Vascular decomposition using weighted approximate convex decomposition
Ashirwad Chowriappa, T. Kesavadas, Maxim Mokin, Peter Kan, Sarthak Salunke, Sabareesh K. Natarajan & Peter D. Scott International Journal of Computer Assisted Radiology and Surgery A journal for interdisciplinary research, development and applications of image guided diagnosis and therapy ISSN 1861-6410 Volume 8 Number 2 Int J CARS (2013) 8:207-219 DOI 10.1007/s11548-012-0766-6
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Author's personal copy Int J CARS (2013) 8:207–219 DOI 10.1007/s11548-012-0766-6
ORIGINAL ARTICLE
Vascular decomposition using weighted approximate convex decomposition Ashirwad Chowriappa · T. Kesavadas · Maxim Mokin · Peter Kan · Sarthak Salunke · Sabareesh K. Natarajan · Peter D. Scott
Received: 10 January 2012 / Accepted: 21 May 2012 / Published online: 13 June 2012 © CARS 2012
Abstract Objective Stroke treatment often requires analysis of vascular pathology evaluated using computed tomography (CT) angiography. Due to vascular variability and complexity, finding precise relationships between vessel geometries and arterial pathology is difficult. A new convex shape decomposition strategy was developed to understand complex vascular structures and synthesize a weighted approximate convex decomposition (WACD) method for vascular decomposition in computer-aided diagnosis. Materials and methods The vascular tree is decomposed into optimal number of components (determined by an expert). The decomposition is based on two primary features of vascular structures: (i) the branching factor that allows structural decomposition and (ii) the concavity over the vessel surface seen primarily at the site of an aneurysm. Such surfaces are decomposed into subcomponents. Vascular sections are reconstructed using CT angiograms. Next the dual graph is constructed, and edge weights for the graph are computed from shape indices. Graph vertices are iteratively clustered by a mesh decimation operator, while minimizing a cost function related to concavity. Results The method was validated by first comparing results with an approximate convex decomposition (ACD) method A. Chowriappa (B) · P. D. Scott Department of Computer Science and Engineering, The State University of New York, Buffalo, NY, USA e-mail:
[email protected] T. Kesavadas · S. Salunke Department of Mechanical and Aerospace Engineering, The State University of New York, Buffalo, NY, USA M. Mokin · P. Kan · S. K. Natarajan Department of Neuroscience, Millard Fillmore Gates Hospital, Buffalo, NY, USA
and next on vessel sections (n = 177) whose number of clusters (ground truth) was predetermined by an expert. In both cases, WACD produced promising results with 84.7 % of the vessel sections correctly clustered and when compared with ACD produced a more effective decomposition. Next, the algorithm was validated in a longitudinal study data of 4 subjects where volumetric and surface area comparisons were made between expert segmented sections and WACD decomposed sections that contained aneurysms. The results showed a mean error rate of 7.8 % for volumetric comparisons and 10.4 % for surface area comparisons. Conclusion Decomposition of the cerebral vasculature from CT angiograms into a geometrically optimal set of convex regions may be useful for computer-assisted diagnosis. A new WACD method capable of decomposing complex vessel structures, including bifurcations and aneurysms, was developed and tested with promising results. Keywords Convex decomposition · Vascular segmentation · Aneurysm · Bifurcation segmentation · Curvature · Shape index · Computer-aided diagnosis Introduction Stroke remains the third most common cause of death in industrialized nations and the single most common reason for permanent adult disability. One of the common causes of stroke being ruptured aneurysms that may lead to subarachnoid hemorrhage (SAH), a severe condition associated with high mortality and morbidity. Each year, approximately 795,000 Americans experience a new or recurrent stroke [1]. The diagnosis and management of such vascular conditions represents a challenge to emergency physicians, neuroradiologists, neurologists, and neurosurgeons. With increased storage [2] and spatial resolution of
