Series Elastic Component Research Articles

Modeling of mechanical dysfunction in regional stunned myocardium of the left ventricle

Reversible mechanical dysfunction of the myocardium after a single or multiple episode(s) of coronary artery occlusion has been observed in previous studies and is termed myocardial stunning. The hypothesis that stunning could be represented by a decrease in maximum available muscle force in the stunned region was examined by means of a mathematical model that incorporates series viscoelastic elements. A canine experimental model was also employed to demonstrate depressed contractility and a consistent delay of shortening in the stunned region. The mechanical model of the left ventricle was designed to include a normal and stunned region, for which the stunned region was allowed to have variable size. Each region consisted of a volume and time dependent force generator in parallel with a passive elastic force element. The passive elastic element was placed in series with a constant viscosity component and a series elastic component. The model was solved by means of a computer. Passive and active properties of each region could be altered independently. The typical regional measures of muscle performance such as percent shortening, percent bulge, percent thickening, delay of shortening, percent increase in end-diastolic length and other hemodynamic measures were computed. These results were similar to those observed in animal models of stunning. In addition, a nearly linear relationship with end-diastolic length and delay of shortening was predicted by the model. It was concluded that a decrease in the peak isovolumic elastance and augmentation of viscosity effect of creep during stunning can explain mechanical abnormalities of stunned myocardium.

New models of the oculomotor mechanics based on data obtained with chronic muscle force transducers.

Several phenomenological models of the oculomotor mechanics that produce saccadic eye movements have been developed. These models have been based on measurements of macroscopic muscle and orbital tissue properties and measurements of eye kinematics during saccades. We recorded the forces generated by the medial and lateral recti during saccades in an alert, behaving monkey using chronically implanted force transducers. With this new data, we tested the ability of the classic saccade models to generate realistic muscle force profiles. Errors in the predictions of the classic saccade models led to a reexamination of the current models of extraocular muscle. Both a phenomenological, Hill-type muscle model and an approximation to Huxley's molecular level muscle model based on the cross-bridge mechanism of contraction (distribution moment model) were derived and studied for monkey extraocular muscle. Simulations of the distribution moment model led to insights suggesting (i) specific modifications in the lumped force/velocity relationship in the Hill-type model that resulted in this type of phenomenological model being able to generate realistic dynamics in extraocular muscle during saccades; (ii) the distribution of activity in the different fiber types in extraocular muscle may be central to the characteristics exhibited by the muscle during saccades; (iii) the transient properties of lengthening muscle such as yielding are not significant during saccades; and (iv) the series elastic component in active muscle may be predominantly generated by the elastic properties of the cross-bridges.

Cardiac quick-release contraction mechanoenergetics analysis using a cardiac muscle cross-bridge model.

Huxley's sliding filament cross-bridge muscle model coupled with parallel and series elastic components was simulated to examine the conflicting reports on the amount of energy saved by quick release at the peak contraction time. Cross-bridge energy utilization was determined by considering the ATP hydrolysis for the cross-bridge cycling. The quick-release cases were simulated by letting the muscle fiber suddenly shorten to the resting fiber length at peak systole, and then the contraction was allowed to continue at the resting length. Simulation results demonstrated that, using realistic parameter values, typically approximately 15% of the muscle fiber energy is used after peak systole (and approximately 30% of the cross-bridge energy), but this is also a function of the muscle fiber properties characterized by cross-bridge association and dissociation rate constants. Increasing the kinetic rate constants, the series elasticity, the initial fiber length, or the time of peak intracellular calcium will increase the amount of energy left, which may explain some of the discrepancies in the literature. Cardiac muscle hypertrophy will increase the fraction of muscle fiber energy left after peak systole to approximately 30%. The strongest indicator of the percent energy left at peak systole was the time the fiber reached peak systole, and as the fiber reached peak systole faster, the amount of energy saved by quick release increased.

Changes in stiffness induced by hindlimb suspension in rat soleus muscle.

