Simulation of steady state and transient cardiac muscle response experiments with a Huxley-based contraction model

Feed Forward Modeling: an efficient approach for mathematical modeling of the force frequency relationship in the rabbit isolated ventricular myocyte

Background and Objective. This study addresses the Force-Frequency relationship, a fundamental characteristic of cardiac muscle influenced byβ1-adrenergic stimulation. This relationship reveals that heart rate (HR) changes at the sinoatrial node lead to alterations in ventricular cell contractility, increasing the force and decreasing relaxation time for higher beat rates. Traditional models lacking this relationship offer an incomplete physiological depiction, impacting the interpretation of in silico experiment results. To improve this, we propose a new mathematical model for ventricular myocytes, named 'Feed Forward Modeling' (FFM).Methods. FFM adjusts model parameters like channel conductance and Ca2+pump affinity according to stimulation frequency, in contrast to fixed parameter values. An empirical sigmoid curve guided the adaptation of each parameter, integrated into a rabbit ventricular cell electromechanical model. Model validation was achieved by comparing simulated data with experimental current-voltage (I-V) curves for L-type Calcium and slow Potassium currents.Results. FFM-enhanced simulations align more closely with physiological behaviors, accurately reflecting inotropic and lusitropic responses. For instance, action potential duration at 90% repolarization (APD90) decreased from 206 ms at 1 Hz to 173 ms at 4 Hz using FFM, contrary to the conventional model, where APD90 increased, limiting high-frequency heartbeats. Peak force also showed an increase with FFM, from 8.5 mN mm-2at 1 Hz to 11.9 mN mm-2at 4 Hz, while it barely changed without FFM. Relaxation time at 50% of maximum force (t50) similarly improved, dropping from 114 ms at 1 Hz to 75.9 ms at 4 Hz with FFM, a change not observed without the model.Conclusion. The FFM approach offers computational efficiency, bypassing the need to model all beta-adrenergic pathways, thus facilitating large-scale simulations. The study recommends that frequency change experiments include fractional dosing of isoproterenol to better replicate heart conditionsin vivo.

Mathematical analysis of left ventricular elastance with respect to afterload change during ejection phase

Since the left ventricle (LV) has pressure (Plv) and volume (Vlv), we can define LV elastance from the ratio between Plv and Vlv, termed as "instantaneous elastance." On the other hand, end-systolic elastance (Emax) is known to be a good index of LV contractility, which is measured by the slope of several end-systolic Plv-Vlv points obtained by using different loads. The word Emax originates from the assumption that LV elastance increases during the ejection phase and attains its maximum at the end-systole. From this concept, we can define another elastance determined by the slope of isochronous Plv-Vlv points, that is Plv-Vlv points at a certain time after the ejection onset time by using different loads. We refer to this elastance as "load-dependent elastance." To reveal the relation between these two elastances, we used a hemodynamic model that included a detailed ventricular myocyte contraction model. From the simulation results, we found that the isochronous Plv-Vlv points lay in one line and that the line slope corresponding to the load-dependent elastance slightly decreased during the ejection phase, which is quite different from the instantaneous elastance. Subsequently, we analyzed the mechanism determining these elastances from the model equations. We found that instantaneous elastance is directly related to contraction force generated by the ventricular myocyte, but the load-dependent elastance is determined by two factors: one is the transient characteristics of the cardiac cell, i.e., the velocity-dependent force drops characteristics in instantaneous shortening. The other is the force-velocity relation of the cardiac cell. We also found that the linear isochronous pressure-volume relation is based on the approximately linear relation between the time derivative of the cellular contraction force and the cellular shortening velocity that results from the combined characteristics of LV and aortic compliances.

Ionic Mechanisms of Propagated Repolarization in a One-Dimensional Strand of Human Ventricular Myocyte Model

Although repolarization has been suggested to propagate in cardiac tissue both theoretically and experimentally, it has been challenging to estimate how and to what extent the propagation of repolarization contributes to relaxation because repolarization only occurs in the course of membrane excitation in normal hearts. We established a mathematical model of a 1D strand of 600 myocytes stabilized at an equilibrium potential near the plateau potential level by introducing a sustained component of the late sodium current (INaL). By applying a hyperpolarizing stimulus to a small part of the strand, we succeeded in inducing repolarization which propagated along the strand at a velocity of 1~2 cm/s. The ionic mechanisms responsible for repolarization at the myocyte level, i.e., the deactivation of both the INaL and the L-type calcium current (ICaL), and the activation of the rapid component of delayed rectifier potassium current (IKr) and the inward rectifier potassium channel (IK1), were found to be important for the propagation of repolarization in the myocyte strand. Using an analogy with progressive activation of the sodium current (INa) in the propagation of excitation, regenerative activation of the predominant magnitude of IK1 makes the myocytes at the wave front start repolarization in succession through the electrical coupling via gap junction channels.

