Incorporating Inductances in Tissue-Scale Models of Cardiac Electrophysiology

In standard models of cardiac electrophysiology, including the bidomain and monodomain models, local perturbations can propagate at infinite speed. We address this unrealistic property by developing a hyperbolic bidomain model that is based on a generalization of Ohm's law with a Cattaneo-type model for the fluxes. Further, we obtain a hyperbolic monodomain model in the case that the intracellular and extracellular conductivity tensors have the same anisotropy ratio. In one spatial dimension, the hyperbolic monodomain model is equivalent to a cable model that includes axial inductances, and the relaxation times of the Cattaneo fluxes are strictly related to these inductances. A purely linear analysis shows that the inductances are negligible, but models of cardiac electrophysiology are highly nonlinear, and linear predictions may not capture the fully nonlinear dynamics. In fact, contrary to the linear analysis, we show that for simple nonlinear ionic models, an increase in conduction velocity is obtained for small and moderate values of the relaxation time. A similar behavior is also demonstrated with biophysically detailed ionic models. Using the Fenton-Karma model along with a low-order finite element spatial discretization, we numerically analyze differences between the standard monodomain model and the hyperbolic monodomain model. In a simple benchmark test, we show that the propagation of the action potential is strongly influenced by the alignment of the fibers with respect to the mesh in both the parabolic and hyperbolic models when using relatively coarse spatial discretizations. Accurate predictions of the conduction velocity require computational mesh spacings on the order of a single cardiac cell. We also compare the two formulations in the case of spiral break up and atrial fibrillation in an anatomically detailed model of the left atrium, and [...].Comment: 20 pages, 12 figure

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

Derivation of the bidomain equations for a beating heart with a general microstructure

A novel multiple scales method is formulated that can be applied to problems which have an almost\ud periodic microstructure not in Cartesian coordinates but in a general curvilinear coordinate system.\ud The method is applied to a model of the electrical activity of cardiac myocytes and used to derive a\ud version of the bidomain equations describing the macroscopic electrical activity of cardiac tissue. The\ud treatment systematically accounts for the non-uniform orientation of the cells within the tissue and for\ud deformations of the tissue occurring as a result of the heart beat

A Multidomain Model for Ionic Electrodiffusion and Osmosis with an Application to Cortical Spreading Depression

Ionic electrodiffusion and osmotic water flow are central processes in many physiological systems. We formulate a system of partial differential equations that governs ion movement and water flow in biological tissue. A salient feature of this model is that it satisfies a free energy identity, ensuring the thermodynamic consistency of the model. A numerical scheme is developed for the model in one spatial dimension and is applied to a model of cortical spreading depression, a propagating breakdown of ionic and cell volume homeostasis in the brain.Comment: submitted for publication, Aug. 28, 201

A parallel solver for reaction-diffusion systems in computational electrocardiology

In this work, a parallel three-dimensional solver for numerical simulations in computational electrocardiology is introduced and studied. The solver is based on the anisotropic Bidomain %(AB) cardiac model, consisting of a system of two degenerate parabolic reaction-diffusion equations describing the intra and extracellular potentials of the myocardial tissue. This model includes intramural fiber rotation and anisotropic conductivity coefficients that can be fully orthotropic or axially symmetric around the fiber direction. %In case of equal anisotropy ratio, this system reduces to The solver also includes the simpler anisotropic Monodomain model, consisting of only one reaction-diffusion equation. These cardiac models are coupled with a membrane model for the ionic currents, consisting of a system of ordinary differential equations that can vary from the simple FitzHugh-Nagumo (FHN) model to the more complex phase-I Luo-Rudy model (LR1). The solver employs structured isoparametric $Q_1$ finite elements in space and a semi-implicit adaptive method in time. Parallelization and portability are based on the PETSc parallel library. Large-scale computations with up to $O(10^7)$ unknowns have been run on parallel computers, simulating excitation and repolarization phenomena in three-dimensional domains

Chaste: a test-driven approach to software development for biological modelling

Chaste (‘Cancer, heart and soft-tissue environment’) is a software library and a set of test suites for computational simulations in the domain of biology. Current functionality has arisen from modelling in the fields of cancer, cardiac physiology and soft-tissue mechanics. It is released under the LGPL 2.1 licence.\ud \ud Chaste has been developed using agile programming methods. The project began in 2005 when it was reasoned that the modelling of a variety of physiological phenomena required both a generic mathematical modelling framework, and a generic computational/simulation framework. The Chaste project evolved from the Integrative Biology (IB) e-Science Project, an inter-institutional project aimed at developing a suitable IT infrastructure to support physiome-level computational modelling, with a primary focus on cardiac and cancer modelling

A multiresolution space-time adaptive scheme for the bidomain model in electrocardiology

This work deals with the numerical solution of the monodomain and bidomain models of electrical activity of myocardial tissue. The bidomain model is a system consisting of a possibly degenerate parabolic PDE coupled with an elliptic PDE for the transmembrane and extracellular potentials, respectively. This system of two scalar PDEs is supplemented by a time-dependent ODE modeling the evolution of the so-called gating variable. In the simpler sub-case of the monodomain model, the elliptic PDE reduces to an algebraic equation. Two simple models for the membrane and ionic currents are considered, the Mitchell-Schaeffer model and the simpler FitzHugh-Nagumo model. Since typical solutions of the bidomain and monodomain models exhibit wavefronts with steep gradients, we propose a finite volume scheme enriched by a fully adaptive multiresolution method, whose basic purpose is to concentrate computational effort on zones of strong variation of the solution. Time adaptivity is achieved by two alternative devices, namely locally varying time stepping and a Runge-Kutta-Fehlberg-type adaptive time integration. A series of numerical examples demonstrates thatthese methods are efficient and sufficiently accurate to simulate the electrical activity in myocardial tissue with affordable effort. In addition, an optimalthreshold for discarding non-significant information in the multiresolution representation of the solution is derived, and the numerical efficiency and accuracy of the method is measured in terms of CPU time speed-up, memory compression, and errors in different norms.Comment: 25 pages, 41 figure

Numerical Simulations of Fractionated Electrograms and Pathological Cardiac Action Potential

The aim of this work is twofold. First we focus on the complex phenomenon of electrogram fractionation, due to the presence of discontinuities in the conduction properties of the cardiac tissue in a bidomain model. Numerical simulations of paced activation may help to understand the role of the membrane ionic currents and of the changes in cellular coupling in the formation of conduction blocks and fractionation of the electrogram waveform. In particular, we show that fractionation is independent ofINAalterations and that it can be described by the bidomain model of cardiac tissue. Moreover, some deflections in fractionated electrograms may give nonlocal information about the shape of damaged areas, also revealing the presence of inhomogeneities in the intracellular conductivity of the medium at a distance.The second point of interest is the analysis of the effects of space–time discretization on numerical results, especially during slow conduction in damaged cardiac tissue. Indeed, large discretization steps can induce numerical artifacts such as slowing down of conduction velocity, alteration in extracellular and transmembrane potential waveforms or conduction blocks, which are not predicted by the continuous bidomain model. Several possible numerical and physiological explanations of these effects are given. Essentially, the discrete system obtained at the end of the approximation process may be interpreted as a discrete model of the cardiac tissue made up of isopotential cells where the effective intracellular conductivity tensor depends on the space discretization steps; the increase of these steps results in an increase of the effective intracellular resistance and can induce conduction blocks if a certain critical value is exceeded