A Finite Element Approach for Skeletal Muscle using a Distributed Moment Model of Contraction

Abstract

The present paper describes a geometrically and physically nonlinear continuum model to study the mechanical behaviour of passive and active skeletal muscle. The contraction is described with a Huxley type model. A Distributed Moments approach is used to convert the Huxley partial differential equation in a set of ordinary differential equations. An isoparametric brick element is developed to solve the field equations numerically. Special arrangements are made to deal with the combination of highly nonlinear effects and the nearly incompressible behaviour of the muscle. For this a Natural Penalty Method (NPM) and an Enhanced Stiffness Method (ESM) are tested. Finally an example of an analysis of a contracting tibialis anterior muscle of a rat is given. The DM-method proved to be an efficient tool in the numerical solution process. The ESM showed the best performance in describing the incompressible behaviour.     

Activation in a Skeletal Muscle Contraction Model with a Modification for Insect Fibrillar Muscle

A sliding filament model for muscle contraction is extended by including an activation mechanism based on the hypothesis that the binding of calcium by a regulating protein in the myofibrils must occur before the rate constant governing the making of interactions between cross-bridges and thin filament sites can take on nonzero values. The magnitude of the rate constant is proportional to the amount of bound calcium. The model's isometric twitch and rise of force in an isometric tetanus are similar to the curves produced by real muscles. It redevelops force after a quick release in an isometric tetanus faster than the initial rise. Quick release experiments on the model during an isometric twitch show that the “active state” curve produced is different from the postulated calcium binding curve. The force developed by the model can be increased by a small quick stretch delivered soon after activation to values near the maximum generated in an isometric tetanus. Following the quick stretch, the force remains near the tetanic maximum for a long time even though the calcium binding curve rises to a peak and subsequently decays by about 50%. The model satisfies the constraint of shortening with a constant velocity under a constant load. Modifications can be made in the model so that it produces the delayed force changes following step length changes characteristic of insect fibrillar muscle.     

Mathematical Model for Skeletal Muscle to Simulate the Concentric and Eccentric Contraction

Skeletal muscles are responsible for the relative motion of the bones at the joints and provide the required strength. They exhibit highly nonlinear mechanical behaviour and are described by nonlinear hyperelastic constitutive relations. It is distinct from other biological soft tissue. Its hyperelastic or viscoelastic behaviour is modelled by using CE, SEE, and PEE. Contractile element simulates the behaviour of skeletal muscle when it is subjected to eccentric and concentric contraction. This research aims to estimate the stress induced in skeletal muscle in eccentric and concentric contraction with respect to the predefined strain. With the use of mathematical model for contraction of skeletal muscle for eccentric and concentric contraction, the stress induced in the skeletal muscle is estimated in this research. Mathematical model is developed for the muscle using EMG signals and Force-velocity relationship calculated. With the use of force-velocity of contraction of muscle, mathematical model is developed. This can be useful to understand the mechanical behaviour of skeletal muscles in eccentric and concentric contraction with clinical relevance. Authors are further working to develop the mathematical model with torsion force with proper activation function of muscle and experimentation for extraction of the anisotropic mechanical properties of skeletal muscle.     

Local Control Model of Excitation–Contraction Coupling in Skeletal Muscle

The voltage-activated H⁺ selective conductance of rat alveolar epithelial cells was studied using whole-cell and excised-patch voltage-clamp techniques. The effects of substituting deuterium oxide, D₂O, for water, H₂O, on both the conductance and the pH dependence of gating were explored. D⁺ was able to permeate proton channels, but with a conductance only about 50% that of H⁺. The conductance in D₂O was reduced more than could be accounted for by bulk solvent isotope effects (i.e., the lower mobility of D⁺ than H⁺), suggesting that D⁺ interacts specifically with the channel during permeation. Evidently the H⁺ or D⁺ current is not diffusion limited, and the H⁺ channel does not behave like a water-filled pore. This result indirectly strengthens the hypothesis that H⁺ (or D⁺) and not OH⁻ is the ionic species carrying current. The voltage dependence of H⁺ channel gating characteristically is sensitive to pH_o and pH_i and was regulated by pD_o and pD_i in an analogous manner, shifting 40 mV/U change in the pD gradient. The time constant of H⁺ current activation was about three times slower (τ_act was larger) in D₂O than in H₂O. The size of the isotope effect is consistent with deuterium isotope effects for proton abstraction reactions, suggesting that H⁺ channel activation requires deprotonation of the channel. In contrast, deactivation (τ_tail) was slowed only by a factor ≤1.5 in D₂O. The results are interpreted within the context of a model for the regulation of H⁺ channel gating by mutually exclusive protonation at internal and external sites (Cherny, V.V., V.S. Markin, and T.E. DeCoursey. 1995. J. Gen. Physiol. 105:861–896). Most of the kinetic effects of D₂O can be explained if the pK _a of the external regulatory site is ∼0.5 pH U higher in D₂O.     

