Passive forces in mammalian skeletal muscle: a freely-jointed and worm-like chain.

The passive mechanical properties of whole muscle in active and nonactive states are compared. The experimental results are presented as stress-strain curves, which are analyzed in the framework of the current theoretical background [viz. the freely-jointed chain model (FJCM) and the worm-like chain model (WLCM)] in a semi-quantitative fashion. This analysis shows that both models can explain the mechanical behavior of whole muscle in non-active state. In the active state, the presence of crossbridges alters the mechanical response, leading to a markedly different behavior, as expected. A discussion of the mechanisms involved and the interpretation of the parameters required for the fitting of the stress-strain curves is also presented.     

A mathematical analysis of mechanical factors in the forced expiration

A one compartment, mechanical model of the human lung-thorax system is presented and mathematically analyzed. The equation relating the thoracic muscular stress to the expired air volume is developed and investigated. Assuming that the pressure drop along the airways is a linear function of air flow rate and that the effective lung-thorax compliance is constant, a form for the muscular stress as a function of time is developed. This is used to predict volume-time and flow-volume curves, which are compared to those measured on a normal individual. It appears that these theoretical results have the essential characteristics of the experimental curves. These results, coupled with the one-to-one correspondence between the parameters of the model and those of the prototype, suggest that this model should have great utility in the study of ventilatory mechanics.     

Analysis of models for the activation and contraction of muscle

Huxley (1957) proposed a sliding filament model of muscular contraction to which Julian (1969) added equations for the activation produced by cations. Each parameter in the combined Huxley-Julian model has been varied systematically to determine its effect on the predicted twitches. The slower rate constant for Ca activation has a predominant effect on the relaxation phase of the twitch. The series elasticity and the rate constants for the making and the breaking of cross-bridges all strongly affect the contraction phase of the twitch. Further experimental work is required to determine which factor is rate limiting under a given set of conditions.Taking the Fourier transform of the twitch gives a prediction for the frequency response of the model. The predicted frequency response curves are well-fitted by those of a simple, second order system, in agreement with recent experiments (Mannard &; Stein, 1973). The parameters of the best-fitting, second-order, frequency response curves vary experimentally with mean stimulus rate. This variation probably results from a saturation at higher stimulus rates of the pXocesses for reuptake of Ca into the sarcoplasmic reticulum. The saturation of Ca reuptake, together with the saturation of the myofilaments by Ca at higher stimulus rates, can account qualitatively for the sigmoid rate-tension curves found experimentally.     

Prediction of internal pressure profile of compression bandages using stress relaxation parameters

The efficacy of compression therapy using compression bandages is highly dependent on the level of compression applied and the sustenance of the pressure during the course of treatment. This study attempts to predict the pressure profile generated by compression bandages using constitutive equations describing relaxation behavior of viscoelastic materials. It is observed that this pressure profile is highly correlated with the stress relaxation behavior of the bandage. To model the pressure profile, the stress relaxation behavior of compression bandages was studied and modeled using three mechanical models: the Maxwell model, the standard linear solid model and the two-component Maxwell model with a nonlinear spring. It was observed that the models with more component values explained the experimental relaxation curves better. The parameters used for modelling relaxation behavior were used to describe the pressure profile, which is significantly dependent on the longitudinal stress relaxation behavior of the bandage, using the modified Laplace's law equation. This approach thus helps in evaluating the bandage performance with time during compression therapy as novel wound care management.     

A Stereoselective Synthesis of (±)-endo-Brevicomin,a Pheromone Inhibitor Produced by Dendroctonus Bark Beetles

Prediction of elution curves in gel chromatography was attempted on the basis of a mass balance model which gave consideration to gel phase diffusion and longitudinal dispersion in a column. The basic differential equations for the model were solved by means of Laplace transformation, and then the solution in Laplace domain was inverted into time domain numerically. The calculated elution curves were in good agreement with the experimental ones of NaCl and myoglobin with various Sephadex gel columns. This indicated the validity of the calculation method and the model employed in this study.

Furthermore, the elution curves were calculated tentatively for various combinations of the parameters appearing in the mass balance model. Then, the magnitude of peak asymmetry, the shift of peak position and the maximum peak height of the elution curves were correlated with various parameters and operational variables. These correlations might permit prediction of suitable operational conditions for gel chromatography, especially for molecular weight determination.     

