Overview
Computational geometry describes how to represent smooth biological shapes in computationally efficient manners. Along with representing a given shape, we want to compute geometric descriptors accurately such as curvature and even geodesics.
Technical Formulation
We represent geometries through analysis suitable splines, so the mesh created can be used directly in a computational pipeline. More specifically, we utilize truncated hierarchical (TH) B-Splines because of their local refinement allows for higher geometric resolutions with fewer degrees of freedom. Care must be taken when constructing spline representations because non-trivial topologies — such as closed ventricular surfaces — cannot be represented by a single patch, requiring multi-patch constructions that must be carefully assembled to avoid gaps or overlaps.
Clinical Application
The clinical applications are representing patient-specific geometries to run simulations on. The local refinement of THB-Splines allow for capturing local anatomical variations across patients. The smoothness of splines is very important for running clinical simulations because sharp angles typically show stress concentrations, which may alter clinical-decisions even though its really an artifact of the mesh.
References & Resources
- The NURBS Book - The definitive reference for NURBS construction and algorithms.
- THB-splines: The truncated basis for hierarchical splines - Giannelli, Jüttler, Speleers. Core mathematical construction.
- Differential Geometry and Lie Groups A Computational Perspective - Basics of differential geometry from a computational viewpoint.