Fig. 5: Analysis of brain blood vessels using TDA.

a To extract the geometric features, persistent homology (PH), a main method of topological data analysis (TDA), is used. This method can evaluate structures by making virtual circles from the classified signals as a starting point. When the size of concentric circles is increased little by little, the larger circle will appear which connects them (birth time of circle). When the size of concentric circles become even larger, the appeared circles will disappear (dead time of circle). The radius of concentric circles at the birth time is defined as r = r_b, and the radius at the dead time is defined as r = r_d. These values (r_b and r_d) are plotted as persistent diagram. To compare two areas based on their features, the distance between the two point clouds shown in PD is calculated by the Sliced Wasserstein kernel. Using the distance matrix obtained by calculating the distances between the point clouds in all pairs of areas by the Sliced Wasserstein kernel as an input, the proximity between areas from a geometric point of view is visualized by multidimensional scaling (MDS). b, c MDS plots showing blood vessels in the brain. The features of α-SMA⁺ mature blood vessels (b) or VE-cad⁺ blood capillaries (c) in each brain area were extracted with PH (14 brain areas: cerebellum (CB), cortical subplate (CTXsp), fiber tracts (fiber), hippocampal formation (HPF), hypothalamus (HY), isocortex (ISO), midbrain (MB), medulla (MY), olfactory areas (OLF), pons (P), pallidum (PAL), striatum (STR), thalamus (TH), and ventricular systems (VS)). The geometric features of α-SMA⁺ mature blood vessels or VE-cad⁺ blood capillaries in each area are shown in MDS. Four independent mouse brains (female, 6–15 months) were used for analysis. Source data are provided as a Source Data file. d, e The 3D brain images with α-SMA⁺ signals (d) and VE-cad⁺ signals (e). The representative 3D images of the isocortex area, showing the featured vascular structures inside the brain, are shown.