Gradient-based parameter optimization method to determine membrane ionic current composition in human induced pluripotent stem cell-derived cardiomyocytes

Premature cardiac myocytes derived from human induced pluripotent stem cells (hiPSC-CMs) show heterogeneous action potentials (APs), probably due to different expression patterns of membrane ionic currents. We developed a method for determining expression patterns of functional channels in terms of whole-cell ionic conductance (G_x) using individual spontaneous AP configurations. It has been suggested that apparently identical AP configurations can be obtained using different sets of ionic currents in mathematical models of cardiac membrane excitation. If so, the inverse problem of G_x estimation might not be solved. We computationally tested the feasibility of the gradient-based optimization method. For a realistic examination, conventional 'cell-specific models' were prepared by superimposing the model output of AP on each experimental AP recorded by conventional manual adjustment of G_xs of the baseline model. G_xs of 4-6 major ionic currents of the 'cell-specific models' were randomized within a range of ± 5-15% and used as an initial parameter set for the gradient-based automatic G_xs recovery by decreasing the mean square error (MSE) between the target and model output. Plotting all data points of the MSE-G_x relationship during optimization revealed progressive convergence of the randomized population of G_xs to the original value of the cell-specific model with decreasing MSE. The absence of any other local minimum in the global search space was confirmed by mapping the MSE by randomizing G_xs over a range of 0.1-10 times the control. No additional local minimum MSE was obvious in the whole parameter space, in addition to the global minimum of MSE at the default model parameter.

Lower troponin expression in the right ventricle of rats explains interventricular differences in E-C coupling

Despite distinctive functional and anatomic differences, a precise understanding of the cardiac interventricular differences in excitation–contraction (E–C) coupling mechanisms is still lacking. Here, we directly compared rat right and left cardiomyocytes (RVCM and LVCM). Whole-cell patch clamp, the IonOptix system, and fura-2 fluorimetry were used to measure electrical properties (action potential and ionic currents), single-cell contractility, and cytosolic Ca²⁺ ([Ca²⁺]_i), respectively. Myofilament proteins were analyzed by immunoblotting. RVCM showed significantly shorter action potential duration (APD) and higher density of transient outward K⁺ current (I_to). However, the triggered [Ca²⁺]_i change (Ca²⁺ transient) was not different, while the decay rate of the Ca²⁺ transient was slower in RVCM. Although the relaxation speed was also slower, the sarcomere shortening amplitude (ΔSL) was smaller in RVCM. SERCA activity was ∼60% lower in RVCM, which is partly responsible for the slower decay of the Ca²⁺ transient. Immunoblot analysis revealed lower expression of the cardiac troponin complex (cTn) in RVCM, implying a smaller Ca²⁺ buffering capacity (κ_S), which was proved by in situ analysis. The introduction of these new levels of cTn, I_to, and SERCA into a mathematical model of rat LVCM reproduced the similar Ca²⁺ transient, slower Ca²⁺ decay, shorter APD, and smaller ΔSL of RVCM. Taken together, these data show reduced expression of cTn proteins in the RVCM, which provides an explanation for the interventricular difference in the E–C coupling kinetics.

Troponin levels make a difference

JGP study reveals that lower troponin expression in the right ventricle underlies interventricular differences in excitation-contraction coupling.

A mathematical model of hiPSC cardiomyocytes electromechanics

Human induced pluripotent stem cell-derived cardiomyocytes (hiPSC-CMs) are becoming instrumental in cardiac research, human-based cell level cardiotoxicity tests, and developing patient-specific care. As one of the principal functional readouts is contractility, we propose a novel electromechanical hiPSC-CM computational model named the hiPSC-CM-CE. This model comprises a reparametrized version of contractile element (CE) by Rice et al., 2008, with a new passive force formulation, integrated into a hiPSC-CM electrophysiology formalism by Paci et al. in 2020. Our simulated results were validated against in vitro data reported for hiPSC-CMs at matching conditions from different labs. Specifically, key action potential (AP) and calcium transient (CaT) biomarkers simulated by the hiPSC-CM-CE model were within the experimental ranges. On the mechanical side, simulated cell shortening, contraction-relaxation kinetic indices (RT50 and RT25 ), and the amplitude of tension fell within the experimental intervals. Markedly, as an inter-scale analysis, correct classification of the inotropic effects due to non-cardiomyocytes in hiPSC-CM tissues was predicted on account of the passive force expression introduced to the CE. Finally, the physiological inotropic effects caused by Verapamil and Bay-K 8644 and the aftercontractions due to the early afterdepolarizations (EADs) were simulated and validated against experimental data. In the future, the presented model can be readily expanded to take in pharmacological trials and genetic mutations, such as those involved in hypertrophic cardiomyopathy, and study arrhythmia trigger mechanisms.

