Mathematical modelling of trabecular bone tissue remodelling under load

V. Tverier, A. Kichenko, Y. Nyashin, V. Lokhov

Abstract: Bone tissue is a heterogeneous, anisotropic material; structural features of trabecular bone tissue can be described by means of the fabric tensor. Biomechanical modelling tasks demand to study the history of the formation of bone structures in time under both physiological and pathological loadings. This is possible to implement if there is both a constitutive relation which allows us to connect the stress tensor, the strain tensor, and the fabric tensor, and evolution equations which allows us to describe the evolution of the fabric tensor and bone density. These equations and relations for trabecular bone are used; an initial boundary value problem on the trabecular bone tissue remodelling is stated. The solution algorithm is developed and trabecular bone tissue evolution is demonstrated on the model example when the stress–strain state is changed. The computing experiment is carried out to determine coefficients in evolution equations; it was considered that bone tissue adaptation should take place for 160 days. The results demonstrate different character of influence of changes of loading conditions on process of structure formation which according to Wolff’s law.

Keywords: Biomechanical modelling; bone tissue remodelling; bone tissue structure; constitutive relation; evolution equation; fabric tensor; state of homeostasis (physiological equilibrium); trabecular (cancellous) bone tissue; Wolffs law

Date published: 2015-12-10