[Mathematical model of the bone callus tissue obtained during distraction osteosynthesis]

After some surgical interventions it's need to regenerate a part of bone. The new bone formation in the external fixation apparatus is the result of elongation and maturation of the bone callus, originated in pre-destructive period between bone fragments. The mathematical model of bone callus is proposed. Its main structural elements are osteogeneous and cartilage cells, osteoblasts, matrix, blood vessels. The derived equations take account of matrix production, dependent on the stress in tissue, and transformation of one kind of cells into another in accordance with O2 concentration is more or less than some critical value. The material of the callus is considered as a growing elastic body with growth coefficients, dependent on the structural tensor and Ca concentration. The structural tensor represents the anisotropy of fibrils orientation that take place during deformation. The concentrations of O2 and Ca are connected with the density of blood vessels whose propagation is related to the diffusion of a hypothetical substance. The formulated boundary problem allows to define changes in the growing zone of the callus and to define its size as a function of the rate of elongation.