A microstructurally driven model for pulmonary artery tissue

A new constitutive model for elastic, proximal pulmonary artery tissue is presented here, called the total crimped fiber model. This model is based on the material and microstructural properties of the two main, passive, load-bearing components of the artery wall, elastin, and collagen. Elastin matrix proteins are modeled with an orthotropic neo-Hookean material. High stretch behavior is governed by an orthotropic crimped fiber material modeled as a planar sinusoidal linear elastic beam, which represents collagen fiber deformations. Collagen-dependent artery orthotropy is defined by a structure tensor representing the effective orientation distribution of collagen fiber bundles. Therefore, every parameter of the total crimped fiber model is correlated with either a physiologic structure or geometry or is a mechanically measured material property of the composite tissue. Further, by incorporating elastin orthotropy, this model better represents the mechanics of arterial tissue deformation. These advancements result in a microstructural total crimped fiber model of pulmonary artery tissue mechanics, which demonstrates good quality of fit and flexibility for modeling varied mechanical behaviors encountered in disease states.     

Self-crimping, biodegradable, electrospun polymer microfibers

Semicrystalline poly(l-lactide-co-ε-caprolactone) (P(LLA-CL)) was used to produce electrospun fibers with diameters on the subcellular scale. P(LLA-CL) was chosen because it is biocompatible and its chemical and physical properties are easily tunable. The use of a rotating wire mandrel as a collection device in the electrospinning process, along with high collection speeds, was used to align electrospun fibers. Upon removal of the fibers from the mandrel, the fibers shrunk in length, producing a crimp pattern characteristic of collagen fibrils in soft connective tissues. The crimping effect was determined to be a result of the residual stresses resident in the fibers due to the fiber alignment process and the difference between the operating temperature (T(op)) and the glass-transition temperature (T(g)) of the polymer. The electrospun fibers could be induced to crimp by adjusting the operating temperature to be greater than that of the polymer glass-transition temperature. Moreover, the crimped fibers exhibited a toe region in their stress-strain profile that is characteristic of collagen present in tendons and ligaments. The crimp pattern was retained during in vitro degradation over 4 weeks. Primary bovine fibroblasts seeded onto these crimped fibers attached, proliferated, and deposited extracellular matrix (ECM) molecules on the surface of the fiber mats. These self-crimping fibers hold great promise for use in tissue engineering scaffolds for connective tissues that require fibers similar in structure to that of crimped collagen fibrils.     

A composite micromechanical model for connective tissues: Part I--Theory.

A micromechanical model has been developed to study and predict the mechanical behavior of fibrous soft tissues. The model uses the theorems of least work and minimum potential energy to predict upper and lower bounds on material behavior based on the structure and properties of tissue components. The basic model consists of a composite of crimped collagen fibers embedded in an elastic glycosaminoglycan matrix. Upper and lower bound aggregation rules predict composite material behavior under the assumptions of uniform strain and uniform stress, respectively. Input parameters consist of the component material properties and the geometric configuration of the fibers. The model may be applied to a variety of connective tissue structures and is valuable in giving insight into material behavior and the nature of interactions between tissue components in various structures. Application of the model to rat tail tendon and cat knee joint capsule is described in a companion paper [2].     

Collagen fiber alignment does not explain mechanical anisotropy in fibroblast populated collagen gels

Many load-bearing soft tissues exhibit mechanical anisotropy. In order to understand the behavior of natural tissues and to create tissue engineered replacements, quantitative relationships must be developed between the tissue structures and their mechanical behavior. We used a novel collagen gel system to test the hypothesis that collagen fiber alignment is the primary mechanism for the mechanical anisotropy we have reported in structurally anisotropic gels. Loading constraints applied during culture were used to control the structural organization of the collagen fibers of fibroblast populated collagen gels. Gels constrained uniaxially during culture developed fiber alignment and a high degree of mechanical anisotropy, while gels constrained biaxially remained isotropic with randomly distributed collagen fibers. We hypothesized that the mechanical anisotropy that developed in these gels was due primarily to collagen fiber orientation. We tested this hypothesis using two mathematical models that incorporated measured collagen fiber orientations: a structural continuum model that assumes affine fiber kinematics and a network model that allows for nonaffine fiber kinematics. Collagen fiber mechanical properties were determined by fitting biaxial mechanical test data from isotropic collagen gels. The fiber properties of each isotropic gel were then used to predict the biaxial mechanical behavior of paired anisotropic gels. Both models accurately described the isotropic collagen gel behavior. However, the structural continuum model dramatically underestimated the level of mechanical anisotropy in aligned collagen gels despite incorporation of measured fiber orientations; when estimated remodeling-induced changes in collagen fiber length were included, the continuum model slightly overestimated mechanical anisotropy. The network model provided the closest match to experimental data from aligned collagen gels, but still did not fully explain the observed mechanics. Two different modeling approaches showed that the level of collagen fiber alignment in our uniaxially constrained gels cannot explain the high degree of mechanical anisotropy observed in these gels. Our modeling results suggest that remodeling-induced redistribution of collagen fiber lengths, nonaffine fiber kinematics, or some combination of these effects must also be considered in order to explain the dramatic mechanical anisotropy observed in this collagen gel model system.     

