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Mathematical model of the process of internal adaptable reorganization cancellous and cortical human bone tissue

Akulich Yu.V.

#### Abstract:

The aim of this study is to develop a mathematical model of the process of adaptation of internal restructuring spongy and cortical bone tissues, which allows them to predict the changes in the mechanical properties of real-time are important for medical practice cases. In contrast to this topic proposed a mathematical model describes the adaptation of a particular case where there is a proportional change in the load on the bone: the components of the tensor of small elastic strains vary in proportion to one parameter. In this case, we can assume that the type of anisotropy of bone tissue in the process of restructuring remains unchanged and therefore not changed and accepted model of the structure of bone tissue. The model uses a classical constitutive relation is linear elastic porous medium obtained Hegedus and Cowin [1], the kinetic equation and the proposed internal restructuring, establishing a linear relationship between the rate of change of stiffness tensor and the scalar bone deformation stimulus activity of bone cells. As a proportionality coefficient which characterizes the sensitivity of bone adaptation to changing physiological strain, used in the equation of the fourth-rank tensor. The substantiation of the structure built by the kinetic equation by comparing the new phenomenological equation of poroelastic medium adaptable to the same equation, as follows from the theory of Hegedus and Cowin [1-3]. It is shown that under the assumption of a linear dependence of the activity of bone cells from the strain stimulus all the independent components of the tensor of adaptation as a function of the parameters of the structure can be determined analytically. For both types of bone tissue obtained expressions of these functions. The results of the test calculation of the adaptation of the homogeneous porous rod made of spongy bone tissue under uniaxial compression. The resulting time-monotonous character of the establishment of all the parameters of the adaptation process is a qualitative confirmation of the validity of the developed model.

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