Monodomain Research Articles

In silico validation of electrocardiographic imaging to reconstruct the endocardial and epicardial repolarization pattern using the equivalent dipole layer source model

The solution of the inverse problem of electrocardiology allows the reconstruction of the spatial distribution of the electrical activity of the heart from the body surface electrocardiogram (electrocardiographic imaging, ECGI). ECGI using the equivalent dipole layer (EDL) model has shown to be accurate for cardiac activation times. However, validation of this method to determine repolarization times is lacking. In the present study, we determined the accuracy of the EDL model in reconstructing cardiac repolarization times, and assessed the robustness of the method under less ideal conditions (addition of noise and errors in tissue conductivity). A monodomain model was used to determine the transmembrane potentials in three different excitation-repolarization patterns (sinus beat and ventricular ectopic beats) as the gold standard. These were used to calculate the body surface ECGs using a finite element model. The resulting body surface electrograms (ECGs) were used as input for the EDL-based inverse reconstruction of repolarization times. The reconstructed repolarization times correlated well (COR > 0.85) with the gold standard, with almost no decrease in correlation after adding errors in tissue conductivity of the model or noise to the body surface ECG. Therefore, ECGI using the EDL model allows adequate reconstruction of cardiac repolarization times.Graphical abstractValidation of electrocardiographic imaging for repolarization using forward calculated body surface ECGs from simulated activation-repolarization sequences.

Dynamics of a neuron\u2013glia system: the occurrence of seizures and the influence of electroconvulsive stimuli

In this paper, we investigate the dynamics of a neuron–glia cell system and the underlying mechanism for the occurrence of seizures. For our mathematical and numerical investigation of the cell model we will use bifurcation analysis and some computational methods. It turns out that an increase of the potassium concentration in the reservoir is one trigger for seizures and is related to a torus bifurcation. In addition, we will study potassium dynamics of the model by considering a reduced version and we will show how both mechanisms are linked to each other. Moreover, the reduction of the potassium leak current will also induce seizures. Our study will show that an enhancement of the extracellular potassium concentration, which influences the Nernst potential of the potassium current, may lead to seizures. Furthermore, we will show that an external forcing term (e.g. electroshocks as unidirectional rectangular pulses also known as electroconvulsive therapy) will establish seizures similar to the unforced system with the increased extracellular potassium concentration. To this end, we describe the unidirectional rectangular pulses as an autonomous system of ordinary differential equations. These approaches will explain the appearance of seizures in the cellular model. Moreover, seizures, as they are measured by electroencephalography (EEG), spread on the macro–scale (cm). Therefore, we extend the cell model with a suitable homogenised monodomain model, propose a set of (numerical) experiment to complement the bifurcation analysis performed on the single–cell model. Based on these experiments, we introduce a bidomain model for a more realistic modelling of white and grey matter of the brain. Performing similar (numerical) experiment as for the monodomain model leads to a suitable comparison of both models. The individual cell model, with its seizures explained in terms of a torus bifurcation, extends directly to corresponding results in both the monodomain and bidomain models where the neural firing spreads almost synchronous through the domain as fast traveling waves, for physiologically relevant paramenters.

A coupled monodomain solver with optimal memory usage for the simulation of cardiac wave propagation

The monodomain model is a common description for explaining the electrical activity of the heart. The model is nonlinear and consists of an ODE system that describes electrochemical reactions in the cardiac cells and a parabolic PDE that express the diffusion of the electrical signal. FEM approaches for the numerical solution of the monodomain equations can be classified into those that simultaneously update the state variables with respect to time by solving a coupled system and those that perform the temporal update in a decoupled manner. While coupled strategies are known to yield more accurate results, they so far have not been applied to physiological cell models due to their enormous memory requirements and computational times. In this paper, we tackle these challenges and suggest a novel computational strategy, that exploits the sparsity of local matrices in the assembly of global FEM matrices and hence features optimal usage of memory. We demonstrate the practicability of our coupled approach by employing three popular physiological cell models (Luo-Rudy phase-I, Ten Tusscher 2006 and O’Hara-Rudy 2011) and compare it with a commonly used decoupled strategy. Qualitatively different results obtained for the simulation of cardiac reentry underline the potential relevance of our work for future studies of cardiac pathology.

