On a phenomenological model for fatigue effects in skeletal muscles

This work deals with the development and implementation of a new fatigue model for simulating fatigue effects in skeletal muscles. Basic idea of this modelling strategy is an approach that divides the fibres of a muscle into three groups: fibres in the active state, those that are already fatigued and fibres in the resting state. All fibres are able to switch between the different groups by defining adequate rates. In this way a continuous transfer of fibres between those three states has been described. Rooted on an incompressible, hyperelastic constitutive law with transversely isotropic characteristics the fatigue model has been implemented in the framework of the finite element method. Numerical examples are given in order to illustrate the ability of this model. Further, we validate the model by fatigue experiments of the rat soleus muscle. In doing so, it proves that the model is able to predict physiological observations and mechanical test results.     

A three-dimensional muscle model: a quantified relation between form and function of skeletal muscles

A three-dimensional muscle model with complex geometry is described and tested against experimental data. Using this model, several muscles were constructed. These muscles have equal optimum length but differ in architecture. The force exerted by the constructed muscles, in relation to their actual length and velocity of shortening, is discussed. Generally speaking, the constructed muscles with considerable pennation have great fiber angles, a great physiological cross section, a narrow active and steep passive length-force relation, and a low maximal velocity of shortening. The maximal power (force times velocity) delivered by the constructed muscles is shown to be almost independent of the architecture of the muscles. The steepness of the passive length-force relation is determined mainly by the shortest fibers within the group of constructed muscles, whereas maximal velocity of shortening and the width of the active length-force relation are determined mainly by the longest fibers. The validity of the three-dimensional muscle model with respect to some morphological and functional characteristics is tested. Length-force relations of constructed muscles are compared with the actual length-force relations of mm. gastrocnemii mediales and mm. semimembranosi of male Wistar rats. Moreover, actual fiber angle, fiber length, and muscle thickness of three mm. gastrocnemii mediales are compared with values found for constructed muscles. It is concluded that the three-dimensional muscle model closely approximates the actual muscle form and function.     

A three-dimensional constitutive model for the stress relaxation of articular ligaments

A new nonlinear constitutive model for the three-dimensional stress relaxation of articular ligaments is proposed. The model accounts for finite strains, anisotropy, and strain-dependent stress relaxation behavior exhibited by these ligaments. The model parameters are identified using published uniaxial stress–stretch and stress relaxation data on human medial collateral ligaments (MCLs) subjected to tensile tests in the fiber and transverse to the fiber directions (Quapp and Weiss in J Biomech Eng Trans ASME 120:757–763, 1998; Bonifasi-Lista et al. in J Orthop Res 23(1):67–76, 2005). The constitutive equation is then used to predict the nonlinear elastic and stress relaxation response of ligaments subjected to shear deformations in the fiber direction and transverse to the fiber direction, and an equibiaxial extension. A direct comparison with stress relaxation data collected by subjecting human MCLs to shear deformation in the fiber direction is presented in order to demonstrate the predictive capabilities of the model.     

A finite-element model for the mechanical analysis of skeletal muscles

In the present paper, a finite-element model for simulating muscle mechanics is described. Based on nonlinear continuum mechanics an algorithm is proposed that includes the contractile active and passive properties of skeletal muscle. Stress in the muscle is assumed to result from the superposition of a passive and an active part. The passive properties are described by a hyperelastic constitutive material law whereas the active part depends on the fibre length, shortening velocity and an activation function. The constraint of approximate incompressibility of the muscle element is satisfied as a property of the constitutive equations. Because of the nonlinear behaviour of the material and the highly dynamical performance an incremental procedure including iterative methods is used. The advantage of the model over previous formulations is the possibility to integrate the element into an engineering standard finite-element programme ANSYS using advanced numerical tools. The model allows simulations of muscle recruitment, calculations of stress and strain distributions and predictions of muscle shape. Other possible applications are studies of the muscle architecture, the effect of inertia and impacts. First, simple examples are presented.     

