Cardiac Bidomain Model Research Articles

Newton–Krylov-BDDC deluxe solvers for non-symmetric fully implicit time discretizations of the bidomain model

A novel theoretical convergence rate estimate for a Balancing Domain Decomposition by Constraints algorithm is proven for the solution of the cardiac bidomain model, describing the propagation of the electric impulse in the cardiac tissue. The non-linear system arises from a fully implicit time discretization and a monolithic solution approach. The preconditioned non-symmetric operator is constructed from the linearized system arising within the Newton–Krylov approach for the solution of the non-linear problem; we theoretically analyze and prove a convergence rate bound for the Generalised Minimal Residual iterations’ residual. The theory is confirmed by extensive parallel numerical tests, widening the class of robust and efficient solvers for implicit time discretizations of the bidomain model.

Reduced order method for finite difference modeling of cardiac propagation

Abstract Efficient numerical simulation of cardiac electrophysiology is crucial for studying the electrical properties of the heart tissue. The cardiac bidomain model is the most widely accepted representation of the electrical behaviour of the heart muscle. The bidomain model offers fast cardiac potential variation, which can lead to high computational cost due to the required large grid sizes and small time steps. In this paper, the complexity of the finite difference approximation of the bidomain equations is reduced with the model order reduction technique. Proper orthogonal decomposition, a projection-based algorithm, is used to efficiently approximate the original high fidelity cardiac bidomain model with a low-dimensional system of equations. The low-dimensional basis functions are computed first from the ‘snapshots,’ which contain the solutions of the full-order system for different temporal and spatial parameters. Galerkin projection of the original cardiac bidomain system onto the subspace of the reduced order basis functions reduces the size of the linear system. Numerical results confirm the efficiency of the proposed reduced order modeling technique, reducing the simulation time by a factor of 9.54, while maintaining an RMS error of 0.769 mV between the original full order solution and the reduced order POD solution.

Stochastically forced cardiac bidomain model

The bidomain system of degenerate reaction–diffusion equations is a well-established spatial model of electrical activity in cardiac tissue, with “reaction” linked to the cellular action potential and “diffusion” representing current flow between cells. The purpose of this paper is to introduce a “stochastically forced” version of the bidomain model that accounts for various random effects. We establish the existence of martingale (probabilistic weak) solutions to the stochastic bidomain model. The result is proved by means of an auxiliary nondegenerate system and the Faedo–Galerkin method. To prove convergence of the approximate solutions, we use the stochastic compactness method and Skorokhod–Jakubowski a.s. representations. Finally, via a pathwise uniqueness result, we conclude that the martingale solutions are pathwise (i.e., probabilistic strong) solutions.

DUAL-PRIMAL METHODS FOR THE CARDIAC BIDOMAIN MODEL

The cardiac Bidomain model consists in a reaction–diffusion system of partial differential equations which is often discretized by low-order finite elements in space and implicit–explicit methods in time; the resulting linear systems are very ill-conditioned and they must be solved at each time step of a cardiac beat simulation. In this paper we will construct and analyze Balancing Domain Decomposition by Constraints and Finite Element Tearing and Interconnecting Dual-Primal methods for the Bidomain operator. Proven theoretical estimates show that the proposed methods are scalable, quasi-optimal and robust with respect to possible coefficient discontinuities of the Bidomain operator and the time step. The results of extensive parallel numerical tests in three dimensions confirm the convergence rates predicted by the theory; large numerical simulations up to 400 millions of degrees of freedom on 27K cores of BlueGene/Q are also provided.