Simulation of Skeletal Muscles in Real-Time with Parallel Computing in GPU

Modeling and simulation of the skeletal muscles are usually solved using the Finite Element method (FEM) which, although accurate, commonly needs a complex mesh and the solution is not processed in real-time. In this work, a meshfree model that simulates skeletal muscles considering their functioning and control based on electrical activity, their structure based on biological tissue, and that computes in real-time, is presented. Meshfree methods were used because they are able to surpass most of the limitations that are present in mesh-based methods. The muscular belly was modelled as a particle-based viscoelastic fluid, which is controlled using the monodomain model and shape matching. The smoothed particle hydrodynamics (SPH) method was used to solve both the fluid dynamics and the electrophysiological model. To analyze the accuracy of the method, a similar model was implemented with FEM. Both FEM and SPH methods provide similar solutions of the models in terms of pressure and displacement, with an error of around 0.09, with up to a 10% difference between them. Through the use of General-purpose computing on graphics processing units (GPGPU), real-time simulations that offer a viable alternative to mesh-based models for interactive biological tissue simulations was achieved.

Adaptive-step methods for Markov-based membrane models

This work presents new adaptive time-step schemes based on the Uniformization method for solving stiff electrophysiology models of excitable cells. Uniformization is a numerical method for continuous-time Markov chains that was proven to be effective in solving modern membrane models that have ion channels described by Markov chains. The new methods were tested using three highly stiff cell models previously published. In addition to the simulation of single cells, simulations of well connected cellular networks were performed using the monodomain model. Our results suggest that the proposed adaptive numerical methods significantly outperform previous techniques and are up to 27 times faster than the classical adaptive time-step Euler method.

A posteriori error estimates for the monodomain model in cardiac electrophysiology

We consider the monodomain model, a system of a parabolic semilinear reaction–diffusion equation coupled with a nonlinear ordinary differential equation, arising from the (simplified) mathematical description of the electrical activity of the heart. We derive a posteriori error estimators accounting for different sources of error (space/time discretization and linearization). We prove reliability and efficiency (this latter under a suitable assumption) of the error indicators. Finally, numerical experiments assess the validity of the theoretical results.

On the existence–uniqueness and computation of solution of a class of cardiac electric activity models

In this study, we consider a system of coupled PDEs–ODEs which govern the electric activity in heart with a diffusion term modeling the potential in the surrounding tissue and the nonlinear ionic model proposed by the different reduced cell level models. We prove the existence and uniqueness of the bidomain model with the Morris and Lecar ionic model. The global existence of solution is established based on regularization argument using Fedo–Galerkin/compactness approach. The uniqueness of the solution is shown based on Gronwell’s lemma upon some special treatment of nonlinear terms. The computational realization of the monodomain model with different reduced ionic test models is presented. Also, the coupled parabolic PDE and ODE system modeling the cardiac electric activity in a monodomain system representation of cardiac tissue with different ionic models (at cell level), viz. FitzHugh–Nagumo model, Roger–McCulloch model (RMM), Aliev–Panfilov model and Mitchell–Schaeffer model (MSM), is numerically analyzed by Galerkin linear finite elements in space and the backward Euler method in time. Simulations are done using FreeFem++ open source, and the evolution pattern of the solution has been investigated. MSM and RMM models are unconditionally stable with respect to applied stimulus and initial state, although other models are conditionally stable.

