A bone remodelling model based on generalised thermodynamic potentials and optimisation applied to a trabecula with cyclic loading

BACKGROUND: Bone diseases caused by an imbalance of bone turnover represent a major public health concern worldwide. Studies involving bone remodelling mechanisms can assist in the treatment of osteoporosis, osteopenia and in cases of fractures. In recent decades several authors have developed bone remodelling models. AIM: The aim of this study is to propose a model based on the thermodynamic framework to describe the process of bone remodelling. A secondary aim is to model a trabecula subjected to cyclic loading and calibrate the model with experimental data. METHODS: Thermodynamic potentials are used to generate the functions of state based on internal scalar variables. The evolution of the variables in time is determined by dissipation potentials, which are created through the use of convex analysis. Constitutive equations are solved with mathematical programming algorithms and the numerical implementation of this theory uses the Finite Elements Method for spatial discretization. RESULTS: The proposed theory was applied to a one-dimensional example, and two situations an undamaged material and an initially damaged material were simulated. The one-dimensional example shows a microscopic view of a trabecula under the influence of a growing load cycle throughout 1200 days. This dynamic process may represent the rehabilitation of an athlete, starting with light exercises up to a very heavy physical activity. CONCLUSIONS: The model was able to represent one bone remodelling cycle in the trabecula. Although it is not yet possible to obtain an experimental curve of a traction test in vivo, the in silico model showed a process of damage that is similar to the static test of the literature. The results also suggest a modification in the equation adopted for the Helmholtz potential shown here. This study presents a consistent thermodynamic formalism for bone remodelling, which may allow further contributions as the incorporation of chemical reactions, mass transference and anisotropic damage.

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