Biomechanical modelling of trabecular bone tissue in remodelling equilibrium

In the 19th century, J. Wolff noted that the bone of a healthy person or animal adapts to the loads under which it is placed. At the tissue level, weakly porous regions of compact tissue and strongly porous regions of the cancellous (spongy) tissue are seen in the bone. It is known that in the cancellous bone tissue the adaptation mechanism is realized by aligning the trabeculae (bone bunches forming a solid matrix) along the principal stress trajectories. When the cancellous tissue reaches the optimal structure for the existing loading in the local area, the bone passes into a state of equilibrium (homeostasis). In his works, S. Cowin proposed to describe the position of the trabeculae at each discrete instant of time by the principal trajectories of the fabric tensor calculated from the solution of the system of rate equations. On the basis of the proposed relationships, many problems of the dynamics of cancellous bone tissue are solved, the results of which are functions of a certain quantity that depends on time. For example, the change in porosity, the components of the deviator of the fabric tensor. In this work, rate equations are used to determine the structure, stress state or elastic properties of a bone sample in equilibrium. The mathematical statement is presented by the anisotropic elasticity problem. All numerical calculations were performed using the ANSYS Mechanical APDL software on example of a rectangular plate tension. The results of the problem obtained by the analytical approach and the finite element method are compared. It is assumed that the obtained relationships can be useful for predicting of the stress-strain state in the bone of known structure, or vice versa, to predict the optimal bone structure for the given loading conditions.