Models of cardiac tissue electrophysiology: progress, challenges and open questions

Personalized Evaluation of Atrial Complexity of Patients Undergoing Atrial Fibrillation Ablation: A Clinical Computational Study

Current clinical guidelines establish Pulmonary Vein (PV) isolation as the indicated treatment for Atrial Fibrillation (AF). However, AF can also be triggered or sustained due to atrial drivers located elsewhere in the atria. We designed a new simulation workflow based on personalized computer simulations to characterize AF complexity of patients undergoing PV ablation, validated with non-invasive electrocardiographic imaging and evaluated at one year after ablation. We included 30 patients using atrial anatomies segmented from MRI and simulated an automata model for the electrical modelling, consisting of three states (resting, excited and refractory). In total, 100 different scenarios were simulated per anatomy varying rotor number and location. The 3 states were calibrated with Koivumaki action potential, entropy maps were obtained from the electrograms and compared with ECGi for each patient to analyze PV isolation outcome. The completion of the workflow indicated that successful AF ablation occurred in patients with rotors mainly located at the PV antrum, while unsuccessful procedures presented greater number of driving sites outside the PV area. The number of rotors attached to the PV was significantly higher in patients with favorable long-term ablation outcome (1-year freedom from AF: 1.61 ± 0.21 vs. AF recurrence: 1.40 ± 0.20; p-value = 0.018). The presented workflow could improve patient stratification for PV ablation by screening the complexity of the atria.

Sex Differences in Drug-Induced Arrhythmogenesis

The electrical activity in the heart varies significantly between men and women and results in a sex-specific response to drugs. Recent evidence suggests that women are more than twice as likely as men to develop drug-induced arrhythmia with potentially fatal consequences. Yet, the sex-specific differences in drug-induced arrhythmogenesis remain poorly understood. Here we integrate multiscale modeling and machine learning to gain mechanistic insight into the sex-specific origin of drug-induced cardiac arrhythmia at differing drug concentrations. To quantify critical drug concentrations in male and female hearts, we identify the most important ion channels that trigger male and female arrhythmogenesis, and create and train a sex-specific multi-fidelity arrhythmogenic risk classifier. Our study reveals that sex differences in ion channel activity, tissue conductivity, and heart dimensions trigger longer QT-intervals in women than in men. We quantify the critical drug concentration for dofetilide, a high risk drug, to be seven times lower for women than for men. Our results emphasize the importance of including sex as an independent biological variable in risk assessment during drug development. Acknowledging and understanding sex differences in drug safety evaluation is critical when developing novel therapeutic treatments on a personalized basis. The general trends of this study have significant implications on the development of safe and efficacious new drugs and the prescription of existing drugs in combination with other drugs.

Theoretical Investigation of the Mechanism by which A Gain-of-Function Mutation of the TRPM4 Channel Causes Conduction Block

In the heart, TRPM4 is most abundantly distributed in the conduction system. Previously, a single mutation, 'E7K', was identified in its distal N-terminus to cause conduction disorder because of enhanced cell-surface expression. It remains, however, unclear how this expression increase leads to conduction failure rather than abnormally enhanced cardiac excitability. To address this issue theoretically, we mathematically formulated the gating kinetics of the E7K-mutant TRPM4 channel by a combined use of voltage jump analysis and ionomycin-perforated cell-attached recording technique and incorporated the resultant rate constants of opening and closing into a human Purkinje fiber single-cell action potential (AP) model (Trovato model) to perform 1D-cable simulations. The results from TRPM4 expressing HEK293 cells showed that as compared with the wild-type, the open state is much preferred in the E7K mutant with increased voltage-and Ca²⁺-sensitivities. These theoretical predictions were confirmed by power spectrum and single channel analyses of expressed wild-type and E7K-mutant TRPM4 channels. In our modified Trovato model, the facilitated opening of the E7K mutant channel markedly prolonged AP duration with concomitant depolarizing shifts of the resting membrane potential in a manner dependent on the channel density (or maximal activity). This was, however, little evident in the wild-type TRPM4 channel. Moreover, 1D-cable simulations with the modified Trovato model revealed that increasing the density of E7K (but not of wild-type) TRPM4 channels progressively reduced AP conduction velocity eventually culminating in complete conduction block. These results clearly suggest the brady-arrhythmogenicity of the E7K mutant channel which likely results from its pathologically enhanced activity.

Understanding PITX2-Dependent Atrial Fibrillation Mechanisms through Computational Models

Atrial fibrillation (AF) is a common arrhythmia. Better prevention and treatment of AF are needed to reduce AF-associated morbidity and mortality. Several major mechanisms cause AF in patients, including genetic predispositions to AF development. Genome-wide association studies have identified a number of genetic variants in association with AF populations, with the strongest hits clustering on chromosome 4q25, close to the gene for the homeobox transcription PITX2. Because of the inherent complexity of the human heart, experimental and basic research is insufficient for understanding the functional impacts of PITX2 variants on AF. Linking PITX2 properties to ion channels, cells, tissues, atriums and the whole heart, computational models provide a supplementary tool for achieving a quantitative understanding of the functional role of PITX2 in remodelling atrial structure and function to predispose to AF. It is hoped that computational approaches incorporating all we know about PITX2-related structural and electrical remodelling would provide better understanding into its proarrhythmic effects leading to development of improved anti-AF therapies. In the present review, we discuss advances in atrial modelling and focus on the mechanistic links between PITX2 and AF. Challenges in applying models for improving patient health are described, as well as a summary of future perspectives.

A space-fractional bidomain framework for cardiac electrophysiology: 1D alternans dynamics

Cardiac electrophysiology modeling deals with a complex network of excitable cells forming an intricate syncytium: the heart. The electrical activity of the heart shows recurrent spatial patterns of activation, known as cardiac alternans, featuring multiscale emerging behavior. On these grounds, we propose a novel mathematical formulation for cardiac electrophysiology modeling and simulation incorporating spatially non-local couplings within a physiological reaction-diffusion scenario. In particular, we formulate, a space-fractional electrophysiological framework, extending and generalizing similar works conducted for the monodomain model. We characterize one-dimensional excitation patterns by performing an extended numerical analysis encompassing a broad spectrum of space-fractional derivative powers and various intra- and extracellular conductivity combinations. Our numerical study demonstrates that (i) symmetric properties occur in the conductivity parameters' space following the proposed theoretical framework, (ii) the degree of non-local coupling affects the onset and evolution of discordant alternans dynamics, and (iii) the theoretical framework fully recovers classical formulations and is amenable for parametric tuning relying on experimental conduction velocity and action potential morphology.

A modified approach to determine the six cardiac bidomain conductivities

Accurate values for the six cardiac bidomain conductivities are crucial for meaningful computational studies of conduction in cardiac tissue, and are yet to be determined by experimental means. Although previous studies have proposed an approach using a multi-electrode array to measure potentials, from which the conductivities can be determined, it has been found that the conductivities cannot be retrieved consistently when the noise in the potentials varies. This paper presents a protocol, which not only has been shown to retrieve the conductivities to a reasonable accuracy, but does so under the presence of a more appropriate additive Gaussian noise model, while using fewer computational resources. Through repetitions of the protocol, a comparison of two pre-fabricated 128 electrode arrays, one array with a square arrangement of electrodes and the other with a rectangular arrangement, was made against a 75-electrode array proposed in previous studies. Results indicated that the two pre-fabricated arrays were generally more capable of obtaining the cardiac conductivities to a higher degree of accuracy than the 75-electrode array. The 128-electrode rectangular array was orientated such that the length of the array first ran along the direction of the fibres, then was reorientated such that the length of the array ran perpendicular to the direction of the fibres. The 128-electrode rectangular array, when orientated in this manner, was more capable of retrieving the conductivities than the remainder of the arrays tested, and thus we suggest this arrangement be used during experimental trials.

