Objective characterization of the course of the parasellar internal carotid artery using mathematical tools

Along the internal carotid artery (ICA), atherosclerotic plaques are often located in its cavernous sinus (parasellar) segments (pICA). Studies indicate that the incidence of pre-atherosclerotic lesions is linked with the complexity of the pICA; however, the pICA shape was never objectively characterized. Our study aims at providing objective mathematical characterizations of the pICA shape.

Three-dimensional (3D) computer models, reconstructed from contrast enhanced computed tomography (CT) data of 30 randomly selected patients (60 pICAs) were analyzed with modern visualization software and new mathematical algorithms. As objective measures for the pICA shape complexity, we provide calculations of curvature energy, torsion energy, and total complexity of 3D skeletons of the pICA lumen. We further measured the posterior knee of the so-called "carotid siphon" with a virtual goniometer and performed correlations between the objective mathematical calculations and the subjective angle measurements.

Firstly, our study provides mathematical characterizations of the pICA shape, which can serve as objective reference data for analyzing connections between pICA shape complexity and vascular diseases. Secondly, we provide an objective method for creating such data. Thirdly, we evaluate the usefulness of subjective goniometric measurements of the angle of the posterior knee of the carotid siphon.