Mathematical modeling of electrocardiograms: a numerical study

Comparative analysis of parametric B-spline and Hermite cubic spline based methods for accurate ECG signal modeling

Analyzing Electrocardiogram (ECG) signals is imperative for diagnosing cardiovascular diseases. However, evaluating ECG analysis techniques faces challenges due to noise and artifacts in actual signals. Machine learning for automatic diagnosis encounters data acquisition hurdles due to medical data privacy constraints. Addressing these issues, ECG modeling assumes a crucial role in biomedical and parametric spline-based methods have garnered significant attention for their ability to accurately represent the complex temporal dynamics of ECG signals. This study conducts a comparative analysis of two parametric spline-based methods-B-spline and Hermite cubic spline-for ECG modeling, aiming to identify the most effective approach for accurate and reliable ECG representation. The Hermite cubic spline serves as one of the most effective interpolation methods, while B-spline is an approximation method. The comparative analysis includes both qualitative and quantitative evaluations. Qualitative assessment involves visually inspecting the generated spline-based models, comparing their resemblance to the original ECG signals, and employing power spectrum analysis. Quantitative analysis incorporates metrics such as root mean square error (RMSE), Percentage Root Mean Square Difference (PRD) and cross correlation, offering a more objective measure of the model's performance. Preliminary results indicate promising capabilities for both spline-based methods in representing ECG signals. However, the analysis unveils specific strengths and weaknesses for each method. The B-spline method offers greater flexibility and smoothness, while the cubic spline method demonstrates superior waveform capturing abilities with the preservation of control points, a critical aspect in the medical field. Presented research provides valuable insights for researchers and practitioners in selecting the most appropriate method for their specific ECG modeling requirements. Adjustments to control points and parameterization enable the generation of diverse ECG waveforms, enhancing the versatility of this modeling technique. This approach has the potential to extend its utility to other medical signals, presenting a promising avenue for advancing biomedical research.

A reduced complexity ECG imaging model for regularized inversion optimization

The resolution of the inverse problem of electrocardiography represents a major interest in the diagnosis and catheter-based therapy of cardiac arrhythmia. In this context, the ability to simulate several cardiac electrical behaviors was crucial for evaluating and comparing the performance of inversion methods. For this application, existing models are either too complex or do not produce realistic cardiac patterns. In this work, a low-resolution heart-torso model generating realistic whole heart cardiac mappings and electrocardiograms in healthy and pathological cases is designed. This model was built upon a simplified heart-torso geometry and implements the monodomain formalism by using the finite element method. In addition, a model reduction step through a sensitivity analysis was proposed where parameters were identified using an evolutionary optimization approach. Finally, the study illustrates the usefulness of the proposed model by comparing the performance of different variants of Tikhonov-based inversion methods for the determination of the regularization parameter in healthy, ischemic and ventricular tachycardia scenarios. First, results of the sensitivity analysis show that among 58 parameters only 25 are influent. Note also that the level of influence of the parameters depends on the heart region. Besides, the synthesized electrocardiograms globally present the same characteristic shape compared to the reference once with a correlation value that reaches 88%. Regarding inverse problem, results highlight that only Robust Generalized Cross Validation and Discrepancy Principle provide best performance, with a quasi-perfect success rate for both, and a respective relative error, between the generated electrocardiograms to the reference one, of 0.75 and 0.62.

Electric Field-Based Spatial Analysis of Noncontact Unipolar Electrograms to Map Regional Activation-Repolarization Intervals

BACKGROUND

Spatial heterogeneity in repolarization plays an important role in generating and sustaining cardiac arrhythmias. Reliable determination of repolarization times remains challenging.

OBJECTIVES

The goal of this study was to improve processing of densely sampled noncontact unipolar electrograms to yield reliable high-resolution activation and repolarization maps.

METHODS

Endocardial noncontact unipolar electrograms were both simulated and recorded in pig left ventricle. Electrical activity on the endocardial surface was processed in terms of a pseudo-electric field. Activation and repolarization times were calculated by using an amplitude-weighted average on QRS and T waves (ie, the E-field method). This was compared vs the conventional Wyatt method on unipolar electrograms. Timing maps were validated against timing on endocardial action potentials in a simulation study. In vivo, activation and repolarization times determined by using this alternative E-field method were validated against simultaneously recorded endocardial monophasic action potentials (MAPs).

