Bone surface remodelling in the diaphysis using strain energy density as controlled variable: Theoretical predictions

Theoretical predictions of surface bone remodelling in the diaphysis of a long bone under a constant compressive load are made using strain energy density as controlled variable. The bone is modeled as a hollow circular cylinder. The non-linear differential equations governing movement of the periosteal and endosteal surface was deduced. It is shown that there are eight possible types of solutions to the problem. The particular type of response depends upon the magnitude of the applied compressive load and the values of the surface remodelling rate coefficients. For which there are fortyfive different types. Our solution include the solution got by Cowin using strain as controlled variable. Although the limitation of the theoretical model is exit, it can still consist with the typical animal experimental result. So reliability of adaptive bone remodeling using strain energy density as controlled variable was confirmed in the theory.