Periosteal and endosteal control of bone remodeling under torsional loading

The shape changes that occur in the mid-diaphysis of a long bone due to adaptive remodeling induced by increasing or decreasing the axial and/or torsional loading of the bone are investigated using a simple model. In this model the mid-diaphysis of a long bone is represented as a hollow thick-walled right-circular cylinder, and different optimal strategies for bone remodeling are considered. It is shown that if such a thick-walled right-circular cylinder capable of surface remodeling is subjected to an axial compressive load and a twisting torque, then the remodeling patterns depend on whether the periosteal surface or the endosteal surface controls the limits of the remodeling process. It is shown that the effect of increasing the torque is always opposite to the effect of increasing the compressive load. Thus, similar remodeling patterns are obtained by increasing one type of loading and decreasing the other. Aside from the restriction of idealized cylindrical geometry, the only assumptions made are that the bone tissue is linearly elastic and that there exists a finite range of remodeling equilibrium stresses. Only those loading situations which maintain the bone in remodeling equilibrium are considered in this work. It follows that the results presented are independent of the specific type of rule governing the temporal evolution of the bone shape, since any such rule applies only in situations where there is active remodeling and, hence, no remodeling equilibrium.