[Interaction of Membrane and Calcium Oscillators in Cardiac Pacemaker Cells: Mathematical Modeling].

An integrative model of the calcium dynamics in cardiac pacemaker cells is developed taking into account a synergetic effect of the interaction between an outer membrane oscillator and an intracellular calcium oscillator ("membrane and Ca(2+)-clock"). The main feature of the model is a description of the stochastic dynamics of Ca2+ release units within the electron-conformational mechanism of the functioning of ryanodine-sensitive calcium channels. It is shown that interaction of two cellular oscillators provides a stable action potential generation in the cardiac pacemaker cells even in the case of the stochastic Ca2+ dynamics. We studied in detail the effect of ryanodine channels sensitivity to an increase in the intracellular calcium concentration in sarcoplasmic reticulum and in the dyadic space on the behavior of calcium-release system. A parametric analysis of the integrative model of pacemaker cells is performed.

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.

[Modeling of disturbances in electrical and mechanical function of cardiomyocytes under acute ischemia].

The effect of acute myocardial ischemia on the electrical and mechanical function of cardiomyocytes was studied in the framework of a mathematical model of a single cardiomyocyte. Acute ischemia consequences were simulated via a combination of two factors--a reduction of intracellular ATP concentration and an increase in extracellular potassium concentration, which affect the kinetics of ATP-sensitive potassium current and other potassium currents. In accord with experimental data, ischemic models produce action potential shortening and diastolic depolarization, which reduce contractile, ability of cardiomyocytes, Utilizing a 'difference-current integral' approach, we assessed quantitative contribution of ionic currents to changes in the action potential generation during ischemic injuries. It has been shown that an increase in the amplitude of inward rectifier potassium current I(K1) with increased extracellular potassium concentration has most essential contribution to the changes in the action potential duration under ischemia.

Minimum Information about a Cardiac Electrophysiology Experiment (MICEE): standardised reporting for model reproducibility, interoperability, and data sharing.

Cardiac experimental electrophysiology is in need of a well-defined Minimum Information Standard for recording, annotating, and reporting experimental data. As a step towards establishing this, we present a draft standard, called Minimum Information about a Cardiac Electrophysiology Experiment (MICEE). The ultimate goal is to develop a useful tool for cardiac electrophysiologists which facilitates and improves dissemination of the minimum information necessary for reproduction of cardiac electrophysiology research, allowing for easier comparison and utilisation of findings by others. It is hoped that this will enhance the integration of individual results into experimental, computational, and conceptual models. In its present form, this draft is intended for assessment and development by the research community. We invite the reader to join this effort, and, if deemed productive, implement the Minimum Information about a Cardiac Electrophysiology Experiment standard in their own work.

[Mathematical modelling in physiology].

The article illustrates the method of mathematical modelling in physiology as a unique tool to study physiological processes. A number of demonstrated examples appear as a result of long-term experience in mathematical modelling of electrical and mechanical phenomena in the heart muscle. These examples are presented here to show that the modelling provides insight into mechanisms underlying these phenomena and is capable to predict new ones that were previously unknown. While potentialities of the mathematical modelling are analyzed with regard to the myocardium, they are quite universal to deal with any physiological processes.

[Contribution of cooperative mechanisms of the thin filament activation to the myocardium contractile function. Assessment by a mathematical model].

We applied a mathematical model to compare the contributions to the myocardium mechanical activity of two a priori feasible variants of the cooperative effect of myosin cross-bridges on the calcium activation of sarcomere thin filaments. One of these variants implies that cross-bridge cooperative influence on the troponin C affinity to calcium is localized within the same functional cluster A7TmTn where the cross-bridge is attached. The second variant is based on the assumption that cross-bridges may affect the affinity of troponin C to calcium in other functional clusters A7TmTn as well (so that the nearer to the cross-bridge a cluster lies, the stronger the cross-bridge affects the affinity of CaTnC complex in this cluster). Each of these two variants and its contribution to the active mechanical performance of the heart muscle during the contraction-relaxation cycle were alternatively assessed in the model. It was found that only the second variant correctly simulates the mechanical activity of the muscle. Thus, the modeling suggests that just this variant of the cooperativity seems to be more feasible.

[Spatio-temporal heterogeneity of human left ventricle contractions in norm and under ischemic heart disease].

2D-ultrasonic images of the left ventricle are analyzed and spatio-temporal heterogeneity of regional wall motion is characterized quantitatively in norm and in ischemic heart disease. Negative correlation between regional heterogeneity and global ejection fraction is revealed. Regional heterogeneity is shown to increase significantly in a group of patients as compared to healthy people. Role of apex in ventricular pump function is specified. Diagnostic indicator of regional wall motion abnormality is suggested. The data allow one to challenge a new paradigm of morpho-functional organization of heart based on the functional heterogeneity of modules (segments) that compose ventricular wall and differ in biomechanical characteristics and timing of both electrical and mechanical activation.