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imaging technologies [3] such as rotational angiography (RA), computed tomography (CT), and magnetic resonance (MR), so have the capabilities of computer-assisted tools for diagnosis and prognosis. To perform pre- or post-stroke assessment/treatment, some form of analysis of the vasculature [4] has to be carried out, particularly in comparative assessment for establishing meaningful relationships among and between patient vasculatures, in the case of populationbased comparisons, analysis in non-invasive pathophysiological studies, and in diagnosis of neurovascular diseases [5,6]. However, due to the large variability of anatomies among patients, such relationships between vascular geometry and physiopathology are difficult to make. In this paper, we look at shape decomposition as a fundamental approach to understand the nature of complex vascular structures in computer-aided diagnosis (CAD) and propose a novel methodology for vascular decomposition. Our main contribution in this area is convex decomposition. To our knowledge, this is the first time that this methodology has been used to address the problem of 3D vascular analysis. Vascular decomposition involves decomposing the complex vessel tree into meaningful subparts. Our algorithm first computes the decomposition of the vascular tree into optimal (meaningful) components. We identify meaningfulness of a decomposition based on two primary factors in view of vascular structures. The first being the branching factor that we consider as structural decomposition. If the vessel structure has branches, then we decompose the shape into subcomponents. The second is due to the concavity over the vessel surface seen primarily at umbilic and hyperbolic points. These are saddle regions formed at the sites of an aneurysm ostium (neck) and undulations over aneurysm surfaces that form convex regions. We refer to clustering due to convexity over such regions as geometrical decomposition of the vessel. In our approach, both structural and geometric decomposition are used to obtain a meaningful decomposition of vessel structures. This decomposition can be particularly valuable for recognizing vascular structures especially for CAD assistance and analysis systems [7,8]. By simplifying a complex structure into a set of simple components, various other operations such as shape recognition (matching) [9], shape retrieval [10], skeleton extraction [11,12], and shape analysis [13,14] can be performed on a component-based approach. Background The literature shows several 2D/3D vascular segmentation and decomposition methods [15–19]. For an extended review on vessel segmentation algorithms, we refer the reader to the following surveys [3,20–22]. Some most commonly used methods for vascular segmentation and decomposition use the notion of centerline for vessel tracking, whereas others are based on deformable models. In [23], the authors pro-
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pose a 3D vessel tracking algorithm that directly generates a continuous model of the segmented vessel. The quantification of geometric relationships between elements of a vascular network that is based on the definition of centerlines is presented in [24]. Here, the authors present a framework for geometric analysis of vascular structures. A centerline-tracking approach based on the analysis of moments is described in [25]. Also, in [26], the authors propose a centerline method that robustly decomposes a vessel surface into its constituent branches. Recently, several methods based on deformable models have been proposed for the segmentation of vessels or, more generally, tube-like structures. These model-based techniques can naturally capture the physics and geometry of vascular structures that vary spatially. A geometric deformable model for segmenting tubular structures is presented in [27]. The main advantage of this technique is its ability to segment twisted, convoluted, and occluded structures without the need for user interaction. In [28], the authors discuss a deformable model for detecting bifurcations and providing structural analysis. For volume-based models, a discussion on approaches that use volume decomposition for branch segmentation can be found in [12,29,30]. In [12], the authors develop a relationship between shape decomposition and skeletonization for volume data and ways to decompose the volume. Their work is, however, limited to branched volumes rather than a general model for vascular shapes. Methods for volume decomposition for shape analysis are discussed in [29] where the authors proposed an algorithm for volume decomposition and hierarchical skeletonization for branched objects. The method that we propose is a model-based approach that provides for a more global representation than the previously described methods. Our approach for vascular decomposition is based on identifying the