15 research outputs found
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 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 unknowns have been run
on parallel computers, simulating excitation and repolarization phenomena in
three-dimensional domains
Modeling Excitable Tissue
This open access volume presents a novel computational framework for understanding how collections of excitable cells work. The key approach in the text is to model excitable tissue by representing the individual cells constituting the tissue. This is in stark contrast to the common approach where homogenization is used to develop models where the cells are not explicitly present. The approach allows for very detailed analysis of small collections of excitable cells, but computational challenges limit the applicability in the presence of large collections of cells
Modeling Excitable Tissue
This open access volume presents a novel computational framework for understanding how collections of excitable cells work. The key approach in the text is to model excitable tissue by representing the individual cells constituting the tissue. This is in stark contrast to the common approach where homogenization is used to develop models where the cells are not explicitly present. The approach allows for very detailed analysis of small collections of excitable cells, but computational challenges limit the applicability in the presence of large collections of cells
Cut finite element discretizations of cell-by-cell EMI electrophysiology models
The EMI (Extracellular-Membrane-Intracellular) model describes electrical
activity in excitable tissue, where the extracellular and intracellular spaces
and cellular membrane are explicitly represented. The model couples a system of
partial differential equations in the intracellular and extracellular spaces
with a system of ordinary differential equations on the membrane. A key
challenge for the EMI model is the generation of high-quality meshes conforming
to the complex geometries of brain cells. To overcome this challenge, we
propose a novel cut finite element method (CutFEM) where the membrane geometry
can be represented independently of a structured and easy-to-generated
background mesh for the remaining computational domain.
Starting from a Godunov splitting scheme, the EMI model is split into
separate PDE and ODE parts. The resulting PDE part is a non-standard elliptic
interface problem, for which we devise two different CutFEM formulations: one
single-dimensional formulation with the intra/extracellular electrical
potentials as unknowns, and a multi-dimensional formulation that also
introduces the electrical current over the membrane as an additional unknown
leading to a penalized saddle point problem. Both formulations are augmented by
suitably designed ghost penalties to ensure stability and convergence
properties that are insensitive to how the membrane surface mesh cuts the
background mesh. For the ODE part, we introduce a new unfitted discretization
to solve the membrane bound ODEs on a membrane interface that is not aligned
with the background mesh. Finally, we perform extensive numerical experiments
to demonstrate that CutFEM is a promising approach to efficiently simulate
electrical activity in geometrically resolved brain cells.Comment: 25 pages, 7 figure
Enhancing multi-scale cardiac simulations by coupling electrophysiology and mechanics: a flexible high performance approach to cardiac electromechanics
This work focuses on the development of computational methods for the simulation of the propagation of the electrical potential in the heart and of the resulting mechanical contraction. The interaction of these two physical phenomena is described by an electromechanical model which consists of the monodomain system, which describes the propagation of the action potential in the cardiac tissue, and the equations of incompressible elasticity, which describe its mechanical response. In fully-coupled electromechanical simulations, two main computational challenges are usually identified in literature: the time integration of the monodomain system and the efficient solution of the equations of incompressible elasticity. These two are the actual bottlenecks in the realization of accurate and efficient fully-coupled electromechanical simulations. The first computational challenge arises from the discretization in time of the equations that describe the electrical activation of cardiac tissue. The monodomain system should be discretized employing both fine spatial grids and small time-steps, to capture the spatial steep gradients typical of the action potential and the behavior of the stiff gating variables, respectively. To obtain an accurate and computationally-cheap numerical solution, we propose a novel method based on coupling high-order backward differentiation formulae with high-order exponential time stepping schemes for the time integration of the monodomain system. We propose a novel quasi-Newton approach for the implicit discretization of the monodomain equation. We also compare this latter approach against a complex step differentiation-based approach. As a result, we show by means of numerical tests the accuracy of the developed strategies and how the use of high-order time integration schemes affects the simulation of post- processed quantities of clinical relevance such as the conduction velocity. The second computational challenge is due to the structure the discretization of the equations of incompressible elasticity. Due to the incompressibility constraint, the arising linear system has a saddle point structure for which standard solution methods such as multigrid or domain de- composition do not provide optimal convergence if not properly adapted. In order to overcome this problematic, we propose a segregated multigrid preconditioned solution method. The segregated approach allows to recast the saddle-point problem into two elliptic problems for which multigrid methods are shown to provide optimal convergence. The electromechanical model is employed to evaluate the effects of geometrical changes due to the contraction of the heart on simulated electrocardiograms. Finally, the effect of different electrical activations on the resulting pressure-volume loops is investigated by coupling the electromechanical model with a lumped model of the circulatory system
Algunos aspectos de la actividad eléctrica en el tejido cardiaco utilizando elementos finitos
En el siglo pasado fueron muchos los avances que se dieron en las ciencias biológicas, en particular, en la fisiología humana con la utilización de los modelos matemáticos para acercarnos a la comprensión de su fenomenología fisiológica, en especial con las ecuaciones diferenciales tanto las ordinarias como las parciales. Sólo por mencionar algunos de los principales investigadores en este campo tenemos a Hodkin y Huxley en (1952), [25, 26, 27, 29]; FitzHugh-Nagumo (1961), Hirota, Satsamo (1987), quienes han colaborado con una serie de modelos matemáticos de gran aplicabilidad en la fisiología humana, mostrando el acercamiento con otras disciplinas de las ciencias exactas para la interpretación y esclarecimiento del fenómeno afrontado. Los avances en la computación hacen posible simulaciones en tiempo real de los fenómenos y esclarecer los procesos óptimos que utiliza la fisiología para completar sus procesos funcionales [19].iv, 80 p.Contenido parcial: Fisiología del corazón -- La dinámica de las células excitables -- El modelo Hodgkin-Huxley -- La actividad eléctrica del tejido cardiaco -- El método de los elementos finitos