Introduction

Bone is a living material which is continuously reorganized by bone cells in response to their mechanical and biochemical environment. This process, known as bone remodeling, is of major importance in everyday life, in case of fractures and to allow osseointegration phenomena around implants.

Bone remodeling can be described as a stress- and chemistry-driven evolution of the mechanical properties of bone tissue. The interplay between mechanics and biochemistry, as well as the multiple scales involved in this process, represent serious challenges for the development of realistic bone adaptation models.

This study describes a novel, thermodynamically sound model of bone remodeling which, while being mechanistic in nature, is able to account for the above issues.

Methods

The theory of material remodeling [1] offers a firm basis to build a model of bone remodeling. Bone is described as an orthotropic elastic medium whose elastic properties evolve in time according to the prevailing mechanical and chemical stimuli. In this study, we focus on a special class of evolution, namely the stress-driven rotation of the elastic principal axes of the bony material. Our model is based on balance laws derived through a suitable statement of the virtual power principle, as well as the description of a constitutive theory. The latter is based on the definition of a strain-energy density depending only on the elastic strain, and on a formulation of the dissipation principle incorporating the dissipation due to remodeling. This modeling framework leads to a remodeling evolution law giving an explicit relationship coupling the dissipation related to the remodeling, the stress and strain tensors, the rotation of the material axes and its evolution.

Remodeling equilibrium is achieved when material properties no longer evolve, corresponding to a stationary state of the rotation. It is worth noting that this model predicts the principal axes of the strain and stress tensors to be collinear at the remodeling equilibrium [2]. Thus, a physically sound condition for remodeling equilibrium [3] is recovered without any ad-hoc assumption.

Results and Discussion

The model was studied in 2D where the rotation of the elastic principal axes is parameterized by an angle.

In case of uniform stress/strain conditions, stable remodeling equilibrium states were found to correspond to strain energy minima (imposed displacements).

Finite element simulations were also performed to study the evolution of the elastic principal axes resulting from different boundary conditions. The prediction of the rotation of the principal axes of the material in simple loading configurations is consistent with the superimposed boundary conditions confirming the ability of the model to simulate the material response to non-uniform stress configurations.

Conclusion

A novel, thermodynamically sound model of bone remodeling was proposed. Preliminary results obtained in 2D are promising and show the potential of this approach. Model predictions for in vivo biomechanical loading configurations need to be further tested. Suitable experimental data will be identified. The particular case of tissue surrounding an implant will be also studied.

Our model can also integrate the mechanobiological phenomena regulating bone remodeling. However, this would require a reliable description of the biochemical stimuli of bone remodeling. This matter is out of the scope of this paper and will be addressed in future works.

Acknowledgement

This project has received funding from the European Research Council under the European Union's Horizon 2020 research and innovation program (grant agreement No 682001, project ERC Consolidator Grant 2015 BoneImplant).

References

[1] DiCarlo A, Naili S, and Quiligotti S (2006) C. R. Mecanique, 334:651-661

[2] Sansalone V, Naili S, and Di Carlo A (2011) Comput Methods Biomech Biomed Engin, 14(s1):203-204

[3] Cowin S (1986) J Biomech Eng, 108:83-88