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

Three-dimensional histology: tools and application to quantitative assessment of cell-type distribution in rabbit heart

Aims

Cardiac histo-anatomical organization is a major determinant of function. Changes in tissue structure are a relevant factor in normal and disease development, and form targets of therapeutic interventions. The purpose of this study was to test tools aimed to allow quantitative assessment of cell-type distribution from large histology and magnetic resonance imaging- (MRI) based datasets.

Methods and results

Rabbit heart fixation during cardioplegic arrest and MRI were followed by serial sectioning of the whole heart and light-microscopic imaging of trichrome-stained tissue. Segmentation techniques developed specifically for this project were applied to segment myocardial tissue in the MRI and histology datasets. In addition, histology slices were segmented into myocytes, connective tissue, and undefined. A bounding surface, containing the whole heart, was established for both MRI and histology. Volumes contained in the bounding surface (called ‘anatomical volume’), as well as that identified as containing any of the above tissue categories (called ‘morphological volume’), were calculated. The anatomical volume was 7.8 cm³ in MRI, and this reduced to 4.9 cm³ after histological processing, representing an ‘anatomical’ shrinkage by 37.2%. The morphological volume decreased by 48% between MRI and histology, highlighting the presence of additional tissue-level shrinkage (e.g. an increase in interstitial cleft space). The ratio of pixels classified as containing myocytes to pixels identified as non-myocytes was roughly 6:1 (61.6 vs. 9.8%; the remaining fraction of 28.6% was ‘undefined’).

Conclusion

Qualitative and quantitative differentiation between myocytes and connective tissue, using state-of-the-art high-resolution serial histology techniques, allows identification of cell-type distribution in whole-heart datasets. Comparison with MRI illustrates a pronounced reduction in anatomical and morphological volumes during histology processing.

Human cardiac systems electrophysiology and arrhythmogenesis: iteration of experiment and computation

Human cardiac electrophysiology (EP) is a unique system for computational modelling at multiple scales. Due to the complexity of the cardiac excitation sequence, coordinated activity must occur from the single channel to the entire myocardial syncytium. Thus, sophisticated computational algorithms have been developed to investigate cardiac EP at the level of ion channels, cardiomyocytes, multicellular tissues, and the whole heart. Although understanding of each functional level will ultimately be important to thoroughly understand mechanisms of physiology and disease, cardiac arrhythmias are expressly the product of cardiac tissue-containing enough cardiomyocytes to sustain a reentrant loop of activation. In addition, several properties of cardiac cellular EP, that are critical for arrhythmogenesis, are significantly altered by cell-to-cell coupling. However, relevant human cardiac EP data, upon which to develop or validate models at all scales, has been lacking. Thus, over several years, we have developed a paradigm for multiscale human heart physiology investigation and have recovered and studied over 300 human hearts. We have generated a rich experimental dataset, from which we better understand mechanisms of arrhythmia in human and can improve models of human cardiac EP. In addition, in collaboration with computational physiologists, we are developing a database for the deposition of human heart experimental data, including thorough experimental documentation. We anticipate that accessibility to this human heart dataset will further human EP computational investigations, as well as encourage greater data transparency within the field of cardiac EP.

Bayesian Sensitivity Analysis of a Cardiac Cell Model Using a Gaussian Process Emulator

Models of electrical activity in cardiac cells have become important research tools as they can provide a quantitative description of detailed and integrative physiology. However, cardiac cell models have many parameters, and how uncertainties in these parameters affect the model output is difficult to assess without undertaking large numbers of model runs. In this study we show that a surrogate statistical model of a cardiac cell model (the Luo-Rudy 1991 model) can be built using Gaussian process (GP) emulators. Using this approach we examined how eight outputs describing the action potential shape and action potential duration restitution depend on six inputs, which we selected to be the maximum conductances in the Luo-Rudy 1991 model. We found that the GP emulators could be fitted to a small number of model runs, and behaved as would be expected based on the underlying physiology that the model represents. We have shown that an emulator approach is a powerful tool for uncertainty and sensitivity analysis in cardiac cell models.

AC Electric Field-Induced Trapping of Microparticles in Pinched Microconfinements

The trapping of charged microparticles under confinement in a converging-diverging microchannel, under a symmetric AC field of tunable frequency, is studied. We show that at low frequencies, the trapping characteristics stem from the competing effects of positive dielectrophoresis and the linear electrokinetic phenomena of electroosmosis and electrophoresis. It is found, somewhat unexpectedly, that electroosmosis and electrophoresis significantly affect the concentration profile of the trapped analyte, even for a symmetric AC field. However, at intermediate frequencies, the microparticle trapping mechanism is predominantly a consequence of positive dielectrophoresis. We substantiate our experimental results for the microparticle concentration distribution, along the converging-diverging microchannel, with a detailed theoretical analysis that takes into account all of the relevant frequency-dependent electrokinetic phenomena. This study should be useful in understanding the response of biological components such as cells to applied AC fields. Moreover, it will have potential applications in the design of efficient point-of-care diagnostic devices for detecting biomarkers and also possibly in some recent strategies in cancer therapy using AC fields.

Unstable spiral waves and local Euclidean symmetry in a model of cardiac tissue

This paper investigates the properties of unstable single-spiral wave solutions arising in the Karma model of two-dimensional cardiac tissue. In particular, we discuss how such solutions can be computed numerically on domains of arbitrary shape and study how their stability, rotational frequency, and spatial drift depend on the size of the domain as well as the position of the spiral core with respect to the boundaries. We also discuss how the breaking of local Euclidean symmetry due to finite size effects as well as the spatial discretization of the model is reflected in the structure and dynamics of spiral waves. This analysis allows identification of a self-sustaining process responsible for maintaining the state of spiral chaos featuring multiple interacting spirals.