Isolating Patterns in Open Reaction-Diffusion Systems

Realistic examples of reaction-diffusion phenomena governing spatial and spatiotemporal pattern formation are rarely isolated systems, either chemically or thermodynamically. However, even formulations of 'open' reaction-diffusion systems often neglect the role of domain boundaries. Most idealizations of closed reaction-diffusion systems employ no-flux boundary conditions, and often patterns will form up to, or along, these boundaries. Motivated by boundaries of patterning fields related to the emergence of spatial form in embryonic development, we propose a set of mixed boundary conditions for a two-species reaction-diffusion system which forms inhomogeneous solutions away from the boundary of the domain for a variety of different reaction kinetics, with a prescribed uniform state near the boundary. We show that these boundary conditions can be derived from a larger heterogeneous field, indicating that these conditions can arise naturally if cell signalling or other properties of the medium vary in space. We explain the basic mechanisms behind this pattern localization and demonstrate that it can capture a large range of localized patterning in one, two, and three dimensions and that this framework can be applied to systems involving more than two species. Furthermore, the boundary conditions proposed lead to more symmetrical patterns on the interior of the domain and plausibly capture more realistic boundaries in developmental systems. Finally, we show that these isolated patterns are more robust to fluctuations in initial conditions and that they allow intriguing possibilities of pattern selection via geometry, distinct from known selection mechanisms.

Enabling forward uncertainty quantification and sensitivity analysis in cardiac electrophysiology by reduced order modeling and machine learning

We present a new, computationally efficient framework to perform forward uncertainty quantification (UQ) in cardiac electrophysiology. We consider the monodomain model to describe the electrical activity in the cardiac tissue, coupled with the Aliev-Panfilov model to characterize the ionic activity through the cell membrane. We address a complete forward UQ pipeline, including both: (i) a variance-based global sensitivity analysis for the selection of the most relevant input parameters, and (ii) a way to perform uncertainty propagation to investigate the impact of intra-subject variability on outputs of interest depending on the cardiac potential. Both tasks exploit stochastic sampling techniques, thus implying overwhelming computational costs because of the huge amount of queries to the high-fidelity, full-order computational model obtained by approximating the coupled monodomain/Aliev-Panfilov system through the finite element method. To mitigate this computational burden, we replace the full-order model with computationally inexpensive projection-based reduced-order models (ROMs) aimed at reducing the state-space dimensionality. Resulting approximation errors on the outputs of interest are finally taken into account through artificial neural network (ANN)-based models, enhancing the accuracy of the whole UQ pipeline. Numerical results show that the proposed physics-based ROMs outperform regression-based emulators relying on ANNs built with the same amount of training data, in terms of both numerical accuracy and overall computational efficiency.

A numerical study on the effects of spatial and temporal discretization in cardiac electrophysiology

Millions of degrees of freedom are often required to accurately represent the electrophysiology of the myocardium due to the presence of discretization effects. This study seeks to explore the influence of temporal and spatial discretization on the simulation of cardiac electrophysiology in conjunction with changes in modeling choices. Several finite element analyses are performed to examine how discretization affects solution time, conduction velocity and electrical excitation. Discretization effects are considered along with changes in the electrophysiology model and solution approach. Two action potential models are considered: the Aliev-Panfilov model and the ten Tusscher-Noble-Noble-Panfilov model. The solution approaches consist of two time integration schemes and different treatments for solving the local system of ordinary differential equations. The efficiency and stability of the calculation approaches are demonstrated to be dependent on the action potential model. The dependency of the conduction velocity on the element size and time step is shown to be different for changes in material parameters. Finally, the discrepancies between the wave propagation in coarse and fine meshes are analyzed based on the temporal evolution of the transmembrane potential at a node and its neighboring Gauss points. Insight obtained from this study can be used to suggest new methods to improve the efficiency of simulations in cardiac electrophysiology.

Scaling digital twins from the artisanal to the industrial

Mathematical modeling and simulation are moving from being powerful development and analysis tools towards having increased roles in operational monitoring, control and decision support, in which models of specific entities are continually updated in the form of a digital twin. However, current digital twins are largely the result of bespoke technical solutions that are difficult to scale. We discuss two exemplar applications that motivate challenges and opportunities for scaling digital twins, and that underscore potential barriers to wider adoption of this technology.

Parameter variations in personalized electrophysiological models of human heart ventricles

The objectives of this study were to evaluate the accuracy of personalized numerical simulations of the electrical activity in human ventricles by comparing simulated electrocardiograms (ECGs) with real patients’ ECGs and analyzing the sensitivity of the model output to variations in the model parameters. We used standard 12-lead ECGs and up to 224 unipolar body-surface ECGs to record three patients with cardiac resynchronization therapy devices and three patients with focal ventricular tachycardia. Patient-tailored geometrical models of the ventricles, atria, large vessels, liver, and spine were created using computed tomography data. Ten cases of focal ventricular activation were simulated using the bidomain model and the TNNP 2006 cellular model. The population-based values of electrical conductivities and other model parameters were used for accuracy analysis, and their variations were used for sensitivity analysis. The mean correlation coefficient between the simulated and real ECGs varied significantly (from r = 0.29 to r = 0.86) among the simulated cases. A strong mean correlation (r > 0.7) was found in eight of the ten model cases. The accuracy of the ECG simulation varied widely in the same patient depending on the localization of the excitation origin. The sensitivity analysis revealed that variations in the anisotropy ratio, blood conductivity, and cellular apicobasal heterogeneity had the strongest influence on transmembrane potential, while variation in lung conductivity had the greatest influence on body-surface ECGs. Futhermore, the anisotropy ratio predominantly affected the latest activation time and repolarization time dispersion, while the cellular apicobasal heterogeneity mainly affected the dispersion of action potential duration, and variation in lung conductivity mainly led to changes in the amplitudes of ECGs and cardiac electrograms. We also found that the effects of certain parameter variations had specific regional patterns on the cardiac and body surfaces. These observations are useful for further developing personalized cardiac models.

A Framework for the generation of digital twins of cardiac electrophysiology from clinical 12-leads ECGs

Cardiac digital twins (Cardiac Digital Twin (CDT)s) of human electrophysiology (Electrophysiology (EP)) are digital replicas of patient hearts derived from clinical data that match like-for-like all available clinical observations. Due to their inherent predictive potential, CDTs show high promise as a complementary modality aiding in clinical decision making and also in the cost-effective, safe and ethical testing of novel EP device therapies. However, current workflows for both the anatomical and functional twinning phases within CDT generation, referring to the inference of model anatomy and parameters from clinical data, are not sufficiently efficient, robust and accurate for advanced clinical and industrial applications. Our study addresses three primary limitations impeding the routine generation of high-fidelity CDTs by introducing; a comprehensive parameter vector encapsulating all factors relating to the ventricular EP; an abstract reference frame within the model allowing the unattended manipulation of model parameter fields; a novel fast-forward electrocardiogram (Electrocardiogram (ECG)) model for efficient and bio-physically-detailed simulation required for parameter inference. A novel workflow for the generation of CDTs is then introduced as an initial proof of concept. Anatomical twinning was performed within a reasonable time compatible with clinical workflows (<4h) for 12 subjects from clinically-attained magnetic resonance images. After assessment of the underlying fast forward ECG model against a gold standard bidomain ECG model, functional twinning of optimal parameters according to a clinically-attained 12 lead ECG was then performed using a forward Saltelli sampling approach for a single subject. The achieved results in terms of efficiency and fidelity demonstrate that our workflow is well-suited and viable for generating biophysically-detailed CDTs at scale.

