considered, the Fung model had the overall best fitting results; its applications are limited because of its phenomenological nature. Although a structurally motivated model using the nonlinear fiber stress–strain relation with the neo-Hookean term provided fits comparably as good as the Fung model, the values of its model parameters exhibited large within-group variations. Prescribing the dispersion of fiber orientation in the structurally motivated model, however, reduced the variations without compromising the fits and was thus considered to be the best structurally motivated model for the scaffolds. It appeared that the structurally motivated models could be further improved for fitting the mechanical behavior of the electrospun scaffold; fiber interactions are suggested to be considered in future models.     

Computational modeling of the arterial wall based on layer-specific histological data

Arterial walls typically have a heterogeneous structure with three different layers (intima, media, and adventitia). Each layer can be modeled as a fiber-reinforced material with two families of relatively stiff collagenous fibers symmetrically arranged within an isotropic soft ground matrix. In this paper, we present two different modeling approaches, the embedded fiber (EF) approach and the angular integration (AI) approach, to simulate the anisotropic behavior of individual arterial wall layers involving layer-specific data. The EF approach directly incorporates the microscopic arrangement of fibers that are synthetically generated from a random walk algorithm and captures material anisotropy at the element level of the finite element formulation. The AI approach smears fibers in the ground matrix and treats the material as homogeneous, with material anisotropy introduced at the constitutive level by enhancing the isotropic strain energy with two anisotropic terms. Both approaches include the influence of fiber dispersion introduced by fiber angular distribution (departure of individual fibers from the mean orientation) and take into consideration the dispersion caused by fiber waviness, which has not been previously considered. By comparing the numerical results with the published experimental data of different layers of a human aorta, we show that by using histological data both approaches can successfully capture the anisotropic behavior of individual arterial wall layers. Furthermore, through a comprehensive parametric study, we establish the connections between the AI phenomenological material parameters and the EF parameters having straightforward physical or geometrical interpretations. This study provides valuable insight for the calibration of phenomenological parameters used in the homogenized modeling based on the fiber microscopic arrangement. Moreover, it facilitates a better understanding of individual arterial wall layers, which will eventually advance the study of the structure–function relationship of arterial walls as a whole.     

Tissue mechanical behavior of tendinous fibers under statistical and dynamic requirements

With a device for dynamical processes tension tests were performed on bundles of collagen fibres of human and bovin origin. Part of the studies were performed under statical condition with an universal materials testing machine, equipped with a closed loop feed back control system. If the force is to be kept constant subsequently to a strain process on a collagen fiber bundle (isotonic condition), the fiber length must increase (creep phenomenon, retardation). The force decreases under constant length after a preceding strain process (relaxation). In analogy to the statical relaxation, statical isorheological line, and statical force recovery curve the dynamical (cyclic) relaxation, dynamical (cyclic) isorheological curve, and dynamical (cyclic) force recovery curve are described. The mechanical-rheological properties collagen fiber bundles are discussed in relation to functional anatomy.     

Influence of domain orientation on the mechanical properties of regenerated cellulose fibers

The determination of the crystal orientation of regenerated cellulose fibers produced under different drawing regimes is presented. Orientation is determined by using wide-angle X-ray diffraction from a synchrotron source and by measuring the azimuthal width of equatorial reflections. The orientation parameter theta is then determined to compare fiber samples. By using a 500 nm beam size, clear differences between the crystal orientations of the skin and the core of the fibers are reported for a range of differently processed fibers for the first time. These results are shown to have implications for the mechanical properties of regenerated cellulose fibers. By applying tensile deformation to fiber bundles it is shown that the most misoriented samples undergo rapid decreases in the orientation parameter, which is an indication of crystal reorientation. However, the more highly oriented fibers undergo little reorientation. An average shear modulus for these fibers is determined by placing the data on a master curve and fitting with a model equation. By using another model for the fibers of low orientation and the shear modulus from the master curve analysis, it is shown that the deformation of less oriented fibers is dominated by shear between crystals, whereas the more oriented filaments are likely to undergo more significant chain deformation. By using a new model for fibers of low orientation, a parameter ksigma is introduced that gives the proportion of the fiber stress that is due to crystal shear. Systematic differences between this parameter for fibers of increasing initial orientation are reported. Moreover it is shown that the fibers of initially lower average orientation are governed by uniform strain, in agreement with the new model, whereas more highly oriented fibers deform under uniform stress. Furthermore, the model that we propose for misoriented domains and the use of a new factor dictating the proportion of shear stress may have general applications in materials engineering.     

Structural and mechanical aspects of the skin of Bufo marinus (Anura, Amphibia)

Specific biomechanical characters and some structures possibly related to them were investigated in the skin of the toad Bufo marinus using tensile testing techniques (at constant strain till rupture) as well as morphological methods (histological, immunohistochemical and electronmicroscopical). Mechanical parameters of the native skin varied considerably according to sex, individual variability and/or site of specimen collection. In skin strips of males and females excised from different parts of the body thickness ranged from 0.45 to 0.87 mm, strain (epsilonf) from 96.52 to 211.03, tensile strength (sigmam) from 5.72 to 9.38 MPa, and stiffness (E-modulus) from 5.76 to 6.73. The dermis of B. marinus is provided with a collagenous stratum compactum of considerable thickness, a stratum spongiosum with loosely arranged fibres and a marked calcified layer (substantia amorpha). Collagen appears to be the main determinant of skin mechanics. However, the slope of the J-shaped static stress-strain curves indicates elastin to be responsible for the high values of strain. Contrary to van Gieson and orcein staining, immunostaining with a monoclonal antibody against elastin revealed very few elastic fibers between collagen bundles and in the vertical fiber tracts (perforating bundles), but a considerable amount in the tela subcutanea. This was partly confirmed at the ultrastructural level by tannic acid staining.     

