The importance of mechanical conditions in the testing of excitation abnormalities in a population of electro-mechanical models of human ventricular cardiomyocytes

Background: Populations of in silico electrophysiological models of human cardiomyocytes represent natural variability in cell activity and are thoroughly calibrated and validated using experimental data from the human heart. The models have been shown to predict the effects of drugs and their pro-arrhythmic risks. However, excitation and contraction are known to be tightly coupled in the myocardium, with mechanical loads and stretching affecting both mechanics and excitation through mechanisms of mechano-calcium-electrical feedback. However, these couplings are not currently a focus of populations of cell models. Aim: We investigated the role of cardiomyocyte mechanical activity under different mechanical conditions in the generation, calibration, and validation of a population of electro-mechanical models of human cardiomyocytes. Methods: To generate a population, we assumed 11 input parameters of ionic currents and calcium dynamics in our recently developed TP + M model as varying within a wide range. A History matching algorithm was used to generate a non-implausible parameter space by calibrating the action potential and calcium transient biomarkers against experimental data and rejecting models with excitation abnormalities. The population was further calibrated using experimental data on human myocardial force characteristics and mechanical tests involving variations in preload and afterload. Models that passed the mechanical tests were validated with additional experimental data, including the effects of drugs with high or low pro-arrhythmic risk. Results: More than 10% of the models calibrated on electrophysiological data failed mechanical tests and were rejected from the population due to excitation abnormalities at reduced preload or afterload for cell contraction. The final population of accepted models yielded action potential, calcium transient, and force/shortening outputs consistent with experimental data. In agreement with experimental and clinical data, the models demonstrated a high frequency of excitation abnormalities in simulations of Dofetilide action on the ionic currents, in contrast to Verapamil. However, Verapamil showed a high frequency of failed contractions at high concentrations. Conclusion: Our results highlight the importance of considering mechanoelectric coupling in silico cardiomyocyte models. Mechanical tests allow a more thorough assessment of the effects of interventions on cardiac function, including drug testing.

In silico analysis of the contribution of cardiomyocyte-fibroblast electromechanical interaction to the arrhythmia

Although fibroblasts are about 5–10 times smaller than cardiomyocytes, their number in the ventricle is about twice that of cardiomyocytes. The high density of fibroblasts in myocardial tissue leads to a noticeable effect of their electromechanical interaction with cardiomyocytes on the electrical and mechanical functions of the latter. Our work focuses on the analysis of the mechanisms of spontaneous electrical and mechanical activity of the fibroblast-coupled cardiomyocyte during its calcium overload, which occurs in a variety of pathologies, including acute ischemia. For this study, we developed a mathematical model of the electromechanical interaction between cardiomyocyte and fibroblasts and used it to simulate the impact of overloading cardiomyocytes. In contrast to modeling only the electrical interaction between cardiomyocyte and fibroblasts, the following new features emerge in simulations with the model that accounts for both electrical and mechanical coupling and mechano-electrical feedback loops in the interacting cells. First, the activity of mechanosensitive ion channels in the coupled fibroblasts depolarizes their resting potential. Second, this additional depolarization increases the resting potential of the coupled myocyte, thus augmenting its susceptibility to triggered activity. The triggered activity associated with the cardiomyocyte calcium overload manifests itself in the model either as early afterdepolarizations or as extrasystoles, i.e., extra action potentials and extra contractions. Analysis of the model simulations showed that mechanics contribute significantly to the proarrhythmic effects in the cardiomyocyte overloaded with calcium and coupled with fibroblasts, and that mechano-electrical feedback loops in both the cardiomyocyte and fibroblasts play a key role in this phenomenon.

