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.     

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.     

A 3D skeletal muscle model coupled with active contraction of muscle fibres and hyperelastic behaviour

This paper presents a three-dimensional finite element model of skeletal muscle which was developed to simulate active and passive non-linear mechanical behaviours of the muscle during lengthening or shortening under either quasi-static or dynamic condition. Constitutive relation of the muscle was determined by using a strain energy approach, while active contraction behaviour of the muscle fibre was simulated by establishing a numerical algorithm based on the concept of the Hill's three-element muscle model. The proposed numerical algorithm could be used to predict concentric, eccentric, isometric and isotonic contraction behaviours of the muscle. The proposed numerical algorithm and constitutive model for the muscle were derived and implemented into a non-linear large deformation finite element programme ABAQUS by using user-defined material subroutines. A number of scenarios have been used to demonstrate capability of the model for simulating both quasi-static and dynamic response of the muscle. Validation of the proposed model has been performed by comparing the simulated results with the experimental ones of frog gastrocenemius muscle deformation. The effects of the fusiform muscle geometry and fibre orientation on the stress and fibre stretch distributions of frog muscle during isotonic contraction have also been investigated by using the proposed model. The predictability of the present model for dynamic response of the muscle has been demonstrated by simulating the extension of a squid tentacle during a strike to catch prey.     

Generating fibre orientation maps in human heart models using Poisson interpolation

Smoothly varying muscle fibre orientations in the heart are critical to its electrical and mechanical function. From detailed histological studies and diffusion tensor imaging, we now know that fibre orientations in humans vary gradually from approximately ? 70° in the outer wall to +80° in the inner wall. However, the creation of fibre orientation maps for computational analyses remains one of the most challenging problems in cardiac electrophysiology and cardiac mechanics. Here, we show that Poisson interpolation generates smoothly varying vector fields that satisfy a set of user-defined constraints in arbitrary domains. Specifically, we enforce the Poisson interpolation in the weak sense using a standard linear finite element algorithm for scalar-valued second-order boundary value problems and introduce the feature to be interpolated as a global unknown. User-defined constraints are then simply enforced in the strong sense as Dirichlet boundary conditions. We demonstrate that the proposed concept is capable of generating smoothly varying fibre orientations, quickly, efficiently and robustly, both in a generic bi-ventricular model and in a patient-specific human heart. Sensitivity analyses demonstrate that the underlying algorithm is conceptually able to handle both arbitrarily and uniformly distributed user-defined constraints; however, the quality of the interpolation is best for uniformly distributed constraints. We anticipate our algorithm to be immediately transformative to experimental and clinical settings, in which it will allow us to quickly and reliably create smooth interpolations of arbitrary fields from data-sets, which are sparse but uniformly distributed.     

Three-dimensional finite element modelling of muscle forces during mastication

This paper presents a three-dimensional finite element model of human mastication. Specifically, an anatomically realistic model of the masseter muscles and associated bones is used to investigate the dynamics of chewing. A motion capture system is used to track the jaw motion of a subject chewing standard foods. The three-dimensional nonlinear deformation of the masseter muscles are calculated via the finite element method, using the jaw motion data as boundary conditions. Motion-driven muscle activation patterns and a transversely isotropic material law, defined in a muscle-fibre coordinate system, are used in the calculations. Time-force relationships are presented and analysed with respect to different tasks during mastication, e.g. opening, closing, and biting, and are also compared to a more traditional one-dimensional model. The results strongly suggest that, due to the complex arrangement of muscle force directions, modelling skeletal muscles as conventional one-dimensional lines of action might introduce a significant source of error.     

Mechanics of cranial sutures using the finite element method

To investigate how cranial suture morphology and the arrangement of sutural collagen fibres respond to compressive and tensile loads, an idealised bone–suture–bone complex was analysed using a two-dimensional finite element model. Three suture morphologies were simulated with an increasing interdigitation index (I.I.): butt-ended, moderate interdigitated, and complex interdigitated. The collagen matrix within all sutures was modelled as an isotropic material, and as an orthotropic material in the interdigitated sutures with fibre alignment as reported in studies of miniature pigs. Static uniform compressive or tensile loading was applied to the complex. In interdigitated sutures with isotropic material properties, the orientation of the maximum (tensile) principal stresses within the suture matched the collagen fibre orientation observed in compressed and tensed sutures of miniature pigs. This suggests that randomly arranged sutural collagen fibres could optimise to an orientation most appropriate to withstand the predominant type of loading. A compression-resistant fibre arrangement imparted the highest suture strain energy relative to the isotropic and tension-resistant arrangements, indicating that this configuration maximises energy storage. A comparison across the different suture morphologies indicated that bone strain energy generally decreased with a decrease in I.I., irrespective of the sutural fibre arrangement. However, high bone stress at the interdigitation apices shifted to the limbs of the suture with an increase in I.I. These combined findings highlight the importance of suture morphology and anisotropy as properties having a significant influence on sutural mechanics.     

