Computational approaches to investigating the electromechanics of healthy and diseased hearts are becoming essential for the comprehensive understanding of cardiac function. In this article, we first present a brief review of existing image-based computational models of cardiac structure. We then provide a detailed explanation of a processing pipeline which we have recently developed for constructing realistic computational models of the heart from high resolution structural and diffusion tensor (DT) magnetic resonance (MR) images acquired ex vivo. The presentation of the pipeline incorporates a review of the methodologies that can be used to reconstruct models of cardiac structure. In this pipeline, the structural image is segmented to reconstruct the ventricles, normal myocardium, and infarct. A finite element mesh is generated from the segmented structural image, and fiber orientations are assigned to the elements based on DTMR data. The methods were applied to construct seven different models of healthy and diseased hearts. These models contain millions of elements, with spatial resolutions in the order of hundreds of microns, providing unprecedented detail in the representation of cardiac structure for simulation studies.

Significant advancements in imaging technology and the dramatic increase in computer power over the last few years broke the ground for the construction of anatomically realistic models of the heart at an unprecedented level of detail. To effectively make use of high-resolution imaging datasets for modeling purposes, the imaged objects have to be discretized. This procedure is trivial for structured grids. However, to develop generally applicable heart models, unstructured grids are much preferable. In this study, a novel image-based unstructured mesh generation technique is proposed. It uses the dual mesh of an octree applied directly to segmented 3-D image stacks. The method produces conformal, boundary-fitted, and hexahedra-dominant meshes. The algorithm operates fully automatically with no requirements for interactivity and generates accurate volume-preserving representations of arbitrarily complex geometries with smooth surfaces. The method is very well suited for cardiac electrophysiological simulations. In the myocardium, the algorithm minimizes variations in element size, whereas in the surrounding medium, the element size is grown larger with the distance to the myocardial surfaces to reduce the computational burden. The numerical feasibility of the approach is demonstrated by discretizing and solving the monodomain and bidomain equations Correspondence to: Gernot Plank, gernot.plank@medunigraz.at. on the generated grids for two preparations of high experimental relevance, a left ventricular wedge preparation, and a papillary muscle. NIH Public Access

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Electrical activity in cardiac tissue can be described by the bidomain equations whose solution for large scale simulations still remains a computational challenge. Therefore, improvements in the discrete formulation of the problem which decrease computational and/or memory demands are highly desirable. In this study, we propose a novel technique for computing shape functions of finite elements. The technique generates macro finite elements (MFEs) based on the local decomposition of elements into tetrahedral sub-elements with linear shape functions. Such an approach necessitates the direct use of hybrid meshes composed of different types of elements. MFEs are compared to classic standard finite elements with respect to accuracy and RAM memory usage under different scenarios of cardiac modeling including bidomain and monodomain simulations in 2D and 3D for simple and complex tissue geometries. In problems with analytical solutions, MFEs displayed the same numerical accuracy of standard linear triangular and tetrahedral elements. In propagation simulations, conduction velocity and activation times agreed very well with those computed with standard finite elements. However, MFEs offer a significant decrease in memory requirements. We conclude that hybrid meshes composed of MFEs are well suited for solving problems in cardiac computational electrophysiology.

In the engineering literature, the so‐called Global Extraction Element‐by‐Element (GE‐EBE) Methods have been successfully used for solving real‐life mechanical problems, especially, in 3D [2], although the EBE preconditioners C haven't any asymptotically preconditioning effects on the stiffness matrix K, i. e. the spectral condition number k(C‐1K) behaves like O(h‐2), say, for elasticity problems. However, the method has quite good scaling properties. From a theoretical point of view, the GE‐EBE preconditioner is nothing else than an Additive Schwarz Method (ASM) preconditioner generated by an overlapping splitting of the finite element space into subspaces spanned by the element ansatz functions. The authors propose to use the GE‐EBE methods and their patch counterpart (GE‐PBP) either as smoothers, especially, in their multiplicative version, or as preconditioners in their multilevel additive version. In the multilevel version, such preconditioners are asymptotically optimal. The GE‐PBP smoother is especially suited for the use in Algebraic Multi‐Grid (AMG) Methods based on matrix graph coarsening strategies. The numerical results presented confirm the good smoothing properties of GE‐PBP smoothers as well as the preconditioning properties of multilevel additive GE‐PBP preconditioners. Finally, the authors discuss parallelization aspects.

The efficient solution of large scaled finite element (f. e.) equations from industrial problems is a challenging task. Typically, such problems are quite complex due to the multiscale behavior of the geometry and the data. This is the reason why standard algorithms for mesh generation and solution are inefficient or can even fail. In this context, robust methods for treating multiscale problems are needed. The mesh generator should produce rather fine meshes catching the geometry with an asymptotically optimal complexity. Such a generator is proposed in two and tree dimensions. Usually, the meshes for geometries with microscales contain a large number of nodes. Thus, fast solvers for the f. e. equations are needed, where no hierarchical mesh structure is available. Such solvers are presented for the Potential and the Linear Elasticity Problems in two and three space dimensions within the framework of Algebraic Multigrid Methods. Together, these algorithms form an efficient tool for the Numerical Simulation of Multiscale Problems and can be easily integrated in existing f. e. codes including commercial codes.