Computer simulations for the mechanical analysis of skeletal muscles

The structure of a skeletal muscle can be seen as a complex hierarchical organisation in which thousands of muscle fibers are arranged within a connective tissue network. Inside of the single muscle fibre many force-producing cells, known as sarcomeres, are connected and take care of the contraction of the whole muscle. The material behaviour of muscles is nonlinear. Due to the fact that muscles can have large deformations in space, geometrical non-linearities must additionally be taken into account. For the simulation of such a behaviour the finite element method is used in the present approach. The material behaviour of the muscle is split into a so-called active and a passive part. To describe the passive part special unit cells consisting of one tetrahedral element and six truss elements have been derived. Additionally to these unit cells other truss elements are attached representing bundles of muscle fibers and therefore the active part of the material behaviour. The contractile behaviour of the muscle is mainly in.uenced by the stretch of the muscle fibres, the shortening velocity and the activation status of the muscle. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A mathematical study of a muscle contraction model in which the fibre is a continuum of elements

In the framework of the sliding-filament theory of muscle contraction, we introduce and study a model which assumes the muscle fibre to be a continuum of elastic and contractile elements. Using the contracting mapping principle, we prove existence and uniqueness of the solution of the nonlinear and nonlocal hyperbolic equation related to the model.     

Experimental and numerical investigations in skeletal muscle modelling

In the present paper, a method to compare displacement results of a muscle contraction simulation with results of optical experiments is proposed. A human skeletal muscle has been reconstructed to a volume element out of real two-dimensional MRI data. The surrounding tissue has also been taken into account in order to describe the interaction with other components in a realistic way. These regions of the interface have been supported by several spring stiffness. The numerical model has been fitted by this stiffness on one significant node. The results shown satisfied agreement with the optical experiments. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bridging scales: An attempt to incorporate cellular responses within a three-dimensional FEM model of active muscle contraction

A mathematical model of the cellular responses of skeletal muscles has been integrated within a three-dimensional biomechanical Finite Element (FEM) model. The FEM model is based on a tri-cubic Hermite Finite Element discretisation of the governing equations of finite elasticity theory and a transversely isotropic constitutive law. To incorporate the cellular information, homogenised values of key physiological parameters, e.g. the pre- and post-power stroke concentration of crossbridge attachments, are computed at the Gauss points of the FEMintegration scheme. These values are then used to modify the stress tensor in such a way that it resembles the contractile response. The advantages of such an improved three-dimensional FEM model are far reaching. These models can be used, for example, to investigate and study local muscle contraction, muscle recruitment patterns, force generation, or fatigue response of skeletal muscles. As an illustrative example, one twitch of the tibialis anterior, in which 25% of the muscle fibres are excited by a nerve stimulus, is simulated. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A pseudo active kinematic constraint for a biological living soft tissue: An effect of the collagen network

Recent studies in mammalian hearts show that left ventricular wall thickening is an important mechanism for systolic ejection, and that during contraction the cardiac muscle develops significant stresses in the muscular cross-fiber direction. We suggested that the collagen network surrounding the muscular fibers could account for these mechanical behaviors. To test this hypothesis we develop a model for large deformation response of active, incompressible, nonlinear elastic and transversely isotropic living soft tissue (such as cardiac or arteries tissues) in which we include a coupling effect between the connective tissue and the muscular fibers. Then, a three-dimensional finite element formulation including this internal pseudo-active kinematic constraint is derived. Analytical and finite element solutions are in a very good agreement. The numerical results show this wall thickening effect with an order of magnitude compatible with the experimental observations.     

Theoretical study of some features of the dynamic behavior of skeletal muscle as a unidimensional viscoelastic medium

The results of a theoretical (model) study of the behavior of a muscle treated as a unidimensional viscoelastic bundle of fibers of finite length are presented. Solution of a second-order differential equation which takes account of the wave processes in the medium under consideration provides an explanation for several of the dynamic effects which are observed in experimental investigations into muscle mechanics (the jumps in the stresses during the linear extension of a muscle, the rapid weakening during its contraction, and so on). The analysis shows that the empirical equation for a muscle due to Hill may be considered as a relationship which is a consequence of its viscoelastic properties and wave phenomena.     

