A model of cardiac electrical activity incorporating ionic pumps and concentration changes

Computational modeling of cardiac electrophysiology and arrhythmogenesis: toward clinical translation

The complexity of cardiac electrophysiology, involving dynamic changes in numerous components across multiple spatial (from ion channel to organ) and temporal (from milliseconds to days) scales, makes an intuitive or empirical analysis of cardiac arrhythmogenesis challenging. Multiscale mechanistic computational models of cardiac electrophysiology provide precise control over individual parameters, and their reproducibility enables a thorough assessment of arrhythmia mechanisms. This review provides a comprehensive analysis of models of cardiac electrophysiology and arrhythmias, from the single cell to the organ level, and how they can be leveraged to better understand rhythm disorders in cardiac disease and to improve heart patient care. Key issues related to model development based on experimental data are discussed, and major families of human cardiomyocyte models and their applications are highlighted. An overview of organ-level computational modeling of cardiac electrophysiology and its clinical applications in personalized arrhythmia risk assessment and patient-specific therapy of atrial and ventricular arrhythmias is provided. The advancements presented here highlight how patient-specific computational models of the heart reconstructed from patient data have achieved success in predicting risk of sudden cardiac death and guiding optimal treatments of heart rhythm disorders. Finally, an outlook toward potential future advances, including the combination of mechanistic modeling and machine learning/artificial intelligence, is provided. As the field of cardiology is embarking on a journey toward precision medicine, personalized modeling of the heart is expected to become a key technology to guide pharmaceutical therapy, deployment of devices, and surgical interventions.

Models of the cardiac L-type calcium current: A quantitative review

The L-type calcium current (I CaL ) plays a critical role in cardiac electrophysiology, and models ofI CaL are vital tools to predict arrhythmogenicity of drugs and mutations. Five decades of measuring and modelingI CaL have resulted in several competing theories (encoded in mathematical equations). However, the introduction of new models has not typically been accompanied by a data-driven critical comparison with previous work, so that it is unclear which model is best suited for any particular application. In this review, we describe and compare 73 published mammalianI CaL models and use simulated experiments to show that there is a large variability in their predictions, which is not substantially diminished when grouping by species or other categories. We provide model code for 60 models, list major data sources, and discuss experimental and modeling work that will be required to reduce this huge list of competing theories and ultimately develop a community consensus model ofI CaL . This article is categorized under: Cardiovascular Diseases > Computational Models Cardiovascular Diseases > Molecular and Cellular Physiology.

The virtual sinoatrial node: What did computational models tell us about cardiac pacemaking?

Since its discovery, the sinoatrial node (SAN) has represented a fascinating and complex matter of research. Despite over a century of discoveries, a full comprehension of pacemaking has still to be achieved. Experiments often produced conflicting evidence that was used either in support or against alternative theories, originating intense debates. In this context, mathematical descriptions of the phenomena underlying the heartbeat have grown in importance in the last decades since they helped in gaining insights where experimental evaluation could not reach. This review presents the most updated SAN computational models and discusses their contribution to our understanding of cardiac pacemaking. Electrophysiological, structural and pathological aspects - as well as the autonomic control over the SAN - are taken into consideration to reach a holistic view of SAN activity.

Thermal ablation effects on rotors that characterize functional re-entry cardiac arrhythmia

Thermal ablation is a well-established successful treatment for cardiac arrhythmia, but it still presents limitations that require further studies and developments. In the rotor-driven functional re-entry arrhythmia, tissue heterogeneity results on the generation of spiral/scroll waves and wave break dynamics that may cause dangerous sustainable fibrillation. The selection of the target region to perform thermal ablation to mitigate this type of arrhythmia is challenging, since it considerably affects the local electrophysiology dynamics. This work deals with the numerical simulation of the thermal ablation of a cardiac muscle tissue and its effects on the dynamics of rotor-driven functional re-entry arrhythmia. A non-homogeneous two-dimensional rectangular region is used in the present numerical analysis, where radiofrequency ablation is performed. The electrophysiology problem for the propagation of the action potential in the cardiac tissue is simulated with the Fenton-Karma model. Thermal damage caused to the tissue by the radiofrequency heating is modeled by the Arrhenius equation. The effects of size and position of a heterogeneous region in the original muscle tissue were first analyzed, in order to verify the possible existence of the functional re-entry arrhythmia during the time period considered in the simulations. For each case that exhibited re-entry arrhythmia, six different ablation procedures were analyzed, depending on the position of the radiofrequency electrode and heating time. The obtained results revealed the effects of different model parameters on the existence and possible mitigation of the functional re-entry arrhythmia.

A Parameter Representing Missing Charge Should Be Considered when Calibrating Action Potential Models

Computational models of the electrical potential across a cell membrane are longstanding and vital tools in electrophysiology research and applications. These models describe how ionic currents, internal fluxes, and buffering interact to determine membrane voltage and form action potentials (APs). Although this relationship is usually expressed as a differential equation, previous studies have shown it can be rewritten in an algebraic form, allowing direct calculation of membrane voltage. Rewriting in this form requires the introduction of a new parameter, called Γ₀ in this manuscript, which represents the net concentration of all charges that influence membrane voltage but are not considered in the model. Although several studies have examined the impact of Γ₀ on long-term stability and drift in model predictions, there has been little examination of its effects on model predictions, particularly when a model is refit to new data. In this study, we illustrate how Γ₀ affects important physiological properties such as action potential duration restitution, and examine the effects of (in)correctly specifying Γ₀ during model calibration. We show that, although physiologically plausible, the range of concentrations used in popular models leads to orders of magnitude differences in Γ₀, which can lead to very different model predictions. In model calibration, we find that using an incorrect value of Γ₀ can lead to biased estimates of the inferred parameters, but that the predictive power of these models can be restored by fitting Γ₀ as a separate parameter. These results show the value of making Γ₀ explicit in model formulations, as it forces modellers and experimenters to consider the effects of uncertainty and potential discrepancy in initial concentrations upon model predictions.

Rate dependence of early afterdepolarizations in the His-Purkinje system: A simulation study

BACKGROUND AND OBJECTIVE

Early afterdepolarizations (EADs) are associated with a variety of arrhythmias and have the property of rate dependence. EADs can occur in Purkinje cells while the effect of rate dependence of EADs in the His-Purkinje system has not been fully investigated. In order to reveal the rate dependence of EADs in the His-Purkinje system and its effect on ventricular electrical activities, the simulation research was carried out in this manuscript.

METHODS

This manuscript first studied the relationship between the occurrence of EADs and stimulation cycle length on the DiFranNoble cell model. Then, the relationship between the rate dependence of EADs and the conduction block of the His-Purkinje system at slow heart rates was studied on the rabbit whole ventricular model including the His-Purkinje system, and its mechanism was analyzed from multiple angles.

RESULTS

① The rate dependence of EADs is related to the inconsistency of EADs occurrence in the His-Purkinje system. When the stimulation cycle length is long or short enough, EADs either occur or not occur stably in the His-Purkinje system, while in a certain stimulation cycle length window, the chaotic state of EADs will be observed. ② The key subcellular factors x-gate is an important mechanism involved to the rate dependence of EADs in the His-Purkinje system. ③ The discrete distribution of x-gate values and the "source-sink" mechanism lead to the inconsistency of EADs in the His-Purkinje system. The prolonged action potential duration caused by EADs can lead to conduction block at slow heart rates.

CONCLUSION

The rate dependence of EADs in Purkinje system can lead to disordered ventricular electrical activity.

