Biomedical Engineering Reference
In-Depth Information
Fig. 17.3 Electrophoresis analysis for a triphasic tissue with monovalent counterions. The finite
element mesh is shown at left , in its reference configuration. For this analysis, h
1mm,the
ambient pressure is zero on both sides , the ambient concentration of cation and anion is 150 mM
on both sides ,and ψ 0 = 10 mV on the left (upstream face), and zero on the right (downstream)
face. The fixed charge density in the reference configuration is c r =− 200 mEq / L. The solution
at steady state ( 4 panels at right ) demonstrates that the tissue is in a swollen state (swelling to
the left ) due to the Donnan osmotic pressure in the interstitial fluid (0 . 114 MPa). The cation flux
(electrophoresis), current density (electric conduction), and fluid flux (electroosmosis) are directed
from left to right , whereas the anion flux (electrophoresis) is in the opposite direction. The electric
potential ψ inside the tissue differs from the externally applied potential ψ by the value of the
Donnan potential
=
the tissue and is resisted by the solid matrix stress. Therefore, under traction-free
swelling conditions, triphasic materials are not generally in a stress-free state, unless
the fixed charge density is zero, or the external bath has an osmolarity far exceeding
the tissue fixed charge density (theoretically, if the bath osmolarity is infinite).
In a finite element analysis, the initial state of the solid is stress-free (this is the
reference state). Therefore, to achieve a traction-free swelling condition prior to
subsequent loading, it is necessary to first perform an analysis whereby the fixed
charged density is changed from zero to the desired value (or the bath osmolarity
is decreased from a very high value down to the desired ambient conditions). Once
this free-swelling equilibrium has been achieved, the desired electric potential dif-
ference may be applied between upstream and downstream baths. At steady-state,
the following phenomena are observed (Fig. 17.3 ): (i) The positively charged cation
flows from the higher to the lower electric potential, whereas the negatively charged
anion flows in the opposite direction. This solute flux under the action of an exter-
nally applied electric potential is called electrophoresis. The cation and anion fluxes
do not cancel out, as a net transport of charge is observed from the upstream to the
downstream bath; this net flow is manifested by a non-zero current density I e ,flow-
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