A mathematical study of a continuum-state cross-bridge model of muscle contraction

In the framework of the sliding-filament theory of muscle contraction, we develop a model which describes physioiogical cross-bridge activity with a continuum of states, from those which are detached and refractory to those which are attached. The latter are kinetically more important. An analytical study of the non-linear partial differential equations of the mathematical model shows existence and uniqueness of the solution.     

Three-dimensional reconstruction of M. gastrocnemius contraction

There exist many different approaches investigating the contraction mechanisms of skeletal muscles. Thereby, the mechanical behavior, such as force generation in association with kinematic and microstructure, play an important role in modeling of muscle behavior. Besides the mechanical behaviour, the validation of muscle models requires the geometrical environment, too. The geometry of a muscle can be divided into macrostructure, existing of aponeurosis-tendon-complex (ATC) and muscle tissue (MT), as well as the fascicle architecture, representing the microstructure of the MT. In this study, the macrostructure of the isolated M. gastrocnemius was observed during isometric contraction by using three-dimensional optical measurement systems in combination with mechanical measurement techniques. The surface deformation was reconstructed at specific force and length relationships and further the muscle tissue, aponeurosis, and tendon were distinguished, building up a macroscopic geometrical dataset. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Soliton dynamics in the helix polypeptide chains

The coupling model between the three hydrogen-bonded chains to describe the helix polypeptide chain is investigated. The kink group, envelope-soliton and breather groups are obtained, in which the symmetric and asymmetric kink groups correspond to the local contraction (or expansion) and twisting deformed regions of helix polypeptide chain, respectively. The piezoelectric stress coefficient to describe the piezoelectric phenomenon arising from the local twisting deformation of the helix polypeptide chain is defined. The symmetric breather group will occur in the dipole radiation. The motion of asymmetric envelope-soliton group will result in the wriggle of the helix polypeptide chain, just like that of an earthworm. We show that (i) the twisting deformed structure of the helix polypeptide chain is more stationary than the uniform helix structure. (ii) The piezoelectric effect of collagen fibril of bone originates from the piezoelectricity of the helix polypeptide chain. (iii) The static attraction between the intrinsical charges arising from deformation of the helix polypeptide chain is advantageous to fold protein chain into the tertiary structure. A “gear model” ( or “molecule motor model”) to explain the mechanism of muscle contraction is suggested.     

Force enhancement and stability of finite element muscle models

A continuum-mechanical finite element model of skeletal muscle contractions that includes force enhancement based on actin-titin interaction is presented. This model can simulate muscles with a descending limb in the total force-length relation, which has previously led to unstable behaviour. The model predictions are in agreement with results of active stretch experiments. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical method for a model of inhomogeneous muscle fibers

The numerical solution of a nonlinear and nonlocal system of integro-differential equations that models the contraction of a single heterogeneous muscle fiber is studied. The method proposed is a form of the finite difference method of characteristics. Optimal error bounds are established for the resulting approximation. © 1993 John Wiley & Sons, Inc.     

A mathematical model of the systemic circulatory system with logistically defined nervous system regulatory mechanisms

A mathematical model is developed that accurately describes the pressure, volume and flow dynamics of the systemic circulatory system over the full physiological range of human pressures and volumes. At the heart of this model are mathematical representations for the autonomic and central nervous system reflexes which maintain arterial pressure, cardiac output and cerebral blood flow. These representations involve functions in which a maximum effect and a minimum effect are smoothly connected by a logistic transition. A new approach to modelling the pressure – volume relationship in a vessel with smooth muscle contraction is also presented. To test the model, simulations of cardiac arrest and various haemorrhagic situations were conducted, and predicted results were compared with clinical observations. Near-perfect agreement was obtained between predicted and observed values of the mean circulatory filling pressure, cardiac output and arterial pressure decay in the face of significant haemorrhage, and the critical values delineating progressive from non-progressive hypovolaemic shock.     

