Multiscale Approach to Understand the Multiphysics Phenomena in Bone Adaptation - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 56

neglected and classical poro-elasticity seems appropriate to describe the averaged
hydro-mechanical phenomena of cortical tissue.
A limitation of this optimistic answer to the first question appears under the
light of the more recent descriptions of the lacuno-canalicular structure. Indeed,
when considering the typical fiber-to-fiber distance in the pericellular matrix that
partially occupies the canalicular space [ 166 ] or the canalicular structures
described by Anderson and Knothe Tate [ 6 ] or McNamara et al. [ 97 ] the pore size
may be nanometric, inducing a Donnan pressure of the same order of magnitude as
the hydraulic pressure.
4.2 Second Question: Is the Classical Darcy Law Sufficient
to Describe Interstitial Fluid Flow?
In parallel to the Biot law that is involved in the strain-induced interstitial fluid
flow estimation, the possible implications of multiphysics phenomena in the fluid
transport have to be investigated. In the classical Biot theory, the ability of bone to
transmit fluid is quantified through the Darcy law by the intrinsic permeability
(m 2 ). In anisotropic media, the permeability is characterized by a second-order
tensor. This tensor is both symmetric and positive definite. For isotropic media,
this tensor is characterized by an unique scalar j : This scalar is a textural
parameter only depending on the porous network geometry [ 78 ]. The more precise
the quantification of this parameter at the osteocyte scale (lacuno-canalicular
permeability), the better the understanding of the mechano-sensation process
becomes. However, as visible in Table 2 , the determination of this parameter is
quite controversial.
To understand the uncertainties on the values of this parameter, according to
our multiphysics description of the Darcy law, some coupled effects such as
electro-osmosis could generate an opposite flow, resulting in a decrease in the
apparent permeability [ 3 , 78 ]. Using the coupled Darcy law given by ( 22 ), it has
been shown that, for physiological conditions, the osmotic and electro-osmotic
parts of the macroscopic Darcian velocity introduced represent less than 7 % of the
whole interstitial macroscopic flow (see Fig. 7 ), depending on the value of the
pericellular fibers permeability k f quantifying the volume viscous force F b induced
by the fibrous matrix that partially occupies the canalicular space [ 85 ].
Thus, a classical purely hydraulic Darcy law is sufficient to roughly describe the
bone fluid macroscopic movement. A strong limitation of this result is that the
pericellular matrix is made of glycans proteins that do also present a charge
density. This indicates that the meaningful pore size that should be considered for
the electro-chemical transport could be the typical pericellular fiber-to-fiber dis-
tance instead of the canalicular radius. In particular, it was shown that for so thin
pores of a few nanometers, the electrically induced flow could compensate the
hydraulic flow, resulting in an apparent permeability decrease [ 78 ].