Changes in the Cardiotoxic Effects of Lead Intoxication in Rats Induced by Muscular Exercise

Exposure to lead is associated with an increased risk of cardiovascular diseases. Outbred white male rats were injected with lead acetate intraperitoneally three times a week and/or were forced to run at a speed of 25 m/min for 10 min 5 days a week. We performed noninvasive recording of arterial pressure, electrocardiogram and breathing parameters, and assessed some biochemical characteristics. Electrophoresis in polyacrylamide gel was used to determine the ratio of myosin heavy chains. An in vitro motility assay was employed to measure the sliding velocity of regulated thin filaments on myosin. Isolated multicellular preparations of the right ventricle myocardium were used to study contractility in isometric and physiological modes of contraction. Exercise under lead intoxication normalized the level of calcium and activity of the angiotensin-converting enzyme in the blood serum, normalized the isoelectric line voltage and T-wave amplitude on the electrocardiogram, increased the level of creatine kinase-MB and reduced the inspiratory rate. Additionally, the maximum sliding velocity and the myosin heavy chain ratio were partly normalized. The effect of exercise under lead intoxication on myocardial contractility was found to be variable. In toto, muscular loading was found to attenuate the effects of lead intoxication, as judged by the indicators of the cardiovascular system.

Cardioinotropic Effects in Subchronic Intoxication of Rats with Lead and/or Cadmium Oxide Nanoparticles

Subchronic intoxication was induced in outbred male rats by repeated intraperitoneal injections with lead oxide (PbO) and/or cadmium oxide (CdO) nanoparticles (NPs) 3 times a week during 6 weeks for the purpose of examining its effects on the contractile characteristics of isolated right ventricle trabeculae and papillary muscles in isometric and afterload contractions. Isolated and combined intoxication with these NPs was observed to reduce the mechanical work produced by both types of myocardial preparation. Using the in vitro motility assay, we showed that the sliding velocity of regulated thin filaments drops under both isolated and combined intoxication with CdO–NP and PbO–NP. These results correlate with a shift in the expression of myosin heavy chain (MHC) isoforms towards slowly cycling β–MHC. The type of CdO–NP + PbO–NP combined cardiotoxicity depends on the effect of the toxic impact, the extent of this effect, the ratio of toxicant doses, and the degree of stretching of cardiomyocytes and muscle type studied. Some indices of combined Pb–NP and CdO–NP cardiotoxicity and general toxicity (genotoxicity included) became fully or partly normalized if intoxication developed against background administration of a bioprotective complex.

Mathematical modelling of the mechano-electric coupling in the human cardiomyocyte electrically connected with fibroblasts

Cardiac fibroblasts are interspersed within mammalian cardiac tissue. Fibroblasts are mechanically passive; however, they may communicate electrically with cardiomyocytes via gap junctions and thus affect the electrical and mechanical activity of myocytes. Several in-silico studies at both cellular (0D) and ventricular (3D) levels analysed the effects of fibroblasts on the myocardial electrical function. However, none of them addressed possible effects of fibroblast-myocyte electrical coupling to cardiomyocyte mechanical activity. In this paper, we propose a mathematical model for studying both electrical and mechanical responses of the human cardiomyocyte to its electrotonic interaction with cardiac fibroblasts. Our simulations have revealed that electrotonic interaction with fibroblasts affects not only the mechanical activity of the cardiomyocyte, comprising either moderate or significant reduction of contractility, but also the mechano-calcium and mechano-electric feedback loops, and all these effects are enhanced as the number of coupled fibroblasts is increased. Obtained results suggest that moderate values of the myocyte-fibroblast gap junction conductance (less than 1 nS) can be attributed to physiological conditions, contrasting to the higher values (2 nS and higher) proper rather for pathological situations (e.g. for infarct and/or border zones), since all mechanical indexes falls down dramatically in the case of such high conductance.

Force-velocity characteristics of isolated myocardium preparations from rats exposed to subchronic intoxication with lead and cadmium acting separately or in combination

