Derivation, validation, and prediction of loading-induced mineral apposition rates at endocortical and periosteal bone surfaces based on fluid velocity and pore pressure

The capacity of bone to optimize its structure in response to mechanical loads has been widely observed. The mechanical load acting on a bone at the macroscopic level influences the bone cells, particularly osteocytes within the lacunae canalicular network (LCN). Osteocytes are responsive to a range of physical signals, including strain, interstitial fluid flow, and pore pressure. However, physiological tissue strain is known to be typically smaller than that required to directly induce bone formation. On the other hand, as per evidence provided by this study from the literature, models based on fluid flow alone cannot simultaneously predict bone formation at both the periosteal and endocortical surfaces. This suggests that another component of the osteocyte's mechanical environment, such as pore pressure, may play an essential role in bone adaptation, either alone or in combination with other stimuli, such as tissue strain and/or interstitial fluid flow. In vitro experiments have also confirmed that osteocytes respond to cyclic pore pressure and, thus, have a mechanism to sense the pressure, possibly because of its viscoelasticity. In this work, dissipation energy density, being irreversible work done per unit volume, has been successfully used as a greater stimulus to incorporate all of the parameters of mechanical environments of the LCN, such as waveforms of both fluid velocity and pore pressure, number of loading cycles. Mineral Apposition Rate (MAR) has also been mathematically derived to be proportional to the square root of the dissipation energy density minus its reference value. A hypothesis is accordingly proposed and successfully tested/validated for both endocortical and periosteal surfaces with respect to an in-vivo study on mouse tibia available in the literature. The constant of proportionality and the reference/threshold value of the dissipation energy density are determined through a nonlinear curve fitting. The mathematical/computational method thus developed is then successfully used to predict MAR at both endocortical and periosteal surfaces induced by a different loading condition. Computational implementation of the mathematical model has been done through a poroelastic finite element analysis of bone, where bone is assumed to be porous and filled with fluid, with a boundary condition that the periosteum is impermeable to the fluid and the endosteal surface maintains a reference zero pressure. This work also provides evidence for these assumptions to be true based on the state-of-the-art literature on related experimental studies. The currently developed model shows that the bone uses these conditions (assumptions) to its advantage, as the greater stimulus, i.e., the dissipation energy due to both fluid flow and pore pressure, are of a similar order at both the surfaces, and hence osteogenesis of the same order at both the surfaces. As a bottom line, the resulting model is the first of its kind as it has been able to correctly predict MAR at both endocortical and periosteal surfaces. This study thus significantly advances the modeling of cortical bone adaptation to exogenous mechanical loading.