Soleus muscle atrophy was induced by hind-limb suspension of rats for 3 weeks with the intention of inducing a relative increase in the percentage of fast-twitch fibres and assessing modifications in muscle stiffness. A method of dual controlled releases was used to obtain tension/extension curves and force/velocity relationships characterizing the mechanical behaviour of the soleus. Fibre typing was achieved by myofibrillar adenosine 5'-triphosphatase staining. Results showed that hindlimb suspension decreased the percentage of slow-twitch fibres (-31%) to the profit of fast-twitch fibres (+370%) and intermediate fibres (+255%). This led to an increase in maximal shortening velocity. Tension/extension curves indicated a decrease in soleus stiffness after 3 weeks of unloading. Changes in elastic properties are interpreted in terms of modifications occurring in the active part and the passive part of the so-called series elastic component. These changes also suggest that the parameters derived from a twitch are inappropriate to account for modifications in speed-related properties of muscle.

Skeletal muscle stiffness in static and dynamic contractions

Series elastic stiffness of rat gastrocnemius medialis muscle was determined by means of sinusoidal movements (180 Hz, 0.25% of muscle length) for various contraction conditions. The effects of muscle length, activation level, velocity, prestretch, and temperature on the force-stiffness relationship were investigated. All force-stiffness curves were transformed to a linear force-α curve (Ettema and Huijing, 1993; Morgan, 1977) to distinguish mathematically two series elastic components; a force dependent and force independent compliance. For all isometric conditions a typical force-stiffness curve was found, where stiffness increased with force, and this increase levelled off at higher forces. Stiffness in dynamic shortening and lengthening contractions is related to force in a completely different way than in isometric condition. An increase in temperature caused a decrease in muscle stiffness for a given force, and the effects of muscle length, activation level, and prestretch were small. It was concluded that the series elastic component of skeletal-muscle-tendon complex is probably located in more than two morphologically identifiable elements. Furthermore, we concluded that using a single series elastic element in muscle modelling is not appropriate to describe muscle behaviour under all conditions that occur during in vivo activation.

Effects of stretch-shortening cycle training on mechanical properties and fibre type transition in the rat soleus muscle.

The effects of exercise training on mechanical properties and fibre type transitions have been investigated in rat soleus muscles. The exercise was a repetition of stretch-shortening cycles. A method of dual controlled releases was applied to obtain tension/extension curves, which characterize the elastic behaviour of the series elastic component (SEC), and the force/velocity relationship characterizing the contractile elements. Other contractile measurements included: contraction time (tc), half-relaxation time (t1/2) and twitch/tetanus ratio (Pt/Po). Additionally, the muscle fibre type composition was determined by a classical histochemical method. A 12-week period of training induced a significantly higher percentage of fast-twitch fibres and a lower percentage of slow-twitch fibres (P < 0.01). This fibre adaptation led to a significant (P < 0.01) decrease in tc and an increase in maximum shortening velocity (Vmax). An increase in compliance of the SEC was also observed. This elastic adaptation is interpreted in terms of modification of the active components of the SEC. All the histochemical and mechanical data presented in this study show that rat soleus muscles trained by stretch-shortening cycles acquired faster characteristics. Thus the results confirm that a slow-twitch to a fast-twitch fibre transition is obtainable in mature rats.

On the Solutions of Huxley-type Models in Cardiac Muscle Fiber Contractions

Huxley's sliding filament crossbridge muscle model coupled with parallel and series elastic components was simulated to examine the effect of various solution techniques in cardiac contractions. Solutions of both isometric and isotonic contraction cases showed that the force versus time curves were not significantly altered by solving the three-element Hill model with Huxley's Equation written as either an ordinary or partial differential equation (ODE or PDE), but this makes a difference in the solution time required. Various theoretical studies have used either the ODE or PDE representation. The crossbridge cycles at the end of a contraction showed approximately 25% and 15% difference in the isometric and isotonic cases when Huxley's Equation was written as either an ODE or PDE. Examination of the crossbridge distribution (distribution among states of reach) showed that assuming that the crossbridge distribution is a Gaussian function is a poor approximation since the shape changes considerably between the cardiac contracting and expanding phases, and using a technique such as a distribution moment approximation is questionable. Recent experimental studies have demonstrated that solving Huxley-type relations as ordinary differential equations gives good agreement with cardiac data, implying that as a first approximation, this can be successfully used.

