A tri-phasic mixture model of bone resorption: theoretical investigations

In this paper, for the first time, a tri-phasic model of bone resorption using a mixture with chemical reactions is proposed. Three constituents (matrix, fluid, and cells) are considered. Conservation equations and entropy inequality are provided. The dependent variables in the constitutive equations, such as the rate of resorption, are assumed to be a function of temperature, deformation gradient, and the extent of the chemical reactions. Using constitutive equations in the second law of thermodynamics, a criterion for the thermodynamic equilibrium state is obtained which contains a bio-chemo-mechanical affinity. Using the proposed model, one can find a theoretical explanation for some clinically observed behavior of bone, for instance for the greater rate of bone resorption in cortical than cancellous bone, using the conservation equations and/or consistency requirements of continuum mixture theory. This work can be seen as a first step towards establishing a new theoretical framework which could be developed in the future by collaborative work, and with the hope of shedding some light on the multidisciplinary and complex process of bone resorption.