A constitutive model for vascular tissue that integrates fibril, fiber and continuum levels with application to the isotropic and passive properties of the infrarenal aorta

A fundamental understanding of the mechanical properties of the extracellular matrix (ECM) is critically important to quantify the amount of macroscopic stress and/or strain transmitted to the cellular level of vascular tissue. Structural constitutive models integrate histological and mechanical information, and hence, allocate stress and strain to the different microstructural components of the vascular wall. The present work proposes a novel multi-scale structural constitutive model of passive vascular tissue, where collagen fibers are assembled by proteoglycan (PG) cross-linked collagen fibrils and reinforce an otherwise isotropic matrix material. Multiplicative kinematics account for the straightening and stretching of collagen fibrils, and an orientation density function captures the spatial organization of collagen fibers in the tissue. Mechanical and structural assumptions at the collagen fibril level define a piece-wise analytical stress-stretch response of collagen fibers, which in turn is integrated over the unit sphere to constitute the tissue's macroscopic mechanical properties. The proposed model displays the salient macroscopic features of vascular tissue, and employs the material and structural parameters of clear physical meaning. Likewise, the constitutive concept renders a highly efficient multi-scale structural approach that allows for the numerical analysis at the organ level. Model parameters were estimated from isotropic mean-population data of the normal and aneurysmatic aortic wall and used to predict in-vivo stress states of patient-specific vascular geometries, thought to demonstrate the robustness of the particular Finite Element (FE) implementation. The collagen fibril level of the multi-scale constitutive formulation provided an interface to integrate vascular wall biology and to account for collagen turnover.     

A multiaxial constitutive law for mammalian left ventricular myocardium in steady-state barium contracture or tetanus.

The constitutive law of the material comprising any structure is essential for mechanical analysis since this law enables calculation of the stresses from the deformations and vice versa. To date, there is no constitutive law for actively contracting myocardial tissue. Using 2,3-butanedione monoxime to protect the myocardium from mechanical trauma, we subjected thin midwall slices of rabbit myocardium to multiaxial stretching first in the passive state and then during steady-state barium contracture or during tetani in ryanodine-loaded tissue. Assuming transverse isotropy in both the passive and active conditions, we used our previously described methods (Humphrey et al., 1990a) to obtain both passive and active constitutive laws. The major results of this study are: (1) This is the first multiaxial constitutive law for actively contracting mammalian myocardium. (2) The functional forms of the constitutive law for barium contracture and ryanodine-induced tetani are the same but differ from those in the passive state. Hence, one cannot simply substitute differing values for the coefficients of the passive law to describe the active tissue properties. (3) There are significant stresses developed in the cross-fiber direction (more than 40 percent of those in the fiber direction) that cannot be attributed to either deformation effects or nonparallel muscle fibers. These results provide the foundation for future mechanical analyses of the heart.     

A structural constitutive model considering angular dispersion and waviness of collagen fibres of rabbit facial veins

Background  

Structural constitutive models of vascular wall integrate information on composition and structural arrangements of tissue. In blood vessels, collagen fibres are arranged in coiled and wavy bundles and the individual collagen fibres have a deviation from their mean orientation. A complete structural constitutive model for vascular wall should incorporate both waviness and orientational distribution of fibres. We have previously developed a model, for passive properties of vascular wall, which considers the waviness of collagen fibres. However, to our knowledge there is no structural model of vascular wall which integrates both these features.     

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.     

Experimental investigation of collagen waviness and orientation in the arterial adventitia using confocal laser scanning microscopy

Mechanical properties of the adventitia are largely determined by the organization of collagen fibers. Measurements on the waviness and orientation of collagen, particularly at the zero-stress state, are necessary to relate the structural organization of collagen to the mechanical response of the adventitia. Using the fluorescence collagen marker CNA38-OG488 and confocal laser scanning microscopy, we imaged collagen fibers in the adventitia of rabbit common carotid arteries ex vivo. The arteries were cut open along their longitudinal axes to get the zero-stress state. We used semi-manual and automatic techniques to measure parameters related to the waviness and orientation of fibers. Our results showed that the straightness parameter (defined as the ratio between the distances of endpoints of a fiber to its length) was distributed with a beta distribution (mean value 0.72, variance 0.028) and did not depend on the mean angle orientation of fibers. Local angular density distributions revealed four axially symmetric families of fibers with mean directions of 0°, 90°, 43° and ?43°, with respect to the axial direction of the artery, and corresponding circular standard deviations of 40°, 47°, 37° and 37°. The distribution of local orientations was shifted to the circumferential direction when measured in arteries at the zero-load state (intact), as compared to arteries at the zero-stress state (cut-open). Information on collagen fiber waviness and orientation, such as obtained in this study, could be used to develop structural models of the adventitia, providing better means for analyzing and understanding the mechanical properties of vascular wall.     