Reduced Models of Cardiomyocytes Excitability: Comparing Karma and FitzHugh-Nagumo

Since Noble adapted in 1962 the model of Hodgkin and Huxley to fit Purkinje fibres, the refinement of models for cardiomyocytes has continued. Most of these models are high-dimensional systems of coupled equations so that the possible mathematical analysis is quite limited, even numerically. This has inspired the development of reduced, phenomenological models that preserve qualitatively the main feature of cardiomyocyte's dynamics. In this paper, we present a systematic comparison of the dynamics between two notable low-dimensional models, the FitzHugh-Nagumo model (FitzHugh in Bull Math Biophys 17:257-269, 1955, J Gen Physiol 43:867-896, 1960, Biophys J 1:445-466, 1961) as a prototype of excitable behaviour and a polynomial version of the Karma model (Karma in Phys Rev Lett 71(7):16, 1993, Chaos 4:461, 1994) which is specifically developed to fit cardiomyocyte's behaviour well. We start by introducing the models and considering their pure ODE versions. We analyse the ODEs employing the main ideas and steps used in the setting of geometric singular perturbation theory. Next, we turn to the spatially extended models, where we focus on travelling wave solutions in 1D. Finally, we perform numerical simulations of the 1D PDE Karma model varying model parameters in order to systematically investigate the impact on wave propagation velocity and shape. In summary, our study provides a reference regarding key similarities as well as key differences of the two models.

An in silico hiPSC-Derived Cardiomyocyte Model Built With Genetic Algorithm

The formulation of in silico biophysical models generally requires optimization strategies for reproducing experimentally observed phenomena. In electrophysiological modeling, robust nonlinear regressive methods are often crucial for guaranteeing high fidelity models. Human induced pluripotent stem cell-derived cardiomyocytes (hiPSC-CMs), though nascent, have proven to be useful in cardiac safety pharmacology, regenerative medicine, and in the implementation of patient-specific test benches for investigating inherited cardiac disorders. This study demonstrates the potency of heuristic techniques at formulating biophysical models, with emphasis on a hiPSC-CM model using a novel genetic algorithm (GA) recipe we proposed. The proposed GA protocol was used to develop a hiPSC-CM biophysical computer model by fitting mathematical formulations to experimental data for five ionic currents recorded in hiPSC-CMs. The maximum conductances of the remaining ionic channels were scaled based on recommendations from literature to accurately reproduce the experimentally observed hiPSC-CM action potential (AP) metrics. Near-optimal parameter fitting was achieved for the GA-fitted ionic currents. The resulting model recapitulated experimental AP parameters such as AP durations (APD₅₀, APD₇₅, and APD₉₀), maximum diastolic potential, and frequency of automaticity. The outcome of this work has implications for validating the biophysics of hiPSC-CMs in their use as viable substitutes for human cardiomyocytes, particularly in cardiac safety pharmacology and in the study of inherited cardiac disorders. This study presents a novel GA protocol useful for formulating robust numerical biophysical models. The proposed protocol is used to develop a hiPSC-CM model with implications for cardiac safety pharmacology.

Computational models of atrial fibrillation: achievements, challenges, and perspectives for improving clinical care

Despite significant advances in its detection, understanding and management, atrial fibrillation (AF) remains a highly prevalent cardiac arrhythmia with a major impact on morbidity and mortality of millions of patients. AF results from complex, dynamic interactions between risk factors and comorbidities that induce diverse atrial remodelling processes. Atrial remodelling increases AF vulnerability and persistence, while promoting disease progression. The variability in presentation and wide range of mechanisms involved in initiation, maintenance and progression of AF, as well as its associated adverse outcomes, make the early identification of causal factors modifiable with therapeutic interventions challenging, likely contributing to suboptimal efficacy of current AF management. Computational modelling facilitates the multilevel integration of multiple datasets and offers new opportunities for mechanistic understanding, risk prediction and personalized therapy. Mathematical simulations of cardiac electrophysiology have been around for 60 years and are being increasingly used to improve our understanding of AF mechanisms and guide AF therapy. This narrative review focuses on the emerging and future applications of computational modelling in AF management. We summarize clinical challenges that may benefit from computational modelling, provide an overview of the different in silico approaches that are available together with their notable achievements, and discuss the major limitations that hinder the routine clinical application of these approaches. Finally, future perspectives are addressed. With the rapid progress in electronic technologies including computing, clinical applications of computational modelling are advancing rapidly. We expect that their application will progressively increase in prominence, especially if their added value can be demonstrated in clinical trials.

Evolution of mathematical models of cardiomyocyte electrophysiology

Advanced computational techniques and mathematical modeling have become more and more important to the study of cardiac electrophysiology. In this review, we provide a brief history of the evolution of cardiomyocyte electrophysiology models and highlight some of the most important ones that had a major impact on our understanding of the electrical activity of the myocardium and associated transmembrane ion fluxes in normal and pathological states. We also present the use of these models in the study of various arrhythmogenesis mechanisms, particularly the integration of experimental pharmacology data into advanced humanized models for in silico proarrhythmogenic risk prediction as an essential component of the Comprehensive in vitro Proarrhythmia Assay (CiPA) drug safety paradigm.

OPTIMIZATION OF RABBIT VENTRICULAR ELECTROPHYSIOLOGICAL MODEL AND SIMULATION OF SYNTHETIC ELECTROCARDIOGRAM

This study aimed to use computer simulation method to study the mechanism of cardiac electrical activities. We optimized an electrophysiological rabbit ventricular model, including myocardial segmentation, heterogeneity and a realistic His-Purkinje network. Simulations of normal state, several types of ventricular premature contractions (VPC), conduction system pacing and right ventricular apical pacing were performed and the detailed cardiac electrical activities were studied from cell level to electrocardiogram (ECG) level. A detailed multiscale optimized ventricular model was obtained. The model effectively simulated various types of electrical activities. The synthetic ECG results were very similar to the real clinical ECG. The duration of QRS of typical VPC is 58[Formula: see text]ms, 71% longer than that of a normal-state synthetic QRS and the amplitude of the QRS is 35% larger, while the QRS duration and amplitude of the real clinical ECG of typical VPC are 69% longer and 36% larger than those of the real normal QRS. The duration of QRS of ventricular fusion beat is 31[Formula: see text]ms, 91% of that of a normal-state synthetic QRS and the amplitude of the QRS is 36% larger, while the QRS duration of the real clinical ECG of a ventricular fusion beat is 92% of the real normal QRS and the amplitude is 37% larger. Therefore, the results indicate that this model is effective and reliable in studying the detailed process of cardiac excitation and pacing.

Composite Backward Differentiation Formula for the Bidomain Equations

The bidomain equations have been widely used to model the electrical activity of cardiac tissue. While it is well-known that implicit methods have much better stability than explicit methods, implicit methods usually require the solution of a very large nonlinear system of equations at each timestep which is computationally prohibitive. In this work, we present two fully implicit time integration methods for the bidomain equations: the backward Euler method and a second-order one-step two-stage composite backward differentiation formula (CBDF2) which is an L-stable time integration method. Using the backward Euler method as fundamental building blocks, the CBDF2 scheme is easily implementable. After solving the nonlinear system resulting from application of the above two fully implicit schemes by a nonlinear elimination method, the obtained nonlinear global system has a much smaller size, whose Jacobian is symmetric and possibly positive definite. Thus, the residual equation of the approximate Newton approach for the global system can be efficiently solved by standard optimal solvers. As an alternative, we point out that the above two implicit methods combined with operator splittings can also efficiently solve the bidomain equations. Numerical results show that the CBDF2 scheme is an efficient time integration method while achieving high stability and accuracy.

