A three-dimensional finite element model of human atrial anatomy: new methods for cubic Hermite meshes with extraordinary vertices

Abstract

High-order cubic Hermite finite elements have been valuable in modeling cardiac geometry, fiber orientations, biomechanics, and electrophysiology, but their use in solving three-dimensional problems has been limited to ventricular models with simple topologies. Here, we utilized a subdivision surface scheme and derived a generalization of the "local-to-global" derivative mapping scheme of cubic Hermite finite elements to construct bicubic and tricubic Hermite models of the human atria with extraordinary vertices from computed tomography images of a patient with atrial fibrillation. To an accuracy of 0.6 mm, we were able to capture the left atrial geometry with only 142 bicubic Hermite finite elements, and the right atrial geometry with only 90. The left and right atrial bicubic Hermite meshes were G1 continuous everywhere except in the one-neighborhood of extraordinary vertices, where the mean dot products of normals at adjacent elements were 0.928 and 0.925. We also constructed two biatrial tricubic Hermite models and defined fiber orientation fields in agreement with diagrammatic data from the literature using only 42 angle parameters. The meshes all have good quality metrics, uniform element sizes, and elements with aspect ratios near unity, and are shared with the public. These new methods will allow for more compact and efficient patient-specific models of human atrial and whole heart physiology.

Figure 1

A schematic diagram for construction of a biatrial tricubic Hermite hexahedral mesh and intermediate bicubic Hermite surface meshes from a segmented computed tomography study.

Figure 2

One linear quadrilateral is subdivided twice by the Li-Kobbelt scheme into 16 linear quadrilaterals containing 25 vertices. The coordinates of 16 of the new vertices (red) are used to solve Eq.2 and calculate the 16 Hermite parameters for the original unsubdivided quadrilateral.

Figure 3

An extraordinary vertex lies at the interface of the right superior pulmonary vein (light grey), posterior left atrial wall (darker grey), and a third region between the right pulmonary veins (darkest grey), at a saddle point. Ensemble coordinates s₁ and s₂ (solid arrows) coincide with parametric coordinates ξ₁ and ξ₂ (broken arrows) of one element but have unit magnitude. A neighboring element has parametric coordinates ξ₁ and ξ₂ relatable to ensemble coordinates by linear transformation. Geometric continuity of adjoining patches in the one-neighborhood of an extraordinary vertex was evaluated along the shared contours at three points per contour (asterisks).

Figure 4

Anterolateral views of bicubic Hermite mesh RA-90 (A) and bilinear mesh RA-360 (B), and septal views of bicubic Hermite mesh LA-142 (C) and bilinear mesh LA-568 (D) after geometric optimization with Eq.11. Valence 5 vertices (teal), valence 3 vertices (red), and one valence 6 vertex (black) were used. The tricuspid and mitral valve rings are shaded dark grey for visual contrast. See Results, Section 3.2 for details. SVC = Superior Vena Cava; IVC = Inferior Vena Cava; RAA = Right atrial appendage; TVO = Tricuspid valve orifice; CS = Coronary sinus ostium; RSPV = Right superior pulmonary vein; RIPV = Right inferior pulmonary vein; LAA = Left atrial appendage; MVO = Mitral valve orifice.

Figure 5

Lateral views of bicubic Hermite meshes LA-142 (A) and LA-412 (B). Compared to LA-142, LA-412 cannot capture the geometric detail of the left atrial appendage (LAA), in spite of more geometric degrees of freedom. The additional extraordinary vertices in LA-142 are needed to capture the geometric shape of the LAA and preserve element quality. Cartographic projections of LA-142 and LA-412 are displayed in (C) and (D). The distorted quadrilaterals in the cartographic projection (C) become regular in LA-142 (A), whereas the regular quadrilaterals in cartographic projection (D) become distorted in LA-412 (B) as they are deformed to capture the irregular shape of the LAA. Colors in (C) and (D) are used to indicate distinct topology regions. Valence 5 vertices are teal in (A) and (B), and are indicated by asterisks (*) in (C) and (D). Valence 3 vertices are red in (A) and (B), and are indicated by the symbol ⊗ in (C) and (D). Four valence 3 vertices at the tip of the LAA are indicated by the symbol ⊕ in (C). In (A) and (B), the mitral valve is shaded dark grey for visual contrast. LAA = Left atrial appendage; MVO = Mitral valve orifice; LSPV = Left superior pulmonary vein; LIPV = Left inferior pulmonary vein.

Figure 6

Fiber orientations in a tricubic Hermite model. Images (A) and (B) are depictions of typical fiber orientations from explanted human atria, reprinted from Wang, Br. Heart J., 1995 with permission. The regions indicated by boxes are enlarged in (C) and (D), in which a qualitatively-matching fiber pattern is displayed. Each block of color indicates a region with consistent coordinate axes (a single topological region). See Results, Section 3.3 for details. Images (E) and (F) are equivalent views of the tricubic Hermite hexahedral model. The epicardial surface is colored brown and the endocardial surface is colored white. Valence 3 vertices are colored red, and valence 5 vertices are colored teal. SVC = Superior Vena Cava; BB = Bachmann's bundle; TVO = Tricuspid valve orifice; pLAw = Posterior left atrial wall; MVO = Mitral valve orifice; RIPV = Right inferior pulmonary vein; CS = Coronary sinus; IR = Intercaval region; PM = Pectinate muscles, IPB = Inferoposterior bridge.