Biomedical Engineering Reference
In-Depth Information
22.3 Mathematical Modeling and Anatomy Acquisition
22.3.1 Background
The primary theoretical development of the relationship between the pressure of a
fluid permeating a solid matrix and its constituent displacement was established by
Biot (
1941
). It is well known that a soil under an applied load does not conform to
an instantaneous deflection. Instead, it settles gradually (due to the soil adapting to
that load) at a variable rate. This settlement of soil to the load variation is termed
consolidation. Consolidation is analogous to an elastic 'sponge' saturated with wa-
ter deforming under load. The consequence of an applied load onto this system
would yield a gradual settlement proportional to the rate of water being squeezed
out of the voids of the elastic skeleton. Terzaghi (
1943
) initially applied the afore-
mentioned principles on a restricted one-dimensional case of a column under an
applied constant load. Biot extended Terzaghi's one-dimensional work to portray
the number of physical constants necessary to determine the properties of the soil,
along with the general equations necessary to predict the settlements and stresses in
a three dimensional form. In doing so, he made the following assumptions regarding
the soil: (i) isotropic; (ii) under the final equilibrium conditions, the stress-strain re-
lationships (assumed linear) can be reversed, in addition, small strains are assumed;
(iii) water in the pores is assumed incompressible; (iv) Darcy's law applies to wa-
ter flowing through the porous skeleton. It must be noted that the assumption of
isotropy is not a stringent one, and may be replaced by anisotropy. Assumptions (ii)
and (iii) on the other hand warrant more attention. Surface tension forces allocate a
soil grain configuration of minimum potential energy. This occurs since any changes
would take place at an infinitely slow rate, and so the final state of the soil is indepen-
dent of the path taken to reach this stage. This concept ties in well with assumption
(ii), since if one considered the potential energy negligible in this case; energy could
theoretically be drawn out of applied loading/unloading within a closed cycle. Re-
versibility is assigned on the basis of small strains, since a slight perturbation in the
macroscopic sense is unlikely to disturb the existing grain configuration.
22.3.2 Multiple-Network Poroelastic Theory (MPET)
Regulation of CSF and tissue displacement within the cerebral environment was
modeled by considering a spatio-temporal model. MPET theory is actively used in
geotechnical engineering to model materials which have naturally fractured fluid
passages of various sizes and which possess both storage and transport porosities.
These reservoirs intrinsically contain various subdivisions of permeabilities and are,
therefore, aptly suited to a multiple-poroelastic network (Bai et al.,
1993
). MPET
amalgamates fundamental conservation of mass and momentum principles, stress-
strain relationships, the Terzaghi effective stress principle and porous flow laws. In
the system in question, a solid matrix (
s
) is permeated by
a
=
1
,...,A
number