Cardiac transmembrane ion channels and action potentials: cellular physiology and arrhythmogenic behavior

Cardiac arrhythmias are among the leading causes of mortality. They often arise from alterations in the electrophysiological properties of cardiac cells and their underlying ionic mechanisms. It is therefore critical to further unravel the pathophysiology of the ionic basis of human cardiac electrophysiology in health and disease. In the first part of this review, current knowledge on the differences in ion channel expression and properties of the ionic processes that determine the morphology and properties of cardiac action potentials and calcium dynamics from cardiomyocytes in different regions of the heart are described. Then the cellular mechanisms promoting arrhythmias in congenital or acquired conditions of ion channel function (electrical remodeling) are discussed. The focus is on human-relevant findings obtained with clinical, experimental, and computational studies, given that interspecies differences make the extrapolation from animal experiments to human clinical settings difficult. Deepening the understanding of the diverse pathophysiology of human cellular electrophysiology will help in developing novel and effective antiarrhythmic strategies for specific subpopulations and disease conditions.

An Understandable, Extensible, and Reusable Implementation of the Hodgkin-Huxley Equations Using Modelica

The Hodgkin-Huxley model of the squid giant axon has been used for decades as the basis of many action potential models. These models are usually communicated using just a list of equations or a circuit diagram, which makes them unnecessarily complicated both for novices and for experts. We present a modular version of the Hodgkin-Huxley model that is more understandable than the usual monolithic implementations and that can be easily reused and extended. Our model is written in Modelica using software engineering concepts, such as object orientation and inheritance. It retains the electrical analogy, but names and explains individual components in biological terms. We use cognitive load theory to measure understandability as the amount of items that have to be kept in working memory simultaneously. The model is broken down into small self-contained components in human-readable code with extensive documentation. Additionally, it features a hybrid diagram that uses biological symbols in an electrical circuit and that is directly tied to the model code. The new model design avoids many redundancies and reduces the cognitive load associated with understanding the model by a factor of 6. Extensions can be easily applied due to an unifying interface and inheritance from shared base classes. The model can be used in an educational context as a more approachable introduction to mathematical modeling in electrophysiology. Additionally the modeling approach and the base components can be used to make complex Hodgkin-Huxley-type models more understandable and reusable.

A Mathematical Model of the Mouse Atrial Myocyte With Inter-Atrial Electrophysiological Heterogeneity

Biophysically detailed mathematical models of cardiac electrophysiology provide an alternative to experimental approaches for investigating possible ionic mechanisms underlying the genesis of electrical action potentials and their propagation through the heart. The aim of this study was to develop a biophysically detailed mathematical model of the action potentials of mouse atrial myocytes, a popular experimental model for elucidating molecular and cellular mechanisms of arrhythmogenesis. Based on experimental data from isolated mouse atrial cardiomyocytes, a set of mathematical equations for describing the biophysical properties of membrane ion channel currents, intracellular Ca²⁺ handling, and Ca²⁺-calmodulin activated protein kinase II and β-adrenergic signaling pathways were developed. Wherever possible, membrane ion channel currents were modeled using Markov chain formalisms, allowing detailed representation of channel kinetics. The model also considered heterogeneous electrophysiological properties between the left and the right atrial cardiomyocytes. The developed model was validated by its ability to reproduce the characteristics of action potentials and Ca²⁺ transients, matching quantitatively to experimental data. Using the model, the functional roles of four K⁺ channel currents in atrial action potential were evaluated by channel block simulations, results of which were quantitatively in agreement with existent experimental data. To conclude, this newly developed model of mouse atrial cardiomyocytes provides a powerful tool for investigating possible ion channel mechanisms of atrial electrical activity at the cellular level and can be further used to investigate mechanisms underlying atrial arrhythmogenesis.

Calibration of ionic and cellular cardiac electrophysiology models

Cardiac electrophysiology models are among the most mature and well-studied mathematical models of biological systems. This maturity is bringing new challenges as models are being used increasingly to make quantitative rather than qualitative predictions. As such, calibrating the parameters within ion current and action potential (AP) models to experimental data sets is a crucial step in constructing a predictive model. This review highlights some of the fundamental concepts in cardiac model calibration and is intended to be readily understood by computational and mathematical modelers working in other fields of biology. We discuss the classic and latest approaches to calibration in the electrophysiology field, at both the ion channel and cellular AP scales. We end with a discussion of the many challenges that work to date has raised and the need for reproducible descriptions of the calibration process to enable models to be recalibrated to new data sets and built upon for new studies. This article is categorized under: Analytical and Computational Methods > Computational Methods Physiology > Mammalian Physiology in Health and Disease Models of Systems Properties and Processes > Cellular Models.

Anti-arrhythmic properties of non-antiarrhythmic medications

Traditional anti-arrhythmic drugs are classified by the Vaughan-Williams classification scheme based on their mechanisms of action, which includes effects on receptors and/or ion channels. Some known anti-arrhythmic drugs do not perfectly fit into this classification scheme. Other medications/molecules with established non-anti-arrhythmic indications have shown anti-arrhythmic properties worth exploring. In this narrative review, we discuss the molecular mechanisms and evidence base for the anti-arrhythmic properties of traditional non-antiarrhythmic drugs such as inhibitors of the renin angiotensin system (RAS), statins and polyunsaturated fatty acids (PUFAs). In summary, RAS antagonists, statins and PUFAs are 'upstream target modulators' that appear to have anti-arrhythmic roles. RAS blockers prevent the downstream arrhythmogenic effects of angiotensin II - the main effector peptide of RAS - and the angiotensin type 1 receptor. Statins have pleiotropic effects including anti-inflammatory, immunomodulatory, modulation of autonomic nervous system, anti-proliferative and anti-oxidant actions which appear to underlie their anti-arrhythmic properties. PUFAs have the ability to alter ion channel function and prevent excessive accumulation of calcium ions in cardiac myocytes, which might explain their benefits in certain arrhythmic conditions. Clearly, whilst a number of anti-arrhythmic drugs exist, there is still a need for randomised trials to establish whether additional agents, including those already in clinical use, have significant anti-arrhythmic effects.

Mathematical model of the ventricular action potential and effects of isoproterenol-induced cardiac hypertrophy in rats

Mathematical action potential (AP) modeling is a well-established but still-developing area of research to better understand physiological and pathological processes. In particular, changes in AP mechanisms in the isoproterenol (ISO) -induced hypertrophic heart model are incompletely understood. Here we present a mathematical model of the rat AP based on recordings from rat ventricular myocytes. In our model, for the first time, all channel kinetics are defined with a single type of function that is simple and easy to apply. The model AP and channels dynamics are consistent with the APs recorded from rats for both Control (absence of ISO) and ISO-treated cases. Our mathematical model helps us to understand the reason for the prolongation in AP duration after ISO application while ISO treatment helps us to validate our mathematical model. We reveal that the smaller density and the slower gating kinetics of the transient K⁺ current help explain the prolonged AP duration after ISO treatment and the increasing amplitude of the rapid and the slow inward rectifier currents also contribute to this prolongation alongside the flux in Ca²⁺ currents. ISO induced an increase in the density of the Na⁺ current that can explain the faster upstroke. We believe that AP dynamics from rat ventricular myocytes can be reproduced very well with this mathematical model and that it provides a powerful tool for improved insights into the underlying dynamics of clinically important AP properties such as ISO application.

