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.     

Integrin organization: linking adhesion ligand nanopatterns with altered cell responses

This paper presents a modelling framework in which the mechanochemical properties of smooth muscle cells may be studied. The activation of smooth muscles is considered in a three-dimensional continuum model which is key to realistically capture the function of hollow organs such as blood vessels. On the basis of a general thermodynamical framework the mechanical and chemical phases are specialized in order to quantify the coupled mechanochemical process. A free-energy function is proposed as the sum of a mechanical energy stored in the passive tissue, a coupling between the mechanical and chemical kinetics and an energy related purely to the chemical kinetics and the calcium ion concentration. For the chemical phase it is shown that the cross-bridge model of Hai and Murphy [1988. Am. J. Physiol. Cell Physiol. 254, C99-C106] is included in the developed evolution law as a special case. In order to show the specific features and the potential of the proposed continuum model a uniaxial extension test of a tissue strip is analysed in detail and the related kinematics and stress-stretch relations are derived. Parameter studies point to coupling phenomena; in particular the tissue response is analysed in terms of the calcium ion level. The model for smooth muscle contraction may significantly contribute to current modelling efforts of smooth muscle tissue responses.     

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.     

Mathematical models for the study of electromechanical and mechanoelectrical coupling in the myocardium

Mathematical models have been developed to describe interactions of electrical, mechanical and chemical processes in cardiomyocytes. The models simulate wide range of experimental data on excitation-contraction coupling and, more importantly, on mechanoelectric feedback in heart muscle. The model results clearly show that mechano-dependence of intracellular calcium handling due to cooperative effects of contractile proteins activation plays a key role in cardiac mechanoelectric coupling. At the same time, mechanosensitive currents can also contribute to action potential responses to mechanical perturbations. Using this model to study the heterogeneous myocardium we have shown that temporal and functional electromechanical heterogeneity of coupled cardiomyocytes can essentially determine the myocardium contractility. Optimization of the electromechanical function of contractile system emerges from the fine coordination between the activation sequence of cardiomyocytes, their local electromechanical properties and the mechanical interaction during contraction.     

The computation of simulated endplate potentials

A method is described for computing simulated subthreshold responses (.e. endplate potentials) for the Falk-Fatt (1964) cable model of a muscle fibre due to a punctate change in ionic conductance. The method has been applied to a comparison of the Falk-Fatt and classical models. Comparison with experimental results suggests that the model is adequate to account for the response of the mouse muscle fibre, in which the endplate is highly localized. For frog muscle there are larger discrepancies, which may be due to the extended endplate in this species.     

Orientation and length of mammalian skeletal myocytes in response to a unidirectional stretch

Effects of mechanical forces exerted on mammalian skeletal muscle cells during development were studied using an in vitro model to unidirectionally stretch cultured C2C12 cells grown on silastic membrane. Previous models to date have not studied these responses of the mammalian system specifically. The silastic membrane upon which these cells were grown exhibited linear strain behavior over the range of 3.6-14.6% strain, with a Poisson's ratio of approximately 0.5. To mimic murine in utero long bone growth, cell substrates were stretched at an average strain rate of 2.36%/day for 4 days or 1.77%/day for 6 days with an overall membrane strain of 9.5% and 10.6%, respectively. Both control and stretched fibers stained positively for the contractile protein, alpha-actinin, demonstrating muscle fiber development. An effect of stretch on orientation and length of myofibers was observed. At both strain rates, stretched fibers aligned at a smaller angle relative to the direction of stretch and were significantly longer compared to randomly oriented control fibers. There was no effect of duration of stretch on orientation or length, suggesting the cellular responses are independent of strain rate for the range tested. These results demonstrate that, under conditions simulating mammalian long bone growth, cultured myocytes respond to mechanical forces by lengthening and orienting along the direction of stretch.     

Anisotropic effects of the levator ani muscle during childbirth

Pelvic floor dysfunction and pelvic organ prolapse have been associated with damage to the levator ani (LA) muscle, but the exact mechanisms linking them remain unknown. It has been postulated that factors such as vaginal birth and ageing may contribute to long-term, irreversible LA muscle damage. To investigate the biomechanical significance of the LA muscle during childbirth, researchers and clinicians have used finite element models to simulate the second stage of labour. One of the challenges is to represent the anisotropic mechanical response of the LA muscle. In this study, we investigated the effects of anisotropy by varying the relative stiffness between the fibre and the matrix components, whilst maintaining the same overall stress–strain response in the fibre direction. A foetal skull was passed through two pelvic floor models, which incorporated the LA muscle with different anisotropy ratios. Results showed a substantial decrease in the magnitude of the force required for delivery as the fibre anisotropy was increased. The anisotropy ratio markedly affected the mechanical response of the LA muscle during a simulated vaginal delivery. It is apparent that we need to obtain experimental data on muscle mechanics in order to better approximate the LA muscle mechanical properties for quantitative analysis. These models may advance our understanding of the injury mechanisms of pelvic floor during childbirth.     

