Efficient and Accurate 3D Thickness Measurement in Vessel Wall Imaging: Overcoming Limitations of 2D Approaches Using the Laplacian Method

Abstract

The clinical significance of measuring vessel wall thickness is widely acknowledged. Recent advancements have enabled high-resolution 3D scans of arteries and precise segmentation of their lumens and outer walls; however, most existing methods for assessing vessel wall thickness are 2D. Despite being valuable, reproducibility and accuracy of 2D techniques depend on the extracted 2D slices. Additionally, these methods fail to fully account for variations in wall thickness in all dimensions. Furthermore, most existing approaches are difficult to be extended into 3D and their measurements lack spatial localization and are primarily confined to lumen boundaries. We advocate for a shift in perspective towards recognizing vessel wall thickness measurement as inherently a 3D challenge and propose adapting the Laplacian method as an outstanding alternative. The Laplacian method is implemented using convolutions, ensuring its efficient and rapid execution on deep learning platforms. Experiments using digital phantoms and vessel wall imaging data are conducted to showcase the accuracy, reproducibility, and localization capabilities of the proposed approach. The proposed method produce consistent outcomes that remain independent of centerlines and 2D slices. Notably, this approach is applicable in both 2D and 3D scenarios. It allows for voxel-wise quantification of wall thickness, enabling precise identification of wall volumes exhibiting abnormal wall thickness. Our research highlights the urgency of transitioning to 3D methodologies for vessel wall thickness measurement. Such a transition not only acknowledges the intricate spatial variations of vessel walls, but also opens doors to more accurate, localized, and insightful diagnostic insights.

Keywords: Laplace’s equation; convolutions; vessel wall 3D thickness measurement; vessel wall imaging.

Figure 1

(a,c) Thickness measurement for parallel (double arrows) versus non-parallel (double arrows with $??$) boundaries. (b,d) Subdividing the space between the two boundaries into smaller regions using parallel sublayers. By summing up the distances between these parallel sublayers, the thickness can be computed. The Laplace’s Equation (${\nabla }^2\varphi =0$), and Dirichlet (${\varphi |}_{{\mathcal{S}}_0}{=0.00,\varphi |}_{{\mathcal{S}}_1}=1.00$) and Neumann (${\nabla \varphi |}_{caps}=\overrightarrow{0}$) boundary conditions are shown. At any given point, the thickness is determined by the summation of the lengths of two paths: a positive path (illustrated in red) extending from the specified point to the boundary with the higher potential, and a negative path (depicted in blue) extending from the specified point to the boundary with the lower potential.

Figure 2

(First row) 2D digital phantoms with circular and elliptical ring geometries. (Second row) Solutions to the Laplace’s Equation. (Third row) Subsets of derived streamlines. (Fourth row) Thickness maps.

Figure 3

Thickness measurement errors for 2D digital phantoms as spatial resolution is increased.

Figure 4

(First column) Histology images of two carotid samples and manual annotation of ROIs. (Second column) Vessel wall segmentation masks. (Third column) Solutions to the Laplace’s Equation. (Fourth column) Contours of potential fields and subsets of corresponding streamlines. (Fifth column) Vessel wall thickness maps. (Sixth column) Distance-to-lumen maps.

Figure 5

(First–third columns) 3D masks (volumes) of three digital phantoms with isotropic spatial resolutions of 2, 5, and 10 voxel/mm, respectively. (Fourth column) Equipotential surfaces of the potential field (solutions to the Laplace’s equation) and subsets of corresponding orthogonal streamlines for a spatial resolution of 10 voxel/mm. (Fifth column) Thickness maps on the inner surfaces of the digital phantoms for a spatial resolution of 10 voxel/mm.

Figure 6

Thickness measurement errors for 3D digital phantoms as spatial resolution is increased.

Figure 7

(a,c) Axial views of 3D MERGE MR imaging data for the baseline and follow-up scans. (b,d) The corresponding vessel wall thickness maps superimposed on the axial views. (e,f) Extracted centerlines, bifurcation coordinate systems, and vessel wall thickness maps on the luminal surfaces in baseline and follow-up scans for the right (e) and left (f) carotid arteries.

Figure 8

3D vessel wall thickness measurements on luminal surfaces of the left and right carotid arteries in the baseline and follow-up scans. These regular 2D grids represent rectangular patches (derived from bifurcation coordinate system) on the luminal surfaces in an unwrapped and flattened format. Vertical red lines separate 2D grids associated with CCA and ICA branches. Each thickness value on the flattened grid is computed by averaging thickness measurements within the corresponding patch on the luminal surface.

Figure 9

Average 3D vessel wall thickness measurement, luminal radius, and the normalized wall volume index of the left and right carotid arteries along their centerlines for the baseline and follow-up scans. The solid-line and dash-line curves correspond to the baseline and follow-up scans, respectively. Carotid artery branches are color-coded, with orange indicating the CCA, magenta indicating the ICA, and cyan indicating the bifurcation region.