Generalized minimal principle for rotor filaments

To a reaction-diffusion medium with an inhomogeneous anisotropic diffusion tensor D, we add a fourth spatial dimension such that the determinant of the diffusion tensor is constant in four dimensions. We propose a generalized minimal principle for rotor filaments, stating that the scroll wave filament strives to minimize its surface area in the higher-dimensional space. As a consequence, stationary scroll wave filaments in the original 3D medium are geodesic curves with respect to the metric tensor G=det(D)D(-1). The theory is confirmed by numerical simulations for positive and negative filament tension and a model with a non-stationary spiral core. We conclude that filaments in cardiac tissue with positive tension preferentially reside or anchor in regions where cardiac cells are less interconnected, such as portions of the cardiac wall with a large number of cleavage planes.

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

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

A brief history of tissue models for cardiac electrophysiology

The last four decades have produced a number of significant advances in the developments of computer models to simulate and investigate the electrical activity of cardiac tissue. The tissue descriptions that underlie these simulations have been built from a combination of clever insight and careful comparison with measured data at multiple scales. Tissue models have not only led to greater insights into the mechanisms of life-threatening arrhythmias but have been used to engineer new therapies to treat the consequences of cardiac disease. This paper is a look back at the early years in the cardiac modeling and the challenges facing the field as models move toward the clinic.

Simple model for identifying critical regions in atrial fibrillation

Atrial fibrillation (AF) is the most common abnormal heart rhythm and the single biggest cause of stroke. Ablation, destroying regions of the atria, is applied largely empirically and can be curative but with a disappointing clinical success rate. We design a simple model of activation wave front propagation on an anisotropic structure mimicking the branching network of heart muscle cells. This integration of phenomenological dynamics and pertinent structure shows how AF emerges spontaneously when the transverse cell-to-cell coupling decreases, as occurs with age, beyond a threshold value. We identify critical regions responsible for the initiation and maintenance of AF, the ablation of which terminates AF. The simplicity of the model allows us to calculate analytically the risk of arrhythmia and express the threshold value of transversal cell-to-cell coupling as a function of the model parameters. This threshold value decreases with increasing refractory period by reducing the number of critical regions which can initiate and sustain microreentrant circuits. These biologically testable predictions might inform ablation therapies and arrhythmic risk assessment.

The virtual heart as a platform for screening drug cardiotoxicity

To predict the safety of a drug at an early stage in its development is a major challenge as there is a lack of in vitro heart models that correlate data from preclinical toxicity screening assays with clinical results. A biophysically detailed computer model of the heart, the virtual heart, provides a powerful tool for simulating drug–ion channel interactions and cardiac functions during normal and disease conditions and, therefore, provides a powerful platform for drug cardiotoxicity screening. In this article, we first review recent progress in the development of theory on drug–ion channel interactions and mathematical modelling. Then we propose a family of biomarkers that can quantitatively characterize the actions of a drug on the electrical activity of the heart at multi‐physical scales including cellular and tissue levels. We also conducted some simulations to demonstrate the application of the virtual heart to assess the pro‐arrhythmic effects of cisapride and amiodarone. Using the model we investigated the mechanisms responsible for the differences between the two drugs on pro‐arrhythmogenesis, even though both prolong the QT interval of ECGs. Several challenges for further development of a virtual heart as a platform for screening drug cardiotoxicity are discussed.

Linked Articles

This article is part of a themed section on Chinese Innovation in Cardiovascular Drug Discovery. To view the other articles in this section visit http://dx.doi.org/10.1111/bph.2015.172.issue-23

Fractional diffusion models of cardiac electrical propagation: role of structural heterogeneity in dispersion of repolarization

Impulse propagation in biological tissues is known to be modulated by structural heterogeneity. In cardiac muscle, improved understanding on how this heterogeneity influences electrical spread is key to advancing our interpretation of dispersion of repolarization. We propose fractional diffusion models as a novel mathematical description of structurally heterogeneous excitable media, as a means of representing the modulation of the total electric field by the secondary electrical sources associated with tissue inhomogeneities. Our results, analysed against in vivo human recordings and experimental data of different animal species, indicate that structural heterogeneity underlies relevant characteristics of cardiac electrical propagation at tissue level. These include conduction effects on action potential (AP) morphology, the shortening of AP duration along the activation pathway and the progressive modulation by premature beats of spatial patterns of dispersion of repolarization. The proposed approach may also have important implications in other research fields involving excitable complex media.

Computer-based prediction of the drug proarrhythmic effect: problems, issues, known and suspected challenges

It is likely that computer modelling and simulations will become an element of comprehensive cardiac safety testing. Their role would be primarily the integration and the interpretation of previously gathered data. There are still unanswered questions and issues which we list and describe below. They include sources of data used for the development of the models as well as data utilized as input information, which can come from the in vitro studies and the quantitative structure-activity relationship models. The pharmacokinetics of the drugs in question play a crucial role as their active concentration should be considered, yet the question remains where is the right place to assess it. The pharmacodynamic angle includes complications coming from multiple drugs (i.e. active metabolites) acting in parallel as well as the type of interaction with (potentially) multiple affected channels. Once established, the model and the methodology of its use should be further validated, optimistically against individual data reported at the clinical level as the physiological, anatomical, and genetic parameters play a crucial role in the drug-triggered arrhythmia induction. All the abovementioned issues should be at least considered and-hopefully-resolved, to properly utilize the mathematical models for a cardiac safety assessment.

Computational cardiac electrophysiology: implementing mathematical models of cardiomyocytes to simulate action potentials of the heart

Mathematical models are now an important tool for studying cardiac electrophysiology and arrhythmias. Utilizing these models to quantify behavior and make predictions requires solving the models computationally using numerical schemes. We discuss different solution methods and other computational considerations for simulating cardiac action potentials in single cells and multicellular preparations.

