Computational analyses of the morphological and topological parameters of the vascular network

CC Chih-Chiang Chang AC Alison Chu SM Scott Meyer YD Yichen Ding MS Michel M. Sun PA Parinaz Abiri KB Kyung In Baek VG Varun Gudapati XD Xili Ding PG Pierre Guihard KB Kristina I. Bostrom SL Song Li LG Lynn K. Gordon JZ Jie J. Zheng TH Tzung K. Hsiai

This protocol is extracted from research article:

Three-dimensional Imaging Coupled with Topological Quantification Uncovers Retinal Vascular Plexuses Undergoing Obliteration

**
Theranostics**,
Jan 1, 2021;
DOI:
10.7150/thno.53073

Three-dimensional Imaging Coupled with Topological Quantification Uncovers Retinal Vascular Plexuses Undergoing Obliteration

DOI:
10.7150/thno.53073

Procedure

The raw data were preprocessed in ImageJ to remove stationary noise ^{69}^{, }^{70} and the background was reduced by rolling ball background subtraction (20 pixels in radius). To provide the axial visualization or the depth of the 3-D vascular network, we applied the depth-coded plugin in ImageJ to enhance visualization of the superficial and deep capillaries. In addition, 3-D rendering and semi-automated filament tracing were performed and processed in Amira 6.1. The results of filament tracing were used to quantify both the morphological (Euler-Poincaré characteristic) and topological parameters (clustering coefficients). The Euler number, χ, of the 3-D object was defined as follows ^{56}^{, }^{58}:

where n_{0,} n_{1,} n_{2,} and n_{3} are the numbers of vertices (V), edges (E), faces (F), and the individual voxels contained in a 3-D object, respectively. In the setting of increasing connected edges, the Euler number decreases while the network becomes well-connected. We provided an analysis of the changes in Euler number in response to the morphological changes in Figure S4A. The Euler number remains the same when the new connections form a branching structure (first row of Figure S4A). In addition, the Euler number is reduced when a new connection forms loops in a reticular-like structure (second row in Figure S4A), whereas the Euler number is increased when the new connection forms disconnected objects (third row in Figure S4A). Thus, two factors are the main contributors to the Euler number: 1) the number of loops (holes), and 2) the numbers of disconnected objects ^{16}. In this context, we defined the connectivity for the vascular network as ^{59}:

In comparison to the Euler number, the clustering coefficient (C) reflects the local connectivity of individual vertices or nodes in the vasculature ^{63}^{-}^{65}^{, }^{71}. The clustering coefficient of each vertex and the average clustering coefficient C are defined as follows, respectively:

The local clustering coefficient *C*_{i} for a vertex v_{i} is given by the proportion of links between the vertices within its neighborhood divided by all possible connections between its neighbors (Figure S4B). For detailed mathematical calculation, a graph G (Equation 4) consists of a set of edges, ** E,** and vertices,

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