Is the callus shape an optimal response to a mechanobiological stimulus?

After bone trauma, the natural response to restore bone function is the formation of a callus around the fracture. Although several bone healing models have been developed, none have effectively perceived early callus formation and shape as the result of an optimal response to a mechanobiological stimulus. In this paper, we investigate which stimulus regulates early callus formation. An optimal design problem is formulated, and several objective functions are defined, each using a different mechanobiological stimulus. The following stimuli were analysed: the interfragmentary strain, the second invariant of the deviatoric strain tensor and a generic inflammatory factor. Different regions for callus formation were also evaluated, such as the gap region, the periosteum and the periosteum border. Each stimulus was computed using the finite element method, and the callus shape was optimised using the steepest descent method. The results demonstrated that the inflammatory factor approach, the interfragmentary strain and the second invariant of the deviatoric strain tensor over the inner gap provided the best results when compared with histological callus shapes. Therefore, this work suggests that callus growth can be an optimal mechanobiological response to either local mechanical instability and/or local inflammatory reaction.