Approximate complex electrical potential distribution in the monodomain model with unequal conductivity and relative permittivity anisotropy ratios

Objective: Electrical conductivity and relative permittivity are properties that indicate muscle health and they have different values parallel and perpendicular to the direction of the myofiber, a concept known as anisotropy. When the intrinsic electrical properties of muscle have ratios of anisotropy that are different then there is no analytical solution that can describe the electrical potential distribution in the tissue. Approach: Here, we present approximate analytical solutions to monodomain equations with unequal anisotropy ratios. For this, we base our analysis on perturbation theory where the electrical potential is approximated by the sum of the zeroth- and first-order terms of an infinite series. Main results: The validity of the approach is confirmed using experimental data for healthy and diseased muscle available online. Significance: A better understanding of electrical potential distribution in anisotropic skeletal muscle tissue will allow the development of improved diagnostic tools for neuromuscular diseases.

Simulation of action potential propagation based on the ghost structure method

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

Modeling and simulation of cardiac electric activity in a human cardiac tissue with multiple ischemic zones.

In this work, a human ventricular model (ten Tusscher and Panfilov model) coupled with the tissue level monodomain model is used to analyze the influence of multiple myocardial ischemia on the human cardiac tissue. The existence and uniqueness of the ischemic model comprising the monodomain model with a discontinuous ionic model for the human cardiac tissue is discussed. The coupled system of partial differential equation and ordinary differential equations are solved numerically using [Formula: see text] finite elements in space and Backward Euler finite difference scheme in time. The apriori finite element error estimate for the numerical scheme has been shown to be of [Formula: see text]. Essentially, we evaluate the impact of the increasing size of the ischemic region and the presence of the multiple ischemic regions having equal or different intensities on the neighboring healthy part of the cardiac tissue. We examine both the individual and the combined influence of two types of ischemia, Hyperkalemia (with the variation of the extracellular potassium ion concentration, [Formula: see text]) and Hypoxia (with the variation of intracellular Adenosine triphosphate (ATP) concentration via parameter [Formula: see text]) on the cardiac electrical activity of cardiac tissue. We observe that with the increase in the ischemic region size by a factor five times, there is an additional almost 10% drop in the action potential duration (APD) in the neighboring healthy regions. The combined effect of Hyperkalemia and Hypoxia brings an additional 12% drop in APD in the ischemic subregions and an additional 5% drop in APD in the neighboring healthy part of the cardic tissue in comparison to the only Hyperkalemic ischemia. When the Hyperkalemic and/or Hypoxic degeneracy of a ischemic zone is non-uniform then innercore degeneracy has greater influence on resting potential and APD of outercore of variable intensity ischemic zone than the other way. Also, increasing the number of ischemic subregions from 2 to 4 leads to a 4% drop in APD.

A comparison of fourth-order operator splitting methods for cardiac simulations

Operator splitting methods constitute a powerful strategy to solve problems in the field of cardiac electrophysiology. In particular, it has been shown that operator splitting methods of order higher than two are more efficient than lower-order methods when solving problems involving simulations of the bidomain and monodomain models. An example of such problems is given by the Niederer problem, a well-defined benchmark in cardiac tissue electrophysiology. In this study, we present some fourth-order operator splitting methods, having real and complex coefficients, and compare their performance when applied to the Niederer benchmark as well as a variant with a stiffer cell model.

Properties of cardiac conduction in a cell-based computational model.

The conduction of electrical signals through cardiac tissue is essential for maintaining the function of the heart, and conduction abnormalities are known to potentially lead to life-threatening arrhythmias. The properties of cardiac conduction have therefore been the topic of intense study for decades, but a number of questions related to the mechanisms of conduction still remain unresolved. In this paper, we demonstrate how the so-called EMI model may be used to study some of these open questions. In the EMI model, the extracellular space, the cell membrane, the intracellular space and the cell connections are all represented as separate parts of the computational domain, and the model therefore allows for study of local properties that are hard to represent in the classical homogenized bidomain or monodomain models commonly used to study cardiac conduction. We conclude that a non-uniform sodium channel distribution increases the conduction velocity and decreases the time delays over gap junctions of reduced coupling in the EMI model simulations. We also present a theoretical optimal cell length with respect to conduction velocity and consider the possibility of ephaptic coupling (i.e. cell-to-cell coupling through the extracellular potential) acting as an alternative or supporting mechanism to gap junction coupling. We conclude that for a non-uniform distribution of sodium channels and a sufficiently small intercellular distance, ephaptic coupling can influence the dynamics of the sodium channels and potentially provide cell-to-cell coupling when the gap junction connection is absent.

