Validation of Image-Based Method for Extraction of Coronary Morphometry

Abstract

An accurate analysis of the spatial distribution of blood flow in any organ must be based on detailed morphometry (diameters, lengths, vessel numbers, and branching pattern) of the organ vasculature. Despite the significance of detailed morphometric data, there is relative scarcity of data on 3D vascular anatomy. One of the major reasons is that the process of morphometric data collection is labor intensive. The objective of this study is to validate a novel segmentation algorithm for semi-automation of morphometric data extraction. The utility of the method is demonstrated in porcine coronary arteries imaged by computerized tomography (CT). The coronary arteries of five porcine hearts were injected with a contrast-enhancing polymer. The coronary arterial tree proximal to 1 mm was extracted from the 3D CT images. By determining the centerlines of the extracted vessels, the vessel radii and lengths were identified for various vessel segments. The extraction algorithm described in this paper is based on a topological analysis of a vector field generated by normal vectors of the extracted vessel wall. With this approach, special focus is placed on achieving the highest accuracy of the measured values. To validate the algorithm, the results were compared to optical measurements of the main trunk of the coronary arteries with microscopy. The agreement was found to be excellent with a root mean square deviation between computed vessel diameters and optical measurements of 0.16 mm (<10% of the mean value) and an average deviation of 0.08 mm. The utility and future applications of the proposed method to speed up morphometric measurements of vascular trees are discussed.

Appendix

Computer-Assisted Extraction of Morphometric Data from CT Volumetric Images

The algorithm for extraction of curve-skeletons and determining morphometric data from volumetric images consists of several steps. A detailed description of all steps involved in the algorithm can be found below along with the theoretical framework for the methodology.

Topological Analysis of Vector Fields

The algorithm utilized in this study uses the topology of a vector field defined on the faces of a tetrahedralized set of points. Thus, the vector field is defined by three vectors located at the vertices of a triangle. The vector field inside the triangles is interpolated linearly by computing the barycentric coordinates of the point within the triangle. These coordinates are then used as weights for linearly combining the three vectors defined at the vertices of the triangle to compute the interpolated vector. The advantage of such a linear interpolation is an easier classification of topological features which is briefly described below.

In topological analysis, the zeros of the interpolating vector field are of interest. Synonyms for these zeros are singularities or critical points. Based on the eigenvalues of the matrix of the interpolating vector field, these critical points can be separated into different groups. Within each group, the vector field assumes similar characteristics. Very detailed analysis of these groups and their characteristics can be found in the literature.16,44 In order to identify points on the centerline, singularities where the vectors point toward that specific point are of interest. These types of singularities are attracting node and focus singularities (both eigenvalues of matrix A are negative), as well as attracting spiral singularities (eigenvalues of matrix A have non-zero imaginary part) as depicted in Fig. 6a–d.

Figure 6

Types of singularities that are relevant for topological analysis and for identifying centerlines: (a) saddle singularity of a vector field including surrounding flow depicted by arrow glyphs, (b) node singularity of a vector field including surrounding flow depicted by arrow glyphs, (c) focus singularity of a vector field including surrounding flow depicted by arrow glyphs, and (d) spiral singularity of a vector field including surrounding flow depicted by arrow glyphs

Methodology for Extraction of Quantitative Information

The algorithm for determination of the curve-skeleton consists of several steps. Since the object is given as a volumetric CT-scanned image, the object boundary must to be extracted first. A vector field is then computed that is orthogonal to the object boundary surface. Once the vector field is computed, the curve-skeleton can be determined by applying a topological analysis to this vector field. As a last step, gaps between segments of the curve-skeleton can be closed automatically and vessel diameters can be computed. The following subsections explain these steps in detail.

Extraction of Object Boundary

The CT-scanned vasculature is defined by a volumetric image. A volumetric image consists of voxels aligned along a regular 3D grid. It is generally not likely that the boundary of the vessels is exactly located at these voxels. Hence, better precision can be achieved by finding the exact location in between a set of voxels. Since an accurate representation of the object boundary is crucial to the algorithm, improvement of the precision is an essential step. The method used within the described system uses similar techniques as described by Canny’s non-maxima suppression8 but extended to three dimensions.

First, the image gradient is computed for every voxel. Using an experimentally determined threshold, all voxels with a gradient length below this threshold are neglected. The gradients of the remaining voxels are then compared to their neighbors to identify local maxima along the gradient. In 3D, the direct neighborhood of a single voxel generally consists of 26 voxels forming a cube that surrounds the current voxel. In order to find the local maximum along the current gradient, the gradients of the neighboring voxels in positive and negative directions have to be determined. When using 2D images, nearest neighbor interpolation of these gradients17 may work but yield incorrect results in a 3D volumetric image. Therefore, the gradients on the boundary of the cube formed by the neighboring voxels are interpolated linearly to determine a better approximation of the desired gradients. Figure 7a explains this in more detail where the current voxel marked as a triangle and the direct neighbors forming a cube are shown. The current gradient is extended to the faces of the cube starting at the current voxel. The resulting intersections, marked as diamonds, define the locations for which the gradients have to be interpolated so that the maximal gradient can be determined. The current implementation of the described system uses linear interpolation. Using this method, only very few cases require a pre-filtering to remove noise in data sets. The best results can be achieved by the use of an anisotropic diffusion filter. The five data sets used in this study were not pre-filtered.