Comparison of the cardiac force-time integral with energetics using a cardiac muscle model

Several investigators have found experimentally that the force-time integral varies non-linearly with energy expenditure over the course of a cardiac contraction. Also, recent research findings have indicated that the crossbridge cycle to ATP hydrolysis ratio in muscle fiber systems may not be coupled with a one-to-one ratio. In order to investigate these findings, Huxley's sliding filament crossbridge muscle model coupled with parallel and series elastic components was simulated to examine the behavior of the crossbridge energy utilization and force-time integral vs time. Crossbridge (CB) energy utilization was determined by considering the ATP hydrolysis for the crossbridge cycling, and this CB energy was compared with the force-length energy in a contraction. This CB energy was calculated in both isometric and isotonic contractions as a function of contraction time and compared to the force-time integral. Simulation results demonstrated that the ratio of the force-time integral to CB energy varies strongly throughout the cardiac cycle for both isometric and isotonic cases, as has been observed experimentally. Simulations also showed that using the force-length energy component of energy vs the CB energy gave a better correlation between the total energetic predictions and the force-time integral, agreeing with recent finding that the crossbridge cycle to ATP hydrolysis ratio may not be coupled one-to-one, especially at lower force levels.

The contractile mechanism as an approach to building left ventricular pump models.

The aim was to construct a model linking a simplified interpretation of the contractile process at the myofilament level to the mechanical behaviour of the left ventricle to improve the ability of elastic-resistive models to represent the pumping response of the left ventricle. The mechanical model, consisting of an elastic component connected in series with a contractile component and an elastic component parallel to both the series elastic and contractile components, is able to develop pressure by the binding of a structural substance T to an excitatory substance C, the behaviour of which is a simplification of miofibrillar Ca2+ kinetics. Theoretically, the model was validated for its ability to reproduce by computer simulation, experiments that described the pumping properties of the left ventricle--namely, elasticity, resistivity, deactivating and positive effect of ejection, and the behaviour of intracellular Ca2+. Experimentally, the model was tested to fit intraventricular pressure (P(t)) and volume (V(t)) of single ejective beats in nine open chest dogs fitted with a pressure microtransducer to measure intraventricular P(t) and an aortic flowprobe to measure ventricular outflow and calculate V(t). Parameters were estimated up to maximum negative dP/dt adjusting P(t) or V(t) data of the ejective beats, and the goodness of the fit was evaluated through the root mean square error normalised with respect to the corresponding mean P(t) or V(t) in the fitting interval (NE). Descriptive validation of the model showed that the mean NE for the ejective P(t) fit was 0.03(SD 0.005) and for the V(t) fit 0.014(0.003). Predictive validation of P(t) and V(t) data of beats with partial occlusion of the aorta was performed up to end ejection, with parameters estimated from the P(t) or V(t) fit of the preceding ejective beat. Results gave a mean NE equal to 0.05(0.02) for predicted P(t) and 0.02(0.007) for predicted V(t), from either source of estimated parameters. Explanative validation showed that all the estimated parameters were in the same range used in simulation and that derived indexes [isovolumic maximum pressure (Pmax) = 166(13) mm Hg, time to maximum pressure (TPmax) = 0.186(0.012) s and the slope of the end systolic pressure volume relation (Emax) = 5.45(1.5) mm Hg.ml-1] were within reported experimental values. Finally, the model responded to increased inotropic state [dobutamine (5-35 micrograms.kg-1.min-1)] causing the estimated Pmax and Emax to increase by 33% and 25%, respectively, and TPmax to decrease by 10%. This model represented an improvement over previous pump models because (1) the model was able to represent behaviours other than purely elastic-resistive ones, such as the deactivation and positive effect of ejection; (2) left ventricular properties were the response of model behaviour and not constitutive elements of its structure; and (3) it adequately fulfilled model validation procedures.

