A model of cardiac muscle mechanics and energetics

A short history of the development of mathematical models of cardiac mechanics

Cardiac mechanics plays a crucial role in atrial and ventricular function, in the regulation of growth and remodelling, in the progression of disease, and the response to treatment. The spatial scale of the critical mechanisms ranges from nm (molecules) to cm (hearts) with the fastest events occurring in milliseconds (molecular events) and the slowest requiring months (growth and remodelling). Due to its complexity and importance, cardiac mechanics has been studied extensively both experimentally and through mathematical models and simulation. Models of cardiac mechanics evolved from seminal studies in skeletal muscle, and developed into cardiac specific, species specific, human specific and finally patient specific calculations. These models provide a formal framework to link multiple experimental assays recorded over nearly 100 years into a single unified representation of cardiac function. This review first provides a summary of the proteins, physiology and anatomy involved in the generation of cardiac pump function. We then describe the evolution of models of cardiac mechanics starting with the early theoretical frameworks describing the link between sarcomeres and muscle contraction, transitioning through myosin-level models to calcium-driven systems, and ending with whole heart patient-specific models.

An in silico case study of idiopathic dilated cardiomyopathy via a multi-scale model of the cardiovascular system

Mathematical modelling has been used to comprehend the pathology and the assessment of different treatment techniques such as heart failure and left ventricular assist device therapy in the cardiovascular field. In this study, an in-silico model of the heart is developed to understand the effects of idiopathic dilated cardiomyopathy (IDC) as a pathological scenario, with mechanisms described at the cellular, protein and organ levels. This model includes the right and left atria and ventricles, as well as the systemic and pulmonary arteries and veins. First, a multi-scale model of the whole heart is simulated for healthy conditions. Subsequently, the model is modified at its microscopic and macroscopic spatial scale to obtain the characteristics of IDC. The extracellular calcium concentration, the binding affinity of calcium binding proteins and the maximum and minimum elastances have been identified as key parameters across all relevant scales. The modified parameters cause a change in (a) intracellular calcium concentration characterising cellular properties, such as calcium channel currents or the action potential, (b) the proteins being involved in the sliding filament mechanism and the proportion of the attached crossbridges at the protein level, as well as (c) the pressure and volume values at the organ level. This model allows to obtain insight and understanding of the effects of the treatment techniques, from a physiological and biological point of view.

Computational physiology and the Physiome Project

Bioengineering analyses of physiological systems use the computational solution of physical conservation laws on anatomically detailed geometric models to understand the physiological function of intact organs in terms of the properties and behaviour of the cells and tissues within the organ. By linking behaviour in a quantitative, mathematically defined sense across multiple scales of biological organization--from proteins to cells, tissues, organs and organ systems--these methods have the potential to link patient-specific knowledge at the two ends of these spatial scales. A genetic profile linked to cardiac ion channel mutations, for example, can be interpreted in relation to body surface ECG measurements via a mathematical model of the heart and torso, which includes the spatial distribution of cardiac ion channels throughout the myocardium and the individual kinetics for each of the approximately 50 types of ion channel, exchanger or pump known to be present in the heart. Similarly, linking molecular defects such as mutations of chloride ion channels in lung epithelial cells to the integrated function of the intact lung requires models that include the detailed anatomy of the lungs, the physics of air flow, blood flow and gas exchange, together with the large deformation mechanics of breathing. Organizing this large body of knowledge into a coherent framework for modelling requires the development of ontologies, markup languages for encoding models, and web-accessible distributed databases. In this article we review the state of the field at all the relevant levels, and the tools that are being developed to tackle such complexity. Integrative physiology is central to the interpretation of genomic and proteomic data, and is becoming a highly quantitative, computer-intensive discipline.

