Application of a fiber-reinforced continuum theory to multiple deformations of the annulus fibrosus.

Accurate tissue stress predictions for the annulus fibrosus are essential for understanding the factors that cause or contribute to disc degeneration and mechanical failure. Current computational models used to predict in vivo disc stresses utilize material laws for annular tissue that are not rigorously validated against experimental data. Consequently, predictions of disc stress resulting from physical activities may be inaccurate and therefore unreliable as a basis for defining mechanical-biologic injury criteria. To address this need we present a model for the annulus as an isotropic ground substance reinforced with two families of collagen fibers, and an approach for determining the material constants by simultaneous consideration of multiple experimental data sets. Two strain energy functions for the annulus are proposed and used in the theory to derive the constitutive equations relating the stress to pure stretch deformations. These equations are applied to four distinct experimental protocols and the material constants are determined from a simultaneous, nonlinear regression analysis. Good agreement between theory and experiment is achieved when the invariants are included within multiple, separate exponentials in the strain energy function.     

中亚热带喀斯特常绿落叶阔叶混交林典型树种的木质部解剖与功能特征分析

树木木质部主要由导管、纤维和薄壁组织组成, 分别具有运输、支撑和贮存的生理功能。由于木质部空间限制, 一种组织比例的增加会导致其他组织比例的降低, 因而可能表现出权衡关系。分析木质部组织比例和权衡关系有助于了解植物的生理生态适应性。该研究选择中亚热带喀斯特常绿落叶阔叶混交林21种典型树种(10种落叶树种, 11种常绿树种), 测定枝条木质部各组织比例, 计算水力相关指标并分析性状之间的相关性。结果表明: (1)与全球木质部解剖数据对比分析, 喀斯特树种木质部趋向具有较高比例的薄壁组织; (2)喀斯特树种导管组织比例与薄壁和纤维组织比例之间没有显著的相关性, 但是薄壁和纤维组织比例之间有显著的权衡关系; (3)常绿和落叶树种的木质部水力运输安全性(导管壁加固系数)和效率性(理论导水率)均具有显著的权衡关系, 但是这两个类群线性回归的截距存在显著差异, 即在相同的理论导水率条件下, 落叶树种比常绿树种具有较高的导管壁加固系数(安全性), 可能与常绿树种具有更多的轴向薄壁组织有关。喀斯特树种木质部解剖特征表明薄壁组织的贮存功能对喀斯特树种(尤其是常绿树种)的干旱适应具有重要作用。     

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.     

Simulation of planar soft tissues using a structural constitutive model: Finite element implementation and validation

Computational implementation of physical and physiologically realistic constitutive models is critical for numerical simulation of soft biological tissues in a variety of biomedical applications. It is well established that the highly nonlinear and anisotropic mechanical behaviors of soft tissues are an emergent behavior of the underlying tissue microstructure. In the present study, we have implemented a structural constitutive model into a finite element framework specialized for membrane tissues. We noted that starting with a single element subjected to uniaxial tension, the non-fibrous tissue matrix must be present to prevent unrealistic tissue deformations. Flexural simulations were used to set the non-fibrous matrix modulus because fibers have little effects on tissue deformation under three-point bending. Multiple deformation modes were simulated, including strip biaxial, planar biaxial with two attachment methods, and membrane inflation. Detailed comparisons with experimental data were undertaken to insure faithful simulations of both the macro-level stress–strain insights into adaptations of the fiber architecture under stress, such as fiber reorientation and fiber recruitment. Results indicated a high degree of fidelity and demonstrated interesting microstructural adaptions to stress and the important role of the underlying tissue matrix. Moreover, we apparently resolve a discrepancy in our 1997 study (Billiar and Sacks, 1997. J. Biomech. 30 (7), 753–756) where we observed that under strip biaxial stretch the simulated fiber splay responses were not in good agreement with the experimental results, suggesting non-affine deformations may have occurred. However, by correctly accounting for the isotropic phase of the measured fiber splay, good agreement was obtained. While not the final word, these simulations suggest that affine fiber kinematics for planar collagenous tissues is a reasonable assumption at the macro level. Simulation tools such as these are imperative in the design and simulation of native and engineered tissues.     