RESULTS

Simulation showed that the E-field method provides viable measurements of local endocardial action potential activation and repolarization times. In vivo, correlation of E-field activation times with MAP activation times (rE = 0.76; P < 0.001) was similar to those of Wyatt (rWyatt = 0.80, P < 0.001; P[h1:rE > rWyatt] = 0.82); for repolarization times, correlation improved significantly (rE = 0.96, P < 0.001; rWyatt = 0.82, P < 0.001; P[h1:rE > rWyatt] < 0.00001). This resulted in improved correlations of activation-repolarization intervals to endocardial action potential duration on MAP (rE = 0.96, P < 0.001; rWyatt = 0.86, P < 0.001; P[h1:rE > rWyatt] < 0.00001). Spatial beat-to-beat variation of repolarization could only be calculated by using the E-field methodology and correlated well with the MAP beat-to-beat variation of repolarization (rE = 0.76; P = 0.001).

CONCLUSIONS

The E-field method substantially enhances information from endocardial noncontact electrogram data, allowing for dense maps of activation and repolarization times and derived parameters.

GPU accelerated digital twins of the human heart open new routes for cardiovascular research

The recruitment of patients for rare or complex cardiovascular diseases is a bottleneck for clinical trials and digital twins of the human heart have recently been proposed as a viable alternative. In this paper we present an unprecedented cardiovascular computer model which, relying on the latest GPU-acceleration technologies, replicates the full multi-physics dynamics of the human heart within a few hours per heartbeat. This opens the way to extensive simulation campaigns to study the response of synthetic cohorts of patients to cardiovascular disorders, novel prosthetic devices or surgical procedures. As a proof-of-concept we show the results obtained for left bundle branch block disorder and the subsequent cardiac resynchronization obtained by pacemaker implantation. The in-silico results closely match those obtained in clinical practice, confirming the reliability of the method. This innovative approach makes possible a systematic use of digital twins in cardiovascular research, thus reducing the need of real patients with their economical and ethical implications. This study is a major step towards in-silico clinical trials in the era of digital medicine.

Nonlinear model-based cardiac arrhythmia diagnosis using the optimization-based inverse problem solution

This study investigates a nonlinear model-based feature extraction approach for the accurate classification of four types of heartbeats. The features are the morphological parameters of ECG signal derived from the nonlinear ECG model using an optimization-based inverse problem solution. In the model-based methods, high feature extraction time is a crucial issue. In order to reduce the feature extraction time, a new structure was employed in the optimization algorithms. Using the proposed structure has considerably increased the speed of feature extraction. In the following, the effectiveness of two types of optimization methods (genetic algorithm and particle swarm optimization) and the McSharry ECG model has been studied and compared in terms of speed and accuracy of diagnosis. In the classification section, the adaptive neuro-fuzzy inference system and fuzzy c-mean clustering methods, along with the principal component analysis data reduction method, have been utilized. The obtained results reveal that using an adaptive neuro-fuzzy inference system with data obtained from particle swarm optimization will have the shortest process time and the best diagnosis, with a mean accuracy of 99% and a mean sensitivity of 99.11%.

Numerical study of POD-Galerkin-DEIM reduced order modeling of cardiac monodomain formulation

The three-dimensional cardiac monodomain model with inhomogeneous and anisotropic conductivity characterizes a complicated system that contains spatial and temporal approximation coefficients along with a nonlinear ionic current term. These complexities make its numerical modeling computationally challenging, and therefore, the formation of an efficient computational approximation is important for studying cardiac propagation. In this paper, a reduced order modeling approach has been developed for the simplified cardiac monodomain model, which yields a significant reduction of the full order dynamics of the cardiac tissue, reducing the required computational resources. Additionally, the discrete empirical interpolation technique has been implemented to accurately estimate the nonlinearity of the ionic current of the cardiac monodomain scheme. The proper orthogonal decomposition technique has been utilized, which transforms a given dataset called 'snapshots' to a new coordinate system. The snapshots are computed first from the original system, and they encapsulate all the information observed over both time and parameter variations. Next, the proper orthogonal decomposition provides a reduced order basis for projecting the original solution onto a low-dimensional orthonormal subspace. Finally, a reduced set of unknowns of the forward problem is obtained for which the solution involves significant computational savings compared to that for the original system of unknowns. The efficiency of the model order reduction technique for finite difference solution of cardiac electrophysiology is examined concerning simulation time, error potential, activation time, maximum temporal derivative, and conduction velocity. Numerical results for the monodomain show that its solution time can be reduced by a significant factor, with only 0.474 mV RMS error between the full order and reduced dimensions solution.