A finite element simulation scheme for biological muscular hydrostats

An explicit finite element scheme is developed for biological muscular hydrostats such as squid tentacles, octopus arms and elephant trunks. The scheme is implemented by embedding muscle fibers in finite elements. In any given element, the fiber orientation can be assigned arbitrarily and multiple muscle directions can be simulated. The mechanical stress in each muscle fiber is the sum of active and passive parts. The active stress is taken to be a function of activation state, muscle fiber shortening velocity and fiber strain; while the passive stress depends only on the strain. This scheme is tested by simulating extension of a squid tentacle during prey capture; our numerical predictions are in close correspondence with existing experimental results. It is shown that the present finite element scheme can successfully simulate more complex behaviors such as torsion of a squid tentacle and the bending behavior of octopus arms or elephant trunks.     

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.     

Measuring morphological parameters of the pelvic floor for finite element modelling purposes

The goal of this study was to obtain a complete data set needed for studying the complex biomechanical behaviour of the pelvic floor muscles using a computer model based on the finite element (FE) theory. The model should be able to predict the effect of surgical interventions and give insight into the function of pelvic floor muscles. Because there was a lack of any information concerning morphological parameters of the pelvic floor muscle structures, we performed an experimental measurement to uncover those morphological parameters. Geometric parameters as well as muscle parameters of the pelvic floor muscles were measured on an embalmed female cadaver. A three-dimensional (3D) geometric data set of the pelvic floor including muscle fibre directions was obtained using a palpator device. A 3D surface model based on the experimental data, needed for mathematical modelling of the pelvic floor, was created. For all parts of the diaphragma pelvis, the optimal muscle fibre length was determined by laser diffraction measurements of the sarcomere length. In addition, other muscle parameters such as physiological cross-sectional area and total muscle fibre length were determined. Apart from these measurements we obtained a data set of the pelvic floor structures based on nuclear magnetic resonance imaging (MRI) on the same cadaver specimen. The purpose of this experiment was to discover the relationship between the MRI morphology and geometrical parameters obtained from the previous measurements. The produced data set is not only important for biomechanical modelling of the pelvic floor muscles, but it also describes the geometry of muscle fibres and is useful for functional analysis of the pelvic floor in general. By the use of many reference landmarks all these morphologic data concerning fibre directions and optimal fibre length can be morphed to the geometrical data based on segmentation from MRI scans.These data can be directly used as an input for building a mathematical model based on FE theory.     

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.     

A transversely isotropic viscoelastic constitutive equation for brainstem undergoing finite deformation

The objective of this study was to define the constitutive response of brainstem undergoing finite shear deformation. Brainstem was characterized as a transversely isotropic viscoelastic material and the material model was formulated for numerical implementation. Model parameters were fit to shear data obtained in porcine brainstem specimens undergoing finite shear deformation in three directions: parallel, perpendicular, and cross sectional to axonal fiber orientation and determined using a combined approach of finite element analysis (FEA) and a genetic algorithm (GA) optimizing method. The average initial shear modulus of brainstem matrix of 4-week old pigs was 12.7 Pa, and therefore the brainstem offers little resistance to large shear deformations in the parallel or perpendicular directions, due to the dominant contribution of the matrix in these directions. The fiber reinforcement stiffness was 121.2 Pa, indicating that brainstem is anisotropic and that axonal fibers have an important role in the cross-sectional direction. The first two leading relative shear relaxation moduli were 0.8973 and 0.0741, respectively, with corresponding characteristic times of 0.0047 and 1.4538 s, respectively, implying rapid relaxation of shear stresses. The developed material model and parameter estimation technique are likely to find broad applications in neural and orthopaedic tissues.     

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%).     

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.     

A fibre reorientation model for orthotropic multiplicative growth

The main goal of this contribution consists in the development of a remodelling framework for orthotropic continua whereby the underlying symmetry group is incorporated via two fibre families. Special emphasis is placed on the modelling of biological tissues at finite deformations. Besides the incorporation of a referential mass source, anisotropic growth is addressed by means of a multiplicative decomposition of the overall deformation gradient into an elastic and a growth distortion. Projected quantities of a configurational growth stress tensor are advocated as driving forces for time-dependent saturation–type evolution of the principal values of the growth distortion. Moreover, the reorientation of both fibre families, which directly affects the strain energy as well as the growth distortion itself, is guided by analyzing critical energy points. In particular, a time-dependent formulation is developed which aligns the fibre directions according to the principal stretch directions. Finally, the proposed framework is embedded into a finite element context so that representative numerical examples, examining growth and resorption in volume and density together with fibre reorientation, close this study.     

