How do bones grow?: A mathematical description of the mechanobiological behavior of the epiphyseal plate

Abstract

Growth modulation is an emerging method for the treatment of skeletal deformities originating in the long bones or the vertebral bodies. It requires the controlled application of mechanical loads to the affected bone, causing an alteration of the growth and ossification process occurring in a cartilaginous region called epiphyseal growth plate or physis. In order to avoid the possibility of under- or over-correction, quantification of the applied forces is necessary. Pursuing this goal, herewe propose a phenomenological model of mechanobiological effects on the epiphyseal growth plate, based on the observed similarity between the mechanobiologically induced growth and viscoelastic material behavior. The model incorporates mechanical loading effects on growth direction, growth rate and ossification speed; it also allows to evaluate the occurrenceof transient effects. Model consistency was tested against a rather large set of experiments existing in the literature. A generic simplified geometrical model of bones was established for this. Analytical solutions for growth and ossification evolution were obtained for different loading conditions, allowing to test the ability of the model to describe bone growth under various kinds of mechanical loading conditions. Model-predicted changes regarding epiphyseal growth plate thickness as well as longitudinal growth speed are consistent with experiments in which static tension or compression were applied to long bones. Results suggest that when the mechanical load is sinusoidally variable, conflicting data existing in the literature could be explained by a previously unconsidered effect of the the applied load initial phase. The model can accurately fitdata regarding torsional loads effects on growth. Mechanobiological data for humans is very scarce. For this reason, when possible, the model parameters values were estimated, for the proposed generic geometry, after growth measurements in animal models available in the literature. Although it is not possible to assert their validity for humans, the proposed modelalong with the obtained parameters values give a rational foundation to be used in more advanced computational studies.