How different two almost identical action potentials can be: a model study on cardiac repolarization

Cell-specific models of hiPSC-CMs developed by the gradient-based parameter optimization method fitting two different action potential waveforms

Parameter optimization (PO) methods to determine the ionic current composition of experimental cardiac action potential (AP) waveform have been developed using a computer model of cardiac membrane excitation. However, it was suggested that fitting a single AP record in the PO method was not always successful in providing a unique answer because of a shortage of information. We found that the PO method worked perfectly if the PO method was applied to a pair of a control AP and a model output AP in which a single ionic current out of six current species, such as IKr, ICaL, INa, IKs, IKur or IbNSC was partially blocked in silico. When the target was replaced by a pair of experimental control and IKr-blocked records of APs generated spontaneously in human induced pluripotent stem cell-derived cardiomyocytes (hiPSC-CMs), the simultaneous fitting of the two waveforms by the PO method was hampered to some extent by the irregular slow fluctuations in the Vm recording and/or sporadic alteration in AP configurations in the hiPSC-CMs. This technical problem was largely removed by selecting stable segments of the records for the PO method. Moreover, the PO method was made fail-proof by running iteratively in identifying the optimized parameter set to reconstruct both the control and the IKr-blocked AP waveforms. In the lead potential analysis, the quantitative ionic mechanisms deduced from the optimized parameter set were totally consistent with the qualitative view of ionic mechanisms of AP so far described in physiological literature.

Neural network emulation of the human ventricular cardiomyocyte action potential for more efficient computations in pharmacological studies

Computer models of the human ventricular cardiomyocyte action potential (AP) have reached a level of detail and maturity that has led to an increasing number of applications in the pharmaceutical sector. However, interfacing the models with experimental data can become a significant computational burden. To mitigate the computational burden, the present study introduces a neural network (NN) that emulates the AP for given maximum conductances of selected ion channels, pumps, and exchangers. Its applicability in pharmacological studies was tested on synthetic and experimental data. The NN emulator potentially enables massive speed-ups compared to regular simulations and the forward problem (find drugged AP for pharmacological parameters defined as scaling factors of control maximum conductances) on synthetic data could be solved with average root-mean-square errors (RMSE) of 0.47 mV in normal APs and of 14.5 mV in abnormal APs exhibiting early afterdepolarizations (72.5% of the emulated APs were alining with the abnormality, and the substantial majority of the remaining APs demonstrated pronounced proximity). This demonstrates not only very fast and mostly very accurate AP emulations but also the capability of accounting for discontinuities, a major advantage over existing emulation strategies. Furthermore, the inverse problem (find pharmacological parameters for control and drugged APs through optimization) on synthetic data could be solved with high accuracy shown by a maximum RMSE of 0.22 in the estimated pharmacological parameters. However, notable mismatches were observed between pharmacological parameters estimated from experimental data and distributions obtained from the Comprehensive in vitro Proarrhythmia Assay initiative. This reveals larger inaccuracies which can be attributed particularly to the fact that small tissue preparations were studied while the emulator was trained on single cardiomyocyte data. Overall, our study highlights the potential of NN emulators as powerful tool for an increased efficiency in future quantitative systems pharmacology studies.

The electrical restitution of the non-propagated cardiac ventricular action potential

Sudden changes in pacing cycle length are frequently associated with repolarization abnormalities initiating cardiac arrhythmias, and physiologists have long been interested in measuring the likelihood of these events before their manifestation. A marker of repolarization stability has been found in the electrical restitution (ER), the response of the ventricular action potential duration to a pre- or post-mature stimulation, graphically represented by the so-called ER curve. According to the restitution hypothesis (ERH), the slope of this curve provides a quantitative discrimination between stable repolarization and proneness to arrhythmias. ER has been studied at the body surface, whole organ, and tissue level, and ERH has soon become a key reference point in theoretical, clinical, and pharmacological studies concerning arrhythmia development, and, despite criticisms, it is still widely adopted. The ionic mechanism of ER and cellular applications of ERH are covered in the present review. The main criticism on ERH concerns its dependence from the way ER is measured. Over the years, in fact, several different experimental protocols have been established to measure ER, which are also described in this article. In reviewing the state-of-the art on cardiac cellular ER, I have introduced a notation specifying protocols and graphical representations, with the aim of unifying a sometime confusing nomenclature, and providing a physiological tool, better defined in its scope and limitations, to meet the growing expectations of clinical and pharmacological research.