This investigation continues our study of the effects of Pb-Cd poisoning on the heart, extending the enquiry from isometric to auxotonic contractions, thereby examining the effect on the ability of myocardial tissues to perform mechanical work. Different shifts were revealed in myocardial force-velocity relations following subchronic exposure of rats to lead acetate and cadmium chloride acting separately, in combination, or in combination with a bioprotective complex (BPC). The experiments were conducted on isolated preparations of trabecules and papillary muscles of the right ventricle in physiological loading conditions and on isolated heart muscle contractile proteins examined by the in vitro motility assay. The results of the latter correlate with the shifts in the ratio of cardiac myosin isoforms. The amount of work performed by the myocardium was calculated on the basis of the tension-shortening loop area and was found to be similar in the preparations from all experimental groups. This fact presumably reflects adaptive capacity of the myocardial function even when contractility is damaged due to the metallic intoxication of a moderate severity. Some characteristics of rat myocardium altered by the impact of lead-cadmium intoxication became fully or partly normalized if intoxication developed against background administration of a bioprotective complex (BPC). Together with previously reported results obtained in the isometric mode of contractility, all these results strengthen the scientific foundations of risk assessment and risk management projects in the occupational and environmental conditions characterized by human exposure to lead and/or cadmium.

Changes in rat myocardium contractility under subchronic intoxication with lead and cadmium salts administered alone or in combination

Subchronic intoxications induced in male rats by repeated intraperitoneal injections of lead acetate and cadmium chloride, administered either alone or in combination, are shown to affect the biochemical, cytological and morphometric parameters of blood, liver, heart and kidneys. The single twitch parameters of myocardial trabecular and papillary muscle preparations were measured in the isometric regime to identify changes in the heterometric (length-force) and chronoinotropic (frequency-force) contractility regulation systems. Differences in the responses of these systems in trabecules and papillary muscles to the above intoxications are shown. A number of myocardium mechanical characteristics changing in rats under the effect of a combined lead-cadmium intoxication and increased proportion of α-myosin heavy chains were observed to normalize fully or partially if such intoxication was induced against background administration of a proposed bioprotective complex. Based on the experimental results and literature data, some assumptions are suggested concerning the mechanisms of the cardiotoxic effects produced by lead and cadmium.

Mechano-calcium and mechano-electric feedbacks in the human cardiomyocyte analyzed in a mathematical model

Experiments on animal hearts (rat, rabbit, guinea pig, etc.) have demonstrated that mechano-calcium feedback (MCF) and mechano-electric feedback (MEF) are very important for myocardial self-regulation because they adjust the cardiomyocyte contractile function to various mechanical loads and to mechanical interactions between heterogeneous myocardial segments in the ventricle walls. In in vitro experiments on these animals, MCF and MEF manifested themselves in several basic classical phenomena (e.g., load dependence, length dependence of isometric twitches, etc.), and in the respective responses of calcium transients and action potentials. However, it is extremely difficult to study simultaneously the electrical, calcium, and mechanical activities of the human heart muscle in vitro. Mathematical modeling is a useful tool for exploring these phenomena. We have developed a novel model to describe electromechanical coupling and mechano-electric feedbacks in the human cardiomyocyte. It combines the ‘ten Tusscher–Panfilov’ electrophysiological model of the human cardiomyocyte with our module of myocardium mechanical activity taken from the ‘Ekaterinburg–Oxford’ model and adjusted to human data. Using it, we simulated isometric and afterloaded twitches and effects of MCF and MEF on excitation–contraction coupling. MCF and MEF were found to affect significantly the duration of the calcium transient and action potential in the human cardiomyocyte model in response to both smaller afterloads as compared to bigger ones and various mechanical interventions applied during isometric and afterloaded twitches.

Further analysis of rat myocardium contractility changes associated with a subchronic lead intoxication

A moderate subchronic lead intoxication was observed in male rats after repeated intraperitoneal injections of lead acetate. Right ventricular trabeculae and papillary muscles were isolated for in vitro studying of the contraction-relaxation cycle under isotonic and physiological loading. The contractile function of the myocardium was also assessed by measuring the velocity of thin filament movement over myosin. Lead intoxication led in papillary muscles to a decrease in the maximal rate of isotonic shortening for all afterloads and a decrease in the thin filament sliding velocity. Papillary muscles from lead-exposed rats displayed marked changes in most of the main characteristics of afterload contraction-relaxation cycles, but in trabeculae these changes were less pronounced. The reported changes were attenuated to some extent in rats treated with a Ca-containing bioprotector. The amount of work produced by both types of heart muscle preparations was not changed by lead. Only in papillary muscles the load-dependent relaxation index was significantly increased in the lead-treated groups. Thus subchronic lead intoxication affects the peak rate of force development and relaxation properties of cardiac muscle contracting in isotonic/physiological regimes rather than the total amount of mechanical work, which may reflect adaptive changes in the myocardial function under decreased contractility.