Accelerating the convergence to a limit cycle in 3D cardiac electromechanical simulations through a data-driven 0D emulator

The results of numerical simulations of cardiac electromechanics are typically characterized by a long transient before reaching a periodic solution known as limit cycle. This yields a serious computational overhead, as the only clinically relevant output is associated with such limit cycle. To accelerate the convergence to the limit cycle, we propose a strategy based on a surrogate model, wherein the computationally demanding 3D components are replaced by a 0D emulator, built through an automated data-driven algorithm on the basis of pressure-volume transients of as few as three heartbeats simulated with the 3D model. The 0D emulator, consisting of a time-dependent pressure-volume relationship, can provide the 3D model with an initial guess, such that in just two heartbeats a solution is reached that is as close to the limit cycle as the one obtained after more than 20 heartbeats with the 3D model. The 0D emulator is also recommended in many-query settings (e.g. when performing sensitivity analysis, parameter estimation and uncertainty quantification), that call for the repeated solution of the model for different values of the parameters. Indeed, the construction of the emulator does not have to be repeated when the parameters of the circulation model it is coupled with vary. Finally, should the parameters of the 3D electromechanical model vary as well, we propose a parametric emulator, obtained by interpolation of emulators constructed for given values of the parameters. This paper is accompanied by a Python library implementing the proposed algorithm, open to integration with existing cardiac solvers.

ECG Localization Method Based on Volume Conductor Model and Kalman Filtering

The 12-lead electrocardiogram was invented more than 100 years ago and is still used as an essential tool in the early detection of heart disease. By estimating the time-varying source of the electrical activity from the potential changes, several types of heart disease can be noninvasively identified. However, most previous studies are based on signal processing, and thus an approach that includes physics modeling would be helpful for source localization problems. This study proposes a localization method for cardiac sources by combining an electrical analysis with a volume conductor model of the human body as a forward problem and a sparse reconstruction method as an inverse problem. Our formulation estimates not only the current source location but also the current direction. For a 12-lead electrocardiogram system, a sensitivity analysis of the localization to cardiac volume, tilted angle, and model inhomogeneity was evaluated. Finally, the estimated source location is corrected by Kalman filter, considering the estimated electrocardiogram source as time-sequence data. For a high signal-to-noise ratio (greater than 20 dB), the dominant error sources were the model inhomogeneity, which is mainly attributable to the high conductivity of the blood in the heart. The average localization error of the electric dipole sources in the heart was 12.6 mm, which is comparable to that in previous studies, where a less detailed anatomical structure was considered. A time-series source localization with Kalman filtering indicated that source mislocalization could be compensated, suggesting the effectiveness of the source estimation using the current direction and location simultaneously. For the electrocardiogram R-wave, the mean distance error was reduced to less than 7.3 mm using the proposed method. Considering the physical properties of the human body with Kalman filtering enables highly accurate estimation of the cardiac electric signal source location and direction. This proposal is also applicable to electrode configuration, such as ECG sensing systems.

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.

Computational prediction of drug response in short QT syndrome type 1 based on measurements of compound effect in stem cell-derived cardiomyocytes

Short QT (SQT) syndrome is a genetic cardiac disorder characterized by an abbreviated QT interval of the patient's electrocardiogram. The syndrome is associated with increased risk of arrhythmia and sudden cardiac death and can arise from a number of ion channel mutations. Cardiomyocytes derived from induced pluripotent stem cells generated from SQT patients (SQT hiPSC-CMs) provide promising platforms for testing pharmacological treatments directly in human cardiac cells exhibiting mutations specific for the syndrome. However, a difficulty is posed by the relative immaturity of hiPSC-CMs, with the possibility that drug effects observed in SQT hiPSC-CMs could be very different from the corresponding drug effect in vivo. In this paper, we apply a multistep computational procedure for translating measured drug effects from these cells to human QT response. This process first detects drug effects on individual ion channels based on measurements of SQT hiPSC-CMs and then uses these results to estimate the drug effects on ventricular action potentials and QT intervals of adult SQT patients. We find that the procedure is able to identify IC50 values in line with measured values for the four drugs quinidine, ivabradine, ajmaline and mexiletine. In addition, the predicted effect of quinidine on the adult QT interval is in good agreement with measured effects of quinidine for adult patients. Consequently, the computational procedure appears to be a useful tool for helping predicting adult drug responses from pure in vitro measurements of patient derived cell lines.

