Incorporation of experimentally-derived fiber orientation into a structural constitutive model for planar collagenous tissues

Structural constitutive models integrate information on tissue composition and structure, avoiding ambiguities in material characterization. However, critical structural information (such as fiber orientation) must be modeled using assumed statistical distributions, with the distribution parameters estimated from fits to the mechanical test data. Thus, full realization of structural approaches continues to be limited without direct quantitative structural information for direct implementation or to validate model predictions. In the present study, fiber orientation information obtained using small angle light scattering (SALS) was directly incorporated into a structural constitutive model based on work by Lanir (J. Biomech., v. 16, pp. 1-12, 1983). Demonstration of the model was performed using existing biaxial mechanical and fiber orientation data for native bovine pericardium (Sacks and Chuong, ABME, v.26, pp. 892-902, 1998). The structural constitutive model accurately predicted the complete measured biaxial mechanical response. An important aspect of this approach is that only a single equibiaxial test to determine the effective fiber stress-strain response and the SALS-derived fiber orientation distribution were required to determine the complete planar biaxial mechanical response. Changes in collagen fiber crimp under equibiaxial strain suggest that, at the meso-scale, fiber deformations follow the global tissue strains. This result supports the assumption of affine strain to estimate the fiber strains. However, future evaluations will have to be performed for tissue subjected to a wider range of strain to more fully validate the current approach.     

Characterizing local collagen fiber re-alignment and crimp behavior throughout mechanical testing in a mature mouse supraspinatus tendon model

BackgroundCollagen fiber re-alignment and uncrimping are two postulated mechanisms of tendon structural response to load. Recent studies have examined structural changes in response to mechanical testing in a postnatal development mouse supraspinatus tendon model (SST), however, those changes in the mature mouse have not been characterized. The objective of this study was to characterize collagen fiber re-alignment and crimp behavior throughout mechanical testing in a mature mouse SST.Method of approachA tensile mechanical testing set-up integrated with a polarized light system was utilized for alignment and mechanical analysis. Local collagen fiber crimp frequency was quantified immediately following the designated loading protocol using a traditional tensile set up and a flash-freezing method. The effect of number of preconditioning cycles on collagen fiber re-alignment, crimp frequency and mechanical properties in midsubstance and insertion site locations were examined.ResultsDecreases in collagen fiber crimp frequency were identified at the toe-region of the mechanical test at both locations. The insertion site re-aligned throughout the entire test, while the midsubstance re-aligned during preconditioning and the test's linear-region. The insertion site demonstrated a more disorganized collagen fiber distribution, lower mechanical properties and a higher cross-sectional area compared to the midsubstance location.ConclusionsLocal collagen fiber re-alignment, crimp behavior and mechanical properties were characterized in a mature mouse SST model. The insertion site and midsubstance respond differently to mechanical load and have different mechanisms of structural response. Additionally, results support that collagen fiber crimp is a physiologic phenomenon that may explain the mechanical test toe-region.     

Directional dependence of osteoblastic calcium response to mechanical stimuli

In adaptive bone remodeling, mechanical signals such as stress/strain caused by loading/deformation are believed to play important roles as regulators of the process in which osteoclastic resorption and osteoblastic formation are coordinated under a local mechanical environment. The mechanism by which cells sense and transduce mechanical signals to the intracellular biochemical signaling cascade is still unclear, however to address this issue, the present study investigated the characteristic response of a single osteoblastic cell, MC3T3-E1, to a well-defined mechanical stimulus and the involvement of the cytoskeletal actin fiber structure in the mechanotransduction pathway. First, by mechanically perturbing to a single cell using a microneedle, a change in the intracellular calcium ion concentration [Ca²⁺]_i was observed as a primal signaling response to a mechanical stimulus, and the threshold value of the perturbation as the mechanical stimulus was evaluated quantitatively. Second, to study directional dependence of the response to the mechanical stimulus, the effect of actin fiber orientation on the threshold value of the calcium response was investigated at various magnitudes and directions of the stimulus. It was found that the osteoblastic response to the perturbation exhibited a directional dependence. That is, the sensitivity of osteoblastic cells to a mechanical stimulus depends on the angle of the applied deformation with respect to the cytoskeletal actin fiber orientation. This finding is phenomenological evidence that cytoskeletal actin fiber structures are involved in the mechanotransduction mechanism, which may be related to cell polarization behaviors such as cellular alignment caused by mechanical stimulation.     

