**5. Model of surface growth with application to bone remodeling**

The present model aims at describing radial bone remodeling, accounting for chemical and mechanical influences from the surrounding. Our approach of bone growth typically follows the streamlines of continuum mechanical models of bone adaptation, including the time-dependent description of the external geometry of cortical bone surfaces in the spirit of free boundary value problems – a process sometimes called net 'surface remodeling' - and of the bone material properties, sometimes coined net 'internal remodeling' (Cowin, 2001).

### **5.1 Material driving forces for surface growth**

In the sequel, the framework for surface growth elaborated in (Ganghoffer, 2010) will be applied to describe bone modeling and remodeling. As a prerequisite, we recall the identification of the driving forces for surface growth. We consider a tissue element under grow submitted to a surface force field *S***f** (surface density) and to line densities *p* , *p* defined as the projections onto the unit vectors , *g g* **τ ν** resp. along the contour of the open growing surface *Sg* (Figure 5); hence, those line densities are respectively tangential and normal to the surface *Sg* (forces acting in the tangent plane).

Fig. 5. Tissue element under growth: elements of differential geometry.

Focusing on the surface behavior, the potential energy of the growing tissue element is set as the expression

$$\begin{split} V &= \int\_{\Omega\_{\mathcal{S}}} \mathcal{W}\_{0} \{ \mathbf{F} \} dx\_{\mathcal{S}} + \int\_{\mathcal{S}\_{\mathcal{S}}} \nu^{S} \{ \tilde{\mathbf{F}}, \mathbf{N}; \mathbf{X}\_{\mathcal{S}} \} d\sigma\_{\mathcal{S}} + \int\_{\mathcal{S}\_{\mathcal{S}}} \mu\_{k} n\_{k}^{\sigma} d\sigma\_{\mathcal{S}} \\ &- \int\_{\mathcal{S}\_{\mathcal{S}}} \mathbf{f}\_{\mathcal{S}} \tilde{\mathbf{x}} d\sigma\_{\mathcal{S}} - \int\_{\tilde{\varepsilon} \mathcal{S}\_{\mathcal{S}}} p\_{\tau} \tilde{\mathbf{x}} \mathbf{r}\_{\mathcal{S}} dl\_{\mathcal{S}} - \int\_{\mathcal{E} \mathcal{S}\_{\mathcal{S}}} p\_{\nu} \tilde{\mathbf{x}} \mathbf{v}\_{\mathcal{S}} dl\_{\mathcal{S}} \end{split} \tag{5.1}$$

Thereby, the growing solid surface is supposed to be endowed with a volumetric density *W*<sup>0</sup> **F** depending upon the transformation gradient : **F x** *<sup>X</sup>* , a surface energy with density , ; *<sup>S</sup>* **FNX***<sup>S</sup>* per unit reference surface, depending upon the surface gradient **F** , the unit normal vector **N** to *Sg* , and possibly explicitly upon the surface position vector

Thermodynamics of Surface Growth with Application to Bone Remodeling 393

basing on the *surface stress* **T** . The Lagrangian curvature tensor is defined as : **K N** *<sup>R</sup>* . The chemical potential as the partial derivative of the surface energy density with respect to

, ,

The contributions arising from the domain variation due to surface growth are considered

The material surface driving force (for surface growth) triggers the motion of the surface of the growing solid; it is identified from the material variation of *V* as the vector acting on the

*<sup>g</sup> <sup>S</sup> N k Sk S n*

Bone is considered as a homogeneous single phase continuum material; from a microstructural viewpoint, bone consists mainly of hydroxyapatite, a type-I collagen, providing the structural rigidity. The collageneous fraction will be discarded, as the mineral carries most of the strain energy (Silva and Ulm, 2002). The ultrastructure may be considered as a continuum, subjected to a portion of its boundary to the chemical activity generated by osteoclasts, generating an overall change of mass of the solid (the mineral

*g g*

 *gV d MJd <sup>N</sup>* 

with *VN* the normal surface velocity, *M* the bone mineral molar mass, and / *J VM*

the molar influx of minerals (positive in case of bone apposition, and negative when resorption occurs). Clearly, the previous expression shows that the knowledge of the normal surface growth velocity determines the molar influx of minerals. Estimates of the order of magnitude of the dissolution rate given in (Christoffersen at al., 1997), for a pH of 7.2 (although much higher compared to the pH for which bone resorption takes place) and at a temperature of 310K, are indicative of values of the molar influx in the interval 9 8 12 *<sup>J</sup>* 10 ,1.8.10 . . *mol s m* . The osteoclasts responsible for bone resorption attach to the bone surface, remove the collageneous fraction of the material by transport phenomena, and diffuse within the material. This osteoclasts activity occurs at a typical scale of about 50

*dt*

 **f** (5.7)

.

   

therein represents the molar flux of bone material being dissolved,

*gg g S g S <sup>d</sup> dx <sup>d</sup>*

**V N**

**Σ N Σ P K**

*S k k k k XFN*

*n*

:

*<sup>F</sup>* **<sup>T</sup>** 

: . . ..*t S*

itself built from *the surface stress* : *<sup>S</sup>*

(5.6)

, and on the curvature tensor : **K N** *<sup>R</sup>* in the

*g g* (5.8)

*g N*

> *m* ,

*n*

 

the superficial concentration

variation of the surface position

referential configuration.

**5.2 Bone remodeling** 

fraction) given by

hence

The quantity . *<sup>g</sup> <sup>S</sup> <sup>g</sup>* **V N***d*

as irreversible.

**X***S* on *Sg* (no tilda notation is adopted here since the support of **X***S* is strictly restricted to the surface *Sg* ), and chemical energy *k k n* , with *<sup>k</sup>* the chemical potential of the surface concentration of species *nk* . The surface gradient **F** maps material lengths (or material tangent vectors) onto the deformed surface; it is elaborated as the surface projection of **F** (onto the tangent plane to *<sup>a</sup>* ), viz