A computational remodeling approach to predict the physiological architecture of the collagen fibril network in corneo-scleral shells

Organized collagen fibrils form complex networks that introduce strong anisotropic and highly nonlinear attributes into the constitutive response of human eye tissues. Physiological adaptation of the collagen network and the mechanical condition within biological tissues are complex and mutually dependent phenomena. In this contribution, a computational model is presented to investigate the interaction between the collagen fibril architecture and mechanical loading conditions in the corneo-scleral shell. The biomechanical properties of eye tissues are derived from the single crimped fibril at the micro-scale via the collagen network of distributed fibrils at the meso-scale to the incompressible and anisotropic soft tissue at the macro-scale. Biomechanically induced remodeling of the collagen network is captured on the meso-scale by allowing for a continuous re-orientation of preferred fibril orientations and a continuous adaptation of the fibril dispersion. The presented approach is applied to a numerical human eye model considering the cornea and sclera. The predicted fibril morphology correlates well with experimental observations from X-ray scattering data.     

Hyperelastic modeling of the combined effects of tissue swelling and deformation-related collagen renewal in fibrous soft tissue

A continuum mechanics constitutive model is presented for the interaction between swelling and collagen remodeling in biological soft tissue. The model is inherently two-way: swelling stretches the collagen fibers which affects their rate of degradation—the remodeled fibrous microarchitecture provides selective directional stiffening that causes the swollen tissue to expand more in the unreinforced directions. The constitutive model specifically treats stretch-stabilization wherein the rate of enzymatic-induced degradation of collagen is a decreasing function of fiber stretch. New collagen replacement takes place in a generally swollen environment, and this synthesis is tracked as a function of time by means of a time integration scheme that accounts for the historical sequence of collagen recreation. The model allows for the specification of the collagen pre-stretch at the time of first synthesis, thus allowing for the consideration of either initially limp replacement fiber or initially pre-tensioned replacement fiber. Loading and swelling that occurs on time scales that are commensurate with the natural time scales for fiber degradation and replacement lead to the consideration of time-integral constitutive equations. Loading and swelling that take place on time scales that are very different from that of the remodeling time scales provide a simplified treatment in which there are definite notions of a short-time instantaneous response and also a large-time approach to a steady-state condition of homeostasis.     

The use of small angle light scattering in assessing strain induced collagen degradation in arterial tissue ex vivo

Collagen is the predominant load bearing component in many soft tissues including arterial tissue and is therefore critical in determining the mechanical integrity of such tissues. Degradation of collagen fibres is hypothesized to be a strain dependent process whereby the rate of degradation is affected by the magnitude of strain applied to the collagen fibres. The aim of this study is to investigate the ability of small angle light scattering (SALS) imaging to identify strain dependent degradation of collagen fibres in arterial tissue ex vivo, and determine whether a strain induced protection mechanism exists in arterial tissue as observed in pure collagen and other collagenous tissues. SALS was used in combination with histological and second harmonic generation (SHG) analysis to determine the collagen fibre architecture in arterial tissue subjected to strain directed degradation. SALS alignment analysis identified statistically significant differences in fibre alignment depending on the strain magnitude applied to the tissue. These results were also observed using histology and SHG. Our findings suggest a strain protection mechanism may exist for arterial collagen at intermediate strain magnitudes between 0% and 25%. These findings may have implications for the onset and progression of arterial disease where changes in the mechanical environment of arterial tissue may lead to changes in the collagen degradation rate.     

Dependence of invadopodia function on collagen fiber spacing and cross-linking: computational modeling and experimental evidence

Invadopodia are subcellular organelles thought to be critical for extracellular matrix (ECM) degradation and the movement of cells through tissues. Here we examine invadopodia generation, turnover, and function in relation to two structural aspects of the ECM substrates they degrade: cross-linking and fiber density. We set up a cellular automaton computational model that simulates ECM penetration and degradation by invadopodia. Experiments with denatured collagen (gelatin) were used to calibrate the model and demonstrate the inhibitory effect of ECM cross-linking on invadopodia degradation and penetration. Incorporation of dynamic invadopodia behavior into the model amplified the effect of cross-linking on ECM degradation, and was used to model feedback from the ECM. When the model was parameterized with spatial fibrillar dimensions that closely matched the organization, in real life, of native ECM collagen into triple-helical monomers, microfibrils, and macrofibrils, little or no inhibition of invadopodia penetration was observed in simulations of sparse collagen gels, no matter how high the degree of cross-linking. Experimental validation, using live-cell imaging of invadopodia in cells plated on cross-linked gelatin, was consistent with simulations in which ECM cross-linking led to higher rates of both invadopodia retraction and formation. Analyses of invadopodia function from cells plated on cross-linked gelatin and collagen gels under standard concentrations were consistent with simulation results in which sparse collagen gels provided a weak barrier to invadopodia. These results suggest that the organization of collagen, as it may occur in stroma or in vitro collagen gels, forms gaps large enough so as to have little impact on invadopodia penetration/degradation. By contrast, dense ECM, such as gelatin or possibly basement membranes, is an effective obstacle to invadopodia penetration and degradation, particularly when cross-linked. These results provide a novel framework for further studies on ECM structure and modifications that affect invadopodia and tissue invasion by cells.     

Mechanisms of Plastic Deformation in Collagen Networks Induced by Cellular Forces

Contractile cells can reorganize fibrous extracellular matrices and form dense tracts of fibers between neighboring cells. These tracts guide the development of tubular tissue structures and provide paths for the invasion of cancer cells. Here, we studied the mechanisms of the mechanical plasticity of collagen tracts formed by contractile premalignant acinar cells and fibroblasts. Using fluorescence microscopy and second harmonic generation, we quantified the collagen densification, fiber alignment, and strains that remain within the tracts after cellular forces are abolished. We explained these observations using a theoretical fiber network model that accounts for the stretch-dependent formation of weak cross-links between nearby fibers. We tested the predictions of our model using shear rheology experiments. Both our model and rheological experiments demonstrated that increasing collagen concentration leads to substantial increases in plasticity. We also considered the effect of permanent elongation of fibers on network plasticity and derived a phase diagram that classifies the dominant mechanisms of plasticity based on the rate and magnitude of deformation and the mechanical properties of individual fibers. Plasticity is caused by the formation of new cross-links if moderate strains are applied at small rates or due to permanent fiber elongation if large strains are applied over short periods. Finally, we developed a coarse-grained model for plastic deformation of collagen networks that can be employed to simulate multicellular interactions in processes such as morphogenesis, cancer invasion, and fibrosis.     

A physically motivated constitutive model for cell-mediated compaction and collagen remodeling in soft tissues

Collagen is the main load-bearing component of many soft tissues and has a large influence on the mechanical behavior of tissues when exposed to mechanical loading. Therefore, it is important to increase our understanding of collagen remodeling in soft tissues to understand the mechanisms behind pathologies and to control the development of the collagen network in engineered tissues. In the present study, a constitutive model was developed by coupling a recently developed model describing the orientation and contractile stresses exerted by cells in response to mechanical stimuli to physically motivated collagen remodeling laws. In addition, cell-mediated contraction of the collagen fibers was included as a mechanism for tissue compaction. The model appeared to be successful in predicting a range of experimental observations, which are (1) the change in transition stretch of periosteum after remodeling at different applied stretches, (2) the compaction and alignment of collagen fibers in tissue-engineered strips, (3) the fiber alignment in cruciform gels with different arm widths, and (4) the alignment of collagen fibers in engineered vascular grafts. Moreover, by changing the boundary conditions, the model was able to predict a helical architecture in the vascular graft without assuming the presence of two helical fiber families a priori. Ultimately, this model may help to increase our understanding of collagen remodeling in physiological and pathological conditions, and it may provide a tool for determining the optimal experimental conditions for obtaining native-like collagen architectures in engineered tissues.     

Diffusion anisotropy in collagen gels and tumors: the effect of fiber network orientation

The interstitial matrix is comprised of cross-linked collagen fibers, generally arranged in nonisotropic orientations. Spatial alignment of matrix components within the tissue can affect diffusion patterns of drugs. In this study, we developed a methodology for the calculation of diffusion coefficients of macromolecules and nanoparticles in collagenous tissues. The tissues are modeled as three-dimensional, stochastic, fiber networks with varying degrees of alignment. We employed a random walk approach to simulate diffusion and a Stokesian dynamics method to account for hydrodynamic hindrance. We performed our analysis for four different structures ranging from nearly isotropic to perfectly aligned. We showed that the overall diffusion coefficient is not affected by the orientation of the network. However, structural anisotropy results in diffusion anisotropy, which becomes more significant with increase in the degree of alignment, the size of the diffusing particle, and the fiber volume fraction. To test our model predictions we performed diffusion measurements in reconstituted collagen gels and tumor xenografts. We measured fiber alignment and diffusion with second harmonic generation and multiphoton fluorescent recovery after photobleaching techniques, respectively. The results showed for the first time in tumors that the structure and orientation of collagen fibers in the extracellular space leads to diffusion anisotropy.     

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.     

Remodelling of the angular collagen fiber distribution in cardiovascular tissues

Understanding collagen fiber remodelling is desired to optimize the mechanical conditioning protocols in tissue-engineering of load-bearing cardiovascular structures. Mathematical models offer strong possibilities to gain insight into the mechanisms and mechanical stimuli involved in these remodelling processes. In this study, a framework is proposed to investigate remodelling of angular collagen fiber distribution in cardiovascular tissues. A structurally based model for collagenous cardiovascular tissues is extended with remodelling laws for the collagen architecture, and the model is subsequently applied to the arterial wall and aortic valve. For the arterial wall, the model predicts the presence of two helically arranged families of collagen fibers. A branching, diverging hammock-type fiber architecture is predicted for the aortic valve. It is expected that the proposed model may be of great potential for the design of improved tissue engineering protocols and may give further insight into the pathophysiology of cardiovascular diseases.     

Contributions of Glycosaminoglycans to Collagen Fiber Recruitment in Constitutive Modeling of Arterial Mechanics

The contribution of glycosaminoglycans (GAGs) to the biological and mechanical functions of biological tissue has emerged as an important area of research. GAGs provide structural basis for the organization and assembly of extracellular matrix (ECM). The mechanics of tissue with low GAG content can be indirectly affected by the interaction of GAGs with collagen fibers, which have long been known to be one of the primary contributors to soft tissue mechanics. Our earlier study showed that enzymatic GAG depletion results in straighter collagen fibers that are recruited at lower levels of stretch, and a corresponding shift in earlier arterial stiffening (Mattson et al., 2016). In this study, the effect of GAGs on collagen fiber recruitment was studied through a structure-based constitutive model. The model incorporates structural information, such as fiber orientation distribution, content, and recruitment of medial elastin, medial collagen, and adventitial collagen fibers. The model was first used to study planar biaxial tensile stress-stretch behavior of porcine descending thoracic aorta. Changes in elastin and collagen fiber orientation distribution, and collagen fiber recruitment were then incorporated into the model in order to predict the stress-stretch behavior of GAG depleted tissue. Our study shows that incorporating early collagen fiber recruitment into the model predicts the stress-stretch response of GAG depleted tissue reasonably well (rms = 0.141); considering further changes of fiber orientation distribution does not improve the predicting capability (rms = 0.149). Our study suggests an important role of GAGs in arterial mechanics that should be considered in developing constitutive models.     