Cardiac energetics: sense and nonsense

1. The background to current ideas in cardiac energetics is outlined and, in the genomic era, the need is stressed for detailed knowledge of mouse heart mechanics and energetics. 2. The mouse heart is clearly different to the rat in terms of its excitation-contraction (EC) coupling and the common assumption that heart rate difference between mice and humans will account for the eightfold difference in myocardial oxygen consumption is wrong, because the energy per beat of the mouse heart is approximately one-third that of the human heart. 3. In vivo evidence suggests that there may well be an eightfold species difference in the non-beating metabolism of mice and human hearts. It is speculated that the magnitude of basal metabolism in the heart is regulatable and that, in the absence of perfusion, it falls to approximately one-quarter of its in vivo rate and that in clinical conditions, such as hibernation, it probably decreases; its magnitude may be controlled by the endothelium. 4. The active energy balance sheet is briefly discussed and it is suggested that the activation heat accounts for 20-25% of the active energy per beat and cross-bridge turnover accounts for the balance. It is argued that force, not shortening, is the major determinant of cardiac energy usage. 5. The outcome of recent cardiac modelling with variants of the Huxley and Hill/Eisenberg models is described. It has been necessary to invoke 'loose coupling' to replicate the low cardiac energy flux measured at low afterloads (medium to high velocities of shortening). 6. Lastly, some of the unexplained or 'nonsense' energetic data are outlined and eight unsolved problems in cardiac energetics are discussed.

A distribution-moment model of deactivation in cardiac muscle

The Distribution Moment (DM) model has simulated experimental data on skeletal muscle, but it has not been used previously to study the mechanics of active contraction in cardiac muscle. In contrast to previous models of striated muscle contraction, all parameters have physical meaning and assumptions concerning biophysical events within the cell are consistent with available data. In order to simulate cardiac muscle deactivation using the DM model it was necessary to make the cross-bridge detachment rates large for large displacements from the neutral equilibrium position of a cross-bridge. To examine the effect of cooperativity on cardiac muscle contraction, we used the DM model's tight coupling scheme with binding of one or two calcium sites regulating contraction. As observed experimentally, our model predicted a reduction of isometric tension development following rapid shortening lengthening transients when contraction is regulated by either one or two calcium binding sites. The predicted deactivating effect increased if the transient was applied late in the twitch when contraction is regulated by two calcium binding sites, but not when it is regulated by one site. This is the first study in which deactivation has been simulated without making any provisions for length-dependent calcium trononin dissociation.

Modelling the mechanical properties of cardiac muscle

A model of passive and active cardiac muscle mechanics is presented, suitable for use in continuum mechanics models of the whole heart. The model is based on an extensive review of experimental data from a variety of preparations (intact trabeculae, skinned fibres and myofibrils) and species (mainly rat and ferret) at temperatures from 20 to 27 degrees C. Experimental tests include isometric tension development, isotonic loading, quick-release/restretch, length step and sinusoidal perturbations. We show that all of these experiments can be interpreted with a four state variable model which includes (i) the passive elasticity of myocardial tissue, (ii) the rapid binding of Ca2+ to troponin C and its slower tension-dependent release, (iii) the kinetics of tropomyosin movement and availability of crossbridge binding sites and the length dependence of this process and (iv) the kinetics of crossbridge tension development under perturbations of myofilament length.

Modelling and simulation of motility in actomyosin systems

We present a model mechanism for simulating the diffusive motion and fluctuations inherent in myofibrillar sarcomere and its subunits at the molecular level. The model couples Langevin dynamics with Huxley kinetics to reproduce the transient patterns of momentum transfer, force generation and resulting motility due to the interactive activities of actin and myosin crossbridges. When myosin is detached from actin, our model predicts Brownian displacements centered at 0 +/- 8 nm (mean +/- SD, n = 265,308) and it is broadly distributed due to the Brownian noise. Attachment events produced displacements with step sizes of approximately 8 +/- 6 nm (mean +/- SD, n = 34,693), which is in agreement with some recent optical-tweezers transducer experimental results. The proposed model could form the basis for a complete qualitative and quantitative description of the evolving complex interactions of the molecular proteins--actin and myosin--in the overall framework of muscular contraction studies.