Effects of subchronic lead intoxication of rats on the myocardium contractility

Outbred male rats were repeatedly injected IP with sub-lethal doses of lead acetate 3 times a week during 5 weeks. They developed an explicit, even if moderate, lead intoxication characterized by typical hematological and some other features. The next day after the last injection the heart of each animal was excised, and the trabecules and papillary muscles from the right ventricle were used for modeling in vitro isometric (with varying starting length of the preparation) regimes of the contraction-relaxation cycle with different preloads. Several well-established parameters of this model were found changed compared with the preparations taken from the hearts of healthy control rats. Background in vivo calcium treatment attenuated both systemic and cardiotoxic effects of lead to an extent. We show for the first time that subchronic intoxication with lead caused myocardial preparations in a wide range of lengths to respond by a decrease in the time and speed parameters of the isometric contraction while maintaining its amplitude and by a decrease in the passive stiffness of trabecules. The responses of the various heart structures are outlined, and the isomyosin ratio is shown to have shifted towards the slow isoform. Mechanistic and toxicological inferences from the results are discussed.

Mechano-electric heterogeneity of the myocardium as a paradigm of its function

Myocardial heterogeneity is well appreciated and widely documented, from sub-cellular to organ levels. This paper reviews significant achievements of the group, led by Professor Vladimir S. Markhasin, Russia, who was one of the pioneers in studying and interpreting the relevance of cardiac functional heterogeneity.

Mechano-electric feedback in one-dimensional model of myocardium

We utilized our earlier developed 1D mathematical model of the heart muscle strand to study contribution of the bilateral interactions between excitation and contraction on the cellular and tissue levels to the local and global myocardium function. Numerical experiments on the model showed that an initially uniform strand, formed on the inherently identical cells, became functionally heterogeneous due to the asynchronous excitation via the electrical wave spread. Mechanical interactions between the cells and the mechano-electric feedback beat-to-beat affect the functional characteristics of coupled cardiomyocytes further, adjusting their electrical and mechanical heterogeneity to the activation timing. Model simulations showed that functional heterogeneity increases with an enlarged spatial extension of the myocardial strand (in terms of the longer slack length not a higher stretch of the strand), demonstrating a special role of the heart size in its function. Model analysis suggests that cooperative mechanisms of myofilament calcium activation contribute essentially to the generation of cellular functional heterogeneity in contracting cardiac tissue.

The cardiac muscle duplex as a method to study myocardial heterogeneity

This paper reviews the development and application of paired muscle preparations, called duplex, for the investigation of mechanisms and consequences of intra-myocardial electro-mechanical heterogeneity. We illustrate the utility of the underlying combined experimental and computational approach for conceptual development and integration of basic science insight with clinically relevant settings, using previously published and new data. Directions for further study are identified.

Electrical wave propagation in an anisotropic model of the left ventricle based on analytical description of cardiac architecture

We develop a numerical approach based on our recent analytical model of fiber structure in the left ventricle of the human heart. A special curvilinear coordinate system is proposed to analytically include realistic ventricular shape and myofiber directions. With this anatomical model, electrophysiological simulations can be performed on a rectangular coordinate grid. We apply our method to study the effect of fiber rotation and electrical anisotropy of cardiac tissue (i.e., the ratio of the conductivity coefficients along and across the myocardial fibers) on wave propagation using the ten Tusscher–Panfilov (2006) ionic model for human ventricular cells. We show that fiber rotation increases the speed of cardiac activation and attenuates the effects of anisotropy. Our results show that the fiber rotation in the heart is an important factor underlying cardiac excitation. We also study scroll wave dynamics in our model and show the drift of a scroll wave filament whose velocity depends non-monotonically on the fiber rotation angle; the period of scroll wave rotation decreases with an increase of the fiber rotation angle; an increase in anisotropy may cause the breakup of a scroll wave, similar to the mother rotor mechanism of ventricular fibrillation.

