Modelling skeletal muscle fibre orientation arrangement

Skeletal muscle tissues have complex geometries. In addition, the complex fibre orientation arrangement makes it quite difficult to create an accurate finite element muscle model. There are many possible ways to specify the complex fibre orientations in a finite element model, for example defining a local element coordinate system. In this paper, an alternative method using ABAQUS, which is combination of the finite element method and the non-uniform rational B-spline solid representation, is proposed to calculate the initial fibre orientations. The initial direction of each muscle fibre is specified as the tangent direction of the NURBS curve which the fibre lies on, and the directions of the deformed fibres are calculated from the initial fibre directions, the deformation gradients and the fibre stretch ratios. Several examples are presented to demonstrate the ability of the proposed method. Results show that the proposed method is able to characterise both the muscle complex fibre orientation arrangement and its complex mechanical response.     

Modelling skeletal muscle fibre orientation arrangement

Skeletal muscle tissues have complex geometries. In addition, the complex fibre orientation arrangement makes it quite difficult to create an accurate finite element muscle model. There are many possible ways to specify the complex fibre orientations in a finite element model, for example defining a local element coordinate system. In this paper, an alternative method using ABAQUS, which is combination of the finite element method and the non-uniform rational B-spline solid representation, is proposed to calculate the initial fibre orientations. The initial direction of each muscle fibre is specified as the tangent direction of the NURBS curve which the fibre lies on, and the directions of the deformed fibres are calculated from the initial fibre directions, the deformation gradients and the fibre stretch ratios. Several examples are presented to demonstrate the ability of the proposed method. Results show that the proposed method is able to characterise both the muscle complex fibre orientation arrangement and its complex mechanical response.     

A computational model for collagen fibre remodelling in the arterial wall

As the interaction between tissue adaptation and the mechanical condition within tissues is complex, mathematical models are desired to study this interrelation. In this study, a mathematical model is presented to investigate the interplay between collagen architecture and mechanical loading conditions in the arterial wall. It is assumed that the collagen fibres align along preferred directions, situated in between the principal stretch directions. The predicted fibre directions represent symmetrically arranged helices and agree qualitatively with morphometric data from literature. At the luminal side of the arterial wall, the fibres are oriented more circumferentially than at the outer side. The discrete transition of the fibre orientation at the media-adventitia interface can be explained by accounting for the different reference configurations of both layers. The predicted pressure-radius relations resemble experimentally measured sigma-shaped curves. As there is a strong coupling between the collagen architecture and the mechanical loading condition within the tissue, we expect that the presented model for collagen remodelling is useful to gain further insight into the processes involved in vascular adaptation, such as growth and smooth muscle tone adaptation.     

A two-muscle,continuum-mechanical forward simulation of the upper limb

By following the common definition of forward-dynamics simulations, i.e. predicting movement based on (neural) muscle activity, this work describes, for the first time, a forward-dynamics simulation framework of a musculoskeletal system, in which all components are represented as continuous, three-dimensional, volumetric objects. Within this framework, the mechanical behaviour of the entire muscle–tendon complex is modelled as a nonlinear hyperelastic material undergoing finite deformations. The feasibility and the full potential of the proposed forward-dynamics simulation framework is demonstrated on a two-muscle, three-dimensional, continuum-mechanical model of the upper limb. The musculoskeletal model consists of three bones, i.e. humerus, ulna, and radius, an one-degree-of-freedom elbow joint, and an antagonistic muscle pair, i.e. the biceps and triceps brachii, and takes into consideration the contact between the skeletal muscles and the humerus. Numerical studies have shown that the proposed upper limb model is capable of predicting realistic moment arms and muscle forces for the entire range of activation and motion. Within the limitations of the model, the presented simulations provide, for the first time, insights into existing contact forces and their influence on the muscle fibre stretch. Based on the presented simulations, the overall change in fibre stretch is typically less than 3%, despite the fact that the contact forces reach up to 71% of the exerted muscle force. Movement-predicting simulations are achieved by minimising a nonlinear moment equilibrium equation. Based on the forward-dynamics simulation approach, an iterative solution procedures for position-driven (inverse dynamics) and force-driven scenarios have been proposed accordingly. Applying these methodologies to time-dependent scenarios demonstrates that the proposed methods can be linked to state-of-the-art control algorithms predicting time-dependent muscle activation levels based on principles of forward dynamics.     