A re-examination of calcium activation in the Huxley cross-bridge model

This paper investigates mathematical relations between models of calcium activation kinetics and Huxley-type models of cross-bridge dynamics in muscle. It is found that different calcium-activation schemes lead to the same form of generalized Huxley rate equation with calcium activation. [formula: see text] if it is assumed that calcium-troponin interaction rates are fast compared to the rates of transition associated with force-generating cross-bridge states. Calcium affects cross-bridge dynamics by modifying the bonding rate f, but does not affect the number of interacting cross bridges a or the unbonding rate g; this occurs through the appearance in the equation of an activation factor, r, which is a pure function of sarcoplasmic free calcium concentration. In particular, it is shown that both the "tight-coupling" and "loose-coupling" calcium-activation schemes introduced by Zahalak and Ma [1] lead to the same rate equation with the same activation factor; the difference between them appears in the calcium mass-balance equation. While both of these activation models can be made to fit simple twitch and force-velocity data equally well, experimentally observed load-dependent shifts in the free calcium concentration are compatible with the right-coupling scheme, but not with loose coupling.

Theoretical treatment of striated muscle: Dynamic extension of four-state model

We constructed a muscle model, based on the model first proposed by Gray and Gonda [6,7), that simulates the twitch contraction of striated muscle. Their original model postulated four basic states in the contraction cycle and predicted the properties of steady state contraction in striated muscle. Using the relationship between steady state tension and calcium concentration, we described several rate constants as functions of calcium concentration and calculated the number of attached crossbridges at various calcium concentration values. The results for both skeletal and cardiac muscle were approximately consistent with those of X-ray studies. Assuming that rate constants change immediately with the phasic alteration of intracellular calcium concentration, we estimated the time course of crossbridge distribution during twitch contraction; these findings were also consistent with those of X-ray studies. We also simulated the effects of calcium concentration and sarcomere length on the magnitude of twitch tension. These simulations suggest that the major determinants of crossbridge distribution during twitch contraction are the time courses of calcium transients and the rate constants of crossbridge kinetics. Our findings suggest that the model used in this study provides a theoretical basis for interpreting the characteristics of cardiac muscle encountered in the clinical setting.

Regional fibre stress-fibre strain area as an estimate of regional blood flow and oxygen demand in the canine heart

1. In the present study the relation between regional left ventricular contractile work, regional myocardial blood flow and oxygen uptake was assessed during asynchronous electrical activation. 2. In analogy to the use of the pressure-volume area for the estimation of global oxygen demand, the fibre stress-fibre strain area, as assessed regionally, was used to estimate regional oxygen demand. The more often used relation between the pressure-sarcomere length area and regional oxygen demand was also assessed. 3. Experiments were performed in six anaesthetized dogs with open chests. Regional differences in mechanical work were generated by asynchronous electrical activation of the myocardial wall. The ventricles were paced from the right atrium, the left ventricular free wall, the left ventricular apex or the right ventricular outflow tract. Regional fibre strain was measured at the epicardial anterior left ventricular free wall with a two-dimensional video technique. 4. Regional fibre stress was estimated from left ventricular pressure, the ratio of left ventricular cavity volume to wall volume, and regional deformation. Total mechanical power (TMP) was calculated from the fibre stress-fibre strain area (SSA) and the duration of the cardiac cycle (tcycle) using the equation: TMP = SSA/tcycle. Regional myocardial blood flow was measured with radioactive microspheres. Regional oxygen uptake was estimated from regional myocardial blood flow values and arteriovenous differences in oxygen content. 5. During asynchronous electrical activation, total mechanical power, pressure-sarcomere length area, myocardial blood flow and oxygen uptake were significantly lower in early than in late activated regions (P < 0.05). 6. Within the experiments, the correlation between the pressure-sarcomere length area and regional oxygen uptake was not significantly lower than the one between total mechanical power (TMP) and regional oxygen uptake (VO2,reg). However, variability of this relation between the experiments was less for total mechanical power. Pooling all experimental data revealed: VO2,reg = k1 TMP+k2, with k1 = 4.94 +/- 0.31 mol J-1 k2 = 24.2 +/- 1.9 mmol m-3 s-1 (means +/- standard error of the estimate). 7. This relation is in quantitative agreement with previously reported relations between the pressure-volume area and global oxygen demand. The results indicate that asynchronous electrical activation causes a redistribution of mechanical work and oxygen demand and that regional total mechanical power is a better and more general estimate of regional oxygen demand than the regional pressure-sarcomere length area.

