A mathematical model of the mouse ventricular myocyte contraction

A compartmentalized mathematical model of the β1- and β2-adrenergic signaling systems in ventricular myocytes from mouse in heart failure

Mouse models of heart failure are extensively used to research human cardiovascular diseases. In particular, one of the most common is the mouse model of heart failure resulting from transverse aortic constriction (TAC). Despite this, there are no comprehensive compartmentalized mathematical models that describe the complex behavior of the action potential, [Ca²⁺]_i transients, and their regulation by β₁- and β₂-adrenergic signaling systems in failing mouse myocytes. In this paper, we develop a novel compartmentalized mathematical model of failing mouse ventricular myocytes after TAC procedure. The model describes well the cell geometry, action potentials, [Ca²⁺]_i transients, and β₁- and β₂-adrenergic signaling in the failing cells. Simulation results obtained with the failing cell model are compared with those from the normal ventricular myocytes. Exploration of the model reveals the sarcoplasmic reticulum Ca²⁺ load mechanisms in failing ventricular myocytes. We also show a larger susceptibility of the failing myocytes to early and delayed afterdepolarizations and to a proarrhythmic behavior of Ca²⁺ dynamics upon stimulation with isoproterenol. The mechanisms of the proarrhythmic behavior suppression are investigated and sensitivity analysis is performed. The developed model can explain the existing experimental data on failing mouse ventricular myocytes and make experimentally testable predictions of a failing myocyte's behavior.

A new model of myofibroblast-cardiomyocyte interactions and their differences across species

Although coupling between cardiomyocytes and myofibroblasts is well known to affect the physiology and pathophysiology of cardiac tissues across species, relating these observations to humans is challenging because the effect of this coupling varies across species and because the sources of these effects are not known. To identify the sources of cross-species variation, we built upon previous mathematical models of myofibroblast electrophysiology and developed a mechanoelectrical model of cardiomyocyte-myofibroblast interactions as mediated by electrotonic coupling and transforming growth factor-β1. The model, as verified by experimental data from the literature, predicted that both electrotonic coupling and transforming growth factor-β1 interaction between myocytes and myofibroblast prolonged action potential in rat myocytes but shortened action potential in human myocytes. This variance could be explained by differences in the transient outward K⁺ current associated with differential Kv4.2 gene expression across species. Results are useful for efforts to extrapolate the results of animal models to the predicted effects in humans and point to potential therapeutic targets for fibrotic cardiomyopathy.

The ForceLAB simulator: Application to the comparison of current models of cardiomyocyte contraction

Mathematical models are useful tools in the study of physiological phenomena. However, due to differences in assumptions and formulations, discrepancy in simulations may occur. Among the models for cardiomyocyte contraction based on Huxley's cross-bridge cycling, those proposed by Negroni and Lascano (NL) and Rice et al. (RWH) are the most frequently used. This study was aimed at developing a computational tool, ForceLAB, which allows implementing different contraction models and modifying several functional parameters. As an application, electrically-stimulated twitches triggered by an equal Ca²⁺ input and steady-state force x pCa relationship (pCa = -log of the molar free Ca²⁺ concentration) simulated with the NL and RWH models were compared. The equilibrium Ca²⁺-troponin C (TnC) dissociation constant (K_d) was modified by changing either the association (k_on) or the dissociation (k_off) rate constant. With the NL model, raising K_d by either maneuver decreased monotonically twitch amplitude and duration, as expected. With the RWH model, in contrast, the same K_d variation caused increase or decrease of peak force depending on which rate constant was modified. Additionally, force x pCa curves simulated using Ca²⁺ binding constants estimated in cardiomyocytes bearing wild-type and mutated TnC were compared to curves previously determined in permeabilized fibers. Mutations increased k_on and k_off, and decreased K_d. Both models produced curves fairly comparable to the experimental ones, although sensitivity to Ca²⁺ was greater, especially with RWH model. The NL model reproduced slightly better the qualitative changes associated with the mutations. It is expected that this tool can be useful for teaching and investigation.