Improving anterior deltoid activity in a musculoskeletal shoulder model – an analysis of the torque-feasible space at the sternoclavicular joint

Modelling the shoulder's musculature is challenging given its mechanical and geometric complexity. The use of the ideal fibre model to represent a muscle's line of action cannot always faithfully represent the mechanical effect of each muscle, leading to considerable differences between model-estimated and in vivo measured muscle activity. While the musculo–tendon force coordination problem has been extensively analysed in terms of the cost function, only few works have investigated the existence and sensitivity of solutions to fibre topology. The goal of this paper is to present an analysis of the solution set using the concepts of torque-feasible space (TFS) and wrench-feasible space (WFS) from cable-driven robotics. A shoulder model is presented and a simple musculo–tendon force coordination problem is defined. The ideal fibre model for representing muscles is reviewed and the TFS and WFS are defined, leading to the necessary and sufficient conditions for the existence of a solution. The shoulder model's TFS is analysed to explain the lack of anterior deltoid (DLTa) activity. Based on the analysis, a modification of the model's muscle fibre geometry is proposed. The performance with and without the modification is assessed by solving the musculo–tendon force coordination problem for quasi-static abduction in the scapular plane. After the proposed modification, the DLTa reaches 20% of activation.     

Muscle spindle receptors

Summary The depolarization of the sensory terminals of muscle spindle primary endings is studied in terms of a simplified core conductor model of the system. The terminals branching from a group Ia afferent fibre have different geometrical structures, depending on whether they innervate the nuclear bag or the nuclear chain intrafusal fibres of a muscle spindle.The depolarization of the two different structures as a function of space and time is analysed by digital simulation technique. The studies indicate that the tapered core conductor model of the bag-fibre terminals responds faster than a uniform system similar to the terminals innervating the nuclear chain fibres. The steady state response, on the other hand, appears to supply a more effective depolarization in the case of a uniform system with otherwise equivalent parameters.The results suggest that the bag fibre terminals are best suited for converting dynamical stimuli into an electrical signal, whilst the chain fibre terminals seem to yield a more effective steady state response. The results are discussed in relation to the functional properties of the mammalian muscle spindles and in relation to the earlier proposed models of the mechanical system represented by the two types of intrafusal muscle fibres.     

Strain-induced reorientation of an intramuscular connective tissue network: implications for passive muscle elasticity

The most abundant intramuscular connective tissue component, the perimysium, of bovine M. sternomandibularis muscle was shown to be a crossed-ply arrangement of crimped collagen fibres which reorientate and decrimp on changing muscle fibre sarcomere length. Reorientation of perimysial strands was observed by light microscopy and identification of these strands as collagen fibres was confirmed by high-angle X-ray diffraction. Mean collagen fibre direction with respect to the muscle fibres ranged from approximately 80 degrees at sarcomere length = 1.1 micron to approximately 20 degrees at 3.9 microns. This behaviour was well described by a model of a crimped planar network surrounding a muscle fibre bundle of constant volume but varying length. Modelling of the mechanical properties of the perimysium at different sarcomere lengths produced a load-sarcomere length curve which was in good agreement with the passive elastic properties of the muscle, especially at long sarcomere lengths. It is concluded that the role of the perimysial collagen network is to prevent over-stretching of the muscle fibre bundles.     

A role of calcium in altered sodium ion transport of hypertensives?

Research on the etiology of essential hypertension has led to many reports of altered ion transport in cells from hypertensive patients and animal models. Abnormalities in sodium and calcium ion gradients and transport in vascular smooth muscle, neuronal tissue, cardiac muscle as well as erythrocytes have been extensively investigated. It is not clear whether these abnormalities are of primary or secondary nature. The current knowledge of sodium and calcium ion transport in essential hypertension is briefly reviewed here. Furthermore, evidence is presented which suggests a role of calcium in the regulation of sodium transport activity.     