most concave L-ring neighborhood (Figs. 2, 3) in the decomposition of a surface (manifold-2) and partition it in order to reduce its concavity. Convex decomposition is a topic that has been significantly researched [13,32,33]. However, it is not well suited for complex shapes such as vascular structures. Since an exact decomposition of a complex shape can lead to a large number of components that may not have optimal (meaningful) relationships in a global sense. To overcome this problem, we propose a weighted approximate convex decomposition approach (WACD) that is well suited for vessel decomposition illustrated in Fig. 1b, c. It can be seen that (c) produces an optimal representation of the vessel structure than (b). Approximate convex decomposition was addressed by Lien and Amato [31,34]. In [34], the authors exploit a divide-andconquer strategy that consists of iteratively dividing a mesh until the concavity of each subpart was lesser than a specified threshold. At each iterative step, the vertex with the highest concavity was selected, and the cluster to which it belonged was subdivided into two clusters. We evaluate our algorithm
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Fig. 1 a Reconstructed surface mesh of an aneurysm head and inlets, b exact convex decomposition with 146 clusters, c weighted approximate convex decomposition with 15 clusters. Here clusters are shown as different-colored convex hulls of varying sizes
by comparing it with the decomposition produced by an expert (ground truth) and next on longitudinal study data. The remainder of this paper is addressed as follows: in section “Methodology”, we introduce surface mesh generation from clinical data and compute the duel graph. Next, curvature parameters that drive a decomposition cost function are defined in section “Mesh decomposition”, followed by mesh portioning in section “Mesh partitioning”. Finally, the WACD methodology is validated and results are presented in section “Experimental setup and results” followed by the conclusion in section “Conclusion”. Methodology Overview The proposed WACD approach for vascular decomposition proceeds as follows. We construct the dual graph from the manifold-2 surface mesh of vascular sections reconstructed from CTA (computed tomography angiography) data sets. Next, edge weights for the graph are computed from shape indices. We then iteratively cluster vertices by a mesh decimation operator, while minimizing a cost function related to concavity and the aspect ratio of the generated clusters. Data pre-processing and mesh generation The clinical data set consisting of contrast-enhanced CTA data (slick thickness: 1.25 mm. stored using DICOM standards) was used for mesh generation. Maximum intensity projections were computed, and the volume of interest around the vascular structures of interest was identified. More specifically, the following vessel sections were reconstructed: basilar arteries (BA), anterior communicant (ACoA), posterior communicant (PCoA), middle cerebral (MCA), internal carotid (ICA), and superior cerebellar (SCA). Reconstruction was performed by first segmenting the vessel contours by means of level set evolution, and a 3D model of the vessel was obtained as the iso-surface of intensity zero from the
result of the level set evolution, using the marching cubes algorithm [35]. Mesh decomposition From the computed triangulated surface representation of the vasculature, convex decomposition is employed to partition the mesh into a minimal set of convex subsurfaces, S = {s1 , s2 , ..sn }. This is achieved by first constructing a dual graph D∗ . The manifold-2 surface S is defined in R3 as V = {v1 , v2 , v3 , .., vi }, T = {t1 , t2 , t3 , .., t j }, and E = {e1 , e2 , e3 , . . . , ek } (where V is the vertex set, T the set of triangles, and E is the edge set). We define the dual graph D ∗ associated with the surface mesh S as follows: (i) each vertex in D∗ corresponds to a triangle in T , (ii) two vertices in D ∗ are joined by an edge if their respective triangles share an edge. (iii) Next, edge weights for D ∗ are computed from shape indices (section “Computing edge weight”) to favor certain features over others. In mesh decimation (section “Mesh decimation”), vertices of D∗ are iteratively clustered by applying a decimation operator that minimizes a weighted cost function. Shape index The curvature of a point on the surface can be defined by its maximum and minimum curvatures (k1, k2 ). Using these curvature measures, we determine whether the given point lies on a concave, convex, ridge, or saddle region (Fig. 2). Saddle regions are characterized by being concave on one plane and convex from another (e.g., horse saddle) and most frequently correspond to surfaces at vessel bifurcations. Ridges on the other hand can be found on aneurysm heads and ostiums (neck). The curvature of a surface can be calculated either directly from first and second derivatives or indirectly as the rate of change of normal orientations in a certain local context region. The conventional approach of using the pair of Gaussian curvature K and mean curvature H provides a poor shape
