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

Abstract

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 bone’s 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 bone’s adaptation to mechanical loading.

Predicting Cortical Bone Resorption in Disuse Condition Caused by Transient Muscle Paralysis

“…The developed disuse model showed the linear relation between MRR and the √Ø (i.e., the square root of the dissipation energy density), which is in accordance with the earlier work by Sugiyama et al (Sugiyama et al, 2012). Note that the dissipation energy density is proportional to the square of strain and, hence, MRR being proportional to √Ø is akin to bone formation rate (BFR) being proportional to the strain (Prasad, 2023). A work by Singh et al (Singh et al, 2023) has also shown a similar kind of trend, i.e., the rate of new bone formation is directly proportional to the √Ø.…”

“…The axial load associated with physiological loading condition was obtained from the literature (Prasad et al 2010). In accordance with the literature, the periosteal and endocortical bone surfaces were assumed to be completely impermeable and permeable, respectively, (Singh, Singh, and Prasad 2023) (Gatti et al 2021) (Fan et al 2016) implemented through zero-flow and zero-pressure boundary conditions respectively (Fig. 2(b)).…”

“…The finite element problem was solved using the commercial software ABAQUS (Dassault System). The existing literature assumes that the periosteal and endocortical surfaces of the bone were impermeable and permeable, respectively (Singh, Singh, and Prasad 2023) (Fan et al 2016). To incorporate such assumptions, a zero-pressure boundary condition was applied at the endocortical surface, whereas the fluid velocity normal to the periosteal surface was prescribed as zero Fig 2(b).…”

“…According to Kumar et al,(Kumar, Jasiuk, and Dantzig 2011) the dissipation energy density ( DED ), which is a function of fluid velocity, can be calculated as: Where n p is the LCN porosity, v fl is the fluid velocity. To capture the effect of the non-local behavior of osteocytes, a spherical zone of influence (ZOI) was considered (as shown in Fig 6) similar to the earlier study by Singh et al (Singh, Singh, and Prasad 2023). Considering the ZOI, the dissipation energy density was calculated using Eq.…”

“…The dissipated energy, which is a function of fluid velocity, is assumed to be a stimulus to the mechano-sensing cell, i.e., osteocyte (Kumar, Jasiuk, and Dantzig 2011). It is evident from the literature that in the case of a loaded bone, the resultant fluid velocity is higher at the endocortical surface while negligible at the periosteal surface (Singh, Singh, and Prasad 2023) due to the permeable and impermeable nature of the respective surfaces (Fan et al 2016). Botox-induced calf muscle paralysis pauses the locomotion, resulting in zero strain on bone and, thus, no fluid flow inside the LCN Fig 1(d).…”