Figure 7

(a) Determination of the maximum gradient with sub-voxel precision of a voxel (marked as triangle) and its neighboring voxels: the gradient direction is shown combined with the locations of the interpolated gradients at the intersection of the current gradient direction with the cube defined by the neighboring voxels marked as diamonds. (b) Computation of the local maximum of the gradient (symbolically shown for one coefficient of the gradient vector). The gradient is marked as a triangle and the two interpolated gradients at the edge of the cube are shown as diamonds. The maximal gradient (circle) is determined by computing the zero of the first derivative resulting in the maximum gradient

Once the neighboring gradients in positive and negative direction of the current gradient are computed, these are compared in order to find the local maxima. Thus, if the length of the current gradient is larger than the length of both of its neighbors, the local maximum can be calculated similar to the 2D case. When interpolated quadratically, the three gradients together form a parabolic curve along the direction of the current gradient as shown in Fig. 7b. In general, the current gradient is larger than the interpolated neighbors since only local maxima are considered in this step. Hence, the local maximum can be identified by determining the zero of the first derivative of the parabolic curve. Determining all local maxima within the volumetric image in this fashion then results in a more accurate and smoother approximation of the object boundary with sub-voxel precision. Once all points on the boundary are extracted from the volumetric image using this gradient approach with sub-voxel precision, the resulting point cloud can be further processed in order to identify the curve-skeleton.

Determination of the Vector Field

The proposed method computes a curve-skeleton by applying a topological analysis to a vector field that is determined based on the geometric configuration of the object of which the curve-skeleton is to be determined. The vector field is computed at the identified points on the vessel boundary in such a way that the vectors are orthogonal to the vessel boundary surface. Based on these vectors, the vector field inside the vessels is computed using linear interpolation.

Different approaches are possible for calculating such a vector field. A repulsive force field can be determined that uses the surrounding points on the boundary surface.11 The repulsive force is defined similarly to the repulsive force of a generalized potential field.1,18 The basic idea is to simulate a potential field that is generated by the force field inside the object by electrically charging the object boundary. Alternatively, we may define a normal vector and the respective plane. The normal of this plane then defines the orthogonal vector corresponding to the current point.25

Since these are volumetric data sets, the image gradients can be used to define the vectors on the boundary surface. These image gradients are previously determined as they are needed for extraction of the boundary. Since the points are only moved along the direction of the image gradient when determining the sub-voxel precision, this image gradient is still orthogonal to the boundary surface and therefore represents a good approximation for the desired vector field.

The proposed software system uses a Gaussian matrix to compute the image gradients. Therefore, the resulting gradients are smoothed to reduce any remaining noise in the boundary representation. This also reduces the error that occurs whenever gradients are computed close to gaps within the vessel boundary. Due to the use of vector field topology methods for determining center points, the algorithm tends to be less sensitive to errors in the gradients as compared to methods that project the boundary onto the center points directly.25 In our analysis, gaps within the vessel boundary only occurred for very small vessels with diameters close to the size of a voxel due to partial volume effects. It should be noted that all three methods result in vectors pointing to the inside of the object.

Determination of the Curve-Skeleton

In order to determine the curve-skeleton of the object, a tetrahedrization of all points on the object boundary is computed first. For this, Si’s34 fast implementation of a Delaunay tetrahedrization algorithm is used. This algorithm results in a tetrahedrization of the entire convex hull defined by the set of boundary points. Thus, it includes tetrahedra that are located completely inside the vessels but also tetrahedra that are completely outside of the vessels and connect two vessels. By using the previously computed vectors that point to the inside of the vasculature, outside tetrahedra can be distinguished from tetrahedra that are located inside the vessels. Hence, all outside tetrahedra can be removed, leaving a Delaunay tetrahedrization of the inside of the vasculature only. Note that this step also closes small gaps that may exist since tetrahedra covering these gaps will still have vectors attached to the vertices which point inward. Since vectors are known for each vertex of every tetrahedron, the complete vector field can be computed using this tetrahedrization by linear interpolation within each tetrahedron. This vector field is then used to identify points of the curve-skeleton which are then connected with each other. The vectors of the remaining tetrahedra are non-zero (the tetrahedron would be removed otherwise). Thus, the trivial vector field where the vectors are zero inside the entire tetrahedron does not occur. Figure 8a shows an example of the tetrahedrization with outside tetrahedra removed as previously described for a small vessel with a diameter of about three voxels. Based on this tetrahedrization and associated vector field, the center lines can be identified.