Assessing the mechanical energy costs of various tissue reshaping mechanisms

Early-stage embryos must reshape the tissues of which they are made into organs and other life-sustaining structures; and if non-mammalian embryos fail to complete these tasks before the energy provided by their yolk runs out, they die. The aim of this study is to use a cell-level computational model to investigate the energetic cost of a variety of mechanisms that can drive an in-plane reshaping pattern known as convergent extension—a motif in which a tissue narrows in one in-plane direction and expands in another. Mechanisms considered include oriented lamellipodia, directed mitosis, stress fibers, and anisotropic external tension. Both isolated patches of tissue and actively contracting tissues that deform adjacent passive areas are considered. The cell-level finite element model used here assumes that the cell membrane and its associated proteins generate a net tension γ along each cell–cell interface and that the cytoplasm and its embedded networks and structures have an effective viscosity?μ. Work costs are based exclusively on mechanical considerations such as edge lengths and tensions, and because a traditional mechanical efficiency cannot be calculated, mechanisms are compared on the basis of the work they must do to the tissue to cause a specified rate of in-plane reshaping. Although the model contains a number of simplifications compared to real embryonic tissues, it is able to show that the work requirements for tissue reshaping by mitoses and by lamellipodia are of the same order. Lamellipodia are energetically most effective when their tensions are approximately twice as large as the interfacial tensions in the surrounding cells. The model also shows that stress fibers or other direct stretch or compression mechanisms are at least five times more efficient for tissue reshaping than are mitoses or lamellipodia and that the work needed to deform a typical cellular tissue is more than thirty times greater than if it did not contain cell boundaries. Collectively, these findings indicate that common tissue reshaping mechanisms have mechanical efficiencies of less than one percent and that mechanical efficiency is not the primary determinant of which mechanism(s) an embryo uses to reshape its tissues.     

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.     

A closed-form structural model of planar fibrous tissue mechanics

Structural models of tissue mechanics, in which the tissue is represented as a sum or integral of fiber contributions for a distribution of fiber orientations, are a popular tool to represent the complex mechanical behavior of soft tissues. A significant practical challenge, however, is evaluation of the integral that defines the stress. Numerical integration is accurate but computationally demanding, posing an impediment to incorporation of structural models into large-scale finite-element simulations. In this paper, a closed-form analytic evaluation of the integral is derived for fibers distributed according to a von Mises distribution and an exponential fiber stress–strain law.     

Simulation of soft tissue failure using the material point method

Understanding the factors that control the extent of tissue damage as a result of material failure in soft tissues may provide means to improve diagnosis and treatment of soft tissue injuries. The objective of this research was to develop and test a computational framework for the study of the failure of anisotropic soft tissues subjected to finite deformation. An anisotropic constitutive model incorporating strain-based failure criteria was implemented in an existing computational solid mechanics software based on the material point method (MPM), a quasi-meshless particle method for simulations in computational mechanics. The constitutive model and the strain-based failure formulations were tested using simulations of simple shear and tensile mechanical tests. The model was then applied to investigate a scenario of a penetrating injury: a low-speed projectile was released through a myocardial material slab. Sensitivity studies were performed to establish the necessary grid resolution and time-step size. Results of the simple shear and tensile test simulations demonstrated the correct implementation of the constitutive model and the influence of both fiber family and matrix failure on predictions of overall tissue failure. The slab penetration simulations produced physically realistic wound tracts, exhibiting diameter increase from entrance to exit. Simulations examining the effect of bullet initial velocity showed that the anisotropy influenced the shape and size of the exit wound more at lower velocities. Furthermore, the size and taper of the wound cavity was smaller for the higher bullet velocity. It was concluded that these effects were due to the amount of momentum transfer. The results demonstrate the feasibility of using MPM and the associated failure model for large-scale numerical simulations of soft tissue failure.     

A structural model for the flexural mechanics of nonwoven tissue engineering scaffolds

The development of methods to predict the strength and stiffness of biomaterials used in tissue engineering is critical for load-bearing applications in which the essential functional requirements are primarily mechanical. We previously quantified changes in the effective stiffness (E) of needled nonwoven polyglycolic acid (PGA) and poly-L-lactic acid (PLLA) scaffolds due to tissue formation and scaffold degradation under three-point bending. Toward predicting these changes, we present a structural model for E of a needled nonwoven scaffold in flexure. The model accounted for the number and orientation of fibers within a representative volume element of the scaffold demarcated by the needling process. The spring-like effective stiffness of the curved fibers was calculated using the sinusoidal fiber shapes. Structural and mechanical properties of PGA and PLLA fibers and PGA, PLLA, and 50:50 PGA/PLLA scaffolds were measured and compared with model predictions. To verify the general predictive capability, the predicted dependence of E on fiber diameter was compared with experimental measurements. Needled nonwoven scaffolds were found to exhibit distinct preferred (PD) and cross-preferred (XD) fiber directions, with an E ratio (PD/XD) of approximately 3:1. The good agreement between the predicted and experimental dependence of E on fiber diameter (R2 = 0.987) suggests that the structural model can be used to design scaffolds with E values more similar to native soft tissues. A comparison with previous results for cell-seeded scaffolds (Engelmayr, G. C., Jr., et al., 2005, Biomaterials, 26(2), pp. 175-187) suggests, for the first time, that the primary mechanical effect of collagen deposition is an increase in the number of fiber-fiber bond points yielding effectively stiffer scaffold fibers. This finding indicated that the effects of tissue deposition on needled nonwoven scaffold mechanics do not follow a rule-of-mixtures behavior. These important results underscore the need for structural approaches in modeling the effects of engineered tissue formation on nonwoven scaffolds, and their potential utility in scaffold design.     