Chronic Sympathetic Hyperactivity Triggers Electrophysiological Remodeling and Disrupts Excitation-Contraction Coupling in Heart

The sympathetic nervous system is essential for maintenance of cardiac function via activation of post-junctional adrenergic receptors. Prolonged adrenergic receptor activation, however, has deleterious long-term effects leading to hypertrophy and the development of heart failure. Here we investigate the effect of chronic adrenergic receptors activation on excitation-contraction coupling (ECC) in ventricular cardiomyocytes from a previously characterized mouse model of chronic sympathetic hyperactivity, which are genetically deficient in the adrenoceptor α2A and α2C genes (ARDKO). When compared to wild-type (WT) cardiomyocytes, ARDKO displayed reduced fractional shortening (~33%) and slower relaxation (~20%). Furthermore, ARDKO cells exhibited several electrophysiological changes such as action potential (AP) prolongation (~50%), reduced L-type calcium channel (LCC) current (~33%), reduced outward potassium (K+) currents (~30%), and increased sodium/calcium exchanger (NCX) activity (~52%). Consistent with reduced contractility and calcium (Ca2+) currents, the cytosolic Ca2+ ([Ca2+]i) transient from ARDKO animals was smaller and decayed slower. Importantly, no changes were observed in membrane resting potential, AP amplitude, or the inward K+ current. Finally, we modified our existing cardiac ECC computational model to account for changes in the ARDKO heart. Simulations suggest that cellular changes in the ARDKO heart resulted in variable and dyssynchronous Ca2+-induced Ca2+ release therefore altering [Ca2+]i transient dynamics and reducing force generation. In conclusion, chronic sympathetic hyperactivity impairs ECC by changing the density of several ionic currents (and thus AP repolarization) causing altered Ca2+ dynamics and contractile activity. This demonstrates the important role of ECC remodeling in the cardiac dysfunction secondary to chronic sympathetic activity.

The Effects of Mechanical Preload on Transmural Differences in Mechano-Calcium-Electric Feedback in Single Cardiomyocytes: Experiments and Mathematical Models

Transmural differences in ventricular myocardium are maintained by electromechanical coupling and mechano-calcium/mechano-electric feedback. In the present study, we experimentally investigated the influence of preload on the force characteristics of subendocardial (Endo) and subepicardial (Epi) single ventricular cardiomyocytes stretched by up to 20% from slack sarcomere length (SL) and analyzed the results with the help of mathematical modeling. Mathematical models of Endo and Epi cells, which accounted for regional heterogeneity in ionic currents, Ca²⁺ handling, and myofilament contractile mechanisms, showed that a greater slope of the active tension–length relationship observed experimentally in Endo cardiomyocytes could be explained by greater length-dependent Ca²⁺ activation in Endo cells compared with Epi ones. The models also predicted that greater length dependence of Ca²⁺ activation in Endo cells compared to Epi ones underlies, via mechano-calcium-electric feedback, the reduction in the transmural gradient in action potential duration (APD) at a higher preload. However, the models were unable to reproduce the experimental data on a decrease of the transmural gradient in the time to peak contraction between Endo and Epi cells at longer end-diastolic SL. We hypothesize that preload-dependent changes in viscosity should be involved alongside the Frank–Starling effects to regulate the transmural gradient in length-dependent changes in the time course of contraction of Endo and Epi cardiomyocytes. Our experimental data and their analysis based on mathematical modeling give reason to believe that mechano-calcium-electric feedback plays a critical role in the modulation of electrophysiological and contractile properties of myocytes across the ventricular wall.

