Relating strain amplitude, strain threshold and bone formation rate to exogenous forcing frequency

The literature supports the existence of a strain threshold, above which cortical bone adapts to exogenous mechanical loading by forming new bone. This strain threshold, however, varies with loading conditions, locations, waveforms, frequency etc. and there is a need to mathematically express the strain threshold in terms of these parameters. There have been several parametric, mathematical or numerical models in the literature for the cortical bones adaptation to mechanical loading, which may be already fitting some of the experimental data; however, they may not be easily and confidently derived from the first principles. To fill the gap, this work has attempted to derive the corresponding bone formation rate (BFR) rather from the first principles, namely using the energy principles. The derived model has been compared to the existing parametric models and validated with respect to the diverse experimental data available in the literature. The developed model is able to not only predict the BFR, but also helps to understand the nature and possible mathematical form of the strain threshold for cortical bones adaptation to mechanical loading.