Passive mechanics of canine internal abdominal muscles.

The internal abdominal muscles are biaxially loaded in vivo, and therefore length-tension relations along and transverse to the directions of the muscle fibers are important in understanding their mechanical properties. We hypothesized that 1) internal oblique and transversus abdominis form an internal abdominal composite muscle with altered compliance than that of either muscle individually, and 2) anisotropy, different compliances in orthogonal directions, of internal abdominal composite muscle is less pronounced than that of its individual muscles. To test these hypotheses, in vitro mechanical testing was performed on 5 x 5 cm squares of transversus abdominis, internal oblique, and the two muscles together as a composite. These tissues were harvested from the left lateral side of abdominal muscles of eleven mongrel dogs (15-23 kg) and placed in a bath of oxygenated Krebs solution. Each tissue strip was attached to a biaxial mechanical testing device. Each muscle was passively lengthened and shortened along muscle fibers, transverse to fibers, or simultaneously along and transverse to muscle fibers. Both transversus abdominis and internal oblique muscles demonstrated less extensibility in the direction transverse to muscle fibers than along fibers. Biaxial loading caused a stiffening effect that was greater in the direction along the fibers than transverse to the fibers. Furthermore, the abdominal muscle composite was less compliant than either muscle alone in the direction of the muscle fibers. Taken together, our data suggested that the internal abdominal composite tissue has complex mechanical properties that are dependent on the mechanical properties of internal oblique and transversus abdominis muscles.     

The influence of fiber orientation on the equilibrium properties of neutral and charged biphasic tissues

Constitutive models facilitate investigation into load bearing mechanisms of biological tissues and may aid attempts to engineer tissue replacements. In soft tissue models, a commonly made assumption is that collagen fibers can only bear tensile loads. Previous computational studies have demonstrated that radially aligned fibers stiffen a material in unconfined compression most by limiting lateral expansion while vertically aligned fibers buckle under the compressive loads. In this short communication, we show that in conjunction with swelling, these intuitive statements can be violated at small strains. Under such conditions, a tissue with fibers aligned parallel to the direction of load initially provides the greatest resistance to compression. The results are further put into the context of a Benninghoff architecture for articular cartilage. The predictions of this computational study demonstrate the effects of varying fiber orientations and an initial tare strain on the apparent material parameters obtained from unconfined compression tests of charged tissues.     

Elastic modulus of hard tissues

This work aims at evaluating the elastic modulus of hard biological tissues by considering their staggered platelet micro-structure. An analytical expression for the effective modulus along the stagger direction is formulated using three non-dimensional structural variables. Structures with a single staggered hierarchy (e.g. collagen fibril) are first studied and predictions are compared with the experimental results and finite element simulations from the literature. A more complicated configuration, such as an array of fibrils, is analyzed next. Finally, a mechanical model is proposed for tooth dentin, in which variations in the multi-scale structural hierarchy are shown to significantly affect the macroscopic mechanical properties.     

Molecular assembly and mechanical properties of the extracellular matrix: A fibrous protein perspective

The extracellular matrix is an integral and dynamic component of all tissues. Macromolecular compositions and structural architectures of the matrix are tissue-specific and typically are strongly influenced by the magnitude and direction of biomechanical forces experienced as part of normal tissue function. Fibrous extracellular networks of collagen and elastin provide the dominant response to tissue mechanical forces. These matrix proteins enable tissues to withstand high tensile and repetitive stresses without plastic deformation or rupture. Here we provide an overview of the hierarchical molecular and supramolecular assembly of collagens and elastic fibers, and review their capacity for mechanical behavior in response to force. This article is part of a Special Issue entitled: Fibrosis: Translation of basic research to human disease.     