Electrocardiogram signal generation using electrical model of cardiac cell: application in cardiac ischemia

One of the most common causes of heart failure is ischaemia. In this disease, the heart muscles die due to the lack or insufficiency of the blood in the cardiac veins. As a result of such a phenomenon, the action potential in that part of the heart would fade. In this article, using the electric model of the cardiac cell and the mechanism of producing an ECG signal in the heart, the process of producing cardiac electrical potential has been modelled. In this regard, the basic constituent signals of the ECG are generated. Afterward, by accumulating these signals, the final ECG is reproduced. In addition, by variation of the presented model parameters, the cardiac ischaemic signal is simulated in a way that the influence of ventricle ischaemia on the ventricular tissues is considered. The results of such a simulation demonstrate a sufficient match between the model output and the reported changes of the cardiac arrhythmia including ischaemic failures. Here, we report the 91% match between the simulated signal and the considered clinical data.

Evaluation of Fifteen Algorithms for the Resolution of the Electrocardiography Imaging Inverse Problem Using ex-vivo and in-silico Data

The electrocardiographic imaging inverse problem is ill-posed. Regularization has to be applied to stabilize the problem and solve for a realistic solution. Here, we assess different regularization methods for solving the inverse problem. In this study, we assess (i) zero order Tikhonov regularization (ZOT) in conjunction with the Method of Fundamental Solutions (MFS), (ii) ZOT regularization using the Finite Element Method (FEM), and (iii) the L1-Norm regularization of the current density on the heart surface combined with FEM. Moreover, we apply different approaches for computing the optimal regularization parameter, all based on the Generalized Singular Value Decomposition (GSVD). These methods include Generalized Cross Validation (GCV), Robust Generalized Cross Validation (RGCV), ADPC, U-Curve and Composite REsidual and Smoothing Operator (CRESO) methods. Both simulated and experimental data are used for this evaluation. Results show that the RGCV approach provides the best results to determine the optimal regularization parameter using both the FEM-ZOT and the FEM-L1-Norm. However for the MFS-ZOT, the GCV outperformed all the other regularization parameter choice methods in terms of relative error and correlation coefficient. Regarding the epicardial potential reconstruction, FEM-L1-Norm clearly outperforms the other methods using the simulated data but, using the experimental data, FEM based methods perform as well as MFS. Finally, the use of FEM-L1-Norm combined with RGCV provides robust results in the pacing site localization.

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

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

Scalable and Accurate ECG Simulation for Reaction-Diffusion Models of the Human Heart

Realistic electrocardiogram (ECG) simulation with numerical models is important for research linking cellular and molecular physiology to clinically observable signals, and crucial for patient tailoring of numerical heart models. However, ECG simulation with a realistic torso model is computationally much harder than simulation of cardiac activity itself, so that many studies with sophisticated heart models have resorted to crude approximations of the ECG. This paper shows how the classical concept of electrocardiographic lead fields can be used for an ECG simulation method that matches the realism of modern heart models. The accuracy and resource requirements were compared to those of a full-torso solution for the potential and scaling was tested up to 14,336 cores with a heart model consisting of 11 million nodes. Reference ECGs were computed on a 3.3 billion-node heart-torso mesh at 0.2 mm resolution. The results show that the lead-field method is more efficient than a full-torso solution when the number of simulated samples is larger than the number of computed ECG leads. While the initial computation of the lead fields remains a hard and poorly scalable problem, the ECG computation itself scales almost perfectly and, even for several hundreds of ECG leads, takes much less time than the underlying simulation of cardiac activity.

Sensitivity analysis of ventricular activation and electrocardiogram in tailored models of heart-failure patients

Cardiac resynchronization therapy is not effective in a variable proportion of heart failure patients. An accurate knowledge of each patient's electroanatomical features could be helpful to determine the most appropriate treatment. The goal of this study was to analyze and quantify the sensitivity of left ventricular (LV) activation and the electrocardiogram (ECG) to changes in 39 parameters used to tune realistic anatomical-electrophysiological models of the heart. Electrical activity in the ventricles was simulated using a reaction-diffusion equation. To simulate cellular electrophysiology, the Ten Tusscher-Panfilov 2006 model was used. Intracardiac electrograms and 12-lead ECGs were computed by solving the bidomain equation. Parameters showing the highest sensitivity values were similar in the six patients studied. QRS complex and LV activation times were modulated by the sodium current, the cell surface-to-volume ratio in the LV, and tissue conductivities. The T-wave was modulated by the calcium and rectifier-potassium currents, and the cell surface-to-volume ratio in both ventricles. We conclude that homogeneous changes in ionic currents entail similar effects in all ECG leads, whereas the effects of changes in tissue properties show larger inter-lead variability. The effects of parameter variations are highly consistent between patients and most of the model tuning could be performed with only ~10 parameters.