Generative adversarial networks for construction of virtual populations of mechanistic models: simulations to study Omecamtiv Mecarbil action

Biophysical models are increasingly used to gain mechanistic insights by fitting and reproducing experimental and clinical data. The inherent variability in the recorded datasets, however, presents a key challenge. In this study, we present a novel approach, which integrates mechanistic modeling and machine learning to analyze in vitro cardiac mechanics data and solve the inverse problem of model parameter inference. We designed a novel generative adversarial network (GAN) and employed it to construct virtual populations of cardiac ventricular myocyte models in order to study the action of Omecamtiv Mecarbil (OM), a positive cardiac inotrope. Populations of models were calibrated from mechanically unloaded myocyte shortening recordings obtained in experiments on rat myocytes in the presence and absence of OM. The GAN was able to infer model parameters while incorporating prior information about which model parameters OM targets. The generated populations of models reproduced variations in myocyte contraction recorded during in vitro experiments and provided improved understanding of OM’s mechanism of action. Inverse mapping of the experimental data using our approach suggests a novel action of OM, whereby it modifies interactions between myosin and tropomyosin proteins. To validate our approach, the inferred model parameters were used to replicate other in vitro experimental protocols, such as skinned preparations demonstrating an increase in calcium sensitivity and a decrease in the Hill coefficient of the force–calcium (F–Ca) curve under OM action. Our approach thereby facilitated the identification of the mechanistic underpinnings of experimental observations and the exploration of different hypotheses regarding variability in this complex biological system.

Frank-Starling mechanism, fluid responsiveness, and length-dependent activation: Unravelling the multiscale behaviors with an in silico analysis

The Frank-Starling mechanism is a fundamental regulatory property which underlies the cardiac output adaptation to venous filling. Length-dependent activation is generally assumed to be the cellular origin of this mechanism. At the heart scale, it is commonly admitted that an increase in preload (ventricular filling) leads to an increased cellular force and an increased volume of ejected blood. This explanation also forms the basis for vascular filling therapy. It is actually difficult to unravel the exact nature of the relationship between length-dependent activation and the Frank-Starling mechanism, as three different scales (cellular, ventricular and cardiovascular) are involved. Mathematical models are powerful tools to overcome these limitations. In this study, we use a multiscale model of the cardiovascular system to untangle the three concepts (length-dependent activation, Frank-Starling, and vascular filling). We first show that length-dependent activation is required to observe both the Frank-Starling mechanism and a positive response to high vascular fillings. Our results reveal a dynamical length dependent activation-driven response to changes in preload, which involves interactions between the cellular, ventricular and cardiovascular levels and thus highlights fundamentally multiscale behaviors. We show however that the cellular force increase is not enough to explain the cardiac response to rapid changes in preload. We also show that the absence of fluid responsiveness is not related to a saturating Frank-Starling effect. As it is challenging to study those multiscale phenomena experimentally, this computational approach contributes to a more comprehensive knowledge of the sophisticated length-dependent properties of cardiac muscle.

A Multiple Step Active Stiffness Integration Scheme to Couple a Stochastic Cross-Bridge Model and Continuum Mechanics for Uses in Both Basic Research and Clinical Applications of Heart Simulation

In a multiscale simulation of a beating heart, the very large difference in the time scales between rapid stochastic conformational changes of contractile proteins and deterministic macroscopic outcomes, such as the ventricular pressure and volume, have hampered the implementation of an efficient coupling algorithm for the two scales. Furthermore, the consideration of dynamic changes of muscle stiffness caused by the cross-bridge activity of motor proteins have not been well established in continuum mechanics. To overcome these issues, we propose a multiple time step scheme called the multiple step active stiffness integration scheme (MusAsi) for the coupling of Monte Carlo (MC) multiple steps and an implicit finite element (FE) time integration step. The method focuses on the active tension stiffness matrix, where the active tension derivatives concerning the current displacements in the FE model are correctly integrated into the total stiffness matrix to avoid instability. A sensitivity analysis of the number of samples used in the MC model and the combination of time step sizes confirmed the accuracy and robustness of MusAsi, and we concluded that the combination of a 1.25 ms FE time step and 0.005 ms MC multiple steps using a few hundred motor proteins in each finite element was appropriate in the tradeoff between accuracy and computational time. Furthermore, for a biventricular FE model consisting of 45,000 tetrahedral elements, one heartbeat could be computed within 1.5 h using 320 cores of a conventional parallel computer system. These results support the practicality of MusAsi for uses in both the basic research of the relationship between molecular mechanisms and cardiac outputs, and clinical applications of perioperative prediction.

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.

