Biomedical Engineering Reference
In-Depth Information
(Deen,
1987
; Mackie and Meares,
1955
) as well as enhanced convection under
dynamic loading (Mauck et al.,
2003
; Albro et al.,
2008
). (iii) Changes in tissue
and cell volume due to alterations in their osmotic environment (Albro et al.
2007
,
2009a
). (iv) Partial solute exclusion from pore spaces due to steric volume and short-
range electrostatic effects (Albro et al.,
2009a
), which may depend on solid matrix
deformation and solute concentration (Lazzara and Deen,
2004
; Albro et al.,
2009b
).
(v) Deviation of the physicochemical responses of solutions from ideal behavior
with varying solute concentration and solid matrix deformation. These features were
incorporated into FEBio, under a framework described as a biphasic-solute material.
This framework has now been extended to also include triphasic materials (Lai et al.,
1991
), as described below.
17.2 Mixture Models for Solutes in Porous Media
Mixture theory is able to describe the interaction of any number of solid of fluid
constituents. Solutes may be modeled as fluid constituents in a mixture that also
contains a porous solid and a solvent; when sufficiently dilute, the volume fraction
of solutes may be neglected relative to that of the solid and solvent. In mixture the-
ory, the axioms of mass balance and momentum balance must be satisfied for each
constituent. Interactions among the constituents are represented by supply terms
that represent exchanges of mass or momentum. In the mass balance equation, mass
supplies occur only when chemical reactions take place among the mixture con-
stituents which produce mass exchanges between reactants and products (Eringen
and Ingram,
1965
). In the linear momentum balance equation, momentum supplies
may represent a combination of passive mechanisms, such as friction between con-
stituents, and active mechanisms such as momentum supply from motors (Bowen,
1976
; Ateshian et al.,
2010
).
In most applications of mixture theory for biological tissues, it has been as-
sumed that each constituent of the mixture is intrinsically incompressible (Mow
et al.,
1980
; Bowen,
1980
; Lai et al.,
1991
). This assumption implies that the true
density of each constituent is invariant in space and time, though heterogeneous
mixtures may exhibit a spatial variation in mixture density. A saturated mixture of
intrinsically incompressible constituents does not change in volume when subjected
to a hydrostatic pressure. However, when a mixture includes a porous solid matrix, a
material region defined on this solid may change in volume as fluid enters or leaves
the pore space, depending on loading and boundary conditions.
One of the challenges of presenting mixture models is the choice of notation and
the variety of ways that dependent variables describing the state of the mixture may
be represented. Part of this challenge arises from the fact that mixture theory com-
bines the classical fields of fluid mechanics, solid mechanics and chemistry. Each of
these fields may have a preferred representation for the functions of state. Another
challenge arises from the fact that intensive variables may be normalized in differ-
ent, but equally valid ways when dealing with a mixture of multiple constituents.