A Finite Element Model Approach to Determine the Influence of Electrode Design and Muscle Architecture on Myoelectric Signal Properties

Introduction

Surface electromyography (sEMG) is the measurement of the electrical activity of the skeletal muscle tissue detected at the skin’s surface. Typically, a bipolar electrode configuration is used. Most muscles have pennate and/or curved fibres, meaning it is not always feasible to align the bipolar electrodes along the fibres direction. Hence, there is a need to explore how different electrode designs can affect sEMG measurements.

Method

A three layer finite element (skin, fat, muscle) muscle model was used to explore different electrode designs. The implemented model used as source signal an experimentally recorded intramuscular EMG taken from the biceps brachii muscle of one healthy male. A wavelet based intensity analysis of the simulated sEMG signal was performed to analyze the power of the signal in the time and frequency domain.

Results

The model showed muscle tissue causing a bandwidth reduction (to 20-92- Hz). The inter-electrode distance (IED) and the electrode orientation relative to the fibres affected the total power but not the frequency filtering response. The effect of significant misalignment between the electrodes and the fibres (60°- 90°) could be reduced by increasing the IED (25–30 mm), which attenuates signal cancellation. When modelling pennated fibres, the muscle tissue started to act as a low pass filter. The effect of different IED seems to be enhanced in the pennated model, while the filtering response is changed considerably only when the electrodes are close to the signal termination within the model. For pennation angle greater than 20°, more than 50% of the source signal was attenuated, which can be compensated by increasing the IED to 25 mm.

Conclusion

Differences in tissue filtering properties, shown in our model, indicates that different electrode designs should be considered for muscle with different geometric properties (i.e. pennated muscles).     

Distributed Representations for Actin-Myosin Interaction in the Oscillatory Contraction of Muscle

In this paper we suggest and test a specific hypothesis relating the attachment-detachment cycle of cross bridges between actin (I) and myosin (A) filaments to the measured length-tension dynamics of active insect fibrillar flight muscle. It is first shown that if local A-filament strain perturbs the rate constants in the cross-bridge cycle appropriately, then exponentially delayed tension changes can follow imposed changes of length; the latter phenomenon is sufficient for the work-producing property of fibrillar muscle, as measured with small-signal forcing of length and at low Ca²⁺ concentration, and possibly for related effects described recently in frog striated muscle. It is not clear a priori that the above explanation of work production by fibrillar muscle will remain tenable when the viscoelastic complexity of the heterogeneous sarcomere is taken into account. However, White's (1967) recent mechanical and electron microscope study of the passive dynamics of glycerinated fibrillar muscle has produced a model of the distributed viscoeleastic structure sufficiently explicit that alternative schemes for cross-bridge force generation in this muscle can now be tested more critically than previously. Therefore, we derive and solve third-order partial-differential equations which relate local interfilament shear forces associated with the perturbed cross-bridge cycles to the over-all length-tension dynamics of an idealized sarcomere. We then show (a) that the starting hypothesis can account approximately for the small-signal dynamics of glycerinated muscle in the work-producing state over two decades of frequency and (b) that the rate constants for cross-bridge formation and breakage, restricted solely by fitting of the model to the mechanical data, determine a cycling rate of cross bridges in the model compatible with recent measurements of ATP hydrolysis rate vs. stretch in this muscle. Finally, the formulation is extended tentatively to the large-signal nonlinear case, and shown to compare favorably with previous suggestions for the origin of the work-producing dynamics of fibrillar flight muscle.     

