Multiscale Approach to Understand the Multiphysics Phenomena in Bone Adaptation - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 47

3.1.2 Electrokinetics in the Fluid Phase of Bone Tissue

The multiphysics description of the transport phenomena in the bone interstitial
fluid requires to combine electrostatics, fluid movement and ionic transport.

• Double-layer and streaming potentials The interstitial fluid, which is composed
by a Newtonian and incompressible water-like solvent containing biochemical
agents, is a dielectric material characterized by a spherical permittivity tensor
e f ¼ e f I ; I being the unit second-order tensor. For simplification purpose, if we
represent all the charged species in the fluid by monovalent ions (seen as point
charges) characterized by their molar concentration n þ and n ; the charge

density in the fluid q f is then expressed by:

q f ¼ F ð n þ n Þ; ð 5 Þ

where F is the Faraday constant. When advected by the strain-induced
interstitial fluid velocity, the mobile charge population of the double-layer
generates the streaming currents. In parallel, to conserve charge, the move-
ment of the net charge generates an electric potential, often referred to as
streaming potential W b :
As shown in Fig. 4 , the surface charge of the pores induces asymmetric
Boltzmann distributions of the cationic and anionic concentrations which are
governed by the reduced double-layer potential u:

n ¼ n b exp ð u Þ: ð 6 Þ

Note that the reduction of electric potentials I involves the Faraday constant
F ; the ideal gas constant R and the absolute temperature T ; so that I ¼
FI = RT : In summary, the electric potential in the fluid / f is decomposed into
the sum of the double layer potential u and the streaming potential W b [ 86 ]:

/ f ¼ u þ W b : ð 7 Þ

Furthermore, the double-layer potential u obeys the Poisson-Boltzmann equation
[ 52 , 74 ]:

rrð W b þ u Þ¼ 1
L D

sinh u : ð 8 Þ

The Debye length L D ¼ e f RT =ð 2F 2 n p characterises the thickness of the diffuse

ionic layer. When the pore size is large when compared to the Debye length, the
hyperbolic sine of Eq. ( 8 ) can be linearized (sinh u ' u) to obtain the Debye-
Hueckel approximation [ 84 ].

• Fluid movement If the interstitial fluid is assumed Newtonian and incompress-
ible, its constitutive law is [ 74 ]: