An invertible computer model for bone's adaptation to mechanical environment

Abstract

Bone adapts to its strain environment. There are several existing in vivo and in silico models of bone adaptation. These models provide some insight into the mechanisms of bone adaptation. For example, induced strain, strain gradients, and interstitial fluid flow are considered osteogenic stimuli for the purpose of computing bone growth. However, none of these existing in silico models can predict loading conditions for a desired or prescribed bone growth. Also, the reported models have been validated using mostly only one kind of loading conditions. This thesis presents a new invertible bone adaptation model that predicts bone adaptation to various loading conditions and can be easily inverted to predict loading conditions for desired bone growth. The model has been developed in three steps: (i) testing of existing models, (ii) development of new model for predicting overall Bone Formation Rate (BFR), and (iii) development of new model for predicting site-specific Mineral Apposition Rate (MAR). Accordingly, Carter’s (1987) bone adaptation models have been tested using different loading conditions first to identify its strengths and limitations. These models considered strain energy density, fatigue damage accumulation, or induced stress as osteogenic stimuli for predicting overall bone growth at a bone section. This test identified the fatigue damage accumulation as relatively better stimulus for predicting overall bone growth. However, Carter’s fatigue model could not predict enhanced bone growth for rest-inserted loading cases in comparison with continuous loading cases. This anomaly has been addressed in the second step by incorporating viscoelastic property of bone. The Kelvin-Voigt model of viscoelasticity has been considered assuming approximate incorporation of the effect of the viscous fluid flow within the bone matrix. The stimulus of the new model has been derived using fatigue failure theory. Unlike Carter’s model, the new model considers the peak-to-trough induced strain amplitude rather than the amplitude of the applied stress. Moreover, the total number of loading cycles has been considered rather than the number of loading cycles per day for deriving the mechanical stimulus. The error between the average bone formation rate (BFR) predicted by the model and that according to the experimental data available in the literature, has been minimized by optimization of the model parameters. The new model is quickly invertible, e.g., it can predict the peak strain required for the desired or prescribed bone formation rate (BFR) at the tibial diaphysis. The overall bone adaptation model has been further updated for predicting the site-specific lamellar bone adaptation by simulating the inter-cellular flow of the secondary messenger Ca2+ ions within an idealized lacunar-canalicular network of osteoblasts and osteocytes. The ion concentration within an osteocyte in the network has been considered proportional to the corresponding mechanical stimulus. The Ca2+ ions are considered fully utilized for new bone formation by osteoblasts and hence the concentration of the secondary messenger has been taken as zero. Fick’s first law of diffusion has been used to compute ion flow rate to the osteoblasts. Mineral apposition rate (MAR) at each osteoblast is assumed to be proportional to the stimulus flow rate at that osteoblast. The developed model predicts site-specific lamellar bone for various loading conditions, distribution of which is not significantly different from the experimental results. The model has then been inverted to predict loading conditions for prescribed new bone formation. The results confirm to the experimental results existing in the literature.