Biomedical Engineering Reference
In-Depth Information
time course of displacement and pressure experienced in the column. Ini-
tially, there is instant drainage and deformation at the surface, with the load
supported by the interstitial fluid at depth. This generates a pressure gradient
acting directionally opposite to the deformation source (i.e., high pressure at
depth and low pressure at the surface). Over time as more fluid drains (i.e.,
gradual reduction in the gradient over time), the load is transferred from the
interstitial fluid to the solid matrix (i.e., gradual increase in deformation at
depth). The behavior exemplified here is often compared to that of deforming
a saturated sponge.
There is little doubt that the microscale events occurring during surgical
loading of the brain are complex. However, consolidation theory provides a
framework which captures the bulk deformation and hydraulic behavior
associated with surgical deformations. Although a viscoelastic response was
not explicitly incorporated in the above description, alteration of the consti-
tutive equations can be readily accomplished to include these effects. How-
ever, further investigation in the context of surgical loading is needed to
better understand which modeling terms are most important for nonrigid
intraoperative registration of preoperative data. In the next section, we
briefly discuss the issues involved in solving equations such as (15.1a-b) on
complex spatial domains.
15.4
Finite Element Methodology
The finite element (FE) method is a classical engineering analysis technique
that produces solutions to PDEs which describe complex systems and pro-
cesses and are spatially distributed. It has been widely used in structural
and continuum mechanics, and has become very popular in biomedical
applications as well. In essence, the FE strategy divides the domain of inter-
est (e.g., the brain) into an interconnected set of subregions or elements
which fill the volume of interest. Discrete approximations to the PDEs that
govern the physical processes to be simulated (e.g., consolidation theory)
are developed on each element which can possess its own local properties
(e.g., gray versus white matter), thereby allowing complicated geometries
and tissue heterogeneties to be represented through a simple building block
structure. In the limit of vanishingly small elements, the FE approximation
to the PDE solution converges to its analytical continuum, provided that the
principles of numerical consistency and stability are satisfied (e.g., see ref-
erence 26). Thus, given sufficient resolution of the geometry of interest, FE
methods produce highly accurate solutions to complex equations under
realistic conditions.
Following the development by Paulsen et al.,
19
Galerkin weighed resid-
ual discretization of Equation (15.1) begins with volume integration after