A novel in vitro three-dimensional skeletal muscle model

A novel three-dimensional (3D) skeletal muscle model composed of C2C12 mouse myoblasts is described. This model was generated by cultivating myoblasts in suspension using the rotary cell culture system (RCCS), a unique culture environment. Single-cell suspensions of myoblasts were seeded at 5 × 10⁵/ml in growth medium without exogenous support structures or substrates. Cell aggregation occurred in both RCCS and suspension control (SC) conditions within 12 h but occurred more rapidly in the SC at all time intervals examined. RCCS-cultured myoblasts fused and differentiated into a 3D construct without serum deprivation or alterations. Syncitia were quantified at 3 and 6+ d in stained thin sections. A significantly greater number of syncitia was found at 6+ d in the RCCS cultures compared to the SC. The majority of syncitia were localized to the periphery of the cell constructs for all treatments. The expression of sarcomeric myosin heavy chain (MHC) was localized at or near the periphery of the 3D construct. The majority of MHC was associated with the large cells (syncitia) of the 6+-d aggregates. These results show, for the first time, that myoblasts form syncitia and express MHC in the presence of growth factors and without the use of exogenous supports or substrates. This model test system is useful for investigating initial cell binding, myoblast fusion and syncitia formation, and differentiation processes.     

A predictive model of fatigue in human skeletal muscles.

Fatigue is a major limitation to the clinical application of functional electrical stimulation. The activation pattern used during electrical stimulation affects force and fatigue. Identifying the activation pattern that produces the greatest force and least fatigue for each patient is, therefore, of great importance. Mathematical models that predict muscle forces and fatigue produced by a wide range of stimulation patterns would facilitate the search for optimal patterns. Previously, we developed a mathematical isometric force model that successfully identified the stimulation patterns that produced the greatest forces from healthy subjects under nonfatigue and fatigue conditions. The present study introduces a four-parameter fatigue model, coupled with the force model that predicts the fatigue induced by different stimulation patterns on different days during isometric contractions. This fatigue model accounted for 90% of the variability in forces produced by different fatigue tests. The predicted forces at the end of fatigue testing differed from those observed by only 9%. This model demonstrates the potential for predicting muscle fatigue in response to a wide range of stimulation patterns.     

Simple constitutive model for a cortical bone

This short study presents a simple, one-dimensional constitutive model for the cortical bone with haversian structure. The model is developed within the general framework of the continuum damage theory. The kinetic equation is derived (rather than assumed a priori) through consideration of the irreversible changes of the mesostructure. As a consequence the analytical results closely approximate experimental measurements even though the theory does not introduce a single experimentally unidentifiable material parameter.     

Structural three-dimensional constitutive law for the passive myocardium

A three-dimensional constitutive law is proposed for the myocardium. Its formulation is based on a structural approach in which the total strain energy of the tissue is the sum of the strain energies of its constituents: the muscle fibers, the collagen fibers and the fluid matrix which embeds them. The ensuing material law expresses the specific structural and mechanical properties of the tissue, namely, the spatial orientation of the comprising fibers, their waviness in the unstressed state and their stress-strain behavior when stretched. Having assumed specific functional forms for the distribution of the fibers spatial orientation and waviness, the results of biaxial mechanical tests serve for the estimation of the material constants appearing in the constitutive equations. A very good fit is obtained between the measured and the calculated stresses, indicating the suitability of the proposed model for describing the mechanical behavior of the passive myocardium. Moreover, the results provide general conclusions concerning the structural basis for the tissue overall mechanical properties, the main of which is that the collagen matrix, though comprising a relatively small fraction of the whole tissue volume, is the dominant component accounting for its stiffness.     

Palisade endings and proprioception in extraocular muscles: a comparison with skeletal muscles

This article describes current views on motor and sensory control of extraocular muscles (EOMs) based on anatomical data. The special morphology of EOMs, including their motor innervation, is described in comparison to classical skeletal limb and trunk muscles. The presence of proprioceptive organs is reviewed with emphasis on the palisade endings (PEs), which are unique to EOMs, but the function of which is still debated. In consideration of the current new anatomical data about the location of cell bodies of PEs, a hypothesis on the function of PEs in EOMs and the multiply innervated muscle fibres they are attached to is put forward.     

