A simple model of cardiac mitochondrial respiration with experimental validation.

Cardiac mitochondria are intracellular organelles that play an important role in energy metabolism and cellular calcium regulation. In particular, they influence the excitation-contraction cycle of the heart cell. A large number of mathematical models have been proposed to better understand the mitochondrial dynamics, but they generally show a high level of complexity, and their parameters are very hard to fit to experimental data. We derived a model based on historical free energy-transduction principles, and results from the literature. We proposed simple expressions that allow to reduce the number of parameters to a minimum with respect to the mitochondrial behavior of interest for us. The resulting model has thirty-two parameters, which are reduced to twenty-three after a global sensitivity analysis of its expressions based on Sobol indices. We calibrated our model to experimental data that consists of measurements of mitochondrial respiration rates controlled by external ADP additions. A sensitivity analysis of the respiration rates showed that only seven parameters can be identified using these observations. We calibrated them using a genetic algorithm, with five experimental data sets. At last, we used the calibration results to verify the ability of the model to accurately predict the values of a sixth dataset. Results show that our model is able to reproduce both respiration rates of mitochondria and transitions between those states, with very low variability of the parameters between each experiment. The same methodology may apply to recover all the parameters of the model, if corresponding experimental data were available.

Modelling the action potential propagation in a heart with structural heterogeneities: From high-resolution MRI to numerical simulations.

Mathematical modelling and numerical simulation in cardiac electrophysiology have already been studied extensively. However, there is a clear lack of techniques and methodologies for studying the propagation of action potential in a heart with structural defects. In this article, we present a modified version of the bidomain model, derived using homogenisation techniques with the assumption of existence of diffusive inclusions in the cardiac tissue. The diffusive inclusions represent regions without electrically active myocytes, for example, fat, fibrosis, and so forth. We present an application of this model to a rat heart. Starting from high-resolution MRI, the geometry of the heart is built and meshed using image processing techniques. We perform a study of the effects of tissue heterogeneities induced by diffusive inclusions on the velocity and shape of the depolarisation wavefront. We present several test cases with different geometries of diffusive inclusions. We reach the conclusion that the conduction velocity is not affected in the best cases, while it is affected by up to 76% in the worst case scenario. Additionally, the shape of the wavefront was affected in some cases.

Considering New Regularization Parameter-Choice Techniques for the Tikhonov Method to Improve the Accuracy of Electrocardiographic Imaging.

The electrocardiographic imaging (ECGI) inverse problem highly relies on adding constraints, a process called regularization, as the problem is ill-posed. When there are no prior information provided about the unknown epicardial potentials, the Tikhonov regularization method seems to be the most commonly used technique. In the Tikhonov approach the weight of the constraints is determined by the regularization parameter. However, the regularization parameter is problem and data dependent, meaning that different numerical models or different clinical data may require different regularization parameters. Then, we need to have as many regularization parameter-choice methods as techniques to validate them. In this work, we addressed this issue by showing that the Discrete Picard Condition (DPC) can guide a good regularization parameter choice for the two-norm Tikhonov method. We also studied the feasibility of two techniques: The U-curve method (not yet used in the cardiac field) and a novel automatic method, called ADPC due its basis on the DPC. Both techniques were tested with simulated and experimental data when using the method of fundamental solutions as a numerical model. Their efficacy was compared with the efficacy of two widely used techniques in the literature, the L-curve and the CRESO methods. These solutions showed the feasibility of the new techniques in the cardiac setting, an improvement of the morphology of the reconstructed epicardial potentials, and in most of the cases of their amplitude.

Optical Imaging of Ventricular Action Potentials in a Torso Tank: A New Platform for Non-Invasive Electrocardiographic Imaging Validation.