An Introductory Overview of Image-Based Computational Modeling in Personalized Cardiovascular Medicine

Cardiovascular diseases account for the number one cause of deaths in the world. Part of the reason for such grim statistics is our limited understanding of the underlying mechanisms causing these devastating pathologies, which is made difficult by the invasiveness of the procedures associated with their diagnosis (e.g., inserting catheters into the coronal artery to measure blood flow to the heart). Likewise, it is also difficult to design and test assistive devices without implanting them in vivo. However, with the recent advancements made in biomedical scanning technologies and computer simulations, image-based modeling (IBM) has arisen as the next logical step in the evolution of non-invasive patient-specific cardiovascular medicine. Yet, due to its novelty, it is still relatively unknown outside of the niche field. Therefore, the goal of this manuscript is to review the current state-of-the-art and the limitations of the methods used in this area of research, as well as their applications to personalized cardiovascular investigations and treatments. Specifically, the modeling of three different physics – electrophysiology, biomechanics and hemodynamics – used in the cardiovascular IBM is discussed in the context of the physiology that each one of them describes and the mechanisms of the underlying cardiac diseases that they can provide insight into. Only the “bare-bones” of the modeling approaches are discussed in order to make this introductory material more accessible to an outside observer. Additionally, the imaging methods, the aspects of the unique cardiac anatomy derived from them, and their relation to the modeling algorithms are reviewed. Finally, conclusions are drawn about the future evolution of these methods and their potential toward revolutionizing the non-invasive diagnosis, virtual design of treatments/assistive devices, and increasing our understanding of these lethal cardiovascular diseases.

Modulating the tension-time integral of the cardiac twitch prevents dilated cardiomyopathy in murine hearts

Dilated cardiomyopathy (DCM) is often associated with sarcomere protein mutations that confer reduced myofilament tension–generating capacity. We demonstrated that cardiac twitch tension-time integrals can be targeted and tuned to prevent DCM remodeling in hearts with contractile dysfunction. We employed a transgenic murine model of DCM caused by the D230N-tropomyosin (Tm) mutation and designed a sarcomere-based intervention specifically targeting the twitch tension-time integral of D230N-Tm hearts using multiscale computational models of intramolecular and intermolecular interactions in the thin filament and cell-level contractile simulations. Our models predicted that increasing the calcium sensitivity of thin filament activation using the cardiac troponin C (cTnC) variant L48Q can sufficiently augment twitch tension-time integrals of D230N-Tm hearts. Indeed, cardiac muscle isolated from double-transgenic hearts expressing D230N-Tm and L48Q cTnC had increased calcium sensitivity of tension development and increased twitch tension-time integrals compared with preparations from hearts with D230N-Tm alone. Longitudinal echocardiographic measurements revealed that DTG hearts retained normal cardiac morphology and function, whereas D230N-Tm hearts developed progressive DCM. We present a computational and experimental framework for targeting molecular mechanisms governing the twitch tension of cardiomyopathic hearts to counteract putative mechanical drivers of adverse remodeling and open possibilities for tension-based treatments of genetic cardiomyopathies.

Tuning the molecular mechanisms that govern the twitch tension of cardiomyopathic hearts counteracts mechanical drivers of adverse remodeling.

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.

Drug-induced shortening of the electromechanical window is an effective biomarker for in silico prediction of clinical risk of arrhythmias

BACKGROUND AND PURPOSE

Early identification of drug-induced cardiac adverse events is key in drug development. Human-based computer models are emerging as an effective approach, complementary to in vitro and animal models. Drug-induced shortening of the electromechanical window has been associated with increased risk of arrhythmias. This study investigates the potential of a cellular surrogate for the electromechanical window (EMw) for prediction of pro-arrhythmic cardiotoxicity, and its underlying ionic mechanisms, using human-based computer models.

EXPERIMENTAL APPROACH

In silico drug trials for 40 reference compounds were performed, testing up to 100-fold the therapeutic concentrations (EFTPCmax ) and using a control population of human ventricular action potential (AP) models, optimised to capture pro-arrhythmic ionic profiles. EMw was calculated for each model in the population as the difference between AP and Ca2+ transient durations at 90%. Drug-induced changes in the EMw and occurrence of repolarisation abnormalities (RA) were quantified.

KEY RESULTS

Drugs with clinical risk of Torsade de Pointes arrhythmias induced a concentration-dependent EMw shortening, while safe drugs lead to increase or small change in EMw. Risk predictions based on EMw shortening achieved 90% accuracy at 10× EFTPCmax , whereas RA-based predictions required 100× EFTPCmax to reach the same accuracy. As it is dependent on Ca2+ transient, the EMw was also more sensitive than AP prolongation in distinguishing between pure hERG blockers and multichannel compounds also blocking the calcium current.

CONCLUSION AND IMPLICATIONS

The EMw is an effective biomarker for in silico predictions of drug-induced clinical pro-arrhythmic risk, particularly for compounds with multichannel blocking action.