Increased extravasation of macromolecules in skeletal muscles of the Zucker rat model

Objective: Assess whether changes in permeability of the muscle regional microcirculation occur in the obese Zucker rat model. Research Methods and Procedures: Capillary permeability to albumin was assessed in vivo in Zucker rats (n = 15) and lean controls (n = 15) by quantifying the extravasation of albumin‐bound Evans Blue (EB) in different organs. Unanaesthetized animals were injected with EB 20 mg/kg in the caudal vein, and EB was extracted by formamide from selected organs collected after exsanguination. Results: Relative to control animals, Zucker rats had higher body weight (Δ = +33%; p < 0.001), plasma triglycerides (Δ = +244%; p < 0.001), and insulin (Δ = +240%; p < 0.001) concentrations. Plasma glucose concentrations were not different between the two groups (p = not significant). Using the EB technique, we showed a 30% to 50% (p < 0.01) increase in the extravasation of EB in the obese rats, regardless of the skeletal muscle group studied. This increase in skeletal muscle vasopermeability was not paralleled by any increase in the expression of the muscle endothelium—nitric oxide (NO) system because the total NO synthase (NOS) activity in skeletal muscle of the obese Zucker rat was significantly lower (p < 0.001), as was the endothelial NOS immunoreactive mass (p < 0.001), compared with lean controls. Discussion: In conclusion, there seems to be dissociation between capillary permeability and local regulation of microcirculation in skeletal muscles of the obese Zucker rat. It is suggested that the increase in skeletal muscle vasopermeability (extravasation of macromolecules) is a compensation for the loss of NO‐dependent vasodilation and capillary recruitment noted in this model of obesity and insulin resistance.     

Analysis of a model for antagonistic muscles

In the present work a linear model for a pair of antogonistic muscles is analysed. Each constituent muscle in this model is identical to ones considered previously (Stein and Ouztöreli, 1976). Analytical properties of the antagonistic muscles and dynamics of the system are described and some numerical results are discussed. The natural modes of the system are determined by a fourth order polynomial, which most commonly has one pair of conjugate complex roots and two negative real roots. The filtering of neural inputs through the active state properties of the muscle increases the order of the system to fifth order for these inputs.This work was partly supported by the National Scientific and Engineering Research Council of Canada Grant NRC-A4345 and by the Medical Research Council of Canada Grant MRC-MT-3307 through the University of Alberta     

Development of a finite element musculoskeletal model with the ability to predict contractions of three-dimensional muscles

Representation of realistic muscle geometries is needed for systematic biomechanical simulation of musculoskeletal systems. Most of the previous musculoskeletal models are based on multibody dynamics simulation with muscles simplified as one-dimensional (1D) line-segments without accounting for the large muscle attachment areas, spatial fibre alignment within muscles and contact and wrapping between muscles and surrounding tissues. In previous musculoskeletal models with three-dimensional (3D) muscles, contractions of muscles were among the inputs rather than calculated, which hampers the predictive capability of these models. To address these issues, a finite element musculoskeletal model with the ability to predict contractions of 3D muscles was developed. Muscles with realistic 3D geometry, spatial muscle fibre alignment and muscle-muscle and muscle-bone interactions were accounted for. Active contractile stresses of the 3D muscles were determined through an efficient optimization approach based on the measured kinematics of the lower extremity and ground force during gait. This model also provided stresses and strains of muscles and contact mechanics of the muscle-muscle and muscle-bone interactions. The total contact force of the knee predicted by the model corresponded well to the in vivo measurement. Contact and wrapping between muscles and surrounding tissues were evident, demonstrating the need to consider 3D contact models of muscles. This modelling framework serves as the methodological basis for developing musculoskeletal modelling systems in finite element method incorporating 3D deformable contact models of muscles, joints, ligaments and bones.