Presynaptic and postsynaptic competition in models for the development of neuromuscular connections

In the establishment of connections between nerve and muscle there is an initial stage when each muscle fibre is innervated by several different motor axons. Withdrawal of connections then takes place until each fibre has contact from just a single axon. The evidence suggests that the withdrawal process involves competition between nerve terminals. We examine in formal models several types of competitive mechanism that have been proposed for this phenomenon. We show that a model which combines competition for a presynaptic resource with competition for a postsynaptic resource is superior to others. This model accounts for many anatomical and physiological findings and has a biologically plausible implementation. Intrinsic withdrawal appears to be a side effect of the competitive mechanism rather than a separate non-competitive feature. The model's capabilities are confirmed by theoretical analysis and full scale computer simulations.     

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.     

Stretch-induced Increase in Activation of Skinned Muscle Fibres by Calcium

IT is well known that the active tetanic tension of a living striated muscle fibre decreases linearly with increase of fibre length beyond its slack length^1,2 and this has been explained² by the decreased number of interacting sites between thick (myosin-containing) and thin (actin-containing) filaments. Skinned fibres³ have been shown to behave similarly at high concentrations of calcium⁴. But examination of the mechanical properties, of partially activated skinned muscle fibres showed that the isometric tension increased with the increase of fibre length beyond its slack length if contraction was induced by a low concentration of calcium ions.     

Model for the β motor stimulation in chelonian muscle spindles

A previously proposed model to simulate the behaviour of the chelonian muscle spindle during mechanical stretch has been extended to include the properties of the spindle during activation of the intrafusal muscle fibres. It is assumed that the overall transfer function of the non-activated spindle can be entirely ascribed to the visco-elastic properties of its intrafusal fibres. It is found that the activated spindle can then be simulated by incorporating a force generator into the visco-elastic model and by accepting stepwise changes in its parameter values at the onset and at the end of fusimotor stimulation. The influence of extrafusal fibre contraction has been accounted for by inserting the Voigt muscle model in parallel with the spindle model.     

3D finite element models of shoulder muscles for computing lines of actions and moment arms

Accurate representation of musculoskeletal geometry is needed to characterise the function of shoulder muscles. Previous models of shoulder muscles have represented muscle geometry as a collection of line segments, making it difficult to account for the large attachment areas, muscle–muscle interactions and complex muscle fibre trajectories typical of shoulder muscles. To better represent shoulder muscle geometry, we developed 3D finite element models of the deltoid and rotator cuff muscles and used the models to examine muscle function. Muscle fibre paths within the muscles were approximated, and moment arms were calculated for two motions: thoracohumeral abduction and internal/external rotation. We found that muscle fibre moment arms varied substantially across each muscle. For example, supraspinatus is considered a weak external rotator, but the 3D model of supraspinatus showed that the anterior fibres provide substantial internal rotation while the posterior fibres act as external rotators. Including the effects of large attachment regions and 3D mechanical interactions of muscle fibres constrains muscle motion, generates more realistic muscle paths and allows deeper analysis of shoulder muscle function.     

Calcium-induced calcium release mechanism from the sarcoplasmic reticulum in skinned crab muscle fibres

Mechanically skinned skeletal muscle fibres of the crab Carcinus maenas have been used to investigate the mechanism of calcium release from the sarcoplasmic reticulum. Calcium release has been monitored by the amplitude and kinetics of the tension developed by the fibre. Results show that a very low calcium concentration, insufficient to directly activate contractile proteins, induces a release of calcium from the SR. This release is stimulated by low concentrations of caffeine and inhibited by small amounts of EGTA. Thus, a graded calcium-induced calcium release mechanism dependent on extrareticular calcium concentration has been demonstrated in skinned crab muscle fibre.     

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%).     

A model of multicomponent cardiac fibre

In order to simulate the contraction of a cardiac myofibre, a multicomponent fibre model has been developed. This model is composed of a series of segments which are activated in succession. Each segment is represented by the Hill's three component model of the sarcomere. The contractile element behaviour is described by the Huxley's theory and the time dependence agrees with the activation factor proposed by Julian for skeletal muscle, and modified by Wong for cardiac muscle. The two elastic elements have non-linear exponential characteristics. The isometric contraction of the multicomponent fibre has been simulated by means of a computer program. The results show the tension generated by the fibre, the propagation of the contraction along the fibre and the different contribution of each segment depending on its position inside the fibre.