Mathematical model of the anatomy and fibre orientation field of the left ventricle of the heart

Background

One of the main factors affecting propagation of electrical waves and contraction in ventricles of the heart is anisotropy of cardiac tissue. Anisotropy is determined by orientation of myocardial fibres. Determining fibre orientation field and shape of the heart is important for anatomically accurate modelling of electrical and mechanical function of the heart. The aim of this paper is to introduce a theoretical rule-based model for anatomy and fibre orientation of the left ventricle (LV) of the heart and to compare it with experimental data. We suggest explicit analytical formulae that allow us to obtain the left ventricle form and its fibre direction field. The ventricle band concept of cardiac architecture given by Torrent-Guasp is chosen as the model postulate.

Methods

In our approach, anisotropy of the heart is derived from some general principles. The LV is considered as a set of identical spiral surfaces, each of which can be produced from the other by rotation around one vertical axis. Each spiral surface is filled with non-intersecting curves which represent myocardial fibres.

For model verification, we use experimental data on fibre orientation in human and canine hearts.

Results

LV shape and anisotropy are represented by explicit analytical expressions in a curvilinear 3-D coordinate system. The derived fibre orientation field shows good qualitative agreement with experimental data. The model reveals the most thorough quantitative simulation of fibre angles at the LV middle zone.

Conclusions

Our analysis shows that the band concept can generate realistic anisotropy of the LV. Our model shows good qualitative agreement between the simulated fibre orientation field and the experimental data on LV anisotropy, and the model can be used for various numerical simulations to study the effects of anisotropy on cardiac excitation and mechanical function.

Contribution of mechanical factors to arrhythmogenesis in calcium overloaded cardiomyocytes: model predictions and experiments

It is well-known that Ca²⁺ overload in cardiomyocytes may underlie arrhythmias. However, the possible contribution of mechanical factors to rhythm disturbances in Ca²⁺ overloaded myocytes has not been sufficiently investigated. We used a mathematical model of the electrical and mechanical activity of cardiomyocytes to reveal an essential role of the mechanisms of cardiac mechano-electric feedback in arrhythmogenesis in Ca²⁺ overloaded myocardium. In the model, the following mechanical factors increased Ca²⁺ overload in contracting cardiomyocytes and promoted rhythm disturbances: i) a decrease in the mechanical load for afterloaded contractions; and ii) a decrease in the initial length of sarcomeres for isometric twitches. In exact accordance with the model predictions, in experiments on papillary muscles from the right ventricle of guinea pigs with Ca²⁺ overloaded cardiomyocytes (using 0.5-1 μM of ouabain), we found that emergence of rhythm disturbances and extrasystoles depends on the mechanical conditions of muscle contraction.

Role of myocardial viscoelasticity in disturbances of electrical and mechanical activity in calcium overloaded cardiomyocytes: mathematical modeling

Cardiomyocyte Ca(2+) overload is closely linked to cardiac arrhythmias. We have earlier shown in a mathematical model that myocardium mechanical activity may contribute to rhythm disturbances induced by Ca(2+) overload in cardiomyocytes with reduced Na(+)-K(+) pump work (Sulman et al., 2008). The same model is used here to address possible contribution of the passive mechanical properties of cardiac muscle (i.e. myocardial viscous and elastic properties) to the arrhythmogenesis. In a series of contractions at regular pacing rate of 75 beats/min a model with higher viscosity demonstrated essentially earlier appearance of extrasystoles due to a faster cardiomyocyte Ca(2+) loading up to a level triggering spontaneous Ca(2+) releases from the sarcoplasmic reticulum. The model predicts that myocardial elasticity also may affect arrhythmogenesis in cardiomyocytes overloaded with Ca(2+). Contribution of the mechanical properties of the myocardial tissue to the arrhythmia has been analyzed for wide ranges of both viscosity and elasticity coefficients. The results suggest that myocardial viscoelastic properties may be a factor affecting Ca(2+) handling in cardiomyocytes and contributing to cardiac mechano-electric feedback in arrhythmogenesis.