Human ventricular activation sequence and the simulation of the electrocardiographic QRS complex and its variability in healthy and intraventricular block conditions

Aims

To investigate how variability in activation sequence and passive conduction properties translates into clinical variability in QRS biomarkers, and gain novel physiological knowledge on the information contained in the human QRS complex.

Methods and results

Multiscale bidomain simulations using a detailed heart-torso human anatomical model are performed to investigate the impact of activation sequence characteristics on clinical QRS biomarkers. Activation sequences are built and validated against experimentally-derived ex vivo and in vivo human activation data. R-peak amplitude exhibits the largest variability in terms of QRS morphology, due to its simultaneous modulation by activation sequence speed, myocardial intracellular and extracellular conductivities, and propagation through the human torso. QRS width, however, is regulated by endocardial activation speed and intracellular myocardial conductivities, whereas QR intervals are only affected by the endocardial activation profile. Variability in the apico-basal location of activation sites on the anterior and posterior left ventricular wall is associated with S-wave progression in limb and precordial leads, respectively, and occasional notched QRS complexes in precordial derivations. Variability in the number of early activation sites successfully reproduces pathological abnormalities of the human conduction system in the QRS complex.

Conclusion

Variability in activation sequence and passive conduction properties captures and explains a large part of the clinical variability observed in the human QRS complex. Our physiological insights allow for a deeper interpretation of human QRS biomarkers in terms of QRS morphology and location of early endocardial activation sites. This might be used to attain a better patient-specific knowledge of activation sequence from routine body-surface electrocardiograms.

Using physiologically based models for clinical translation: predictive modelling, data interpretation or something in-between?

Heart disease continues to be a significant clinical problem in Western society. Predictive models and simulations that integrate physiological understanding with patient information derived from clinical data have huge potential to contribute to improving our understanding of both the progression and treatment of heart disease. In particular they provide the potential to improve patient selection and optimisation of cardiovascular interventions across a range of pathologies. Currently a significant proportion of this potential is still to be realised. In this paper we discuss the opportunities and challenges associated with this realisation. Reviewing the successful elements of model translation for biophysically based models and the emerging supporting technologies, we propose three distinct modes of clinical translation. Finally we outline the challenges ahead that will be fundamental to overcome if the ultimate goal of fully personalised clinical cardiac care is to be achieved.

3D Finite Element Electrical Model of Larval Zebrafish ECG Signals

Assessment of heart function in zebrafish larvae using electrocardiography (ECG) is a potentially useful tool in developing cardiac treatments and the assessment of drug therapies. In order to better understand how a measured ECG waveform is related to the structure of the heart, its position within the larva and the position of the electrodes, a 3D model of a 3 days post fertilisation (dpf) larval zebrafish was developed to simulate cardiac electrical activity and investigate the voltage distribution throughout the body. The geometry consisted of two main components; the zebrafish body was modelled as a homogeneous volume, while the heart was split into five distinct regions (sinoatrial region, atrial wall, atrioventricular band, ventricular wall and heart chambers). Similarly, the electrical model consisted of two parts with the body described by Laplace's equation and the heart using a bidomain ionic model based upon the Fitzhugh-Nagumo equations. Each region of the heart was differentiated by action potential (AP) parameters and activation wave conduction velocities, which were fitted and scaled based on previously published experimental results. ECG measurements in vivo at different electrode recording positions were then compared to the model results. The model was able to simulate action potentials, wave propagation and all the major features (P wave, R wave, T wave) of the ECG, as well as polarity of the peaks observed at each position. This model was based upon our current understanding of the structure of the normal zebrafish larval heart. Further development would enable us to incorporate features associated with the diseased heart and hence assist in the interpretation of larval zebrafish ECGs in these conditions.

Numerical simulation of electrocardiograms for full cardiac cycles in healthy and pathological conditions

This work is dedicated to the simulation of full cycles of the electrical activity of the heart and the corresponding body surface potential. The model is based on a realistic torso and heart anatomy, including ventricles and atria. One of the specificities of our approach is to model the atria as a surface, which is the kind of data typically provided by medical imaging for thin volumes. The bidomain equations are considered in their usual formulation in the ventricles, and in a surface formulation on the atria. Two ionic models are used: the Courtemanche-Ramirez-Nattel model on the atria and the 'minimal model for human ventricular action potentials' by Bueno-Orovio, Cherry, and Fenton in the ventricles. The heart is weakly coupled to the torso by a Robin boundary condition based on a resistor-capacitor transmission condition. Various electrocardiograms (ECGs) are simulated in healthy and pathological conditions (left and right bundle branch blocks, Bachmann's bundle block, and Wolff-Parkinson-White syndrome). To assess the numerical ECGs, we use several qualitative and quantitative criteria found in the medical literature. Our simulator can also be used to generate the signals measured by a vest of electrodes. This capability is illustrated at the end of the article. Copyright © 2015 John Wiley & Sons, Ltd.