Mathematical model for β1-adrenergic regulation of the mouse ventricular myocyte contraction

The β1-adrenergic regulation of cardiac myocyte contraction plays an important role in regulating heart function. Activation of this system leads to an increased heart rate and stronger myocyte contraction. However, chronic stimulation of the β1-adrenergic signaling system can lead to cardiac hypertrophy and heart failure. To understand the mechanisms of action of β1-adrenoceptors, a mathematical model of cardiac myocyte contraction that includes the β1-adrenergic system was developed and studied. The model was able to simulate major experimental protocols for measurements of steady-state force-calcium relationships, cross-bridge release rate and force development rate, force-velocity relationship, and force redevelopment rate. It also reproduced quite well frequency and isoproterenol dependencies for intracellular Ca2+ concentration ([Ca2+]i) transients, total contraction force, and sarcomere shortening. The mathematical model suggested the mechanisms of increased contraction force and myocyte shortening on stimulation of β1-adrenergic receptors is due to phosphorylation of troponin I and myosin-binding protein C and increased [Ca2+]i transient resulting from activation of the β1-adrenergic signaling system. The model was used to simulate work-loop contractions and estimate the power during the cardiac cycle as well as the effects of 4-aminopyridine and tedisamil on the myocyte contraction. The developed mathematical model can be used further for simulations of contraction of ventricular myocytes from genetically modified mice and myocytes from mice with chronic cardiac diseases.

NEW & NOTEWORTHY A new mathematical model of mouse ventricular myocyte contraction that includes the β1-adrenergic system was developed. The model simulated major experimental protocols for myocyte contraction and predicted the effects of 4-aminopyridine and tedisamil on the myocyte contraction. The model also allowed for simulations of work-loop contractions and estimation of the power during the cardiac cycle.

Transmural cellular heterogeneity in myocardial electromechanics

Myocardial heterogeneity is an attribute of the normal heart. We have developed integrative models of cardiomyocytes from the subendocardial (ENDO) and subepicardial (EPI) ventricular regions that take into account experimental data on specific regional features of intracellular electromechanical coupling in the guinea pig heart. The models adequately simulate experimental data on the differences in the action potential and contraction between the ENDO and EPI cells. The modeling results predict that heterogeneity in the parameters of calcium handling and myofilament mechanics in isolated ENDO and EPI cardiomyocytes are essential to produce the differences in Ca2+ transients and contraction profiles via cooperative mechanisms of mechano-calcium-electric feedback and may further slightly modulate transmural differences in the electrical properties between the cells. Simulation results predict that ENDO cells have greater sensitivity to changes in the mechanical load than EPI cells. These data are important for understanding the behavior of cardiomyocytes in the intact heart.

A finite element model of myocardial infarction using a composite material approach

Computational models are effective tools to study cardiac mechanics under normal and pathological conditions. They can be used to gain insight into the physiology of the heart under these conditions while they are adaptable to computer assisted patient-specific clinical diagnosis and therapeutic procedures. Realistic cardiac mechanics models incorporate tissue active/passive response in conjunction with hyperelasticity and anisotropy. Conventional formulation of such models leads to mathematically-complex problems usually solved by custom-developed non-linear finite element (FE) codes. With a few exceptions, such codes are not available to the research community. This article describes a computational cardiac mechanics model developed such that it can be implemented using off-the-shelf FE solvers while tissue pathologies can be introduced in the model in a straight-forward manner. The model takes into account myocardial hyperelasticity, anisotropy, and active contraction forces. It follows a composite tissue modeling approach where the cardiac tissue is decomposed into two major parts: background and myofibers. The latter is modelled as rebars under initial stresses mimicking the contraction forces. The model was applied in silico to study the mechanics of infarcted left ventricle (LV) of a canine. End-systolic strain components, ejection fraction, and stress distribution attained using this LV model were compared quantitatively and qualitatively to corresponding data obtained from measurements as well as to other corresponding LV mechanics models. This comparison showed very good agreement.

The effects of load on transmural differences in contraction of isolated mouse ventricular cardiomyocytes

