Peter K. Moore

Anode/cathode make and break phenomena in a model of defibrillation

The goal of this simulation study is to examine, in a sheet of myocardium, the contribution of anode and cathode break phenomena in terminating a spiral wave reentry by the defibrillation shock. The tissue is represented as a homogeneous bidomain with unequal anisotropy ratios. Two case studies are presented in this article: tissue that can electroporate at high levels of transmembrane potential, and model tissue that does not support electroporation. In both cases, the spiral wave is initiated via cross-field stimulation of the bidomain sheet. The extracellular defibrillation shock is delivered via two small electrodes located at opposite tissue boundaries. Modifications in the active membrane kinetics enable the delivery of high-strength defibrillation shocks. Numerical solutions are obtained using an efficient semi-implicit predictor-corrector scheme that allows one to execute the simulations within reasonable time. The simulation results demonstrate that anode and/or cathode break excitations contribute significantly to the activity during and after the shock. For a successful defibrillation shock, the virtual electrodes and the break excitations restrict the spiral wave and render the tissue refractory so it cannot further maintain the reentry. The results also indicate that electroporation alters the anode/cathode break phenomena, the major impact being on the timing of the cathode-break excitations. Thus, electroporation results in different patterns of transmembrane potential distribution after the shock. This difference in patterns may or may not result in change of the outcome of the shock.

A numerically efficient model for simulation of defibrillation in an active bidomain sheet of myocardium

Presented here is an efficient algorithm for solving the bidomain equations describing myocardial tissue with active membrane kinetics. An analysis of the accuracy shows advantages of this numerical technique over other simple and therefore popular approaches. The modular structure of the algorithm provides the critical flexibility needed in simulation studies: fiber orientation and membrane kinetics can be easily modified. The computational tool described here is designed specifically to simulate cardiac defibrillation, i. e., to allow modeling of strong electric shocks applied to the myocardium extracellularly. Accordingly, the algorithm presented also incorporates modifications of the membrane model to handle the high transmembrane voltages created in the immediate vicinity of the defibrillation electrodes.

Success and failure of the defibrillation shock: insights from a simulation study.

Mechanisms for Shock Failure. Introduction: This simulation study presents a further inquiry into the mechanisms by which a strong electric shock fails to halt life‐threatening; cardiac arrhythmins.

A posteriori error estimation with finite element semi- and fully discrete methods for nonlinear parabolic equations in one space dimension

A posteriors error estimates for semi- and fully discrete finite element methods using a pth degree polynomial basis are considered for nonlinear parabolic equations. The error estimates are obtained by solving local parabolic or elliptic equations for corrections to the solution on each element using a

Adaptive local overlapping grid methods for parabolic systems in two space dimensions

p + 1

High-order adaptive methods for parabolic systems

st degree polynomial, which is zero at the nodes. Singly implicit Runge–Kutta (SIRK) and backward difference formula (BDF) methods are considered in the fully discrete case. Local errors are defined in an analogous manner to the respective methods for ordinary differential equations. The a posteriors error estimates are shown to converge to the local errors, even in case the local correction is missing from the nonlinear term. Computational examples compare and verify the theoretical results.

Integrated space-time adaptive hp -refinement methods for parabolic systems

Abstract Adaptive mesh refinement techniques are described for two-dimensional systems of parabolic partial differential equations. Solutions are calculated using Galerkins method with a piecewise bilinear basis in space and backward Euler integration in time. A posteriori estimates of the local discretization error of piecewise bilinear finite element solutions are obtained by a p -refinement technique. These error estimates are used to control a local h -refinement strategy where finer grids are recursively introduced in regions where a prescribed tolerance is exceeded. Fine grids at a given level of refinement may overlap each other and independent solutions are generated on each of them. A version of the Schwarz alternating principle is used to coordinate solutions between overlapping fine grids. Computational results demonstrating the performance of the adaptive procedure on linear and nonlinear problems and apparent convergence of the error estimate for linear heat conduction problem and uniform global refinement is presented.

A local refinement finite-element method for one-dimensional parabolic systems

Abstract We consider the adaptive solution of parabolic partial differential systems in one and two space dimensions by finite element procedures that automatically refine and coarsen computational meshes, vary the degree of the piecewise polynomial basis and, in one dimension, move the computational mesh. Two-dimensional meshes of triangular, quadrilateral, or a mixture of triangular and quadrilateral elements are generated using a finite quadtree procedure that is also used for data management. A posteriori estimates, used to control adaptive enrichment, are generated from the hierarchical polynomial basis. Temporal integration, within a method-of-lines framework, uses either backward difference methods or a variant of the singly implicit Runge-Kutta (SIRK) methods. A high-level user interface facilitates use of the adaptive software.

Comparison of adaptive methods for one-dimensional parabolic systems

Abstract We describe adaptive integrated space-time hp-refinement algorithms for one-dimensional vector systems of parabolic partial differential equations. Solutions are calculated using Galerkins method with a piecewise polynomial hierarchical basis in space and singly implicit Runge-Kutta (SIRK) methods in time. A posteriori estimates of local spatial and temporal discretization errors are used with a priori error estimates to control spatial and temporal enrichment. New techniques simplify spatial error estimation with high-order approximation; integrate spatial and temporal discretization and enrichment; and enable the selection of future meshes and acceptance of partial time steps. A base hp-refinement strategy and several variants are developed and compared using a number of linear and nonlinear examples.

Interpolation error-based a posteriori error estimation for two-point boundary value problems and parabolic equations in one space dimension

Vector systems of parabolic partial differential equations in one space dimension are solved by an adaptive local mesh refinement Galerkin finite-element procedure. Piecewise linear polynomials are used for the spatial representation of the solution and the backward Euler method is used for temporal integration. A local error indicator based on an estimate of the local discretization error is used to control an adaptive-feedback mesh refinement strategy, where finer space-time meshes are recursively added to coarser ones in regions where greater solution resolution is needed. A posteriori estimates of the local discretization error are obtained by a p-refinement procedure that uses piecewise quadratic hierarchic finite-element approximations in space and trapezoidal rule integration in time. Superconvergence properties of the finite-element method are used to neglect errors at nodes and thereby increase the computational efficiency of the error estimation procedure. Further improvements in computational e...