Stability analysis of the POD reduced order method for solving the bidomain model in cardiac electrophysiology

In this paper we show the numerical stability of the Proper Orthogonal Decomposition (POD) reduced order method used in cardiac electrophysiology applications. The difficulty of proving the stability comes from the fact that we are interested in the bidomain model, which is a system of degenerate parabolic equations coupled to a system of ODEs representing the cell membrane electrical activity. The proof of the stability of this method is based on a priori estimates controlling the gap between the reduced order solution and the Galerkin finite element one. We present some numerical simulations confirming the theoretical results. We also combine the POD method with a time splitting scheme allowing a faster solving of the bidomain problem and show numerical results. Finally, we conduct numerical simulation in 2D illustrating the stability of the POD method in its sensitivity to the ionic model parameters. We also perform 3D simulation using a massively parallel code. We show the computational gain using the POD reduced order model. We also show that this method has a better scalability than the full finite element method.

Simulation of Cardiac Arrhythmias Using a 2D Heterogeneous Whole Heart Model

Simulation studies of cardiac arrhythmias at the whole heart level with electrocardiogram (ECG) gives an understanding of how the underlying cell and tissue level changes manifest as rhythm disturbances in the ECG. We present a 2D whole heart model (WHM2D) which can accommodate variations at the cellular level and can generate the ECG waveform. It is shown that, by varying cellular-level parameters like the gap junction conductance (GJC), excitability, action potential duration (APD) and frequency of oscillations of the auto-rhythmic cell in WHM2D a large variety of cardiac arrhythmias can be generated including sinus tachycardia, sinus bradycardia, sinus arrhythmia, sinus pause, junctional rhythm, Wolf Parkinson White syndrome and all types of AV conduction blocks. WHM2D includes key components of the electrical conduction system of the heart like the SA (Sino atrial) node cells, fast conducting intranodal pathways, slow conducting atriovenctricular (AV) node, bundle of His cells, Purkinje network, atrial, and ventricular myocardial cells. SA nodal cells, AV nodal cells, bundle of His cells, and Purkinje cells are represented by the Fitzhugh-Nagumo (FN) model which is a reduced model of the Hodgkin-Huxley neuron model. The atrial and ventricular myocardial cells are modeled by the Aliev-Panfilov (AP) two-variable model proposed for cardiac excitation. WHM2D can prove to be a valuable clinical tool for understanding cardiac arrhythmias.

Detailed Anatomical and Electrophysiological Models of Human Atria and Torso for the Simulation of Atrial Activation

Atrial arrhythmias, and specifically atrial fibrillation (AF), induce rapid and irregular activation patterns that appear on the torso surface as abnormal P-waves in electrocardiograms and body surface potential maps (BSPM). In recent years both P-waves and the BSPM have been used to identify the mechanisms underlying AF, such as localizing ectopic foci or high-frequency rotors. However, the relationship between the activation of the different areas of the atria and the characteristics of the BSPM and P-wave signals are still far from being completely understood. In this work we developed a multi-scale framework, which combines a highly-detailed 3D atrial model and a torso model to study the relationship between atrial activation and surface signals in sinus rhythm. Using this multi scale model, it was revealed that the best places for recording P-waves are the frontal upper right and the frontal and rear left quadrants of the torso. Our results also suggest that only nine regions (of the twenty-one structures in which the atrial surface was divided) make a significant contribution to the BSPM and determine the main P-wave characteristics.

Regional segmentation of ventricular models to achieve repolarization dispersion in cardiac electrophysiology modeling

The electrocardiogram (ECG) is one of the most significant outputs of a computational model of cardiac electrophysiology because it relates the numerical results to clinical data and is a universal tool for diagnosing heart diseases. One key features of the ECG is the T-wave, which is caused by longitudinal and transmural heterogeneity of the action potential duration (APD). Thus, in order to model a correct wave of repolarization, different cell properties resulting in different APDs must be assigned across the ventricular wall and longitudinally from apex to base. To achieve this requirement, a regional parametrization of the heart is necessary. We propose a robust approach to obtain the transmural and longitudinal segmentation in a general heart geometry without relying on ad hoc procedures. Our approach is based on auxiliary harmonic lifting analyses, already used in the literature to generate myocardial fiber orientations. Specifically, the solution of a sequence of Laplace boundary value problems allows parametrically controlled segmentation of both heart ventricles. The flexibility and simplicity of the proposed method is demonstrated through several representative examples, varying the locations and extents of the epicardial, midwall, and endocardial layers. Effects of the control parameters on the T-wave morphology are illustrated via computed ECGs.

