Kinetic equations of trabecular bone tissue remodelling in Ilyushin's space

Abstract

In the 19th century, Wolff noted that the bone of a healthy person or animal adapts to the stresses to which it is subjected. It is known that in the trabecular bone tissue, the adaptation mechanism is implemented through the alignment of trabeculae (bone beams) along the lines of action of the principal stresses. When the trabecular tissue reaches the optimal structure for a specific load in the local area, the bone goes into a state of equilibrium (homeostasis). In his works, Cowin proposed to describe the position of trabeculae at each moment in time by the principal directions of the fabric tensor, which are sought from the solution of the system of kinetic equations proposed by him for each component of this tensor. Unfortunately at present, it is impossible to experimentally trace the evolution of the tensor of the structure in vivo, that is, to get the opportunity to estimate the values of the constants of the kinetic equations. The paper proposes a kinetic equation constructed in the Ilyushin's deviator space directly reflecting the law of Wolff. The equation has one material constant. The kinetic equation is consistent with the Cowin's constitutive equation, which allowed us to determine not only the vector, but also the scalar properties of the fabric tensor. The verification of the proposed equations using the example of structure remodelling for the problem of uniform compression showed good accuracy.