Biology Reference
In-Depth Information
Finite-Element Modeling of Cardiac Ventricular Anatomy
The structure of the cardiac ventricles can be modeled using finite-
element modeling (FEM) methods developed by Nielsen et al. [78]. The
geometry of the heart to be modeled is described initially using a
predefined mesh with six circumferential elements and four axial ele-
ments. Elements use a cubic Hermite interpolation in the transmural
coordinate (h), and bilinear interpolation in the longitudinal (m) and
circumferential (q) coordinates. Voxels in the 3D DTMR images identi-
fied as being on the epicardial and endocardial surfaces by the
semiautomated contouring algorithms described above are used to
deform this initial FEM template. Deformation of the initial mesh is
performed to minimize an objective function F(
n
):
2
D
2
2
2
F
()
n
=
γ
d
v
( )
e
−
v
+
{
a
∇+
n
b
(
∇∂
n
)}
e
∑
∫
(7)
d
d
ℜ
2
d
=
1
where
n
is a vector of mesh nodal values,
v
d
are the surface voxel data,
v
(e
d
) are the projections of the surface voxel data on the mesh, and a
and b are user-defined constants. This objective function consists of
two terms. The first describes distance between each surface image
voxel (
v
d
) and its projection onto the mesh
v
(e
d
). The second, known
as the weighed Sobelov norm, limits the stretching (first derivative
terms) and the bending (second derivative terms) of the surface. The
parameters a and b control the degree of deformation of each element.
The weighted Sobelov norm is particularly useful in cases where there
is an uneven distribution of surface voxels across the elements. A linear
least-squares algorithm is used to minimize this objective function.
After the geometric mesh is fitted to DTMRI data, the fiber field is
defined for the model. Principal eigenvectors lying within the bound-
aries of the mesh computed above are transformed into the local
geometric coordinates of the model using the following transformation:
V
G
= [
FG H
]
T
[
R
]
V
S
(8)
where
R
is a rotation matrix that transforms a vector from scanner
coordinates (
V
S
) into the FEM model coordinates
V
G
, and
F
,
G
,
H
are
orthogonal geometric unit vectors computed from the ventricular
geometry as described by LeGrice et al [86]. Once the fiber vectors are
represented in geometric coordinates, DTMRI inclination and imbrication
angles (a and f) are fitted using a bilinear interpolation in the local e
1
and e
2
coordinates, and a cubic Hermite interpolation in the
ε
3
coordinate.
A graphical user interface for fitting FEMs to both the ventricular sur-
faces and fiber field data has been implemented using the MatLab
programming language (available at
www.ccbm.jhu.edu).
An example
of an FEM fitted to the epicardial surface of a reconstructed normal
canine heart obtained using this software tool was shown in figure 9.7a.
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