Systems approaches in integrative cardiac biology: illustrations from cardiac heterocellular signalling studies

Understanding the complexity of cardiac physiology requires system-level studies of multiple cardiac cell types. Frequently, however, the end result of published research lacks the detail of the collaborative and integrative experimental design process, and the underlying conceptual framework. We review the recent progress in systems modelling and omics analysis of the heterocellular heart environment through complementary forward and inverse approaches, illustrating these conceptual and experimental frameworks with case studies from our own research program. The forward approach begins by collecting curated information from the niche cardiac biology literature, and connecting the dots to form mechanistic network models that generate testable system-level predictions. The inverse approach starts from the vast pool of public omics data in recent cardiac biological research, and applies bioinformatics analysis to produce novel candidates for further investigation. We also discuss the possibility of combining these two approaches into a hybrid framework, together with the benefits and challenges. These interdisciplinary research frameworks illustrate the interplay between computational models, omics analysis, and wet lab experiments, which holds the key to making real progress in improving human cardiac wellbeing.

In Silico Organ Modelling in Predicting Efficacy and Safety of New Medicines

The development of new medicines faces important challenges due to difficulties in the assessment of their efficacy and their safety in the targeted human population. In silico approaches through the use of mathematical modelling and computer simulations are increasingly being used to overcome some of the limitations of current experimental methods used in the development of new medicines. This chapter describes state-of-the-art in silico approaches for the evaluation of the safety and efficacy of medicines targeting important causes of mortality such as cardiovascular disease. Firstly, we describe the in silico multi-scale mathematical models and simulation techniques required to describe drug-induced effects on physiological systems such as the heart from the subcellular to the whole organ level. Then we illustrate the power of in silico approaches used to augment experimental and clinical investigations, by providing the framework to unravel multi-scale mechanisms underlying variability in the response to medicines and to focus on effects in human rather than animal models. We devote the last part of the chapter to discussing the process of validation of in silico models and simulations, which is key in building up their credibility.

Computational generation of the Purkinje network driven by clinical measurements: the case of pathological propagations

To properly describe the electrical activity of the left ventricle, it is necessary to model the Purkinje fibers, responsible for the fast and coordinate ventricular activation, and their interaction with the muscular propagation. The aim of this work is to propose a methodology for the generation of a patient-specific Purkinje network driven by clinical measurements of the activation times related to pathological propagations. In this case, one needs to consider a strongly coupled problem between the network and the muscle, where the feedback from the latter to the former cannot be neglected as in a normal propagation. We apply the proposed strategy to data acquired on three subjects, one of them suffering from muscular conduction problems owing to a scar and the other two with a muscular pre-excitation syndrome (Wolff-Parkinson-White). To assess the accuracy of the proposed method, we compare the results obtained by using the patient-specific Purkinje network generated by our strategy with the ones obtained by using a non-patient-specific network. The results show that the mean absolute errors in the activation time is reduced for all the cases, highlighting the importance of including a patient-specific Purkinje network in computational models.

Using waveform information in nonlinear data assimilation

Information in measurements of a nonlinear dynamical system can be transferred to a quantitative model of the observed system to establish its fixed parameters and unobserved state variables. After this learning period is complete, one may predict the model response to new forces and, when successful, these predictions will match additional observations. This adjustment process encounters problems when the model is nonlinear and chaotic because dynamical instability impedes the transfer of information from the data to the model when the number of measurements at each observation time is insufficient. We discuss the use of information in the waveform of the data, realized through a time delayed collection of measurements, to provide additional stability and accuracy to this search procedure. Several examples are explored, including a few familiar nonlinear dynamical systems and small networks of Colpitts oscillators.

Fibroblast proliferation alters cardiac excitation conduction and contraction: a computational study

In this study, the effects of cardiac fibroblast proliferation on cardiac electric excitation conduction and mechanical contraction were investigated using a proposed integrated myocardial-fibroblastic electromechanical model. At the cellular level, models of the human ventricular myocyte and fibroblast were modified to incorporate a model of cardiac mechanical contraction and cooperativity mechanisms. Cellular electromechanical coupling was realized with a calcium buffer. At the tissue level, electrical excitation conduction was coupled to an elastic mechanics model in which the finite difference method (FDM) was used to solve electrical excitation equations, and the finite element method (FEM) was used to solve mechanics equations. The electromechanical properties of the proposed integrated model were investigated in one or two dimensions under normal and ischemic pathological conditions. Fibroblast proliferation slowed wave propagation, induced a conduction block, decreased strains in the fibroblast proliferous tissue, and increased dispersions in depolarization, repolarization, and action potential duration (APD). It also distorted the wave-front, leading to the initiation and maintenance of re-entry, and resulted in a sustained contraction in the proliferous areas. This study demonstrated the important role that fibroblast proliferation plays in modulating cardiac electromechanical behaviour and which should be considered in planning future heart-modeling studies.

Evolution and pharmacological modulation of the arrhythmogenic wave dynamics in canine pulmonary vein model

Aims

Atrial fibrillation (AF), the commonest cardiac arrhythmia, has been strongly linked with arrhythmogenic sources near the pulmonary veins (PVs), but underlying mechanisms are not fully understood. We aim to study the generation and sustenance of wave sources in a model of the PV tissue.

Methods and results

A previously developed biophysically detailed three-dimensional canine atrial model is applied. Effects of AF-induced electrical remodelling are introduced based on published experimental data, as changes of ion channel currents (I_CaL, I_K1, I_to, and I_Kur), the action potential (AP) and cell-to-cell coupling levels. Pharmacological effects are introduced by blocking specific ion channel currents. A combination of electrical heterogeneity (AP tissue gradients of 5–12 ms) and anisotropy (conduction velocities of 0.75–1.25 and 0.21–0.31 m/s along and transverse to atrial fibres) can results in the generation of wave breaks in the PV region. However, a long wavelength (171 mm) prevents the wave breaks from developing into re-entry. Electrical remodelling leads to decreases in the AP duration, conduction velocity and wavelength (to 49 mm), such that re-entry becomes sustained. Pharmacological effects on the tissue heterogeneity and vulnerability (to wave breaks and re-entry) are quantified to show that drugs that increase the wavelength and stop re-entry (I_K1 and I_Kur blockers) can also increase the heterogeneity (AP gradients of 26–27 ms) and the likelihood of wave breaks.

Conclusion

Biophysical modelling reveals large conduction block areas near the PVs, which are due to discontinuous fibre arrangement enhanced by electrical heterogeneity. Vulnerability to re-entry in such areas can be modulated by pharmacological interventions.

