Theoretical modeling and numerical simulation of bone reconstruction using a second gradient approach with evolutive microstructure

Ph.D. director : Daniel GEORGE, Associate Professor, george@unistra.fr
Laboratory : Laboratoire des Sciences de l'Ingénieur, de l'Informatique et de l'Imagerie (ICUBE) - UMR 7357
University : Université de Strasbourg.
Collaborations : INSA de Lyon, Université La Sapienza Rome, Polish Academy of Science, Institut Hospitalo-universitaire Strasbourg (IHU), Bioingénierie et Bioimagerie Ostéo-Articulaire (B2OA) Paris 7, MATEIS UMR 5510 CNRS, LTDS UMR 5513 CNRS (Equipex IVTV).

Note :  : A Master internship is be proposed before the start of the PhD.

Subject :
This proposed subject is looking into problems of bone reconstruction through a bioresorbable material. When doing surgery for fractures or artificial joints, non-living materials are often used to reinforce the weak bone structure in order to enhance reconstruction or replace bone. Nowadays, micro or nano-structured porous materials are developed in order to be progressively and fully resorbed while replaced by natural bone. Numerous works already exist on the modeling of bone reconstruction taking into account the biological effects occurring. Some of these models use continuous approach of high gradient theories in order to account for the microstructure phenomena, to some extend. For example, the works from the Rome and Polish Teams, and INSA Lyon have conducted to original and efficient simulations [Lekszycki 1999, 2005 et Madeo 2011, 2012]. However, existing models are limited to isotropic uni/bi-directional cases and do not account for specific microstructural effects, its evolution and anisotropy. The proposed subject is integrated within a fundamental study following the works by the cited teams and in collaboration with them.
We are principally interested in the integration of the material microstructure and bone anisotropy during its reconstruction and using theoretical, numerical approach initially introduced by the works of Germain (1972, 1973), Mindlin (1964, 1968) and Toupin (1962). This theoretical approach enables, through the variational principle, to integrate local microstructural effects using the second gradient of displacement. These effects will require to be anisotropically locally integrated and accounting for the anisotropic bone density evolution as a function of the biological environment and external applied forces. A local homogenization technique will be required in order to be integrated within the second gradient macroscopic model. This subject is a fundamental research ahead of immediate medical applications. However, a strong potential application is foreseen since there exits, at this moment, no numerical model able to propose a 'real' anisotropic bone reconstruction within its biological environment. The subject will require 2D and 3D theoretical and numerical modeling coupling resorption of the bioresorbable material and replacement by natural bone. The main difficulty in this work will be the validation of the numerical model and identification of a number of modeling parameters which could be obtained experimentally in-vivo by our partners. In addition, our integration within the GDR AMORE (Advanced Model Order Reduction in Engineering and Sciences) will enable to use numerical models such as the one developed by F. Chinesta et al. (2011) in order to account for all the parameters integrated within the model.
The proposed subject is strongly integrated within the team activity on materials, microstructure and biomechanics applications. Furthermore, the collaborations with INSA Lyon, University La Sapienza - Rome, the Medical University of Warsaw together with Bone Bioengineery Laboratory (Paris 7) and the Bone Biomechanics Group (Marseille) enable us to foresee some real applications of the developed model for its validation in in-vivo situations. The subject requires good knowledge of theoretical mechanics, physics and numerical physics. Ideally, the retained PhD. candidate will be orientated applied mathematics, numerical physics / mechanics.

National and international collaborations :
· INSA Lyon (Dr. Angela Madeo)
· GDR AMORE (Prof. Francisco Chinesta, Centrale Nantes)
· Laboratoire de bioingénierie et bioimagerie ostéo-articulaire (Prof. Hervé Petite, B2OA, Paris 7)
· Groupe interdisciplinaire en biomécanique ostéoarticulaire et cardiovasculaire (Prof. Patrick Chabrand, Université Aix-Marseille)
· Université La Sapienza, Rome (Prof. Francesco d’ell Isola)
· Université médicale de Varsovie (Prof. Tomasz Lekszycki)
· Bioingénierie et Bioimagerie Ostéo-Articulaire (Prof. Hervé Petite)
· Laboratoire de Tribologie et Dynamique des Systèmes – Equipex IVTV (Prof. Thierry Hoc)
· Laboratoire Matériaux Ingénierie et Science - MATEIS

References :
· Chinesta, F., Ammar, A., Leygue, A. and Keunings, R., 2011, An overview of the proper generalized decomposition with applications in computational rheology, Journal of Non-Newtonian Fluid Mechanics, vol. 166, pp. 578-592.
· Germain, P., 1972, Sur l’application de la méthode des puissances virtuelles en mécanique des milieux continus, Comptes Rendus de l’Académie des Sciences, vol. 274, pp. 1051-1055.
· Germain, P., 1973, La méthode des puissances virtuelles en mécanique des milieux continus : théorie du second gradient, Journal de Mécanique, vol. 12 (2), pp. 235-274.
· Lekszycki, T., 1999, Optimality conditions in modeling of bone adaptation phenomenon, Journal of Theoretical Applied Mechanics, vol. 37 (3), pp. 607-624.
· Lekszycki, T., 2005, Functional adaptation of bone as an optimal control problem, Journal of Theoretical Applied Mechanics, vol. 43 (3), pp. 120-140.
· Madeo, A., George, D., Lekszycki, T., Nierenberger, M., Rémond, Y., 2012, A second gradient continuum model accounting for some effects of micro-structure on reconstructed bone remodeling, Comptes Rendus Mécanique, vol. 340, pp. 575-589.
· Madeo, A., Lekszycki, T., Dell'Isola, F., 2011, A continuum model for the bio-mechanical interactions between living tissue and bioresorbable graft after bone reconstructive surgery, Comptes Rendus Mécanique, vol. 339, pp. 625-640.
· Mindlin, R. D., 1964, Micro-structure in linear elasticity, Archive for Rational Mechanics and Analysis, vol. 16, pp. 51-78.
· Mindlin, R. D., Eshel, N. N., 1968, On first strain-gradient theories in linear elasticity, International Journal of Solids and Structures, vol. 4, pp. 109-124.
· Toupin, R. A., 1962, Elastic materials with couple-stresses, Archive for Rational Mechanics and Analysis, vol. 11, pp. 385-414.