Nonlinearities make a difference: comparison of two common Hill-type models with real muscle

Compared to complex structural Huxley-type models, Hill-type models phenomenologically describe muscle contraction using only few state variables. The Hill-type models dominate in the ever expanding field of musculoskeletal simulations for simplicity and low computational cost. Reasonable parameters are required to gain insight into mechanics of movement. The two most common Hill-type muscle models used contain three components. The series elastic component is connected in series to the contractile component. A parallel elastic component is either connected in parallel to both the contractile and the series elastic component (model [CC+SEC]), or is connected in parallel only with the contractile component (model [CC]). As soon as at least one of the components exhibits substantial nonlinearities, as, e.g., the contractile component by the ability to turn on and off, the two models are mechanically different. We tested which model ([CC+SEC] or [CC]) represents the cat soleus better. Ramp experiments consisting of an isometric and an isokinetic part were performed with an in situ cat soleus preparation using supramaximal nerve stimulation. Hill-type models containing force–length and force–velocity relationship, excitation–contraction coupling and series and parallel elastic force–elongation relations were fitted to the data. To test which model might represent the muscle better, the obtained parameters were compared with experimentally determined parameters. Determined in situations with negligible passive force, the force–velocity relation and the series elastic component relation are independent of the chosen model. In contrast to model [CC+SEC], these relations predicted by model [CC] were in accordance with experimental relations. In conclusion model [CC] seemed to better represent the cat soleus contraction dynamics and should be preferred in the nonlinear regression of muscle parameters and in musculoskeletal modeling.     

Influence of the Mechanical Properties of the Muscle–tendon Unit on Force Generation in Runners with Different Running Economy

In earlier studies, we found more economical runners having a more compliant quadriceps femoris (QF) tendon at low force levels, and a higher contractile strength and stiffness at the triceps surae (TS). To better understand how these differences influence force generation economy and energy recovery, we simulated contractions using a Hill-type muscle model and the previously determined muscle properties as input parameters. For eight different activation levels, we simulated isovelocity concentric contractions preceded by an isovelocity stretch. The length changes and contraction velocities imposed to the muscle–tendon units (MTU) corresponded to those happening whilst running. The main results of the simulations were: (a) a more compliant tendon at low force levels (QF) led to an advantage in force-generation due to a decrease in shortening velocity of the CE, (b) a higher contractile strength and higher stiffness at the TS led to a disadvantage in force-generation at high activation levels and to an advantage at low activation levels. In addition at the high economy runners both MTUs showed an advantageous energy release during shortening, which at the QF was mainly due to a higher elongation of the SEE and at the TS mainly to the higher contractile strength. Especially at low activation levels both MTUs showed an advantageous force generation per activation and a higher energy release as compared to the low economy runners.     

Contraction dynamics and function of the muscle-tendon complex depend on the muscle fibre-tendon length ratio: a simulation study

Experimental studies show different muscle-tendon complex (MTC) functions (e.g. motor or spring) depending on the muscle fibre-tendon length ratio. Comparing different MTC of different animals examined experimentally, the extracted MTC functions are biased by, for example, MTC-specific pennation angle and fibre-type distribution or divergent experimental protocols (e.g. influence of temperature or stimulation on MTC force). Thus, a thorough understanding of variation of these inner muscle fibre-tendon length ratios on MTC function is difficult. In this study, we used a hill-type muscle model to simulate MTC. The model consists of a contractile element (CE) simulating muscle fibres, a serial element (SE) as a model for tendon, and a parallel elastic element (PEE) modelling tissue in parallel to the muscle fibres. The simulation examines the impact of length variations of these components on contraction dynamics and MTC function. Ensuring a constant overall length of the MTC by \(L_\mathrm{MTC} = L_\mathrm{SE} + L_\mathrm{CE}\), the SE rest length was varied over a broad physiological range from 0.1 to 0.9 MTC length. Five different MTC functions were investigated by simulating typical physiological experiments: the stabilising function with isometric contractions, the motor function with contractions against a weight, the capability of acceleration with contractions against a small inertial mass, the braking function by decelerating a mass, and the spring function with stretch-shortening cycles. The ratio of SE and CE mainly determines the MTC function. MTC with comparably short tendon generates high force and maximal shortening velocity and is able to produce maximal work and power. MTC with long tendon is suitable to store and release a maximum amount of energy. Variation of muscle fibre-tendon ratio yielded two peaks for MTC’s force response for short and long SE lengths. Further, maximum work storage capacity of the SE is at long \(\mathrm{rel}L_\mathrm{SE,0}\). Impact of fibre-tendon length ratio on MTC functions will be discussed. Considering a constant set of MTC parameters, quantitative changes in MTC performance (work, stiffness, force, energy storage, dissipation) depending on varying muscle fibre-tendon length ratio were provided, which enables classification and grading of different MTC designs.     