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.

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.

A model of stress relaxation in cross-bridge systems: effect of a series elastic element

Many experimental protocols employed in the study of muscle mechanics use tension transients as a probe of the magnitudes of the kinetic rates in the underlying cross-bridge dynamics. These transients could potentially be modified by the elastic elements that exist both within the fiber and at the points of attachment to the experimental apparatus. To better understand the magnitude of such modifications, we have used computer simulation to investigate the transients that would be expected for cross bridges acting on an actin filament attached to an elastic element. The original model of cross-bridge mechanics by A.F. Huxley was used (Prog. Biophys. 7: 255-318, 1957). After an isometric equilibrium is achieved, a tension transient is produced by changing the dissociation rate constant, g1, while holding the attachment rate constant, f1, fixed. This decreases the number of attached, force-producing cross bridges. We find that the tension transients are markedly slowed by the presence of even a few (> or = 2) nanometers of series elastic strain per half-sarcomere. Thus some rate constants inferred from mechanical transients (e.g., those induced by caged ligands) may underestimate the actual kinetic rates of the cross-bridge processes.

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.

Cooperative effects due to calcium binding by troponin and their consequences for contraction and relaxation of cardiac muscle under various conditions of mechanical loading

A mathematical model for the regulation of mechanical activity in cardiac muscle has been developed based on a three-element rheological model of this muscle. The contractile element has been modeled taking into account the results of extensive mechanical tests that involved the recording of length-force and force-velocity relations and muscle responses to short-time deformations during various phases of the contraction-relaxation cycle. The best agreement between the experimental and the mathematical modeling results was obtained when a postulate stating two types of cooperativity to regulate the calcium binding by troponin was introduced into the model. Cooperativity of the first type is due to the dependence of the affinity of troponin C for Ca2+ on the concentration of myosin crossbridges in the vicinity of a given troponin C. Cooperativity of the second type assumes an increase in the affinity of a given troponin C for Ca2+ when the latter is bound by molecules neighboring troponin.

Combining transmural left ventricular mechanics and energetics to predict oxygen demand

This study relates to our earlier study which predicts the transmural distribution as well as the global left ventricular (LV) function and oxygen demand, based on the LV structure, geometry and sarcomere function. Here, we test the predicted global oxygen demand against experimental data in anesthetized, open chest dogs under changing working conditions. The experimental oxygen demand was calculated from the arterio-venous difference in oxygen content times the measured coronary flow. LV load was manipulated by a combination of a pressurized chamber connected to the femoral artery, phenylephrine infusion and an adjustable arteriovenous shunt. The heart was paced in two present heart rates. The study demonstrates that the global predictions, based on the local distributed oxygen demand model, are comparable to those obtained by other methods of global metabolic predictions. However, unlike other global methods, the distributed model gives spatial information and predicts an endo/epi ratio of oxygen demand ranging between 1.05 to 1.14, depending on the loading conditions, which is comparable to available experimental data. For the experimental conditions studied here (stroke volume, heart rate, aortic pressure), the theoretical analysis shows that only the end diastolic volume is significantly correlated to the endo/epi ratio of the transmural oxygen demand.