Effects of cardiac myosin binding protein-C on the regulation of interaction of cardiac myosin with thin filament in an in vitro motility assay

Modulatory role of whole cardiac myosin binding protein-C (сMyBP-C) in regulation of cardiac muscle contractility was studied in the in vitro motility assay with rabbit cardiac myosin as a motor protein. The effects of cMyBP-C on the interaction of cardiac myosin with regulated thin filament were tested in both in vitro motility and ATPase assays. We demonstrate that the addition of cMyBP-C increases calcium regulated Mg-ATPase activity of cardiac myosin at submaximal calcium. The Hill coefficient for 'pCa-velocity' relation in the in vitro motility assay decreased and the calcium sensitivity increased when сMyBP-C was added. Results of our experiments testifies in favor of the hypothesis that сMyBP-C slows down cross-bridge kinetics when binding to actin.

Study of the interaction between rabbit cardiac contractile and regulatory proteins. An in vitro motility assay

A series of experiments was performed in an in vitro motility assay with reconstructed thin filaments to obtain pCa-force relationships for cardiac isomyosins V1 and V3. Two concentrations of each isomyosin (200 and 300 microg/ml) on the surface of a flow cell were tested. Isometric force was estimated as the amount of actin-binding protein, alpha-actinin, stopping thin filament movement. It was found that the amount of alpha-actinin stopping the movement at saturating calcium concentration for V3 was twice higher than for V1 at both concentrations of isoforms. Hill coefficients of cooperativity (h) were determined for pCa-force relationships. The value of h did not differ significantly for isoforms at 300 microg/ml of protein (h was 1.56 for V1 and 1.54 for V3). However, the Hill coefficient was higher for V3 isoform at 200 microg/ml (h = 2.00 and 1.76 for V3 and V1, respectively). Importantly, the Hill coefficient increased for both isoenzymes when their concentrations were decreased. The connection between Hill coefficient and cooperative interactions between cardiac contractile and regulatory proteins is analyzed in detail.

Mathematical modeling of mechanically modulated rhythm disturbances in homogeneous and heterogeneous myocardium with attenuated activity of na+ -k+ pump

A mathematical model of the cardiomyocyte electromechanical function is used to study contribution of mechanical factors to rhythm disturbances in the case of the cardiomyocyte calcium overload. Particular attention is paid to the overload caused by diminished activity of the sodium-potassium pump. It is shown in the framework of the model, where mechano-calcium feedback is accounted for that myocardium mechanics may significantly enhance arrhythmogenicity of the calcium overload. Specifically, a role of cross-bridge attachment/detachment processes, a role of mechanical conditions of myocardium contractions (length, load), and a role of myocardium viscosity in the case of simulated calcium overload have been revealed. Underlying mechanisms are analyzed. Several approaches are designed in the model and compared to each other for recovery of the valid myocardium electrical and mechanical performance in the case of the partially suppressed sodium-potassium pump.

Activation sequence as a key factor in spatio-temporal optimization of myocardial function

Using one-dimensional models of myocardial tissue, implemented as chains of virtual ventricular muscle segments that are kinematically connected in series, we studied the role of the excitation sequence in spatio-temporal organization of cardiac function. Each model element was represented by a well-verified mathematical model of cardiac electro-mechanical activity. We found that homogeneous chains, consisting of identical elements, respond to non-simultaneous stimulation by generation of complex spatio-temporal heterogeneities in element deformation. These are accompanied by the establishment of marked gradients in local electro-mechanical properties of the elements (heterogeneity in action potential duration, Ca2+ transient characteristics and sarcoplasmic reticulum Ca2+ loading). In heterogeneous chains, composed of elements simulating fast and slow contracting cardiomyocytes from different transmural layers, we found that only activation sequences where stimulation of the slower elements preceded that of faster ones gave rise to optimization of the system's electro-mechanical function, which was confirmed experimentally. Based on the results obtained, we hypothesize that the sequence of activation of cardiomyocytes in different ventricular layers is one of the key factors of spatio-temporal organization of myocardium. Moreover, activation sequence and regional differences in intrinsic electro-mechanical properties of cardiac muscle must be matched in order to optimize myocardial function.