The Intracellular Site of Calcium Activation of Contraction in Frog Skeletal Muscle

Radioautography has been used to localize ⁴⁵Ca in isotopically labeled frog skeletal muscle fibers which had been quickly frozen during a maintained tetanus, a declining tetanus, or during the period immediately following a tetanus or a contracture. During a tetanus almost all of the myofibrillar ⁴⁵Ca is localized in the region of the sarcomere occupied by the thin filaments. The amount varies with the tension being developed by the muscle. The movement of calcium within the reticulum from the tubular portion to the terminal cisternae during the posttetanic period has a half-time of about 9 sec at room temperature and a Q₁₀ of about 1.7. Repolarization is not necessary for this movement. Evidence is given to support the notion that most calcium efflux from the cell occurs from the terminal cisternae into the transverse tubules.     

A Finite Element Model for Ice Ball Evolution in a Multi-probe Cryosurgery

The ice formation in a water body is examined for the computation of temperature field, phase change and a moving ice–water interface whose location is not known à priori. This is classically referred to as the Stefan problem [Rubinstein, L.I. (1971) The Stefan Problem (American Mathematical Society, Providence, Rhode Island 02904]. Based on the Duvaut [Duvaut, G. (1973) “Résolution d'un problème de Stefan” C.R. Acad. Sci. Paris 276, 1461–1463] transformation, the governing equations for heat conduction are formulated within a variational principle that is readily amenable to a standard finite element solution without remeshing. Numerical simulation results pertaining to the freezing of tumour tissue in a multi-cryoprobe cryosurgery are presented. These results lend both quantitative and graphical support to the current empirical standards of “effective therapy” in view of refining clinical applications.     

A Novel Three-Filament Model of Force Generation in Eccentric Contraction of Skeletal Muscles

We propose and examine a three filament model of skeletal muscle force generation, thereby extending classical cross-bridge models by involving titin-actin interaction upon active force production. In regions with optimal actin-myosin overlap, the model does not alter energy and force predictions of cross-bridge models for isometric contractions. However, in contrast to cross-bridge models, the three filament model accurately predicts history-dependent force generation in half sarcomeres for eccentric and concentric contractions, and predicts the activation-dependent forces for stretches beyond actin-myosin filament overlap.     

痉挛型大鼠骨骼肌不同功能状态下表面肌电特征性变化的研究

目的:研究痉挛型瘫痪大鼠骨骼肌不同功能状态对其表面肌电特征性的影响。方法:选用健康5日龄新生wistar大鼠60只随机分为两组即:痉挛型瘫痪大鼠模型组和正常饲养组。复制痉挛型瘫痪大鼠模型成功后饲养30天,根据肌肉三种功能状态分为三组,分别为A组放松状态组、B向心性收缩状态组、C离心收缩状态组。每组包括痉挛型瘫痪大鼠10只,正常大鼠10只。检测工具采用Bio Trace+Software进行表面肌电的测试和分析;检测肌肉为伸膝肌群;检测指标为表面肌电均方根值(RMS);检测方式为电针刺激诱发不同收缩状态。结果采用SPSS17.0统计软件进行数据分析。结果:放松状态下痉挛大鼠RMS(2.76±0.09)v,正常大鼠RMS(2.82±0.07)v,独立样本的t检验P=0.1260.05;向心性收缩状态痉挛大鼠RMS(10.25±0.35)v,正常大鼠RMS(11.07±0.81)v,独立样本的t检验P=0.0120.05;离心性收缩状态痉挛大鼠RMS(3.32±0.27)v,正常大鼠RMS(4.0±3.045)v,独立样本的t检验P=0.0010.05。结论:痉挛型骨骼肌收缩时肌纤维的募集异于正常骨骼肌,表面肌电对鉴别肌痉挛有效。     

A Derivative Matching Approach to Moment Closure for the Stochastic Logistic Model