Multiscale simulation of the effects of atrioventricular block and valve diseases on heart performance

A new mathematical model of the cardiovascular system is proposed. The left ventricle is described by an axisymmetric multiscale model where myocardium is treated as an incompressible transversely isotropic medium with a realistic distribution of fibre orientation. Active tension and its regulation by Ca²⁺ ions are described by our recent kinetic model. A lumped parameter model is used for the simulation of blood circulation, in which the left and right atria and the right ventricle are described by a system of ordinary differential equations for active pressure-volume relationships. The stress and strain of the left ventricle myocardium were calculated by the finite element method implemented by the authors. The changes in the haemodynamics upon changes in preload of a healthy heart, upon physical exercise, and in case of atrioventricular block with different types of arrhythmias were simulated. To simulate the effect of stenosis or regurgitation of the aortic or mitral valves, the hydraulic and inertial flow resistances of the heart valves were set as functions of their orifice areas. The model reproduced a number of phenomena observed in clinical practice, including the classification of the severity of valve disease.

A new myofilament contraction model with ATP consumption for ventricular cell model

A new contraction model of cardiac muscle was developed by combining previously described biochemical and biophysical models. The biochemical component of the new contraction model represents events in the presence of Ca2+-crossbridge attachment and power stroke following inorganic phosphate release, detachment evoked by the replacement of ADP by ATP, ATP hydrolysis, and recovery stroke. The biophysical component focuses on Ca2+ activation and force (F b) development assuming an equivalent crossbridge. The new model faithfully incorporates the major characteristics of the biochemical and biophysical models, such as F b activation by transient Ca2+ ([Ca2+]-F b), [Ca2+]-ATP hydrolysis relations, sarcomere length-F b, and F b recovery after jumps in length under the isometric mode and upon sarcomere shortening after a rapid release of mechanical load under the isotonic mode together with the load-velocity relationship. ATP consumption was obtained for all responses. When incorporated in a ventricular cell model, the contraction model was found to share approximately 60% of the total ATP usage in the cell model.

Experimental assessment of a myocyte-based multiscale model of cardiac contractile dysfunction

Cardiac contractile dysfunction (CD) is a multifactorial syndrome caused by different acute or progressive diseases which hamper assessing the role of the underlying mechanisms characterizing a defined pathological condition. Mathematical modeling can help to understand the processes involved in CD and analyze their relative impact in the overall response. The aim of this study was thus to use a myocyte-based multiscale model of the circulatory system to simulate the effects of halothane, a volatile anesthetic which at high doses elicits significant acute CD both in isolated myocytes and intact animals. Ventricular chambers built using a human myocyte model were incorporated into a whole circulatory system represented by resistances and capacitances. Halothane-induced decreased sarco(endo)plasmic reticulum Ca²⁺ (SERCA2a) reuptake pump, transient outward K⁺ (I_to), Na⁺-Ca²⁺ exchanger (I_NCX) and L-type Ca²⁺ channel (I_CaL) currents, together with ryanodine receptor (RyR2) increased open probability (P_o) and reduced myofilament Ca²⁺ sensitivity, reproduced equivalent decreased action potential duration at 90% repolarization and intracellular Ca²⁺ concentration at the myocyte level reported in the literature. In the whole circulatory system, model reduction in mean arterial pressure, cardiac output and regional wall thickening fraction was similar to experimental results in open-chest sheep subjected to acute halothane overdose. Effective model performance indicates that the model structure could be used to study other changes in myocyte targets eliciting CD.

Automatic Optimization of an in Silico Model of Human iPSC Derived Cardiomyocytes Recapitulating Calcium Handling Abnormalities

The growing importance of human induced pluripotent stem cell-derived cardiomyoyctes (hiPSC-CMs), as patient-specific and disease-specific models for studying cellular cardiac electrophysiology or for preliminary cardiotoxicity tests, generated better understanding of hiPSC-CM biophysical mechanisms and great amount of action potential and calcium transient data. In this paper, we propose a new hiPSC-CM in silico model, with particular attention to Ca²⁺ handling. We used (i) the hiPSC-CM Paci2013 model as starting point, (ii) a new dataset of Ca²⁺ transient measurements to tune the parameters of the inward and outward Ca²⁺ fluxes of sarcoplasmic reticulum, and (iii) an automatic parameter optimization to fit action potentials and Ca²⁺ transients. The Paci2018 model simulates, together with the typical hiPSC-CM spontaneous action potentials, more refined Ca²⁺ transients and delayed afterdepolarizations-like abnormalities, which the old Paci2013 was not able to predict due to its mathematical formulation. The Paci2018 model was validated against (i) the same current blocking experiments used to validate the Paci2013 model, and (ii) recently published data about effects of different extracellular ionic concentrations. In conclusion, we present a new and more versatile in silico model, which will provide a platform for modeling the effects of drugs or mutations that affect Ca²⁺ handling in hiPSC-CMs.

