Active Free Surface Density Maps

Percolation problems were occupied to many physical problems after their establishment in 1957 by Broadbent and Hammersley. They can be used to solve complex systems such as bone remodeling. Volume fraction method was adopted to set some algorithms in the literature. However, different rate of osteoporosis could be observed for different microstructures which have the same mass density, mechanical stimuli, hormonal stimuli and nutrition. Thus it was emphasized that the bone might have identical porosity with different specific surfaces. Active free surface density of bone refers the used total area for its effective free surface. The purpose of this manuscript is to consolidate a mathematical approach which can be called as "active free surface density maps" for different surface patterns and derive their formulations. Active free surface density ratios were calculated for different Archimedean lattice models according to Helmholtz free energy and they were compared with their site and bond percolation thresholds from the background studies to derive their potential probability for bone remodeling.