Background: Non-invasive electrocardiographic imaging (ECGI) is a promising tool to provide high-resolution panoramic imaging of cardiac electrical activity noninvasively from body surface potential measurements. Current experimental methods for ECGI validation are limited to comparison with unipolar electrograms and the relatively low spatial resolution of cardiac mapping arrays. We aim to develop a novel experimental set up combining a human shaped torso tank with high-resolution optical mapping allowing the validation of ECGI reconstructions. Methods: Langendorff-perfused pig hearts (n = 3) were suspended in a human torso-shaped tank, with the left anterior descending artery (LAD) cannulated on a separate perfusion. Electrical signals were recorded from an 108-electrode epicardial sock and 128 electrodes embedded in the tank surface. Simultaneously, optical mapping of the heart was performed through the anterior surface of the tank. Recordings were made in sinus rhythm and ventricular pacing (n = 55), with activation and repolarization heterogeneities induced by perfusion of hot and cold solutions as well as Sotalol through the LAD. Fluoroscopy provided 3D cardiac and electrode geometries in the tank that were transformed to the 2D optical mapping window using an optimization algorithm. Epicardial unipolar electrograms were reconstructed from torso potentials using ECGI and validated using optical activation and repolarization maps. Results: The transformation and alignment of the 3D geometries onto the 2D optical mapping window was good with an average correlation of 0.87 ± 0.10 and error of 7.7 ± 3.1 ms with activation derived from the sock. The difference in repolarization times were more substantial (error = 17.4 ± 3.7 ms) although the sock and optical repolarization patterns themselves were very similar (correlation = 0.83 ± 0.13). Validation of ECGI reconstructions revealed ECGI accurately captures the pattern of activation (correlation = 0.79 ± 0.11) and identified regions of late and/or early repolarization during different perfusions through LAD. ECGI also correctly demonstrated gradients in both activation and repolarization, although in some cases these were under or over-estimated or shifted slightly in space. Conclusion: A novel experimental setup has been developed, combining a human-shaped torso tank with optical mapping, which can be effectively used in the validation of ECGI techniques; including the reconstruction of activation and repolarization patterns and gradients.

Impact of the Endocardium in a Parameter Optimization to Solve the Inverse Problem of Electrocardiography.

Electrocardiographic imaging aims at reconstructing cardiac electrical events from electrical signals measured on the body surface. The most common approach relies on the inverse solution of the Laplace equation in the torso to reconstruct epicardial potential maps from body surface potential maps. Here we apply a method based on a parameter identification problem to reconstruct both activation and repolarization times. From an ansatz of action potential, based on the Mitchell-Schaeffer ionic model, we compute body surface potential signals. The inverse problem is reduced to the identification of the parameters of the Mitchell-Schaeffer model. We investigate whether solving the inverse problem with the endocardium improves the results or not. We solved the parameter identification problem on two different meshes: one with only the epicardium, and one with both the epicardium and the endocardium. We compared the results on both the heart (activation and repolarization times) and the torso. The comparison was done on validation data of sinus rhythm and ventricular pacing. We found similar results with both meshes in 6 cases out of 7: the presence of the endocardium slightly improved the activation times. This was the most visible on a sinus beat, leading to the conclusion that inclusion of the endocardium would be useful in situations where endo-epicardial gradients in activation or repolarization times play an important role.

In silico assessment of the effects of various compounds in MEA/hiPSC-CM assays: Modeling and numerical simulations.

We propose a mathematical approach for the analysis of drugs effects on the electrical activity of human induced pluripotent stem cell-derived cardiomyocytes (hiPSC-CMs) based on multi-electrode array (MEA) experiments. Our goal is to produce an in silico tool able to simulate drugs action in MEA/hiPSC-CM assays. The mathematical model takes into account the geometry of the MEA and the electrodes' properties. The electrical activity of the stem cells at the ion-channel level is governed by a system of ordinary differential equations (ODEs). The ODEs are coupled to the bidomain equations, describing the propagation of the electrical wave in the stem cells preparation. The field potential (FP) measured by the MEA is modeled by the extracellular potential of the bidomain equations. First, we propose a strategy allowing us to generate a field potential in good agreement with the experimental data. We show that we are able to reproduce realistic field potentials by introducing different scenarios of heterogeneity in the action potential. This heterogeneity reflects the differentiation atria/ventricles and the age of the cells. Second, we introduce a drug/ion channels interaction based on a pore block model. We conduct different simulations for five drugs (mexiletine, dofetilide, bepridil, ivabradine and BayK). We compare the simulation results with the field potential collected from experimental measurements. Different biomarkers computed on the FP are considered, including depolarization amplitude, repolarization delay, repolarization amplitude and depolarization-repolarization segment. The simulation results show that the model reflect properly the main effects of these drugs on the FP.

Identification of Ion Currents Components Generating Field Potential Recorded in MEA From hiPSC-CM.

OBJECTIVE: Multi electrodes arrays (MEAs) combined with cardiomyocytes derived from human-induced pluripotent stem cells (hiPSC-CMs) can enable high- or medium-throughput drug screening in safety pharmacology. This technology has recently attracted a lot of attention, in particular from an international initiative named CiPA. But it is currently limited by the difficulty to analyze the measured signals. We propose a strategy to analyze the signals acquired by the MEA and to automatically deduce the channels affected by the drug. METHODS: Our method is based on the bidomain equations, a model for the MEA electrodes, and an inverse problem strategy. RESULTS: in silico MEA signals are obtained for two commercial devices and an example of early after depolarization is presented. Then, by processing real signals obtained for four different compounds, our algorithm was able to provide dose-response curves for potassium, sodium, and calcium channels. For ivabradine and moxifloxacin, the IC50 and dose-response curves are in very good agreement with known values. SIGNIFICANCE: The proposed strategy offers a possible answer to a major question raised by the community of safety pharmacology. By allowing a more automated analysis of the signals, our approach could contribute to promote the technology based on MEA and hiPSC-CMs and, therefore, improve reliability and efficiency of drug screening.