Continuous-time birth-death Markov processes serve as useful models in population biology. When the birth-death rates are nonlinear, the time evolution of the first n order moments of the population is not closed, in the sense that it depends on moments of order higher than n. For analysis purposes, the time evolution of the first n order moments is often made to be closed by approximating these higher order moments as a nonlinear function of moments up to order n, which we refer to as the moment closure function. In this paper, a systematic procedure for constructing moment closure functions of arbitrary order is presented for the stochastic logistic model. We obtain the moment closure function by first assuming a certain separable form for it, and then matching time derivatives of the exact (not closed) moment equations with that of the approximate (closed) equations for some initial time and set of initial conditions. The separable structure ensures that the steady-state solutions for the approximate equations are unique, real and positive, while the derivative matching guarantees a good approximation, at least locally in time. Explicit formulas to construct these moment closure functions for arbitrary order of truncation n are provided with higher values of n leading to better approximations of the actual moment dynamics. A host of other moment closure functions previously proposed in the literature are also investigated. Among these we show that only the ones that achieve derivative matching provide a close approximation to the exact solution. Moreover, we improve the accuracy of several previously proposed moment closure functions by forcing derivative matching.     

A Three-Dimensional Finite Element Model for Biomechanical Analysis of the Hip

The objective of this study was to construct a three-dimensional (3D) finite element model of the hip. The images of the hip were obtained from Chinese visible human dataset. The hip model includes acetabular bone, cartilage, labrum, and bone. The cartilage of femoral head was constructed using the AutoCAD and Solidworks software. The hip model was imported into ABAQUS analysis system. The contact surface of the hip joint was meshed. To verify the model, the single leg peak force was loaded, and contact area of the cartilage and labrum of the hip and pressure distribution in these structures were observed. The constructed 3D hip model reflected the real hip anatomy. Further, this model reflected biomechanical behavior similar to previous studies. In conclusion, this 3D finite element hip model avoids the disadvantages of other construction methods, such as imprecision of cartilage construction and the absence of labrum. Further, it provides basic data critical for accurately modeling normal and abnormal loads, and the effects of abnormal loads on the hip.     

Capsiate Supplementation Reduces Oxidative Cost of Contraction in Exercising Mouse Skeletal Muscle In Vivo

Chronic administration of capsiate is known to accelerate whole-body basal energy metabolism, but the consequences in exercising skeletal muscle remain very poorly documented. In order to clarify this issue, the effect of 2-week daily administration of either vehicle (control) or purified capsiate (at 10- or 100-mg/kg body weight) on skeletal muscle function and energetics were investigated throughout a multidisciplinary approach combining in vivo and in vitro measurements in mice. Mechanical performance and energy metabolism were assessed strictly non-invasively in contracting gastrocnemius muscle using magnetic resonance (MR) imaging and 31-phosphorus MR spectroscopy (³¹P-MRS). Regardless of the dose, capsiate treatments markedly disturbed basal bioenergetics in vivo including intracellular pH alkalosis and decreased phosphocreatine content. Besides, capsiate administration did affect neither mitochondrial uncoupling protein-3 gene expression nor both basal and maximal oxygen consumption in isolated saponin-permeabilized fibers, but decreased by about twofold the K _m of mitochondrial respiration for ADP. During a standardized in vivo fatiguing protocol (6-min of repeated maximal isometric contractions electrically induced at a frequency of 1.7 Hz), both capsiate treatments reduced oxidative cost of contraction by 30-40%, whereas force-generating capacity and fatigability were not changed. Moreover, the rate of phosphocreatine resynthesis during the post-electrostimulation recovery period remained unaffected by capsiate. Both capsiate treatments further promoted muscle mass gain, and the higher dose also reduced body weight gain and abdominal fat content. These findings demonstrate that, in addition to its anti-obesity effect, capsiate supplementation improves oxidative metabolism in exercising muscle, which strengthen this compound as a natural compound for improving health.     