Hybrid duplex: a novel method to study the contractile function of heterogeneous myocardium

In an earlier study, we experimentally mimicked the effects of mechanical interaction between different regions of the ventricular wall by allowing pairs of independently maintained cardiac muscle fibers to interact mechanically in series or in parallel. This simple physiological model of heterogeneous myocardium, which has been termed “duplex,” has provided new insight into basic effects of cardiac electromechanical heterogeneity. Here, we present a novel “hybrid duplex,” where one of the elements is an isolated cardiac muscle and the other a “virtual cardiac muscle.” The virtual muscle is represented by a computational model of cardiomyocyte electromechanical activity. We present in detail the computer-based digital control system that governs the mechanical interaction between virtual and biological muscle, the software used for data analysis, and working implementations of the model. Advantages of the hybrid duplex method are discussed, and experimental recordings are presented for illustration and as proof of the principle.

Influence of viscosity on myocardium mechanical activity: a mathematical model

We have previously proposed and validated a mathematical model of myocardium contraction-relaxation cycle based on current knowledge of regulatory role of Ca2+ and cross-bridge kinetics in cardiac cell. That model did not include viscous elements. Here we propose a modification of the model, in which two viscous elements are added, one in parallel to the contractile element, and one more in parallel to the series elastic element. The modified model allowed us to simulate and explain some subtle experimental data on relaxation velocity in isotonic twitches and on a mismatch between the time course of sarcomere shortening/lengthening and the time course of active force generation in isometric twitches. Model results were compared with experimental data obtained from 28 rat LV papillary muscles contracting and relaxing against various loads. Additional model analysis suggested contribution of viscosity to main inotropic and lusitropic characteristics of myocardium performance.

Mechano-electric interactions in heterogeneous myocardium: development of fundamental experimental and theoretical models

The heart is structurally and functionally a highly non-homogenous organ, yet its main function as a pump can only be achieved by the co-ordinated contraction of millions of ventricular cells. This apparent contradiction gives rise to the hypothesis that 'well-organised' inhomogeneity may be a pre-requisite for normal cardiac function. Here, we present a set of novel experimental and theoretical tools for the study of this concept. Heterogeneity, in its most condensed form, can be simulated using two individually controlled, mechanically interacting elements (duplex). We have developed and characterised three different types of duplexes: (i) biological duplex, consisting of two individually perfused biological samples (like thin papillary muscles or a trabeculae), (ii) virtual duplex, made-up of two interacting mathematical models of cardiac muscle, and (iii) hybrid duplex, containing a biological sample that interacts in real-time with a virtual muscle. In all three duplex types, in-series or in-parallel mechanical interaction of elements can be studied during externally isotonic, externally isometric, and auxotonic modes of contraction and relaxation. Duplex models, therefore, mimic (patho-)physiological mechano-electric interactions in heterogeneous myocardium at the multicellular level, and in an environment that allows one to control mechanical, electrical and pharmacological parameters. Results obtained using the duplex method show that: (i) contractile elements in heterogeneous myocardium are not 'independent' generators of tension/shortening, as their ino- and lusitropic characteristics change dynamically during mechanical interaction-potentially matching microscopic contractility to macroscopic demand, (ii) mechanical heterogeneity contributes differently to action potential duration (APD) changes, depending on whether mechanical coupling of elements is in-parallel or in-series, which may play a role in mechanical tuning of distant tissue regions, (iii) electro-mechanical activity of mechanically interacting contractile elements is affected by their activation sequence, which may optimise myocardial performance by smoothing intrinsic differences in APD. In conclusion, we present a novel set of tools for the experimental and theoretical investigation of cardiac mechano-electric interactions in healthy and/or diseased heterogeneous myocardium, which allows for the testing of previously inaccessible concepts.