Neural network emulation of the human ventricular cardiomyocyte action potential: a tool for more efficient computation in pharmacological studies

Computer models of the human ventricular cardiomyocyte action potential (AP) have reached a level of detail and maturity that has led to an increasing number of applications in the pharmaceutical sector. However, interfacing the models with experimental data can become a significant computational burden. To mitigate the computational burden, the present study introduces a neural network (NN) that emulates the AP for given maximum conductances of selected ion channels, pumps, and exchangers. Its applicability in pharmacological studies was tested on synthetic and experimental data. The NN emulator potentially enables massive speed-ups compared to regular simulations and the forward problem (find drugged AP for pharmacological parameters defined as scaling factors of control maximum conductances) on synthetic data could be solved with average root-mean-square errors (RMSE) of 0.47mV in normal APs and of 14.5mV in abnormal APs exhibiting early afterdepolarizations (72.5% of the emulated APs were alining with the abnormality, and the substantial majority of the remaining APs demonstrated pronounced proximity). This demonstrates not only very fast and mostly very accurate AP emulations but also the capability of accounting for discontinuities, a major advantage over existing emulation strategies. Furthermore, the inverse problem (find pharmacological parameters for control and drugged APs through optimization) on synthetic data could be solved with high accuracy shown by a maximum RMSE of 0.21 in the estimated pharmacological parameters. However, notable mismatches were observed between pharmacological parameters estimated from experimental data and distributions obtained from the Comprehensive in vitro Proarrhythmia Assay initiative. This reveals larger inaccuracies which can be attributed particularly to the fact that small tissue preparations were studied while the emulator was trained on single cardiomyocyte data. Overall, our study highlights the potential of NN emulators as powerful tool for an increased efficiency in future quantitative systems pharmacology studies.

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.

Analysis of sloppiness in model simulations: Unveiling parameter uncertainty when mathematical models are fitted to data

This work introduces a comprehensive approach to assess the sensitivity of model outputs to changes in parameter values, constrained by the combination of prior beliefs and data. This approach identifies stiff parameter combinations strongly affecting the quality of the model-data fit while simultaneously revealing which of these key parameter combinations are informed primarily by the data or are also substantively influenced by the priors. We focus on the very common context in complex systems where the amount and quality of data are low compared to the number of model parameters to be collectively estimated, and showcase the benefits of this technique for applications in biochemistry, ecology, and cardiac electrophysiology. We also show how stiff parameter combinations, once identified, uncover controlling mechanisms underlying the system being modeled and inform which of the model parameters need to be prioritized in future experiments for improved parameter inference from collective model-data fitting.

Ion channel model reduction using manifold boundaries

Mathematical models of voltage-gated ion channels are used in basic research, industrial and clinical settings. These models range in complexity, but typically contain numerous variables representing the proportion of channels in a given state, and parameters describing the voltage-dependent rates of transition between states. An open problem is selecting the appropriate degree of complexity and structure for an ion channel model given data availability. Here, we simplify a model of the cardiac human Ether-à-go-go related gene (hERG) potassium ion channel, which carries cardiac I_Kr, using the manifold boundary approximation method (MBAM). The MBAM approximates high-dimensional model-output manifolds by reduced models describing their boundaries, resulting in models with fewer parameters (and often variables). We produced a series of models of reducing complexity starting from an established five-state hERG model with 15 parameters. Models with up to three fewer states and eight fewer parameters were shown to retain much of the predictive capability of the full model and were validated using experimental hERG1a data collected in HEK293 cells at 37°C. The method provides a way to simplify complex models of ion channels that improves parameter identifiability and will aid in future model development.

Arrhythmogenic Mechanisms in Hypokalaemia: Insights From Pre-clinical Models

Potassium is the predominant intracellular cation, with its extracellular concentrations maintained between 3. 5 and 5 mM. Among the different potassium disorders, hypokalaemia is a common clinical condition that increases the risk of life-threatening ventricular arrhythmias. This review aims to consolidate pre-clinical findings on the electrophysiological mechanisms underlying hypokalaemia-induced arrhythmogenicity. Both triggers and substrates are required for the induction and maintenance of ventricular arrhythmias. Triggered activity can arise from either early afterdepolarizations (EADs) or delayed afterdepolarizations (DADs). Action potential duration (APD) prolongation can predispose to EADs, whereas intracellular Ca²⁺ overload can cause both EADs and DADs. Substrates on the other hand can either be static or dynamic. Static substrates include action potential triangulation, non-uniform APD prolongation, abnormal transmural repolarization gradients, reduced conduction velocity (CV), shortened effective refractory period (ERP), reduced excitation wavelength (CV × ERP) and increased critical intervals for re-excitation (APD-ERP). In contrast, dynamic substrates comprise increased amplitude of APD alternans, steeper APD restitution gradients, transient reversal of transmural repolarization gradients and impaired depolarization-repolarization coupling. The following review article will summarize the molecular mechanisms that generate these electrophysiological abnormalities and subsequent 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.