Modeling and simulation of hypothermia effects on cardiac electrical dynamics.

Previous experimental evidence has shown the effect of temperature on the action potential duration (APD). It has also been demonstrated that regional cooling of the heart can prolong the APD and promote the termination of ventricular tachycardia. The aim of this study is to demonstrate the effect of hypothermia in suppressing cardiac arrhythmias using numerical modeling. For this purpose, we developed a mathematical model that couples Pennes’ bioheat equation and the bidomain model to simulate the effect of heat on the cardiac action potential. The simplification of the proposed heat–bidomain model to the heat–monodomain model is provided. A suitable numerical scheme for this coupling, based on a time adaptive mesh finite element method, is also presented. First, we performed two-dimensional numerical simulations to study the effect of heat on a regular electrophysiological wave, with the comparison of the calculated and experimental values of Q10. Then, we demonstrated the effect of global hypothermia in suppressing single and multiple spiral waves.

Parallel and space-time adaptivity for the numerical simulation of cardiac action potentials

In this work, the study of parallel and space-time adaptivity for the numerical simulation of action potential propagation in cardiac electrophysiology is presented. The monodomain and bidomain models are employed for the numerical realization. This work is devoted to investigate the spatial adaptivity which is realized within multi finite element methods by using a gradient type a posteriori error estimator and the temporal adaptivity determined by a linearly implicit time integration techniques. In addition, the parallelization of such space-time adaptivity for the simulation of spiral wave dynamics based on non-overlapping domain decomposition techniques and dynamic load balancing is discussed. The numerical results demonstrate that the parallel simulations have been accelerated reasonably good using the adaptive space and time algorithm, with no significant loss in accuracy, for the single action potential wave propagation as well as reentry situations.

Gating-enhanced IMEX splitting methods for cardiac monodomain simulation

The electrical activity in excitable cardiac tissue can be simulated using the so-called monodomain model. The monodomain model is a continuum-based multi-scale model that consists of non-linear ordinary differential equations describing the electrical activity at the cellular scale along with a semi-linear parabolic partial differential equation describing electrical propagation at the tissue scale. The standard “scale-based” splitting method for simulating the monodomain model is to split the tissue and cell models, applying different integrators to each. Typically, the tissue model is simulated with an implicit time-integration method, and the cell model is simulated with an explicit or explicit-exponential one. We demonstrate that the application of implicit-explicit (IMEX) linear multistep or Runge–Kutta methods to this splitting can have poor stability properties when the cell model is stiff. We propose a novel “gating-enhanced” IMEX splitting that treats the tissue variable and the (typically stiff) cell model gating variables together implicitly. The performance of 14 different IMEX methods using both splittings is measured in a variety of one- and two-dimensional experiments. The low incremental overhead combined with the substantially improved stability of the gating-enhanced splitting is shown to result in a performance increase of approximately a factor of four for simulations of the monodomain model with the stiff ten Tusscher–Panfilov model of human endocardial cells.

Numerical Simulation in Electrocardiology Using an Explicit Generalized Finite Difference Method

In this paper we present a fast, accurate and conditionally stable algorithm to solve a monodomain model in 2D, which describes the electrical activity in the heart. The model consists of a parabolic anisotropic Partial Differential Equation (PDE)...