Turbulent electrical activity at sharp-edged inexcitable obstacles in a model for human cardiac tissue

Wave propagation around various geometric expansions, structures, and obstacles in cardiac tissue may result in the formation of unidirectional block of wave propagation and the onset of reentrant arrhythmias in the heart. Therefore, we investigated the conditions under which reentrant spiral waves can be generated by high-frequency stimulation at sharp-edged obstacles in the ten Tusscher-Noble-Noble-Panfilov (TNNP) ionic model for human cardiac tissue. We show that, in a large range of parameters that account for the conductance of major inward and outward ionic currents of the model [fast inward Na+ current ( INa), L—type slow inward Ca2+ current ( ICaL), slow delayed-rectifier current ( IKs), rapid delayed-rectifier current ( IKr), inward rectifier K+ current ( IK1)], the critical period necessary for spiral formation is close to the period of a spiral wave rotating in the same tissue. We also show that there is a minimal size of the obstacle for which formation of spirals is possible; this size is ∼2.5 cm and decreases with a decrease in the excitability of cardiac tissue. We show that other factors, such as the obstacle thickness and direction of wave propagation in relation to the obstacle, are of secondary importance and affect the conditions for spiral wave initiation only slightly. We also perform studies for obstacle shapes derived from experimental measurements of infarction scars and show that the formation of spiral waves there is facilitated by tissue remodeling around it. Overall, we demonstrate that the formation of reentrant sources around inexcitable obstacles is a potential mechanism for the onset of cardiac arrhythmias in the presence of a fast heart rate.

Virtual ablation for atrial fibrillation in personalized in-silico three-dimensional left atrial modeling: comparison with clinical catheter ablation

BACKGROUND

Although catheter ablation is an effective rhythm control strategy for atrial fibrillation (AF), empirically-based ablation has a substantial recurrence rate. The purposes of this study were to develop a computational platform for patient-specific virtual AF ablation and to compare the anti-fibrillatory effects of 5 different virtual ablation protocols with empirically chosen clinical ablations.

METHODS

We included 20 patients with AF (65% male, 60.1 ± 10.5 years old, 80% persistent AF [PeAF]) who had undergone empirically-based catheter ablation: circumferential pulmonary vein isolation (CPVI) for paroxysmal AF (PAF) and additional posterior box lesion (L1) and anterior line (L2) for PeAF. Using patient-specific three-dimensional left atrial (LA) geometry, we generated a finite element model and tested the AF termination rate after 5 different virtual ablations: CPVI alone, CPVI + L1, CPVI + L1,2, CPVI with complex fractionated atrial electrogram (CFAE) ablation, and CFAE ablation alone.

RESULTS

1. Virtual CPVI + L1,2 ablation showed the highest AF termination rate in overall patients (55%) and PeAF patients (n = 16, 62.5%). 2. The virtual AF maintenance duration was shortest in the case of virtual CPVI + L1,2 ablation in overall patients (2.19 ± 1.28 vs. 2.91 ± 1.04 s, p = 0.009) and in patients with PeAF (2.05 ± 1.23 vs. 2.93 ± 10.2 s, p = 0.004) compared with other protocols.

CONCLUSION

Virtual AF ablation using personalized in-silico model of LA is feasible. Virtual ablation with CPVI + L1,2 shows the highest antifibrillatory effect, concordant with the empirical ablation protocol in patients with PeAF.

Caveolae in ventricular myocytes are required for stretch-dependent conduction slowing

Mechanical stretch of cardiac muscle modulates action potential propagation velocity, causing potentially arrhythmogenic conduction slowing. The mechanisms by which stretch alters cardiac conduction remain unknown, but previous studies suggest that stretch can affect the conformation of caveolae in myocytes and other cell types. We tested the hypothesis that slowing of action potential conduction due to cardiac myocyte stretch is dependent on caveolae. Cardiac action potential propagation velocities, measured by optical mapping in isolated mouse hearts and in micropatterned mouse cardiomyocyte cultures, decreased reversibly with volume loading or stretch, respectively (by 19±5% and 26±4%). Stretch-dependent conduction slowing was not altered by stretch-activated channel blockade with gadolinium or by GsMTx-4 peptide, but was inhibited when caveolae were disrupted via genetic deletion of caveolin-3 (Cav3 KO) or membrane cholesterol depletion by methyl-β-cyclodextrin. In wild-type mouse hearts, stretch coincided with recruitment of caveolae to the sarcolemma, as observed by electron microscopy. In myocytes from wild-type but not Cav3 KO mice, stretch significantly increased cell membrane capacitance (by 98±64%), electrical time constant (by 285±149%), and lipid recruitment to the bilayer (by 84±39%). Recruitment of caveolae to the sarcolemma during physiologic cardiomyocyte stretch slows ventricular action potential propagation by increasing cell membrane capacitance.

Electrophysiological and structural remodeling in heart failure modulate arrhythmogenesis. 1D simulation study

Background

Heart failure is a final common pathway or descriptor for various cardiac pathologies. It is associated with sudden cardiac death, which is frequently caused by ventricular arrhythmias. Electrophysiological remodeling, intercellular uncoupling, fibrosis and autonomic imbalance have been identified as major arrhythmogenic factors in heart failure etiology and progression.

Objective

In this study we investigate in silico the role of electrophysiological and structural heart failure remodeling on the modulation of key elements of the arrhythmogenic substrate, i.e., electrophysiological gradients and abnormal impulse propagation.

Methods

Two different mathematical models of the human ventricular action potential were used to formulate models of the failing ventricular myocyte. This provided the basis for simulations of the electrical activity within a transmural ventricular strand. Our main goal was to elucidate the roles of electrophysiological and structural remodeling in setting the stage for malignant life-threatening arrhythmias.

