Cancellous bone tissue remodelling: mathematical modelling

Bone tissue is a heterogeneous, anisotropic material and consists of compact and cancellous bone tissues. The structure of cancellous bone tissue in different parts of the skeleton is not the same. It conforms to Wolff’s law: it aims to become optimal for the loading which acts on the corresponding bone; the bone remodelling by means of osteosynthesis and resorption mechanisms. The modern problems of biomechanics demand research of the history of formation of bone structures in the course of time at both physiological and pathological loadings. The mandible is one of the most liable to external and internal bone changes. It is liable to physiological changes that occur during the ontogenesis of the organism. 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 modelling of the cancellous bone tissue behavior in the human maxillodental system is one of the topical problems of biomechanics and medicine. Structural features of cancellous bone tissue 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. The phenomenological Cowin’s equations are chosen and analyzed in detail as such ones. An initial boundary value problem of the cancellous bone tissue remodelling is stated. The solution of this problem allows us to trace changes in the stress-strain state at the trabecular structure formation according to Wolff’s law. The effective numerical algorithm allows us 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 the process of structure formation, which follows from Wolff’s law.