Hill-type muscle model with serial damping and eccentric force–velocity relation

Hill-type muscle models are commonly used in biomechanical simulations to predict passive and active muscle forces. Here, a model is presented which consists of four elements: a contractile element with force–length and force–velocity relations for concentric and eccentric contractions, a parallel elastic element, a series elastic element, and a serial damping element. With this, it combines previously published effects relevant for muscular contraction, i.e. serial damping and eccentric force–velocity relation. The model is exemplarily applied to arm movements. The more realistic representation of the eccentric force–velocity relation results in human-like elbow-joint flexion. The model is provided as ready to use Matlab ®$\circledR$ and Simulink ®$\circledR$ code.     

A macroscopic ansatz to deduce the Hill relation

In this study, we derive the hyperbolic force-velocity relation of concentric muscular contraction, first formulated empirically by A.V. Hill in 1938, from three essential model assumptions: (1) the structural assembly of three well-known elements - i.e. active, parallel damping, and serial - fulfilling a force equilibrium, (2) the parallel damping coefficient explicitly depending on muscle force output and three parameters, and (3) the kinematic gearing ratio between active and serial element being assigned to a parameter. The energy source within the muscle represented by the force of the active element is an additional fifth parameter. As a result we find the Hill “constants” A and B as functions of our five model parameters. Using A and B values from literature on experimental data, we predict heat power release of our model. By calculating enthalpy rate and mechanical efficiency, we compare the model heat power to predictions from another Hill-type model, to Hill's original findings, and to findings from modern muscle heat measurements. We reconsider why the biggest share of heat rate during isometric contractions (maintenance heat) and the velocity-dependent heat rate during concentric contractions in addition to maintenance heat rate (shortening heat rate) may be traced back to the same mechanism represented by the kinematic gearing ratio. Namely, we suggest that the serial element transfers attachment-detachment fluctuations of actin-myosin crossbridges within one sarcomere to others in the same sarcomere and to those in parallel and in series. Numerically, in case of negligible passive muscular damping, we find the ratio between A and isometric force (relative A) to depend exclusively on the kinematic gearing ratio, whereas the maintenance heat rate scales with the square of relative A. Moreover, this mechanical coupling internal to the muscle fibres may also be behind the macroscopic force dependency of the overall parallel damping coefficient.     

A theory on auto-oscillation and contraction in striated muscle

It is widely accepted that muscle cells take either force-generating or relaxing state in an all-or-none fashion through the so-called excitation–contraction coupling. On the other hand, the membrane-less contractile apparatus takes the third state, i.e., the auto-oscillation (SPOC) state, at the activation level that is intermediate between full activation and relaxation. Here, to explain the dynamics of all three states of muscle, we construct a novel theoretical model based on the balance of forces not only parallel but also perpendicular to the long axis of myofibrils, taking into account the experimental fact that the spacing of myofilament lattice changes with sarcomere length and upon contraction. This theory presents a phase diagram composed of several states of the contractile apparatus and explains the dynamic behavior of SPOC, e.g., periodical changes in sarcomere length with the saw-tooth waveform. The appropriate selection of the constant of the molecular friction due to the cross-bridge formation can explain the difference in the SPOC periods observed under various activating conditions and in different muscle types, i.e., skeletal and cardiac. The theory also predicts the existence of a weak oscillation state at the boundary between SPOC and relaxation regions in the phase diagram. Thus, the present theory comprehensively explains the characteristics of auto-oscillation and contraction in the contractile system of striated muscle.     