A model of multicomponent cardiac fibre

In order to simulate the contraction of a cardiac myofibre, a multicomponent fibre model has been developed. This model is composed of a series of segments which are activated in succession. Each segment is represented by the Hill's three component model of the sarcomere. The contractile element behaviour is described by the Huxley's theory and the time dependence agrees with the activation factor proposed by Julian for skeletal muscle, and modified by Wong for cardiac muscle. The two elastic elements have non-linear exponential characteristics. The isometric contraction of the multicomponent fibre has been simulated by means of a computer program. The results show the tension generated by the fibre, the propagation of the contraction along the fibre and the different contribution of each segment depending on its position inside the fibre.

The dynamic twisting of the left ventricle: a computer study

A mathematical analysis which relates the dynamic twisting motion of the heart around its longitudinal axis to the mechanical function of the left ventricle (LV) is presented. The study thus extends our earlier model which relates the micro-scale sarcomere dynamics, the fibrous structure of the myocardium, and the electrical transmural activation wave to the global LV function. The analysis demonstrates that although the angular twisting motion of the heart moderates the sarcomere length (SL) and the strain rate distributions throughout the myocardium, the global characteristics of the LV function are almost independent of the twisting phenomenon. The endocardial sarcomeres are nevertheless subjected to higher strains and higher (negative) strain rates than the corresponding (positive) epicardial sarcomeres. Utilizing the sarcomere stress length area to predict oxygen demand, it is shown that the twisting motion of the heart produces the metabolic gradient across the LV wall. In spite of the moderating effect of the twist, a larger than normal gradient in oxygen demand is predicted for cases of concentric hypertrophy.

Left ventricular mechanics related to the local distribution of oxygen demand throughout the wall

The complex interactions between left ventricular mechanics and the oxygen demand is studied by relating the left ventricular transmural oxygen demand to the myocardial structural and dynamic characteristics. The study utilizes a recent model of left ventricular contraction, which is based on a nested shell spheroidal geometry, a fan-like fibrous structure, the twisting motion of the left ventricle over its long axis, a transmural electrical activation propagation and the basic laws of sarcomere dynamics. The local "axial" stress (in the direction of the fibers) and the instantaneous sarcomere length are used to calculate the spatial distribution of the intramural oxygen demand per beat Vo2(y), where y is the distance from the endocardium. The normalized local sarcomere stress-length area SLAn(y) is related linearly to Vo2(y) by: Vo2(y) = K1 X SLAn(y) + K2, where K1 and K2 are constants. The calculations show a transmural metabolic gradient which is characterized by higher values of Vo2(y) in the endocardial layers than in the epicardial layers. Shorter endocardial sarcomeres and the twisting motion of the left ventricle around the long axis decrease the metabolic gradient across the wall, while a slow transmural electrical propagation wave as well as a wider angle of distribution of the fan-like fiber architecture increases the transmural metabolic gradient. Integration of the local oxygen demand across the left ventricular wall yields global values in agreement with those based on Suga's pressure-volume area approach. The model thus provides a qualitative and quantitative tool to assess the relation of the local and global oxygen demand to the complex left ventricular structure, fiber mechanics, and the dynamics of contraction.

Continuum rheology of muscle contraction and its application to cardiac contractility

A set of constitutive equations is proposed to describe the mechanics of contraction of skeletal and heart muscle. Fiber tension is assumed to depend on the degree of chemical activation, the stretch ratio, and the rate of stretching of the fibers. The time rate of change of activation is governed by a differential equation. The proposed constitutive equations are used to model the time courses of isotonic and isometric twitches during contraction and relaxation phases of the muscle response to stimulation. Various contractility indices of the left ventricle are considered next by using the proposed constitutive equations. The present analysis introduces a new interpretation of the index of contractility (dP/dt)/P used in cardiac literature. It is shown that this index may not be related at all to the maximum speed of shortening and that it may be dependent on both preload and afterload. The development of pressure during isovolumetric contraction of the left ventricle is shown to be governed by a differential equation describing the time rate of change of tension during isometric contraction of myocardium fibers.