Results

Simulation results illustrate how the presence of M cells and heterogeneous electrophysiological remodeling in the human failing ventricle modulate the dispersion of action potential duration and repolarization time. Specifically, selective heterogeneous remodeling of expression levels for the Na⁺/Ca²⁺ exchanger and SERCA pump decrease these heterogeneities. In contrast, fibroblast proliferation and cellular uncoupling both strongly increase repolarization heterogeneities. Conduction velocity and the safety factor for conduction are also reduced by the progressive structural remodeling during heart failure.

Conclusion

An extensive literature now establishes that in human ventricle, as heart failure progresses, gradients for repolarization are changed significantly by protein specific electrophysiological remodeling (either homogeneous or heterogeneous). Our simulations illustrate and provide new insights into this. Furthermore, enhanced fibrosis in failing hearts, as well as reduced intercellular coupling, combine to increase electrophysiological gradients and reduce electrical propagation. In combination these changes set the stage for arrhythmias.

Continuous-time control of alternans in long Purkinje fibers

Alternans-an arrhythmic response of cardiac tissue to periodic pacing-often serves as a precursor to a more dangerous, and potentially lethal, state of fibrillation. Suppression of alternans using feedback control may be a plausible method to prevent fibrillation. Several approaches based on impulsive control have been proposed previously, where feedback is applied for a brief instance of time during each pacing interval. This paper presents a continuous-time approach, where feedback current is applied at all times, which is capable of suppressing alternans in fibers of significantly greater length (up to at least 4 cm), compared with impulsive control (less than 1 cm), and for a wide range of pacing cycle lengths.

Images as drivers of progress in cardiac computational modelling

Computational models have become a fundamental tool in cardiac research. Models are evolving to cover multiple scales and physical mechanisms. They are moving towards mechanistic descriptions of personalised structure and function, including effects of natural variability. These developments are underpinned to a large extent by advances in imaging technologies. This article reviews how novel imaging technologies, or the innovative use and extension of established ones, integrate with computational models and drive novel insights into cardiac biophysics. In terms of structural characterization, we discuss how imaging is allowing a wide range of scales to be considered, from cellular levels to whole organs. We analyse how the evolution from structural to functional imaging is opening new avenues for computational models, and in this respect we review methods for measurement of electrical activity, mechanics and flow. Finally, we consider ways in which combined imaging and modelling research is likely to continue advancing cardiac research, and identify some of the main challenges that remain to be solved.

Computational model of erratic arrhythmias in a cardiac cell network: the role of gap junctions

Cardiac morbidity and mortality increases with the population age. To investigate the underlying pathological mechanisms, and suggest new ways to reduce clinical risks, computational approaches complementing experimental and clinical investigations are becoming more and more important. Here we explore the possible processes leading to the occasional onset and termination of the (usually) non-fatal arrhythmias widely observed in the heart. Using a computational model of a two-dimensional network of cardiac cells, we tested the hypothesis that an ischemia alters the properties of the gap junctions inside the ischemic area. In particular, in agreement with experimental findings, we assumed that an ischemic episode can alter the gap junctions of the affected cells by reducing their average conductance. We extended these changes to include random fluctuations with time, and modifications in the gap junction rectifying conductive properties of cells along the edges of the ischemic area. The results demonstrate how these alterations can qualitatively give an account of all the main types of non-fatal arrhythmia observed experimentally, and suggest how premature beats can be eliminated in three different ways: a) with a relatively small surgical procedure, b) with a pharmacological reduction of the rectifying conductive properties of the gap-junctions, and c) by pharmacologically decreasing the gap junction conductance. In conclusion, our model strongly supports the hypothesis that non-fatal arrhythmias can develop from post-ischemic alteration of the electrical connectivity in a relatively small area of the cardiac cell network, and suggests experimentally testable predictions on their possible treatments.

Comparison of the relation between left ventricular anatomy and QRS duration in patients with cardiomyopathy with versus without left bundle branch block

QRS duration (QRSd) is used to diagnose left bundle branch block (LBBB) and is important to determine cardiac resynchronization therapy eligibility. The same QRSd thresholds established decades ago are used for all patients. However, significant interpatient variability of normal QRSd exists, and individualized QRSd thresholds might improve diagnosis and intervention strategies. Previous work reported left ventricular (LV) mass and papillary muscle location predicted QRSd in healthy subjects, but the relation in diseased ventricles is unknown. The aim of the present study was to determine the association between LV anatomy and QRSd in patients with cardiomyopathy. Patients referred for primary prevention implantable defibrillators (n = 166) received cardiac magnetic resonance imaging, and those with normal conduction (without bundle branch or fascicular block) and LBBB were studied. The LV mass, length, internal diameter, LV end-diastolic volume, septal and lateral wall thicknesses, and papillary muscle location were measured. In patients with normal conduction, LV length (r = 0.35, p <0.001), mass (r = 0.32, p <0.001), diameter (r = 0.20, p = 0.03), and septal wall thickness (r = 0.20, p = 0.03) had positive correlations with QRSd. In patients with LBBB, LV length (r = 0.32, p = 0.03), mass (r = 0.39, p = 0.01), diameter (r = 0.34, p = 0.02), and LV end-diastolic volume (r = 0.32, p = 0.04) had positive correlations with QRSd. Contrary to previous studies in healthy subjects, papillary muscle angle (location) was not associated with QRSd in cardiomyopathy patients with normal conduction or LBBB. In conclusion, increasing LV anatomical measurements were associated with increasing QRSd in patients with cardiomyopathy. Future work should investigate the use of LV anatomical measurements in developing individualized QRSd thresholds for diagnosing conduction abnormalities such as LBBB and identifying candidates for cardiac resynchronization therapy.

Electrical wave propagation in an anisotropic model of the left ventricle based on analytical description of cardiac architecture

We develop a numerical approach based on our recent analytical model of fiber structure in the left ventricle of the human heart. A special curvilinear coordinate system is proposed to analytically include realistic ventricular shape and myofiber directions. With this anatomical model, electrophysiological simulations can be performed on a rectangular coordinate grid. We apply our method to study the effect of fiber rotation and electrical anisotropy of cardiac tissue (i.e., the ratio of the conductivity coefficients along and across the myocardial fibers) on wave propagation using the ten Tusscher–Panfilov (2006) ionic model for human ventricular cells. We show that fiber rotation increases the speed of cardiac activation and attenuates the effects of anisotropy. Our results show that the fiber rotation in the heart is an important factor underlying cardiac excitation. We also study scroll wave dynamics in our model and show the drift of a scroll wave filament whose velocity depends non-monotonically on the fiber rotation angle; the period of scroll wave rotation decreases with an increase of the fiber rotation angle; an increase in anisotropy may cause the breakup of a scroll wave, similar to the mother rotor mechanism of ventricular fibrillation.