Linking hyperelastic theoretical models and experimental data of vaginal tissue through histological data

Mechanical characterization of living tissues and computer-based simulations related to medical issues, has become increasingly important to improve diagnostic processes and treatments evaluation. This work proposes a link between the mechanical testing and the material model predictions through histological data of vaginal tissue. Histological data was used to link tensile testing experiments with material-dependent parameters; the approach was adequate to capture the nonlinear response of ovine vaginal tissue over a large strain range.The experimental data obtained on a previous study, has two main components: tensile testing and histological analysis of the ovine vaginal tissue. Uniaxial tensile test data and histological data were collected from three sheep groups: virgins, pregnant and parous. The distal part of vaginal wall was selected since it is prone to tears induced by vaginal delivery.The HGO (Holzapfel-Gasser-Ogden) model parameters were fitted using a stochastic approach, namely the Simple Genetic Algorithm (SGA). The SGA was able to fit the experimental data successfully (R² > 0.986). The dimensionless coefficient ξ, was highly correlated with histological data. The ratio was seen to increase linearly with increasing collagen content.Coefficient ξ brings a new way of interpreting and understanding experimental data; it connects the nonlinear mechanical behaviour (tensile test) with tissue’s morphology (histology). It can be used as an ‘inverse’ (approximate) method to estimate the mechanical properties without direct experimental measurements, through basic histology.In this context, the proposed methodology appears very promising in estimating the response of the tissue via histological information.     

Elastic deformation of mineralized collagen fibrils: an equivalent inclusion based composite model

Mineralized collagen fibrils are the basic building blocks of bone tissue at the supramolecular level. Several disease states, manipulation of the expression of specific proteins involved in biomineralization, and treatment with different agents alter the extent of mineralization as well as the morphology of mineral crystals which in turn affect the mechanical function of bone tissue. An experimental assessment of mineralized fibers' mechanical properties is challenged by their small size, leaving analytical and computational models as a viable alternative for investigation of the fibril-level mechanical properties. In the current study the variation of the elastic stiffness tensor of mineralized collagen fibrils with changing mineral volume fraction and mineral aspect ratios was predicted via a micromechanical model. The partitioning of applied stresses between mineral and collagen phases is also predicted for normal and shear loading of fibrils. Model predictions resulted in transversely isotropic collagen fibrils in which the modulus along the longer axis of the fibril was the greatest. All the elastic moduli increased with increasing mineral volume fraction whereas Poisson's ratios decreased with the exception of v12 (=v21). The partitioning of applied stresses were such that the stresses acting on mineral crystals were about 1.5, 15, and 3 times greater than collagen stresses when fibrils were loaded transversely, longitudinally, and in shear, respectively. In the overall the predictions were such that: (a) greatest modulus along longer axis; (b) the greatest mineral/collagen stress ratio along the longer axis of collagen fibers (i.e., greatest relief of stresses acting on collagen); and (c) minimal lateral contraction when fibers are loaded along the longer axis. Overall, the pattern of mineralization as put forth in this model predicts a superior mechanical function along the longer axis of collagen fibers, the direction which is more likely to experience greater stresses.     

Mapping complex myoarchitecture in the bovine tongue with diffusion-spectrum magnetic resonance imaging

The ability to resolve complex fiber populations in muscular tissues is important for relating tissue structure with mechanical function. To address this issue in the case of tongue, we employed diffusion spectrum imaging (DSI), an MRI method for determining three-dimensional myoarchitecture where myofiber populations are variably aligned. By specifically varying gradient field strength, molecular displacement in a tissue can be determined by Fourier-transforming the echo intensity against gradient strength at fixed gradient pulse spacing. The displacement profiles are visualized by graphing three-dimensional isocontour icons for each voxel, with the isocontour shape and size representing the magnitude and direction of the constituting fiber populations. To validate this method, we simulated a DSI experiment within the constraints of arbitrary crossing fibers, and determined that DSI accurately depicts the angular relationships between these fibers. Considering the fiber relationships in the whole bovine tongue, we compared the images obtained by DSI with those obtained by diffusion tensor imaging in an anterior slice of the lingual core, a region known to possess extensive fiber crossing. In contrast to diffusion tensor imaging, which depicts the anterior core solely as a region with low anisotropy due to the presence of mixed-orientation fiber populations, DSI shows two distinct fiber populations, with an explicit orthogonal relationship to each other. In imaging the whole lingual tissue, we discerned arrays of crossing and noncrossing fibers involving the intrinsic and extrinsic muscles, which merged at regions of interface. We conclude that DSI has the capacity to determine three-dimensional fiber orientation in structurally complex muscular tissues.     

