An introduction to mathematical and numerical modeling in heart electrophysiology

The electrical activation of the heart is the biological process that regulates the contraction of the cardiac muscle, allowing it to pump blood to the whole body. In physiological conditions, the pacemaker cells of the sinoatrial node generate an action potential (a sudden variation of the cell transmembrane potential) which, following preferential conduction pathways, propagates throughout the heart walls and triggers the contraction of the heart chambers. The action potential propagation can be mathematically described by coupling a model for the ionic currents, flowing through the membrane of a single cell, with a macroscopical model that describes the propagation of the electrical signal in the cardiac tissue. The most accurate model available in the literature for the description of the macroscopic propagation in the muscle is the Bidomain model, a degenerate parabolic system composed of two non-linear partial differential equations for the intracellular and extracellular potential. In this paper, we present an introduction to the fundamental aspects of mathematical modeling and numerical simulation in cardiac electrophysiology

Spectral analysis of a block-triangular preconditioner for the bidomain system in electrocardiology

In this paper we analyze in detail the spectral properties of the block-triangular preconditioner introduced by Gerardo-Giorda et al. [J. Comput. Phys., 228 (2009), pp. 3625-3639] for the Bidomain system in non-symmetric form. We show that the conditioning of the preconditioned problem is bounded in the Fourier space independently of the frequency variable, ensuring quasi-optimality with respect to the mesh size. We derive an explicit formula to optimize the preconditioner performance by identifying a parameter that depends only on the coefficients of the problem and is easy to compute. We provide numerical tests in three dimensions that confirm the optimality of the parameter and the substantial independence of the mesh size. Copyrigh

Is minimising the convergence rate the best choice for efficient Optimized Schwarz preconditioning in heterogeneous coupling? The Stokes-Darcy case

Optimized Schwarz Methods (OSM) are domain decomposition techniques based on Robin-type interface condition that have became increasingly popular in the last two decades. Ensuring convergence also on non-overlapping decompositions, OSM are naturally advocated for the heterogeneous coupling of multi-physics problems. Classical approaches optimize the coefficients in the Robin condition by minimizing the effective convergence rate of the resulting iterative algorithm. However, when OSM are used as preconditioners for Krylov solvers of the resulting interface problem, such parameter optimization does not necessarily guarantee the fastest convergence. This drawback is already known for homogeneous decomposition, but in the case of heterogeneous decomposition, the poor performance of the classical optimization approach becomes utterly evident. In this paper, we highlight this drawback for the Stokes/Darcy problem and we propose a more effective alternative optimization procedure.European Union Seventh Framework Programme (FP7/2007-2013; grant 294229) to M. Discacciat

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

A computational multiscale model of cortical spreading depression propagation

Cortical Spreading Depression (CSD) is a disruption of the brain hemostasis that, originating in the visual cortex and traveling towards the frontal lobe, temporarily impairs the normal functioning of neurons. Although, as of today, little is known about the mechanisms that can trigger or stop such phenomenon, CSD is commonly accepted as a correlate of migraine visual aura. In this paper, we introduce a multiscale PDE-ODE model that couples the propagation of the depolarization wave associated to CSD with a detailed electrophysiological model for the neuronal activity to capture both macroscopic and microscopic dynamics

Optimized Schwarz Methods in the Stokes-Darcy Coupling

This article studies optimized Schwarz methods for the Stokes–Darcy problem. Robin transmission conditions are introduced, and the coupled problem is reduced to a suitable interface system that can be solved using Krylov methods. Practical strategies to compute optimal Robin coefficients are proposed, which take into account both the physical parameters of the problem and the mesh size. Numerical results show the effectiveness of our approach.European Union Seventh Framework Programme (FP7/2007-2013; grant 294229) to M. Discacciat

Optimized schwarz methods for the bidomain system in electrocardiology

The propagation of the action potential in the heart chambers is accurately described by the Bidomain model, which is commonly accepted and used in the specialistic literature. However, its mathematical structure of a degenerate parabolic system entails high computational costs in the numerical solution of the associated linear system. Domain decomposition methods are a natural way to reduce computational costs, and Optimized Schwarz Methods have proven in the recent years their effectiveness in accelerating the convergence of such algorithms. The latter are based on interface matching conditions more efficient than the classical Dirichlet or Neumann ones. In this paper we analyze an Optimized Schwarz approach for the numerical solution of the Bidomain problem. We assess the convergence of the iterative method by means of Fourier analysis, and we investigate the parameter optimization in the interface conditions. Numerical results in 2D and 3D are given to show the effectiveness of the method

Parallelizing the Kolmogorov-Fokker-Planck Equation

We design two parallel schemes, based on Schwarz Waveform Relaxation (SWR) procedures, for the numerical solution of the Kolmogorov equation. The latter is a simplified version of the Fokker-Planck equation describing the time evolution of the probability density of the velocity of a particle. SWR procedures decompose the spatio- temporal computational domain into subdomains and solve (in parallel) subproblems, that are coupled through suitable conditions at the interfaces to recover the solution of the global problem. We consider coupling conditions of both Dirichlet (Classical SWR) and Robin (Optimized SWR) types. We prove well-posedeness of the schemes subproblems and convergence for the proposed algorithms. We corroborate our findings with some numerical tests

Incorporating landscape heterogeneities in the spread of an epidemics in wildlife

One of the main difficulties in the modeling and numerical simulation of the spread of an infectious disease in wildlife resides in properly taking into account the heterogeneities of the landscape. Forests, plains and mountains present different levels of hospitality, while large interstates, lakes and major waterways can provide strong natural barriers to the epidemic spread. A canonical approach has been to discretize both population and geography into geopolitical units and consider the movement of individuals from unit to unit [4]. This approach, however, does not well represent the biological realities of animal movement, since animals do not move at the scale of geopolitical units. We combine a standard SEI epidemiological model with a diffusion process to account for movement as a continuous process across a continuous region [1]. This results in a system of parabolic reaction-diffusion equations with nonlinear reaction term. Landscape heterogeneities are accounted for by including in the computational domain the significant geographical features of the area. We discretize the resulting model in time by an IMEX scheme and in space by finite elements. To show the effectiveness of the method, we present numerical simulation for rabies epidemics among raccoons in New York State

Numerical simulation of a susceptible-exposed-infectious space-continuous model for the spread of rabies in raccoons across a realistic landscape

We introduce a numerical model for the spread of a lethal infectious disease in wildlife. The reference model is a Susceptible-Exposed-Infectious system where the spatial component of the dynamics is modelled by a diffusion process. The goal is to develop a model to be used for real geographical scenarios, so we do not rely upon simplifying assumptions on the shape of the region of interest. For this reason, space discretization is carried out with the finite element method on an unstructured triangulation. A diffusion term is designed to take into account landscape heterogeneities such as mountains and waterways. Numerical simulations are carried out for rabies epidemics among raccoons in New York state. A qualitative comparison of numerical results to available data from real-world epidemics is discussed