[Mathematical models for the study of electromechanical and mechanoelectrical coupling in the myocardium].

Mathematical models have been developed to describe interactions of electrical, mechanical and chemical processes in cardiomyocytes. The models simulate wide range of experimental data on excitation-contraction coupling and, more importantly, on mechanoelectric feedback in heart muscle. The model results clearly show that mechano-dependence of intracellular calcium handling due to cooperative effects of contractile proteins activation plays a key role in cardiac mechanoelectric coupling. At the same time, mechanosensitive currents can also contribute to action potential responses to mechanical perturbations. Using this model to study the heterogeneous myocardium we have shown that temporal and functional electromechanical heterogeneity of coupled cardiomyocytes can essentially determine the myocardium contractility. Optimization of the electromechanical function of contractile system emerges from the fine coordination between the activation sequence of cardiomyocytes, their local electromechanical properties and the mechanical interaction during contraction.

[Mechanisms of electromechanical function disturbances in cardiomyocytes overloaded with calcium. The theoretical study].

Arrhythmias and mechanical disturbances are simulated in a mathematical model of cardiomyocyte electromechanical activity. The simulated pattern is similar to that observed for acute heart failure associated with calcium overloading of myocardium cells. Special attention was paid to the calcium overloading resulting from the reduced Na+,K+ pump activity. In the framework of the model, it was shown that mechanical factors could promote arrhythmia initiation when the pump activity reduced. Different approaches to electrical and mechanical function restoration during acute heart failure associated with calcium overloading were suggested and analyzed in the model.

[Modeling of mechanoelectric coupling in cardiomyocytes in norm and pathology].

We developed mathematical models of the electromechanical function of cardiomyocytes and the simplest mechanically heterogeneous myocardial systems, muscle duplexes. By means of these models we studied the contribution of mechanoelectric feedbacks to the contractile activity of the myocardium in norm and pathology. In particular, we simulated and clarified the effects of mechanical conditions on both the form and the duration of the action potential during contractions. From this standpoint different kinds of myocardium mechanical heterogeneity were analyzed. As we have established, the latter can play both a positive and a negative role, depending on the distribution of mechanical nonuniformity and the sequence of activation of heterogeneous myocardium system elements. By means of the same models, we studied the contribution of mechanical factors to the arrhythmogenicity in the case of the cardiomyocyte calcium overload caused by the attenuation of the sodium-potassium pump and outlined the ways for correcting the contractile function in these disturbances.

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.

Electron-conformational model of ryanodine receptor lattice dynamics.

We propose a simple, physically reasonable electron-conformational model for the ryanodine receptor (RyR) and, on that basis, present a theory to describe RyR lattice responses to L-type channel triggering as an induced non-equilibrium phase transition. Each RyR is modelled with a single open and a single closed (electronic) state only, described utilizing a s=12 pseudospin approach. In addition to the fast electronic degree of freedom, the RyR channel is characterized by a slow classical conformational coordinate, Q, which specifies the RyR channel calcium conductance and provides a multimodal continuum of possible RyR states. The cooperativity in the RyR lattice is assumed to be determined by inter-channel conformational coupling. Given a threshold sarcoplasmic reticulum (SR) calcium load, the RyR lattice fires due to a nucleation process with a step-by-step domino-like opening of a fraction of lattice channels, providing for a sufficient release to generate calcium sparks. The optimal mode of RyR lattice functioning during calcium-induced calcium release implies a fractional release with a robust termination due to a decrease in SR calcium load, accompanied by a respective change in effective conformational strain of the lattice. SR calcium overload is shown to result in excitation of RyR lattice auto-oscillations with spontaneous RyR channel opening and closure.

[Electromechanical heterogeneity of the myocardium]. / Elektromekhanicheskaia neodnorodnost' miokarda.

Herein we discuss modem data showing that ventricle's working myocardium is highly heterogeneous. Significant transmural differences in electrophysiological and biomechanical properties of cardiomyocytes are reviewed. The reviewed evidence of myocardial heterogeneity constitutes the basis for modem assessment of segmental kinetics of different regions in intact heart. We used muscle duplexes as condensed models of a heterogeneous myocardial system. Experimental data, presented here were obtained both in biological duplexes formed by isolated myocardial preparations and in mathematical models of muscle duplexes. We showed that specific functional heterogeneity of cardiomyocytes, related to their excitation sequence, allowed the myocardium to optimise its contractile function and smooth dispersion of repolarisation.

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.