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Fig. 2 The curvature ranges for corresponding surface characteristics of an aneurysm
Table 1 Shape types for SI range
Fig. 3 L-ring neighborhood of node v j , where L = 6
representation, since the values are strongly correlated. For each node in the graph D ∗ , we define a set of rings around the node as follows: the ith ring around node v j is defined as the set of vertices v ∗ ∈ V for which there exists a shortest path from v j to v ∗ containing i edges. The L-ring neighborhood of node v j is defined as the set of rings i < L about node v j (Fig. 3). To capture the shape of the L-ring neighborhood (in our implementation a 3-ring neighborhood), we use the shape index introduced by Kawata et al. [36], Cees et al. [37], where the shape index SI derived from the principal curvatures is given as: SI =
1 kmin + kmax 1 − arctan 2 π kmin − kmax
(1)
For each vertex vi in S, we estimate its differential properties in a L-ring neighborhood written as a graph of a bivariate function ψ(x, y) given in Monge form by Eq. (2). ψ (x, y) = ω1 k1 x 2 + k2 y 2 + ω2 b0 x 3 + 3b1 x 2 y + 3b2 x y 2 + b3 x 3 + ω3 c0 x 4 + 4c1 x 3 y + 6c2 x 2 y 2 + 4c3 x y 3 (2) +c4 y 4 + h.o.t.
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Shape type
Shape index range
Cup
[0.0–0.25]
Rut
[0.25–0.5]
Saddle
[0.5–0.75]
Ridge
[0.75–1.0]
Cap
[1.0>]
where the coefficients k1 , k2 are the principal curvatures, b0 , b3 are the directional derivatives of the principal curvature along their curvature lines, and ω1 , ω2 , ω3 are the coefficients of power series expansion of the function ψ. From the d- jet expansion of Eq. (2), we obtain, kmin the minimum principal curvature and kmax the maximum principal curvature. The values of SI varies in the closed interval [0.0, 1.0], and every distinct surface shape corresponds to a unique value of SI (except for planar surfaces, which will be mapped to the value 0.5, together with saddle shapes). The shape index is invariant to translations, but due to the limited resolution, noise is introduced in the presence of rotations and scale changes. Based on the shape index ranges, we define the following well-known shape types (Table 1). Computing edge weight Edge weights are then computed from the SI ranges as follows. Two neighboring vertices connected by an edge in the dual graph D ∗ are assigned an edge weight λ determined by the following criteria: ⎧ ⎨ θ1 , 0 < SI ≤ 0.5 (umbilic points) (3) λ = θ2 , 0.5 < SI ≤ 0.75(hyperbolic points) ⎩ θ3 , 0.75 < SI ≤ 1.0 One advantage of using shape indices as edge weights is that transition from one shape type to another is continuous;
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C (u, v) = αAR (u, v) + λ
Con(S (u, v)) N
(5)
where Con (S(u, v)) is the concavity of surface S (u, v), and N is the normalization factor, which is set as the diagonal of the bounding box of S(u, v). The parameters α and λ control contributions of the aspect ratio (AR) and concavity (Con(S (u, v)). Where α ≥ 2 favor the generation of compact disks in which case the cost is unity, and the aspect ratio AR of the surface S (u, v) is defined by Eq. (6). AR (u, v) = Fig. 4 Portioned sections encapsulated in bounding spheres for setting of weights that favoured branch decomposition
γ (S (u, v))2 2π ∗ σ (S (u, v))
(6)
where γ is the perimeter and σ is the area of the surface S(u, v). Concavity measure
hence, they can be used to describe subtle shape variations. Parameters θ1 , θ2 , and θ3 were set such that a negative value of θ influenced partitioning (such as bifurcations and aneurysm ostiums); on the other hand, a positive θ prevented subdivisions (while minimizing a cost function, section “Mesh decimation”). In our experimental setup, strongly saddleshaped regions corresponded to a larger negative λ, hence favoring partitioning of such regions (Fig. 4). This provides a mechanism to focus on key structural features such as branching and lobulations, while ignoring less significant features such as localized ridges and surface depressions.