Towards Personalized Cardiology: Multi-Scale Modeling of the Failing Heart

BACKGROUND

Despite modern pharmacotherapy and advanced implantable cardiac devices, overall prognosis and quality of life of HF patients remain poor. This is in part due to insufficient patient stratification and lack of individualized therapy planning, resulting in less effective treatments and a significant number of non-responders.

METHODS AND RESULTS

State-of-the-art clinical phenotyping was acquired, including magnetic resonance imaging (MRI) and biomarker assessment. An individualized, multi-scale model of heart function covering cardiac anatomy, electrophysiology, biomechanics and hemodynamics was estimated using a robust framework. The model was computed on n=46 HF patients, showing for the first time that advanced multi-scale models can be fitted consistently on large cohorts. Novel multi-scale parameters derived from the model of all cases were analyzed and compared against clinical parameters, cardiac imaging, lab tests and survival scores to evaluate the explicative power of the model and its potential for better patient stratification. Model validation was pursued by comparing clinical parameters that were not used in the fitting process against model parameters.

CONCLUSION

This paper illustrates how advanced multi-scale models can complement cardiovascular imaging and how they could be applied in patient care. Based on obtained results, it becomes conceivable that, after thorough validation, such heart failure models could be applied for patient management and therapy planning in the future, as we illustrate in one patient of our cohort who received CRT-D implantation.

A two layers monodomain model of cardiac electrophysiology of the atria

Numerical simulations of the cardiac electrophysiology in the atria are often based on the standard bidomain or monodomain equations stated on a two-dimensional manifold. These simulations take advantage of the thinness of the atrial tissue, and their computational cost is reduced, as compared to three-dimensional simulations. However, these models do not take into account the heterogeneities located in the thickness of the tissue, like discontinuities of the fiber direction, although they can be a substrate for atrial arrhythmia (Hocini et al., Circulation 105(20):2442-2448, 2002; Ho et al., Cardiovasc Res 54(2):325-336, 2002; Nattel, Nature 415(6868):219-226, 2002). We investigate a two-dimensional model with two coupled, superimposed layers that allows to introduce three-dimensional heterogeneities, but retains a reasonable computational cost. We introduce the mathematical derivation of this model and error estimates with respect to the three-dimensional model. We give some numerical illustrations of its interest: we numerically show its convergence for vanishing thickness, introduce an optimization process of the coupling coefficient and assess its validity on physiologically relevant geometries. Our model would be an efficient tool to test the influence of three-dimensional fiber direction heterogeneities in reentries or atrial arrhythmia without using three-dimensional models.

Towards computational modelling of the human foetal electrocardiogram: normal sinus rhythm and congenital heart block

AIMS

We aim to engineer a computational model of propagation during normal sinus rhythm in the foetal human heart, by modifying models for adult cardiac tissue to match foetal electrocardiogram (fECG) characteristics. The model will be partially validated by fECG data, and applied to explore possible mechanisms of arrhythmogenesis in the foetal heart.

METHODS AND RESULTS

Foetal electrocardiograms have been recorded during pregnancy, with P- and T-waves, and the QRS complex, identified by averaging and signal processing. Intervals of the fECG are extracted and used to modify currently available human adult cardiomyocyte models. RR intervals inform models of the pacemaking cells by constraining their rate, the QT interval and its rate dependence constrain models of ventricular cells, and the width of the P-wave, the QR and PR intervals constrain propagation times, conduction velocities, and intercellular coupling. These cell models are coupled into a one-dimensional (1D) model of propagation during normal sinus rhythm in the human foetal heart. We constructed a modular, heterogeneous 1D model for propagation in the foetal heart, and predicted the effects of reduction in L-type Ca(++) current. These include bradycardia and atrioventricular conduction blocks. These may account quantitatively for congenital heart block produced by positive IgG antibodies.

CONCLUSION

The fECG can be interpreted mechanistically and quantitatively by using a simple computational model for propagation. After further validation, by clinical recordings of the fECG and the electrophysiological experiments on foetal cardiac cells and tissues, the model may be used to predict the effects of maternally administered pharmaceuticals on the fECG.

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.