A three-dimensional collagen-fiber network model of the extracellular matrix for the simulation of the mechanical behaviors and micro structures

The extracellular matrix (ECM) provides structural and biochemical support to cells and tissues, which is a critical factor for modulating cell dynamic behavior and intercellular communication. In order to further understand the mechanisms of the interactive relationship between cell and the ECM, we developed a three-dimensional (3D) collagen-fiber network model to simulate the micro structure and mechanical behaviors of the ECM and studied the stress–strain relationship as well as the deformation of the ECM under tension. In the model, the collagen-fiber network consists of abundant random distributed collagen fibers and some crosslinks, in which each fiber is modeled as an elastic beam and a crosslink is modeled as a linear spring with tensile limit, it means crosslinks will fail while the tensile forces exceed the limit of spring. With the given parameters of the beam and the spring, the simulated tensile stress–strain relation of the ECM highly matches the experimental results including damaged and failed behaviors. Moreover, by applying the maximal inscribed sphere method, we measured the size distribution of pores in the fiber network and learned the variation of the distribution with deformation. We also defined the alignment of the collagen-fibers to depict the orientation of fibers in the ECM quantitatively. By the study of changes of the alignment and the damaged crosslinks against the tensile strain, this paper reveals the comprehensive mechanisms of four stages of ‘toe’, ‘linear’, ‘damage’ and ‘failure’ in the tensile stress–strain relation of the ECM which can provide further insight in the study of cell-ECM interaction.     

The influence of specimen length on the tensile failure properties of tendon collagen

Connective tissues such as ligament, tendon and skin are composites of strength-bearing collagen fibers embedded in a hydrated matrix. The tensile response and failure properties of rat-tail tendon are thought to represent those of the collagen fiber itself. In this study, the tensile failure properties of rat-tail tendon (tendon collagen) were determined for specimens of various test length. The experimental results indicated that failure strain, based on the test grip-to-grip dimension, and failure strain energy density decreased as specimen length increased. The failure stress, on the other hand, did not change appreciably with specimen length. Thus, tensile failure data cannot simply be normalized by the grip-to-grip length of the test specimen. Experimental data from various laboratories must clearly document the length of the test specimen.     

Nonlinear Mechanical Properties of Prestressed Branched Fibrous Networks

Random fiber networks constitute the solid skeleton of many biological materials such as the cytoskeleton of cells and extracellular matrix of soft tissues. These random networks show unique mechanical properties such as nonlinear shear strain-stiffening and strain softening when subjected to preextension and precompression, respectively. In this study, we perform numerical simulations to characterize the influence of axial prestress on the nonlinear mechanical response of random network structures as a function of their micromechanical and geometrical properties. We build our numerical network models using the microstructure of disordered hexagonal lattices and quantify their nonlinear shear response as a function of uniaxial prestress strain. We consider three different material models for individual fibers and fully characterize their influence on the mechanical response of prestressed networks. Moreover, we investigate both the influence of geometric disorder keeping the network connectivity constant and the influence of the randomness in the stiffness of individual fibers keeping their mean stiffness constant. The effects of network connectivity and bending rigidity of fibers are also determined. Several important conclusions are made, including that the tensile and compressive prestress strains, respectively, increase and decrease the initial network shear stiffness but have no effect on the maximal shear modulus. We discuss the findings in terms of microstructural properties such as the local strain energy distribution.     

Fiber-based modeling of in situ ankle ligaments with consideration of progressive failure

Ligament sprains account for a majority of injuries to the foot and ankle complex among athletic populations. The infeasibility of measuring the in situ response and load paths of individual ligaments has precluded a complete characterization of their mechanical behavior via experiment. In the present study a fiber-based modeling approach of in situ ankle ligaments was developed and validated for determining the heterogeneous force-elongation characteristics and the consequent injury patterns. Nine major ankle ligaments were modeled as bundles of discrete elements, corresponding functionally to the structure of collagen fibers. To incorporate the progressive nature of ligamentous injury, the limit strain at the occurrence of fiber failure was described by a distribution function ranging from 12% to 18% along the width of the insertion site. The model was validated by comparing the structural kinetic and kinematic response obtained experimentally and computationally under well-controlled foot rotations. The simulation results replicated the 6 degree-of-freedom bony motion and ligamentous injuries and, by implication, the in situ deformations of the ligaments. Gross stiffness of the whole ligament derived from the fibers was comparable to existing experimental data. The present modeling approach provides a biomechanically realistic, interpretable and computationally efficient way to characterize the in situ ligament slack, sequential and heterogeneous uncrimping of collagen fascicles and failure propagation as the external load is applied. Applications of this model include functional ankle joint mechanics, injury prevention and countermeasure design for athletes.     

Micromechanical Modeling Study of Mechanical Inhibition of Enzymatic Degradation of Collagen Tissues

This study investigates how the collagen fiber structure influences the enzymatic degradation of collagen tissues. We developed a micromechanical model of a fibrous collagen tissue undergoing enzymatic degradation based on two central hypotheses. The collagen fibers are crimped in the undeformed configuration. Enzymatic degradation is an energy activated process and the activation energy is increased by the axial strain energy density of the fiber. We determined the intrinsic degradation rate and characteristic energy for mechanical inhibition from fibril-level degradation experiments and applied the parameters to predict the effect of the crimped fiber structure and fiber properties on the degradation of bovine cornea and pericardium tissues under controlled tension. We then applied the model to examine the effect of the tissue stress state on the rate of tissue degradation and the anisotropic fiber structures that developed from enzymatic degradation.