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This paper presents a novel method for the generation of myocardial wall surface meshes from segmented 3D MR images, which typically have strongly anisotropic voxels. 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 same algorithm is independently used to generate the 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.

Surface models of the epicardium and endocardium of the heart chambers are used in a number of biomedical applications for visualization [

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 8.0 mm slice thickness. The irregular triangles are a consequence of the voxel anisotropy. The surface mesh has pronounced terracing artifacts, and the number of triangles is directly related to the number of voxels in the image.

In this paper we present a method for generation of myocardial wall surface meshes from segmented MRI. The meshes are smooth and have prespecified number of triangles and close-to-regular triangles despite the highly anisotropic voxels. Since the marching cubes are the most widely used method, either as a stand-alone method or as a part of other methods, we compare the proposed method to the marching cubes.

The presented method is designed for surface mesh generation of the endocardium of the four cardiac chambers and of the endocardium from segmented cardiac MRI. The four endocardial surfaces are, if the valves are ignored, topologically equivalent to a sphere. We also assume that the segmentation of the entire heart does not include other structures, which makes its outer surface topologically equivalent to a sphere. The main idea is to generate a triangulated mesh on a sphere and then map it independently to the five surfaces. For each segmented object we construct its surface in implicit form and then map the mesh from the sphere to the surface using the gradient field of the solution of the Laplace equation between the surface and the sphere. Each step of the method is explained in the following sections, and Figure

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.

It can be shown that a sphere cannot be triangulated with an arbitrary number of equilateral triangles. In fact, there are only three configurations of a triangulated sphere with equilateral triangles: regular tetrahedron (4 equilateral triangles), regular octahedron (8 equilateral triangles), and regular icosahedron (20 equilateral triangles) [

In order to construct the surface mesh, we define a homeomorphic mapping from the sphere to the surface and apply it to the mesh on the sphere. There are infinitely many ways to construct such a mapping, and here we define a scalar field

While expression (

To map the mesh from the sphere to the object surface, we propagate each mesh vertex along the gradient of

To uniformly place the singularities inside the object relatively close to the object surface, we first approximately uniformly sample the sphere (as explained in Section

To increase the accuracy of the fitting of the object surface to the boundary points, instead of propagating the mesh from the sphere to the level set of

To represent the stopping function we use pseudothin plate spline model on the sphere proposed by Wahba [

At this point the singularity locations as well as coefficient

Coefficient

We propagate the mesh vertices from the sphere to the object surface according to (

Instead of using the value of

To quantify the closeness of the meshes generated by the marching cubes and the proposed method we compute the in-slice distance between the respective mesh cross-sections in a given slice (results given in Section

The method was tested using anatomical cardiac MRI scans of five healthy volunteers. The scans were acquired using steady-state free-precession short axis cine imaging with a 1.5 T clinical MRI scanner (Intera, Philips Medical Systems, Best, The Netherlands). The scans had 12-17 contiguous short axis slices with 256

The presented method has three parameters: the number of singularities (

In this section we analyze the effect of the parameter values on the resulting mesh. The studies were done on the right ventricle of one of the subjects, since the right ventricle is more curved than the other three chambers and is the only chamber that has both convex and concave regions.

In the first study we generated a sequence of surface meshes by increasing the number of singularities. Then we computed the average distance between consecutive meshes in the sequence to quantify the change the mesh undergoes as

Average distance between consecutive meshes as a function of the number of singularities.

In the second study we generated a sequence of surface meshes by increasing the number of control points. Then we computed the average distance between consecutive meshes in the sequence to quantify the change the mesh undergoes as

Average distance between consecutive meshes as a function of the number of control points.

In the third study we generated a sequence of surface meshes by increasing the number of mesh vertices. Then we computed the average distance between consecutive meshes in the sequence to quantify the change the mesh undergoes as

Average distance between consecutive meshes as a function of the number of mesh vertices.

To measure the mesh quality, we use a triangle quality index suggested in [

We generated a sequence of surface meshes by increasing the number of mesh vertices. Then we computed the average quality index for each mesh in the sequence (Figure

Average triangle quality index as a function of the number of mesh vertices.

