A novel mathematical model of bone remodelling cycles for trabecular bone at the cellular level

Abstract

After an initial phase of growth and development, bone undergoes a continuous cycle of repair, renewal and optimisation by a process called remodelling. This paper describes a novel mathematical model of the trabecular bone remodelling cycle. It is essentially formulated to simulate a remodelling event at a fixed position in the bone, integrating bone removal by osteoclasts and formation by osteoblasts. The model is developed to construct the variation in bone thickness at a particular point during the remodelling event, derived from standard bone histomorphometric analyses. The novelties of the approach are the adoption of a predator-prey model to describe the dynamic interaction between osteoclasts and osteoblasts, using a genetic algorithm-based solution; quantitative reconstruction of the bone remodelling cycle; and the introduction of a feedback mechanism in the bone formation activity to co-regulate bone thickness. The application of the model is first demonstrated by using experimental data recorded for normal (healthy) bone remodelling to predict the temporal variation in the number of osteoblasts and osteoclasts. The simulated histomorphometric data and remodelling cycle characteristics compare well with the specified input data. Sensitivity studies then reveal how variations in the model's parameters affect its output; it is hoped that these parameters can be linked to specific biochemical factors in the future. Two sample pathological conditions, hypothyroidism and primary hyperparathyroidism, are examined to demonstrate how the model could be applied more broadly, and, for the first time, the osteoblast and osteoclast populations are predicted for these conditions. Further data are required to fully validate the model's predictive capacity, but this work shows it has potential, especially in the modelling of pathological conditions and the optimisation of the treatment of those conditions.