Variation and robustness of the mechanics of gastrulation: the role of tissue mechanical properties during morphogenesis

Diverse mechanisms of morphogenesis generate a wide variety of animal forms. In this work, we discuss two ways that the mechanical properties of embryonic tissues could guide one of the earliest morphogenetic movements in animals, gastrulation. First, morphogenetic movements are a function of both the forces generated by cells and the mechanical properties of the tissues. Second, cells could change their behavior in response to their mechanical environment. Theoretical studies of gastrulation indicate that different morphogenetic mechanisms differ in their inherent sensitivity to tissue mechanical properties. Those few empirical studies that have investigated the mechanical properties of amphibian and echinoderm gastrula-stage embryos indicate that there could be high embryo-to-embryo variability in tissue stiffness. Such high embryo-to-embryo variability would imply that gastrulation is fairly robust to variation in tissue stiffness. Cell culture studies demonstrate a wide variety of cellular responses to the mechanical properties of their microenvironment. These responses are likely to be developmentally regulated, and could either increase or decrease the robustness of gastrulation movements depending on which cells express which responses. Hence both passive physical and mechanoregulatory processes will determine how sensitive gastrulation is to tissue mechanics. Addressing these questions is important for understanding the significance of diverse programs of early development, and how genetic or environmental perturbations influence development. We discuss methods for measuring embryo-to-embryo variability in tissue mechanics, and for experimentally perturbing those mechanical properties to determine the sensitivity of gastrulation to tissue mechanics.     

Micromechanical modeling of the epimysium of the skeletal muscles

A micromechanical model has been developed to investigate the mechanical properties of the epimysium. In the present model, the collagen fibers in the epimysium are embedded randomly in the ground substance. Two parallel wavy collagen fibers and the surrounding ground substance are used as the repeat unit (unit cell), and the epimysium is considered as an aggregate of unit cells. Each unit cell is distributed in the epimysium with some different angle to the muscle fiber direction. The model allows the progressive straightening of the collagen fiber as well as the effects of fiber reorientation. The predictions of the model compare favorably against experiment. The effects of the collagen fiber volume fraction, collagen fiber waviness at the rest length and the mechanical properties of the collagen fibers and the ground substance are analyzed. This model allows the analysis of mechanical behavior of most soft tissues if appropriate experimental data are available.     

Measurement of surface deformation of soft tissue

A method is described for measuring the surface shape and deformations of soft tissue in three dimensions. The method uses close range stereophotogrammetry to record the three-dimensional locations of miniature optical targets applied to the tissue surface. This has been applied to study of human lumbar intervertebral disc. Measurements of the strain along surface annular fibers have been made under varying loads. In this case the maximum expected errors are about 0.15 mm, which corresponds to a strain of less than 1%. Preliminary findings have differed from predictions made in published mathematical models for the disc in that they show very little strain of the annulus in compression loading, but confirm axial torsional loading as liable to produce mechanical disruption of the disc annulus.     

Phase transformations in a model mesenchymal tissue

Connective tissues, the most abundant tissue type of the mature mammalian body, consist of cells suspended in complex microenvironments known as extracellular matrices (ECMs). In the immature connective tissues (mesenchymes) encountered in developmental biology and tissue engineering applications, the ECMs contain varying amounts of randomly arranged fibers, and the physical state of the ECM changes as the fibers secreted by the cells undergo fibril and fiber assembly and organize into networks. In vitro composites consisting of assembling solutions of type I collagen, containing suspended polystyrene latex beads ( approximately 6 microm in diameter) with collagen-binding surface properties, provide a simplified model for certain physical aspects of developing mesenchymes. In particular, assembly-dependent topological (i.e., connectivity) transitions within the ECM could change a tissue from one in which cell-sized particles (e.g., latex beads or cells) are mechanically unlinked to one in which the particles are part of a mechanical continuum. Any particle-induced alterations in fiber organization would imply that cells could similarly establish physically distinct microdomains within tissues. Here we show that the presence of beads above a critical number density accelerates the sol-gel transition that takes place during the assembly of collagen into a globally interconnected network of fibers. The presence of this suprathreshold number of beads also dramatically changes the viscoelastic properties of the collagen matrix, but only when the initial concentration of soluble collagen is itself above a critical value. Our studies provide a starting point for the analysis of phase transformations of more complex biomaterials including developing and healing tissues as well as tissue substitutes containing living cells.