Mechanical properties of cardiomyocytes from different transmural regions are heterogeneous in the left ventricular wall. The cardiomyocyte mechanical environment affects this heterogeneity because of mechano-electric feedback mechanisms. In the present study, we investigated the effects of the mechanical load (preload and afterload) on transmural differences in contraction of subendocardial (ENDO) and subepicardial (EPI) single cells isolated from the murine left ventricle. Various preloads imposed via axial stretch and afterloads (unloaded and heavy loaded conditions) were applied to the cells using carbon fiber techniques for single myocytes. To simulate experimentally obtained results and to predict mechanisms underlying the cellular response to change in load, our mathematical models of the ENDO and EPI cells were used. Our major findings are the following. Our results show that ENDO and EPI cardiomyocytes have different mechanical responses to changes in preload to the cells. Under auxotonic contractions at low preload (unstretched cells), time to peak contraction (T_max) and the time constant of [Ca²⁺]_i transient decay were significantly longer in ENDO cells than in EPI cells. An increase in preload (stretched cells) prolonged T_max in both cell types; however, the prolongation was greater in EPI cells, resulting in a decrease in the transmural gradient in T_max at high preload. Comparing unloaded and heavy loaded (isometric) contractions of the cells we found that transmural gradient in the time course of contraction is independent of the loading conditions. Our mathematical cell models were able to reproduce the experimental results on the distinct cellular responses to changes in the mechanical load when we accounted for an ENDO/EPI difference in the parameters of cooperativity of calcium activation of myofilaments.

Simulation of the effects of moderate stimulation/inhibition of the β1-adrenergic signaling system and its components in mouse ventricular myocytes

The β1-adrenergic signaling system is one of the most important protein signaling systems in cardiac cells. It regulates cardiac action potential duration, intracellular Ca2+concentration ([Ca2+]i) transients, and contraction force. In this paper, a comprehensive experimentally based mathematical model of the β1-adrenergic signaling system for mouse ventricular myocytes is explored to simulate the effects of moderate stimulations of β1-adrenergic receptors (β1-ARs) on the action potential, Ca2+and Na+dynamics, as well as the effects of inhibition of protein kinase A (PKA) and phosphodiesterase of type 4 (PDE4). Simulation results show that the action potential prolongations reach saturating values at relatively small concentrations of isoproterenol (∼0.01 μM), while the [Ca2+]itransient amplitude saturates at significantly larger concentrations (∼0.1–1.0 μM). The differences in the response of Ca2+and Na+fluxes to moderate stimulation of β1-ARs are also observed. Sensitivity analysis of the mathematical model is performed and the model limitations are discussed. The investigated model reproduces most of the experimentally observed effects of moderate stimulation of β1-ARs, PKA, and PDE4 inhibition on the L-type Ca2+current, [Ca2+]itransients, and the sarcoplasmic reticulum Ca2+load and makes testable predictions for the action potential duration and [Ca2+]itransients as functions of isoproterenol concentration.

Ryanodine receptor sensitivity governs the stability and synchrony of local calcium release during cardiac excitation-contraction coupling

Calcium-induced calcium release is the principal mechanism that triggers the cell-wide [Ca(2+)]i transient that activates muscle contraction during cardiac excitation-contraction coupling (ECC). Here, we characterize this process in mouse cardiac myocytes with a novel mathematical action potential (AP) model that incorporates realistic stochastic gating of voltage-dependent L-type calcium (Ca(2+)) channels (LCCs) and sarcoplasmic reticulum (SR) Ca(2+) release channels (the ryanodine receptors, RyR2s). Depolarization of the sarcolemma during an AP stochastically activates the LCCs elevating subspace [Ca(2+)] within each of the cell's 20,000 independent calcium release units (CRUs) to trigger local RyR2 opening and initiate Ca(2+) sparks, the fundamental unit of triggered Ca(2+) release. Synchronization of Ca(2+) sparks during systole depends on the nearly uniform cellular activation of LCCs and the likelihood of local LCC openings triggering local Ca(2+) sparks (ECC fidelity). The detailed design and true SR Ca(2+) pump/leak balance displayed by our model permits investigation of ECC fidelity and Ca(2+) spark fidelity, the balance between visible (Ca(2+) spark) and invisible (Ca(2+) quark/sub-spark) SR Ca(2+) release events. Excess SR Ca(2+) leak is examined as a disease mechanism in the context of "catecholaminergic polymorphic ventricular tachycardia (CPVT)", a Ca(2+)-dependent arrhythmia. We find that that RyR2s (and therefore Ca(2+) sparks) are relatively insensitive to LCC openings across a wide range of membrane potentials; and that key differences exist between Ca(2+) sparks evoked during quiescence, diastole, and systole. The enhanced RyR2 [Ca(2+)]i sensitivity during CPVT leads to increased Ca(2+) spark fidelity resulting in asynchronous systolic Ca(2+) spark activity. It also produces increased diastolic SR Ca(2+) leak with some prolonged Ca(2+) sparks that at times become "metastable" and fail to efficiently terminate. There is a huge margin of safety for stable Ca(2+) handling within the cell and this novel mechanistic model provides insight into the molecular signaling characteristics that help maintain overall Ca(2+) stability even under the conditions of high SR Ca(2+) leak during CPVT. Finally, this model should provide tools for investigators to examine normal and pathological Ca(2+) signaling characteristics in the heart.