A new constitutive model for multi-layered collagenous tissues

Collagenous tissues such as the aneurysmal wall or the aorta are multi-layered structures with the mean fibre alignments distinguishing one layer from another. A constitutive representation of the multiple collagen layers is not yet developed, and hence the aim of the present study. The proposed model is based on the constitutive theory of finite elasticity and is characterized by an anisotropic strain-energy function which takes the material structure into account. The passive tissue behaviour is modelled and the related mechanical response is assumed to be dominated by elastin and collagen. While elastin is modelled by the neo-Hookean material the constitutive response of collagen is assumed to be transversely isotropic for each individual layer and based on an exponential function. The proposed constitutive function is polyconvex which ensures material stability. The model has five independent material parameters, each of which has a clear physical interpretation: the initial stiffnesses of the collagen fabric in the two principal directions, the shear modulus pertaining to the non-collagenous matrix material, a parameter describing the level of nonlinearity of the collagen fabric, and the angle between the principal directions of the collagen fabric and the reference coordinate system. An extension-inflation test of the adventitia of a human femoral artery is simulated by means of the finite element method and an error function is minimized by adjusting the material parameters yielding a good agreement between the model and the experimental data.     

Prediction of fibre architecture and adaptation in diseased carotid bifurcations

Many studies have used patient-specific finite element models to estimate the stress environment in atherosclerotic plaques, attempting to correlate the magnitude of stress to plaque vulnerability. In complex geometries, few studies have incorporated the anisotropic material response of arterial tissue. This paper presents a fibre remodelling algorithm to predict the fibre architecture, and thus anisotropic material response in four patient-specific models of the carotid bifurcation. The change in fibre architecture during disease progression and its affect on the stress environment in the plaque were predicted. The mean fibre directions were assumed to lie at an angle between the two positive principal strain directions. The angle and the degree of dispersion were assumed to depend on the ratio of principal strain values. Results were compared with experimental observations and other numerical studies. In non-branching regions of each model, the typical double helix arterial fibre pattern was predicted while at the bifurcation and in regions of plaque burden, more complex fibre architectures were found. The predicted change in fibre architecture in the arterial tissue during plaque progression was found to alter the stress environment in the plaque. This suggests that the specimen-specific anisotropic response of the tissue should be taken into account to accurately predict stresses in the plaque. Since determination of the fibre architecture in vivo is a difficult task, the system presented here provides a useful method of estimating the fibre architecture in complex arterial geometries.     

An orthotropic viscoelastic material model for passive myocardium: theory and algorithmic treatment

This contribution presents a novel constitutive model in order to simulate an orthotropic rate-dependent behaviour of the passive myocardium at finite strains. The motivation for the consideration of orthotropic viscous effects in a constitutive level lies in the disagreement between theoretical predictions and experimentally observed results. In view of experimental observations, the material is deemed as nearly incompressible, hyperelastic, orthotropic and viscous. The viscoelastic response is formulated by means of a rheological model consisting of a spring coupled with a Maxwell element in parallel. In this context, the isochoric free energy function is decomposed into elastic equilibrium and viscous non-equilibrium parts. The baseline elastic response is modelled by the orthotropic model of Holzapfel and Ogden [Holzapfel GA, Ogden RW. 2009. Constitutive modelling of passive myocardium: a structurally based framework for material characterization. Philos Trans Roy Soc A Math Phys Eng Sci. 367:3445–3475]. The essential aspect of the proposed model is the account of distinct relaxation mechanisms for each orientation direction. To this end, the non-equilibrium response of the free energy function is constructed in the logarithmic strain space and additively decomposed into three anisotropic parts, denoting fibre, sheet and normal directions each accompanied by a distinct dissipation potential governing the evolution of viscous strains associated with each orientation direction. The evolution equations governing the viscous flow have an energy-activated nonlinear form. The energy storage in the Maxwell branches has a quadratic form leading to a linear stress–strain response in the logarithmic strain space. On the numerical side, the algorithmic aspects suitable for the implicit finite element method are discussed in a Lagrangian setting. The model shows excellent agreement compared to experimental data obtained from the literature. Furthermore, the finite element simulations of a heart cycle carried out with the proposed model show significant deviations in the strain field relative to the elastic solution.     

Predicting bone remodeling around tissue- and bone-level dental implants used in reduced bone width

The objective of this study was to predict time-dependent bone remodeling around tissue- and bone-level dental implants used in patients with reduced bone width. The remodeling of bone around titanium tissue-level, and titanium and titanium–zirconium alloy bone-level implants was studied under 100 N oblique load for one month by implementing the Stanford theory into three-dimensional finite element models. Maximum principal stress, minimum principal stress, and strain energy density in peri-implant bone and displacement in x- and y- axes of the implant were evaluated. Maximum and minimum principal stresses around tissue-level implant were higher than bone-level implants and both bone-level implants experienced comparable stresses. Total strain energy density in bone around titanium implants slightly decreased during the first two weeks of loading followed by a recovery, and the titanium–zirconium implant showed minor changes in the axial plane. Total strain energy density changes in the loading and contralateral sides were higher in tissue-level implant than other implants in the cortical bone at the horizontal plane. The displacement values of the implants were almost constant over time. Tissue-level implants were associated with higher stresses than bone-level implants. The time-dependent biomechanical outcome of titanium–zirconium alloy bone-level implant was comparable to the titanium implant.     