All-Optical Electrophysiology Refines Populations of In Silico Human iPSC-CMs for Drug Evaluation

High-throughput in vitro drug assays have been impacted by recent advances in human induced pluripotent stem cell-derived cardiomyocyte (hiPSC-CM) technology and by contact-free all-optical systems simultaneously measuring action potentials (APs) and Ca2+ transients (CaTrs). Parallel computational advances have shown that in silico simulations can predict drug effects with high accuracy. We combine these in vitro and in silico technologies and demonstrate the utility of high-throughput experimental data to refine in silico hiPSC-CM populations and to predict and explain drug action mechanisms. Optically obtained hiPSC-CM APs and CaTrs were used from spontaneous activity and under optical pacing in control and drug conditions at multiple doses. An updated version of the Paci2018 model was developed to refine the description of hiPSC-CM spontaneous electrical activity; a population of in silico hiPSC-CMs was constructed and calibrated using simultaneously recorded APs and CaTrs. We tested in silico five drugs (astemizole, dofetilide, ibutilide, bepridil, and diltiazem) and compared the outcomes to in vitro optical recordings. Our simulations showed that physiologically accurate population of models can be obtained by integrating AP and CaTr control records. Thus, constructed population of models correctly predicted the drug effects and occurrence of adverse episodes, even though the population was optimized only based on control data and in vitro drug testing data were not deployed during its calibration. Furthermore, the in silico investigation yielded mechanistic insights; e.g., through simulations, bepridil's more proarrhythmic action in adult cardiomyocytes compared to hiPSC-CMs could be traced to the different expression of ion currents in the two. Therefore, our work 1) supports the utility of all-optical electrophysiology in providing high-content data to refine experimentally calibrated populations of in silico hiPSC-CMs, 2) offers insights into certain limitations when translating results obtained in hiPSC-CMs to humans, and 3) shows the strength of combining high-throughput in vitro and population in silico approaches.

A robust multi-objective optimization framework to capture both cellular and intercellular properties in cardiac cellular model tuning: Analyzing different regions of membrane resistance profile in parameter fitting

Mathematical models of cardiac cells have been established to broaden understanding of cardiac function. In the process of developing electrophysiological models for cardiac myocytes, precise parameter tuning is a crucial step. The membrane resistance (Rm) is an essential feature obtained from cardiac myocytes. This feature reflects intercellular coupling and affects important phenomena, such as conduction velocity, and early after-depolarizations, but it is often overlooked during the phase of parameter fitting. Thus, the traditional parameter fitting that only includes action potential (AP) waveform may yield incorrect values for Rm. In this paper, a novel multi-objective parameter fitting formulation is proposed and tested that includes different regions of the Rm profile as additional objective functions for optimization. As Rm depends on the transmembrane voltage (Vm) and exhibits singularities for some specific values of Vm, analyses are conducted to carefully select the regions of interest for the proper characterization of Rm. Non-dominated sorting genetic algorithm II is utilized to solve the proposed multi-objective optimization problem. To verify the efficacy of the proposed problem formulation, case studies and comparisons are carried out using multiple models of human cardiac ventricular cells. Results demonstrate Rm is correctly reproduced by the tuned cell models after considering the curve of Rm obtained from the late phase of repolarization and Rm value calculated in the rest phase as additional objectives. However, relative deterioration of the AP fit is observed, demonstrating trade-off among the objectives. This framework can be useful for a wide range of applications, including the parameters fitting phase of the cardiac cell model development and investigation of normal and pathological scenarios in which reproducing both cellular and intercellular properties are of great importance.