Arrhythmogenic effects of ultra-long and bistable cardiac action potentials

Contemporary accounts of the initiation of cardiac arrhythmias typically rely on after-depolarizations as the trigger for reentrant activity. The after-depolarizations are usually triggered by calcium entry or spontaneous release within the cells of the myocardium or the conduction system. Here we propose an alternative mechanism whereby arrhythmias are triggered autonomously by cardiac cells that fail to repolarize after a normal heartbeat. We investigated the proposal by representing the heart as an excitable medium of FitzHugh-Nagumo cells where a proportion of cells were capable of remaining depolarized indefinitely. As such, those cells exhibit bistable membrane dynamics. We found that heterogeneous media can tolerate a surprisingly large number of bistable cells and still support normal rhythmic activity. Yet there is a critical limit beyond which the medium is persistently arrhythmogenic. Numerical analysis revealed that the critical threshold for arrhythmogenesis depends on both the strength of the coupling between cells and the extent to which the abnormal cells resist repolarization. Moreover, arrhythmogenesis was found to emerge preferentially at tissue boundaries where cells naturally have fewer neighbors to influence their behavior. These findings may explain why atrial fibrillation typically originates from tissue boundaries such as the cuff of the pulmonary vein.

Cardiac fibrillation is a medical condition where normal heart function is compromised as electrical activity becomes disordered. How fibrillation arises spontaneously is not fully understood. It is generally thought to be triggered by premature depolarization of the cardiac action potential in one or more cells. Those premature beats, known as after-depolarizations, subsequently initiate a self-sustaining rotor in the otherwise normal heart tissue. In this study, we propose an alternative mechanism whereby arrhythmias are initiated by cardiac cells that fail to repolarize of their own accord but still operate normally when embedded in functional heart tissue. We find that such cells can act as focal ectopic sources under appropriate conditions of inter-cellular coupling. Moreover, those cells are more prone to initiating arrhythmia when they are located on natural tissue boundaries. This may explain why atrial fibrillation typically originates from the site where the pulmonary vein attaches to the wall of the heart.

The Role of CaMKII Overexpression and Oxidation in Atrial Fibrillation-A Simulation Study

This simulation study aims to investigate how the Calcium/calmodulin-dependent protein kinase II (CaMKII) overexpression and oxidation would influence the cardiac electrophysiological behavior and its arrhythmogenic mechanism in atria. A new-built CaMKII oxidation module and a refitted CaMKII overexpression module were integrated into a mouse atrial cell model for analyzing cardiac electrophysiological variations in action potential (AP) characteristics and intracellular Ca²⁺ cycling under different conditions. Simulation results showed that CaMKII overexpression significantly increased the phosphorylation level of its downstream target proteins, resulting in prolonged AP and smaller calcium transient amplitude, and impaired the Ca²⁺ cycling stability. These effects were exacerbated by extra reactive oxygen species, which oxidized CaMKII and led to continuous high CaMKII activation in both systolic and diastolic phases. Intracellular Ca²⁺ depletion and sustained delayed afterdepolarizations (DADs) were observed under co-existing CaMKII overexpression and oxidation, which could be effectively reversed by clamping the phosphorylation level of ryanodine receptor (RyR). We also found that the stability of RyR release highly depended on a delicate balance between the level of RyR phosphorylation and sarcoplasmic reticulum Ca²⁺ concentration, which was closely related to the genesis of DADs. We concluded that the CaMKII overexpression and oxidation have a synergistic role in increasing the activity of CaMKII, and the unstable RyR may be the key downstream target in the CaMKII arrhythmogenic mechanism. Our simulation provides detailed mechanistic insights into the arrhythmogenic effect of CaMKII overexpression and oxidation, which suggests CaMKII as a promising target in the therapy of atrial fibrillation.

Efficiency of semi-implicit alternating direction implicit methods for solving cardiac monodomain model

It is well known that numerical simulations of the cardiac monodomain model require fine mesh resolution, which increases the computational resources required. In this paper, we construct three operator-splitting alternating direction implicit (ADI) schemes to efficiently solve the nonlinear cardiac monodomain model. The main objective of the proposed methods is to reduce the computational time and memory consumed for solving electrocardiology models, compared to standard numerical methods. The proposed methods have second-order accuracy in both space and time while evaluating the ionic model only once per time-step. Several examples using regular wave, spiral wave reentry, and nonsymmetrical scroll wave are conducted, and the efficiency of the proposed ADI methods is compared to the standard semi-implicit Crank-Nicolson/Adams-Bashforth method. Large-scale two- and three-dimensional simulations are performed.

On the Role of Ionic Modeling on the Signature of Cardiac Arrhythmias for Healthy and Diseased Hearts

Computational cardiology is rapidly becoming the gold standard for innovative medical treatments and device development. Despite a worldwide effort in mathematical and computational modeling research, the complexity and intrinsic multiscale nature of the heart still limit our predictability power raising the question of the optimal modeling choice for large-scale whole-heart numerical investigations. We propose an extended numerical analysis among two different electrophysiological modeling approaches: a simplified phenomenological one and a detailed biophysical one. To achieve this, we considered three-dimensional healthy and infarcted swine heart geometries. Heterogeneous electrophysiological properties, fine-tuned DT-MRI -based anisotropy features, and non-conductive ischemic regions were included in a custom-built finite element code. We provide a quantitative comparison of the electrical behaviors during steady pacing and sustained ventricular fibrillation for healthy and diseased cases analyzing cardiac arrhythmias dynamics. Action potential duration (APD) restitution distributions, vortex filament counting, and pseudo-electrocardiography (ECG) signals were numerically quantified, introducing a novel statistical description of restitution patterns and ventricular fibrillation sustainability. Computational cost and scalability associated with the two modeling choices suggests that ventricular fibrillation signatures are mainly controlled by anatomy and structural parameters, rather than by regional restitution properties. Finally, we discuss limitations and translational perspectives of the different modeling approaches in view of large-scale whole-heart in silico studies.

Approaches for determining cardiac bidomain conductivity values: progress and challenges

Modelling the electrical activity of the heart is an important tool for understanding electrical function in various diseases and conduction disorders. Clearly, for model results to be useful, it is necessary to have accurate inputs for the models, in particular the commonly used bidomain model. However, there are only three sets of four experimentally determined conductivity values for cardiac ventricular tissue and these are inconsistent, were measured around 40 years ago, often produce different results in simulations and do not fully represent the three-dimensional anisotropic nature of cardiac tissue. Despite efforts in the intervening years, difficulties associated with making the measurements and also determining the conductivities from the experimental data have not yet been overcome. In this review, we summarise what is known about the conductivity values, as well as progress to date in meeting the challenges associated with both the mathematical modelling and the experimental techniques. Graphical abstract Epicardial potential distributions, arising from a subendocardial ischaemic region, modelled using conductivity data from the indicated studies.

Deep learning-based reduced order models in cardiac electrophysiology

Predicting the electrical behavior of the heart, from the cellular scale to the tissue level, relies on the numerical approximation of coupled nonlinear dynamical systems. These systems describe the cardiac action potential, that is the polarization/depolarization cycle occurring at every heart beat that models the time evolution of the electrical potential across the cell membrane, as well as a set of ionic variables. Multiple solutions of these systems, corresponding to different model inputs, are required to evaluate outputs of clinical interest, such as activation maps and action potential duration. More importantly, these models feature coherent structures that propagate over time, such as wavefronts. These systems can hardly be reduced to lower dimensional problems by conventional reduced order models (ROMs) such as, e.g., the reduced basis method. This is primarily due to the low regularity of the solution manifold (with respect to the problem parameters), as well as to the nonlinear nature of the input-output maps that we intend to reconstruct numerically. To overcome this difficulty, in this paper we propose a new, nonlinear approach relying on deep learning (DL) algorithms—such as deep feedforward neural networks and convolutional autoencoders—to obtain accurate and efficient ROMs, whose dimensionality matches the number of system parameters. We show that the proposed DL-ROM framework can efficiently provide solutions to parametrized electrophysiology problems, thus enabling multi-scenario analysis in pathological cases. We investigate four challenging test cases in cardiac electrophysiology, thus demonstrating that DL-ROM outperforms classical projection-based ROMs.