Mechanistic,Mathematical Model to Predict the Dynamics of Tissue Genesis in Bone Defects via Mechanical Feedback and Mediation of Biochemical Factors

The link between mechanics and biology in the generation and the adaptation of bone has been well studied in context of skeletal development and fracture healing. Yet, the prediction of tissue genesis within - and the spatiotemporal healing of - postnatal defects, necessitates a quantitative evaluation of mechano-biological interactions using experimental and clinical parameters. To address this current gap in knowledge, this study aims to develop a mechanistic mathematical model of tissue genesis using bone morphogenetic protein (BMP) to represent of a class of factors that may coordinate bone healing. Specifically, we developed a mechanistic, mathematical model to predict the dynamics of tissue genesis by periosteal progenitor cells within a long bone defect surrounded by periosteum and stabilized via an intramedullary nail. The emergent material properties and mechanical environment associated with nascent tissue genesis influence the strain stimulus sensed by progenitor cells within the periosteum. Using a mechanical finite element model, periosteal surface strains are predicted as a function of emergent, nascent tissue properties. Strains are then input to a mechanistic mathematical model, where mechanical regulation of BMP-2 production mediates rates of cellular proliferation, differentiation and tissue production, to predict healing outcomes. A parametric approach enables the spatial and temporal prediction of endochondral tissue regeneration, assessed as areas of cartilage and mineralized bone, as functions of radial distance from the periosteum and time. Comparing model results to histological outcomes from two previous studies of periosteum-mediated bone regeneration in a common ovine model, it was shown that mechanistic models incorporating mechanical feedback successfully predict patterns (spatial) and trends (temporal) of bone tissue regeneration. The novel model framework presented here integrates a mechanistic feedback system based on the mechanosensitivity of periosteal progenitor cells, which allows for modeling and prediction of tissue regeneration on multiple length and time scales. Through combination of computational, physical and engineering science approaches, the model platform provides a means to test new hypotheses in silico and to elucidate conditions conducive to endogenous tissue genesis. Next generation models will serve to unravel intrinsic differences in bone genesis by endochondral and intramembranous mechanisms.     

Computational model predicts cell orientation in response to a range of mechanical stimuli

To build anisotropic, mechanically functioning tissue, it is essential to understand how cells orient in response to mechanical stimuli. Therefore, a computational model was developed which predicts cell orientation, based on the actin stress fiber distribution inside the cell. In the model, the stress fiber distribution evolves dynamically according to the following: (1) Stress fibers contain polymerized actin. The total amount of depolymerized plus polymerized actin is constant. (2) Stress fibers apply tension to their environment. This active tension is maximal when strain rate and absolute strain are zero and reduces with increasing shortening rate and absolute strain. (3) A high active fiber stress in a direction leads to a large amount of fibers in this direction. (4) The cell is attached to a substrate; all fiber stresses are homogenized into a total cell stress, which is in equilibrium with substrate stress. This model predicts that on a substrate of anisotropic stiffness, fibers align in the stiffest direction. Under cyclic strain when the cellular environment is so stiff that no compaction occurs (1 MPa), the model predicts strain avoidance, which is more pronounced with increasing strain frequency or amplitude. Under cyclic strain when the cellular environment is so soft that cells can compact it (10 kPa), the model predicts a preference for the cyclically strained compared to the compacting direction. These model predictions all agree with experimental evidence. For the first time, a computational model predicts cell orientation in response to this range of mechanical stimuli using a single set of parameters.     

