Vibration exercise for treatment of osteoporosis: A theoretical model

Orthopaedic rehabilitation of osteoporosis by muscle vibration exercise is investigated theoretically using Wolff's theory of strain-induced bone ‘remodelling’. The remodelling equation for finite amplitude vibration to be transmitted to the bone via muscle corresponds to a slowly time-varying non-linear dynamic system. This slowly time-varying system is governed by a Riccatti equation with rapidly varying coefficients that oscillate with the frequency of the applied vibration. An averaging technique is used to determine the effective force transmitted to the bone. This force is expressed in terms of the stiffness and damping parameters of the connected muscle. The analytical result predicts that, in order to obtain bone reinforcement, the frequency and amplitude of vibration should not exceed specified levels. Furthermore, low-frequency vibration does not stimulate the bone sufficiently to cause significant remodelling. The theoretical model herein confirms the clinical recommendations regarding vibration exercise and its effects on rehabilitation. In a numerical example, the model predicts that a femur with reduced bone mass as a result of bed rest will be healed completely by vibration consisting of an acceleration of 2g applied at a frequency of 30 Hz over a period of 250 days.