In-silico pace-mapping using a detailed whole torso model and implanted electronic device electrograms for more efficient ablation planning

BACKGROUND

Pace-mapping is a commonly used electrophysiological (EP) procedure which aims to identify exit sites of ventricular tachycardia (VT) by matching ventricular activation patterns (assessed by QRS morphology) at specific pacing locations with activation during VT. However, long procedure durations and the need for VT induction render this technique non-optimal. To demonstrate the potential of in-silico pace-mapping, using stored electrogram (EGM) recordings of clinical VT from implanted devices to guide pre-procedural ablation planning.

METHOD

Six scar-related VT episodes were simulated in a 3D torso model reconstructed from computed tomography (CT) imaging data, including three different infarct anatomies mapped from infarcted porcine imaging data. In-silico pace-mapping was performed to localise VT exit sites and isthmuses by using 12-lead electrocardiogram (ECG) signals and different combinations of EGM sensing vectors from implanted devices, through the creation of conventional correlation maps and reference-less maps.

RESULTS

Our in-silico platform was successful in identifying VT exit sites for a variety of different VT morphologies from both ECG correlation maps and corresponding EGM maps, with the latter dependent upon the number of sensing vectors used. We also showed the added utility of both ECG and EGM reference-less pace-mapping for the identification of slow-conducting isthmuses, uncovering the optimal algorithm parameters. Finally, EGM-based pace-mapping was shown to be more dependent upon the mapped surface (epicardial/endocardial), relative to the VT origin.

CONCLUSIONS

In-silico pace-mapping can be used along with EGMs from implanted devices to localise VT ablation targets in pre-procedural planning.

In-silico study of the cardiac arrhythmogenic potential of biomaterial injection therapy

Biomaterial injection is a novel therapy to treat ischemic heart failure (HF) that has shown to reduce remodeling and restore cardiac function in recent preclinical studies. While the effect of biomaterial injection in reducing mechanical wall stress has been recently demonstrated, the influence of biomaterials on the electrical behavior of treated hearts has not been elucidated. In this work, we developed computational models of swine hearts to study the electrophysiological vulnerability associated with biomaterial injection therapy. The propagation of action potentials on realistic biventricular geometries was simulated by numerically solving the monodomain electrophysiology equations on anatomically-detailed models of normal, HF untreated, and HF treated hearts. Heart geometries were constructed from high-resolution magnetic resonance images (MRI) where the healthy, peri-infarcted, infarcted and gel regions were identified, and the orientation of cardiac fibers was informed from diffusion-tensor MRI. Regional restitution properties in each case were evaluated by constructing a probability density function of the action potential duration (APD) at different cycle lengths. A comparative analysis of the ventricular fibrillation (VF) dynamics for every heart was carried out by measuring the number of filaments formed after wave braking. Our results suggest that biomaterial injection therapy does not affect the regional dispersion of repolarization when comparing untreated and treated failing hearts. Further, we found that the treated failing heart is more prone to sustain VF than the normal heart, and is at least as susceptible to sustained VF as the untreated failing heart. Moreover, we show that the main features of VF dynamics in a treated failing heart are not affected by the level of electrical conductivity of the biogel injectates. This work represents a novel proof-of-concept study demonstrating the feasibility of computer simulations of the heart in understanding the arrhythmic behavior in novel therapies for HF.

Heart failure-induced atrial remodelling promotes electrical and conduction alternans

Heart failure (HF) is associated with an increased propensity for atrial fibrillation (AF), causing higher mortality than AF or HF alone. It is hypothesized that HF-induced remodelling of atrial cellular and tissue properties promotes the genesis of atrial action potential (AP) alternans and conduction alternans that perpetuate AF. However, the mechanism underlying the increased susceptibility to atrial alternans in HF remains incompletely elucidated. In this study, we investigated the effects of how HF-induced atrial cellular electrophysiological (with prolonged AP duration) and tissue structural (reduced cell-to-cell coupling caused by atrial fibrosis) remodelling can have an effect on the generation of atrial AP alternans and their conduction at the cellular and one-dimensional (1D) tissue levels. Simulation results showed that HF-induced atrial electrical remodelling prolonged AP duration, which was accompanied by an increased sarcoplasmic reticulum (SR) Ca²⁺ content and Ca²⁺ transient amplitude. Further analysis demonstrated that HF-induced atrial electrical remodelling increased susceptibility to atrial alternans mainly due to the increased sarcoplasmic reticulum Ca²⁺-ATPase (SERCA) Ca²⁺ reuptake, modulated by increased phospholamban (PLB) phosphorylation, and the decreased transient outward K⁺ current (I_to). The underlying mechanism has been suggested that the increased SR Ca²⁺ content and prolonged AP did not fully recover to their previous levels at the end of diastole, resulting in a smaller SR Ca²⁺ release and AP in the next beat. These produced Ca²⁺ transient alternans and AP alternans, and further caused AP alternans and Ca²⁺ transient alternans through Ca²⁺→AP coupling and AP→Ca²⁺ coupling, respectively. Simulation of a 1D tissue model showed that the combined action of HF-induced ion channel remodelling and a decrease in cell-to-cell coupling due to fibrosis increased the heart tissue’s susceptibility to the formation of spatially discordant alternans, resulting in an increased functional AP propagation dispersion, which is pro-arrhythmic. These findings provide insights into how HF promotes atrial arrhythmia in association with atrial alternans.

Atrial Fibrillation (AF) is the most common arrhythmia in adults, especially in the elderly, with the increased incidence of stroke being a major complication that increases morbidity and mortality. The occurrence of AF is often accompanied by heart failure (HF). AF and HF are also known to have the bidirectional relationship that AF worsens HF and HF promotes AF. HF can induce atrial remodelling, including electrical remodelling, atrial fibrosis, stretch and dilatation, and oxidative stress, in which many factors are associated with arrhythmogenic atrial alternans. HF-induced atrial remodelling varies during various stages and complications of HF, but possible mechanisms underlying their pro-susceptibility to alternans have not been completely elucidated. In this study, we investigated the effects of HF-induced atrial remodelling with prolonged action potential duration (APD) and decreased cell-to-cell coupling on susceptibility to atrial alternans. Simulation results showed that HF-induced an increase in sarcoplasmic reticulum Ca²⁺-ATPase (SERCA) Ca²⁺ reuptake caused by increased phospholamban phosphorylation and a decrease in transient outward K⁺ current played significant roles in the genesis of Ca²⁺ transient alternans and action potential alternans at the single-cell level. The HF-induced decline of cell-to-cell coupling and APD prolongation promoted the genesis of spatially discordant alternans in atrial tissue. This provides insights into how HF facilitates atrial arrhythmia in relation to atrial alternans.

Stabilized hybrid discontinuous Galerkin finite element method for the cardiac monodomain equation

Numerical methods for solving the cardiac electrophysiology model, which describes the electrical activity in the heart, are proposed. The model problem consists of a nonlinear reaction-diffusion partial differential equation coupled to systems of ordinary differential equations that describes electrochemical reactions in cardiac cells. The proposed methods combine an operator splitting technique for the reaction-diffusion equation with primal hybrid methods for spatial discretization considering continuous or discontinuous approximations for the Lagrange multiplier. A static condensation is adopted to form a reduced global system in terms of the multiplier only. Convergence studies exhibit optimal rates of convergence and numerical experiments show that the proposed schemes can be more efficient than standard numerical techniques commonly used in this context when preconditioned iterative methods are used for the solution of linear systems.

Electrophysiological significance of the interatrial conduction including cavo-tricuspid isthmus during atrial fibrillation

KEY POINTS

The interatrial conduction, including Bachmann's bundle, the posterior septal conduction, the anterior septal conduction, and the cavo-tricuspid isthmus, contributes to the maintenance mechanisms of atrial fibrillation in a 3D biatrial model. The interatrial conduction ablation including a cavo-tricuspid isthmus ablation significantly affects the wave dynamics of atrial fibrillation (AF) and facilitates the AF termination or atrial tachycardia conversion of the AF after the circumferential pulmonary vein isolation. Additional cavo-tricuspid isthmus ablation after the circumferential pulmonary vein isolation improves long-term rhythm outcome after clinical AF catheter ablation.