Data-driven estimation of cardiac electrical diffusivity from 12-lead ECG signals

Diagnosis and treatment of dilated cardiomyopathy (DCM) is challenging due to a large variety of causes and disease stages. Computational models of cardiac electrophysiology (EP) can be used to improve the assessment and prognosis of DCM, plan therapies and predict their outcome, but require personalization. In this work, we present a data-driven approach to estimate the electrical diffusivity parameter of an EP model from standard 12-lead electrocardiograms (ECG). An efficient forward model based on a mono-domain, phenomenological Lattice-Boltzmann model of cardiac EP, and a boundary element-based mapping of potentials to the body surface is employed. The electrical diffusivity of myocardium, left ventricle and right ventricle endocardium is then estimated using polynomial regression which takes as input the QRS duration and electrical axis. After validating the forward model, we computed 9500 EP simulations on 19 different DCM patients in just under three seconds each to learn the regression model. Using this database, we quantify the intrinsic uncertainty of electrical diffusion for given ECG features and show in a leave-one-patient-out cross-validation that the regression method is able to predict myocardium diffusion within the uncertainty range. Finally, our approach is tested on the 19 cases using their clinical ECG. 84% of them could be personalized using our method, yielding mean prediction errors of 18.7ms for the QRS duration and 6.5° for the electrical axis, both values being within clinical acceptability. By providing an estimate of diffusion parameters from readily available clinical data, our data-driven approach could therefore constitute a first calibration step toward a more complete personalization of cardiac EP.

Statistical personalization of ventricular fiber orientation using shape predictors

This paper presents a predictive framework for the statistical personalization of ventricular fibers. To this end, the relationship between subject-specific geometry of the left (LV) and right ventricles (RV) and fiber orientation is learned statistically from a training sample of ex vivo diffusion tensor imaging datasets. More specifically, the axes in the shape space which correlate most with the myocardial fiber orientations are extracted and used for prediction in new subjects. With this approach and unlike existing fiber models, inter-subject variability is taken into account to generate latent shape predictors that are statistically optimal to estimate fiber orientation at each individual myocardial location. The proposed predictive model was applied to the task of personalizing fibers in 10 canine subjects. The results indicate that the ventricular shapes are good predictors of fiber orientation, with an improvement of 11.4% in accuracy over the average fiber model.

Mechanistic insights into hypothermic ventricular fibrillation: the role of temperature and tissue size

AIMS

Hypothermia is well known to be pro-arrhythmic, yet it has beneficial effects as a resuscitation therapy and valuable during intracardiac surgeries. Therefore, we aim to study the mechanisms that induce fibrillation during hypothermia. A better understanding of the complex spatiotemporal dynamics of heart tissue as a function of temperature will be useful in managing the benefits and risks of hypothermia.

METHODS AND RESULTS

We perform two-dimensional numerical simulations by using a minimal model of cardiac action potential propagation fine-tuned on experimental measurements. The model includes thermal factors acting on the ionic currents and the gating variables to correctly reproduce experimentally recorded restitution curves at different temperatures. Simulations are implemented using WebGL, which allows long simulations to be performed as they run close to real time. We describe (i) why fibrillation is easier to induce at low temperatures, (ii) that there is a minimum size required for fibrillation that depends on temperature, (iii) why the frequency of fibrillation decreases with decreasing temperature, and (iv) that regional cooling may be an anti-arrhythmic therapy for small tissue sizes however it may be pro-arrhythmic for large tissue sizes.

CONCLUSION

Using a mathematical cardiac cell model, we are able to reproduce experimental observations, quantitative experimental results, and discuss possible mechanisms and implications of electrophysiological changes during hypothermia.

Heterogeneous and anisotropic integrative model of pulmonary veins: computational study of arrhythmogenic substrate for atrial fibrillation

Mechanisms underlying the genesis of re-entrant substrate for the most common cardiac arrhythmia, atrial fibrillation (AF), are not well understood. In this study, we develop a multi-scale three-dimensional computational model that integrates cellular electrophysiology of the left atrium (LA) and pulmonary veins (PVs) with the respective tissue geometry and fibre orientation. The latter is reconstructed in unique detail from high-resolution (approx. 70 μm) contrast micro-computed tomography data. The model is used to explore the mechanisms of re-entry initiation and sustenance in the PV region, regarded as the primary source of high-frequency electrical activity in AF. Simulations of the three-dimensional model demonstrate that an initial break-down of normal electrical excitation wave-fronts can be caused by the electrical heterogeneity between the PVs and LA. High tissue anisotropy is then responsible for the slow conduction and generation of a re-entrant circuit near the PVs. Evidence of such circuits has been seen clinically in AF patients. Our computational study suggests that primarily the combination of electrical heterogeneity and conduction anisotropy between the PVs and LA tissues leads to the generation of a high-frequency (approx. 10 Hz) re-entrant source near the PV sleeves, thus providing new insights into the arrhythmogenic mechanisms of excitation waves underlying AF.

Fibroblast-myocyte electrotonic coupling: does it occur in native cardiac tissue?

Heterocellular electrotonic coupling between cardiac myocytes and non-excitable connective tissue cells has been a long-established and well-researched fact in vitro. Whether or not such coupling exists in vivo has been a matter of considerable debate. This paper reviews the development of experimental insight and conceptual views on this topic, describes evidence in favour of and against the presence of such coupling in native myocardium, and identifies directions for further study needed to resolve the riddle, perhaps less so in terms of principal presence which has been demonstrated, but undoubtedly in terms of extent, regulation, patho-physiological context, and actual relevance of cardiac myocyte–non-myocyte coupling in vivo. This article is part of a Special Issue entitled "Myocyte-Fibroblast Signalling in Myocardium."

