The bone tissue in different parts of the skeleton conforms to Wolff’s law: it aims to become optimal for the loading which acts on the corresponding bone; the bone is remodelling by means of osteosynthesis and resorption mechanisms. The modern problems of biomechanics demand research on the history of formation of bone structures in the course of time at both physiological and pathological loadings. Ever changing loadings of different nature have influence on development and functioning of the trabecular bone tissue. The mandible is one of the most liable to external and internal changes bones. Very often one has to deal with pathological changes caused by incorrect loading of different regions of bone tissue due to dysfunction of a dentition, a temporomandibular joint and so on. For example, the Popov-Godon’s syndrome which connects with tooth loss is accompanied by pathological remodelling of the surrounding bone tissue. Thus, the mathematical modeling of the cancellous bone tissue behavior in the human maxillodental system is one of the most topical problems of biomechanics and medicine. Trabecular bone tissue is a heterogeneous, porous, anisotropic material. Heterogeneity of spongy structure can be described by methods of quantitative stereology. At the same time, structural features of the trabecular bone can be described by means of the fabric tensor. This is possible to implement if there is both a constitutive relation which connects the stress tensor, the fabric tensor, and the strain tensor, and kinetic equations which describe the evolution of the fabric tensor and bone density. An initial boundary value problem on the trabecular bone tissue remodelling is stated. The effective numerical algorithm allowing to solve the problem is developed. This algorithm is implemented as a complex of problem-oriented programs. Verification of the model and identification of its parameters are carried out. All numerical calculations are performed using the ANSYS software. Trabecular bone tissue evolution is demonstrated on the set of model examples when the stress–strain state is changed. The results demonstrate different character of influence of changes of loading conditions on process of structure formation which follows from Wolff’s law.
MATHEMATICAL MODELING OF CANCELLOUS BONE TISSUE ADAPTATION APPLIED TO THE HUMAN MAXILLODENTAL SYSTEM