Biomechanical properties and a kinetic simulation model of the smooth muscle I2 in the buccal mass of Aplysia

The muscle I2 is a smooth muscle from the buccal mass of the marine mollusc Aplysia californica whose neural control, in vivo kinematics, and behavioral role have been extensively analyzed. In this study, we measured the activation and contractile dynamics of the muscle in order to construct a Hill-type kinetic model of the muscle. This is the first study to our knowledge, of Aplysia muscle contractile dynamics. The isometric force-frequency relationship of I2 had a frequency threshold of about 6–8 Hz, and its force output saturated at 20–25 Hz, properties that match the high frequency (20 Hz) bursts generated by the B31/B32 neurons that innervate it. Peak isometric force was generated at about 118% of the in situ relaxed length. These results and I2's estimated in vivo kinematics suggest that it generates maximum force at the onset of protraction. The muscle tension during iso-velocity lengthening and shortening was an asymmetric function of velocity. Short range stiffness and yielding responses were observed in lengthening, whereas muscle tension decreased smoothly in shortening. These visco-elastic properties suggest that the I2 muscle can serve to brake forceful retraction movements. A Hill-type model, parameterized from the measurements, captured many of the mechanical properties of I2. Our results provide a quantitative understanding of the biomechanical significance of the muscle's neural control and provide a basis for simulation studies of the control of feeding behavior. Received: 5 February 1999 / Accepted in revised form: 18 May 1999     

Models of contractile units and their assembly in smooth muscle

It is believed that the contractile filaments in smooth muscle are organized into arrays of contractile units (similar to the sarcomeric structure in striated muscle), and that such an organization is crucial for transforming the mechanical activities of actomyosin interaction into cell shortening and force generation. Details of the filament organization, however, are still poorly understood. Several models of contractile filament architecture are discussed here. To account for the linear relationship observed between the force generated by a smooth muscle and the muscle length at the plateau of an isotonic contraction, a model of contractile unit is proposed. The model consists of 2 dense bodies with actin (thin) filaments attached, and a myosin (thick) filament lying between the parallel thin filaments. In addition, the thick filament is assumed to span the whole contractile unit length, from dense body to dense body, so that when the contractile unit shortens, the amount of overlap between the thick and thin filaments (i.e., the distance between the dense bodies) decreases in exact proportion to the amount of shortening. Assembly of the contractile units into functional contractile apparatus is assumed to involve a group of cells that form a mechanical syncytium. The contractile apparatus is assumed malleable in that the number of contractile units in series and in parallel can be altered to accommodate strains on the muscle and to maintain the muscle's optimal mechanical function.     

Does a two-element muscle model offer advantages when estimating ankle plantar flexor forces during human cycling?

Traditional Hill-type muscle models, parameterized using high-quality experimental data, are often “too weak” to reproduce the joint torques generated by healthy adults during rapid, high force tasks. This study investigated whether the failure of these models to account for different types of motor units contributes to this apparent weakness; if so, muscle-driven simulations may rely on excessively high muscle excitations to generate a given force. We ran a series of forward simulations that reproduced measured ankle mechanics during cycling at five cadences ranging from 60 to 140 RPM. We generated both “nominal” simulations, in which an abstract ankle model was actuated by a 1-element Hill-type plantar flexor with a single contractile element (CE), and “test” simulations, in which the same model was actuated by a 2-element plantar flexor with two CEs that accounted for the force-generating properties of slower and faster motor units. We varied the total excitation applied to the 2-element plantar flexor between 60 and 105% of the excitation from each nominal simulation, and we varied the amount distributed to each CE between 0 and 100% of the total. Within this test space, we identified the excitation level and distribution, at each cadence, that best reproduced the plantar flexor forces generated in the nominal simulations. Our comparisons revealed that the 2-element model required substantially less total excitation than the 1-element model to generate comparable forces, especially at higher cadences. For instance, at 140 RPM, the required excitation was reduced by 23%. These results suggest that a 2-element model, in which contractile properties are “tuned” to represent slower and faster motor units, can increase the apparent strength and perhaps improve the fidelity of simulations of tasks with varying mechanical demands.     

Preserved Close Linkage Between the Genes Encoding Troponin I and Troponin T, Reflecting an Evolution of Adapter Proteins Coupling the Ca2+ Signaling of Contractility