ABSTRACT

Although it is known that atrial fibrillation (AF) is mainly a left atrial (LA) disease, the role of the right atrium (RA) and interatrial conduction (IAC), including the cavo-tricuspid isthmus (CTI), has not been clearly defined. We tested AF wave dynamics with or without IAC in computational modelling and the rhythm outcome of AF catheter ablation (AFCA) including CTI ablation in clinical cohort data. We evaluated the dominant frequency (DF) in 3D biatrial AF simulations integrated with 3D-computed tomograms obtained from 10 patients. The IAC was implemented at Bachmann's bundle, posterior septum and the CTI. After virtual circumferential PV isolation (CPVI), we disconnected IACs one by one, and observed the wave dynamics. We compared the long-term rhythm outcome after CPVI alone and additional CTI ablation in 846 patients with AFCA. LA-DF was higher than RA-DF in AF (P < 0.001). After CPVI, the DF decreased significantly by additional IAC ablation (P = 0.003), especially in the LA (P = 0.016). The amount of DF reduction (P = 0.020) and rates of AF termination (P < 0.001) or AT conversion (P = 0.021) were significantly higher after IAC ablations including CTI than those without. In clinical AFCA, the AF recurrence rate was significantly lower in patients with additional CTI ablation than CPVI alone during 25 ± 20 months' follow-up (hazard ratio 0.60 [0.46-0.79], P < 0.001, Log rank P < 0.001). IAC contributes to the maintenance mechanism of AF, and IAC including CTI ablation affects AF wave dynamics, facilitating AF termination in 3D biatrial modelling. Additional CTI ablation after CPVI improves the long-term rhythm outcome in clinical AFCA, potentially in a paroxysmal type with accompanying atrial flutter, or atrial dimension close to normal.

In silico models for evaluating proarrhythmic risk of drugs

Safety evaluation of drugs requires examination of the risk of generating Torsade de Pointes (TdP) because it can lead to sudden cardiac death. Until recently, the QT interval in the electrocardiogram (ECG) has been used in the evaluation of TdP risk because the QT interval is known to be associated with the development of TdP. Although TdP risk evaluation based on QT interval has been successful in removing drugs with TdP risk from the market, some safe drugs may have also been affected due to the low specificity of QT interval-based evaluation. For more accurate evaluation of drug safety, the comprehensive in vitro proarrhythmia assay (CiPA) has been proposed by regulatory agencies, industry, and academia. Although the CiPA initiative includes in silico evaluation of cellular action potential as a component, attempts to utilize in silico simulation in drug safety evaluation are expanding, even to simulating human ECG using biophysical three-dimensional models of the heart and torso under the effects of drugs. Here, we review recent developments in the use of in silico models for the evaluation of the proarrhythmic risk of drugs. We review the single cell, one-dimensional, two-dimensional, and three-dimensional models and their applications reported in the literature and discuss the possibility of utilizing ECG simulation in drug safety evaluation.

An audit of uncertainty in multi-scale cardiac electrophysiology models

Models of electrical activation and recovery in cardiac cells and tissue have become valuable research tools, and are beginning to be used in safety-critical applications including guidance for clinical procedures and for drug safety assessment. As a consequence, there is an urgent need for a more detailed and quantitative understanding of the ways that uncertainty and variability influence model predictions. In this paper, we review the sources of uncertainty in these models at different spatial scales, discuss how uncertainties are communicated across scales, and begin to assess their relative importance. We conclude by highlighting important challenges that continue to face the cardiac modelling community, identifying open questions, and making recommendations for future studies. This article is part of the theme issue 'Uncertainty quantification in cardiac and cardiovascular modelling and simulation'.

Understanding the transition from paroxysmal to persistent atrial fibrillation

Atrial fibrillation (AF) is the most common cardiac arrhytmia, characterized by the chaotic motion of electrical wavefronts in the atria. In clinical practice, AF is classified under two primary categories: paroxysmal AF, short intermittent episodes separated by periods of normal electrical activity; and persistent AF, longer uninterrupted episodes of chaotic electrical activity. However, the precise reasons why AF in a given patient is paroxysmal or persistent is poorly understood. Recently, we have introduced the percolation-based Christensen-Manani-Peters (CMP) model of AF which naturally exhibits both paroxysmal and persistent AF, but precisely how these differences emerge in the model is unclear. In this paper, we dissect the CMP model to identify the cause of these different AF classifications. Starting from a mean-field model where we describe AF as a simple birth-death process, we add layers of complexity to the model and show that persistent AF arises from reentrant circuits which exhibit an asymmetry in their probability of activation relative to deactivation. As a result, different simulations generated at identical model parameters can exhibit fibrillatory episodes spanning several orders of magnitude from a few seconds to months. These findings demonstrate that diverse, complex fibrillatory dynamics can emerge from very simple dynamics in models of AF.

Key aspects for effective mathematical modelling of fractional-diffusion in cardiac electrophysiology: a quantitative study

Microscopic structural features of cardiac tissue play a fundamental role in determining complex spatio-temporal excitation dynamics at the macroscopic level. Recent efforts have been devoted to the development of mathematical models accounting for non-local spatio-temporal coupling able to capture these complex dynamics without the need of resolving tissue heterogeneities down to the micro-scale. In this work, we analyse in detail several important aspects affecting the overall predictive power of these modelling tools and provide some guidelines for an effective use of space-fractional models of cardiac electrophysiology in practical applications. Through an extensive computational study in simplified computational domains, we highlight the robustness of models belonging to different categories, i.e., physiological and phenomenological descriptions, against the introduction of non-locality, and lay down the foundations for future research and model validation against experimental data. A modern genetic algorithm framework is used to investigate proper parameterisations of the considered models, and the crucial role played by the boundary assumptions in the considered settings is discussed. Several numerical results are provided to support our claims.

Sensitivity and Uncertainty Analysis of Two Human Atrial Cardiac Cell Models Using Gaussian Process Emulators

Biophysically detailed cardiac cell models reconstruct the action potential and calcium dynamics of cardiac myocytes. They aim to capture the biophysics of current flow through ion channels, pumps, and exchangers in the cell membrane, and are highly detailed. However, the relationship between model parameters and model outputs is difficult to establish because the models are both complex and non-linear. The consequences of uncertainty and variability in model parameters are therefore difficult to determine without undertaking large numbers of model evaluations. The aim of the present study was to demonstrate how sensitivity and uncertainty analysis using Gaussian process emulators can be used for a systematic and quantitive analysis of biophysically detailed cardiac cell models. We selected the Courtemanche and Maleckar models of the human atrial action potential for analysis because these models describe a similar set of currents, with different formulations. In our approach Gaussian processes emulate the main features of the action potential and calcium transient. The emulators were trained with a set of design data comprising samples from parameter space and corresponding model outputs, initially obtained from 300 model evaluations. Variance based sensitivity indices were calculated using the emulators, and first order and total effect indices were calculated for each combination of parameter and output. The differences between the first order and total effect indices indicated that the effect of interactions between parameters was small. A second set of emulators were then trained using a new set of design data with a subset of the model parameters with a sensitivity index of more than 0.1 (10%). This second stage analysis enabled comparison of mechanisms in the two models. The second stage sensitivity indices enabled the relationship between the L-type Ca 2+ current and the action potential plateau to be quantified in each model. Our quantitative analysis predicted that changes in maximum conductance of the ultra-rapid K + channel I Kur would have opposite effects on action potential duration in the two models, and this prediction was confirmed by additional simulations. This study has demonstrated that Gaussian process emulators are an effective tool for sensitivity and uncertainty analysis of biophysically detailed cardiac cell models.

