Structural hierarchies and interactions in the tendon composite

Tendon functions by transmitting tensile loads from muscle to bone. Morphologically, it can be described as a macromolecular multicomposite material, basically consisting of collagen fibrils held together by a soft, hydrated matrix material. Recently, tendon has been deformed beyond the "in vivo" elastic limit and by cyclical loading systematically damaged. Using high-resolution electron microscopy, decomposition of the collagen fibril into subfibrils (15 nm diameter) and microfibrils (3.5 nm diameter) has been noted. The interfacial adhesion between such units is strongly dependent on age, and is probably related with crosslinking phenomena observed by biochemical methods. In addition, tendon collagen contains a considerable amount of water throughout the entire structure which strongly affects its overall mechanical behavior. The various bound states of water have been identified using primarily dynamic mechanical spectroscopy coupled with more conventional methods of structural characterization.Published in Mekhanika Polimerov, No. 4, pp. 693–701, July–August, 1976.     

Fatigue damage modeling of fibrous soft tissues

Ultimate tendon failure is often caused by fatigue loading. Recent interventions revealed a three-phase progression of histological changes during cyclic loading of the tendon. It starts from localized kinked fiber deformations, continues with additional fiber delaminations and finally leads to fiber angulations and discontinuities [5, 6]. In the present contribution, we propose a physically motivated constitutive model able to describe fatigue evolution in tendon subject to cyclic loading. The damage of the collagen fibers is elucidated by a successive permanent opening of tropocollagen molecules [7], which represent the basic building blocks of collagen fibrils. The fibril strain increase is triggered by a time-force depending rupture of glycosaminoglycan sidechains of adjacent collagen fibrils. The so obtained model is in line with recent experimental findings available in literature. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Ultrastructural elements of the strength of the muscle-tendon junction

Scanning and transmission electron microscopy have been used for the study of the structural organization of muscle-tendon junctions in humans and rats. It was found that the structural interconnection of muscle fibers with the collagen fibrils of the tendon takes place at the level of the basal membrane of the muscle fibers; the strength of the junction of muscle with tendon is provided by the dense interweaving of the fibrillar component of the basal membrane about the collagen fibrils of the tendon. The amorphous substance of the basal membrane acts as an adhesive substance.     

Relationship of the ultrastructure and mechanical properties in tendinous collagen,a highly ordered macromolecular composite

A relationship has been found between the stress and deformation in tendon and, from an analysis of the data obtained, it was established that there is a major stress-bearing fiber, the diameter of which increases with increasing age of the animal. The relationship of the results obtained with the structure and properties of tendon as well as with the aging process is stressed. Models of the ultrastructure of collagen fibers in rat tail tendon or in other soft connective tissue were prepared by sealing rigid fibers in a considerably softer elastic matrix. Loss of fiber resistance was achieved by contraction of this matrix. These synthetic composite systems exactly reflect a number of the features of the collagen ultra-structure. Models were also used for studying stresses in the matrix about the fibers as this necessary information could not be obtained by a direct study of the biological tissues. Examination of the reaction between tropocollagen and the connective tissue polysaccharides in the process of collagen fiber formation shows that a mechanism exists through which the fibers may lose their resistance also in vivo.     

Experimental and numerical investigations on engineering tissue enhanced by mechanical stimulation

The aim of the paper is to investigate the remodeling phenomenon of a cell-seeded material (collagen type-I) due to collagen type-II newly synthesized by the cells. For the experiments, a cell-seeded condensed collagen gel is mechanically stimulated in a bioreactor for four weeks. The remodeled stiffness of the cell-seeded gel is measured by a compression test and is explained with an evolution law. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Investigation of the dynamic fatigue of composite polymer materials

The dynamic fatigue of composite film materials has been studied as a function of the frequency and amplitude of deformation. The tests were carried out on a vibration apparatus with a frequency range from 10 to 600 cps. The test objects consisted of foil packaging with a polyethylene backing (foil-film).The dependence of the dynamic fatigue of foil-film on amplitude, frequency, and acceleration was studied. It has been shown that in all cases failure of the material is preceded by cracking of the foil and peeling of the foil from the polyethylene backing at the site of the cracks. The material fails as a result of puncture of the polyethylene backing by the broken edge of the cracked foil.Mekhanika Polimerov, Vol. 1, No. 5, pp. 90–94, 1965     