Ca²⁺-regulated motility is essential to numerous cellular functions, including muscle contraction. Systems with troponin C, myosin light chain, or calmodulin as the Ca²⁺ receptor have evolved in striated muscle and other types of cells to transduce the cytoplasm Ca²⁺ signals into allosteric conformational changes of contractile proteins. While these Ca²⁺ receptors are homologous proteins, their coupling to the responding elements is quite different in various cell types. The Ca²⁺ regulatory system in vertebrate striated muscle represents a highly specialized such signal transduction pathway consisting of the troponin complex and tropomyosin associated with the actin filament. To understand the molecular mechanism in the Ca²⁺ regulation of muscle contraction and cell motility, we have revealed a preserved ancestral close linkage between the genes encoding two of the troponin subunits, troponin I and troponin T, in the genome of mouse. The data suggest that the troponin I and troponin T genes may have originated from a single locus and evolved in parallel to encode a striated muscle-specific adapter to couple the Ca²⁺ receptor, troponin C, to the actin–myosin contractile machinery. This hypothesis views the three troponin subunits as two structure–function domains: the Ca²⁺ receptor and the signal transducing adapter. This model may help to further our understanding of the Ca²⁺ regulation of muscle contraction and the structure–function relationship of other potential adapter proteins which are converged to constitute the Ca²⁺ signal transduction pathways governing nonmuscle cell motility. Received: 15 April 1999 / Accepted: 15 July 1999     

Suppression of the autooscillatory contractile activity of <Emphasis Type="Italic">Physarum polycephalum</Emphasis> plasmodium by the inhibitor of the IP<Subscript>3</Subscript>-Induced Ca<Superscript>2+</Superscript> release, 2-aminoethoxydiphenyl borate

In order to elucidate the function of inositol 1,4,5-trisphosphate (IP₃)-activated reticular Ca²⁺ channel (IP₃R) in autooscillatory contractile activity of Physarum polycephalum plasmodium, we applied 2-aminoethoxydiphenyl borate (2-APB), a membrane-permeable inhibitor of IP₃-induced Ca²⁺ release. Taking into account that for the type 1 IP₃ R the inhibitory efficacy of 2-APB decreases with the rise of the IP₃ level [Bilmen, J.G. and Michelangeli, F., Cell Signal., 2002, vol. 14, no. 11, pp. 955–960], 2-APB was applied to plasmodium in normal conditions and after the treatment with glucose or 3-O-methylglucose, the attractants capable to induce an elevation of the IP₃ production. We found that 20–50 μM 2-APB induced a reversible cessation of contractile autooscillations, which occurred in two different modes: as a fast stop and a gradual damping. The damping of oscillations was accompanied by an increase in their period, a prolongation of the contraction phase, and, often, by an increase in the mean level of the contraction force. The number of species responding by the fast stop at a 2-APB concentration of 50 μM was two times greater than at 20 μM 2-APB. In the presence of the attractants in concentrations of 10 and 50 mM, the fast stop was never observed at 20 μM of 2-APB. Moreover, the damping of oscillations was preceded by a period of varying duration, when the regular oscillatory mode was maintained. We conclude that the fast stop results from the direct inter-action of 2-APB with IP₃R of Physarum polycephalum plasmodium and that IP₃R is indispensable for the plasmodial oscillator.     

Blebbistatin specifically inhibits actin-myosin interaction in mouse cardiac muscle

Blebbistatin is a powerful inhibitor of actin-myosin interaction in isolated contractile proteins. To examine whether blebbistatin acts in a similar manner in the organized contractile system of striated muscle, the effects of blebbistatin on contraction of cardiac tissue from mouse were studied. The contraction of paced intact papillary muscle preparations and shortening of isolated cardiomyocytes were inhibited by blebbistatin with inhibitory constants in the micromolar range (1.3–2.8 µM). The inhibition constants are similar to those previously reported for isolated cardiac myosin subfragments showing that blebbistatin action is similar in filamentous myosin of the cardiac contractile apparatus and isolated proteins. The inhibition was not associated with alterations in action potential duration or decreased influx through L-type Ca²⁺ channels. Experiments on permeabilized cardiac muscle preparations showed that the inhibition was not due to alterations in Ca²⁺ sensitivity of the contractile filaments. The maximal shortening velocity was not affected by 1 µM blebbistatin. In conclusion, we show that blebbistatin is an inhibitor of the actin-myosin interaction in the organized contractile system of cardiac muscle and that its action is not due to effects on the Ca²⁺ influx and activation systems. heart; electrophysiology; permeabilized muscle     

Effects of the length ratio between the contractile element and the series elastic element on an explosive muscular performance.

