A Framework for the Automatic Generation of Surface Topologies for Abdominal Aortic Aneurysm Models

Abstract

Patient-specific abdominal aortic aneurysms (AAAs) are characterized by local curvature changes, which we assess using a feature-based approach on topologies representative of the AAA outer wall surface. The application of image segmentation methods yields 3D reconstructed surface polygons that contain low-quality elements, unrealistic sharp corners, and surface irregularities. To optimize the quality of the surface topology, an iterative algorithm was developed to perform interpolation of the AAA geometry, topology refinement, and smoothing. Triangular surface topologies are generated based on a Delaunay triangulation algorithm, which is adapted for AAA segmented masks. The boundary of the AAA wall is represented using a signed distance function prior to triangulation. The irregularities on the surface are minimized by an interpolation scheme and the initial coarse triangulation is refined by forcing nodes into equilibrium positions. A surface smoothing algorithm based on a low-pass filter is applied to remove sharp corners. The optimal number of iterations needed for polygon refinement and smoothing is determined by imposing a minimum average element quality index with no significant AAA sac volume change. This framework automatically generates high-quality triangular surface topologies that can be used to characterize local curvature changes of the AAA wall.

Appendix

3-D shape indices.

Nomenclature	Name	Equation
IPR	Isoperimetric ratio	${\text{IPR}} = {\frac{S}{{V^{{{\raise0.7ex\hbox{$2$} \!\mathord{\left/ {\vphantom {2 3}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$3$}}}} }}}$
NFI	Non-fusiform index	$NFI = {\frac{{{\frac{S}{{V^{{{\raise0.7ex\hbox{$2$} \!\mathord{\left/ {\vphantom {2 3}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$3$}}}} }}}}}{{{\frac{{S_{fusiform} }}{{V_{fusiform}^{{{\raise0.7ex\hbox{$2$} \!\mathord{\left/ {\vphantom {2 3}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$3$}}}} }}}}}} = {\frac{IPR}{{IPR_{fusiform} }}}$

Second-order curvature-based indices (calculation described in detail in Martufi et al. 9)

Nomenclature	Name	Equation
GAA	Area averaged Gaussian curvature	$GAA = {\frac{{\sum_{all\;elements} {K_{j} } S_{j} }}{{\sum_{all\;elements} {S_{j} } }}}$
MAA	Area averaged Mean curvature	$MAA = {\frac{{\sum_{all\;elements} {M_{j} } S_{j} }}{{{{\sum_{all\;elements} {S_{j} } }} }}}$
GLN	L2 norm of the Gaussian curvature	$GLN = {\frac{1}{4\pi }}\sqrt {\sum_{all\;elements} {S_{j} } \cdot \sum_{all\;elements} {\left( {K_{j}^{2} S_{j} } \right)} }$
MLN	L2 norm of the Mean curvature	$MLN = {\frac{1}{4\pi }}\sqrt {\sum_{all\;elements} {\left( {M_{j}^{2} S_{j} } \right)} }$