Tenocyte contraction induces crimp formation in tendon-like tissue

Tendons are composed of longitudinally aligned collagen fibrils arranged in bundles with an undulating pattern, called crimp. The crimp structure is established during embryonic development and plays a vital role in the mechanical behaviour of tendon, acting as a shock-absorber during loading. However, the mechanism of crimp formation is unknown, partly because of the difficulties of studying tendon development in vivo. Here, we used a 3D cell culture system in which embryonic tendon fibroblasts synthesise a tendon-like construct comprised of collagen fibrils arranged in parallel bundles. Investigations using polarised light microscopy, scanning electron microscopy and fluorescence microscopy showed that tendon constructs contained a regular pattern of wavy collagen fibrils. Tensile testing indicated that this superstructure was a form of embryonic crimp producing a characteristic toe region in the stress–strain curves. Furthermore, contraction of tendon fibroblasts was the critical factor in the buckling of collagen fibrils during the formation of the crimp structure. Using these biological data, a finite element model was built that mimics the contraction of the tendon fibroblasts and monitors the response of the Extracellular matrix. The results show that the contraction of the fibroblasts is a sufficient mechanical impulse to build a planar wavy pattern. Furthermore, the value of crimp wavelength was determined by the mechanical properties of the collagen fibrils and inter-fibrillar matrix. Increasing fibril stiffness combined with constant matrix stiffness led to an increase in crimp wavelength. The data suggest a novel mechanism of crimp formation, and the finite element model indicates the minimum requirements to generate a crimp structure in embryonic tendon.     

Monitoring micrometer-scale collagen organization in rat-tail tendon upon mechanical strain using second harmonic microscopy

We continuously monitored the microstructure of a rat-tail tendon during stretch/relaxation cycles. To that purpose, we implemented a new biomechanical device that combined SHG imaging and mechanical testing modalities. This multi-scale experimental device enabled simultaneous visualization of the collagen crimp morphology at the micrometer scale and measurement of macroscopic strain-stress response. We gradually increased the ultimate strain of the cycles and showed that preconditioning mostly occurs in the first stretching. This is accompanied by an increase of the crimp period in the SHG image. Our results indicate that preconditioning is due to a sliding of microstructures at the scale of a few fibrils and smaller, that changes the resting length of the fascicle. This sliding can reverse on long time scales. These results provide a proof of concept that continuous SHG imaging performed simultaneously with mechanical assay allows analysis of the relationship between macroscopic response and microscopic structure of tissues.     

Strain-Induced Alignment in Collagen Gels

Collagen is the most abundant extracellular-network-forming protein in animal biology and is important in both natural and artificial tissues, where it serves as a material of great mechanical versatility. This versatility arises from its almost unique ability to remodel under applied loads into anisotropic and inhomogeneous structures. To explore the origins of this property, we develop a set of analysis tools and a novel experimental setup that probes the mechanical response of fibrous networks in a geometry that mimics a typical deformation profile imposed by cells in vivo. We observe strong fiber alignment and densification as a function of applied strain for both uncrosslinked and crosslinked collagenous networks. This alignment is found to be irreversibly imprinted in uncrosslinked collagen networks, suggesting a simple mechanism for tissue organization at the microscale. However, crosslinked networks display similar fiber alignment and the same geometrical properties as uncrosslinked gels, but with full reversibility. Plasticity is therefore not required to align fibers. On the contrary, our data show that this effect is part of the fundamental non-linear properties of fibrous biological networks.     

