Multiscale Approach to Understand the Multiphysics Phenomena in Bone Adaptation - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 61

authors highlighted the need for a new interpretation of experimental results where
it was found that cyclic loading increases marker penetration within the lacuno-
canalicular system [ 68 ], and proposed that cyclic stress stimulates an active
transport mechanism.
If we take into account the electrical effects on diffusion, a new angle can be
proposed to explain such an active transport. Due to the surface charge charac-
terizing the bone pores, double layers develop inside the pores causing an asym-
metric ionic distribution near their walls (see Fig. 4 ). The influence of these
electrical phenomena depends on both the fluid salinity and the pore size. Indeed,
the lower the salinity, the larger the characteristic length scale of these electrostatic
phenomena (Debye length) becomes. Moreover, the Debye's length being nano-
metric, the electrical effects rapidly fade out for pore sizes as large as the cana-
licular radius. However, when considering typical pore sizes related to the
pericellular matrix typical dimensions, these electrical effects should be important.
Using a home-made code based on the recursive resolution of the Cartesian
Poisson-Boltzmann problem proposed in Derjaguin et al. [ 29 ], a model of straight
channel described in cartesian coordinate (half-size h) is built for a representative
pore filled with a water-saturated electrolyte and presenting a negative surface
charge density of 0 : 2 C.m 2 : The reduced electrical effective diffusion param-
eters (see Remark) are calculated for monovalent ions considering various pore
sizes and salinity values, as shown by Fig. 10 . On the left, considering a typical
fiber-to-fiber distance of the pericellular matrix elements 2h ¼ 10 nm [ 158 , 166 ],
we present the evolution of the reduced effective diffusivity parameters with the
salinity. On the right, the pore size dependance of these parameters is shown
considering a salinity of 0 : 01 M.

Remark The one-dimensional diffusivity parameters derived from Eq. ( 21 ) are
reduced thanks to the diffusion coefficients of the ions D and the porosity g f :
These two graphs show that the electro-chemical effects may strongly affect the
effective diffusion process. The evolutions of the cationic, respectively anionic,
effective diffusion parameters exhibit the permselectivity property of the lacuno-
canalicular system which acts here as a negatively charged nanoporous medium
which tends to enhance the cationic transport and limit the anionic one [ 45 , 122 ].
In particular, for pore sizes corresponding to the pericellular fiber-to-fiber distance,
the electrostatic exclusion-enrichment effect becomes the dominating mass
transport mechanism. Thus, depending on their charge, the chemical species
transport may be enhanced by one or two orders of magnitude. This process could
be a possible explanation of the active transport mechanism in bone claimed by
Petrov and Pollack [ 119 ].
Such a charge effect within the lacuno-canalicular space has been observed in
the in vivo tracer experiments of Tami et al. [ 147 ]. Indeed these authors showed
that the transport mechanisms of negatively charged and neutral dextran particles
were not the same within the lacuno-canalicular pores. If the neutral species
remained confined in the vasculature, the anions did penetrate inside the