The Cardiac Pacemaker Story-Fundamental Role of the Na+/Ca2+ Exchanger in Spontaneous Automaticity

The electrophysiological mechanism of the sinus node automaticity was previously considered exclusively regulated by the so-called "funny current". However, parallel investigations increasingly emphasized the importance of the Ca²⁺-homeostasis and Na⁺/Ca²⁺ exchanger (NCX). Recently, increasing experimental evidence, as well as insight through mechanistic in silico modeling demonstrates the crucial role of the exchanger in sinus node pacemaking. NCX had a key role in the exciting story of discovery of sinus node pacemaking mechanisms, which recently settled with a consensus on the coupled-clock mechanism after decades of debate. This review focuses on the role of the Na⁺/Ca²⁺ exchanger from the early results and concepts to recent advances and attempts to give a balanced summary of the characteristics of the local, spontaneous, and rhythmic Ca²⁺ releases, the molecular control of the NCX and its role in the fight-or-flight response. Transgenic animal models and pharmacological manipulation of intracellular Ca²⁺ concentration and/or NCX demonstrate the pivotal function of the exchanger in sinus node automaticity. We also highlight where specific hypotheses regarding NCX function have been derived from computational modeling and require experimental validation. Nonselectivity of NCX inhibitors and the complex interplay of processes involved in Ca²⁺ handling render the design and interpretation of these experiments challenging.

Cardiomyocyte calcium handling in health and disease: Insights from in vitro and in silico studies

Calcium (Ca²⁺) plays a central role in cardiomyocyte excitation-contraction coupling. To ensure an optimal electrical impulse propagation and cardiac contraction, Ca²⁺ levels are regulated by a variety of Ca²⁺-handling proteins. In turn, Ca²⁺ modulates numerous electrophysiological processes. Accordingly, Ca²⁺-handling abnormalities can promote cardiac arrhythmias via various mechanisms, including the promotion of afterdepolarizations, ion-channel modulation and structural remodeling. In the last 30 years, significant improvements have been made in the computational modeling of cardiomyocyte Ca²⁺ handling under physiological and pathological conditions. However, numerous questions involving the Ca²⁺-dependent regulation of different macromolecular complexes, cross-talk between Ca²⁺-dependent regulatory pathways operating over a wide range of time scales, and bidirectional interactions between electrophysiology and mechanics remain to be addressed by in vitro and in silico studies. A better understanding of disease-specific Ca²⁺-dependent proarrhythmic mechanisms may facilitate the development of improved therapeutic strategies. In this review, we describe the fundamental mechanisms of cardiomyocyte Ca²⁺ handling in health and disease, and provide an overview of currently available computational models for cardiomyocyte Ca²⁺ handling. Finally, we discuss important uncertainties and open questions about cardiomyocyte Ca²⁺ handling and highlight how synergy between in vitro and in silico studies may help to answer several of these issues.

A Brief History of Pacemaking

Cardiac pacemaking is a most fundamental cardiac function, thoroughly investigated for decades with a multiscale approach at organ, tissue, cell and molecular levels, to clarify the basic mechanisms underlying generation and control of cardiac rhythm. Understanding the processes involved in pacemaker activity is of paramount importance for a basic physiological knowledge, but also as a way to reveal details of pathological dysfunctions useful in the perspective of a therapeutic approach. Among the mechanisms involved in pacemaking, the "funny" (I_f) current has properties most specifically fitting the requirements for generation and control of repetitive activity, and has consequently received the most attention in studies of the pacemaker function. Present knowledge of the basic mechanisms of pacemaking and the properties of funny channels has led to important developments of clinical relevance. These include: (1) the successful development of heart rate-reducing agents, such as ivabradine, able to control cardiac rhythm and useful in the treatment of diseases such as coronary artery disease, heart failure and tachyarrhythmias; (2) the understanding of the genetic basis of disorders of cardiac rhythm caused by HCN channelopathies; (3) the design of strategies to implement biological pacemakers based on transfer of HCN channels or of stem cell-derived pacemaker cells expressing I_f, with the ultimate goal to replace electronic devices. In this review, I will give a brief historical account of the discovery of the funny current and the development of the concept of I_f-based pacemaking, in the context of a wider, more complex model of cardiac rhythmic function.

A novel gain-of-function mutation in SCN5A responsible for multifocal ectopic Purkinje-related premature contractions

Recently, four SCN5A mutations have been associated with Multifocal Ectopic Purkinje-related Premature Contractions (MEPPC), a rare cardiac syndrome combining polymorphic ventricular arrhythmia with dilated cardiomyopathy (DCM). Here, we identified a novel heterozygous mutation in SCN5A (c.611C>A, pAla204Glu) in a young woman presenting with polymorphic premature ventricular contractions (PVCs) and DCM. After failure of antiarrhythmic drugs and an attempt of radiofrequency catheter ablation showing three exit-sites of PVCs, all with presystolic Purkinje potentials, a treatment by hydroquinidine was tried, leading to an immediate and spectacular disappearance of all PVCs and normalization of cardiac function. Electrophysiological studies showed that Na_v 1.5-A204E mutant channels exhibited a significant leftward shift of 8 mV of the activation curve, leading to a larger hyperpolarized window current when compared to wild-type. Action potential modeling using Purkinje fiber and ventricular cell models predicted an arrhythmogenic effect predominant in Purkinje fibers for the A204E mutation. Comparison with other MEPPC-associated Na_v 1.5 mutations revealed a common electrophysiological pattern of abnormal voltage-dependence of activation leading to a larger hyperpolarized window current as a shared biophysical mechanism of this syndrome. These features of the mutant sodium channels are likely to be responsible for the hyperexcitability of the fascicular-Purkinje system observed in patients with MEPPC.

Four Ways to Fit an Ion Channel Model

Mathematical models of ionic currents are used to study the electrophysiology of the heart, brain, gut, and several other organs. Increasingly, these models are being used predictively in the clinic, for example, to predict the risks and results of genetic mutations, pharmacological treatments, or surgical procedures. These safety-critical applications depend on accurate characterization of the underlying ionic currents. Four different methods can be found in the literature to fit voltage-sensitive ion channel models to whole-cell current measurements: method 1, fitting model equations directly to time-constant, steady-state, and I-V summary curves; method 2, fitting by comparing simulated versions of these summary curves to their experimental counterparts; method 3, fitting to the current traces themselves from a range of protocols; and method 4, fitting to a single current trace from a short and rapidly fluctuating voltage-clamp protocol. We compare these methods using a set of experiments in which hERG1a current was measured in nine Chinese hamster ovary cells. In each cell, the same sequence of fitting protocols was applied, as well as an independent validation protocol. We show that methods 3 and 4 provide the best predictions on the independent validation set and that short, rapidly fluctuating protocols like that used in method 4 can replace much longer conventional protocols without loss of predictive ability. Although data for method 2 are most readily available from the literature, we find it performs poorly compared to methods 3 and 4 both in accuracy of predictions and computational efficiency. Our results demonstrate how novel experimental and computational approaches can improve the quality of model predictions in safety-critical applications.