Existence of global solutions in a model of electrical activity of the monodomain type for a ventricle

Introduction: A monodomain model of electrical activity for an isolated ventricle is formulated. This model is written as a reaction diffusion PDE coupled to an ODE, The Rogers-Mculloch model is used to represent the electrical activity through the cell membrane. Method: We give a definition of weak and strong solution of the variational Cauchy problem associated to the monodomain model. A sequence of approximate solutions of Faedo-Galerkin type is proposed.Results: It is shown that the sequence of approximate solutions converge to a weak solution according to the proposed definition. Finally, we have that this weak solution is also a strong solution. Conclusion: The monodomain model of electrical activity in an isolated ventricle that is proposed has a weak solution in an appropriate sense. In addition, this weak solution is also a strong solution.

Cardiac Re-entry Dynamics and Self-termination in DT-MRI Based Model of Human Fetal Heart

The effect of human foetal heart geometry and anisotropy on anatomy induced drift and self-termination of cardiac re-entry is studied here in MRI based 2D slice and 3D whole heart computer simulations. Isotropic and anisotropic models of 20 weeks of gestational age human foetal heart obtained from 100μm voxel diffusion tensor MRI data sets were used in the computer simulations. The fiber orientation angles of the heart were obtained from the orientation of the DT-MRI primary eigenvectors. In a spatially homogeneous electrophysiological monodomain model with the DT-MRI based heart geometries, cardiac re-entry was initiated at a prescribed location in a 2D slice, and in the 3D whole heart anatomy models. Excitation was described by simplified FitzHugh-Nagumo kinetics. In a slice of the heart, with propagation velocity twice as fast along the fibres than across the fibers, DT-MRI based fiber anisotropy changes the re-entry dynamics from pinned to an anatomical re-entry. In the 3D whole heart models, the fiber anisotropy changes cardiac re-entry dynamics from a persistent re-entry to the re-entry self-termination. The self-termination time depends on the re-entry’s initial position. In all the simulations with the DT-MRI based cardiac geometry, the anisotropy of the myocardial tissue shortens the time to re-entry self-termination several folds. The numerical simulations depend on the validity of the DT-MRI data set used. The ventricular wall showed the characteristic transmural rotation of the helix angle of the developed mammalian heart, while the fiber orientation in the atria was irregular.

A space-fractional Monodomain model for cardiac electrophysiology combining anisotropy and heterogeneity on realistic geometries

Classical models of electrophysiology do not typically account for the effects of high structural heterogeneity in the spatio-temporal description of excitation waves propagation. We consider a modification of the Monodomain model obtained by replacing the diffusive term of the classical formulation with a fractional power of the operator, defined in the spectral sense. The resulting nonlocal model describes different levels of tissue heterogeneity as the fractional exponent is varied. The numerical method for the solution of the fractional Monodomain relies on an integral representation of the nonlocal operator combined with a finite element discretisation in space, allowing to handle in a natural way bounded domains in more than one spatial dimension. Numerical tests in two spatial dimensions illustrate the features of the model. Activation times, action potential duration and its dispersion throughout the domain are studied as a function of the fractional parameter: the expected peculiar behaviour driven by tissue heterogeneities is recovered.

High-Order Operator Splitting for the Bidomain and Monodomain Models

The bidomain and monodomain models are multiscale reaction-diffusion models that describe the electrical activity in myocardial tissue. Because of the size and specific structure of the system required to produce meaningful data, numerical solutions are often found through the application of operator-splitting (OS) methods. First- and second-order OS methods have been successfully implemented for finding numerical solutions of the bidomain and monodomain models. It is well known that OS methods with order higher than two require backward time integration within each step. Accordingly, one may conclude that splitting methods with order higher than two are not suitable for models that contain deterministic parabolic equations because the necessary backward time integration would cause instabilities. In this paper, we demonstrate that it is indeed possible to obtain stable results from third-order OS methods applied to the bidomain and monodomain models. Furthermore, we demonstrate that the accompanying gain...