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Electrical coupling of cardiomyocytes and fibroblasts is well-established in vitro

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Whether such hetero-cellular coupling exists in vivo has been a matter of debate

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We review the development of experimental and conceptual insight into the topic

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Conclusion 1: hetero-cellular coupling in heart tissue has been shown in principle

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Conclusion 2: extent, regulation, context, and relevance remain to be established

A quantitative comparison of the behavior of human ventricular cardiac electrophysiology models in tissue

Numerical integration of mathematical models of heart cell electrophysiology provides an important computational tool for studying cardiac arrhythmias, but the abundance of available models complicates selecting an appropriate model. We study the behavior of two recently published models of human ventricular action potentials, the Grandi-Pasqualini-Bers (GPB) and the O'Hara-Virág-Varró-Rudy (OVVR) models, and compare the results with four previously published models and with available experimental and clinical data. We find the shapes and durations of action potentials and calcium transients differ between the GPB and OVVR models, as do the magnitudes and rate-dependent properties of transmembrane currents and the calcium transient. Differences also occur in the steady-state and S1–S2 action potential duration and conduction velocity restitution curves, including a maximum conduction velocity for the OVVR model roughly half that of the GPB model and well below clinical values. Between single cells and tissue, both models exhibit differences in properties, including maximum upstroke velocity, action potential amplitude, and minimum diastolic interval. Compared to experimental data, action potential durations for the GPB and OVVR models agree fairly well (although OVVR epicardial action potentials are shorter), but maximum slopes of steady-state restitution curves are smaller. Although studies show alternans in normal hearts, it occurs only in the OVVR model, and only for a narrow range of cycle lengths. We find initiated spiral waves do not progress to sustained breakup for either model. The dominant spiral wave period of the GPB model falls within clinically relevant values for ventricular tachycardia (VT), but for the OVVR model, the dominant period is longer than periods associated with VT. Our results should facilitate choosing a model to match properties of interest in human cardiac tissue and to replicate arrhythmia behavior more closely. Furthermore, by indicating areas where existing models disagree, our findings suggest avenues for further experimental work.

Toward GPGPU accelerated human electromechanical cardiac simulations

In this paper, we look at the acceleration of weakly coupled electromechanics using the graphics processing unit (GPU). Specifically, we port to the GPU a number of components of CHeart--a CPU-based finite element code developed for simulating multi-physics problems. On the basis of a criterion of computational cost, we implemented on the GPU the ODE and PDE solution steps for the electrophysiology problem and the Jacobian and residual evaluation for the mechanics problem. Performance of the GPU implementation is then compared with single core CPU (SC) execution as well as multi-core CPU (MC) computations with equivalent theoretical performance. Results show that for a human scale left ventricle mesh, GPU acceleration of the electrophysiology problem provided speedups of 164 × compared with SC and 5.5 times compared with MC for the solution of the ODE model. Speedup of up to 72 × compared with SC and 2.6 × compared with MC was also observed for the PDE solve. Using the same human geometry, the GPU implementation of mechanics residual/Jacobian computation provided speedups of up to 44 × compared with SC and 2.0 × compared with MC.

Lp-norm regularization in volumetric imaging of cardiac current sources

Advances in computer vision have substantially improved our ability to analyze the structure and mechanics of the heart. In comparison, our ability to observe and analyze cardiac electrical activities is much limited. The progress to computationally reconstruct cardiac current sources from noninvasive voltage data sensed on the body surface has been hindered by the ill-posedness and the lack of a unique solution of the reconstruction problem. Common L2- and L1-norm regularizations tend to produce a solution that is either too diffused or too scattered to reflect the complex spatial structure of current source distribution in the heart. In this work, we propose a general regularization with Lp-norm (1 < p < 2) constraint to bridge the gap and balance between an overly smeared and overly focal solution in cardiac source reconstruction. In a set of phantom experiments, we demonstrate the superiority of the proposed Lp-norm method over its L1 and L2 counterparts in imaging cardiac current sources with increasing extents. Through computer-simulated and real-data experiments, we further demonstrate the feasibility of the proposed method in imaging the complex structure of excitation wavefront, as well as current sources distributed along the postinfarction scar border. This ability to preserve the spatial structure of source distribution is important for revealing the potential disruption to the normal heart excitation.

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.

Excitation-contraction coupling between human atrial myocytes with fibroblasts and stretch activated channel current: a simulation study

Myocytes have been regarded as the main objectives in most cardiac modeling studies and attracted a lot of attention. Connective tissue cells, such as fibroblasts (Fbs), also play crucial role in cardiac function. This study proposed an integrated myocyte-I_sac-Fb electromechanical model to investigate the effect of Fbs and stretch activated ion channel current (I_sac) on cardiac electrical excitation conduction and mechanical contraction. At the cellular level, an active Fb model was coupled with a human atrial myocyte electrophysiological model (including I_sac) and a mechanical model. At the tissue level, electrical excitation conduction was coupled with an elastic mechanical model, in which finite difference method (FDM) was used to solve the electrical excitation equations, while finite element method (FEM) was used for the mechanics equations. The simulation results showed that Fbs and I_sac coupling caused diverse effects on action potential morphology during repolarization, depolarized the resting membrane potential of the human atrial myocyte, slowed down wave propagation, and decreased strains in fibrotic tissue. This preliminary simulation study indicates that Fbs and I_sac have important implications for modulating cardiac electromechanical behavior and should be considered in future cardiac modeling studies.