Polymeric Composites Based on Carboxymethyl Cellulose Cryogel and Conductive Polymers: Synthesis and Characterization

In this study, a super porous polymeric network prepared from a natural polymer, carboxymethyl cellulose (CMC), was used as a scaffold in the preparation of conductive polymers such as poly(Aniline) (PANi), poly(Pyrrole) (PPy), and poly(Thiophene) (PTh). CMC–conductive polymer composites were characterized by Fourier transform infrared (FT-IR) spectroscopy, thermogravimetric analysis (TGA) techniques, and conductivity measurements. The highest conductivity was observed as 4.36 × 10−4 ± 4.63 × 10−5 S·cm−1 for CMC–PANi cryogel composite. The changes in conductivity of prepared CMC cryogel and its corresponding PAN, PPy, and PTh composites were tested against HCl and NH3 vapor. The changes in conductivity values of CMC cryogel upon HCl and NH3 vapor treatment were found to increase 1.5- and 2-fold, respectively, whereas CMC–PANi composites showed a 143-fold increase in conductivity upon HCl and a 12-fold decrease in conductivity upon NH3 treatment, suggesting the use of natural polymer–conductive polymer composites as sensor for these gases.

Accelerating simulations of cardiac electrical dynamics through a multi-GPU platform and an optimized data structure

Simulations of cardiac electrophysiological models in tissue, particularly in 3D require the solutions of billions of differential equations even for just a couple of milliseconds, thus highly demanding in computational resources. In fact, even studies in small domains with very complex models may take several hours to reproduce seconds of electrical cardiac behavior. Today's Graphics Processor Units (GPUs) are becoming a way to accelerate such simulations, and give the added possibilities to run them locally without the need for supercomputers. Nevertheless, when using GPUs, bottlenecks related to global memory access caused by the spatial discretization of the large tissue domains being simulated, become a big challenge. For simulations in a single GPU, we propose a strategy to accelerate the computation of the diffusion term through a data-structure and memory access pattern designed to maximize coalescent memory transactions and minimize branch divergence, achieving results approximately 1.4 times faster than a standard GPU method. We also combine this data structure with a designed communication strategy to take advantage in the case of simulations in multi-GPU platforms. We demonstrate that, in the multi-GPU approach performs, simulations in 3D tissue can be just 4× slower than real time.

Embedding high-dimensional Bayesian optimization via generative modeling: Parameter personalization of cardiac electrophysiological models

The estimation of patient-specific tissue properties in the form of model parameters is important for personalized physiological models. Because tissue properties are spatially varying across the underlying geometrical model, it presents a significant challenge of high-dimensional (HD) optimization at the presence of limited measurement data. A common solution to reduce the dimension of the parameter space is to explicitly partition the geometrical mesh. In this paper, we present a novel concept that uses a generative variational auto-encoder (VAE) to embed HD Bayesian optimization into a low-dimensional (LD) latent space that represents the generative code of HD parameters. We further utilize VAE-encoded knowledge about the generative code to guide the exploration of the search space. The presented method is applied to estimating tissue excitability in a cardiac electrophysiological model in a range of synthetic and real-data experiments, through which we demonstrate its improved accuracy and substantially reduced computational cost in comparison to existing methods that rely on geometry-based reduction of the HD parameter space.

Multi-scale approaches for the simulation of cardiac electrophysiology: II - Tissue-level structure and function

Computational models of the heart, from cell-level models, through one-, two- and three-dimensional tissue-level simplifications, to biophysically-detailed three-dimensional models of the ventricles, atria or whole heart, allow the simulation of excitation and propagation of this excitation, and have provided remarkable insight into the normal and pathological functioning of the heart. In this article we present equations for modelling cellular excitation (i.e. the cell action potential) from both a phenomenological and a biophysical perspective. Hodgkin-Huxley formalism is discussed, along with the current generation of biophysically-detailed cardiac cell models. Alternative Markovian formulations for modelling ionic currents are also presented. Equations describing propagation of this cellular excitation, through one-, two- and three-dimensional idealised or realistic tissues, are then presented. For all types of model, from cell to tissue, methods for discretisation and integration of the underlying equations are discussed. The article finishes with a discussion of two tissue-level experimental imaging techniques - diffusion tensor magnetic resonance imaging and optical imaging - that can be used to provide data for parameterisation and validation of cell- and tissue-level cardiac models.

Smeared Multiscale Finite Element Models for Mass Transport and Electrophysiology Coupled to Muscle Mechanics

Mass transport represents the most fundamental process in living organisms. It includes delivery of nutrients, oxygen, drugs, and other substances from the vascular system to tissue and transport of waste and other products from cells back to vascular and lymphatic network and organs. Furthermore, movement is achieved by mechanical forces generated by muscles in coordination with the nervous system. The signals coming from the brain, which have the character of electrical waves, produce activation within muscle cells. Therefore, from a physics perspective, there exist a number of physical fields within the body, such as velocities of transport, pressures, concentrations of substances, and electrical potential, which is directly coupled to biochemical processes of transforming the chemical into mechanical energy and further internal forces for motion. The overall problems of mass transport and electrophysiology coupled to mechanics can be investigated theoretically by developing appropriate computational models. Due to the enormous complexity of the biological system, it would be almost impossible to establish a detailed computational model for the physical fields related to mass transport, electrophysiology, and coupled fields. To make computational models feasible for applications, we here summarize a concept of smeared physical fields, with coupling among them, and muscle mechanics, which includes dependence on the electrical potential. Accuracy of the smeared computational models, also with coupling to muscle mechanics, is illustrated with simple example, while their applicability is demonstrated on a liver model with tumors present. The last example shows that the introduced methodology is applicable to large biological systems.

Pro-arrhythmic Effects of Hydrogen Sulfide in Healthy and Ischemic Cardiac Tissues: Insight From a Simulation Study

Hydrogen sulfide (H₂S), an ambient air pollutant, has been reported to increase cardiac events in patients with cardiovascular diseases, but the underlying mechanisms remain not elucidated. This study investigated the pro-arrhythmic effects of H₂S in healthy and ischemic conditions. Experimental data of H₂S effects on ionic channels (including the L-type Ca²⁺ channel and ATP-sensitive K⁺ channel) were incorporated into a virtual heart model to evaluate their integral action on cardiac arrhythmogenesis. It was shown that H₂S depressed cellular excitability, abbreviated action potential duration, and augmented tissue’s transmural dispersion of repolarization, resulting in an increase in tissue susceptibility to initiation and maintenance of reentry. The observed effects of H₂S on cardiac excitation are more remarkable in the ischemic condition than in the healthy condition. This study provides mechanistic insights into the pro-arrhythmic effects of air pollution (H₂S), especially in the case with extant ischemic conditions.

Unified mechanism of local drivers in a percolation model of atrial fibrillation

The mechanisms of atrial fibrillation (AF) are poorly understood, resulting in disappointing success rates of ablative treatment. Different mechanisms defined largely by different atrial activation patterns have been proposed and, arguably, this dispute has slowed the progress of AF research. Recent clinical evidence suggests a unifying mechanism of local drivers based on sustained reentrant circuits in the complex atrial architecture. Here, we present a percolation inspired computational model showing spontaneous emergence of AF that strongly supports, and gives a theoretical explanation for, the clinically observed diversity of activation. We show that the difference in surface activation patterns is a direct consequence of the thickness of the discrete network of heart muscle cells through which electrical signals percolate to reach the imaged surface. The model naturally follows the clinical spectrum of AF spanning sinus rhythm, paroxysmal AF, and persistent AF as the decoupling of myocardial cells results in the lattice approaching the percolation threshold. This allows the model to make the prediction that, for paroxysmal AF, reentrant circuits emerge near the endocardium, but in persistent AF they emerge deeper in the bulk of the atrial wall. If experimentally verified, this may go towards explaining the lowering ablation success rate as AF becomes more persistent.

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.