Visualization of Myocardial Strain Pattern Uniqueness with Respect to Activation Time and Contractility: A Computational Study

Speckle tracking echography is used to measure myocardial strain patterns in order to assess the state of myocardial tissue. Because electro-mechanical coupling in myocardial tissue is complex and nonlinear, and because of the measurement errors the uniqueness of strain patterns is questionable. In this study, the uniqueness of strain patterns was visualized in order to revel characteristics that may improve their interpretation. A computational model of sarcomere mechanics was used to generate a database of 1681 strain patterns, each simulated with a different set of sarcomere parameters: time of activation (TA) and contractility (Con). TA and Con ranged from −100 ms to 100 ms and 2% to 202% in 41 steps respectively, thus forming a two-dimensional 41 × 41 parameter space. Uniqueness of the strain pattern was assessed by using a cohort of similar strain patterns defined by a measurement error. The cohort members were then visualized in the parameter space. Each cohort formed one connected component (or blob) in the parameter space; however, large differences in the shape, size, and eccentricity of the blobs were found for different regions in the parameter space. The blobs were elongated along the TA direction (±50 ms) when contractility was low, and along the Con direction (±50%) when contractility was high. The uniqueness of the strain patterns can be assessed and visualized in the parameter space. The strain patterns in the studied database are not degenerated because a cohort of similar strain patterns forms only one connected blob in the parameter space. However, the elongation of the blobs means that estimations of TA when contractility is low and of Con when contractility is high have high uncertainty.

Uniqueness of local myocardial strain patterns with respect to activation time and contractility of the failing heart: a computational study

Background

Myocardial deformation measured by strain is used to detect electro-mechanical abnormalities in cardiac tissue. Estimation of myocardial properties from regional strain patterns when multiple pathologies are present is therefore a promising application of computer modelling. However, if different tissue properties lead to indistinguishable strain patterns (‘degeneracy’), the applicability of any such method will be limited. We investigated whether estimation of local activation time (AT) and contractility from myocardial strain patterns is theoretically possible.

Methods

For four different global cardiac pathologies local myocardial strain patterns for 1025 combinations of AT and contractility were simulated with a computational model (CircAdapt). For each strain pattern, a cohort of similar patterns was found within estimated measurement error using the sum of least-squared differences. Cohort members came from (1) the same pathology only, and (2) all four pathologies. Uncertainty was calculated as accuracy and precision of cohort members in parameter space. Connectedness within the cohorts was also studied.

Results

We found that cohorts drawn from one pathology had parameters with adjacent values although their distribution was neither constant nor symmetrical. In comparison cohorts drawn from four pathologies had disconnected components with drastically different parameter values and accuracy and precision values up to three times higher.

Conclusions

Global pathology must be known when extracting AT and contractility from strain patterns, otherwise degeneracy occurs causing unacceptable uncertainty in derived parameters.

Electronic supplementary material

The online version of this article (10.1186/s12938-018-0614-1) contains supplementary material, which is available to authorized users.

Fitting two human atrial cell models to experimental data using Bayesian history matching

Cardiac cell models are potentially valuable tools for applications such as quantitative safety pharmacology, but have many parameters. Action potentials in real cardiac cells also vary from beat to beat, and from one cell to another. Calibrating cardiac cell models to experimental observations is difficult, because the parameter space is large and high-dimensional. In this study we have demonstrated the use of history matching to calibrate the maximum conductance of ion channels and exchangers in two detailed models of the human atrial action potential against measurements of action potential biomarkers. History matching is an approach developed in other modelling communities, based on constructing fast-running Gaussian process emulators of the model. Emulators were constructed from a small number of model runs (around 10²), and then run many times (>10⁶) at low computational cost, each time with a different set of model parameters. Emulator outputs were compared with experimental biomarkers using an implausibility measure, which took into account experimental variance as well as emulator variance. By repeating this process, the region of non-implausible parameter space was iteratively reduced. Both cardiac cell models were successfully calibrated to experimental datasets, resulting in sets of parameters that could be sampled to produce variable action potentials. However, model parameters did not occupy a small range of values. Instead, the history matching process exposed inputs that can co-vary across a wide range and still be consistent with a particular biomarker. We also found correlations between some biomarkers, indicating a need for better descriptors of action potential shape.

Unlocking data sets by calibrating populations of models to data density: A study in atrial electrophysiology

The understanding of complex physical or biological systems nearly always requires a characterization of the variability that underpins these processes. In addition, the data used to calibrate these models may also often exhibit considerable variability. A recent approach to deal with these issues has been to calibrate populations of models (POMs), multiple copies of a single mathematical model but with different parameter values, in response to experimental data. To date, this calibration has been largely limited to selecting models that produce outputs that fall within the ranges of the data set, ignoring any trends that might be present in the data. We present here a novel and general methodology for calibrating POMs to the distributions of a set of measured values in a data set. We demonstrate our technique using a data set from a cardiac electrophysiology study based on the differences in atrial action potential readings between patients exhibiting sinus rhythm (SR) or chronic atrial fibrillation (cAF) and the Courtemanche-Ramirez-Nattel model for human atrial action potentials. Not only does our approach accurately capture the variability inherent in the experimental population, but we also demonstrate how the POMs that it produces may be used to extract additional information from the data used for calibration, including improved identification of the differences underlying stratified data. We also show how our approach allows different hypotheses regarding the variability in complex systems to be quantitatively compared.