A SimpleModel for Anisotropic Damage with Applications to Soft Tissues

Arteries are reinforced by helically arranged collagen fibers and posses orthotropic elastic properties. In this paper a polyconvex anisotropic energy is used in order to guarantee the existence of minimizers for the purely elastic boundary value problem. Anisotropic discontinuous damage effects, which are induced by decreasing stiffness of particular fibers, are observed in a certain range of overexpansion. A simple thermodynamical consistent anisotropic damage model is constructed, basing on the assumption that damage mainly takes place in the fiber directions as a result of breakage of collagen cross‐links.Finally a cycled overexpansion of a test material from an artery is analyzed in order to show the main characteristics of the proposed model. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Biomechanical modelling of cellular articular cartilage replacement tissues

The aim of the present study is to investigate the strength and damping properties of cellular articular cartilage replacement material. For this purpose, a viscoelastic-diffusion model for the acellular water-saturated condensed collagen gel type I is proposed and validated experimentally. Moreover, a remodelling law for the cell seeded collagen gel is introduced. For an experimental study of the interaction between fibre growth and mechanical stimulation, bioreactors are developed and histological investigations are carried out. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A hyperelastic biphasic fiber reinforced model for articular cartilage considering the distribution and orientation of collagen fibers

Articular cartilage is a multiphase material consisting of fluids and electrolytes, which is described with the Theory of Porous Media. The mechanical characteristics of articular cartilage are porosity, incompressible material behavior combined with transversely isotropic behavior for solid and fluid phases. There are two central points to model articular cartilage: the poro-viscosity of the porous matrix and the visco elasticity, and orientation of the collagen fibers. A numerical example is presented. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Extremal loading of soft fibrous tissues: multi-scale mechanics and constitutive modeling

In the current contribution, we present a multi-scale constitutive model capturing macroscopic inelastic effects (like stress softening and permanent set) in soft tissues under cyclic loading. Soft biological tissues can be described as a biological composite material. The extracellular matrix is hereby reinforced by collagen fibers which themself are an assembly of collagen fibrils embedded in a proteoglycan (PG) rich matrix. Micro-damage induced by cyclic loading is treated by an interaction scenario between the fibrils and the PGs. At the low strain regime PGs promote sliding between fibrils [1] which leads to the yielding of statistical distributed overlapping segments. The breakage of the PG-bridges is defined by a decreasing PG-density. Due to the accumulated damage of the PG connections at high tissue strains, the strains at the fibril level increases. This finally drives the over-stretching of the fibrils, which is associated with a permanent rupture of the hydrogen bonds inside of the tropocollagen molecules [2]. The so obtained model is in line with recent experimental findings [1, 2] and was additionally validated against experimental data available in literature. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling of Anisotropic Damage in Arterial Walls

In a certain range of overexpansion arterial walls are characterized by an orthotropic elastic material behavior. Due to different stabilities of the helically arranged fibers, i.e. breakage of collagen crosslinks between the fibers, damage effects are observed in experiments. Because of the fibrous composition it is assumed that damage mainly occurs in the fiber direction. The proposed damage model is extended to arterial wall applications by introducing a referential damage state. The damage approach is applied to a polyconvex model for the hyperelastic behavior of arteries in order to obtain a materially stable model, which guarantees the existence of solutions of the underlying boundary value problem. The performance of the proposed model is presented in a numerical example, where the overexpansion of an atherosclerotic artery is simulated. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Failure of adhesive bonds between proteoglycans causes viscoelasticity of soft tissues

Soft tissues can be considered as a composite material where a matrix (ground substance) is reinforced by collagen fibers. These fibers consist of fibrils, which are connected by proteoglycan (PG) bridges. The time-dependent properties of soft tissues appear to be mainly caused by proteoglycans [3]. This contribution presents a modeling approach where damage in the PG bridges arises due to the failure of the covalent bonds between two proteoglycans. The breakage of covalent bonds is reversible over time and incorporated using a healing formulation. A high PG density supports interfibrillar sliding and hence leads to a lower fibril stretch [8]. Accordingly, the damage propagation in PG bridges leads to a higher stretch in the fibrils and therefore to a stiffer material response. The strain energy of the fibrils is based on the response of single tropocollagen molecules and takes both, an entropic and an energetic regime into account [5]. Finally, the model is compared against experimental data available in the literature. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simulation of Damage Hysteresis in Soft Biological Tissues