Stiffness-distortion sarcomere model for muscle simulation.

A relatively simple method is presented for incorporating cross-bridge mechanisms into a muscle model. The method is based on representing force in a half sarcomere as the product of the stiffness of all parallel cross bridges and their average distortion. Differential equations for sarcomeric stiffness are derived from a three-state kinetic scheme for the cross-bridge cycle. Differential equations for average distortion are derived from a distortional balance that accounts for distortion entering and leaving due to cross-bridge cycling and for distortion imposed by shearing motion between thick and thin filaments. The distortion equations are unique and enable sarcomere mechanodynamics to be described by only a few ordinary differential equations. Model predictions of small-amplitude step and sinusoidal responses agreed well with previously described experimental results and allowed unique interpretations to be made of various response components. Similarly good results were obtained for model reproductions of force-velocity and large-amplitude step and ramp responses. The model allowed reasonable predictions of contractile behavior by taking into account what is understood to be basic muscle contractile mechanisms.     

Biomechanical analysis of primate bipedal walking by computer simulation

A computer simulation technique was applied to make clear the mechanical characteristics of primate bipedal walking. A primate body and the walking mechanism were modeled mathematically with a set of dynamic equations. Using a digital computer, the following were calculated from these equations by substituting measured displacements and morphological data of each segment of the primate: the acceleration, joint angle, center of gravity, foot force, joint moment, muscular force, transmitted force at the joint, electric activity of the muscle, generated power by the leg and energy expenditure in walking.The model was evaluated by comparing some of the calculated results with the experimental results such as foot force and electromyographic data, and improved in order to obtain the agreement between them.The level bipedal walking of man, chimpanzee and Japanese monkey and several types of synthesized walking were analyzed from the viewpoint of biomechanics.It is concluded that the bipedal walking of chimpanzee is nearer to that of man than to that of the Japanese monkey because of its propulsive mechanism, but it requires large muscular force for supporting the body weight.     

Assessing the accuracy of subject-specific,muscle-model parameters determined by optimizing to match isometric strength

Biomechanical models are sensitive to the choice of model parameters. Therefore, determination of accurate subject specific model parameters is important. One approach to generate these parameters is to optimize the values such that the model output will match experimentally measured strength curves. This approach is attractive as it is inexpensive and should provide an excellent match to experimentally measured strength. However, given the problem of muscle redundancy, it is not clear that this approach generates accurate individual muscle forces. The purpose of this investigation is to evaluate this approach using simulated data to enable a direct comparison. It is hypothesized that the optimization approach will be able to recreate accurate muscle model parameters when information from measurable parameters is given. A model of isometric knee extension was developed to simulate a strength curve across a range of knee angles. In order to realistically recreate experimentally measured strength, random noise was added to the modeled strength. Parameters were solved for using a genetic search algorithm. When noise was added to the measurements the strength curve was reasonably recreated. However, the individual muscle model parameters and force curves were far less accurate. Based upon this examination, it is clear that very different sets of model parameters can recreate similar strength curves. Therefore, experimental variation in strength measurements has a significant influence on the results. Given the difficulty in accurately recreating individual muscle parameters, it may be more appropriate to perform simulations with lumped actuators representing similar muscles.     

肌肉收缩的理论研究

基于肌球蛋白工作循环模型,从物理学的角度出发,利用化学动力学方法,给出肌动蛋白丝的动力学方程,讨论肌球蛋白的运力学行为,发现肌动蛋白运动呈锯齿状,并得到振动周期约为3.0s,与实验结果基本吻合。结论是宏观的肌肉运动是单分子运动的集体协同行为,为肌肉的运动训练和治疗提供理论参考。     

Model of geometrical and smooth muscle tone adaptation of carotid artery subject to step change in pressure

Recent experimental studies have shown significant alterations of the vascular smooth muscle (VSM) tone when an artery is subjected to an elevation in pressure. Therefore, the VSM participates in the adaptation process not only by means of its synthetic activity (fibronectins and collagen) or proliferative activity (hypertrophy and hyperplasia) but also by adjusting its contractile properties and its tone level. In previous theoretical models describing the time evolution of the arterial wall adaptation in response to induced hypertension, the contribution of VSM tone has been neglected. In this study, we propose a new biomechanical model for the wall adaptation to induced hypertension, including changes in VSM tone. On the basis of Hill's model, total circumferential stress is separated into its passive and active components, the active part being the stress developed by the VSM. Adaptation rate equations describe the geometrical adaptation (wall thickening) and the adaptation of active stress (VSM tone). The evolution curves that are derived from the theoretical model fit well the experimental data describing the adaptation of the rat common carotid subjected to a step increase in pressure. This leads to the identification of the model parameters and time constants by characterizing the rapidity of the adaptation processes. The agreement between the results of this simple theoretical model and the experimental data suggests that the theoretical approach used here may appropriately account for the biomechanics underlying the arterial wall adaptation.     