A multiscale mechanobiological model of bone remodelling predicts site-specific bone loss in the femur during osteoporosis and mechanical disuse

We propose a multiscale mechanobiological model of bone remodelling to investigate the site-specific evolution of bone volume fraction across the midshaft of a femur. The model includes hormonal regulation and biochemical coupling of bone cell populations, the influence of the microstructure on bone turnover rate, and mechanical adaptation of the tissue. Both microscopic and tissue-scale stress/strain states of the tissue are calculated from macroscopic loads by a combination of beam theory and micromechanical homogenisation. This model is applied to simulate the spatio-temporal evolution of a human midshaft femur scan subjected to two deregulating circumstances: (i) osteoporosis and (ii) mechanical disuse. Both simulated deregulations led to endocortical bone loss, cortical wall thinning and expansion of the medullary cavity, in accordance with experimental findings. Our model suggests that these observations are attributable to a large extent to the influence of the microstructure on bone turnover rate. Mechanical adaptation is found to help preserve intracortical bone matrix near the periosteum. Moreover, it leads to non-uniform cortical wall thickness due to the asymmetry of macroscopic loads introduced by the bending moment. The effect of mechanical adaptation near the endosteum can be greatly affected by whether the mechanical stimulus includes stress concentration effects or not.     

Stability analysis of the two compartment model of cell proliferation.

The behavior of the G₀ model upon perturbation of its equilibrium state is investigated in some detail. Algebraic expressions for stability conditions of the linearized model are derived. A distribution of mitotic times as well as population-dependent transition probabilities are incorporated into the analysis.     

On the Presence of Affine Fibril and Fiber Kinematics in the Mitral Valve Anterior Leaflet

In this study, we evaluated the hypothesis that the constituent fibers follow an affine deformation kinematic model for planar collagenous tissues. Results from two experimental datasets were utilized, taken at two scales (nanometer and micrometer), using mitral valve anterior leaflet (MVAL) tissues as the representative tissue. We simulated MVAL collagen fiber network as an ensemble of undulated fibers under a generalized two-dimensional deformation state, by representing the collagen fibrils based on a planar sinusoidally shaped geometric model. The proposed approach accounted for collagen fibril amplitude, crimp period, and rotation with applied macroscopic tissue-level deformation. When compared to the small angle x-ray scattering measurements, the model fit the data well, with an r² = 0.976. This important finding suggests that, at the homogenized tissue-level scale of ∼1 mm, the collagen fiber network in the MVAL deforms according to an affine kinematics model. Moreover, with respect to understanding its function, affine kinematics suggests that the constituent fibers are largely noninteracting and deform in accordance with the bulk tissue. It also suggests that the collagen fibrils are tightly bounded and deform as a single fiber-level unit. This greatly simplifies the modeling efforts at the tissue and organ levels, because affine kinematics allows a straightforward connection between the macroscopic and local fiber strains. It also suggests that the collagen and elastin fiber networks act independently of each other, with the collagen and elastin forming long fiber networks that allow for free rotations. Such freedom of rotation can greatly facilitate the observed high degree of mechanical anisotropy in the MVAL and other heart valves, which is essential to heart valve function. These apparently novel findings support modeling efforts directed toward improving our fundamental understanding of tissue biomechanics in healthy and diseased conditions.     

Cardiac chemical constant. Part I: Derivation of the characteristic phenomenological equation

A physiochemical parameter is derived and defined as the cardiac chemical equilibrium dissociation constant (K_D), K_D is based upon a phenomenological model in which the cardiac muscle chemical reaction kinetics describe the interconversion between long and short unils (i.e. the individual sarcomere is fully extended or fully contracted). K_D is defined as the ratio of the number of units in the long state to the number of units in the short state. The mathematical development proceeds through four stages: derivation of the governing differential equation during cardiac systole; simplification of the differential equation to describe the cardiac model; determination of the upper and lower limits and average value of Nt (the total number of units in a hypothetical mid-wall circumferential fibre); definition and calculation of the cardiac chemical constant (K_D). K_D is shown to describe a series of equilibrium points throughout cardiac systole. This requires that each mechanical equilibrium state (a series of static, steady-state intervals over time) is also associated with its own specific chemical equilibrium state.     

