Due to a trade-off between spatio-temporal
resolution and signal-to-noise ratio cardiac
MR images typically have strongly anisotropic
voxels. If such image are segmented and meshed,
the resulting meshes usually have irregular
triangles and pronounced terracing artifacts,
none of which is desirable. An example of these
effects can be seen in Fig. 1, where the
marching cubes methods was used to generate the
left ventricular endocardial surface from a
cardiac MRI scan.

Figure 1: A left ventricular surface
model generated by applying the marching cubes
algorithm to a segmented cardiac MR image with
1.44 mm in-plane resolution and 8 mm slice
thickness. The surface has pronounced terracing
artifacts and irregular triangles. The figure
is from [1] and it is used with permission;
Copyright © 2009 Hindawi Publishing
Corporation; All rights reserved.

To address the above problems we have
developed a novel method for the generation of
myocardial wall surface meshes from segmented
3D MR images. The method maps a premeshed
sphere to the surface of the segmented
object. The mapping is defined by the gradient
field of the solution of the Laplace equation
between the sphere and the surface of the
object. The steps of the method are illustrated in
Fig. 2 and method details are given in [1].

Figure 2:
Mesh generation summary. The input image (a)
is segmented into the object and background,
resulting in a binary image (b). A sphere
enclosing the object is centered at the
object barycenter (c). The sphere is
uniformly sampled with the number of points
equal to the number of singularities. The
binary image is resampled with isotropic
voxels and the Laplace equation is
numerically solved between the sphere
(boundary condition of 0) and the object
(boundary condition of 1). The solution of
the Laplace equation is encoded in the gray
levels in (c) and (d). The binary object is
eroded, and the points are propagated from
the sphere to the eroded object in the
direction of the gradient of the Laplace
equation solution to define the singularity
locations, shown as red squares in (d) and
(e). Boundary points, specified as midpoints
for each pair of neighboring voxel, where
one voxel is in the object and the other is
in the background, are shown as red dots in
(e). The singularity locations as well as
the boundary points are used to specify the
analytic solution of the Laplace
equation. The boundary points are propagated
in the negative gradient direction of the
solution of the Laplace equation from the
object boundary to the sphere (f). Their
values of the underlying solution of the
Laplace equation are interpolated at the
sphere to the define the stopping
function. The number of degrees of freedom
of the stopping function is defined by the
number of control points, which are shown as
blue circles in (g). An approximately
uniform mesh is generated on the sphere. The
vertices of the mesh on the sphere, shown as
black crosses in (g), are propagated from
the sphere in the direction of the gradient
of the solution of the Laplace equation
until the value of the underlying solution
of the Laplace equation is equal to the
corresponding value of the stopping
function. The propagated mesh nodes define
the final mesh, shown in (h). Figures
(a)-(h) are two dimensional for illustration
purposes, while the method is three
dimensional. The figure
is from [1] and it is used with permission;
Copyright © 2009 Hindawi Publishing
Corporation; All rights reserved.

The method allows for the direct control of
the number of mesh vertices and triangles (the
two are directly related for closed
surfaces). This is illustrated in Fig. 3 for
the case of right ventricular endocardial
surface, where the number of mesh vertices was
varied from 200 to 5000.

Figure 3:
Each column shows a mesh on the sphere and
the corresponding right ventricular mesh
obtained by propagating the mesh from the
sphere to the right ventricular surface. The
numbers of mesh vertices are 200 for (a) and
(e), 500 for (b) and (f), 1000 for (c) and
(g), and 5000 for (d) and (h). The figure
is from [1] and it is used with permission;
Copyright © 2009 Hindawi Publishing
Corporation. All rights reserved.

The same algorithm can be
used to generate surface meshes of the
epicardium and endocardium of the four cardiac
chambers. The generated meshes are smooth
despite the strong voxel anisotropy, which is
not the case for the marching cubes and related
methods. While the proposed method generates
more regular mesh triangles than the marching
cubes and allows for a complete control of the
number of triangles, the generated meshes are
still close to the ones obtained by the
marching cubes. The method was tested on 3D
short-axis cardiac MR images with strongly
anisotropic voxels in the long-axis
direction. For the five tested subjects, the
average in-slice distance between the meshes
generated by the proposed method and by the
marching cubes was 0.4 mm.

References:

[1] Skrinjar, O., Bistoquet, A., "Generation of
Myocardial Wall Surface Meshes from Segmented
MRI", International Journal of Biomedical
Imaging, vol. 2009, Article ID 313517, 10
pages, 2009. DOI:10.1155/2009/313517. LINK
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