A computational model of induced pluripotent stem-cell derived cardiomyocytes incorporating experimental variability from multiple data sources

Key points

Induced pluripotent stem cell‐derived cardiomyocytes (iPSC‐CMs) capture patient‐specific genotype–phenotype relationships, as well as cell‐to‐cell variability of cardiac electrical activity

Computational modelling and simulation provide a high throughput approach to reconcile multiple datasets describing physiological variability, and also identify vulnerable parameter regimes

We have developed a whole‐cell model of iPSC‐CMs, composed of single exponential voltage‐dependent gating variable rate constants, parameterized to fit experimental iPSC‐CM outputs

We have utilized experimental data across multiple laboratories to model experimental variability and investigate subcellular phenotypic mechanisms in iPSC‐CMs

This framework links molecular mechanisms to cellular‐level outputs by revealing unique subsets of model parameters linked to known iPSC‐CM phenotypes

Abstract

There is a profound need to develop a strategy for predicting patient‐to‐patient vulnerability in the emergence of cardiac arrhythmia. A promising in vitro method to address patient‐specific proclivity to cardiac disease utilizes induced pluripotent stem cell‐derived cardiomyocytes (iPSC‐CMs). A major strength of this approach is that iPSC‐CMs contain donor genetic information and therefore capture patient‐specific genotype–phenotype relationships. A cited detriment of iPSC‐CMs is the cell‐to‐cell variability observed in electrical activity. We postulated, however, that cell‐to‐cell variability may constitute a strength when appropriately utilized in a computational framework to build cell populations that can be employed to identify phenotypic mechanisms and pinpoint key sensitive parameters. Thus, we have exploited variation in experimental data across multiple laboratories to develop a computational framework for investigating subcellular phenotypic mechanisms. We have developed a whole‐cell model of iPSC‐CMs composed of simple model components comprising ion channel models with single exponential voltage‐dependent gating variable rate constants, parameterized to fit experimental iPSC‐CM data for all major ionic currents. By optimizing ionic current model parameters to multiple experimental datasets, we incorporate experimentally‐observed variability in the ionic currents. The resulting population of cellular models predicts robust inter‐subject variability in iPSC‐CMs. This approach links molecular mechanisms to known cellular‐level iPSC‐CM phenotypes, as shown by comparing immature and mature subpopulations of models to analyse the contributing factors underlying each phenotype. In the future, the presented models can be readily expanded to include genetic mutations and pharmacological interventions for studying the mechanisms of rare events, such as arrhythmia triggers.

Induced pluripotent stem cell‐derived cardiomyocytes (iPSC‐CMs) capture patient‐specific genotype–phenotype relationships, as well as cell‐to‐cell variability of cardiac electrical activity

Computational modelling and simulation provide a high throughput approach to reconcile multiple datasets describing physiological variability, and also identify vulnerable parameter regimes

We have developed a whole‐cell model of iPSC‐CMs, composed of single exponential voltage‐dependent gating variable rate constants, parameterized to fit experimental iPSC‐CM outputs

We have utilized experimental data across multiple laboratories to model experimental variability and investigate subcellular phenotypic mechanisms in iPSC‐CMs

This framework links molecular mechanisms to cellular‐level outputs by revealing unique subsets of model parameters linked to known iPSC‐CM phenotypes

Biological Relativity Requires Circular Causality but Not Symmetry of Causation: So, Where, What and When Are the Boundaries?

Since the Principle of Biological Relativity was formulated and developed there have been many implementations in a wide range of biological fields. The purpose of this article is to assess the status of the applications of the principle and to clarify some misunderstandings. The principle requires circular causality between levels of organization. But the forms of causality are also necessarily different. They contribute in asymmetric ways. Upward causation can be represented by the differential or similar equations describing the mechanics of lower level processes. Downward causation is then best represented as determining initial and boundary conditions. The questions tackled in this article are: (1) where and when do these boundaries exist? and (2) how do they convey the influences between levels? We show that not all boundary conditions arise from higher-level organization. It is important to distinguish those that do from those that don't. Both forms play functional roles in organisms, particularly in their responses to novel challenges. The forms of causation also change according to the levels concerned. These principles are illustrated with specific examples.

Cellular Mechanisms of Sinus Node Dysfunction in Carriers of the SCN5A-E161K Mutation and Role of the H558R Polymorphism

Background: Carriers of the E161K mutation in the SCN5A gene, encoding the Na_V1.5 pore-forming α-subunit of the ion channel carrying the fast sodium current (I_Na), show sinus bradycardia and occasional exit block. Voltage clamp experiments in mammalian expression systems revealed a mutation-induced 2.5- to 4-fold reduction in I_Na peak current density as well as a +19 mV shift and reduced steepness of the steady-state activation curve. The highly common H558R polymorphism in Na_V1.5 limits this shift to +13 mV, but also introduces a -10 mV shift in steady-state inactivation. Aim: We assessed the cellular mechanism by which the E161K mutation causes sinus node dysfunction in heterozygous mutation carriers as well as the potential role of the H558R polymorphism. Methods: We incorporated the mutation-induced changes in I_Na into the Fabbri-Severi model of a single human sinoatrial node cell and the Maleckar et al. human atrial cell model, and carried out simulations under control conditions and over a wide range of acetylcholine levels. Results: In absence of the H558R polymorphism, the E161K mutation increased the basic cycle length of the sinoatrial node cell from 813 to 866 ms. In the simulated presence of 10 and 25 nM acetylcholine, basic cycle length increased from 1027 to 1131 and from 1448 to 1795 ms, respectively. The increase in cycle length was the result of a significant slowing of diastolic depolarization. The mutation-induced reduction in I_Na window current had reduced the contribution of the mutant component of I_Na to the net membrane current during diastolic depolarization to effectively zero. Highly similar results were obtained in presence of the H558R polymorphism. Atrial excitability was reduced, both in absence and presence of the H558R polymorphism, as reflected by an increase in threshold stimulus current and a concomitant decrease in capacitive current of the atrial cell. Conclusion: We conclude that the experimentally identified mutation-induced changes in I_Na can explain the clinically observed sinus bradycardia and potentially the occasional exit block. Furthermore, we conclude that the common H558R polymorphism does not significantly alter the effects of the E161K mutation and can thus not explain the reduced penetrance of the E161K mutation.

A thermodynamic description for physiological transmembrane transport

A general formulation for both passive and active transmembrane transport is derived from basic thermodynamical principles. The derivation takes into account the energy required for the motion of molecules across membranes and includes the possibility of modeling asymmetric flow. Transmembrane currents can then be described by the general model in the case of electrogenic flow. As it is desirable in new models, it is possible to derive other well-known expressions for transmembrane currents as particular cases of the general formulation. For instance, the conductance-based formulation for current turns out to be a linear approximation of the general formula for current. Also, under suitable assumptions, other formulas for current based on electrodiffusion, like the constant field approximation by Goldman, can be recovered from the general formulation. The applicability of the general formulations is illustrated first with fits to existing data, and after, with models of transmembrane potential dynamics for pacemaking cardiocytes and neurons. The general formulations presented here provide a common ground for the biophysical study of physiological phenomena that depend on transmembrane transport.

A Heart for Diversity: Simulating Variability in Cardiac Arrhythmia Research

In cardiac electrophysiology, there exist many sources of inter- and intra-personal variability. These include variability in conditions and environment, and genotypic and molecular diversity, including differences in expression and behavior of ion channels and transporters, which lead to phenotypic diversity (e.g., variable integrated responses at the cell, tissue, and organ levels). These variabilities play an important role in progression of heart disease and arrhythmia syndromes and outcomes of therapeutic interventions. Yet, the traditional in silico framework for investigating cardiac arrhythmias is built upon a parameter/property-averaging approach that typically overlooks the physiological diversity. Inspired by work done in genetics and neuroscience, new modeling frameworks of cardiac electrophysiology have been recently developed that take advantage of modern computational capabilities and approaches, and account for the variance in the biological data they are intended to illuminate. In this review, we outline the recent advances in statistical and computational techniques that take into account physiological variability, and move beyond the traditional cardiac model-building scheme that involves averaging over samples from many individuals in the construction of a highly tuned composite model. We discuss how these advanced methods have harnessed the power of big (simulated) data to study the mechanisms of cardiac arrhythmias, with a special emphasis on atrial fibrillation, and improve the assessment of proarrhythmic risk and drug response. The challenges of using in silico approaches with variability are also addressed and future directions are proposed.