Effects of the length ratio between the contractile element (CE) and the series elastic element (SEE) on the behavior of the muscle tendon complex were investigated during stretch-shortening cycles. A computer simulation model of the Hill-type muscle tendon complex was constructed. The proximal end of the CE was affixed to a point in the gravitational field, and a massless supporting object was affixed to the distal end of the SEE. A mass was held on the supporting object. Initially, the muscle tendon complex was fixed at a certain length, and the CE was activated at 100%. Through this process, the CE contracted as much as the SEE was stretched. Thereafter, the supporting object was released, which caused the muscle tendon complex to propel the mass upward, simulating a stretch-shortening cycle. The length ratio between the CE and the SEE, the size of the mass and the initial length of the CE were sequentially changed. As a result, it was found that a higher performance is obtained with a longer SEE when the mass is small, while with a shorter SEE when the mass is large.     

Work partitioning of transversally loaded muscle: experimentation and simulation

Skeletal muscles are surrounded by other muscles, connective tissue and bones, which may transfer transversal forces to the muscle belly. Simple Hill-type muscle models do not consider transversal forces. Thus, the aim of this study was to examine and model the influence of transversal muscle loading on contraction dynamics, e.g. on the rate of force development and on the maximum isometric muscle force (F_im). Isometric experiments with and without transversal muscle loading were conducted on rat muscles. The muscles were loaded (1.3 N cm^? 2) by a custom-made plunger which was able to move in transversal direction. Then the muscle was fully stimulated, the isometric force was measured at the distal tendon and the movement of the plunger was captured with a high-speed camera. The interaction between the muscle and the transversal load was modelled based on energy balance between the (1) work done by the contractile component (CC) and (2) the work done to lift the load, to stretch the series elastic structures and to deform the muscle. Compared with the unloaded contraction, the force rate was reduced by about 25% and F_im was reduced by 5% both in the experiment and in the simulation. The reduction in F_im resulted from using part of the work done by the CC to lift the load and deform the muscle. The response of the muscle to transversal loading opens a window into the interdependence of contractile and deformation work, which can be used to specify and validate 3D muscle models.     

Tendon crimps and peritendinous tissues responding to tensional forces

Tendons transmit forces generated from muscle to bone making joint movements possible. Tendon collagen has a complex supramolecular structure forming many hierarchical levels of association; its main functional unit is the collagen fibril forming fibers and fascicles. Since tendons are enclosed by loose connective sheaths in continuity with muscle sheaths, it is likely that tendon sheaths could play a role in absorbing/transmitting the forces created by muscle contraction. In this study rat Achilles tendons were passively stretched in vivo to be observed at polarized light microscope (PLM), scanning electron microscope (SEM) and transmission electron microscope (TEM). At PLM tendon collagen fibers in relaxed rat Achilles tendons ran straight and parallel, showing a periodic crimp pattern. Similarly tendon sheaths showed apparent crimps. At higher magnification SEM and TEM revealed that in each tendon crimp large and heterogeneous collagen fibrils running straight and parallel suddenly changed their direction undergoing localized and variable modifications. These fibril modifications were named fibrillar crimps. Tendon sheaths displayed small and uniform fibrils running parallel with a wavy course without any ultrastructural aspects of crimp. Since in passively stretched Achilles tendons fibrillar crimps were still observed, it is likely that during the tendon stretching, and presumably during the tendon elongation in muscle contraction, the fibrillar crimp may be the real structural component of the tendon crimp acting as shock absorber. The peritendinous sheath can be stretched as tendon, but is not actively involved in the mechanism of shock absorber as the fibrillar crimp. The different functional behaviour of tendons and sheaths may be due to the different structural and molecular arrangement of their fibrils.     

A one-parameter neural activation to muscle activation model: estimating isometric joint moments from electromyograms

Nonlinearities have been observed in the isometric EMG-force relationship. However, these are generally not included when using EMG-driven Hill-type muscle models that account for muscle activation dynamics. In this paper, we present a formulation for a one-parameter transformation model (i.e., A-model) that accounts for the type of physiological nonlinearities observed at low levels of force. The general shape for the curvilinear portion of the curve was based on phenomenological data reported by Woods and Bigland-Ritchie. The one-parameter A-model is easy to implement, and when used with an EMG-driven Hill-type model, was shown to provide a better fit of the measured joint moment. Optimization methods were used to determine the appropriate curvature of the relationship for each muscle, and thus introduced a degree of "tuning" to each subject.     