Biomechanics of Cardiac Function

The heart pumps blood to maintain circulation and ensure the delivery of oxygenated blood to all the organs of the body. Mechanics play a critical role in governing and regulating heart function under both normal and pathological conditions. Biological processes and mechanical stress are coupled together in regulating myocyte function and extracellular matrix structure thus controlling heart function. Here, we offer a brief introduction to the biomechanics of left ventricular function and then summarize recent progress in the study of the effects of mechanical stress on ventricular wall remodeling and cardiac function as well as the effects of wall mechanical properties on cardiac function in normal and dysfunctional hearts. Various mechanical models to determine wall stress and cardiac function in normal and diseased hearts with both systolic and diastolic dysfunction are discussed. The results of these studies have enhanced our understanding of the biomechanical mechanism in the development and remodeling of normal and dysfunctional hearts. Biomechanics provide a tool to understand the mechanism of left ventricular remodeling in diastolic and systolic dysfunction and guidance in designing and developing new treatments.

Knock-in mice harboring a Ca(2+) desensitizing mutation in cardiac troponin C develop early onset dilated cardiomyopathy

The physiological consequences of aberrant Ca(2+) binding and exchange with cardiac myofilaments are not clearly understood. In order to examine the effect of decreasing Ca(2+) sensitivity of cTnC on cardiac function, we generated knock-in mice carrying a D73N mutation (not known to be associated with heart disease in human patients) in cTnC. The D73N mutation was engineered into the regulatory N-domain of cTnC in order to reduce Ca(2+) sensitivity of reconstituted thin filaments by increasing the rate of Ca(2+) dissociation. In addition, the D73N mutation drastically blunted the extent of Ca(2+) desensitization of reconstituted thin filaments induced by cTnI pseudo-phosphorylation. Compared to wild-type mice, heterozygous knock-in mice carrying the D73N mutation exhibited a substantially decreased Ca(2+) sensitivity of force development in skinned ventricular trabeculae. Kaplan-Meier survival analysis revealed that median survival time for knock-in mice was 12 weeks. Echocardiographic analysis revealed that knock-in mice exhibited increased left ventricular dimensions with thinner walls. Echocardiographic analysis also revealed that measures of systolic function, such as ejection fraction (EF) and fractional shortening (FS), were dramatically reduced in knock-in mice. In addition, knock-in mice displayed electrophysiological abnormalities, namely prolonged QRS and QT intervals. Furthermore, ventricular myocytes isolated from knock-in mice did not respond to β-adrenergic stimulation. Thus, knock-in mice developed pathological features similar to those observed in human patients with dilated cardiomyopathy (DCM). In conclusion, our results suggest that decreasing Ca(2+) sensitivity of the regulatory N-domain of cTnC is sufficient to trigger the development of DCM.

Structure and function of cardiac troponin C (TNNC1): Implications for heart failure, cardiomyopathies, and troponin modulating drugs