Differences between models of partial thickness and subendocardial ischaemia in terms of sensitivity analyses of ST-segment epicardial potential distributions

Mathematical modelling is a useful technique to help elucidate the connection between non-transmural ischaemia and ST elevation and depression of the ECG. Generally, models represent non-transmural ischaemia using an ischaemic zone that extends from the endocardium partway to the epicardium. However, recent experimental work has suggested that ischaemia typically arises within the heart wall. This work examines the effect of modelling cardiac ischaemia in the left ventricle using two different models: subendocardial ischaemia and partial thickness ischaemia, representing the first and second scenarios, respectively. We found that it is possible, only in the model of subendocardial ischaemia, to see a single minimum on the epicardial surface above the ischaemic region, and this only occurs for low ischaemic thicknesses. This may help to explain the rarity of ST depression that is located over the ischaemic region. It was also found that, in both models, the epicardial potential distribution is most sensitive to the proximity of the ischaemic region to the epicardium, rather than to the thickness of the ischaemic region. Since proximity does not indicate the thickness of the ischaemic region, this suggests a reason why it may be difficult to determine the degree of ischaemia using the ST segment of the ECG.

Investigation of the Role of Myocyte Orientations in Cardiac Arrhythmia Using Image-Based Models

Cardiac electrical excitation-propagation is influenced by myocyte orientations (cellular organization). Quantitatively understanding this relationship presents a significant research challenge, especially during arrhythmias in which excitation patterns become complex. Tissue-scale simulations of cardiac electrophysiology, incorporating both dynamic action potential behavior and image-based myocardial architecture, provide an approach to investigate three-dimensional (3D) propagation of excitation waves in the heart. In this study, we aimed to assess the importance of natural variation in myocyte orientations on cardiac arrhythmogenesis using 3D tissue electrophysiology simulations. Three anatomical models (i.e., describing myocyte orientations) of healthy rat ventricles—obtained using diffusion tensor imaging at 100 μm resolution—were registered to a single biventricular geometry (i.e., a single cardiac shape), in which the myocyte orientations could be represented by each of the diffusion tensor imaging data sets or by an idealized rule-based description. The Fenton-Karma cellular excitation model was modified to reproduce rat ventricular action potential duration restitution to create reaction-diffusion cardiac electrophysiology models. Over 250 3D simulations were performed to investigate the effects of myocyte orientations on the following: 1) ventricular activation, 2) location-dependent arrhythmia induction via rapid pacing, and 3) dynamics of re-entry averaged over multiple episodes. It was shown that 1) myocyte orientation differences manifested themselves in local activation times, but the influence on total activation time was small; 2) differences in myocyte orientations could critically affect the inducibility and persistence of arrhythmias for specific stimulus-location/cycle-length combinations; and 3) myocyte orientations alone could be an important determinant of scroll wave break, although no significant differences were observed in averaged arrhythmia dynamics between the four myocyte orientation scenarios considered. Our results show that myocyte orientations are an important determinant of arrhythmia inducibility, persistence, and scroll wave break. These findings suggest that where specificity is desired (for example, when predicting location-dependent, patient-specific arrhythmia inducibility), subject-specific myocyte orientations may be important.

Modelling the electrical activity of skeletal muscle tissue using a multi-domain approach

Electromyography (EMG) can be used to study the behaviour of the motor neurons and thus provides insights into the physiology of the central nervous system. However, due to the high complexity of neuromuscular control, EMG signals are challenging to interpret. While the exact knowledge of the excitation patterns of a specific muscle within an in vivo experimental setting remains elusive, simulations allow to systematically investigate EMG signals in a controlled environment. Within this context, simulations can provide virtual EMG data, which, for example, can be used to validate and optimise signal analysis methods that aim to estimate the relationship between EMG signals and the output of motor neuron pools. However, since existing methods, which are employed to compute EMG signals, exhibit deficiencies with respect to the physical model itself as well as with respect to numerical aspects, we propose a novel homogenised continuum model that closely resolves the electro-physiological behaviour of skeletal muscle tissue. The proposed model is based on an extension of the well-established bidomain model and includes a biophysically detailed description of the electrical activity within the tissue, which is due to the depolarisation of the muscle fibre membranes. In contrast to all other published EMG models, which assume that the electrical potential field for each muscle fibre can be calculated independently, the proposed model assumes that the electrical potential in the muscle fibres is coupled to the electrical potential in the extracellular space. We show that the newly proposed model is able to simulate realistic EMG signals and demonstrate the potential to employ the predicted virtual EMG signal in order to evaluate the goodness of automated decomposition algorithms.

Spatiotemporal correlation uncovers characteristic lengths in cardiac tissue

Complex spatiotemporal patterns of action potential duration have been shown to occur in many mammalian hearts due to period-doubling bifurcations that develop with increasing frequency of stimulation. Here, through high-resolution optical mapping experiments and mathematical modeling, we introduce a characteristic spatial length of cardiac activity in canine ventricular wedges via a spatiotemporal correlation analysis, at different stimulation frequencies and during fibrillation. We show that the characteristic length ranges from 40 to 20 cm during one-to-one responses and it decreases to a specific value of about 3 cm at the transition from period-doubling bifurcation to fibrillation. We further show that during fibrillation, the characteristic length is about 1 cm. Another significant outcome of our analysis is the finding of a constitutive phenomenological law obtained from a nonlinear fitting of experimental data which relates the conduction velocity restitution curve with the characteristic length of the system. The fractional exponent of 3/2 in our phenomenological law is in agreement with the domain size remapping required to reproduce experimental fibrillation dynamics within a realistic cardiac domain via accurate mathematical models.

Simulation of action potential propagation based on the ghost structure method

In this paper, a ghost structure (GS) method is proposed to simulate the monodomain model in irregular computational domains using finite difference without regenerating body-fitted grids. In order to verify the validity of the GS method, it is first used to solve the Fitzhugh-Nagumo monodomain model in rectangular and circular regions at different states (the stationary and moving states). Then, the GS method is used to simulate the propagation of the action potential (AP) in transverse and longitudinal sections of a healthy human heart, and with left bundle branch block (LBBB). Finally, we analyze the AP and calcium concentration under healthy and LBBB conditions. Our numerical results show that the GS method can accurately simulate AP propagation with different computational domains either stationary or moving, and we also find that LBBB will cause the left ventricle to contract later than the right ventricle, which in turn affects synchronized contraction of the two ventricles.

Comprehensive Uncertainty Quantification and Sensitivity Analysis for Cardiac Action Potential Models

Recent efforts to ensure the reliability of computational model-based predictions in healthcare, such as the ASME V&V40 Standard, emphasize the importance of uncertainty quantification (UQ) and sensitivity analysis (SA) when evaluating computational models. UQ involves empirically determining the uncertainty in model inputs-typically resulting from natural variability or measurement error-and then calculating the resultant uncertainty in model outputs. SA involves calculating how uncertainty in model outputs can be apportioned to input uncertainty. Rigorous comprehensive UQ/SA provides confidence that model-based decisions are robust to underlying uncertainties. However, comprehensive UQ/SA is not currently feasible for whole heart models, due to numerous factors including model complexity and difficulty in measuring variability in the many parameters. Here, we present a significant step to developing a framework to overcome these limitations. We: (i) developed a novel action potential (AP) model of moderate complexity (six currents, seven variables, 36 parameters); (ii) prescribed input variability for all parameters (not empirically derived); (iii) used a single "hyper-parameter" to study increasing levels of parameter uncertainty; (iv) performed UQ and SA for a range of model-derived quantities with physiological relevance; and (v) present quantitative and qualitative ways to analyze different behaviors that occur under parameter uncertainty, including "model failure". This is the first time uncertainty in every parameter (including conductances, steady-state parameters, and time constant parameters) of every ionic current in a cardiac model has been studied. This approach allowed us to demonstrate that, for this model, the simulated AP is fully robust to low levels of parameter uncertainty - to our knowledge the first time this has been shown of any cardiac model. A range of dynamics was observed at larger parameter uncertainty (e.g., oscillatory dynamics); analysis revealed that five parameters were highly influential in these dynamics. Overall, we demonstrate feasibility of performing comprehensive UQ/SA for cardiac cell models and demonstrate how to assess robustness and overcome model failure when performing cardiac UQ analyses. The approach presented here represents an important and significant step toward the development of model-based clinical tools which are demonstrably robust to all underlying uncertainties and therefore more reliable in safety-critical decision-making.