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.     

Viscoelastic properties of passive skeletal muscle in compression: stress-relaxation behaviour and constitutive modelling

The compressive properties of skeletal muscle are important in impact biomechanics, rehabilitation engineering and surgical simulation. However, the mechanical behaviour of muscle tissue in compression remains poorly characterised. In this paper, the time-dependent properties of passive skeletal muscle were investigated using a combined experimental and theoretical approach. Uniaxial ramp and hold compression tests were performed in vitro on fresh porcine skeletal muscle at various rates and orientations of the tissue fibres. Results show that above a very small compression rate, the viscoelastic component plays a significant role in muscle mechanical properties; it represents approximately 50% of the total stress reached at a compression rate of 0.5% s⁻¹. A stiffening effect with compression rate is observed especially in directions closer to the muscle fibres. Skeletal muscle viscoelastic behaviour is thus dependent on compression rate and fibre orientation.

A model is proposed to represent the observed experimental behaviour, which is based on the quasi-linear viscoelasticity framework. A previously developed strain-dependent Young's Moduli formulation was extended with Prony series to account for the tissue viscoelastic properties. Parameters of the model were obtained by fitting to stress-relaxation data obtained in the muscle fibre, cross-fibre and 45° directions. The model then successfully predicted stress-relaxation behaviour at 60° from the fibre direction (errors <25%). Simultaneous fitting to data obtained at compression rates of 0.5% s⁻¹, 1% s⁻¹ and 10% s⁻¹ was performed and the model provided a good fit to the data as well as good predictions of muscle behaviour at rates of 0.05% s⁻¹ and 5% s⁻¹ (errors <25%).     

Towards an analytical model of soft biological tissues

In the past years, soft-tissue modelling research has seen substantial developments, a significant part of which can be ascribed to the refinement of numerical techniques, such as Finite Element analysis. A large class of physico-mechanical properties can be effectively simulated and predictions can be made for a variety of phenomena. However, there is still much that can be conceptually explored by means of fundamental theoretical analysis. In the past few years, driven by our interest in articular cartilage mechanics, we have developed theoretical microstructural models for linear elasticity and permeability that accounted for the presence and arrangement of collagen fibres in cartilage. In this paper, we investigate analytically the non-linear elasticity of soft tissues with collagen fibres arranged according to a given distribution of orientation, a problem that, aside from the case of fibres aligned in a finite number of distinct directions, has been treated exclusively numerically in the literature. We show that, for the case of a tissue with complex fibre arrangement, such as articular cartilage, the theoretical framework commonly used leads to an integral expression of the elastic strain energy potential. The present model is a first attempt in the development of a unified analytical microstructural model for non-linear elasticity and permeability of hydrated, fibre-reinforced soft tissues.     

Cortical microtubules are responsible for gravity resistance in plants

Mechanical resistance to the gravitational force is a principal gravity response in plants distinct from gravitropism. In the final step of gravity resistance, plants increase the rigidity of their cell walls. Here we discuss the role of cortical microtubules, which sustain the function of the cell wall, in gravity resistance. Hypocotyls of Arabidopsis tubulin mutants were shorter and thicker than the wild-type, and showed either left-handed or right-handed helical growth at 1 g. The degree of twisting phenotype was intensified under hypergravity conditions. Hypergravity also induces reorientation of cortical microtubules from transverse to longitudinal directions in epidermal cells. In tubulin mutants, the percentage of cells with longitudinal microtubules was high even at 1 g, and it was further increased by hypergravity. The left-handed helical growth mutants had right-handed microtubule arrays, whereas the right-handed mutant had left-handed arrays. Moreover, blockers of mechanoreceptors suppressed both the twisting phenotype and reorientation of microtubules in tubulin mutants. These results support the hypothesis that cortical microtubules play an essential role in maintenance of normal growth phenotype against the gravitational force, and suggest that mechanoreceptors are involved in signal perception in gravity resistance. Space experiments will confirm whether this view is applicable to plant resistance to 1 g gravity, as to the resistance to hypergravity.Key words: cortical microtubules, gravity, gravity resistance, hypergravity, mechanoreceptor, microgravity, tubulin mutants     

Stress-driven collagen fiber remodeling in arterial walls

A stress-driven model for the relation between the collagen morphology and the loading conditions in arterial walls is proposed. We assume that the two families of collagen fibers in arterial walls are aligned along preferred directions, located between the directions of the two maximal principal stresses. For the determination of these directions an iterative finite element based procedure is developed. As an example the remodeling of a section of a human common carotid artery is simulated. We find that the predicted fiber morphology correlates well with experimental observations. Interesting outcomes of the model including local shear minimization and the possibility of axial compressions due to high blood pressure are revealed and discussed. In memory of Zinaida Hariton, 1926–2002     

