Stimulus Protocol Determines the Most Computationally Efficient Preconditioner for the Bidomain Equations

The efficient solution of the bidomain equations is a fundamental tool in the field of cardiac electrophysiology. When choosing a finite element discretization of the coupled system, one has to deal with the solution of a large, highly sparse system of linear equations. The conjugate gradient algorithm, along with suitable preconditioning, is the natural choice in this scenario. In this study, we identify the optimal preconditioners with respect to both stimulus protocol and mesh geometry. The results are supported by a comprehensive study of the mesh-dependence properties of several preconditioning techniques found in the literature. Our results show that when only intracellular stimulus is considered, incomplete LU factorization remains a valid choice for current cardiac geometries. However, when extracellular shocks are delivered to tissue, preconditioners that take into account the structure of the system minimize execution time and ensure mesh-independent convergence.

[1]  C. Henriquez.  Simulating the electrical behavior of cardiac tissue using the bidomain model. , 1993, Critical reviews in biomedical engineering.

[2]  Luca F. Pavarino,et al.  Multilevel Additive Schwarz Preconditioners for the Bidomain Reaction-Diffusion System , 2008, SIAM J. Sci. Comput..

[3]  V. Simoncini,et al.  Efficient algebraic solution of reaction-diffusion systems for the cardiac excitation process , 2002 .

[4]  G Plank,et al.  Solvers for the cardiac bidomain equations. , 2008, Progress in biophysics and molecular biology.

[5]  H. Elman,et al.  Efficient preconditioning of the linearized Navier-Stokes , 1999 .

[6]  Rodrigo Weber dos Santos,et al.  Parallel multigrid preconditioner for the cardiac bidomain model , 2004, IEEE Transactions on Biomedical Engineering.

[7]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[8]  Alan Garny,et al.  A numerical guide to the solution of the bi-domain equations of cardiac electrophysiology. , 2010, Progress in biophysics and molecular biology.

[9]  W. Krassowska,et al.  The induction of reentry in cardiac tissue. The missing link: How electric fields alter transmembrane potential. , 1998, Chaos.

[10]  Gernot Plank,et al.  Solving the Coupled System Improves Computational Efficiency of the Bidomain Equations , 2009, IEEE Transactions on Biomedical Engineering.

[11]  P. C. Franzone,et al.  A PARALLEL SOLVER FOR REACTION-DIFFUSION SYSTEMS IN COMPUTATIONAL ELECTROCARDIOLOGY , 2004 .

[12]  Luca F. Pavarino,et al.  Multilevel Schwarz and Multigrid Preconditioners for the Bidomain System , 2008 .

[13]  V. Simoncini,et al.  Algebraic multigrid preconditioners for the bidomain reaction--diffusion system , 2009 .

[14]  Alejandro F. Frangi,et al.  Assessing influence of conductivity in heart modelling with the aim of studying cardiovascular diseases , 2008, SPIE Medical Imaging.

[15]  Alexander G. Fletcher,et al.  Chaste: A test-driven approach to software development for biological modelling , 2009, Comput. Phys. Commun..

[16]  Jonathan P. Whiteley,et al.  An Efficient Numerical Technique for the Solution of the Monodomain and Bidomain Equations , 2006, IEEE Transactions on Biomedical Engineering.

[17]  Xing Cai,et al.  On the Computational Complexity of the Bidomain and the Monodomain Models of Electrophysiology , 2006, Annals of Biomedical Engineering.

[18]  Natalia A Trayanova,et al.  Differences Between Left and Right Ventricular Chamber Geometry Affect Cardiac Vulnerability to Electric Shocks , 2005, Circulation research.

[19]  James P. Keener,et al.  Mathematical physiology , 1998 .

[20]  Natalia A. Trayanova,et al.  Computational techniques for solving the bidomain equations in three dimensions , 2002, IEEE Transactions on Biomedical Engineering.

[21]  Gernot Plank,et al.  Development of an anatomically detailed MRI-derived rabbit ventricular model and assessment of its impact on simulations of electrophysiological function , 2009, American journal of physiology. Heart and circulatory physiology.

[22]  Rodrigo Weber dos Santos,et al.  Algebraic Multigrid Preconditioner for the Cardiac Bidomain Model , 2007, IEEE Transactions on Biomedical Engineering.

[23]  David Gavaghan,et al.  The role of transmural ventricular heterogeneities in cardiac vulnerability to electric shocks. , 2008, Progress in biophysics and molecular biology.

[24]  Luca F. Pavarino,et al.  DECOUPLED SCHWARZ ALGORITHMS FOR IMPLICIT DISCRETIZATIONS OF NONLINEAR MONODOMAIN AND BIDOMAIN SYSTEMS , 2009 .

[25]  P. Wesseling.  An Introduction to Multigrid Methods , 1992 .

[26]  Craig S Henriquez,et al.  Electrotonic influences on action potential duration dispersion in small hearts: a simulation study. , 2005, American journal of physiology. Heart and circulatory physiology.

[27]  Bruce H Smaill,et al.  Laminar Arrangement of Ventricular Myocytes Influences Electrical Behavior of the Heart , 2007, Circulation research.