Stability analysis of bone remodeling equation using multi-element models

An improved analytical model was developed for the stability conditions of the bone remodeling equation. An n-element model and an m × n element model were used to study the two nonlinearities in the high-order nonlinear bone remodeling equation due to the exponential form of the remodeling coefficient, B(t), and the order of the non-linear remodeling equation. When the power index relating the density and Young's modulus satisfies -1<r<1, the continuous solution is stable, while for r>1 the system can reach discontinuous solution through an unstable continuous solution. The results show that the stability of the high-order nonlinear bone remodeling equation is mainly determined by the power index, r, while the two nonlinerities in the equation have little effect.