The relationship between crimp pattern and mechanical response of human patellar tendon-bone units

The objectives of this study are twofold. First, to further develop the understanding of the relationship between the observed mechanical response and changes in the crimp pattern in human patellar tendon bone units. This is accomplished through the use of a specially constructed test frame and microscope system that permits observation and measurement of the crimp patterns as a function of load. Second, the results of the experimental study are used to develop a constitutive equation that includes spatial variation in the crimp pattern. The results of both the experimental and analytical study imply that local strain in the proximity of the attachment site is significantly larger than the strain in the central region of the tendon. The experimental and histological results are for specimens taken from four human bone-patellar tendon-bone units.     

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.     

Tensile properties and fiber alignment of human supraspinatus tendon in the transverse direction demonstrate inhomogeneity,nonlinearity, and regional isotropy

A recent study (Lake et al., 2009); reported the properties of human supraspinatus tendon (SST) tested along the predominant fiber direction. The SST was found to have a relatively disperse distribution of collagen fibers, which may represent an adaptation to multiaxial loads imposed by the complex loading environment of the rotator cuff. However, the multiaxial mechanical properties of human SST remain unknown. The objective of this study, therefore, was to evaluate the mechanical properties, fiber alignment, change in alignment with applied load, and structure–function relationships of SST in transverse testing. Samples from six SST locations were tested in uniaxial tension with samples oriented transverse to the tendon long-axis. Polarized light imaging was used to quantify collagen fiber alignment and change in alignment under applied load. The mechanical properties of samples taken near the tendon–bone insertion were much greater on the bursal surface compared to the joint surface (e.g., bursal moduli 15–30 times greater than joint; p<0.001). In fact, the transverse moduli values of the bursal samples were very similar to values obtained from samples tested along the tendon long-axis (Lake et al., 2009). This key and unexpected finding suggests planar mechanical isotropy for bursal surface samples near the insertion, which may be due to complex in vivo loading. Organizationally, fiber distributions became less aligned along the tendon long-axis in the toe-region of the stress–strain response. Alignment changes occurred to a slightly lesser degree in the linear-region, suggesting that movement of collagen fibers may play a role in mechanical nonlinearity. Transverse mechanical properties were significantly correlated with fiber alignment (e.g., for linear-region modulus r_s=0.74, p<0.0001), demonstrating strong structure–function relationships. These results greatly enhance current understanding of the properties of human SST and provide clinicians and scientists with vital information in attempting to treat or replace this complex tissue.     

The dynamics of collagen uncrimping and lateral contraction in tendon and the effect of ionic concentration

Under tensile loading, tendon undergoes a number of unique structural changes that govern its mechanical response. For example, stretching a tendon is known to induce both the progressive “uncrimping” of wavy collagen fibrils and extensive lateral contraction mediated by fluid flow out of the tissue. However, it is not known whether these processes are interdependent. Moreover, the rate-dependence of collagen uncrimping and its contribution to tendon's viscoelastic mechanical properties are unknown. Therefore, the objective of this study was to (a) develop a methodology allowing for simultaneous measurement of crimp, stress, axial strain and lateral contraction in tendon under dynamic loading; (b) determine the interdependence of collagen uncrimping and lateral contraction by testing tendons in different swelling conditions; and (c) assess how the process of collagen uncrimping depends on loading rate. Murine flexor carpi ulnaris (FCU) tendons in varying ionic environments were dynamically stretched to a set strain level and imaged through a plane polariscope with the polarizer and analyzer at a fixed angle. Analysis of the resulting images allowed for direct measurement of the crimp frequency and indirect measurement of the tendon thickness. Our findings demonstrate that collagen uncrimping and lateral contraction can occur independently and interstitial fluid impacts tendon mechanics directly. Furthermore, tensile stress, transverse contraction and degree of collagen uncrimping were all rate-dependent, suggesting that collagen uncrimping plays a role in tendon's dynamic mechanical response. This study is the first to characterize the time-dependence of collagen uncrimping in tendon, and establishes structure–function relationships for healthy tendons that can be used to better understand and assess changes in tendon mechanics after disease or injury.     