Differential roles of two delayed rectifier potassium currents in regulation of ventricular action potential duration and arrhythmia susceptibility

KEY POINTS

Arrhythmias result from disruptions to cardiac electrical activity, although the factors that control cellular action potentials are incompletely understood. We combined mathematical modelling with experiments in heart cells from guinea pigs to determine how cellular electrical activity is regulated. A mismatch between modelling predictions and the experimental results allowed us to construct an improved, more predictive mathematical model. The balance between two particular potassium currents dictates how heart cells respond to perturbations and their susceptibility to arrhythmias.

ABSTRACT

Imbalances of ionic currents can destabilize the cardiac action potential and potentially trigger lethal cardiac arrhythmias. In the present study, we combined mathematical modelling with information-rich dynamic clamp experiments to determine the regulation of action potential morphology in guinea pig ventricular myocytes. Parameter sensitivity analysis was used to predict how changes in ionic currents alter action potential duration, and these were tested experimentally using dynamic clamp, a technique that allows for multiple perturbations to be tested in each cell. Surprisingly, we found that a leading mathematical model, developed with traditional approaches, systematically underestimated experimental responses to dynamic clamp perturbations. We then re-parameterized the model using a genetic algorithm, which allowed us to estimate ionic current levels in each of the cells studied. This unbiased model adjustment consistently predicted an increase in the rapid delayed rectifier K⁺ current and a drastic decrease in the slow delayed rectifier K⁺ current, and this prediction was validated experimentally. Subsequent simulations with the adjusted model generated the clinically relevant prediction that the slow delayed rectifier is better able to stabilize the action potential and suppress pro-arrhythmic events than the rapid delayed rectifier. In summary, iterative coupling of simulations and experiments enabled novel insight into how the balance between cardiac K⁺ currents influences ventricular arrhythmia susceptibility.

Dynamical mechanisms of phase-2 early afterdepolarizations in human ventricular myocytes: insights from bifurcation analyses of two mathematical models

Early afterdepolarization (EAD) is known as a cause of ventricular arrhythmias in long QT syndromes. We theoretically investigated how the rapid (I_Kr) and slow (I_Ks) components of delayed-rectifier K⁺ channel currents, L-type Ca²⁺ channel current (I_CaL), Na⁺/Ca²⁺ exchanger current (I_NCX), Na⁺-K⁺ pump current (I_NaK), intracellular Ca²⁺ (Ca_i) handling via sarcoplasmic reticulum (SR), and intracellular Na⁺ concentration (Na_i) contribute to initiation, termination, and modulation of phase-2 EADs, using two human ventricular myocyte models. Bifurcation structures of dynamical behaviors in model cells were explored by calculating equilibrium points, limit cycles (LCs), and bifurcation points as functions of parameters. EADs were reproduced by numerical simulations. The results are summarized as follows: 1) decreasing I_Ks and/or I_Kr or increasing I_CaL led to EAD generation, to which mid-myocardial cell models were especially susceptible; the parameter regions of EADs overlapped the regions of stable LCs. 2) Two types of EADs (termination mechanisms), I_Ks activation-dependent and I_CaL inactivation-dependent EADs, were detected; I_Ks was not necessarily required for EAD formation. 3) Inhibiting I_NCX suppressed EADs via facilitating Ca²⁺-dependent I_CaL inactivation. 4) Ca_i dynamics (SR Ca²⁺ handling) and Na_i strongly affected bifurcations and EAD generation in model cells via modulating I_CaL, I_NCX, and I_NaK Parameter regions of EADs, often overlapping those of stable LCs, shifted depending on Ca_i and Na_i in stationary and dynamic states. 5) Bradycardia-related induction of EADs was mainly due to decreases in Na_i at lower pacing rates. This study demonstrates that bifurcation analysis allows us to understand the dynamical mechanisms of EAD formation more profoundly.