In-silico modeling of atrial repolarization in normal and atrial fibrillation remodeled state

Atrial fibrillation (AF) is the most common cardiac arrhythmia, and the total number of AF patients is constantly increasing. The mechanisms leading to and sustaining AF are not completely understood yet. Heterogeneities in atrial electrophysiology seem to play an important role in this context. Although some heterogeneities have been used in in-silico human atrial modeling studies, they have not been thoroughly investigated. In this study, the original electrophysiological (EP) models of Courtemanche et al., Nygren et al. and Maleckar et al. were adjusted to reproduce action potentials in 13 atrial regions. The parameter sets were validated against experimental action potential duration data and ECG data from patients with AV block. The use of the heterogeneous EP model led to a more synchronized repolarization sequence in a variety of 3D atrial anatomical models. Combination of the heterogeneous EP model with a model of persistent AF-remodeled electrophysiology led to a drastic change in cell electrophysiology. Simulated Ta-waves were significantly shorter under the remodeling. The heterogeneities in cell electrophysiology explain the previously observed Ta-wave effects. The results mark an important step toward the reliable simulation of the atrial repolarization sequence, give a deeper understanding of the mechanism of atrial repolarization and enable further clinical investigations.

Effects of pacing magnitudes and forms on bistability width in a modeled ventricular tissue

Bistability in periodically paced cardiac tissue is relevant to cardiac arrhythmias and its control. In the present paper, one-dimensional tissue of the phase I Luo-Rudy model is numerically investigated. The effects of various parameters of pacing signals on bistability width are studied. The following conclusions are obtained: (i) Pacing can be classified into two types: pulsatile and sinusoidal types. Pulsatile pacing reduces bistability width as its magnitude is increased. Sinusoidal pacing increases the width as its amplitude is increased. (ii) In a pacing period the hyperpolarizing part plays a more important role than the depolarizing part. Variations of the hyperpolarizing ratio in a period evidently change the width of bistability and its variation tendency. (iii) A dynamical mechanism is proposed to qualitatively explain the phenomena, which reveals the reason for the different effects of pulsatile and sinusoidal pacing on bistability. The methods for changing bistability width by external pacing may help control arrhythmias in cardiology.

Computer models to study uterine activation at labour

Improving our understanding of the initiation of labour is a major aim of modern obstetric research, in order to better diagnose and treat pregnant women in which the process occurs abnormally. In particular, increased knowledge will help us identify the mechanisms responsible for preterm labour, the single biggest cause of neonatal morbidity and mortality. Attempts to improve our understanding of the initiation of labour have been restricted by the inaccessibility of gestational tissues to study during pregnancy and at labour, and by the lack of fully informative animal models. However, computer modelling provides an exciting new approach to overcome these restrictions and offers new insights into uterine activation during term and preterm labour. Such models could be used to test hypotheses about drugs to treat or prevent preterm labour. With further development, an effective computer model could be used by healthcare practitioners to develop personalized medicine for patients on a pregnancy-by-pregnancy basis. Very promising work is already underway to build computer models of the physiology of uterine activation and contraction. These models aim to predict changes and patterns in uterine electrical excitation during term labour. There have been far fewer attempts to build computer models of the molecular pathways driving uterine activation and there is certainly scope for further work in this area. The integration of computer models of the physiological and molecular mechanisms that initiate labour will be particularly useful.

Pro-arrhythmogenic effects of atrial fibrillation-induced electrical remodelling: insights from the three-dimensional virtual human atria

Chronic atrial fibrillation (AF) is associated with structural and electrical remodelling in the atria, which are associated with a high recurrence of AF. Through biophysically detailed computer modelling, this study investigated mechanisms by which AF-induced electrical remodelling promotes and perpetuates AF. A family of Courtemanche–Ramirez–Nattel variant models of human atrial cell action potentials (APs), taking into account of intrinsic atrial electrophysiological properties, was modified to incorporate various experimental data sets on AF-induced changes of major ionic channel currents (I_CaL, I_Kur, I_to, I_K1, I_Ks, I_NaCa) and on intracellular Ca²⁺ handling. The single cell models for control and AF-remodelled conditions were incorporated into multicellular three-dimensional (3D) atrial tissue models. Effects of the AF-induced electrical remodelling were quantified as the changes of AP profile, AP duration (APD) and its dispersion across the atria, and the vulnerability of atrial tissue to the initiation of re-entry. The dynamic behaviour of re-entrant excitation waves in the 3D models was characterised. In our simulations, AF-induced electrical remodelling abbreviated atrial APD non-uniformly across the atria; this resulted in relatively short APDs co-existing with marked regional differences in the APD at junctions of the crista terminalis/pectinate muscle, pulmonary veins/left atrium. As a result, the measured tissue vulnerability to re-entry initiation at these tissue junctions was increased. The AF-induced electrical remodelling also stabilized and accelerated re-entrant excitation waves, leading to rapid and sustained re-entry. Under the AF-remodelled condition, re-entrant scroll waves in the 3D model degenerated into persistent and erratic wavelets, leading to fibrillation. In conclusion, realistic 3D atrial tissue models indicate that AF-induced electrical remodelling produces regionally heterogeneous and shortened APD; these respectively facilitate initiation and maintenance of re-entrant excitation waves.

Reconstruction of retrospective cardiac activity--numerical study in a single cell and in a linear strand

Although computational modeling of the prospective electrical activity in the cardiac tissue is well established and robust, the retrospective extrapolation of this activity has not been explored to date. Here, we establish an algorithm for the backward-in-time extrapolation of electrical activity from measurements taken in the present. Using minimal human cardiac kinetic models and a modified Newton-Raphson algorithm, we demonstrate the feasibility of past activity reconstruction in a single cell and in a linear strand. In a single cell, reconstruction of state variables' shape, the action potential morphology, and the time of stimulation was successful for up to 1300 ms poststimulation and for data with signal-to-noise ratio levels higher than 40 dB. For linear strands, the action potential morphology was reconstructed for 500 ms poststimulation, and the reconstructed conduction velocity remained unaffected for signal-to-noise ratio levels higher than 50 dB. Moreover, tissue restitution properties due to various pacing rates were successfully reconstructed by the backward-in-time algorithm. These preliminary results demonstrate that past cardiac activity may be reconstructed from measurements in the present. We envision that this methodology could be implemented in future clinical applications, for example to trace the location and timing of ectopic foci during ablation procedures.