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.

A bilayer model of human atria: mathematical background, construction, and assessment.

AIMS: Atrial numerical modelling has generally represented the organ as either a surface or tissue with thickness. While surface models have significant computational advantages over tissue models, they cannot fully capture propagation patterns seen in vivo, such as dissociation of activity between endo- and epicardium. We introduce an intermediate representation, a bilayer model of the human atria, which is capable of recreating recorded activation patterns. METHODS AND RESULTS: We simultaneously solved two surface monodomain problems by formalizing an optimization method to set a coupling term between them. Two different asymptotically equivalent numerical implementations of the model are presented. We then built a geometrically and electrophysiologically detailed model of the human atria based on CT data, including two layers of fibre directions, major muscle bundles, and discrete atrial coupling. We adjusted parameters to recreate clinically measured activation times. Activation was compared with a monolayer model. Activation was fit to the physiological range measured over the entire atria. The crista terminalis and pectinate muscles were important for local right atrial activation, but did not significantly affect total activation time. Propagation in the bilayer model was similar to that of a monolayer, but with noticeable difference, due to three-dimensional propagation where fibre direction changed abruptly across the wall, resulting in a slight dissociation of activity. CONCLUSION: Atrial structure plays the dominant role in determining activation. A bilayer model is able to take into account transmural heterogeneities, while maintaining the low computational load associated with surface models.

A bilayer representation of the human atria.

Atrial fibrillation is the most commonly encountered clinical arrhythmia. Despite recent advances in treatment by catheter ablation, its origin is still incompletely understood and it may be difficult to treat. Computer modelling offers an attractive complement to experiment. Simulations of fibrillation, however, are computationally demanding since the phenomenon requires long periods of observation. Because the atria are thin walled structures, they are often modelled as surfaces. However, this may not always be appropriate as the crista terminalis and pectinate muscles are discrete fibrous structures lying on the endocardium and cannot be incorporated into the surface. In the left atrium, there are essentially two layers with an abrupt change in fibre orientation between them. We propose a double layer method, using shell elements to incorporate wall thickness, where fibre direction is independent in each layer and layers are electrically linked. Starting from human multi-detector CT (MDCT) images, we extracted surfaces for the atria and manually added a coronary sinus. Propagation of electrical activity was modelled with the monodomain equation. Results indicate that major features are retained while reducing computation cost considerably. Meshes based on the two layer approach will facilitate studies of AF.

Multifocal ectopic Purkinje-related premature contractions: a new SCN5A-related cardiac channelopathy.

OBJECTIVES: The aim of this study was to describe a new familial cardiac phenotype and to elucidate the electrophysiological mechanism responsible for the disease. BACKGROUND: Mutations in several genes encoding ion channels, especially SCN5A, have emerged as the basis for a variety of inherited cardiac arrhythmias. METHODS: Three unrelated families comprising 21 individuals affected by multifocal ectopic Purkinje-related premature contractions (MEPPC) characterized by narrow junctional and rare sinus beats competing with numerous premature ventricular contractions with right and/or left bundle branch block patterns were identified. RESULTS: Dilated cardiomyopathy was identified in 6 patients, atrial arrhythmias were detected in 9 patients, and sudden death was reported in 5 individuals. Invasive electrophysiological studies demonstrated that premature ventricular complexes originated from the Purkinje tissue. Hydroquinidine treatment dramatically decreased the number of premature ventricular complexes. It normalized the contractile function in 2 patients. All the affected subjects carried the c.665G>A transition in the SCN5A gene. Patch-clamp studies of resulting p.Arg222Gln (R222Q) Nav1.5 revealed a net gain of function of the sodium channel, leading, in silico, to incomplete repolarization in Purkinje cells responsible for premature ventricular action potentials. In vitro and in silico studies recapitulated the normalization of the ventricular action potentials in the presence of quinidine. CONCLUSIONS: A new SCN5A-related cardiac syndrome, MEPPC, was identified. The SCN5A mutation leads to a gain of function of the sodium channel responsible for hyperexcitability of the fascicular-Purkinje system. The MEPPC syndrome is responsive to hydroquinidine.

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.