NEW & NOTEWORTHY

We investigated mechanisms of phase-2 early afterdepolarization (EAD) by bifurcation analyses of human ventricular myocyte (HVM) models. EAD formation in paced HVMs basically depended on bifurcation phenomena in non-paced HVMs, but was strongly affected by intracellular ion concentrations in stationary and dynamic states. EAD generation did not necessarily require I_Ks.

A Parsimonious Model of the Rabbit Action Potential Elucidates the Minimal Physiological Requirements for Alternans and Spiral Wave Breakup

Elucidating the underlying mechanisms of fatal cardiac arrhythmias requires a tight integration of electrophysiological experiments, models, and theory. Existing models of transmembrane action potential (AP) are complex (resulting in over parameterization) and varied (leading to dissimilar predictions). Thus, simpler models are needed to elucidate the "minimal physiological requirements" to reproduce significant observable phenomena using as few parameters as possible. Moreover, models have been derived from experimental studies from a variety of species under a range of environmental conditions (for example, all existing rabbit AP models incorporate a formulation of the rapid sodium current, INa, based on 30 year old data from chick embryo cell aggregates). Here we develop a simple "parsimonious" rabbit AP model that is mathematically identifiable (i.e., not over parameterized) by combining a novel Hodgkin-Huxley formulation of INa with a phenomenological model of repolarization similar to the voltage dependent, time-independent rectifying outward potassium current (IK). The model was calibrated using the following experimental data sets measured from the same species (rabbit) under physiological conditions: dynamic current-voltage (I-V) relationships during the AP upstroke; rapid recovery of AP excitability during the relative refractory period; and steady-state INa inactivation via voltage clamp. Simulations reproduced several important "emergent" phenomena including cellular alternans at rates > 250 bpm as observed in rabbit myocytes, reentrant spiral waves as observed on the surface of the rabbit heart, and spiral wave breakup. Model variants were studied which elucidated the minimal requirements for alternans and spiral wave break up, namely the kinetics of INa inactivation and the non-linear rectification of IK.The simplicity of the model, and the fact that its parameters have physiological meaning, make it ideal for engendering generalizable mechanistic insight and should provide a solid "building-block" to generate more detailed ionic models to represent complex rabbit electrophysiology.

Improving cardiomyocyte model fidelity and utility via dynamic electrophysiology protocols and optimization algorithms

Mathematical models of cardiac electrophysiology are instrumental in determining mechanisms of cardiac arrhythmias. However, the foundation of a realistic multiscale heart model is only as strong as the underlying cell model. While there have been myriad advances in the improvement of cellular-level models, the identification of model parameters, such as ion channel conductances and rate constants, remains a challenging problem. The primary limitations to this process include: (1) such parameters are usually estimated from data recorded using standard electrophysiology voltage-clamp protocols that have not been developed with model building in mind, and (2) model parameters are typically tuned manually to subjectively match a desired output. Over the last decade, methods aimed at overcoming these disadvantages have emerged. These approaches include the use of optimization or fitting tools for parameter estimation and incorporating more extensive data for output matching. Here, we review recent advances in parameter estimation for cardiomyocyte models, focusing on the use of more complex electrophysiology protocols and global search heuristics. We also discuss future applications of such parameter identification, including development of cell-specific and patient-specific mathematical models to investigate arrhythmia mechanisms and predict therapy strategies.

In silico assessment of drug safety in human heart applied to late sodium current blockers

Drug-induced action potential (AP) prolongation leading to Torsade de Pointes is a major concern for the development of anti-arrhythmic drugs. Nevertheless the development of improved anti-arrhythmic agents, some of which may block different channels, remains an important opportunity. Partial block of the late sodium current (I(NaL)) has emerged as a novel anti-arrhythmic mechanism. It can be effective in the settings of free radical challenge or hypoxia. In addition, this approach can attenuate pro-arrhythmic effects of blocking the rapid delayed rectifying K(+) current (I(Kr)). The main goal of our computational work was to develop an in-silico tool for preclinical anti-arrhythmic drug safety assessment, by illustrating the impact of I(Kr)/I(NaL) ratio of steady-state block of drug candidates on "torsadogenic" biomarkers. The O'Hara et al. AP model for human ventricular myocytes was used. Biomarkers for arrhythmic risk, i.e., AP duration, triangulation, reverse rate-dependence, transmural dispersion of repolarization and electrocardiogram QT intervals, were calculated using single myocyte and one-dimensional strand simulations. Predetermined amounts of block of I(NaL) and I(Kr) were evaluated. "Safety plots" were developed to illustrate the value of the specific biomarker for selected combinations of IC(50)s for I(Kr) and I(NaL) of potential drugs. The reference biomarkers at baseline changed depending on the "drug" specificity for these two ion channel targets. Ranolazine and GS967 (a novel potent inhibitor of I(NaL)) yielded a biomarker data set that is considered safe by standard regulatory criteria. This novel in-silico approach is useful for evaluating pro-arrhythmic potential of drugs and drug candidates in the human ventricle.