Multiscale, structure-based modeling for the elastic mechanical behavior of arterial walls

Passive elastic behavior of arterial wall remains difficult to model. Although phenomenological and structural models exist, the question of how the three-dimensional network structure of the collagen in the artery determines its mechanical properties is still open. A model is presented that incorporates a collagen network as well as the noncollagenous material that comprise the artery. The collagen architecture is represented as a network of interconnected fibers, and a neo-Hookean constitutive equation is used to describe the contribution of the noncollagenous matrix. The model is multiscale in that volume-averaging theory is applied to the collagen network, and it is structural in that parameters of the microstructure of the collagen network were considered instead of a macroscopic constitutive law. The computational results provided a good fit to published experimental data for decellularized porcine carotid arteries. The model predicted increased circumferential compliance for increased axial stretch, consistent with previously published reports, and a relatively small sensitivity to open angle. Even at large extensions, the model predicted that the noncollagenous matrix would be in compression, preventing collapse of the collagen network. The incorporation of fiber-fiber interactions led to an accurate model of artery wall behavior with relatively few parameters. The counterintuitive result that the noncollagenous component is in compression during extension and inflation of the tissue suggests that the collagen is important even at small strains, with the noncollagenous components supporting the network, but not resisting the load directly. More accurate representation of the microstructure of the artery wall is needed to explore this issue further.     

A novel constitutive model for passive right ventricular myocardium: evidence for myofiber–collagen fiber mechanical coupling

The function of right ventricle (RV) is recognized to play a key role in the development of many cardiopulmonary disorders, such as pulmonary arterial hypertension (PAH). Given the strong link between tissue structure and mechanical behavior, there remains a need for a myocardial constitutive model that accurately accounts for right ventricular myocardium architecture. Moreover, most available myocardial constitutive models approach myocardium at the length scale of mean fiber orientation and do not explicitly account for different fibrous constituents and possible interactions among them. In the present work, we developed a fiber-level constitutive model for the passive mechanical behavior of the right ventricular free wall (RVFW). The model explicitly separates the mechanical contributions of myofiber and collagen fiber ensembles, and accounts for the mechanical interactions between them. To obtain model parameters for the healthy passive RVFW, the model was informed by transmural orientation distribution measurements of myo- and collagen fibers and was fit to the mechanical testing data, where both sets of data were obtained from recent experimental studies on non-contractile, but viable, murine RVFW specimens. Results supported the hypothesis that in the low-strain regime, the behavior of the RVFW is governed by myofiber response alone, which does not demonstrate any coupling between different myofiber ensembles. At higher strains, the collagen fibers and their interactions with myofibers begin to gradually contribute and dominate the behavior as recruitment proceeds. Due to the use of viable myocardial tissue, the contribution of myofibers was significant at all strains with the predicted tensile modulus of \(\sim \)32 kPa. This was in contrast to earlier reports (Horowitz et al. 1988) where the contribution of myofibers was found to be insignificant. Also, we found that the interaction between myo- and collagen fibers was greatest under equibiaxial strain, with its contribution to the total stress not exceeding 20 %. The present model can be applied to organ-level computational models of right ventricular dysfunction for efficient diagnosis and evaluation of pulmonary hypertension disorder.     

Arterial Extracellular Matrix: A Mechanobiological Study of the Contributions and Interactions of Elastin and Collagen

The complex network structure of elastin and collagen extracellular matrix (ECM) forms the primary load bearing components in the arterial wall. The structural and mechanobiological interactions between elastin and collagen are important for properly functioning arteries. Here, we examined the elastin and collagen organization, realignment, and recruitment by coupling mechanical loading and multiphoton imaging. Two-photon excitation fluorescence and second harmonic generation methods were performed with a multiphoton video-rate microscope to capture real time changes to the elastin and collagen structure during biaxial deformation. Enzymatic removal of elastin was performed to assess the structural changes of the remaining collagen structure. Quantitative analysis of the structural changes to elastin and collagen was made using a combination of two-dimensional fast Fourier transform and fractal analysis, which allows for a more complete understanding of structural changes. Our study provides new quantitative evidence, to our knowledge on the sequential engagement of different arterial ECM components in response to mechanical loading. The adventitial collagen exists as large wavy bundles of fibers that exhibit fiber engagement after 20% strain. The medial collagen is engaged throughout the stretching process, and prominent elastic fiber engagement is observed up to 20% strain after which the engagement plateaus. The fiber orientation distribution functions show remarkably different changes in the ECM structure in response to mechanical loading. The medial collagen shows an evident preferred circumferential distribution, however the fiber families of adventitial collagen are obscured by their waviness at no or low mechanical strains. Collagen fibers in both layers exhibit significant realignment in response to unequal biaxial loading. The elastic fibers are much more uniformly distributed and remained relatively unchanged due to loading. Removal of elastin produces similar structural changes in collagen as mechanical loading. Our study suggests that the elastic fibers are under tension and impart an intrinsic compressive stress on the collagen.     

