A dual adaptive explicit time integration algorithm for efficiently solving the cardiac monodomain equation

The monodomain model is widely used in in-silico cardiology to describe excitation propagation in the myocardium. Frequently, operator splitting is used to decouple the stiff reaction term and the diffusion term in the monodomain model so that they can be solved separately. Commonly, the diffusion term is solved implicitly with a large time step while the reaction term is solved by using an explicit method with adaptive time stepping. In this work, we propose a fully explicit method for the solution of the decoupled monodomain model. In contrast to semi-implicit methods, fully explicit methods present lower memory footprint and higher scalability. However, such methods are only conditionally stable. We overcome the conditional stability limitation by proposing a dual adaptive explicit method in which adaptive time integration is applied for the solution of both the reaction and diffusion terms. We perform a set of numerical examples where cardiac propagation is simulated under physiological and pathophysiological conditions. Results show that the proposed method presents preserved accuracy and improved computational efficiency as compared to standard operator splitting-based methods. © 2021 John Wiley & Sons Ltd

Efficient time splitting schemes for the monodomain equation in cardiac electrophysiology

Approximating the fast dynamics of depolarization waves in the human heart described by the monodomain model is numerically challenging. Splitting methods for the PDE-ODE coupling enable the computation with very fine space and time discretizations. Here, we compare different splitting approaches regarding convergence, accuracy, and efficiency. Simulations were performed for a benchmark problem with the Beeler–Reuter cell model on a truncated ellipsoid approximating the left ventricle including a localized stimulation. For this configuration, we provide a reference solution for the transmembrane potential. We found a semi-implicit approach with state variable interpolation to be the most efficient scheme. The results are transferred to a more physiological setup using a bi-ventricular domain with a complex external stimulation pattern to evaluate the accuracy of the activation time for different resolutions in space and time

Efficient time splitting schemes for the monodomain equation in cardiac electrophysiology

Approximating the fast dynamics of depolarization waves in the human heart described by the monodomain model is numerically challenging. Splitting methods for the PDE-ODE coupling enable the computation with very fine space and time discretizations. Here, we compare different splitting approaches regarding convergence, accuracy and efficiency. Simulations were performed for a benchmark configuration with the Beeler–Reuter cell model on a truncated ellipsoid approximating the left ventricle including a localized stimulation. For this benchmark configuration, we provide a reference solution for the transmembrane potential. We found a semi-implicit approach with state variable interpolation to be the most efficient scheme. The results are transferred to a more physiological setup using a bi-ventricular domain with a complex external stimulation pattern to evaluate the accuracy of the activation time for different resolutions in space and time

Isogeometric approximation of cardiac electrophysiology models on surfaces: An accuracy study with application to the human left atrium

We consider Isogeometric Analysis in the framework of the Galerkin method for the spatial approximation of cardiac electrophysiology models defined on NURBS surfaces; specifically, we perform a numerical comparison between basis functions of degree p ≥ 1 and globally C k -continuous, with k = 0 or p − 1, to find the most accurate approximation of a propagating front with the minimal number of degrees of freedom. We show that B-spline basis functions of degree p ≥ 1, which are C p−1 -continuous capture accurately the front velocity of the transmembrane potential even with moderately refined meshes; similarly, we show that, for accurate tracking of curved fronts, high-order continuous B-spline basis functions should be used. Finally, we apply Isogeometric Analysis to an idealized human left atrial geometry described by NURBS with physiologically sound fiber directions and anisotropic conductivity tensor to demonstrate that the numerical scheme retains its favorable approximation properties also in a more realistic setting

Towards New High-Order Operator Splitting Time-Integration Methods

Operator splitting (OS) methods represent a powerful strategy to solve an extensive range of mathematical models in the form of differential equations. They have a long and celebrated history, having been successfully used for well over half a century to provide efficient numerical solutions to challenging problems. In fact, OS methods are often the only viable way to solve many problems in practice. The simplest, and perhaps, most well-known OS methods are Lie--Trotter--Godunov and the Strang--Marchuk methods. They compute a numerical solution that is first-, and second-order accurate in time, respectively. OS methods can be derived by imposing order conditions using the Campbell--Baker--Hausdorff formula. It follows that, by setting the appropriate order conditions, it is possible to derive OS methods of any desired order. An important observation regarding OS methods with order higher than two is that, according to the Sheng--Suzuki theorem, at least one of their defining coefficients must be negative. Therefore, the time integration with OS methods of order higher than two has not been considered suitable to solve deterministic parabolic problems, because the necessary backward time integration would cause instabilities. Throughout this thesis, we focus our attention on high-order (i.e., order higher than two) OS methods. We successfully assess the convergence properties of some higher-order OS methods on diffusion-reaction problems describing cardiac electrophysiology and on an advection-diffusion-reaction problem describing chemical combustion. Furthermore, we compare the efficiency performance of higher-order methods to second-order methods. For all the cases considered, we confirm an improved efficiency performance compared to methods of lower order. Next, we observe how, when using OS and Runge--Kutta type methods to advance the time integration, we can construct a unique extended Butcher tableau with a similar structure to the ones describing Generalized Additive Runge--Kutta (GARK) methods. We define a combination of methods to be OS-GARK methods. We apply linear stability analysis to OS-GARK methods; this allows us to conveniently analyze the stability properties of any combination of OS and Runge--Kutta methods. Doing so, we are able to perform an eigenvalue analysis to understand and improve numerically unstable solutions