Figure 8

(a) A bifurcation for a small vessel (three voxels in diameter). The extracted center line is shown along with the respective tetrahedrization. (b) Single slice through the tetrahedrization of the phantom data set. The point on the centerline is identified in the center of the image

Once the vector field is defined within the entire object, one could use an approach similar to the one used by Cornea et al. 11 and compute the 3D topological skeleton of the vector field which yields the curve-skeleton of the object. Since singularities are very rare in a 3D vector field, Cornea et al. introduced additional starting points for the separatrices, such as low divergence points and high curvature points, to obtain a good representation of the curve-skeleton. Therefore, a different approach is described in this paper that analyzes the vector field on the faces of the tetrahedra.

In order to perform a topological analysis on the faces of the tetrahedra, the vector field has to be projected onto those faces first. Since tri-linear interpolation is used within the tetrahedra, it is sufficient to project the vectors at the vertices onto each face and then interpolate linearly within the face using these newly computed vectors. Based on the resulting vector field, a topological analysis can be performed on each face of every tetrahedron.

Points on the curve-skeleton can be identified by computing the singularities within the vector field interpolated within every face of the tetrahedrization. For example, for a perfectly cylindrical object, the vector boundary points directly at the center of the cylinder. When examining the resulting vector field at a cross section of the cylinder, a focus singularity is located at the center of the cylinder within this cross section. The location of this focus singularity resembles a point on the curve-skeleton of the cylinder. Hence, a singularity of type node, focus, or spiral within a face of a tetrahedron indicates a point of the curve-skeleton. Since the vectors at the boundary point inwards, only sinks (i.e., attracting singularities) need to be considered in order to identify the curve-skeleton. Since not all objects are cylindrical in shape and given the numerical errors and tolerances, points on the curve-skeleton can be identified from sinks that resemble focus and spiral singularities. Figure 8b illustrates an example for a cylindrical object for which a cross-section (a slice perpendicular to the object) is shown. There are two large triangles that connect two opposite sides of the object. Based on these triangles, which resemble faces of tetrahedra of the tetrahedrization, the center point (shown in red) can be identified based on the topological analysis within these triangles.

Obviously, only faces that are close to being a cross section of the object should be considered in order to identify points on the curve-skeleton. To determine such cross-sectional faces, the vectors at the vertices can be used. If the vectors at the vertices, which are orthogonal to the object boundary, are approximately coplanar with the face, then this face describes a cross section of the object. As a test, the scalar product between the normal vector of the face and the vector at all three vertices can be used. If the result is smaller than a user-defined threshold, this face is used to determine points on the curve-skeleton. If we compute the singularity on one of these faces, then we obtain a point which is part of the curve-skeleton. Note that since linear interpolation is used within the face, only a single singularity can be present in each face. In case of bifurcations, there will be two neighboring tetrahedra which contain a singularity, one for each branch. Additionally, this approach disregards boundary points which are based on noise voxels. In order for a set of boundary points to be considered, they need to have gradient vectors that point toward the center from at least three different directions. Hence, boundary points based on noise voxels are automatically neglected because it is very unlikely that there are other corresponding boundary points in the vicinity with gradient vectors pointing in the direction of the first boundary point.

After computing the center points, the vessel diameters are computed for each center point and all points within the vicinity are identified. From this set of points, only the ones that are within the slice of the vessel used to determine the center point are selected to describe the boundary. The radius is then computed as the average of the distances between the center points and the points on the boundary of the vessel slice.

Once individual points of the curve-skeleton (including the corresponding vessel diameters) are computed by identifying the focus and spiral singularities within the faces of the tetrahedra, this set of points must be connected in order to retrieve the entire curve-skeleton. Since the tetrahedrization describes the topology of the object, the connectivity information of the tetrahedra can be used. Thus, identified points of the curve-skeleton of neighboring tetrahedra are connected with each other forming the curve-skeleton. In some cases, gaps will remain due to the choice of thresholds which can be closed using the method described in the next section.

Closing Gaps within the Curve-Skeleton

Ideally, the method described results in a vascular tree representing the topology of the vasculature exactly. Due to numerical tolerances, however, sometimes gaps may occur between parts of the curve-skeleton which can be filled automatically. Since the tetrahedrization of the points on the boundary describe only the inside of the object, the algorithm can search for loose ends of the curve-skeleton and connect these if they are close to each other. In addition, it can be verified that the connection stays within the object. To test this, those tetrahedra which are close to the line connecting the two candidates and potentially filling a gap are identified. Then, the algorithm computes how much of the line is covered by those tetrahedral; i.e., the fraction of the line contained within the tetrahedra. If all those fractions add up to 1, then the line is completely within the object and it is a valid connection. Otherwise, the connection is rejected since it would introduce an incorrect connection of two independent vessels.