Biomedical Engineering Reference
In-Depth Information
Generally, the smooth geometry of biological tissue has been represented by
defining geometric finite element mesh with appropriate basis function classes.
The concept of basis functions is used to discretize the physical geometric domain
into multiple geometric elements that can be iteratively fitted to the shape of one
part of the organ using the finite element method. For example, the geometric field
value in a one-dimensional geometric field can be interpolated by a linear
Lagrange basis function over a local coordinate system denoted by n, as follows,
u
ð
n
Þ¼ð
1
n
Þ
u
1
þ
nu
2
;
ð
1
Þ
where the field value, u is varied linearly between the nodal values (u
1
at n
¼
0; u
2
at n
¼
1). Higher order geometric basis functions such as quadratic or cubic
functions can also be used to represent the geometric field. Even though the
Lagrange basis functions provide continuity of u across elemental boundaries, they
do not maintain higher order continuity, i.e., the derivatives on the curvature in
terms of geometry, are not continuous, and therefore the smoothness of the bio-
logical tissue is not represented accurately using the Lagrange basis function. One
approach that has been adopted is to define these additional derivative parameters,
such as
ou
on
, and ensure that both the field value and the derivative are continuous
across the elemental boundaries, e.g., in the form of cubic Hermite basis functions
as follows,
u
ð
n
Þ¼
u
1
þ
u
0
1
n
þð
3u
2
3u
1
2u
0
1
u
0
2
Þ
n
2
þð
u
0
1
þ
u
0
2
þ
2u
1
2u
2
Þ
n
3
;
ð
2
Þ
where u
1
, u
0
1
and u
2
, u
0
2
are the nodal values and derivatives, respectively. A pre-
vious study has adopted the cubic Hermite basis functions to track the center-line
of the small intestine based on imaging evidence from the Visible Human Project
[
37
]; the wall of the intestinal model was then projected outward with a constant
radius to create an intestinal lumen space (Fig.
1
a).
Even though the model constructed from the Visible Human Project success-
fully generated the gross anatomy of the human small intestine, a more detailed
representation of the intestinal micro-structure, in particular the muscle layers,
would be required for a more realistic electromechanical model. Morphological
studies of segments of small intestine have found that the radius and wall thickness
are fairly uniform within each segment. Therefore, a simple hollow cylindrical
tube can be used to represent a relatively short segment of the small intestine, in
order to reduce the computation required to apply the anatomical model in an
electromechanical modeling framework.
As the thickness and radial dimensions of intestines vary considerably between
species. We chose to focus on a 40 mm segment of a rat small intestine as the rat
species has been used in a number of experimental studies. Based on imaging and
morphometric evidence [
1
,
19
], the rat intestine has an external radius of 2.3 mm
and a thickness of 0.9 mm. Figure
1
b shows an idealized geometric model of the
small intestine, defined by 216 nodes and 128 geometric elements with a combi-
nation of different types of basis functions. In a three-dimensional space, the
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