Finite element and finite volume-element simulation of pseudo-ECGs and cardiac alternans

In this paper, we are interested in the spatio-temporal dynamics of the transmembrane potential in paced isotropic and anisotropic cardiac tissues. In particular, we observe a specific precursor of cardiac arrhythmias that is the presence of alternans in the action potential duration. The underlying mathematical model consists of a reaction–diffusion system describing the propagation of the electric potential and the nonlinear interaction with ionic gating variables. Either conforming piecewise continuous finite elements or a finite volume-element scheme are employed for the spatial discretization of all fields, whereas operator splitting strategies of first and second order are used for the time integration. We also describe an efficient mechanism to compute pseudo-ECG signals, and we analyze restitution curves and alternans patterns for physiological and pathological cardiac rhythms

Numerical methods for simulation of electrical activity in the myocardial tissue

Mathematical models of electric activity in cardiac tissue are becoming increasingly powerful tools in the study of cardiac arrhythmias. Considered here are mathematical models based on ordinary differential equations (ODEs) and partial differential equations (PDEs) that describe the behaviour of this electrical activity. Generating an efficient numerical solution of these models is a challenging task, and in fact the physiological accuracy of tissue-scale models is often limited by the efficiency of the numerical solution process. In this thesis, we discuss two sets of experiments that test ideas for making the numerical solution process more efficient. In the first set of experiments, we examine the numerical solution of four single cell cardiac electrophysiological models, which consist solely of ODEs. We study the efficiency of using implicit-explicit Runge-Kutta (IMEX-RK) splitting methods to solve these models. We find that variable step-size implementations of IMEX-RK methods (ARK3 and ARK5) that take advantage of Jacobian structure clearly outperform most methods commonly used in practice for two of the models, and they outperform all methods commonly used in practice for the remaining models. In the second set of experiments, we examine the solution of the bidomain model, a model consisting of both ODEs and PDEs that are typically solved separately. We focus these experiments on numerical methods for the solution of the two PDEs in the bidomain model. The most popular method for this task, the Crank-Nicolson method, produces unphysical oscillations; we propose a method based on a second-order L-stable singly diagonally implicit Runge-Kutta (SDIRK) method to eliminate these oscillations. We find that although the SDIRK method is able to eliminate these unphysical oscillations, it is only more efficient for crude error tolerances

Rush-Larsen time-stepping methods of high order for stiff problems in cardiac electrophysiology

To address the issues of stability and accuracy for reaction-diffusion equations, the development of high order and stable time-stepping methods is necessary. This is particularly true in the context of cardiac electrophysiology, where reaction-diffusion equations are coupled with stiff ODE systems. Many research have been led in that way in the past 15 years concerning implicit-explicit methods and exponential integrators. In 2009, Perego and Veneziani proposed an innovative time-stepping method of order 2. In this paper we present the extension of this method to the orders 3 and 4 and introduce the Rush-Larsen schemes of order k (shortly denoted RL\_k). The RL\_k schemes are explicit multistep exponential integrators. They display a simple general formulation and an easy implementation. The RL\_k schemes are shown to be stable under perturbation and convergent of order k. Their Dahlquist stability analysis is performed. They have a very large stability domain provided that the stabilizer associated with the method captures well enough the stiff modes of the problem. The RL\_k method is numerically studied as applied to the membrane equation in cardiac electrophysiology. The RL k schemes are shown to be stable under perturbation and convergent oforder k. Their Dahlquist stability analysis is performed. They have a very large stability domain provided that the stabilizer associated with the method captures well enough the stiff modes of the problem. The RL k method is numerically studied as applied to the membrane equation in cardiac electrophysiology