A comparison of coupled and uncoupled solvers for the cardiac Bidomain model

The aim of this work is to compare a new uncoupled solver for the cardiac Bidomain model with a usual coupled solver. The Bidomain model describes the bioelectric activity of the cardiac tissue and consists of a system of a non-linear parabolic reaction-diffusion partial differential equation (PDE) and an elliptic linear PDE. This system models at macroscopic level the evolution of the transmembrane and extracellular electric potentials of the anisotropic cardiac tissue. The evolution equation is coupled through the non-linear reaction term with a stiff system of ordinary differential equations (ODEs), the so-called membrane model, describing the ionic currents through the cellular membrane. A novel uncoupled solver for the Bidomain system is here introduced, based on solving twice the parabolic PDE and once the elliptic PDE at each time step, and it is compared with a usual coupled solver. Three-dimensional numerical tests have been performed in order to show that the proposed uncoupled method has the same accuracy of the coupled strategy. Parallel numerical tests on structured meshes have also shown that the uncoupled technique is as scalable as the coupled one. Moreover, the conjugate gradient method preconditioned by Multilevel Hybrid Schwarz preconditioners converges faster for the linear systems deriving from the uncoupled method than from the coupled one. Finally, in all parallel numerical tests considered, the uncoupled technique proposed is always about two or three times faster than the coupled approach.

A Simplified 3D Model of Whole Heart Electrical Activity and 12-Lead ECG Generation

We present a computationally efficient three-dimensional bidomain model of torso-embedded whole heart electrical activity, with spontaneous initiation of activation in the sinoatrial node, incorporating a specialized conduction system with heterogeneous action potential morphologies throughout the heart. The simplified geometry incorporates the whole heart as a volume source, with heart cavities, lungs, and torso as passive volume conductors. We placed four surface electrodes at the limbs of the torso: V R, V L, V F and V GND and six electrodes on the chest to simulate the Einthoven, Goldberger-augmented and precordial leads of a standard 12-lead system. By placing additional seven electrodes at the appropriate torso positions, we were also able to calculate the vectorcardiogram of the Frank lead system. Themodel was able to simulate realistic electrocardiogram (ECG) morphologies for the 12 standard leads, orthogonal X, Y, and Z leads, as well as the vectorcardiogram under normal and pathological heart states. Thus, simplified and easy replicable 3D cardiac bidomain model offers a compromise between computational load and model complexity and can be used as an investigative tool to adjust cell, tissue, and whole heart properties, such as setting ischemic lesions or regions of myocardial infarction, to readily investigate their effects on whole ECG morphology.

Balancing Neumann–Neumann methods for the cardiac Bidomain model

Balancing Neumann---Neumann preconditioners are constructed, analyzed and numerically studied for the cardiac Bidomain model in three-dimensions. This reaction---diffusion system is discretized by low-order finite elements in space and implicit---explicit methods in time, yielding very ill-conditioned linear systems that must be solved at each time step. The proposed algorithm is based on decomposing the domain into nonoverlapping subdomains and on solving iteratively the Bidomain Schur complement obtained by implicitly eliminating the degrees of freedom interior to each subdomain. The iteration is preconditioned by a Balancing Neumann---Neumann method employing local Neumann solves on each subdomain and a coarse Bidomain solve. A novel approach for the estimation of the average operator of the nonoverlapping decomposition provides a framework for designing coarse spaces for Balancing Neumann---Neumann methods. The theoretical estimates obtained show that the proposed method is scalable, quasi-optimal and robust with respect to possible coefficient discontinuities of the Bidomain operator. The results of extensive parallel numerical tests in three dimensions confirm the convergence rates predicted by the theory.

Representing Cardiac Bidomain Bath-Loading Effects by an Augmented Monodomain Approach: Application to Complex Ventricular Models