Effect of collagen fibre orientation on intervertebral disc torsion mechanics

The intervertebral disc is a complex fibro-cartilaginous material, consisting of a pressurized nucleus pulposus surrounded by the annulus fibrosus, which has an angle-ply structure. Disc injury and degeneration are noted by significant changes in tissue structure and function, which significantly alters stress distribution and disc joint stiffness. Differences in fibre orientation are thought to contribute to changes in disc torsion mechanics. Therefore, the objective of this study was to evaluate the effect of collagen fibre orientation on internal disc mechanics under compression combined with axial rotation. We developed and validated a finite element model (FEM) to delineate changes in disc mechanics due to fibre orientation from differences in material properties. FEM simulations were performed with fibres oriented at \(\pm 30^{\circ }\) throughout the disc (uniform by region and fibre layer). The initial model was validated by published experimental results for two load conditions, including \(0.48\,\hbox {MPa}\) axial compression and \(10\,\hbox {Nm}\) axial rotation. Once validated, fibre orientation was rotated by \(4^{\circ }\) or \(8^{\circ }\) towards the horizontal plane, resulting in a decrease in disc joint torsional stiffness. Furthermore, we observed that axial rotation caused a sinusoidal change in disc height and radial bulge, which may be beneficial for nutrient transport. In conclusion, including anatomically relevant fibre angles in disc joint FEMs is important for understanding stress distribution throughout the disc and will be important for understanding potential causes for disc injury. Future models will include regional differences in fibre orientation to better represent the fibre architecture of the native disc.     

Viscoelastic properties of passive skeletal muscle in compression—Cyclic behaviour

Skeletal muscle relaxation behaviour in compression has been previously reported, but the anisotropic behaviour at higher loading rates remains poorly understood. In this paper, uniaxial unconfined cyclic compression tests were performed on fresh porcine muscle samples at various fibre orientations to determine muscle viscoelastic behaviour. Mean compression level of 25% was applied and cycles of 2% and 10% amplitude were performed at 0.2–80 Hz. Under cycles of low frequency and amplitude, linear viscoelastic cyclic relaxation was observed. Fibre/cross-fibre results were qualitatively similar, but the cross-fibre direction was stiffer (ratio of 1.2). In higher amplitude tests nonlinear viscoelastic behaviour with a frequency dependent increase in the stress cycles amplitude was found (factor of 4.1 from 0.2 to 80 Hz).The predictive capability of an anisotropic quasi-linear viscoelastic model previously fitted to stress-relaxation data from similar tissue samples was investigated. Good qualitative results were obtained for low amplitude cycles but differences were observed in the stress cycle amplitudes (errors of 7.5% and 31.8%, respectively, in the fibre/cross-fibre directions). At higher amplitudes significant qualitative and quantitative differences were evident. A nonlinear model formulation was therefore developed which provided a good fit and predictions to high amplitude low frequency cyclic tests performed in the fibre/cross-fibre directions. However, this model gave a poorer fit to high frequency cyclic tests and to relaxation tests. Neither model adequately predicts the stiffness increase observed at frequencies above 5 Hz.Together with data previously presented, the experimental data presented here provide a unique dataset for validation of future constitutive models for skeletal muscle in compression.     

A validated model of passive muscle in compression

A better characterisation of soft tissues is required to improve the accuracy of human body models used, amongst other applications, for virtual crash modelling. This paper presents a theoretical model and the results of an experimental procedure to characterise the quasi-static, compressive behaviour of skeletal muscle in three dimensions. Uniaxial, unconstrained compression experiments have been conducted on aged and fresh animal muscle samples oriented at various angles from the fibre direction. A transversely isotropic hyperelastic model and a model using the theory of transverse isotropy and strain dependent Young's moduli (SYM) have been fitted to the experimental data. Results show that the hyperelastic model does not adequately fit the data in all directions of testing. In contrast, the SYM gives a good fit to the experimental data in both the fibre and cross-fibre direction, up to 30% strain for aged samples. The model also yields good prediction of muscle behaviour at 45° from the fibre direction. Fresh samples show a different behaviour than aged tissues at 45° from the fibre direction. However, the SYM is able to capture this difference and gives a good fit to the experimental data in the fibre, the cross-fibre and at 45° from the fibre direction. The model also yields good prediction of muscle behaviour when compressed at 30° and 60° from the fibre direction. The effect of the time of test after death has also been investigated. Significant stiffening of muscle behaviour is noted a few hours after death of the subject.