A Contemporary Look at Biomechanical Models of Myocardium

Understanding and predicting the mechanical behavior of myocardium under healthy and pathophysiological conditions are vital to developing novel cardiac therapies and promoting personalized interventions. Within the past 30 years, various constitutive models have been proposed for the passive mechanical behavior of myocardium. These models cover a broad range of mathematical forms, microstructural observations, and specific test conditions to which they are fitted. We present a critical review of these models, covering both phenomenological and structural approaches, and their relations to the underlying structure and function of myocardium. We further explore the experimental and numerical techniques used to identify the model parameters. Next, we provide a brief overview of continuum-level electromechanical models of myocardium, with a focus on the methods used to integrate the active and passive components of myocardial behavior. We conclude by pointing to future directions in the areas of optimal form as well as new approaches for constitutive modeling of myocardium.

Biomimetic cardiovascular platforms for in vitro disease modeling and therapeutic validation

Bioengineered tissues have become increasingly more sophisticated owing to recent advancements in the fields of biomaterials, microfabrication, microfluidics, genetic engineering, and stem cell and developmental biology. In the coming years, the ability to engineer artificial constructs that accurately mimic the compositional, architectural, and functional properties of human tissues, will profoundly impact the therapeutic and diagnostic aspects of the healthcare industry. In this regard, bioengineered cardiac tissues are of particular importance due to the extremely limited ability of the myocardium to self-regenerate, as well as the remarkably high mortality associated with cardiovascular diseases worldwide. As novel microphysiological systems make the transition from bench to bedside, their implementation in high throughput drug screening, personalized diagnostics, disease modeling, and targeted therapy validation will bring forth a paradigm shift in the clinical management of cardiovascular diseases. Here, we will review the current state of the art in experimental in vitro platforms for next generation diagnostics and therapy validation. We will describe recent advancements in the development of smart biomaterials, biofabrication techniques, and stem cell engineering, aimed at recapitulating cardiovascular function at the tissue- and organ levels. In addition, integrative and multidisciplinary approaches to engineer biomimetic cardiovascular constructs with unprecedented human and clinical relevance will be discussed. We will comment on the implementation of these platforms in high throughput drug screening, in vitro disease modeling and therapy validation. Lastly, future perspectives will be provided on how these biomimetic platforms will aid in the transition towards patient centered diagnostics, and the development of personalized targeted therapeutics.

Smeared multiscale finite element model for electrophysiology and ionic transport in biological tissue

Basic functions of living organisms are governed by the nervous system through bidirectional signals transmitted from the brain to neural networks. These signals are similar to electrical waves. In electrophysiology the goal is to study the electrical properties of biological cells and tissues, and the transmission of signals. From a physics perspective, there exists a field of electrical potential within the living body, the nervous system, extracellular space and cells. Electrophysiological problems can be investigated experimentally and also theoretically by developing appropriate mathematical or computational models. Due to the enormous complexity of biological systems, it would be almost impossible to establish a detailed computational model of the electrical field, even for only a single organ (e.g. heart), including the entirety of cells comprising the neural network. In order to make computational models feasible for practical applications, we here introduce the concept of smeared fields, which represents a generalization of the previously formulated multiscale smeared methodology for mass transport in blood vessels, lymph, and tissue. We demonstrate the accuracy of the smeared finite element computational models for the electric field in numerical examples. The electrical field is further coupled with ionic mass transport within tissue composed of interstitial spaces extracellularly and by cytoplasm and organelles intracellularly. The proposed methodology, which couples electrophysiology and molecular ionic transport, is applicable to a variety of biological systems.

Real-time interactive simulations of large-scale systems on personal computers and cell phones: Toward patient-specific heart modeling and other applications

Cardiac dynamics modeling has been useful for studying and treating arrhythmias. However, it is a multiscale problem requiring the solution of billions of differential equations describing the complex electrophysiology of interconnected cells. Therefore, large-scale cardiac modeling has been limited to groups with access to supercomputers and clusters. Many areas of computational science face similar problems where computational costs are too high for personal computers so that supercomputers or clusters currently are necessary. Here, we introduce a new approach that makes high-performance simulation of cardiac dynamics and other large-scale systems like fluid flow and crystal growth accessible to virtually anyone with a modest computer. For cardiac dynamics, this approach will allow not only scientists and students but also physicians to use physiologically accurate modeling and simulation tools that are interactive in real time, thereby making diagnostics, research, and education available to a broader audience and pushing the boundaries of cardiac science.

Computational Multiscale Solvers for Continuum Approaches

Computational multiscale analyses are currently ubiquitous in science and technology. Different problems of interest-e.g., mechanical, fluid, thermal, or electromagnetic-involving a domain with two or more clearly distinguished spatial or temporal scales, are candidates to be solved by using this technique. Moreover, the predictable capability and potential of multiscale analysis may result in an interesting tool for the development of new concept materials, with desired macroscopic or apparent properties through the design of their microstructure, which is now even more possible with the combination of nanotechnology and additive manufacturing. Indeed, the information in terms of field variables at a finer scale is available by solving its associated localization problem. In this work, a review on the algorithmic treatment of multiscale analyses of several problems with a technological interest is presented. The paper collects both classical and modern techniques of multiscale simulation such as those based on the proper generalized decomposition (PGD) approach. Moreover, an overview of available software for the implementation of such numerical schemes is also carried out. The availability and usefulness of this technique in the design of complex microstructural systems are highlighted along the text. In this review, the fine, and hence the coarse scale, are associated with continuum variables so atomistic approaches and coarse-graining transfer techniques are out of the scope of this paper.

ElectroMap: High-throughput open-source software for analysis and mapping of cardiac electrophysiology

The ability to record and analyse electrical behaviour across the heart using optical and electrode mapping has revolutionised cardiac research. However, wider uptake of these technologies is constrained by the lack of multi-functional and robustly characterised analysis and mapping software. We present ElectroMap, an adaptable, high-throughput, open-source software for processing, analysis and mapping of complex electrophysiology datasets from diverse experimental models and acquisition modalities. Key innovation is development of standalone module for quantification of conduction velocity, employing multiple methodologies, currently not widely available to researchers. ElectroMap has also been designed to support multiple methodologies for accurate calculation of activation, repolarisation, arrhythmia detection, calcium handling and beat-to-beat heterogeneity. ElectroMap implements automated signal segmentation, ensemble averaging and integrates optogenetic approaches. Here we employ ElectroMap for analysis, mapping and detection of pro-arrhythmic phenomena in silico, in cellulo, animal model and in vivo patient datasets. We anticipate that ElectroMap will accelerate innovative cardiac research and enhance the uptake, application and interpretation of mapping technologies leading to novel approaches for arrhythmia prevention.

Response function framework for the dynamics of meandering or large-core spiral waves and modulated traveling waves

In many oscillatory or excitable systems, dynamical patterns emerge which are stationary or periodic in a moving frame of reference. Examples include traveling waves or spiral waves in chemical systems or cardiac tissue. We present a unified theoretical framework for the drift of such patterns under small external perturbations, in terms of overlap integrals between the perturbation and the adjoint critical eigenfunctions of the linearized operator (i.e., response functions). For spiral waves, the finite radius of the spiral tip trajectory and spiral wave meander are taken into account. Different coordinate systems can be chosen, depending on whether one wants to predict the motion of the spiral-wave tip, the time-averaged tip path, or the center of the meander flower. The framework is applied to analyze the drift of a meandering spiral wave in a constant external field in different regimes.