Application of stochastic phenomenological modelling to cell-to-cell and beat-to-beat electrophysiological variability in cardiac tissue

Variability in the action potential of isolated myocytes and tissue samples is observed in experimental studies. Variability is manifested as both differences in the action potential (AP) morphology between cells (extrinsic variability), and also ‘intrinsic’ or beat-to-beat variability of repolarization (BVR) in the AP duration of each cell. We studied the relative contributions of experimentally recorded intrinsic and extrinsic variability to dispersion of repolarization in tissue. We developed four cell-specific parameterizations of a phenomenological stochastic differential equation AP model exhibiting intrinsic variability using APs recorded from isolated guinea pig ventricular myocytes exhibiting BVR. We performed simulations in tissue using the four different model parameterizations in the presence and the absence of both intrinsic and extrinsic variability. We altered the coupling of the tissue to determine how inter-cellular coupling affected the dispersion of the AP duration in tissue. Both intrinsic and extrinsic variability were gradually revealed by reduction of tissue coupling. However, the recorded extrinsic variability between individual myocytes produced a greater degree of dispersion in repolarization in tissue than the intrinsic variability of each myocyte.

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We modelled inter-cell and beat-to-beat repolarization variability in cardiomyocytes.

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We coupled the cells together into cardiac tissue.

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Reducing tissue coupling increased repolarization dispersion in tissue.

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Inter-cell variability had a greater effect on repolarization dispersion.

Bridging the gap between computation and clinical biology: validation of cable theory in humans

Introduction: Computerized simulations of cardiac activity have significantly contributed to our understanding of cardiac electrophysiology, but techniques of simulations based on patient-acquired data remain in their infancy. We sought to integrate data acquired from human electrophysiological studies into patient-specific models, and validated this approach by testing whether electrophysiological responses to sequential premature stimuli could be predicted in a quantitatively accurate manner.

Methods: Eleven patients with structurally normal hearts underwent electrophysiological studies. Semi-automated analysis was used to reconstruct activation and repolarization dynamics for each electrode. This S₂ extrastimuli data was used to inform individualized models of cardiac conduction, including a novel derivation of conduction velocity restitution. Activation dynamics of multiple premature extrastimuli were then predicted from this model and compared against measured patient data as well as data derived from the ten-Tusscher cell-ionic model.

Results: Activation dynamics following a premature S₃ were significantly different from those after an S₂. Patient specific models demonstrated accurate prediction of the S₃ activation wave, (Pearson's R² = 0.90, median error 4%). Examination of the modeled conduction dynamics allowed inferences into the spatial dispersion of activation delay. Further validation was performed against data from the ten-Tusscher cell-ionic model, with our model accurately recapitulating predictions of repolarization times (R² = 0.99).

Conclusions: Simulations based on clinically acquired data can be used to successfully predict complex activation patterns following sequential extrastimuli. Such modeling techniques may be useful as a method of incorporation of clinical data into predictive models.

Exploiting mathematical models to illuminate electrophysiological variability between individuals

Across individuals within a population, several levels of variability are observed, from the differential expression of ion channels at the molecular level, to the various action potential morphologies observed at the cellular level, to divergent responses to drugs at the organismal level. However, the limited ability of experiments to probe complex interactions between components has hitherto hindered our understanding of the factors that cause a range of behaviours within a population. Variability is a challenging issue that is encountered in all physiological disciplines, but recent work suggests that novel methods for analysing mathematical models can assist in illuminating its causes. In this review, we discuss mathematical modelling studies in cardiac electrophysiology and neuroscience that have enhanced our understanding of variability in a number of key areas. Specifically, we discuss parameter sensitivity analysis techniques that may be applied to generate quantitative predictions based on considering behaviours within a population of models, thereby providing novel insight into variability. Our discussion focuses on four issues that have benefited from the utilization of these methods: (1) the comparison of different electrophysiological models of cardiac myocytes, (2) the determination of the individual contributions of different molecular changes in complex disease phenotypes, (3) the identification of the factors responsible for the variable response to drugs, and (4) the constraining of free parameters in electrophysiological models of heart cells. Together, the studies that we discuss suggest that rigorous analyses of mathematical models can generate quantitative predictions regarding how molecular-level variations contribute to functional differences between experimental samples. These strategies may be applicable not just in cardiac electrophysiology, but in a wide range of disciplines.