Analysis of spontaneous oscillations for a three-state power-stroke model

Our study considers the mechanism of the spontaneous oscillations of molecular motors that are driven by the power stroke principle by applying linear stability analysis around the stationary solution. By representing the coupling equation of microscopic molecular motor dynamics and mesoscopic sarcomeric dynamics by a rank-1 updated matrix system, we derived the analytical representations of the eigenmodes of the Jacobian matrix that cause the oscillation. Based on these analytical representations, we successfully derived the essential conditions for the oscillation in terms of the rate constants of the power stroke and the reversal stroke transitions of the molecular motor. Unlike the two-state model, in which the dependence of the detachment rates on the motor coordinates or the applied forces on the motors plays a key role for the oscillation, our three-state power stroke model demonstrates that the dependence of the rate constants of the power and reversal strokes on the strains in the elastic elements in the motor molecules plays a key role, where these rate constants are rationally determined from the free energy available for the power stroke, the stiffness of the elastic element in the molecular motor, and the working stroke size. By applying the experimentally confirmed values to the free energy, the stiffness, and the working stroke size, our numerical model reproduces well the experimentally observed oscillatory behavior. Furthermore, our analysis shows that two eigenmodes with real positive eigenvalues characterize the oscillatory behavior, where the eigenmode with the larger eigenvalue indicates the transient of the system of the quick sarcomeric lengthening induced by the collective reversal strokes, and the smaller eigenvalue correlates with the speed of sarcomeric shortening, which is much slower than lengthening. Applying the perturbation analyses with primal physical parameters, we find that these two real eigenvalues occur on two branches derived from a merge point of a pair of complex-conjugate eigenvalues generated by Hopf bifurcation.

Multiscale model of the human cardiovascular system: Description of heart failure and comparison of contractility indices

A multiscale model of the cardiovascular system is presented. Hemodynamics is described by a lumped parameter model, while heart contraction is described at the cellular scale. An electrophysiological model and a mechanical model were coupled and adjusted so that the pressure and volume of both ventricles are linked to the force and length of a half-sarcomere. Particular attention was paid to the extreme values of the sarcomere length, which must keep physiological values. This model is able to reproduce healthy behavior, preload variations experiments, and ventricular failure. It also allows to compare the relevance of standard cardiac contractility indices. This study shows that the theoretical gold standard for assessing cardiac contractility, namely the end-systolic elastance, is actually load-dependent and therefore not a reliable index of cardiac contractility.

A Tension-Based Model Distinguishes Hypertrophic versus Dilated Cardiomyopathy

The heart either hypertrophies or dilates in response to familial mutations in genes encoding sarcomeric proteins, which are responsible for contraction and pumping. These mutations typically alter calcium-dependent tension generation within the sarcomeres, but how this translates into the spectrum of hypertrophic versus dilated cardiomyopathy is unknown. By generating a series of cardiac-specific mouse models that permit the systematic tuning of sarcomeric tension generation and calcium fluxing, we identify a significant relationship between the magnitude of tension developed over time and heart growth. When formulated into a computational model, the integral of myofilament tension development predicts hypertrophic and dilated cardiomyopathies in mice associated with essentially any sarcomeric gene mutations, but also accurately predicts human cardiac phenotypes from data generated in induced-pluripotent-stem-cell-derived myocytes from familial cardiomyopathy patients. This tension-based model also has the potential to inform pharmacologic treatment options in cardiomyopathy patients.

A human ventricular myocyte model with a refined representation of excitation-contraction coupling

Cardiac Ca(2+)-induced Ca(2+) release (CICR) occurs by a regenerative activation of ryanodine receptors (RyRs) within each Ca(2+)-releasing unit, triggered by the activation of L-type Ca(2+) channels (LCCs). CICR is then terminated, most probably by depletion of Ca(2+) in the junctional sarcoplasmic reticulum (SR). Hinch et al. previously developed a tightly coupled LCC-RyR mathematical model, known as the Hinch model, that enables simulations to deal with a variety of functional states of whole-cell populations of a Ca(2+)-releasing unit using a personal computer. In this study, we developed a membrane excitation-contraction model of the human ventricular myocyte, which we call the human ventricular cell (HuVEC) model. This model is a hybrid of the most recent HuVEC models and the Hinch model. We modified the Hinch model to reproduce the regenerative activation and termination of CICR. In particular, we removed the inactivated RyR state and separated the single step of RyR activation by LCCs into triggering and regenerative steps. More importantly, we included the experimental measurement of a transient rise in Ca(2+) concentrations ([Ca(2+)], 10-15 μM) during CICR in the vicinity of Ca(2+)-releasing sites, and thereby calculated the effects of the local Ca(2+) gradient on CICR as well as membrane excitation. This HuVEC model successfully reconstructed both membrane excitation and key properties of CICR. The time course of CICR evoked by an action potential was accounted for by autonomous changes in an instantaneous equilibrium open probability of couplons. This autonomous time course was driven by a core feedback loop including the pivotal local [Ca(2+)], influenced by a time-dependent decay in the SR Ca(2+) content during CICR.