Myocardial mechanics and the Fenn effect determined from a cardiac muscle crossbridge model

A three-element cardiac muscle fibre model, utilising Huxley's sliding filament theory for the contractile element and coupled with parallel and series elastic components, was simulated to see if it were possible to predict the cardiac Fenn effect. The force/length energy (FLE) was computed in both isometric and isotonic contractions, as a function of muscle fibre length (preload) in the isometric case and afterload in the isotonic contraction case. Simulation results demonstrated that isotonic contractions produced a greater FLE than isometric contractions at every corresponding afterload, with the difference being equal to the work produced in the isotonic case, which is characteristic of the Fenn effect. The maximum energy utilisation was observed at maximum force isometric contractions, as has been experimentally observed in cardiac muscle. Changing the stiffness of the series elastic component did not change the Fenn-effect behaviour. Fenn-effect plots using crossbridge energy predictions showed behaviour similar to the FLE plots, but the FLE: crossbridge energy ratio declined with decreasing force even though the efficiency has been experimentally found to be constant.

Age‐related changes in rat colon mechanics

AbstractTo determine whether myogenic factors are responsible for slowed colonic transit in senescent rats, maximum shortening velocity (V0), compliance of the series elastic component (SEC), measurements of passive force in calcium‐depleted tissue and peak isometric force (F0) were examined in proximal and distal colonic circular smooth muscle from 6‐ and 30‐monthold Fischer rats (n = 5). After mucosa was removed, measurements were made on strips stimulated with 80 m&gt;m KCl in a 2.5 m&gt;m Ca2+ Kreb's solution. Muscle strips were quick released at peak isometric force (F0) to afterloads of 60% of F0. The changes in muscle length from zero to 40 msec and 1 to 2 sec after release during isotonic contraction were used to calculate the SEC and V0 as a fraction of total muscle length. Passive force (Fp) was measured in 2.5 m&gt;m Ca2+ Kreb's solution and in a zero Ca2+, 0.1 m&gt;m EGTA solution to determine the contribution of contractile and passive elements to passive force. The results of these studies indicate there is no difference in the V0 (L0/sec) of adult (8.4 ± 1.5) and aged (7.5 ± 2.0) animals (P ± 0.05). Peak force (F0) in the distal colon of the aged rats was greater than adult rats (1.23 ± 0.1 vs 0.85 ± 0.01 kg/cm2, P = 0.05). The stiffness of the parallel elastic component and the length‐tension relationship were similar in adult and aged animals. Negligible decreases in Fp were observed in zero calcium medium. However, basal contractile tone was elevated in aged animals (P = 0.05). These studies indicate basic differences in aged colonic circular muscle that may contribute to altered bowel transit and function during ageing.

Variable cross-bridge cycling-ATP coupling accounts for cardiac mechanoenergetics

Cardiac twitch contractions were simulated by Huxley's sliding filament cross-bridge muscle model coupled with parallel and series elastic components. The energetics of the contraction were based on the ATP hydrolysis for the cross-bridge cycling. Force-length area (FLA), as a measure of the total mechanical energy, was computed for both isometric and isotonic contractions in a manner similar to the pressure-volume area (PVA) (Suga, H. Physiol. Rev. 70: 247-277, 1990). PVA correlates linearly with cardiac oxygen consumption, and since FLA is analogous to PVA, FLA should correlate with the ATP expended. Simulations comparing FLA with the cross-bridge cycling ATP usage showed that at lower muscle fiber activation levels (shorter initial fiber lengths and lower preload levels) FLA decreased more rapidly than the number of muscle fiber cross-bridge cycles in both isometric and isotonic contraction cases. This suggests that one ATP can cause more than one cross-bridge cycle at lower activation levels as was proposed by Yanagida, Arata, and Oosawa (Nature 316: 366-369, 1985). If the number of cross-bridge cycles to ATP ratio is allowed to increase at lower activation levels as suggested by Yanagida et al., Huxley's model is compatible with the experimental findings on FLA and PVA.

Variable crossbridge cycling-ATP coupling accounts for cardiac mechanoenergetics.

