Automatically generated, anatomically accurate meshes for cardiac electrophysiology problems

Abstract

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 on the generated grids for two preparations of high experimental relevance, a left ventricular wedge preparation, and a papillary muscle.

Fig. 1

Octree-based domain discretization. For the sake of simplicity, the basic principles are outlined by using a quadtree, the 2-D analog of a 3-D octree. (a) Quadtree structure: the tissue (gray quads, material tag “1”) is discretized at a higher resolution, whereas a coarser resolution is used to discretize the bath (white quads, material tag “0”). The solid red line represents the surface of the object. The quadtree is nonconformal, with hanging nodes arising wherever larger cells meet smaller cells. (b) Dual mesh generation: the dual mesh is derived from the primal mesh, i.e., the octree cells, by the following transformation rules. The entities vertex, edge, face, and cell in the primal mesh are translated to cell, face, edge, and vertex in the dual mesh, respectively. (c) Modified dual mesh: in 3-D, the basic dual mesh approach may also yield nonstandard element types. The insertion of nodes at the transition between larger and smaller boxes eliminates this undesirable property, resulting in the generation of “standard” elements only. (d) Element types: the four “standard” elements generated by the algorithm and an example of a “nonstandard” element that could arise in 3-D without modifications of the dual mesh are shown.

Fig. 2

Marking: the image isosurface approximating the surface of the object (red line) subdivides cells of the dual mesh along the object’s boundary. In cells that are not intersected, all vertices pertinent to the cell are marked with the tag assigned to the enclosing cell of the octree. Cutting: to better approximate the object’s surface, cells of the dual mesh that have intersecting edges with the isosurface need to be split. Edges with intersections are subdivided into quarters. If the intersection with a cell edge is located in the second or third quarter of its length, a new vertex is inserted, otherwise the closest vertex is chosen as the new boundary node. The surface of the mesh after splitting (blue line) approximates the object’s surface better than the octree (border between light and dark gray areas).

Fig. 3

Test case for determining mesh quality metrics. (Left panel) Torus geometry is defined by the radius of the generating circle r₁ =11.2, the radius of the revolving circle generating the outer hull r₂ = 5.3, and the radius generating the inner hull of the torus shell r₃ =7.3. The generating circle was chosen to lie in a plane defined by the normal vector n = [1, 1, 1]^T. (Right panel) Example of a generated mesh for H_v = 1 and H_m = 1/2.

Fig. 4

LV wedge and an anterior papillary muscle were selected by defining masks of simple geometric shapes (shown in transparent red): (A) a pie-sector shape for the wedge and (B) a cone-like shape for the papillary muscle. The segmented gray-level image substacks are shown in the rightmost panels (a and b).

Fig. 5

Numerical tests using meshes generated with the presented approach for bidomain simulations (eliciting action potential propagation via field stimulation with electrodes located in the surrounding bath) and monodomain simulations (transmembrane stimulation) for both preparations. Results for the medium mesh resolutions are shown. Locations of the stimulus (I_stim) and grounding (GND) electrodes are indicated (see text for further detail).

Fig. 6

Leftmost panels show meshes of (a) the LV wedge and (b) the left anterior papillary muscle at the medium resolutions of 50 and 35 μm, respectively. The gray rectangles outline the inset, presenting a magnified view of the portion of the mesh in the middle panels. The rightmost panels show adaptive mesh generation in the surrounding bath.

Fig. 7

Effects of reducing the target resolution from 100 μm in (a) down to 50 μm in (b) in the wedge preparation. Both the endocardial surface and a cut through the myocardium showcase the difference in structural detail of the mesh.