A Framework for Structured Modeling of Skeletal Muscle

The aim of this study is to present a detailed continuum mechanics formulation, and the corresponding algorithms, to predict the deformation of skeletal muscle at different structural levels, starting from the muscle fiber level. The model is used to investigate force production and structural changes during isometric and dynamic contractions of the cat medial gastrocnemius. From a comparison with experimental data obtained in our own laboratories, we conclude that the model faithfully predicts all of the observations pertaining to force production, fascicle length and angle of pennation under various test conditions.     

Time Course and Strain Dependence of ADP Release during Contraction of Permeabilized Skeletal Muscle Fibers

A phosphorylated, single cysteine mutant of nucleoside diphosphate kinase, labeled with N-[2-(iodoacetamido)ethyl]-7-diethylaminocoumarin-3-carboxamide (P∼NDPK-IDCC), was used as a fluorescence probe for time-resolved measurement of changes in [MgADP] during contraction of single permeabilized rabbit psoas fibers. The dephosphorylation of the phosphorylated protein by MgADP occurs within the lattice environment of permeabilized fibers with a second-order rate constant at 12°C of 10⁵ M⁻¹ s⁻¹. This dephosphorylation is accompanied by a change in coumarin fluorescence. We report the time course of P∼NDPK-IDCC dephosphorylation during the period of active isometric force redevelopment after quick release of fiber strain at pCa²⁺ of 4.5. After a rapid length decrease of 0.5% was applied to the fiber, the extra NDPK-IDCC produced during force recovery, above the value during the approximately steady state of isometric contraction, was 2.7 ± 0.6 μM and 4.7 ± 1.5 μM at 12 and 20°C, respectively. The rates of P∼NDPK-IDCC dephosphorylation during force recovery were 28 and 50 s⁻¹ at 12 and 20°C, respectively. The time courses of isometric force and P∼NDPK-IDCC dephosphorylation were simulated using a seven-state reaction scheme. Relative isometric force was modeled by changes in the occupancy of strongly bound A.M.ADP.P_i and A.M.ADP states. A strain-sensitive A.M.ADP isomerization step was rate-limiting (3-6 s⁻¹) in the cross-bridge turnover during isometric contraction. At 12°C, the A.M.ADP.P_i and the pre- and postisomerization A.M.ADP states comprised 56%, 38%, and 7% of the isometric force-bearing AM states, respectively. At 20°C, the force-bearing A.M.ADP.P_i state was a lower proportion of the total force-bearing states (37%), whereas the proportion of postisomerization A.M.ADP states was higher (19%). The simulations suggested that release of cross-bridge strain caused rapid depopulation of the preisomerization A.M.ADP state and transient accumulation of MgADP in the postisomerization A.M.ADP state. Hence, the strain-sensitive isomerization of A.M.ADP seems to explain the rate of change of P∼NDPK-IDCC dephosphorylation during force recovery. The temperature-dependent isometric distribution of myosin states is consistent with the previous observation of a small decrease in amplitude of the P_i transient during force recovery at 20°C and the current observation of an increase in amplitude of the ADP-sensitive NDPK-IDCC transient.     

Development of a Finite Element Model of Decompressive Craniectomy

Decompressive craniectomy (DC), an operation whereby part of the skull is removed, is used in the management of patients with brain swelling. While the aim of DC is to reduce intracranial pressure, there is the risk that brain deformation and mechanical strain associated with the operation could damage the brain tissue. The nature and extent of the resulting strain regime is poorly understood at present. Finite element (FE) models of DC can provide insight into this applied strain and hence assist in deciding on the best surgical procedures. However there is uncertainty about how well these models match experimental data, which are difficult to obtain clinically. Hence there is a need to validate any modelling approach outside the clinical setting. This paper develops an axisymmetric FE model of an idealised DC to assess the key features of such an FE model which are needed for an accurate simulation of DC. The FE models are compared with an experimental model using gelatin hydrogel, which has similar poro-viscoelastic material property characteristics to brain tissue. Strain on a central plane of the FE model and the front face of the experimental model, deformation and load relaxation curves are compared between experiment and FE. Results show good agreement between the FE and experimental models, providing confidence in applying the proposed FE modelling approach to DC. Such a model should use material properties appropriate for brain tissue and include a more realistic whole head geometry.