A general framework dedicated to computational morphogenesis Part I - Constitutive equations

In order to understand living organisms, considerable experimental efforts and resources have been devoted to correlate genes and their expressions with cell, tissue, organ and whole organisms' phenotypes. This data driven approach to knowledge discovery has led to many breakthrough in our understanding of healthy and diseased states, and is paving the way to improve the diagnosis and treatment of diseases. Complementary to this data-driven approach, computational models of biological systems based on first principles have been developed in order to deepen our understanding of the multi-scale dynamics that drives normal and pathological biological functions. In this paper we describe the biological, physical and mathematical concepts that led to the design of a Computational Morphogenesis (CM) platform baptized Generic Modeling and Simulating Platform (GMSP). Its role is to generate realistic 3D multi-scale biological tissues from virtual stem cells and the intended target applications include in virtuo studies of normal and abnormal tissue (re)generation as well as the development of complex diseases such as carcinogenesis. At all space-scales of interest, biological agents interact with each other via biochemical, bioelectrical, and mechanical fields that operate in concert during embryogenesis, growth and adult life. The spatio-temporal dependencies of these fields can be modeled by physics-based constitutive equations that we propose to examine in relation to the landmark biological events that occur during embryogenesis.

Numerical approximation of the electromechanical coupling in the left ventricle with inclusion of the Purkinje network

In this work, we consider the numerical approximation of the electromechanical coupling in the left ventricle with inclusion of the Purkinje network. The mathematical model couples the 3D elastodynamics and bidomain equations for the electrophysiology in the myocardium with the 1D monodomain equation in the Purkinje network. For the numerical solution of the coupled problem, we consider a fixed-point iterative algorithm that enables a partitioned solution of the myocardium and Purkinje network problems. Different levels of myocardium-Purkinje network splitting are considered and analyzed. The results are compared with those obtained using standard strategies proposed in the literature to trigger the electrical activation. Finally, we present a numerical study that, although performed in an idealized computational domain, features all the physiological issues that characterize a heartbeat simulation, including the initiation of the signal in the Purkinje network and the systolic and diastolic phases.

Bond graph modelling of the cardiac action potential: implications for drift and non-unique steady states

Mathematical models of cardiac action potentials have become increasingly important in the study of heart disease and pharmacology, but concerns linger over their robustness during long periods of simulation, in particular due to issues such as model drift and non-unique steady states. Previous studies have linked these to violation of conservation laws, but only explored those issues with respect to charge conservation in specific models. Here, we propose a general and systematic method of identifying conservation laws hidden in models of cardiac electrophysiology by using bond graphs, and develop a bond graph model of the cardiac action potential to study long-term behaviour. Bond graphs provide an explicit energy-based framework for modelling physical systems, which makes them well suited for examining conservation within electrophysiological models. We find that the charge conservation laws derived in previous studies are examples of the more general concept of a 'conserved moiety'. Conserved moieties explain model drift and non-unique steady states, generalizing the results from previous studies. The bond graph approach provides a rigorous method to check for drift and non-unique steady states in a wide range of cardiac action potential models, and can be extended to examine behaviours of other excitable systems.

Simultaneous Quantification of Spatially Discordant Alternans in Voltage and Intracellular Calcium in Langendorff-Perfused Rabbit Hearts and Inconsistencies with Models of Cardiac Action Potentials and Ca Transients

Rationale: Discordant alternans, a phenomenon in which the action potential duration (APDs) and/or intracellular calcium transient durations (CaDs) in different spatial regions of cardiac tissue are out of phase, present a dynamical instability for complex spatial dispersion that can be associated with long-QT syndrome (LQTS) and the initiation of reentrant arrhythmias. Because the use of numerical simulations to investigate arrhythmic effects, such as acquired LQTS by drugs is beginning to be studied by the FDA, it is crucial to validate mathematical models that may be used during this process. Objective: In this study, we characterized with high spatio-temporal resolution the development of discordant alternans patterns in transmembrane voltage (Vm) and intracellular calcium concentration ([Cai]+2) as a function of pacing period in rabbit hearts. Then we compared the dynamics to that of the latest state-of-the-art model for ventricular action potentials and calcium transients to better understand the underlying mechanisms of discordant alternans and compared the experimental data to the mathematical models representing Vm and [Cai]+2 dynamics. Methods and Results: We performed simultaneous dual optical mapping imaging of Vm and [Cai]+2 in Langendorff-perfused rabbit hearts with higher spatial resolutions compared with previous studies. The rabbit hearts developed discordant alternans through decreased pacing period protocols and we quantified the presence of multiple nodal points along the direction of wave propagation, both in APD and CaD, and compared these findings with results from theoretical models. In experiments, the nodal lines of CaD alternans have a steeper slope than those of APD alternans, but not as steep as predicted by numerical simulations in rabbit models. We further quantified several additional discrepancies between models and experiments. Conclusions: Alternans in CaD have nodal lines that are about an order of magnitude steeper compared to those of APD alternans. Current action potential models lack the necessary coupling between voltage and calcium compared to experiments and fail to reproduce some key dynamics such as, voltage amplitude alternans, smooth development of calcium alternans in time, conduction velocity and the steepness of the nodal lines of APD and CaD.

Integrative Approach with Electrophysiological and Theoretical Methods Reveals a New Role of S4 Positively Charged Residues in PKD2L1 Channel Voltage-Sensing

Numerical model-based simulations provide important insights into ion channel gating when experimental limitations exist. Here, a novel strategy combining numerical simulations with patch clamp experiments was used to investigate the net positive charges in the putative transmembrane segment 4 (S4) of the atypical, positively-shifted voltage-dependence of polycystic kidney disease 2-like 1 (PKD2L1) channel. Charge-neutralising mutations (K452Q, K455Q and K461Q) in S4 reduced gating charges, positively shifted the Boltzmann-type activation curve [i.e., open probability (P_open)-V curve] and altered the time-courses of activation/deactivation of PKD2L1, indicating that this region constitutes part of a voltage sensor. Numerical reconstruction of wild-type (WT) and mutant PKD2L1-mediated currents necessitated, besides their voltage-dependent gating parameters, a scaling factor that describes the voltage-dependence of maximal conductance, G_max. Subsequent single-channel conductance (γ) measurements revealed that voltage-dependence of G_max in WT can be explained by the inward-rectifying property of γ, which is greatly changed in PKD2L1 mutants. Homology modelling based on PKD2 and Na_VAb structures suggest that such voltage dependence of P_open and γ in PKD2L1 could both reflect the charged state of the S4 domain. The present conjunctive experimental and theoretical approaches provide a framework to explore the undetermined mechanism(s) regulating TRP channels that possess non-classical voltage-dependent properties.

A Biophysical Model for Cytotoxic Cell Swelling

We present a dynamic biophysical model to explain neuronal swelling underlying cytotoxic edema in conditions of low energy supply, as observed in cerebral ischemia. Our model contains Hodgkin-Huxley-type ion currents, a recently discovered voltage-gated chloride flux through the ion exchanger SLC26A11, active KCC2-mediated chloride extrusion, and ATP-dependent pumps. The model predicts changes in ion gradients and cell swelling during ischemia of various severity or channel blockage with realistic timescales. We theoretically substantiate experimental observations of chloride influx generating cytotoxic edema, while sodium entry alone does not. We show a tipping point of Na⁺/K⁺-ATPase functioning, where below cell volume rapidly increases as a function of the remaining pump activity, and a Gibbs-Donnan-like equilibrium state is reached. This precludes a return to physiological conditions even when pump strength returns to baseline. However, when voltage-gated sodium channels are temporarily blocked, cell volume and membrane potential normalize, yielding a potential therapeutic strategy.