Late phase of repolarization is autoregenerative and scales linearly with action potential duration in mammals ventricular myocytes: a model study

Scaling of action potential (AP) duration (APD) in mammals of different size is a rather complex phenomenon, dominated by a regulatory type mechanism of ion channels expression. By means of simulations performed on six published mathematical models of cardiac ventricular APs of different mammals, it is shown that AP repolarization is autoregenerative in its later phase (ARRP) and that the duration of such phase scales linearly with APD. For each AP, a 3-D instantaneous time-voltage-current surface is constructed, which has been recently described in a more simplified model. This representation allows us to measure ARRP and to study the contribution to it for different ion currents. It has been found that the existence of an ARRP is not intrinsic to cardiac models formulation; one out of the six models does not show this phase. A linear correlation between ARRP duration and APD in the remaining models is also found. It is shown that ARRP neither simply depend on AP shape nor on APD. Though I(K1) current seems to be the main responsible for determining and modulating this phase, the mechanism by which ARRP scales linearly with APD remains unclear and raises further questions on the scaling strategies of cardiac repolarization in mammals.

HCN3 contributes to the ventricular action potential waveform in the murine heart

RATIONALE

The hyperpolarization-activated current I(h) that is generated by hyperpolarization-activated cyclic nucleotide-gated channels (HCNs) plays a key role in the control of pacemaker activity in sinoatrial node cells of the heart. By contrast, it is unclear whether I(h) is also relevant for normal function of cardiac ventricles.

OBJECTIVE

To study the role of the HCN3-mediated component of ventricular I(h) in normal ventricular function.

METHODS AND RESULTS

To test the hypothesis that HCN3 regulates the ventricular action potential waveform, we have generated and analyzed a HCN3-deficient mouse line. At basal heart rate, mice deficient for HCN3 displayed a profound increase in the T-wave amplitude in telemetric electrocardiographic measurements. Action potential recordings on isolated ventricular myocytes indicate that this effect was caused by an acceleration of the late repolarization phase in epicardial myocytes. Furthermore, the resting membrane potential was shifted to more hyperpolarized potentials in HCN3-deficient mice. Cardiomyocytes of HCN3-deficient mice displayed approximately 30% reduction of total I(h). At physiological ionic conditions, the HCN3-mediated current had a reversal potential of approximately -35 mV and displayed ultraslow deactivation kinetics.

CONCLUSIONS

We propose that HCN3 together with other members of the HCN channel family confer a depolarizing background current that regulates ventricular resting potential and counteracts the action of hyperpolarizing potassium currents in late repolarization. In conclusion, our data indicate that HCN3 plays an important role in shaping the cardiac action potential waveform.

3D current-voltage-time surfaces unveil critical repolarization differences underlying similar cardiac action potentials: A model study

The number of mathematical models of cardiac cellular excitability is rapidly growing, and compact graphical representations of their properties can make new acquisitions available for a broader range of scientists in cardiac field. Particularly, the intrinsic over-determination of the model equations systems when fitted only to action potential (AP) waveform and the fact that they are frequently tuned on data covering only a relatively narrow range of dynamic conditions, often lead modellers to compare very similar AP profiles, which underlie though quite different excitable properties. In this study I discuss a novel compact 3D representation of the cardiac cellular AP, where the third dimension represents the instantaneous current-voltage profile of the membrane, measured as repolarization proceeds. Measurements of this type have been used previously for in vivo experiments, and are adopted here iteratively at a very high time, voltage, current-resolution on (i) the same human ventricular model, endowed with two different parameters sets which generate the same AP waveform, and on (ii) three different models of the same human ventricular cell type. In these 3D representations, the AP waveforms lie at the intersection between instantaneous time-voltage-current surfaces and the zero-current plane. Different surfaces can share the same intersection and therefore the same AP; in these cases, the morphology of the current surface provides a compact view of important differences within corresponding repolarization dynamics. Refractory period, supernormal excitability window, and extent of repolarization reserve can be visualized at once. Two pivotal dynamical properties can be precisely assessed, i.e. all-or-nothing repolarization window and membrane resistance during recovery. I discuss differences in these properties among the membranes under study, and show relevant implications for cardiac cellular repolarization.