## Connectivity

The number of connected structures in a network can be determined by calculating the Euler characteristic. Trabecular bone is one such network, and its connectivity density (Conn.D) can be calculated by dividing the connectivity estimate by the volume of the sample. The algorithm in BoneJ's Connectivity uses voxel neighbourhoods to calculate the Euler characteristic of the volume and adjusts this to give the contribution of the volume to the connectivity of the structure it was cut from. An assumption is made that there is only one particle in the foreground; to achieve this, run Purify prior to Connectivity. A warning will be displayed if connectivity is negative, which is an unnatural result.

Connectivity works as follows:

1. Purify stack to contain only 1 connected bone phase and 1 connected marrow phase (Purify_ plugin)
2. Iterate through stack, calculating Euler characteristic for each bone voxel (δχ)
3. Calculate the Euler characteristic of the bone sample as though it is floating in space (χ = ∑δχ)
4. Calculate the bone sample's contribution to the Euler characteristic of the bone it was connected to (Δ(χ)) by checking the intersections of voxels and stack edges
5. Calculate connectivity β1 = 1 - Δχ
6. Calculate connectivity density Conn.D = β1 / stack volume

Odgaard A, Gundersen HJG (1993) Quantification of connectivity in cancellous bone, with special emphasis on 3-D reconstructions. Bone 14: 173-182. doi:10.1016/8756-3282(93)90245-6.

Toriwaki J, Yonekura T (2002) Euler number and connectivity indexes of a three dimensional digital picture. Forma 17: 183-209.

• Input
• A binary stack
• Output
• Image: the image title
• Euler ch.: Euler characteristic of the sample as though floating in space (χ)
• Δ(χ): the bone sample's contribution to the Euler characteristic of the bone to which it was connected
• Connectivity: The connectivity of the image (~ number of trabeculae)
• Conn.D: Connectivity density (~ number of trabeculae per unit volume)  This file last modified 1134hrs 13 March 2017 © Michael Doube 2004-2020 :: Designed to be interoperable and standards-compliant