Biomedical Engineering Reference
In-Depth Information
The total streaming current vector
j
is a flux vector similar to the fluid mass flow
rate vector
. Linear irreversible thermodynamics [81-84] provides a structure for
relating these fluxes. For small departures from equilibrium in isotropic materials,
the total streaming current vector
q
j
and the fluid mass flow rate vector
q
are
related by
(
qp
)
(
qV
)
(
jp
)
(
jV
)
q
=−
K
·∇
p
+
K
·∇
V
and
j
=
K
·∇
p
−
K
·∇
V
(9.1)
where the superscripted
matrix coefficients are material properties. These
equations state that the total streaming current vector
K
and fluid mass flow
rate vector
q
are linearly dependent upon the voltage gradient as well as the
pressure gradient. The first equation of Eq. (9.1) when
∇
V
=
0 is Darcy's law (i.e.,
q
=−
K
j
(
qp
)
(
jV
)
·∇
V
)
is Ohm's law in its field version. The Onsager reciprocity theorem relates the
cross-flux coefficients,
K
·∇
p
). The second equation of Eq. (9.1) when
∇
p
=
0(
j
=−
K
(
jp
)
.
The significant and useful connection between pressure and voltage arises from
the recognition that the convective and the conduction currents are equal and
opposite so that there is no net current flow,
(
qV
)
=
K
j
=
0; thus, from the second equation
of Eq. (9.1):
(
jp
)
]
−
1
(
jV
)
∇
p
=
[
K
·
K
∇
V
(9.2)
Integration of this result yields the fact that the pressure must be proportional to
the voltage plus a function of time. This means that when a potential difference
in a bone is measured in connection with an electrokinetic event, the potential
difference is proportional to a pore pressure difference between the same two points
used to measure the potential difference. Since electrodes are much smaller than
pore pressure probes, this theory provides a useful tool for probing the poroelastic
response of the bone. The formulation of the model presented by Salzstein and
Pollack [83] was extended to the Biot poroelastic formalism, and it removed the
incompressibility assumption [1, 2]. The revised model has been applied to study
the mechanosensory system in the bone [1, 2, 85-87].
The anatomical site in the bone tissue that contains the fluid source of the
experimentally observed SGPs was not agreed upon, but it was argued by Cowin
et al
. [2] that it should be the PLC. That argument is summarized here. Earlier, it
had been concluded that the site of SGP creation was the collagen-hydroxyapatite
porosity of the bone mineral, because small pores of approximately 16 nm radius
were consistent with their experimental data if a poroelastic-electrokinetic model
with unobstructed and connected circular pores was assumed [83]. In [2], using
the model of Weinbaum
et al
. [1], it is shown that the published data [83, 88,
89] are also consistent with the argument that the larger pore space (100 nm)
of the PLC is the anatomical source site of the SGPs if the hydraulic drag and
electrokinetic contribution associated with the passage of bone fluid through
the surface matrix (glycocalyx) of the osteocytic process are accounted for. The
mathematical models [1, 83] are similar in that they combine poroelastic and
electrokinetic theories to describe the phase and magnitude of the SGP. The two
theories differ in the description of the interstitial fluid flow and streaming currents