Arterial Extracellular Matrix: A Mechanobiological Study of the Contributions and Interactions of Elastin and Collagen

The complex network structure of elastin and collagen extracellular matrix (ECM) forms the primary load bearing components in the arterial wall. The structural and mechanobiological interactions between elastin and collagen are important for properly functioning arteries. Here, we examined the elastin and collagen organization, realignment, and recruitment by coupling mechanical loading and multiphoton imaging. Two-photon excitation fluorescence and second harmonic generation methods were performed with a multiphoton video-rate microscope to capture real time changes to the elastin and collagen structure during biaxial deformation. Enzymatic removal of elastin was performed to assess the structural changes of the remaining collagen structure. Quantitative analysis of the structural changes to elastin and collagen was made using a combination of two-dimensional fast Fourier transform and fractal analysis, which allows for a more complete understanding of structural changes. Our study provides new quantitative evidence, to our knowledge on the sequential engagement of different arterial ECM components in response to mechanical loading. The adventitial collagen exists as large wavy bundles of fibers that exhibit fiber engagement after 20% strain. The medial collagen is engaged throughout the stretching process, and prominent elastic fiber engagement is observed up to 20% strain after which the engagement plateaus. The fiber orientation distribution functions show remarkably different changes in the ECM structure in response to mechanical loading. The medial collagen shows an evident preferred circumferential distribution, however the fiber families of adventitial collagen are obscured by their waviness at no or low mechanical strains. Collagen fibers in both layers exhibit significant realignment in response to unequal biaxial loading. The elastic fibers are much more uniformly distributed and remained relatively unchanged due to loading. Removal of elastin produces similar structural changes in collagen as mechanical loading. Our study suggests that the elastic fibers are under tension and impart an intrinsic compressive stress on the collagen.     

Electrospinning of collagen nanofibers

Electrospinning is a fabrication process that uses an electric field to control the deposition of polymer fibers onto a target substrate. This electrostatic processing strategy can be used to fabricate fibrous polymer mats composed of fiber diameters ranging from several microns down to 100 nm or less. In this study, we describe how electrospinning can be adapted to produce tissue-engineering scaffolds composed of collagen nanofibers. Optimizing conditions for calfskin type I collagen produced a matrix composed of 100 nm fibers that exhibited the 67 nm banding pattern that is characteristic of native collagen. The structural properties of electrospun collagen varied with the tissue of origin (type I from skin vs type I from placenta), the isotype (type I vs type III), and the concentration of the collagen solution used to spin the fibers. Electrospinning is a rapid and efficient process that can be used to selectively deposit polymers in a random fashion or along a predetermined and defined axis. Toward that end, our experiments demonstrate that it is possible to tailor subtle mechanical properties into a matrix by controlling fiber orientation. The inherent properties of the electrospinning process make it possible to fabricate complex, and seamless, three-dimensional shapes. Electrospun collagen promotes cell growth and the penetration of cells into the engineered matrix. The structural, material, and biological properties of electrospun collagen suggest that this material may represent a nearly ideal tissue engineering scaffold.     

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].     

A microstructurally based continuum model of cartilage viscoelasticity and permeability incorporating measured statistical fiber orientations

The remarkable mechanical properties of cartilage derive from an interplay of isotropically distributed, densely packed and negatively charged proteoglycans; a highly anisotropic and inhomogeneously oriented fiber network of collagens; and an interstitial electrolytic fluid. We propose a new 3D finite strain constitutive model capable of simultaneously addressing both solid (reinforcement) and fluid (permeability) dependence of the tissue’s mechanical response on the patient-specific collagen fiber network. To represent fiber reinforcement, we integrate the strain energies of single collagen fibers—weighted by an orientation distribution function (ODF) defined over a unit sphere—over the distributed fiber orientations in 3D. We define the anisotropic intrinsic permeability of the tissue with a structure tensor based again on the integration of the local ODF over all spatial fiber orientations. By design, our modeling formulation accepts structural data on patient-specific collagen fiber networks as determined via diffusion tensor MRI. We implement our new model in 3D large strain finite elements and study the distributions of interstitial fluid pressure, fluid pressure load support and shear stress within a cartilage sample under indentation. Results show that the fiber network dramatically increases interstitial fluid pressure and focuses it near the surface. Inhomogeneity in the tissue’s composition also increases fluid pressure and reduces shear stress in the solid. Finally, a biphasic neo-Hookean material model, as is available in commercial finite element codes, does not capture important features of the intra-tissue response, e.g., distributions of interstitial fluid pressure and principal shear stress.     

Remodelling of continuously distributed collagen fibres in soft connective tissues

Extracellular matrix remodelling plays an essential role in tissue engineering of load-bearing structures. The goal of this study is to model changes in collagen fibre content and orientation in soft connective tissues due to mechanical stimuli. A theory is presented describing the mechanical condition within the tissue and accounting for the effects of collagen fibre alignment and changes in fibre content. A fibre orientation tensor is defined to represent the continuous distribution of collagen fibre directions. A constitutive model is introduced to relate the fibre configuration to the macroscopic stress within the material. The constitutive model is extended with a structural parameter, the fibre volume fraction, to account for the amount of fibres present within the material. It is hypothesised that collagen fibre reorientation is induced by macroscopic deformations and the amount of collagen fibres is assumed to increase with the mean fibre stretch. The capabilities of the model are demonstrated by considering remodelling within a biaxially stretched cube. The model is then applied to analyse remodelling within a closed stented aortic heart valve. The computed preferred fibre orientation runs from commissure to commissure and resembles the fibre directions in the native aortic valve.     

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.     

Connecting incremental shear modulus and Poisson's ratio of lung tissue with morphology and rheology of microstructure

The width and curvature of the collagen and elastin fiber bundles in the human pulmonary interalveolar septa and alveolar mouths are measured. The data, together with the known mechanical properties of collagen and elastin fibers, are used to derive the incremental elastic moduli of the lung tissue. The constitutive equation for small incremental stress and strain superposed on a homeostatic inflated lung is linear and isotropic, and characterized by two material constants.     

The layered structure of coronary adventitia under mechanical load

The mechanical loading-deformation relation of elastin and collagen fibril bundles is fundamental to understanding the microstructural properties of tissue. Here, we use multiphoton microscopy to obtain quantitative data of elastin and collagen fiber bundles under in situ loading of coronary adventitia. Simultaneous loading-imaging experiments on unstained fresh coronary adventitia allowed morphometric measurements of collagen and elastin fibril bundles and their individual deformation. Fiber data were analyzed at five different distension loading points (circumferential stretch ratio λ_θ = 1.0, 1.2, 1.4, 1.6, and 1.8) at a physiological axial stretch ratio of λ_axial = 1.3. Four fiber geometrical parameters were used to quantify the fibers: orientation angle, waviness, width, and area fraction. The results show that elastin and collagen fibers in inner adventitia form concentric densely packed fiber sheets, and the fiber orientation angle, width, and area fraction vary transmurally. The extent of fiber deformation depends on the initial orientation angle at no-distension state (λ_θ = 1.0 and λ_axial = 1.3). At higher distension loading, the orientation angle and waviness of fibers decrease linearly, but the width of collagen fiber is relatively constant at λ_θ = 1.0–1.4 and then decrease linearly for λ_θ ≥ 1.4. A decrease of the relative dispersion (SD/mean) of collagen fiber waviness suggests a heterogeneous mechanical response to loads. This study provides fundamental microstructural data for coronary artery biomechanics and we consider it seminal for structural models.     

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.     

Constitutive modeling of mouse carotid arteries using experimentally measured microstructural parameters

Changes in the local mechanical environment and tissue mechanical properties affect the biological activity of cells and play a key role in a variety of diseases, such as cancer, arthritis, nephropathy, and cardiovascular disease. Constitutive relations have long been used to predict the local mechanical environment within biological tissues and to investigate the relationship between biological responses and mechanical stimuli. Recent constitutive relations for soft tissues consider both material and structural properties by incorporating parameters that describe microstructural organization, such as fiber distributions, fiber angles, fiber crimping, and constituent volume fractions. The recently developed technique of imaging the microstructure of a single artery as it undergoes multiple deformations provides quantitative structural data that can reduce the number of estimated parameters by using parameters that are truly experimentally intractable. Here, we employed nonlinear multiphoton microscopy to quantify collagen fiber organization in mouse carotid arteries and incorporated measured fiber distribution data into structurally motivated constitutive relations. Microscopy results demonstrate that collagen fibers deform in an affine manner over physiologically relevant deformations. The incorporation of measured fiber angle distributions into constitutive relations improves the model's predictive accuracy and does not significantly reduce the goodness of fit. The use of measured structural parameters rather than estimated structural parameters promises to improve the predictive capabilities of the local mechanical environment, and to extend the utility of intravital imaging methods for estimating the mechanical behavior of tissues using in situ structural information.     

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.