A model of the mechanics of airway narrowing

To examine the interaction between airway smooth muscle shortening and airway wall thickening on changes in pulmonary resistance, we have developed a model of the tracheobronchial tree that allows simulation of the mechanisms involved in airway narrowing. The model is based on the symmetrical dichotomous branching tracheobronchial tree as described by Weibel and uses fluid dynamic equations proposed by Pedley et al. to calculate inspiratory resistance during quiet tidal breathing. To allow for changes in lung volume, we used the airway pressure-area curves developed by Lambert et al. The model is easily implemented with a spreadsheet and personal computer that allows calculation of total and regional pulmonary resistance. At each airway generation in the model, provision is made for airway wall thickness, the maximal airway smooth muscle shortening achievable, and an S-shaped dose-response relationship to describe smooth muscle shortening. To test the validity of the model, we compared pressure-flow curves generated with the model with measurements of pulmonary resistance while normal subjects breathed air and 20% O2-80% He at a variety of lung volumes. By simulating progressive airway smooth muscle shortening, realistic pulmonary resistance vs. dose-response curves were produced. We conclude that this model provides realistic estimates of pulmonary resistance and shows potential for examining the various mechanisms that could produce excessive airway narrowing in disease.     

A myocybernetic control model of skeletal muscle

A mathematical model of skeletal muscle is presented which contains the two physiological control parameters stimulation rate and motor unit recruitment. The model is complete in the sense that it adequately describes all possible contractive states normally occurring in living muscle. The modelling procedure relies entirely on established myo-physiological facts and each assumption made is substantiated by experimental data. Extensive simulation studies reveal that the model is capable of correctly predicting practically all known phenomena of the muscular force-output. A simplified version of the model is also presented, particularly suitable for inclusion as the driving structure in complex musculoskeletal link systems. This version was successfully tested in the prediction of an optimal human motion. The present control model is believed to fill a gap in the literature on models of muscle, and may be expected to provide a sound basis for research into the optimal control aspects of muscular contraction, and to stimulate such research.     

Lysine-sensitive aspartokinase of Escherichia coli K12. Synergy and autosynergy in an allosteric V system.

The interactions of the lysine-sensitive aspartokinase of E. coli K12 with lysine and leucine, as evidenced in the inhibition and binding curves, are well explained by the equations of an allosteric V model. Mathematical treatments of such a model lead to new linearized plots. These representations are applied to our experimental results and allow the direct determination of some parameters of the model (equilibrium constant L' and leucine dissociation constants). The other parameters are obtained by an optimization method. The theoretical curves drawn according to this model account well for the synergistic inhibition between lysine and leucine and for the role of the two nonequivalent lysine binding sites ("autosynergy").     

Fully non-linear hyper-viscoelastic modeling of skeletal muscle in compression

Understanding the behavior of skeletal muscle is critical to implementing computational methods to study how the body responds to compressive loading. This work presents a novel approach to studying the fully nonlinear response of skeletal muscle in compression. Porcine muscle was compressed in both the longitudinal and transverse directions under five stress relaxation steps. Each step consisted of 5% engineering strain over 1 s followed by a relaxation period until equilibrium was reached at an observed change of 1 g/min. The resulting data were analyzed to identify the peak and equilibrium stresses as well as relaxation time for all samples. Additionally, a fully nonlinear strain energy density–based Prony series constitutive model was implemented and validated with independent constant rate compressive data. A nonlinear least squares optimization approach utilizing the Levenberg–Marquardt algorithm was implemented to fit model behavior to experimental data. The results suggested the time-dependent material response plays a key role in the anisotropy of skeletal muscle as increasing strain showed differences in peak stress and relaxation time (p < 0.05), but changes in equilibrium stress disappeared (p > 0.05). The optimizing procedure produced a single set of hyper-viscoelastic parameters which characterized compressive muscle behavior under stress relaxation conditions. The utilized constitutive model was the first orthotropic, fully nonlinear hyper-viscoelastic model of skeletal muscle in compression while maintaining agreement with constitutive physical boundaries. The model provided an excellent fit to experimental data and agreed well with the independent validation in the transverse direction.     