Arterial lumen reductions result in hypertension and possibly rupture of plaque deposits as a consequence of atherosclerotic degeneration which can finally cause a heart attack or a stroke. Hence, therapeutical treatments such as balloon angioplasty and stenting are required. The aim of this work is to propose a new model for the softening hysteresis observed in overstretched arterial tissues and related computer simulations. The physiological loading domain is described by a purely elastic polyconvex anisotropic strain-energy function, which is decoupled into an isotropic part related to the non-collagenous matrix material, and into a transversely isotropic part related to the embedded (collagen) fibers. A scalar-valued damage variable is taken into account for the fibers to cover a saturation function converging to a maximal damage value at a fixed maximum load level. In addition, the model captures remnant strains in the fiber direction after unloading. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A linear theoretical model of cavitation flow past a multihydrofoil system with a finite number of foils

Aiming to study the complicated interference phenomenon of the cavitation flow around a multihydrofoil system with a finite number of foils, we establish a linear theoretical model under the conditions of small angle of attack and small cavitation number. By solving the model numerically, we analyze the effects of different cavitation numbers and foil spacings on cavitation flow. The cavitation shape and hydrodynamics of each foil in the multihydrofoil system are obtained. The shapes of the cavities differ from each other because of the mutual interference phenomenon. Moreover, the change rule for cavitation shapes, which varied with the foil spacing and cavitation number, is very complicated, and no simple mode can be found. The linear theoretical model in this article is suitable for the problem of cavitation flow around multiple hydrofoils with different chord length, camber, and angle of attack.     

Multiscale modelling of skeletal muscle tissue by incorporating microstructural effects

The macroscopic mechanical response of skeletal muscle tissue is mainly influenced by the properties and arrangement of microstructural elements, such as, for example, sarcomeres and connective tissue. Like for many biological materials, the mechanical properties of skeletal muscle tissue can vary quite significantly between different specimens like, for example, different persons or muscle types. Current state-of-the-art continuum-mechanical muscle models often lack the ability to take into account such variations in a natural way. Further, phenomenological constitutive laws face the challenge that appropriate material parameter sets need to be found for each tissue variation. Thus, the present work aims to identify the microstructural features and parameters governing the overall mechanical response and to incorporate them into a macroscopic material model by applying suitable homogenisation methods. The motivation hereby is that the estimation of material parameters for microstructures, such as collagen fibres, can be done in a more reliable and general way and that fluctuations between specimens are included by, for example, adapting the alignment of the collagen fibres inside the muscle. Moreover, instead of computationally expensive homogenisation methods like FE², this work proceeds from well-founded analytical homogenisation techniques in order to keep the model as simple as possible. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Homogenization of bone elasticity based on tissue‐independent (‘universal’) phase properties

As candidates for tissue‐independent phase properties of cortical and trabecular bone we consider (i) hydroxyapatite, (ii) collagen, (iii) ultrastructural water and non‐collagenous proteins, and (iv) marrow (water) filling the Haversian canals and the intertrabecular space. From experiments reported in the literature, we assign stiffness properties to these phases (experimental set I). On the basis of these phase definitions, we develop, within the framework of continuum micromechanics, a two step homogenization procedure: (i) At a length scale of 100 – 200 nm, hydroxyapatite (HA) crystals build up a crystal foam ('polycrystal'), and water and non‐collagenous organic matter fill the intercrystalline space (homogenization step I); (ii) At the ultrastructural scale of mineralized tissues, i.e. 5 to 10 microns, collagen assemblies composed of collagen molecules are embedded into the crystal foam, acting mechanically as cylindrical templates. At an enlarged material scale of 5 to 10 mm, the second homogenization step also accommodates the micropore space as cylindrical pore inclusions (Haversian and Volkmann canals, inter‐trabecular space), that are suitable for both trabecular and cortical bone. The input of this micromechanical model are tissue‐specific volume fractions of HA, collagen, and of the micropore space. The output are tissue‐specific ultrastructural and microstructural (=macroscopic=apparent) elasticity tensors. A second independent experimental set (composition data and experimental stiffness values) is employed to validate the proposed model. We report a a good agreement between model predictions and experimentally determined macroscopic stiffness values. The validation suggests that hydroxyapatite, collagen, and water are tissue‐independent phases, which define, through their mechanical interaction, the elasticity of all bones, whether cortical or trabecular.     

Mechanical modelling and experimental validation of meniscus replacement material

In the present study a condensed collagen gel is investigated for its application as a cartilage replacement material. For this reason, the strength and damping properties are examined experimentally and a theoretical model for predicting specimen deformations and stresses is proposed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)