SIGNIFICANCE STATEMENT

Cytotoxic edema most commonly results from energy shortage, such as in cerebral ischemia, and refers to the swelling of brain cells due to the entry of water from the extracellular space. We show that the principle of electroneutrality explains why chloride influx is essential for the development of cytotoxic edema. With the help of a biophysical model of a single neuron, we show that a tipping point of the energy supply exists, below which the cell volume rapidly increases. We simulate realistic time courses to and reveal critical components of neuronal swelling in conditions of low energy supply. Furthermore, we show that, after transient blockade of the energy supply, cytotoxic edema may be reversed by temporary blockade of Na⁺ channels.

Improved hybrid/GPU algorithm for solving cardiac electrophysiology problems on Purkinje networks

Cardiac Purkinje fibers provide an important pathway to the coordinated contraction of the heart. We present a numerical algorithm for the solution of electrophysiology problems across the Purkinje network that is efficient enough to be used in in silico studies on realistic Purkinje networks with physiologically detailed models of ion exchange at the cell membrane. The algorithm is on the basis of operator splitting and is provided with 3 different implementations: pure CPU, hybrid CPU/GPU, and pure GPU. Compared to our previous work, we modify the explicit gap junction term at network bifurcations to improve its mathematical consistency. Due to this improved consistency of the model, we are able to perform an empirical convergence study against analytical solutions. The study verified that all 3 implementations produce equivalent convergence rates, and shows that the algorithm produces equivalent result across different hardware platforms. Finally, we compare the efficiency of all 3 implementations on Purkinje networks of increasing spatial resolution using membrane models of increasing complexity. Both hybrid and pure GPU implementations outperform the pure CPU implementation, but their relative performance difference depends on the size of the Purkinje network and the complexity of the membrane model used.

Computational analysis of the human sinus node action potential: model development and effects of mutations

KEY POINTS

We constructed a comprehensive mathematical model of the spontaneous electrical activity of a human sinoatrial node (SAN) pacemaker cell, starting from the recent Severi-DiFrancesco model of rabbit SAN cells. Our model is based on electrophysiological data from isolated human SAN pacemaker cells and closely matches the action potentials and calcium transient that were recorded experimentally. Simulated ion channelopathies explain the clinically observed changes in heart rate in corresponding mutation carriers, providing an independent qualitative validation of the model. The model shows that the modulatory role of the 'funny current' (If ) in the pacing rate of human SAN pacemaker cells is highly similar to that of rabbit SAN cells, despite its considerably lower amplitude. The model may prove useful in the design of experiments and the development of heart-rate modulating drugs.

ABSTRACT

The sinoatrial node (SAN) is the normal pacemaker of the mammalian heart.  Over several decades, a large amount of data on the ionic mechanisms underlying the spontaneous electrical activity of SAN pacemaker cells has been obtained, mostly in experiments on single cells isolated from rabbit SAN. This wealth of data has allowed the development of mathematical models of the electrical activity of rabbit SAN pacemaker cells. The present study aimed to construct a comprehensive model of the electrical activity of a human SAN pacemaker cell using recently obtained electrophysiological data from human SAN pacemaker cells.  We based our model on the recent Severi-DiFrancesco model of a rabbit SAN pacemaker cell. The action potential and calcium transient of the resulting model are close to the experimentally recorded values. The model has a much smaller 'funny current' (If ) than do rabbit cells, although its modulatory role is highly similar. Changes in pacing rate upon the implementation of mutations associated with sinus node dysfunction agree with the clinical observations. This agreement holds for both loss-of-function and gain-of-function mutations in the HCN4, SCN5A and KCNQ1 genes, underlying ion channelopathies in If , fast sodium current and slow delayed rectifier potassium current, respectively. We conclude that our human SAN cell model can be a useful tool in the design of experiments and the development of drugs that aim to modulate heart rate.

An approach to expand description of the pump and co-transporter steady-state current

The membrane transporters (pumps and co-transporters) are the main players in maintaining the cell homeostasis. Models of various types, each with their own drawbacks, describe transporter behavior. The aim of this study is to find the link between the biophysically based and empirical models to face and solve their specific problems. Instead of decreasing the number of states and using few complex rate constants as is usually done, we use the number of states as great as possible. Then, each transition in the cycle can represent an elementary process and we can apply the mass action law, according to which if rate constants depend on concentrations the dependence is linear. Thus, the expression for the steady state transporter current can be transformed from a function of rate constants into a function of concentrations. When transporter states form a single cycle, it can be characterized by two modes of action - forward and backward ones. Specific mode is realized depending on the available free energy. Each mode of action is characterized by a set of transporter affinities together with a parameter that describes the maximal turning rate. Except standard affinities corresponding to the substances that are binding to the transporter, affinities for the substances that are released are also defined. Such scheme provides great possibilities to construct approximations as each individual affinity could be estimated from experiments as precisely as possible. The approximations may be used for not only description and study of the transporter current but also in cellular models that attempt to describe wide variety of processes in excitable cells.

Study of the union method of microelectrode array and AFM for the recording of electromechanical activities in living cardiomyocytes

Electrophysiology and mechanics are two essential components in the functions of cardiomyocytes and skeletal muscle cells. The simultaneous recording of electrophysiological and mechanical activities is important for the understanding of mechanisms underlying cell functions. For example, on the one hand, mechanisms under cardiovascular drug effects will be investigated in a comprehensive way by the simultaneous recording of electrophysiological and mechanical activities. On the other hand, computational models of electromechanics provide a powerful tool for the research of cardiomyocytes. The electrical and mechanical activities are important in cardiomyocyte models. The simultaneous recording of electrophysiological and mechanical activities can provide much experimental data for the models. Therefore, an efficient method for the simultaneous recording of the electrical and mechanical data from cardiomyocytes is required for the improvement of cardiac modeling. However, as far as we know, most of the previous methods were not easy to be implemented in the electromechanical recording. For this reason, in this study, a union method of microelectrode array and atomic force microscope was proposed. With this method, the extracellular field potential and beating force of cardiomyocytes were recorded simultaneously with a low root-mean-square noise level of 11.67 μV and 60 pN. Drug tests were conducted to verify the feasibility of the experimental platform. The experimental results suggested the method would be useful for the cardiovascular drug screening and refinement of the computational cardiomyocyte models. It may be valuable for exploring the functional mechanisms of cardiomyocytes and skeletal muscle cells under physiological or pathological conditions.

In Silico Evaluation of the Potential Antiarrhythmic Effect of Epigallocatechin-3-Gallate on Cardiac Channelopathies

Ion channels are transmembrane proteins that allow the passage of ions according to the direction of their electrochemical gradients. Mutations in more than 30 genes encoding ion channels have been associated with an increasingly wide range of inherited cardiac arrhythmias. In this line, ion channels become one of the most important molecular targets for several classes of drugs, including antiarrhythmics. Nevertheless, antiarrhythmic drugs are usually accompanied by some serious side effects. Thus, developing new approaches could offer added values to prevent and treat the episodes of arrhythmia. In this sense, green tea catechins seem to be a promising alternative because of the significant effect of Epigallocatechin-3-Gallate (E3G) on the electrocardiographic wave forms of guinea pig hearts. Thus, the aim of this study was to evaluate the benefits-risks balance of E3G consumption in the setting of ion channel mutations linked with aberrant cardiac excitability phenotypes. Two gain-of-function mutations, Na_v1.5-p.R222Q and Na_v1.5-p.I141V, which are linked with cardiac hyperexcitability phenotypes were studied. Computer simulations of action potentials (APs) show that 30 μM E3G reduces and suppresses AP abnormalities characteristics of these phenotypes. These results suggest that E3G may have a beneficial effect in the setting of cardiac sodium channelopathies displaying a hyperexcitability phenotype.

