**6. Concluding remarks**

Surface growth is by essence a pluridisciplinary field, involving interactions between the physics and mechanics of surfaces and transport phenomena. The literature survey shows different strategies for treating superficial interactions, hence recognizing that no unitary viewpoint yet exists. The present contribution aims at providing a pluridisciplinary approach of surface growth focusing on

Thermodynamics of Surface Growth with Application to Bone Remodeling 401

Gurtin, M.E. (1995). On the nature of configurational forces. *Arch. Rat. Mech. Anal*., Vol. 131,

Gurtin, M.E. (2000). In: *Configurational forces as Basic Concepts of Continuum Physics*. Springer,

Huiskes, R. & Sloff, T.J. (1981). Geometrical and mechanical properties of the human femur.

Huiskes, R.; Weinans, H., Grootenboer, H. J., Dalstra, M., Fudala, B. & Slooff, T.J. (1987). *J.* 

Kessler, D.A. (1990). Coupled-map lattice model for crystal growth. *Phys. Rev*., Vol.A 42,

Kondepudi, D. & Prigogine, I. (1998). *Modern Thermodynamics: From Heat Engines to* 

Langer, J.S. (1980). Instabilities and pattern formation in crystal growth. *Rev. Mod. Phys*.,

Leo, P.H. & Sekerka, R.F. (1989). The effect of surface stress on crystal–melt and crystal–

Linford, R.G. (1973). *Surface thermodynamics of solids. Solid State*. *Surface Science*, Vol.2, pp.1-

Lu, T.W.; O'Connor, J. J., Taylor, S. J. G. & Walker, P. S. (1997). Influence of muscle activity on the forces in the femur: an *in vivo* study. *J. Biomech*. Vol.30, No.11, pp.1101-1106 Maugin, G.A. & Trimarco, C. (1995). The dynamics of configurational forces at phase-

Mindlin, R.D. (1965). Second gradient of strain and surface-tension in linear elasticity. *Int. J.* 

Muller, P. & Kern, R. (2001). In *Stress and Strain in epitaxy*: theoretical concept measurements.

Mullins, W.W. (1963). in *Metals surfaces, structures, energetics, and kinetics*. *Amer. Soc. Metals*.

Reilly, D.T. & Burstein, A.H., 1975. The elastic and ultimate properties of compact bone

Rice, J.C.; Cowin, S.C. & Bowman, J.A. (1988). On the dependence of the elasticity and strength of cancellous bone on apparent density. *J. Biomech*., Vol.21, pp.155-168 Ruimerman, R.; Hilbers, P., Van Rietbergen, B. & Huiskes, R. (2005). A theoretical

Silva, E.C.C.M. & Ulm, F.J. (2002). A bio-chemo-mechanics approach to bone resorption and

Skalak, R., Dasgupta, G. & Moss, M. (1982). Analytical description of growth. *J. Theor. Biol*.,

Skalak, R., Farrow, D.A. & Hoger, A. (1997). Kinematics of surface growth. *J. Math. Biol*.,

framework for strain-related trabecular bone maintenance and adaptation. *J.* 

fracture. Proc. 15th ASCE Engineering Mechanics Confererence. 02-05/06/2002,

crystal equilibrium. *Acta Metall.*, Vol.37, No.12, pp.3119–3138

Maugin, G.A. (1993). *Material Inhomogeneities in elasticity*. Chapman et al., London

transition fronts. *Meccanica* Vol.30, pp.605–619

Ed. M. Handbucken, J.P. Deville, Elsevier

tissue, *J. Biomech*., Vol.8, pp.393-405

Columbia University, New York, USA

*Biomech*., Vol.38, pp.931-941

Vol.94, pp.555-577

Vol.35, pp.869-907

Munster, A. (1970). *Classical Thermodynamics,* John Wiley and Sons

*Solids Struct*., Vol.1, pp.417–438

Metals Park, Ohio, p. 7

Biomechanics VII-A: 7th international congress on biomechanics, Ed. A. Morecki,

pp.67–100

New York

Vol.3A, pp.57-64

pp.6125–6128

Vol.52, pp.1–28

152

*Biomech*., Vol.20, pp.1135-50

*Dissipative Structures*. Wiley

A macroscopic model of bone external remodeling has been developed, basing on the thermodynamics of surfaces and with the identified configurational driving forces promoting surface evolution. The interactions between the surface diffusion of minerals and the mechanical driving factors have been quantified, resulting in a relatively rich model in terms of physical and mechanical parameters. Applications of the developed formalism to real geometries

Works accounting for the multiscale aspect of bone remodeling have emerged in the literature since the late nineteen's considering cell-scale (a few microns) up to bone-scale (a few centimeters) remodeling, showing adaptation of the 3D trabeculae architecture in response to mechanical stimulation, see the recent contributions (Tsubota et al., 2009; Coelho et al., 2009) and the references therein. It is likely that one has in the future to combine models at both micro and macro scales in a hierarchical approach to get deeper insight into the mechanisms of Wolff's law.

The present modeling framework shall serve as a convenient platform for the simulation of bone remodeling with the consideration of real geometries extracted from CT scans. The predictive aspect of those simulations is interesting in a medical context, since it will help doctors in adapting the medical treatment according to short and long term predictions of the simulations.