A generalized DEA model for centralized resource allocation

In this paper, we introduced a new generalized centralized resource allocation model which extends Lozano and Villa’s and Asmild et al.’s models to a more general case. In order to uncover the sources of such total input contraction in the generalized centralized resource allocation model, we applied the structural efficiency to further decompose it into three components: the aggregate technical efficiency, the aggregate allocative efficiency and re-transferable efficiency components. The proposed models are not only flexible enough for the central decision-maker to adjust the inputs and outputs to achieve the total input contraction but also identify the sources of such total input contraction, thereby giving rise to an important interpretation and understanding of the generalized centralized resource allocation model. Finally, an empirical example is used to illustrate the approach.     

Parallel multilevel solvers for the cardiac electro-mechanical coupling

We develop a parallel solver for the cardiac electro-mechanical coupling. The electric model consists of two non-linear parabolic partial differential equations (PDEs), the so-called Bidomain model, which describes the spread of the electric impulse in the heart muscle. The two PDEs are coupled with a non-linear elastic model, where the myocardium is considered as a nearly-incompressible transversely isotropic hyperelastic material. The discretization of the whole electro-mechanical model is performed by Q1 finite elements in space and a semi-implicit finite difference scheme in time. This approximation strategy yields at each time step the solution of a large scale ill-conditioned linear system deriving from the discretization of the Bidomain model and a non-linear system deriving from the discretization of the finite elasticity model. The parallel solver developed consists of solving the linear system with the Conjugate Gradient method, preconditioned by a Multilevel Schwarz preconditioner, and the non-linear system with a Newton–Krylov-Algebraic Multigrid solver. Three-dimensional parallel numerical tests on a Linux cluster show that the parallel solver proposed is scalable and robust with respect to the domain deformations induced by the cardiac contraction.     

On the treatment of active behaviour in continuum muscle mechanics

Two approaches of including active contractile behaviour of muscle tissue written in a continuum-mechanical formulation are presented. One approach relies on the addition of active and passive stress contributions, while the other approach is based on a multiplicative decomposition of the deformation gradient tensor. Both formulations can be stated in a thermodynamically consistent manner, each with different constraints, and both models can reproduce experimental data of passive and fully active muscle. Different behaviours are observed when comparing the active muscle models at submaximal stimulation rates. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Diffusion analogue of a combustion wave in a system with discrete sources

The problem of selfsustaining concentration waves in discrete quasionedimensional system with diffusion and threshold activation of the sources is considered. A number of applications of such models for describing spontaneous contraction waves observed in the course of experiments involving single muscle cells is discussed.     

Stabilizability of two classes of contraction semigroups

This paper studies the strong stabilizability of two classes of Hilbert space contraction semigroups: (i) strict contraction semigroups, which include those with strictly dissipative generators; and (ii) isometric or unitary semigroups. The former class is already weakly stable, while the latter is not strongly stable over the whole space. Our tool is the functional model of Hilbert space contractions; hence, strong stability of the semigroup is studied via stability of its cogenerator. It is shown that a strict contraction semigroup is, in general, not strongly stabilized by the feedback –B*, while an isometric or a unitary semigroup is strongly stabilized by the same feedback, providedB is not compact.     

Modelado numérico del comportamiento del tejido músculo-esquelético

This paper presents a three dimensional computational model to simulate the mechanical behavior of the muscle-tendon unit (MTU). The model has been formulated through a strain energy density function for hyperelastic materials incorporating both the passive and active behavior of the connective tissues, and taking into account preferential directions for both behaviours, not necessarily coincident. The initial stresses, usually present in soft tissues, have been considered when performing the MTU simulation. The geometry used corresponds to the rat tibialis anterior muscle reconstructed from «in-vivo» magnetic resonance images. This paper also studies the effect of the fascia on the MTU passive behaviour and the setting of a hyperelastic model for this tissue by means of experimental tests.