Recruitment of tendon crimp with applied tensile strain

The tensile stress-strain behavior of ligaments and tendons begins with a toe region that is believed to result from the straightening of crimped collagen fibrils. The in situ mechanical function is mostly confined to this toe region and changes in crimp morphology are believed to be associated with pathological conditions. A relatively new imaging technique, optical coherence tomography (OCT), provides a comparatively inexpensive method for nondestructive investigation of tissue ultrastructure with resolution on the order of 15 microm and the potential for use in a clinical setting. The objectives of this work were to assess the utility of OCT for visualizing crimp period, and to use OCT to determine how crimp period changed as a function of applied tensile strain in rat tail tendon fascicles. Fascicles from rat tail tendons were subjected to 0.5 percent strain increments up to 5 percent and imaged at each increment using OCT. A comparison between OCT images and optical microscopy images taken between crossed polarizing lenses showed a visual correspondence between features indicative of crimp pattern. Crimp pattern always disappeared completely before 3 percent axial strain was reached. Average crimp period increased as strain increased, but both elongation and shortening occurred within single crimp periods during the application of increasing strain to the fascicle.     

Structure and torsion in the normal and situs inversus totalis cardiac left ventricle. II. Modeling cardiac adaptation to mechanical load

Mathematical models provide a suitable platform to test hypotheses on the relation between local mechanical stimuli and responses to cardiac structure and geometry. In the present model study, we tested hypothesized mechanical stimuli and responses in cardiac adaptation to mechanical load on their ability to estimate a realistic myocardial structure of the normal and situs inversus totalis (SIT) left ventricle (LV). In a cylindrical model of the LV, 1) mass was adapted in response to myofiber strain at the beginning of ejection and to global contractility (average systolic pressure), 2) cavity volume was adapted in response to fiber strain during ejection, and 3) myofiber orientations were adapted in response to myofiber strain during ejection and local misalignment between neighboring tissue parts. The model was able to generate a realistic normal LV geometry and structure. In addition, the model was also able to simulate the instigating situation in the rare SIT LV with opposite torsion and transmural courses in myofiber direction between the apex and base [Delhaas et al. (6)]. These results substantiate the importance of mechanical load in the formation and maintenance of cardiac structure and geometry. Furthermore, in the model, adapted myocardial architecture was found to be insensitive to fiber misalignment in the transmural direction, i.e., myofiber strain during ejection was sufficient to generate a realistic transmural variation in myofiber orientation. In addition, the model estimates that, despite differences in structure, global pump work and the mass of the normal and SIT LV are similar.     

Electron transfer between the two photosystems. II. Equilibrium constants

(1) The equilibrium constants for the redox reactions occurring between Photosystem (PS) I donors were measured on chloroplasts, dark-adapted in the presence of sodium ascorbate and 3-(3′,4′-dichlorophenyl)-1,1-dimethylurea (DCMU) and then illuminated by d.c. light. The equilibrium constant for the electron transfer between plastocyanin and P-700 is close to 1 and the overall equilibrium constant between cytochrome f and P-700 is about 2.3. As these equilibrium constants do not depend upon the intensity of the d.c. beam, the low values we measured cannot be due to kinetic limitations. (2) The equilibrium constants were measured also in the absence of DCMU using chloroplasts in oxidizing conditions (ferricyanide or far red illumination) illuminated by a saturating flash. During the course of the reduction of PS I donors by plastoquinol molecules formed by the flash, the equilibrium constants are higher than in the preceding conditions: the value for plastocyanin to P-700 is close to 5, and that for cytochrome f to P-700 is about 25. (3) The variations of these equilibrium constants are tentatively interpreted as being due to mutual electrostatic interactions between cytochrome b and f which are included in the same complex. This model implies that the perturbation of the redox properties of cytochrome f by a positive charge located on cytochrome b is identical to the perturbation of the redox properties of cytochrome b by a positive charge located on cytochrome f.