β-adrenergic effects on cardiac myofilaments and contraction in an integrated rabbit ventricular myocyte model

A five-state model of myofilament contraction was integrated into a well-established rabbit ventricular myocyte model of ion channels, Ca(2+) transporters and kinase signaling to analyze the relative contribution of different phosphorylation targets to the overall mechanical response driven by β-adrenergic stimulation (β-AS). β-AS effect on sarcoplasmic reticulum Ca(2+) handling, Ca(2+), K(+) and Cl(-) currents, and Na(+)/K(+)-ATPase properties was included based on experimental data. The inotropic effect on the myofilaments was represented as reduced myofilament Ca(2+) sensitivity (XBCa) and titin stiffness, and increased cross-bridge (XB) cycling rate (XBcy). Assuming independent roles of XBCa and XBcy, the model reproduced experimental β-AS responses on action potentials and Ca(2+) transient amplitude and kinetics. It also replicated the behavior of force-Ca(2+), release-restretch, length-step, stiffness-frequency and force-velocity relationships, and increased force and shortening in isometric and isotonic twitch contractions. The β-AS effect was then switched off from individual targets to analyze their relative impact on contractility. Preventing β-AS effects on L-type Ca(2+) channels or phospholamban limited Ca(2+) transients and contractile responses in parallel, while blocking phospholemman and K(+) channel (IKs) effects enhanced Ca(2+) and inotropy. Removal of β-AS effects from XBCa enhanced contractile force while decreasing peak Ca(2+) (due to greater Ca(2+) buffering), but had less effect on shortening. Conversely, preventing β-AS effects on XBcy preserved Ca(2+) transient effects, but blunted inotropy (both isometric force and especially shortening). Removal of titin effects had little impact on contraction. Finally, exclusion of β-AS from XBCa and XBcy while preserving effects on other targets resulted in preserved peak isometric force response (with slower kinetics) but nearly abolished enhanced shortening. β-AS effects on XBCa and XBcy have greater impact on isometric and isotonic contraction, respectively.

EAD and DAD mechanisms analyzed by developing a new human ventricular cell model

It has long been suggested that the Ca(2+)-mechanisms are largely involved in generating the early afterdepolarization (EAD) as well as the delayed afterdepolarization (DAD). This view was examined in a quantitative manner by applying the lead potential analysis to a new human ventricular cell model. In this ventricular cell model, the tight coupled LCC-RyR model (CaRU) based on local control theory (Hinch et al. 2004) and ion channel models mostly based on human electrophysiological data were included to reproduce realistic Ca(2+) dynamics as well as the membrane excitation. Simultaneously, the Ca(2+) accumulation near the Ca(2+) releasing site was incorporated as observed in real cardiac myocytes. The maximum rate of ventricular repolarization (-1.02 mV/ms) is due to IK1 (-0.55 mV/ms) and the rest is provided nearly equally by INCX (-0.20 mV/ms), INaL (-0.16 mV/ms) and INaT (-0.13 mV/ms). These INaL and INaT components are due to closure of the voltage gate, which remains partially open during the plateau potential. DADs could be evoked by applying high-frequency stimulations supplemented by a partial Na(+)/K(+) pump inhibition, or by a microinjection of Ca(2+). EADs was evoked by retarding the inactivation of INaL. The lead potential (VL) analysis revealed that IK1 and IKr played the primary role to reverse the AP repolarization to depolarizing limb of EAD. ICaL and INCX amplified EAD, while the remaining currents partially antagonized dVL/dt. The maximum rate of rise of EAD was attributable to the rapid activation of both ICaL (45.5%) and INCX (54.5%).

Hill's equation of muscle performance and its hidden insight on molecular mechanisms

Muscles shorten faster against light loads than they do against heavy loads. The hyperbolic equation first used by A.V. Hill over seven decades ago to illustrate the relationship between shortening velocity and load is still the predominant method used to characterize muscle performance, even though it has been regarded as purely empirical and lacking precision in predicting velocities at high and low loads. Popularity of the Hill equation has been sustained perhaps because of historical reasons, but its simplicity is certainly attractive. The descriptive nature of the equation does not diminish its role as a useful tool in our quest to understand animal locomotion and optimal design of muscle-powered devices like bicycles. In this Review, an analysis is presented to illustrate the connection between the historic Hill equation and the kinetics of myosin cross-bridge cycle based on the latest findings on myosin motor interaction with actin filaments within the structural confines of a sarcomere. In light of the new data and perspective, some previous studies of force-velocity relations of muscle are revisited to further our understanding of muscle mechanics and the underlying biochemical events, specifically how extracellular and intracellular environment, protein isoform expression, and posttranslational modification of contractile and regulatory proteins change the interaction between myosin and actin that in turn alter muscle force, shortening velocity, and the relationship between them.