The Influence of Intrinsic Muscle Properties on Musculoskeletal System Stability: A Modelling Study

The objective of this study is to investigate how the intrinsic mechanical properties of muscles will affect the musculoskeletal system stability. A typical musculoskeletal joint driven by a pair of antagonist muscles confined only in the sigittal plane was constructed. The dynamic characteristics of the flexor and extensor muscles induced by neural inputs were represented by three dynamic processes: neural excitation, muscle activation and muscle contraction dynamics. The muscle contraction mechanics was described using a modified Hill's model with a Contractile Element (CE), a parallel elastic element and a serial elastic element. Additionally, the change of muscle Physiological Cross-Sectional Area (PCSA) and pennation angle during muscle contraction were also considered. A set of dynamic simulations were conducted by applying an external impulsive force at the distal part of the musculoskeletal system. Sensitivity analysis was conducted to investigate the effect of the CE's force-length relationship, the CE's force-velocity relationship, the force-length relationship of the serial elastic element, the parallel elastic element and the pennation angle on the system stability. The results show that the muscles with full intrinsic mechanical properties are sufficient to stabilize the system subject to an impulsive force perturbation without reflexive changes in activations. To guarantee a self-stabilizing ability, a proper CE's force-velocity relationship, the existence of a series elastic element and a sufficient muscle co-contraction level are necessary. This study would provide insight into the intrinsic design and function of the musculoskeletal system, and also give implications for the design of bionic actuators, biomimetic robotics and prosthetic devices.     

Accuracy of gastrocnemius muscles forces in walking and running goats predicted by one-element and two-element Hill-type models

Hill-type models are commonly used to estimate muscle forces during human and animal movement—yet the accuracy of the forces estimated during walking, running, and other tasks remains largely unknown. Further, most Hill-type models assume a single contractile element, despite evidence that faster and slower motor units, which have different activation–deactivation dynamics, may be independently or collectively excited. This study evaluated a novel, two-element Hill-type model with “differential” activation of fast and slow contractile elements. Model performance was assessed using a comprehensive data set (including measures of EMG intensity, fascicle length, and tendon force) collected from the gastrocnemius muscles of goats during locomotor experiments. Muscle forces predicted by the new two-element model were compared to the forces estimated using traditional one-element models and to the forces measured in vivo using tendon buckle transducers. Overall, the two-element model resulted in the best predictions of in vivo gastrocnemius force. The coefficient of determination, r², was up to 26.9% higher and the root mean square error, RMSE, was up to 37.4% lower for the two-element model than for the one-element models tested. All models captured salient features of the measured muscle force during walking, trotting, and galloping (r²=0.26–0.51), and all exhibited some errors (RMSE=9.63–32.2% of the maximum in vivo force). These comparisons provide important insight into the accuracy of Hill-type models. The results also show that incorporation of fast and slow contractile elements within muscle models can improve estimates of time-varying, whole muscle force during locomotor tasks.     

A Modified Hai–Murphy Model of Uterine Smooth Muscle Contraction

We extend and analyze the Wang and Politi modified Hai–Murphy model of smooth muscle cell contractions to capture uterine muscle cell response to variations in intracellular calcium concentrations. This model is used to estimate values of unknown parameters in uterine smooth muscle cell cross-bridging. Uterine motility is responsible for carrying out important processes throughout all phases of the nonpregnant female reproductive cycle, including sperm transport, menstruation, and embryo implantation. The modified Hai–Murphy partial differential equation model accounts for the displacement of myosin cross-bridge heads relative to their binding sites. This model was originally developed for the study of airway contractions; we now extended it for use in modeling nonisometric uterine contractions. Our extended model incorporates cross-bridge position and contractile velocity into the original model, resulting in more accurate modeling of the initial stages of contraction and modeling nonisometric contractions. Numerical simulations show that the contraction rate in our extended model is faster than the original Hai–Murphy model. These simulations provide quantitative estimates for the increased level of responsiveness of our extended model to intracellular calcium concentrations. The extended model and new parameter estimates for the cross-bridging can be coupled with uterine flow models to advance our understanding of embryonic motility and intrauterine flow.     

Some proposals in cardiac muscle mechanics and energetics

Based on A. V. Hill's three-component model, mechanical properties of the contractile element (CE), such as velocity and tension, were determined as muscle shortening and loads in quick-release or afterloaded isotonic contraction. The method is applicable for studying cardiac mechanics, to obtain force-velocity data of the same CE length at varous afterloads. Analysis of the energetics of cardiac muscle was based on simulation studies of cardiac mechanics (Wong 1971, 1972). By proper derivation, the conventional contractile element work (CEW) was found to be a minor energy determinant. The tension-time integral and tension-independent heat (Ricchiuti and Gibbs, 1965) represent energy utilization for activation and maintenance of tension, the primary energy determinant.