Although the cardiac bidomain model has been widely used in the simulation of electrical activation, its relatively computationally expensive nature means that monodomain approaches are generally required for long-duration simulations (for example, investigations of arrhythmia mechanisms). However, the presence of a conducting bath surrounding the tissue is known to induce wavefront curvature (surface leading bulk), a phenomena absent in standard monodomain approaches. Here, we investigate the biophysical origin of the bidomain bath-loading induced wavefront curvature and present a novel augmented monodomain-equivalent bidomain approach faithfully replicating all aspects of bidomain wavefront morphology and conduction velocity, but with a fraction of the computational cost. Bath-loading effects are shown to be highly dependent upon specific conductivity parameters, but less dependent upon the thickness or conductivity of the surrounding bath, with even relatively thin surrounding fluid layers (~ 0.1 mm) producing significant wavefront curvature in bidomain simulations. We demonstrate that our augmented monodomain approach can be easily adapted for different conductivity sets and applied to anatomically complex models, thus facilitating fast and accurate simulation of cardiac wavefront dynamics during long-duration simulations, further aiding the faithful comparison of simulations with experiments.

Cardiac simulation on multi-GPU platform

The cardiac bidomain model is a popular approach to study electrical behavior of tissues and simulate interactions between the cells by solving partial differential equations. The iterative and data parallel model is an ideal match for the parallel architecture of Graphic Processing Units (GPUs). In this study, we evaluate the effectiveness of architecture-specific optimizations and fine grained parallelization strategies, completely port the model to GPU, and evaluate the performance of single-GPU and multi-GPU implementations. Simulating one action potential duration (350 msec real time) for a 256×256×256 tissue takes 453 hours on a high-end general purpose processor, while it takes 664 seconds on a four-GPU based system including the communication and data transfer overhead. This drastic improvement (a factor of 2460×) will allow clinicians to extend the time-scale of simulations from milliseconds to seconds and minutes; and evaluate hypotheses in a shorter amount of time that was not feasible previously.

Convergence of discrete duality finite volume schemes for the cardiac bidomain model

We prove convergence of discrete duality finite volume (DDFV) schemes on distorted meshes for a class of simplified macroscopic bidomain models of the electrical activity in the heart. Both time-implicit and linearised time-implicit schemes are treated. A short description is given of the 3D DDFV meshes and of some of the associated discrete calculus tools. Several numerical tests are presented.

Multilevel Additive Schwarz Preconditioners for the Bidomain Reaction-Diffusion System

Multilevel additive Schwarz methods are analyzed and studied numerically for the anisotropic cardiac Bidomain model in three dimensions. This is the most complete model to date of the bioelectrical activity of the heart tissue, consisting of a degenerate parabolic system of nonlinear reaction-diffusion equations coupled with a stiff system of several ordinary differential equations describing the ionic currents through the cellular membrane. Due to the presence of very different scales in both space and time, the numerical discretization of this system by finite elements in space and semi-implicit methods in time produces very ill-conditioned linear systems that must be solved at each time step. The proposed multilevel algorithm employs a hierarchy of nested meshes with overlapping Schwarz preconditioners on each level and is fully additive, hence parallel, within and among levels. Convergence estimates are proved for the resulting multilevel algorithm, showing that its convergence rate is independent of the number of subdomains (scalability), of the mesh sizes of each level and of the number of levels (optimality). Several parallel tests on a Linux cluster confirm the scalability and optimality of the method, as well as its parallel efficiency on both Cartesian and deformed domains in three dimensions.

Estimation of Cardiac Bidomain Parameters from Extracellular Measurement: Two Dimensional Study

Cardiac tissue conductivity measurements can be used to assess the electrical substrate underlying normal and abnormal wavefront propagation. We describe a method of solving the inverse cardiac bidomain model to estimate average longitudinal and transverse intra and extra-cellular conductivities and fiber angle relative to an electrode array placed arbitrarily on the epi- or endocardial surface. A Newton-Raphson reconstruction method and two Tikhonov-type regularizations were able to stably identify conductivities and fiber angles in tissue models having anisotropies similar to those in real cardiac tissue. The reconstruction methods were tested with data from increasingly realistic two dimensional cardiac bidomain models and performed well both when measurement noise was added, and when simulated experimental and forward model matching was diminished. This approach may be a suitable basis for continuous monitoring of myocardial condition in-vivo via a catheter based electrode array.