On the development of an integrated bone remodeling law for orthodontic tooth movements models using the Finite Element Method.

One of the guiding principles in orthodontics is to gradually impose progressive and irreversible bone deformations due to remodeling using specific force systems on the teeth. Bone remodeling leads the teeth into new positions with two tissues having a major influence: the periodontal ligament and the alveolar bone. Their mechanical and biological/physiological reactions to orthodontic forces are tightly linked. This mechanical biological coupling can be treated in biomechanical models, focusing on the mechanics and considering the phenomenological aspects of the biology/physiology. The development of such a model for bone tissue within a Finite Element framework is the core of this work. We propose to reconcile two approaches of bone modeling (small strains linear elasticity for remodeling problems and complex constitutive models for other applications) by writing a constitutive model for trabecular bone at macroscopic level, built on morphological parameters to describe the anisotropy, and accounting for effects such as plasticity of the trabeculae. The continuum parameters such as the stiffness can evolve with morphology as remodeling occurs in the tissue. For this, we extend and enhance Doblaré and Garcia’s remodeling phenomenological model. The remodeling process corresponds to an evolution of a damage tensor representing the bone morphology. To do so, we propose an integration method for an anisotropic Continuum Damage model coupled to plasticity. Adapting Doblaré and Garcia’s remodeling law to our constitutive model, we extend it so that it can be used in the specific case of orthodontic tooth movement, still following Frost’s mechanostat theory. We propose to include the hydrostatic pressure dependency of remodeling, due to the presence of the periodontal ligament, within the bone remodeling law. We finally present a validation method for the mechanical representation of the bone matrix through the knowledge of its morphology, both on engineered cellular solids with bone-like morphology (aluminum and polymeric foams) and on bone (Deer antler) samples. Applying the model on the benchmark problem of the proximal femur remodeling, leads to results that are comparable to other models of the literature. We can therefore assume the way the remodeling model is built is valid. We finally apply the developed model to orthodontic tooth movement simulations. First we propose a model accounting for the nonlinear mechanical response of the PDL through either bilateral contact conditions or spring models. We then present applications of orthodontic tooth movement, either displacement driven or force driven, both 2D and 3D. We thus show we can qualitatively represent the tooth movement, however outlining some of the drawbacks of the models (an unphysiological density distribution can arise due to the poor representation of the actual loads and a strong dependence on the boundary conditions is pointed out). However, we can represent the formation and resorption of hyaline areas, the non-linearity of the force/displacement relationship, and that applying a stepwise increasing force leads to higher displacements than a high initial force as there is no hyaline zone to resorb.