The Cardiac Electrophysiology Web Lab

Computational modeling of cardiac cellular electrophysiology has a long history, and many models are now available for different species, cell types, and experimental preparations. This success brings with it a challenge: how do we assess and compare the underlying hypotheses and emergent behaviors so that we can choose a model as a suitable basis for a new study or to characterize how a particular model behaves in different scenarios? We have created an online resource for the characterization and comparison of electrophysiological cell models in a wide range of experimental scenarios. The details of the mathematical model (quantitative assumptions and hypotheses formulated as ordinary differential equations) are separated from the experimental protocol being simulated. Each model and protocol is then encoded in computer-readable formats. A simulation tool runs virtual experiments on models encoded in CellML, and a website (https://chaste.cs.ox.ac.uk/WebLab) provides a friendly interface, allowing users to store and compare results. The system currently contains a sample of 36 models and 23 protocols, including current-voltage curve generation, action potential properties under steady pacing at different rates, restitution properties, block of particular channels, and hypo-/hyperkalemia. This resource is publicly available, open source, and free, and we invite the community to use it and become involved in future developments. Investigators interested in comparing competing hypotheses using models can make a more informed decision, and those developing new models can upload them for easy evaluation under the existing protocols, and even add their own protocols.

Computational cardiology and risk stratification for sudden cardiac death: one of the grand challenges for cardiology in the 21st century

Risk stratification in the context of sudden cardiac death has been acknowledged as one of the major challenges facing cardiology for the past four decades. In recent years, the advent of high performance computing has facilitated organ-level simulation of the heart, meaning we can now examine the causes, mechanisms and impact of cardiac dysfunction in silico. As a result, computational cardiology, largely driven by the Physiome project, now stands at the threshold of clinical utility in regards to risk stratification and treatment of patients at risk of sudden cardiac death. In this white paper, we outline a roadmap of what needs to be done to make this translational step, using the relatively well-developed case of acquired or drug-induced long QT syndrome as an exemplar case.

Using numeric simulation in an online e-learning environment to teach functional physiological contexts

BACKGROUND AND OBJECTIVES

Mathematical models are suitable to simulate complex biological processes by a set of non-linear differential equations. These simulation models can be used as an e-learning tool in medical education. However, in many cases these mathematical systems have to be treated numerically which is computationally intensive. The aim of the study was to develop a system for numerical simulation to be used in an online e-learning environment.

METHODS

In the software system the simulation is located on the server as a CGI application. The user (student) selects the boundary conditions for the simulation (e.g., properties of a simulated patient) on the browser. With these parameters the simulation on the server is started and the simulation result is re-transferred to the browser.

RESULTS

With this system two examples of e-learning units were realized. The first one uses a multi-compartment model of the glucose-insulin control loop for the simulation of the plasma glucose level after a simulated meal or during diabetes (including treatment by subcutaneous insulin application). The second one simulates the ion transport leading to the resting and action potential in nerves. The student can vary parameters systematically to explore the biological behavior of the system.

CONCLUSIONS

The described system is able to simulate complex biological processes and offers the possibility to use these models in an online e-learning environment. As far as the underlying principles can be described mathematically, this type of system can be applied to a broad spectrum of biomedical or natural scientific topics.

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.

Recent developments in using mechanistic cardiac modelling for drug safety evaluation

•

Modelling and simulation can streamline decision making in drug safety testing.

•

Computational cardiac electrophysiology is a mature technology with a long heritage.

•

There are many challenges and opportunities in using in silico techniques in future.

•

We discuss how models can be used at different stages of drug discovery.

•

CiPA will combine screening platforms, human cell assays and in silico predictions.

On the tenth anniversary of two key International Conference on Harmonisation (ICH) guidelines relating to cardiac proarrhythmic safety, an initiative aims to consider the implementation of a new paradigm that combines in vitro and in silico technologies to improve risk assessment. The Comprehensive In Vitro Proarrhythmia Assay (CiPA) initiative (co-sponsored by the Cardiac Safety Research Consortium, Health and Environmental Sciences Institute, Safety Pharmacology Society and FDA) is a bold and welcome step in using computational tools for regulatory decision making. This review compares and contrasts the state-of-the-art tools from empirical to mechanistic models of cardiac electrophysiology, and how they can and should be used in combination with experimental tests for compound decision making.

Modeling our understanding of the His-Purkinje system

The His-Purkinje System (HPS) is responsible for the rapid electric conduction in the ventricles. It relays electrical impulses from the atrioventricular node to the muscle cells and, thus, coordinates the contraction of ventricles in order to ensure proper cardiac pump function. The HPS has been implicated in the genesis of ventricular tachycardia and fibrillation as a source of ectopic beats, as well as forming distinct portions of reentry circuitry. Despite its importance, it remains much less well characterized, structurally and functionally, than the myocardium. Notably, important differences exist with regard to cell structure and electrophysiology, including ion channels, intracellular calcium handling, and gap junctions. Very few computational models address the HPS, and the majority of organ level modeling studies omit it. This review will provide an overview of our current knowledge of structure and function (including electrophysiology) of the HPS. We will review the most recent advances in modeling of the system from the single cell to the organ level, with considerations for relevant interspecies distinctions.

Cell-specific Dynamic Clamp analysis of the role of funny If current in cardiac pacemaking

We used the Dynamic Clamp technique for i) comparative validation of conflicting computational models of the hyperpolarization-activated funny current, If, and ii) quantification of the role of If in mediating autonomic modulation of heart rate. Experimental protocols based on the injection of a real-time recalculated synthetic If current in sinoatrial rabbit cells were developed. Preliminary results of experiments mimicking the autonomic modulation of If demonstrated the need for a customization procedure to compensate for cellular heterogeneity. For this reason, we used a cell-specific approach, scaling the maximal conductance of the injected current based on the cell's spontaneous firing rate. The pacemaking rate, which was significantly reduced after application of Ivabradine, was restored by the injection of synthetic current based on the Severi-DiFrancesco formulation, while the injection of synthetic current based on the Maltsev-Lakatta formulation did not produce any significant variation. A positive virtual shift of the If activation curve, mimicking the Isoprenaline effects, led to a significant increase in pacemaking rate (+17.3 ± 6.7%, p < 0.01), although of lower magnitude than that induced by real Isoprenaline (+45.0 ± 26.1%). Similarly, a negative virtual shift of the activation curve significantly lowered the pacemaking rate (-11.8 ± 1.9%, p < 0.001), as did the application of real Acetylcholine (-20.5 ± 5.1%). The Dynamic Clamp approach, applied to the If study in cardiomyocytes for the first time and rate-adapted to manage intercellular variability, indicated that: i) the quantitative description of the If current in the Severi-DiFrancesco model accurately reproduces the effects of the real current on rabbit sinoatrial cell pacemaking rate and ii) a significant portion (50-60%) of the physiological autonomic rate modulation is due to the shift of the If activation curve.