Conformal mapping of carotid vessel wall and plaque thickness measured from 3D ultrasound images

Abstract

Measurements of vessel-wall-plus-plaque thickness (VWT) from 3D carotid ultrasound have been shown to be sensitive to the effect of pharmaceutical interventions. Since the geometry of carotid arteries is highly subject-specific, quantitative comparison of the distributions of point-wise VWT measured for different patients or for the same patients at different ultrasound scanning sessions requires the development of a mapping strategy to adjust for the geometric variability of different carotid surface models. In this paper, we present an algorithm mapping each 3D carotid surface to a 2D carotid template with an emphasis on preserving the local geometry of the carotid surface by minimizing local angular distortion. The previously described arc-length scaling (AL) approach was applied to generate an initial 2D VWT map. Using results established in the quasi-conformal theory, a new map was computed to compensate for the angular distortion incurred in AL mapping. As the 2D carotid template lies on an L-shaped non-convex domain, one-to-one correspondence of the mapping operation was not guaranteed. To address this issue, an iterative Beltrami differential chopping and smoothing procedure was developed to enforce bijectivity. Evaluations performed in the 20 carotid surface models showed that the reduction in average angular distortion made by the proposed algorithm was highly significant (P = 2.06 × 10^-5). This study is the first study showing that a bijective conformal map to a non-convex domain can be obtained using the iterative Beltrami differential chopping and smoothing procedure. The improved consistency exhibited in the 2D VWT map generated by the proposed algorithm will allow for unbiased quantitative comparisons of VWT as well as local geometric and hemodynamic quantities in population studies.

Keywords: 2D carotid template; 2D vessel-wall-plus-plaque thickness (VWT) map; 3D ultrasound; Carotid atherosclerosis; Conformal mapping.