Strategies to extend geometric tortuosity from paths to microstructures. Illustrations of various ways to characterize whole microstructures based on geometric tortuosity definition (Eq. (8)). (a) path of minimal length connecting entry (green) and exit (red) planes (Eq. (2)), (b) average of paths connecting entry and exit planes (Eq. (3)), (c) point tortuosities, i.e., tortuosity map, assessed by forward and backward propagations between two opposite planes (Eq. (4)), (d) M-tortuosity averaging local tortuosities between random locations.