Role of CaMKII in post acidosis arrhythmias: a simulation study using a human myocyte model

Postacidotic arrhythmias have been associated to increased sarcoplasmic reticulum (SR) Ca(2+) load and Ca(2+)/calmodulin-dependent protein kinase II (CaMKII) activation. However, the molecular mechanisms underlying these arrhythmias are still unclear. To better understand this process, acidosis produced by CO2 increase from 5% to 30%, resulting in intracellular pH (pHi) change from 7.15 to 6.7, was incorporated into a myocyte model of excitation-contraction coupling and contractility, including acidotic inhibition of L-type Ca(2+) channel (I(CaL)), Na(+)-Ca(2+) exchanger, Ca(2+) release through the SR ryanodine receptor (RyR2) (I(rel)), Ca(2+) reuptake by the SR Ca(2+) ATPase2a (I(up)), Na(+)-K(+) pump, K(+) efflux through the inward rectifier K(+) channel and the transient outward K(+) flow (I(to)) together with increased activity of the Na(+)-H(+) exchanger (I(NHE)). Simulated CaMKII regulation affecting I(rel), I(up), I(CaL), I(NHE) and I(to) was introduced in the model to partially compensate the acidosis outcome. Late Na(+) current increase by CaMKII was also incorporated. Using this scheme and assuming that diastolic Ca(2+) leak through the RyR2 was modulated by the resting state of this channel and the difference between SR and dyadic cleft [Ca(2+)], postacidotic delayed after depolarizations (DADs) were triggered upon returning to normal pHi after 6 min acidosis. The model showed that DADs depend on SR Ca(2+) load and on increased Ca(2+) leak through RyR2. This postacidotic arrhythmogenic pattern relies mainly on CaMKII effect on I(CaL) and I(up), since its individual elimination produced the highest DAD reduction. The model further revealed that during the return to normal pHi, DADs are fully determined by SR Ca(2+) load at the end of acidosis. Thereafter, DADs are maintained by SR Ca(2+) reloading by Ca(2+) influx through the reverse NCX mode during the time period in which [Na(+)]i is elevated.

The force-frequency relationship: insights from mathematical modeling

The force-frequency relationship has intrigued researchers since its discovery by Bowditch in 1871. Many attempts have been made to construct mathematical descriptions of this phenomenon, beginning with the simple formulation of Koch-Wesser and Blinks in 1963 to the most sophisticated ones of today. This property of cardiac muscle is amplified by β-adrenergic stimulation, and, in a coordinated way, the neurohumoral state alters both frequency (acting on the sinoatrial node) as well as force generation (modifying ventricular myocytes). This synchronized tuning is needed to meet new metabolic demands. Cardiac modelers have already linked mechanical and electrical activity in their formulations and showed how those activities feedback on each other. However, now it is necessary to include neurological control to have a complete description of heart performance, especially when changes in frequency are involved. Study of arrhythmias (or antiarrhythmic drugs) based on mathematical models should incorporate this effect to make useful predictions or point out potential pharmaceutical targets.

Cardiac electromechanical models: from cell to organ

The heart is a multiphysics and multiscale system that has driven the development of the most sophisticated mathematical models at the frontiers of computational physiology and medicine. This review focuses on electromechanical (EM) models of the heart from the molecular level of myofilaments to anatomical models of the organ. Because of the coupling in terms of function and emergent behaviors at each level of biological hierarchy, separation of behaviors at a given scale is difficult. Here, a separation is drawn at the cell level so that the first half addresses subcellular/single-cell models and the second half addresses organ models. At the subcellular level, myofilament models represent actin–myosin interaction and Ca-based activation. The discussion of specific models emphasizes the roles of cooperative mechanisms and sarcomere length dependence of contraction force, considered to be the cellular basis of the Frank–Starling law. A model of electrophysiology and Ca handling can be coupled to a myofilament model to produce an EM cell model, and representative examples are summarized to provide an overview of the progression of the field. The second half of the review covers organ-level models that require solution of the electrical component as a reaction–diffusion system and the mechanical component, in which active tension generated by the myocytes produces deformation of the organ as described by the equations of continuum mechanics. As outlined in the review, different organ-level models have chosen to use different ionic and myofilament models depending on the specific application; this choice has been largely dictated by compromises between model complexity and computational tractability. The review also addresses application areas of EM models such as cardiac resynchronization therapy and the role of mechano-electric coupling in arrhythmias and defibrillation.

Increased myofilament Ca2+-sensitivity and arrhythmia susceptibility

Increased myofilament Ca(2+) sensitivity is a common attribute of many inherited and acquired cardiomyopathies that are associated with cardiac arrhythmias. Accumulating evidence supports the concept that increased myofilament Ca(2+) sensitivity is an independent risk factor for arrhythmias. This review describes and discusses potential underlying molecular and cellular mechanisms how myofilament Ca(2+) sensitivity affects cardiac excitation and leads to the generation of arrhythmias. Emphasized are downstream effects of increased myofilament Ca(2+) sensitivity: altered Ca(2+) buffering/handling, impaired energy metabolism and increased mechanical stretch, and how they may contribute to arrhythmogenesis.