Multiscale Approach to Understand the Multiphysics Phenomena in Bone Adaptation - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 51

the ionic concentration n b ½ 0 ; the streaming potential W b ½ 0 ; the fluid pressure p b ½ 0 and
the porosity g f : This corresponds to 16 scalar unknowns.

3.3.2 What Stays Beyond Macroscopic Observation

Due to the scaling laws inherent to cortical bone, both the double layer and the
piezo-electric effects vanish through the homogenization process. Notwithstanding
the fact that the double layer electric potential u and the piezo-electric potential / s
do not filter through the upscaling process, their consequences at the macroscopic
scale can still exist. In particular, as indicated by Ahn and Grodzinsky [ 3 ], the
variations of the piezo-electric potential / s directly modify the pore surface
charge, and thus the double layer potential. As shown hereafter, even if purely
microscopic, this double layer potential can induce important consequences at the
macroscale.

3.3.3 Ionic Electro-Diffusive Transport

In the remodelling process, the paracrine communication between the mechano-
sensors (osteocytes) and the effector cells (osteoclasts and osteoblasts) requires to
develop specific transport processes. Due to the narrow space of the canaliculi,
the convective effect vanishes through the upscaling process. Taking into account
the possible ionic exchanges between the cell and its fluid environment and the
electromigration effects, two macroscopic electro-diffusive Nernst-Planck equa-
tions can be obtained for monovalent ions (see Remark) [ 76 ]:

þ on b ½ 0
ot

o
ot

½ n b ½ 0 g f h exp ð u ½ 0 Þi f h a ½ 0 i int

ð 20 Þ

h i :

D ðr X n b ½ 0 n b ½ 0 r X W b ½ 0 Þ ¼r X

Remark The nabla operator is here indexed X since it corresponds to the mac-
roscopic spatial derivative operator, that is to say with respect to the macroscopic
coordinate X :

This equation exhibits three contributions to the ionic transport: i. the temporal
term involving the influence of the porosity g f ; the averaged double-layer effects
and the surface exchange term a ; ii. a Brownian diffusion term in response to the
salinity gradient; iii. an electromigration term in response to the gradient of the
streaming potential.
These two last terms are quantified using effective diffusion tensors D
involving, in addition to the diffusion coefficients of the ions D ; the porosity g f
and the electro-tortuosity tensors # :