It should be noted that the average

Each row 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 for the four rows are 200, 500, 1000, and 5000. The corresponding mean

The method was tested on the endocardial surfaces of the four cardiac chambers as well as on the epicardial surface of the entire heart for five subjects. The numbers of singularities, control points, and mesh vertices used for the test are reported in Table

The average in-slice distance (

LV endocardium | 227 | 204 | 1180 | 0.4 |

RV endocardium | 246 | 225 | 1220 | 0.3 |

LA endocardium | 62 | 66 | 748 | 0.5 |

RA endocardium | 57 | 58 | 684 | 0.4 |

Epicardium | 422 | 406 | 2472 | 0.3 |

Endocardium surface meshes generated by the proposed method for the left ventricle (red), right ventricle (green), left atrium (blue), and right atrium (yellow).

Epicardium surface mesh generated by the proposed method for the entire myocardium.

Since the marching cubes are the most widely used method, either as a stand-alone method or as a part of other methods, we compared the endocardial and epicardial surface meshes of the five subjects generated by the proposed method to the corresponding surface meshes generated by the marching cubes. The comparison was done in the short-axis slices since the in-plane resolution was 5 times higher than the out-of-plane resolution. Figure

Contours of endocardial meshes generated by the marching cubes (yellow) and the proposed method (red) in short-axis sections for (a) left ventricle, (b) right ventricle, (c) left atrium, and (d) right atrium, and in a long-axis section for (e) left ventricle. The endocardial boundaries are defined by the blood pool segmentation shown in the binary images.

The presented method can be used for the surface mesh generation of any object that is topologically equivalent to a sphere. While the method can be extended to control the triangle size based on the surface curvature, there is no need for such an approach in the case of myocardial wall surfaces since they are not highly curved. The method, unlike other mesh generation methods, allows for a direct control of the number of triangles and vertices in the mesh, which is particularly useful in modeling (e.g., FEM) applications.

The method can be used for the generation of surfaces that are not closed. For example acquired cardiac MRI might not contain slices going through the apex and the base, in such case the corresponding endocardial and epicardial surfaces are not closed. In such cases one can segment the acquired slices, generate the mesh using the proposed method and then clip the bottom and top part of the mesh. This was done for the two atria in Figure

In the examples presented in this paper we generated triangulated surface meshes. However, the method is independent of the type of the mesh; that is, it can be used with any mesh elements as long as the sphere can be meshed with such elements. Once the sphere is meshed, the vertices of the mesh are propagated from the sphere to the surface of the segmented object in the way explained in Section

Figures

While the number of singularities and the number of control points control the smoothness of the underlying implicit continuous surface, the number of mesh vertices affects the triangulation of the continuous surface. From Figure

The reason why the graphs in Figures

The constant and relatively high value of the triangle quality index in Figure

The proposed method generates meshes that are very close to the ones obtained by the marching cubes (Figure

The proposed method can be used with segmented images that have anisotropic voxels. The segmentation boundary points are not strictly interpolated. Instead, they are approximated with an implicit surface that fits them in the least square sense. The surface smoothness versus the goodness of fit is controlled by the number of singularities and number of control points, which can take the same value. If the implicit surface is smooth then the resulting mesh is also smooth. Thus, there is no need for artificial smoothing of the mesh that may shrink or affect the mesh in some other undesired way.

The entire method has been designed to completely avoid numerical optimization and consequently the problem of local extrema.

In the proposed method the segmentation boundary points are approximated with an implicit surface, which is then triangulated by propagating a regular mesh from a sphere to the surface. There are other ways to construct continuous surfaces that interpolate or approximate a given set of points (e.g., [

We note that harmonic functions have already been used to represent shapes [

In the myocardial motion analysis community researchers used a structured volumetric mesh of the left ventricle [

We have developed a novel method for the construction of endocardial and epicardial surface meshes from 3D segmented cardiac MR images with a prespecified number of vertices and triangles. Even when the voxels are strongly anisotropic, the resulting meshes are smooth and have close-to-regular triangles while closely following the segmentation.

This section describes the solution to the Laplace equation over a spherical domain that has a single singularity somewhere within the domain and that is equal to a constant on the boundary of the domain. Let the sphere center be the coordinate system origin,

The authors would like to thank Dr. John Oshinski from the Department of Radiology, Emory University, USA, for providing the images used in this work. This research was supported by the American Heart Association under Grant 0855345E.