Constitutive equations of skeletal muscle based on cross-bridge mechanism

The statistical mechanics of cross-bridge action is considered in order to develop constitutive equations that express fiber tension as a function of degree of activation and time history of speed of contraction. The kinetic equation of A.F. Huxley (1) is generalized to apply to the partially activated state. The rate parameters of attachment and detachment, and cross-bridge compliance are assumed to be step functions of extension, x, with a finite number of discontinuities. This assumption enables integration of the kinetic equation and its moments with respect to x resulting in analytic equations from which x has been eliminated. When the constants in the rate parameters and the force function are chosen so that Hill's force-velocity relation and features of the transient kinetic and tension data can be fitted, the resulting cross-bridge mechanism is quite similar to the one proposed by Podolsky et al. (2). Because the derived constitutive equations simplify mathematical analysis, the influence of various cross-bridge parameters on the mechanical behavior of muscle fibers may be evaluated. For example (a) instantaneous elastic response (T0-T1) and the magnitude of rapid recovery (T2-T1) after a step length change can be adequately explained when the rate of attachment is assumed high for positive x. In that case T2 corresponds to the force generated by cross-bridges in the region of negative x. (b) Kinetic transients occur as a result of the jumps that exist in the distribution of attached cross-bridges during the isometric state. Because of the hyperbolic nature of the kinetic equation, these jumps propagate in the--x direction causing rapid changes in the speed of contraction. (c) When the number of actin sites available for attachment is assumed to depend on the degree of activation, computational results indicate that the speed of shortening is insensitive to the degree of activation at each relative load. (d) It is shown that during sinusoidal oscillation, the mean and second-order harmonics of the experimental force-time curve are strongly dependent on cross-bridge parameters. Therefore, significant information may be lost when the data is expanded into Fourier series and only the first term is considered.

A new approach to the human muscle model

Hill's (1938) two component muscle model is used as basis for digital computer simulation of human muscular contraction by means of an iterative process. The contractile (CC) and series elastic (SEC) components are lumped components of structures which produce and transmit torque to the external environment. The CC is described in angular terms along four dimensions as a series of non-planar torque-angle-angular velocity surfaces stacked on top of each other, each surface being appropriate to a given level of muscular activation. The SEC is described similarly along dimensions of torque, angular stretch, overall muscle angular displacement and activation. The iterative process introduces negligible error and allows the mechanical outcome of a variety of normal muscular contractions to be evaluated parsimoniously. The model allows analysis of many aspects of muscle behaviour as well as optimization studies. Definition of relevant relations should also allow reproduction and prediction of the outcome of contractions in individuals.

Mechanics of myocardial relaxation: application of a model to isometric and isotonic relaxation of rat myocardium

Using a simple model for cardiac muscle relaxation which takes into account muscle length, activation, elasticity and a rate constant for the decay of activation, we are able to use easily measured mechanical parameters to assess the state of the cardiac relaxing system. In isolated trabeculae carneae from the left ventricle of the rat, performing physiologically sequenced contractions, observations have been made (1) at varying preloads and afterloads, (2) with changes in temperature from 23 degrees to 33 degrees C, (3) with changes in bath Ca2+ concentration and (4) with the addition of isoproterenol. During isometric relaxation, the slope (SIM) of the curve relating maximum rate of decline of force (-dF/dtmax) to end-systolic muscle length is load-independent and sensitive to interventions which directly affect the cardiac relaxing system (e.g., temperature, isoproterenol); it is only slightly sensitive to bath calcium concentration. During isotonic relaxation, the maximum velocity of lengthening (+dL/dtmax) is in negative linear proportion to muscle shortening at a given preload, the slope (SIT) of the curve relating +dL/dtmax to end-systolic length is sensitive to the interventions which directly affect the cardiac relaxing system but insensitive to calcium-mediated inotropic interventions. The model provides a theoretical basis for the use of SIM and SIT as measures of the relaxation process.