Investigation on interaction of tannic acid with type I collagen and its effect on thermal,enzymatic, and conformational stability for tissue engineering applications

Collagen is an essential component of tissues, which is the most abundant component in extracellular matrix and highly conserved across the animal kingdom. It can assemble into fiber and play an essential role in cell adhesion and growth and could be extremely useful in tissue engineering. In this study, the effect of tannic acid (TA) on the thermal, enzymatic and conformational stability of type I collagen has been investigated for the development of collagen‐based biomaterials. Interaction of TA with collagen demonstrates the role of hydrogen bonding and hydrophobic interaction in providing the thermal and enzymatic stability. Thermal analysis studies reveal that, hydrothermal stability of collagen increases as well as inhibits the breakdown of collagenase by formation of hydrogen bonds and hydrophobic interactions. TA binds to the collagen with high affinity because the structural flexibility of the collagen compensates for the structural rigidity of the phenolics. Increase in concentration of TA induces significant change in the conformation of triple helix. The free binding energy of TA with collagen‐like peptide was determined to be in the range of ?9.4 to ?11.2 kcal mol^?1, which was calculated by using Autodock Vina software and showed numerous hydrophobic and hydrogen bond interactions. We anticipate that these collagen‐based biomaterials hold great potential for biomedical applications. © 2013 Wiley Periodicals, Inc. Biopolymers 101: 471–483, 2014.     

Remodeling of the collagen fiber architecture due to compaction in small vessels under tissue engineered conditions

Mechanical loading protocols in tissue engineering (TE) aim to improve the deposition of a properly organized collagen fiber network. In addition to collagen remodeling, these conditioning protocols can result in tissue compaction. Tissue compaction is beneficial to tissue collagen alignment, yet it may lead to a loss of functionality of the TE construct due to changes in geometry after culture. Here, a mathematical model is presented to relate the changes in collagen architecture to the local compaction within a TE small blood vessel, assuming that under static conditions, compaction is the main factor responsible for collagen fiber organization. An existing structurally based model is extended to incorporate volumetric tissue compaction. Subsequently, the model is applied to describe the collagen architecture of TE constructs under either strain based or stress based stimulus functions. Our computations indicate that stress based simulations result in a helical collagen fiber distribution along the vessel wall. The helix pitch angle increases from a circumferential direction in the inner wall, over about 45 deg in the middle vessel layer, to a longitudinal direction in the outer wall. These results are consistent with experimental data from TE small diameter blood vessels. In addition, our results suggest a stress dependent remodeling of the collagen, suggesting that cell traction is responsible for collagen orientation. These findings may be of value to design improved mechanical conditioning protocols to optimize the collagen architecture in engineered tissues.     

Improved prediction of the collagen fiber architecture in the aortic heart valve

Living tissues show an adaptive response to mechanical loading by changing their internal structure and morphology. Understanding this response is essential for successful tissue engineering of load-bearing structures, such as the aortic valve. In this study, mechanically induced remodeling of the collagen architecture in the aortic valve was investigated. It was hypothesized that, in uniaxially loaded regions, the fibers aligned with the tensile principal stretch direction. For biaxial loading conditions, on the other hand, it was assumed that the collagen fibers aligned with directions situated between the principal stretch directions. This hypothesis has already been applied successfully to study collagen remodeling in arteries. The predicted fiber architecture represented a branching network and resembled the macroscopically visible collagen bundles in the native leaflet. In addition, the complex biaxial mechanical behavior of the native valve could be simulated qualitatively with the predicted fiber directions. The results of the present model might be used to gain further insight into the response of tissue engineered constructs during mechanical conditioning.     

Scleral anisotropy and its effects on the mechanical response of the optic nerve head

This paper presents a computational modeling study of the effects of the collagen fiber structure on the mechanical response of the sclera and the adjacent optic nerve head (ONH). A specimen-specific inverse finite element method was developed to determine the material properties of two human sclera subjected to full-field inflation experiments. A distributed fiber model was applied to describe the anisotropic elastic behavior of the sclera. The model directly incorporated wide-angle X-ray scattering measurements of the anisotropic collagen structure. The converged solution of the inverse method was used in micromechanical studies of the mechanical anisotropy of the sclera at different scales. The effects of the scleral collagen fiber structure on the ONH deformation were evaluated by progressively filtering out local anisotropic features. It was found that the majority of the midposterior sclera could be described as isotropic without significantly affecting the mechanical response of the tissues of the ONH. In contrast, removing local anisotropic features in the peripapillary sclera produced significant changes in scleral canal expansion and lamina cribrosa deformation. Local variations in the collagen structure of the peripapillary sclera significantly influenced the mechanical response of the ONH.