Solving Optimal Control Problem of Monodomain Model Using Hybrid Conjugate Gradient Methods

We present the numerical solutions for the PDE-constrained optimization problem arising in cardiac electrophysiology, that is, the optimal control problem of monodomain model. The optimal control problem of monodomain model is a nonlinear optimization problem that is constrained by the monodomain model. The monodomain model consists of a parabolic partial differential equation coupled to a system of nonlinear ordinary differential equations, which has been widely used for simulating cardiac electrical activity. Our control objective is to dampen the excitation wavefront using optimal applied extracellular current. Two hybrid conjugate gradient methods are employed for computing the optimal applied extracellular current, namely, the Hestenes-Stiefel-Dai-Yuan (HS-DY) method and the Liu-Storey-Conjugate-Descent (LS-CD) method. Our experiment results show that the excitation wavefronts are successfully dampened out when these methods are used. Our experiment results also show that the hybrid conjugate gradient methods are superior to the classical conjugate gradient methods when Armijo line search is used

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

BPX preconditioners for the Bidomain model of electrocardiology

The aim of this work is to develop a BPX preconditioner for the Bidomain model of electrocardiology. This model describes the bioelectrical activity of the cardiac tissue and consists of a system of a non-linear parabolic reaction\u2013diffusion partial differential equation (PDE) and an elliptic linear PDE, modeling at macroscopic level the evolution of the transmembrane and extracellular electric potentials of the anisotropic cardiac tissue. The evolution equation is coupled through the non-linear reaction term with a stiff system of ordinary differential equations, the so-called membrane model, describing the ionic currents through the cellular membrane. The discretization of the coupled system by finite elements in space and semi-implicit finite differences in time yields at each time step the solution of an ill-conditioned linear system. The goal of the present study is to construct, analyze and numerically test a BPX preconditioner for the linear system arising from the discretization of the Bidomain model. Optimal convergence rate estimates are established and verified by two- and three-dimensional numerical tests on both structured and unstructured meshes. Moreover, in a full heartbeat simulation on a three-dimensional wedge of ventricular tissue, the BPX preconditioner is about 35% faster in terms of CPU times than ILU(0) and an Algebraic Multigrid preconditioner

Integrated Heart - Coupling multiscale and multiphysics models for the simulation of the cardiac function

Mathematical modelling of the human heart and its function can expand our understanding of various cardiac diseases, which remain the most common cause of death in the developed world. Like other physiological systems, the heart can be understood as a complex multiscale system involving interacting phenomena at the molecular, cellular, tissue, and organ levels. This article addresses the numerical modelling of many aspects of heart function, including the interaction of the cardiac electrophysiology system with contractile muscle tissue, the sub-cellular activation-contraction mechanisms, as well as the hemodynamics inside the heart chambers. Resolution of each of these sub-systems requires separate mathematical analysis and specially developed numerical algorithms, which we review in detail. By using specific sub-systems as examples, we also look at systemic stability, and explain for example how physiological concepts such as microscopic force generation in cardiac muscle cells, translate to coupled systems of differential equations, and how their stability properties influence the choice of numerical coupling algorithms. Several numerical examples illustrate three fundamental challenges of developing multiphysics and multiscale numerical models for simulating heart function, namely: (i) the correct upscaling from single-cell models to the entire cardiac muscle, (ii) the proper coupling of electrophysiology and tissue mechanics to simulate electromechanical feedback, and (iii) the stable simulation of ventricular hemodynamics during rapid valve opening and closure

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