Biomedical Engineering Reference
In-Depth Information
presence and persistence of stable spurious oscillations in the pore pressure
which have been attributed to incorrect initial incompressibility constraints
29,30
are in fact controlled by the ratio of time-step size to the square of the space-
step for fixed time integration weightings and physical property selections. In
general, increasing the time-step or decreasing the mesh spacing has a smooth-
ing effect on the discrete solution; however, special cases exist that violate this
generality which can be readily identified through the Von Neumann
approach.
28
The analysis also reveals that explicitly dominated schemes
(
0) and
only become possible through a decoupling of the mechanical equilibrium
(Equation 15.1a) and continuity (Equation 15.1b) equations. In the case of
unsaturated media, a breakdown in the Von Neumann results has been dem-
onstrated to occur due to boundary conditions which also influence numeri-
cal stability.
28
In practice, meshes are not uniform and problem geometries are irregular
and complex; hence, a useful approach for determining the computational
integrity of numerical solutions is to perform a mesh convergence study where
the finite element grid is successfully refined for the actual problem of interest.
This type of study would fall into the category of demonstrating that the finite
element model solutions are mathematically robust. An example of a mesh
convergence study reported by Miga et al. is described later in this section.
23
The second, more important, and difficult question of model validation is to
determine the extent to which the model equations (e.g., the consolidation
Equations (15.1a) and (15.1b)) represent enough of the physics involved in brain
tissue deformation to be useful intraoperatively. Model validation in this con-
text is often confounded by the fact that the rationale for using a computational
model in the first place is that detailed measurements are difficult to obtain in
the setting of interest. Hence, a dilemma often arises and two approaches
emerge: (1) simplify the setting of interest to the point where detailed measure-
ments become possible and assume that the findings extrapolate to the actual
situation, or (2) obtain limited, often less accurate measurements in the setting
of interest and assume that agreement with a sparsely sampled response is
indicative of results that would be obtained throughout the volume if detailed
measurement maps were possible. An example of the former approach is nicely
illustrated by the recent work of Miller.
12,31
Here,
ex vivo
pig brain specimens
were prepared and micromechanical manipulations performed in order to
develop a multiparametered viscoelastic constitutive relationship as a means of
defining and validating a computational model for robotic surgery and virtual
reality surgical systems.
Fortunately, the availability of volumetric noninvasive imaging makes pos-
sible model validation studies which can be carried out
in vivo
in both animal
and human brains where detailed measurement maps of tissue displacements
can be obtained. As a result, there is a rich opportunity to complete model
validation studies in the actual setting of interest. An interesting example of
an initial phase of
in vivo
model validation has been provided in previous
work.
23, 32
A representative quantitative result is presented in Figure 15.3, which
0.5 in Equation 15.5) are not stable for saturated media (
1, 1
S