Geometric versus topological clustering: An insight into conformation mapping

Clustering molecular conformations into “families” is a common procedure in conformational analysis of molecular systems. An implicit assumption which often underlies this clustering approach is that the resulting geometric families reflect the energetic structure of the system's potential energy surface. In a broader context we address the question whether structural similarity is correlated with energy basins, i.e., whether conformations that belong to the same energy basin are also geometrically similar. “Topological mapping” and principal coordinate projections are used here to address this question and to assess the quality of the “family clustering” procedure. Applying the analysis to a small tetrapeptide it was found that the general correlation that exists between energy basins and structural similarity is not absolute. Clusters generated by the geometric “family clustering” procedure do not always reflect the underlying energy basins. In particular it was found that the “family tree” that is generated by the “family clustering” procedure is completely inconsistent with its real topological counterpart, the “disconnectivity” graph of this system. It is also demonstrated that principal coordinate analysis is a powerful visualization technique which, at least for this system, works better when distances are measured in dihedral angle space rather than in cartesian space. © 1997 Wiley-Liss, Inc.     

Relationship between the arrangement of microtubules and chromosome behaviour of syntelic autosomal univalents during prometaphase in crane fly spermatocytes

During meiotic prometaphase of crane fly spermatocytes, syntelic autosomal univalents are able to move between the spindle poles several times. The relationship between the behaviour of chromosomes and the arrangement of microtubules during this stage was studied using a fixation technique (Nicklas et al. 1979) which makes it possible to examine a certain cell in an electron microscope after live observation. After reorientation, when a syntelic univalent had started moving towards the opposite spindle pole, a short chromosome fibre was found. When a univalent had reached the equator, a chromosome fibre could be traced up to the spindle apex. During the movement towards the opposite spindle pole the degree of disorder in the chromosome fibre was high, whereas it was low in the fibre of a motionless univalent. The degree of disorder was determined by the relative proportion of skew fibre microtubules. At the beginning of reorientation a chromosome fibre was still present, but later, it was no longer possible to recognize such a fibre. Instead of a chromosome fibre, a bundle of microtubules laterally associated with the kinetochore was observed. Some microtubules of this bundle had a direct contact with the kinetochore. These observations strongly hint that the laterally associated microtubules have an important function in the reorientation of syntelic autosomal univalents.     

Spatial and Directional Variation of Growth Rates in Arabidopsis Root Apex: A Modelling Study

Growth and cellular organization of the Arabidopsis root apex are investigated in various aspects, but still little is known about spatial and directional variation of growth rates in very apical part of the apex, especially in 3D. The present paper aims to fill this gap with the aid of a computer modelling based on the growth tensor method. The root apex with a typical shape and cellular pattern is considered. Previously, on the basis of two types of empirical data: the published velocity profile along the root axis and dimensions of cell packets formed in the lateral part of the root cap, the displacement velocity field for the root apex was determined. Here this field is adopted to calculate the linear growth rate in different points and directions. The results are interpreted taking principal growth directions into account. The root apex manifests a significant anisotropy of the linear growth rate. The directional preferences depend on a position within the root apex. In the root proper the rate in the periclinal direction predominates everywhere, while in the root cap the predominating direction varies with distance from the quiescent centre. The rhizodermis is distinguished from the neighbouring tissues (cortex, root cap) by relatively high contribution of the growth rate in the anticlinal direction. The degree of growth anisotropy calculated for planes defined by principal growth directions and exemplary cell walls may be as high as 25. The changes in the growth rate variation are modelled.     

Microtubule stabilization leads to growth reorientation in Arabidopsis trichomes

The single-cell trichomes in wild-type Arabidopsis are either unbranched or have two to five branches. Using transgenic Arabidopsis plants expressing a green fluorescent protein-microtubule-associated protein4 fusion protein, which decorates the microtubular cytoskeleton, we observed that during trichome branching, microtubules reorient with respect to the longitudinal growth axis. Considering branching to be a localized microtubule-dependent growth reorientation event, we investigated the effects of microtubule-interacting drugs on branch induction in trichomes. In unbranched trichomes of the mutant stichel, a change in growth directionality, closely simulating branch initiation, could be elicited by a short treatment with paclitaxel, a microtubule-stabilizing drug, but not with microtubule-disrupting drugs. The growth reorientation appeared to be linked to increased microtubule stabilization and to aster formation in the treated trichomes. Taxol-induced microtubule stabilization also led to the initiation of new branch points in the zwichel mutant of Arabidopsis, which is defective in a kinesin-like microtubule motor protein and possesses trichomes that are less branched. Our observations suggest that trichome cell branching in Arabidopsis might be mediated by transiently stabilized microtubular structures, which may form a component of a multiprotein complex required to reorient freshly polymerizing microtubules into new growth directions.