Finite element (FE) models accurately compute the mechanical response of bone and bone-like materials when the models include their detailed microstructure. In order to simulate non-linear behavior, which currently is only feasible at the expense of extremely high computational costs, coarser models can be used if the local morphology has been linked to the apparent mechanical behavior. The aim of this paper is to implement and validate such a constitutive law. This law is able to capture the non-linear structural behavior of bone-like materials through the use of fabric tensors. It also allows for irreversible strains using an elastoplastic material model incorporating hardening. These features are expressed in a constitutive law based on the anisotropic continuum damage theory coupled with isotropic elastoplasticity in a finite strains framework. This material model was implemented into Metafor, a non-linear FE software. The implementation was validated against experimental data of cylindrical samples subjected to compression. Three materials with bone-like microstructure were tested : aluminum foams of variable density (ERG, Oakland, CA), PLA (polylactic acid) foam (CERM, University of Liège) and cancellous bone tissue of a deer antler (Faculty of Veterinary Medicine, University of Liège).

Finite element models accurately compute the mechanical response of bone and bone-like materials when the models include their detailed microstructure. The aim of this paper is to develop and validate a biomechanical model for deer antler cancellous bone tissue based on X-ray microtomographic images. In order to simulate the mechanical behavior under compressive load using a finite element model, images obtained by X-ray microtomography were exported into Metafor, which is an non-linear finite element software initially developed at University of Liège for metal forming processes. This software has recently found biomedical applications. The ultimate goal is to compare model predictions with the mechanical behavior observed experimentally using the Skyscan material testing stage under compression mode. The creation of the biomechanical model mesh from segmented μCT images, its integration into the software Metafor and the simulation of a compression test are described in this paper.

This paper aims at presenting a general versatile time integration scheme applicable to anisotropic damage coupled to elastoplasticity, considering any damage rate and isotropic hardening formulations. For this purpose a staggered time integration scheme in a finite strain framework is presented, together with an analytical consistent tangent operator. The only restrictive hypothesis is to work with an undamaged isotropic material, assumed here to follow a J2 plasticity model. The only anisotropy considered is thus a damage-induced anisotropy. The possibility to couple any damage rate law with the present algorithm is illustrated with a classical ductile damage model for aluminium, and a biological damage-like application. The later proposes an original bone remodelling law coupled to trabecular bone plasticity for the simulation of orthodontic tooth movements. All the developments have been considered in the framework of the implicit non-linear finite element code Metafor (developed at the LTAS/MN2L, University of Liège, Belgium - www.metafor.ltas.ulg.ac.be).

In finite element simulations of orthodontic tooth movement, one of the challenges is to represent long term tooth movement. Large deformation of the periodontal ligament and large tooth displacment due to bone remodelling lead to large distortions of the finite element mesh when a Lagrangian formalism is used. We propose in this work to use an Arbitrary Lagrangian Eulerian (ALE) formalism to delay remeshing operations. A large tooth displacement is obtained including effect of remodelling without the need of remeshing steps but keeping a good-quality mesh. Very large deformations in soft tissues such as the periodontal ligament is obtained using a combination of the ALE formalism used continuously and a remeshing algorithm used when needed. This work demonstrates that the ALE formalism is a very efficient way to delay remeshing operations.

Using morphological data provided by computed tomography, finite element (FE) models can be used to compute the mechanical response of bone and bone-like materials without describing the complex local microarchitecture. A constitutive law is here developed and proposed for this purpose. It captures the non-linear structural behavior of bone-like materials through the use of fabric tensors. It also allows for irreversible strains using a plastic material model, allowing hardening of the yield parameters. These characteristics are expressed in a constitutive law based on the anisotropic continuum damage theory coupled with isotropic elastoplasticity in a finite strains framework. This law is implemented into Metafor, a non-linear FE software. Simulations of cylindrical samples undergoing stepwise compression are presented.

Several authors have proposed mechanical models to predict long term tooth movement, considering both the tooth and its surrounding bone tissue as isotropic linear elastic materials coupled to either an adaptative elasticity behavior or an update of the elasticity constants with density evolution. However, tooth movements obtained through orthodontic appliances result from a complex biochemical process of bone structure and density adaptation to its mechanical environment, called bone remodeling. This process is far from linear reversible elasticity. It leads to permanent deformations due to biochemical actions. The proposed biomechanical constitutive law, inspired from Doblaré and García (2002) [30], is based on a elasto-viscoplastic material coupled with Continuum isotropic Damage Mechanics (Doblaré and García (2002) [30] considered only the case of a linear elastic material coupled with damage). The considered damage variable is not actual damage of the tissue but a measure of bone density. The damage evolution law therefore implies a density evolution. It is here formulated as to be used explicitly for alveolar bone, whose remodeling cells are considered to be triggered by the pressure state applied to the bone matrix. A 2D model of a tooth submitted to a tipping movement, is presented. Results show a reliable qualitative prediction of bone density variation around a tooth submitted to orthodontic forces.

The purpose of this work is to propose an enhancement of Doblaré and García's internal bone remodelling model based on the continuum damage mechanics theory. In their paper, they stated that the evolution of the internal variables of the bone microstructure, and its incidence on the modification of the elastic constitutive parameters, may be formulated following the principles of Continuum Damage Mechanics, although no actual damage was considered. The resorption and apposition criteria (similar to the damage criterion) were expressed in terms of a mechanical stimulus. However, the resorption criterion is lacking a dimensional consistency with the remodelling rate. We here propose an enhancement to this resorption criterion, insuring the dimensional consistency while retaining the physical properties of the original remodelling model. We then analyse the change in the resorption criterion hypersurface in the stress space for a 2D analysis. We finally apply the new formulation to analyse the structural evolution of a 2D femur. This analysis gives results consistent with the original model but with a faster and more stable convergence rate.

Orthodontic tooth movement (OTM) is the result of bone remodeling at the interface with the periodontal ligament (PDL) around a mechanically loaded tooth in response to a biomechanical stimulus. Modeling of the PDL therefore plays an important role in the process of modeling OTM. However when producing a finite element model from clinical computer tomography data, the PDL cannot be segmented and its geometry is approximated by many authors from the root geometry. The aim of this study is to propose alternatives to a geometrical representation of the PDL using either simple spring elements between the teeth and alveolar bone or bilateral sticking contact conditions. Results consist in a comparison of the hydrostatic and Von-Mises stresses in the bone along the root as well as the strain energy used in a bone remodeling algorithm when a 1N force is applied to a single rooted tooth crown. While both models can well represent the pressure (hydrostatic stress) transfer from the tooth to the bone, the bilateral sticking contact conditions show better results to transfer the shear stress as well as the strain energy.

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 non-linear 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.