In striated muscle, the protein troponin complex turns contraction on and off in a calcium-dependent manner. The calcium-sensing component of the complex is troponin C, which is expressed from the TNNC1 gene in both cardiac muscle and slow-twitch skeletal muscle (identical transcript in both tissues) and the TNNC2 gene in fast-twitch skeletal muscle. Cardiac troponin C (cTnC) is made up of two globular EF-hand domains connected by a flexible linker. The structural C-domain (cCTnC) contains two high affinity calcium-binding sites that are always occupied by Ca(2+) or Mg(2+) under physiologic conditions, stabilizing an open conformation that remains anchored to the rest of the troponin complex. In contrast, the regulatory N-domain (cNTnC) contains a single low affinity site that is largely unoccupied at resting calcium concentrations. During muscle activation, calcium binding to cNTnC favors an open conformation that binds to the switch region of troponin I, removing adjacent inhibitory regions of troponin I from actin and allowing muscle contraction to proceed. Regulation of the calcium binding affinity of cNTnC is physiologically important, because it directly impacts the calcium sensitivity of muscle contraction. Calcium sensitivity can be modified by drugs that stabilize the open form of cNTnC, post-translational modifications like phosphorylation of troponin I, or downstream thin filament protein interactions that impact the availability of the troponin I switch region. Recently, mutations in cTnC have been associated with hypertrophic or dilated cardiomyopathy. A detailed understanding of how calcium sensitivity is regulated through the troponin complex is necessary for explaining how mutations perturb its function to promote cardiomyopathy and how post-translational modifications in the thin filament affect heart function and heart failure. Troponin modulating drugs are being developed for the treatment of cardiomyopathies and heart failure.

Three-dimensional cardiac computational modelling: methods, features and applications

The combination of computational models and biophysical simulations can help to interpret an array of experimental data and contribute to the understanding, diagnosis and treatment of complex diseases such as cardiac arrhythmias. For this reason, three-dimensional (3D) cardiac computational modelling is currently a rising field of research. The advance of medical imaging technology over the last decades has allowed the evolution from generic to patient-specific 3D cardiac models that faithfully represent the anatomy and different cardiac features of a given alive subject. Here we analyse sixty representative 3D cardiac computational models developed and published during the last fifty years, describing their information sources, features, development methods and online availability. This paper also reviews the necessary components to build a 3D computational model of the heart aimed at biophysical simulation, paying especial attention to cardiac electrophysiology (EP), and the existing approaches to incorporate those components. We assess the challenges associated to the different steps of the building process, from the processing of raw clinical or biological data to the final application, including image segmentation, inclusion of substructures and meshing among others. We briefly outline the personalisation approaches that are currently available in 3D cardiac computational modelling. Finally, we present examples of several specific applications, mainly related to cardiac EP simulation and model-based image analysis, showing the potential usefulness of 3D cardiac computational modelling into clinical environments as a tool to aid in the prevention, diagnosis and treatment of cardiac diseases.

Modeling and Simulation Approaches for Cardiovascular Function and Their Role in Safety Assessment

Systems pharmacology modeling and pharmacokinetic-pharmacodynamic (PK/PD) analysis of drug-induced effects on cardiovascular (CV) function plays a crucial role in understanding the safety risk of new drugs. The aim of this review is to outline the current modeling and simulation (M&S) approaches to describe and translate drug-induced CV effects, with an emphasis on how this impacts drug safety assessment. Current limitations are highlighted and recommendations are made for future effort in this vital area of drug research.

Numerical evaluation of myofiber orientation and transmural contractile strength on left ventricular function

The left ventricle (LV) of the heart is composed of a complex organization of cardiac muscle fibers, which contract to generate force and pump blood into the body. It has been shown that both the orientation and contractile strength of these myofibers vary across the ventricular wall. The hypothesis of the current study is that the transmural distributions of myofiber orientation and contractile strength interdependently impact LV pump function. In order to quantify these interactions a finite element (FE) model of the LV was generated, which incorporated transmural variations. The influences of myofiber orientation and contractile strength on the Starling relationship and the end-systolic (ES) apex twist of the LV were assessed. The results suggest that reductions in contractile strength within a specific transmural layer amplified the effects of altered myofiber orientation in the same layer, causing greater changes in stroke volume (SV). Furthermore, when the epicardial myofibers contracted the strongest, the twist of the LV apex was greatest, regardless of myofiber orientation. These results demonstrate the important role of transmural distribution of myocardial contractile strength and its interplay with myofiber orientation. The coupling between these two physiologic parameters could play a critical role in the progression of heart failure.