Computational assessment of drug-induced effects on the electrocardiogram: from ion channel to body surface potentials

Background and Purpose

Understanding drug effects on the heart is key to safety pharmacology assessment and anti-arrhythmic therapy development. Here our goal is to demonstrate the ability of computational models to simulate the effect of drug action on the electrical activity of the heart, at the level of the ion-channel, cell, heart and ECG body surface potential.

Experimental Approach

We use the state-of-the-art mathematical models governing the electrical activity of the heart. A drug model is introduced using an ion channel conductance block for the hERG and fast sodium channels, depending on the IC₅₀ value and the drug dose. We simulate the ECG measurements at the body surface and compare biomarkers under different drug actions.

Key Results

Introducing a 50% hERG-channel current block results in 8% prolongation of the APD₉₀ and 6% QT interval prolongation, hERG block does not affect the QRS interval. Introducing 50% fast sodium current block prolongs the QRS and the QT intervals by 12% and 5% respectively, and delays activation times, whereas APD₉₀ is not affected.

Conclusions and Implications

Both potassium and sodium blocks prolong the QT interval, but the underlying mechanism is different: for potassium it is due to APD prolongation; while for sodium it is due to a reduction of electrical wave velocity. This study shows the applicability of in silico models for the investigation of drug effects on the heart, from the ion channel to the ECG-based biomarkers.

Fast data-driven calibration of a cardiac electrophysiology model from images and ECG

Recent advances in computational electrophysiology (EP) models make them attractive for clinical use. We propose a novel data-driven approach to calibrate an EP model from standard 12-lead electrocardiograms (ECG), which are in contrast to invasive or dense body surface measurements widely available in clinical routine. With focus on cardiac depolarization, we first propose an efficient forward model of ECG by coupling a mono-domain, Lattice-Boltzmann model of cardiac EP to a boundary element formulation of body surface potentials. We then estimate a polynomial regression to predict myocardium, left ventricle and right ventricle endocardium electrical diffusion from QRS duration and ECG electrical axis. Training was performed on 4,200 ECG simulations, calculated in aproximately 3 s each, using different diffusion parameters on 13 patient geometries. This allowed quantifying diffusion uncertainty for given ECG parameters due to the ill-posed nature of the ECG problem. We show that our method is able to predict myocardium diffusion within the uncertainty range, yielding a prediction error of less than 5 ms for QRS duration and 2 degree for electrical axis. Prediction results compared favorably with those obtained with a standard optimization procedure, while being 60 times faster. Our data-driven model can thus constitute an efficient preliminary step prior to more refined EP personalization.

Computational modelling of electrocardiograms: repolarisation and T-wave polarity in the human heart

For more than a century, electrophysiologists, cardiologists and engineers have studied the electrical activity of the human heart to better understand rhythm disorders and possible treatment options. Although the depolarisation sequence of the heart is relatively well characterised, the repolarisation sequence remains a subject of great controversy. Here, we study regional and temporal variations in both depolarisation and repolarisation using a finite element approach. We discretise the governing equations in time using an unconditionally stable implicit Euler backward scheme and in space using a consistently linearised Newton-Raphson-based finite element solver. Through systematic parameter-sensitivity studies, we establish a direct relation between a normal positive T-wave and the non-uniform distribution of the controlling parameter, which we have termed refractoriness. To establish a healthy baseline model, we calibrate the refractoriness using clinically measured action potential durations at different locations in the human heart. We demonstrate the potential of our model by comparing the computationally predicted and clinically measured depolarisation and repolarisation profiles across the left ventricle. The proposed framework allows us to explore how local action potential durations on the microscopic scale translate into global repolarisation sequences on the macroscopic scale. We anticipate that our calibrated human heart model can be widely used to explore cardiac excitation in health and disease. For example, our model can serve to identify optimal pacing sites in patients with heart failure and to localise optimal ablation sites in patients with cardiac fibrillation.

Reduced-order modeling for cardiac electrophysiology. Application to parameter identification

A reduced-order model based on proper orthogonal decomposition (POD) is proposed for the bidomain equations of cardiac electrophysiology. Its accuracy is assessed through electrocardiograms in various configurations, including myocardium infarctions and long-time simulations. We show in particular that a restitution curve can efficiently be approximated by this approach. The reduced-order model is then used in an inverse problem solved by an evolutionary algorithm. Some attempts are presented to identify ionic parameters and infarction locations from synthetic electrocardiograms.

A mathematical model of the Purkinje-muscle junctions

This paper is devoted to the construction of a mathematical model of the His-Purkinje tree and the Purkinje-Muscle Junctions (PMJ). A simple numerical scheme is proposed in order to perform some simple numerical experiments.