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 n0, n1, n2, and n3 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 Ci for a vertex vi 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, V, and an edge, eij, connecting vertex vi with vertex vj (Figure S4B). We define the neighborhood set Ni of vertex vi for its directly connected neighbors (Equation 5). By defining ki=| Ni | as the number of neighbors in set Ni for a vertex vi, the clustering coefficient Ci for vertex vi can be calculated (Equation 6).

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