Cardiac twitch contractions were simulated by Huxley's sliding filament crossbridge muscle model. Huxley's model was extended to include cardiac twitch contractions with a model structure having parallel and series elastic components with a crossbridge contractile element. The appropriate crossbridge energetics were added based on the crossbridge cycling rate and the energy of ATP hydrolysis. The force-length area (FLA) as a measure of the total mechanical energy was computed for both isometric and isotonic contractions in a manner similar to the pressure-volume area (PVA), (Suga, H. Physiol. Rev., 70, 247-277, 1990). Experimental studies have demonstrated that the pressure-volume area (PVA) correlates linearly with cardiac oxygen consumption and hence with the energy expenditure of a cardiac contraction. PVA correlates linearly with cardiac oxygen consumption, and since FLA is analogous to PVA, FLA should correlate with the ATP expended. Simulations comparing FLA with the crossbridge cycling ATP usage showed that at lower muscle fiber activation levels (shorter initial fiber lengths and lower preload levels) FLA decreased more rapidly than the number of muscle fiber crossbridge cycles. This could imply that one ATP can cause more than one crossbridge cycle at lower fiber activation levels as was proposed by Yanagida et al. (Nature, 316, 366-369, 1985). If the number of crossbridge cycles to ATP ratio is allowed to increase at lower activation levels, Huxley's model agrees with the experimental findings on FLA and PVA.

Cardiac muscle fiber force versus length determined by a cardiac muscle crossbridge model.

A mathematical model incorporating Huxley's sliding filament crossbridge muscle model coupled with parallel and series elastic components was simulated to examine force-length relations under different external calcium concentrations. Several researchers have determined experimentally in both papillary muscle preparations and in situ heart experiments that the calcium concentration (or effective concentration from inotropic agents) will affect the strength and convexity of the cardiac muscle fiber force-length relations. Simulations were performed over a several-order-of-magnitude range of calcium concentrations in isometric contractions and these showed that the force-length curve convexity was changed. Simulation results demonstrated that increasing the stiffness in the model contractile element or series elasticity element did not change the force-length convexity. Increasing the series elasticity element stiffness did slightly change the shape of the force-length curve. The model predicts that the curve convexity changes as a result of the calcium-troponin interactions.

Isometric trunk extensor endurance in young females

Series elastic stiffness of rat gastrocnemius medialis muscle was determined by means of sinusoidal movements (180 Hz, 0.25% of muscle length) for various contraction conditions. The effects of muscle length, activation level, velocity, prestretch, and temperature on the force-stiffness relationship were investigated. All force-stiffness curves were transformed to a linear force-α curve (Ettema and Huijing, 1993; Morgan, 1977) to distinguish mathematically two series elastic components; a force dependent and force independent compliance. For all isometric conditions a typical force-stiffness curve was found, where stiffness increased with force, and this increase levelled off at higher forces. Stiffness in dynamic shortening and lengthening contractions is related to force in a completely different way than in isometric condition. An increase in temperature caused a decrease in muscle stiffness for a given force, and the effects of muscle length, activation level, and prestretch were small. It was concluded that the series elastic component of skeletal-muscle-tendon complex is probably located in more than two morphologically identifiable elements. Furthermore, we concluded that using a single series elastic element in muscle modelling is not appropriate to describe muscle behaviour under all conditions that occur during in vivo activation.

Series Elastic Properties of Rat Skeletal Muscle: Distinction of Series Elastic Components and Some Implications

Compliance of the series elastic component (SEC) of rat extensor digitorum longus (EDL) and gastrocnemius medialis (GM) muscle-tendon complex was measured using quick length decreases (0.2 mm within 3 ms) during isometric contractions. Extension of tendinous structures at maximal isometric force level was measured by means of photography. These data allowed us to distinguish between series elastic compliance of tendinous structures and muscle fibres (cross-bridges). Using mathematics, similar as in the Alpha method (MORGAN, 1977), a force dependent component and a constant component of series elastic compliance could be distinguished. Extension values of SEC in the cross-bridges of about 1.5% of fibre length were found, which is close to values found for isolated frog muscle fibres. The force dependent compliance was significantly higher (3%). Therefore, it is concluded that part of the force dependent compliance resides within the tendinous structures, even beyond the toe-region. Thus, at high force levels compliance of tendinous structures is not constant. Direct measurements on isolated GM tendon confirmed this conclusion. Functional consequences of these tendinous properties are discussed. For both EDL and GM, about 85% of SEC extension at maximal isometric force (F o ) appeared to be located in the tendinous structures. However, tendinous compliance, normalised with respect to its length, is higher for GM. For GM the free tendon is more compliant than aponeurosis, whilst EDL has a rather uniform distribution of normalised compliance along free tendon and aponeurosis. Differences of tendinous compliance may be related to functional differences between the muscles.