Effects of mechanical interaction between two rabbit cardiac muscles connected in parallel

The hypothesis that myocardium mechanical inhomogeneity produces a substantial effect on mechanical function was tested. Muscle inhomogeneity was studied in isolated papillary muscles or trabeculae excised from rabbit right ventricle and connected in a parallel duplex. Each muscle was placed in a separate perfusion bath. One end of each muscle was fastened to an individual force transducer and the other to the common lever of a servomotor. This arrangement allowed both muscles, being excited independently, to pull jointly a load applied to the lever. Separate electrodes for each perfusion bath allowed to stimulate muscles with a time delay. Tension developed in the individual muscles and their interaction were studied. Developed tension was critically dependent on the timing and sequence of excitation. Using mathematical modeling, patterns of tension distribution experimentally observed in parallel duplexes were simulated. These results suggest that changes both in Ca(2+) transients and in the time course of Ca(2+)-troponin complexion due to the duplexed muscles interaction offset the effect of mechanical inhomogeneity.

Mathematical modeling of the effect of the sarcoplasmic reticulum calcium pump function on load dependent myocardial relaxation

Earlier, we developed a mathematical model of myocardial contraction-relaxation cycle regulation. A great number of mechanical experiments was simulated in the model, the phenomenon of load dependent relaxation (LDR) included. In the present work we used the same model to analyze experimental data revealing that high temperature leads to reduction of LDR. We simulated three main factors arising due to high temperature, which a priori may cause LDR reduction: increasing the cross-bridges cycling rate, decreasing the duration of the Ca transient ascending limb, and increasing Ca pumping rate. Indeed, these factors together result in LDR reduction; i.e., the model correctly simulates the effect of high temperature on LDR in general. At the same time, the sensitivity of LDR to the third factor is much higher than to the first and the second ones; i.e., increasing the rate of Ca pumping is sufficient to induce the observed effect in the framework of the model. This seems to contrast with the result of our previous study dealing with the simulation of LDR disappearance due to increasing Ca pumping rate as it happens during relatively severe cardiac hypertrophy. However, the model analysis shows that the specific mechanism underlying the change in Ca pumping rate in either case is extremely important for the effect on LDR. Particularly, the model predicts that LDR will reduce if this rate increases due to enhanced ATP hydrolysis rate by the Ca pump; and vice versa, if this rate increases due to decreasing retroinhibition of the pump ATPase, it may result in LDR increase. Probably, but the first mechanism is operational due to high temperature and makes LDR to reduce, whereas slowing down Ca pumping due to increasing retroinhibition results in LDR disappearance during severe cardiac hypertrophy.

Mathematical modelling of the contribution of mechanical inhomogeneity in the myocardium to contractile function

Earlier we developed a mathematical model of the cardiac muscle that allowed for inactivation through the effects of cooperativity of contractile proteins. In the present work we used the model to analyze the mechanical function of an inhomogeneous myocardium. To simulate the latter we chose, as the simplest sytstem, a duplex in which muscles with different mechanical properties were connected in series and in parallel. Numerical experiments showed that the basic effect due to the inhomogeneity consists in the non-additivity of the mechanical characteristics of the muscle, e.g., of the relationship between end-systolic length and end-systolic force (Les - Pes). As a rule, non-additivity consists in a negative inotropic effect. The analysis showed that the cause of non-additivity is redistribution of loads between muscles (in a parallel duplex), redistribution of lengths (in a serial duplex), changes in the rate of contraction of each muscle compared to contraction that when working separately, shifts in time to Les. Also, the model predicts that additional inactivation of contractile proteins in a muscle within a duplex against isolation is the substantial mechanism of enhanced non-additivity. Among the factors of inhomogeneity studied the basic determinants are difference in amplitudes between isometric tensions developed by each muscle in isolation and the asynchronism in the development of these tensions.

Mathematical modeling of relations between the kinetics of free intracellular calcium and mechanical function of myocardium

The paper describes a mathematical model of cardiac muscle contraction based on the assumption of two types of co-operativity which control the formation of calcium-troponin complexes and on a simplified scheme of free intracellular calcium kinetics. Calcium transients are shown to be different in isotonic and isometric conditions, being dependent on initial muscle length as well. Numerical experiments and analysis of the model suggest that calcium uptake by the sarcoplasmic reticulum slows down with an increase in the intracellular concentration of free calcium. This suggestion enables the model to explain the disappearance of load-dependent relaxation observed experimentally at cardiac hypertrophy.

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.