肌肉收缩及自发振动的动力学理论

改变实验条件,肌纤维可发生从收缩到自发振动的相变。为了研究这一现象,引进了描述肌纤维内部弹性成分拉伸长度与张力关系的表达式,利用肌肉态方程并考虑肌纤维的特殊结构,给出了描述肌纤维收缩及自发振动的统一动力学方程。从动力学方程出发,肌纤维自发振动的发生条件得到了自然解释,所给出的振动周期和振动曲线同实验结果相符,并给出了一些新的理论结果。这一工作的意义在于,完成了从肌球蛋白单分子性质、肌纤维组织结构到肌纤维功能的信息整合。     

Tonic finite element model of the lower limb

It is widely admitted that muscle bracing influences the result of an impact, facilitating fractures by enhancing load transmission and reducing energy dissipation. However, human numerical models used to identify injury mechanisms involved in car crashes hardly take into account this particular mechanical behavior of muscles. In this context, in this work we aim to develop a numerical model, including muscle architecture and bracing capability, focusing on lower limbs. The three-dimensional (3-D) geometry of the musculoskeletal system was extracted from MRI images, where muscular heads were separated into individual entities. Muscle mechanical behavior is based on a phenomenological approach, and depends on a reduced number of input parameters, i.e., the muscle optimal length and its corresponding maximal force. In terms of geometry, muscles are modeled with 3-D viscoelastic solids, guided in the direction of fibers with a set of contractile springs. Validation was first achieved on an isolated bundle and then by comparing emergency braking forces resulting from both numerical simulations and experimental tests on volunteers. Frontal impact simulation showed that the inclusion of muscle bracing in modeling dynamic impact situations can alter bone stresses to potentially injury-inducing levels.     

Kinetic model for isometric contraction in smooth muscle on the basis of myosin phosphorylation hypothesis.

A kinetic model was proposed to simulate an isometric contraction curve in smooth muscle on the basis of the myosin phosphorylation hypothesis. The Ca2+-calmodulin-dependent activation of myosin light-chain kinase and the phosphorylation-dephosphorylation reaction of myosin were mathematically treated. Solving the kinetic equations at a steady state, we could calculate the relationship between the Ca2+ concentration and the myosin phosphorylation. Assuming that two-head-phosphorylated myosin has an actin-activated Mg2+-ATPase activity and that this state corresponds to an active state, we computed the time courses of the myosin phosphorylation and the active state for various Ca2+ transients. The time course of the active state was converted into that of isometric tension by use of Sandow's model composed of a contractile element and a series elastic component. The model could simulate not only the isometric contraction curves for any given Ca2+ transient but also the following experimental results: the calmodulin-dependent shift of the Ca2+ sensitivity of isometric tension observed in skinned muscle fibers, the disagreement between the Ca2+ sensitivity of myosin phosphorylation and that of isometric tension at a steady state, and the disagreement between the time course of myosin phosphorylation and that of isometric tension development.     

Modeling, simulation, and experimental validation for continuous Cr(VI) removal from aqueous solutions using sawdust as an adsorbent

Continuous adsorption experiments were performed in a fixed-bed adsorption column to evaluate the performance of low-cost adsorbent (sawdust) developed for the removal of Cr(VI) from aqueous solutions. The effects of influencing parameters such as flow rate, mass of adsorbent, initial Cr(VI) concentration were studied and the corresponding breakthrough curves were obtained. The fixed-bed adsorption process parameters such as breakthrough time, total percentage removal of Cr(VI), adsorption exhaustion rate and fraction of unused bed-length were obtained. A mathematical model for fixed-bed adsorption column was proposed by incorporating the effect of velocity variation along the bed-length in the existing model. Pore and solid diffusion models were used to describe the intra-particle mechanism for Cr(VI) adsorption. The proposed mathematical model was validated with the literature data and the experimental data obtained in the present study and the model was found to be good for explaining the behavior of breakthrough curves.