Stretch shorten cycle performance enhancement through flexibility training

Sixteen experienced male powerlifters served as subjects in a training study designed to examine the effect of flexibility training on: (i) the stiffness of the series elastic components (SEC) of the upper body musculature and (ii) rebound and purely concentric bench press performance. Nine of the subjects participated in two sessions of flexibility training twice per week for 8 wk. Prior to and after the training period the subjects' static flexibility, SEC stiffness, rebound bench press (RBP), and purely concentric bench press (PCBP) performance were recorded. The flexibility training induced a significant reduction in the maximal stiffness of the SEC. Furthermore, the experimental subjects produced significantly more work during the initial concentric portion of the RBP lift, enabling a significantly greater load to be lifted in the post-training testing occasion. The benefits to performance achieved by the experimental group consequent to flexibility training were greater during the RBP lift as compared with the PCBP lift. The control subjects exhibited no change in any variable over the training period. These results implied that the RBP performance enhancement observed consequent to flexibility training was directly caused by a reduction in SEC stiffness, increasing the utilization of elastic strain energy during the RBP lift.

The electro-mechanical delay of the erector spinae muscle: influence of rate of force development, fatigue and electrode location

Electro-mechanical delay (EMD) values of the erector spinae muscle were obtained using a technique based on the cross-correlation between the force and the electromyogram (EMG). Seven subjects performed a series of 20 submaximal dynamic isometric contractions in a seated position at two frequencies (0.5 Hz and 1 Hz) to study the influence of the rate of force development on EMD. Mean EMD values of 125.7 (SD 28.1) ms (1 Hz) and 136.8 (SD 28.6) ms (0.5 Hz) were shown to differ significantly (P = 0.02). This finding supports the hypothesis that EMD is inversely related to the rate of force development and implies that the time to stretch the series elastic component is an important factor determining EMD. After performing a series of fatiguing contractions EMD did not differ significantly from the control value. Multiple regression analysis showed that maximal voluntary contraction force (MVC) and endurance time of the fatiguing exercise correlated significantly with EMD. The site from which the EMG signal was recorded had no significant influence on EMD. However, the coefficient of correlation between force and the EMG-signal differed significantly between electrode positions. The magnitude of the EMD values found emphasized the need to account for this delay when interpreting temporal patterns of activation of the muscles in, for example, lifting tasks.

Stiffness changes and fibre type transitions in rat soleus muscle produced by jumping training

Rat soleus muscles were overloaded with intent to induce a relative increase in fast fibres and modifications in muscular stiffness. The overloading technique was a training period consisting of an 11-week vertical jump programme. The method of controlled releases was used to obtain tension/extension curves characterizing the elastic behaviour of the soleus. Fibre typing was made by myofibrillar adenosine 5'-triphosphatase staining. With regard to a control group, training resulted in a relative decrease in type I fibres for the benefit of type II fibres. Training also induced a decrease in muscle stiffness as attested notably by significant differences in maximal extension. These results are interpreted in terms of modifications occurring in the active fraction of the so-called series elastic component.

Regional Comparison of Rat Colon Mechanics

The purpose of this study was to examine the force‐velocity characteristics of colonie muscle and to determine whether these factors contribute to regional specialization of the colon observed in adult Fischer rats. Four measurements were obtained from circular colon muscle strips: maximum shortening velocity, a reflection of crossbridge cycling rate; extension of the parallel elastic component, a measure of passive muscle properties of relaxed muscle; the series elastic component, a measure of passive muscle properties of contracted muscle; and peak isometric force, which is equal to the product of the number of activated crossbridges and the strength of individual crossbridges. Muscle length (L) was expressed in terms of the length of optimal tension development (Lo). Peak isometric force and maximum shortening velocity were 848.9 ± 114.7 g/cm2 and 0.082 ± 0.012 Lo/s for muscle strips from the proximal colon, and 948.0± 138.2 g/cm2 and 0.083 ± 0.014 Lo/s for muscle strips from the distal colon. Shortening velocity, isometric force, and load extension properties of the parallel elastic component and the series elastic component are similar in proximal and distal rat colon. This suggests that regional specialization is not determined at the myofibril level but is most likely determined by extrinsic regulatory factors at the neural or receptor level.