Biomedical Engineering Reference
In-Depth Information
4.3.6. Finite-element mesh generation
Solving the systemof equations in Section 4.3.5 requires the use of a numerical
method such as the FE method. An important initial step in FE analysis is mesh
generation, which involves discretization of the multidimensional domain of the
problem into small elements of simple geometry, such as tetrahedra or hexahedra
for 3D problems. Although hexahedra are known to have higher accuracy than
tetrahedra for the same computational cost, tetrahedra are more popular in mesh
generation for complicated geometries, such as the brain, because they are easier
to generate and automatically refine.
We developed a tetrahedral mesh generator capable of automatically creating
FE meshes from segmented medical images [99]. The input to the mesh generator
is an image that is segmented into brain tissue (both white and gray matter),
ventricular CSF, tumor, and edema, if present. The output meshes satisfy a number
of requirements for accurate FE simulations. For example, tetrahedra in the mesh
conform to the boundaries of the underlying segmented image, have sizes that vary
across the domain according to a user-defined sizing function, and have an element
quality that is sufficient for accurate FE computations. Since ABAQUS was used
in solving the FE problem, the tetrahedral quality measure used by ABAQUS was
adopted. The quality of a tetrahedron
T
is therefore given by
V
T
V
r
q
T
=
,
(11)
where
V
T
is the volume of
T
, and
V
r
is the volume of the equilateral tetrahedron
that can be inscribed in the circumsphere of
T
. For an equilateral tetrahedron,
q
T
is 1.0, and it approaches 0 as the tetrahedron becomes degenerate.
The mesh generation approach starts by casting a regular grid of cubes over the
domain of the input image. The size of the cubes is a user-defined input parameter.
Cubes totally outside the input image labels are removed, and each of the remaining
cubes is tesselated into 5 tetrahedra. Mesh refinement is then performed to make
the tetrahedra satisfy the sizing function, which is defined over the domain of
the input image. Values of the sizing function reflect the maximum acceptable
length of an overlying edge of a tetrahedron. Values of the sizing function may
be specified for each label in the image, derived from the surface curvature, or
defined by the user arbitrarily. Mesh refinement is carried out via the longest edge
propagation path algorithm [100], which guarantees a lower bound on the quality
of the generated tetrahedra after subdivision.
After mesh refinement, tetrahedra that straddle a boundary in the segmented
image are forced to conform to this boundary by attempting a number of local mesh
operations and choosing the one that produces the best quality of the associated
tetrahedra. These local mesh operations are relocation of a mesh node to the
boundary, splitting of an edge at the point where it crosses the boundary, or its
collapse to that point. As the result of these operations, the quality of the elements