Mesh decimation Following the assignment of the edge weights to the dual graph D∗ , convex decomposition is initiated by iteratively applying a half-edge collapse operation on neighboring vertices. A half-edge collapse operator defined as hcol (u, v) when applied to two vertices (u, v) connected by an edge in D∗ merges v with u, and all incident edges on v are connected to u, shown in Fig. 5. To keep track of the unified vertices caused by the decimation operator, we define H(u) to be a vector containing the history of vertex u that is initialized to {∅}. With each operation of hcol (u, v) to a vertex u in D∗ , the vector H(u) is updated as follows: H (u) ← H (u) ∪ H (v) ∪{v}
(4)
The decimation process of half-edge collapse operators is governed by a cost function weighted on λ (section “Shape index”) and minimizes for concavity. The cost associated with hcol (u, v) is given by Eq. (5).
Various definitions have been proposed to measure concavity, for instance [31,32,34]. In [34], the concepts of bridges and pockets are used to measure concavity. We estimate concavity using l2 nor m of the projection |P (vi ) | of a vertex vi onto its convex hull, and for a surface S (u, v), the concavity is given by Eq. (7). con(S (u, v)) = argmaxx |P (vi ) |
(7)
where con (S (u, v)) is the maximum projection distance over all point in S (u, v) shown in Fig. 6. After each edge collapse operation, λ is locally recomputed for surface S (u, v), and the new edge weight λnew obtained from Eq. (8) is used in the update of D ∗ . λnew =
1−δ δ
λ
(8)
An influence parameter δ is used in order to minimize the influence of newly formed surface features caused by the decimation in successive iterations of hcol described next. Mesh partitioning With each iteration of the hcol operator, the lowest mesh simplification cost is applied to a new partition ϕ (n) = {ϕ1n , ϕ2n , ϕ3n , ϕ4n , . . . , ϕ nP(n) } minimizing the cost function
C, ∀k {1, . . . , P (n)} , ϕkn = pkn ∪ H pkn . Where ( pkn ) k {1, . . . , P (n)} represents the dual graph D∗ obtained after n edge collapse operations on P clusters. This procedure is iteratively performed until all edges of D∗ are in clusters with concavity lower than a determined concavity resolution value ε. In Table 2, we describe various decompositions obtained from concavity resolution ranges, 50 < ε < 250.
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Fig. 5 Edge collapse operation where edges are incident on vertex u
Table 2 Decompositions obtained from concavity resolution ranges, 50 < ε < 250 and expert segmentation (ES)
Concavity resolution ε
Branch section WACD parts
ES parts
WACD voxels
ES voxels
WACD parts
ES parts
WACD voxels
ES voxels
50–100
17
7
89
203
14
5
179
529
100–150
11
7
134
203
8
5
327
529
150–200
7
7
192
203
5
5
491
529
200–250
5
7
–
203
2
5
–
529
Fig. 6 The projection P(vi ) of point vi on the convex hull
Algorithm INPUT: A mesh V = {v1 , v2 , v3 , .., vi } and T = {t1 , t2 , t3 , .., t j }, λ and ε FOR EACH S (u, v){ IF Con (S(u, v)) < ε THEN /section “Mesh decimation” RETURN S (u, v) ELSE COMPUTE hcol (u, v) UPDATE H (u) UPDATE λ /section “Concavity Measure” RETURN S (u) } We measured the algorithms computation time on a PC running Windows 7, 64-bits with an i-7 processor clock speed 2.66 GHz and 6 GB of RAM. On average, the estimated
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Aneurysm section
Fig. 7 The computation time for clustering with expected decomposition time indicated by blue line
decomposition time for a triangulated mesh with 3,000 vertices was 250 s (shown in Fig. 7. indicated by a blue line). Parameter estimation and voxelization Estimation of the concavity resolution ε (section “Mesh decimation”) was determined by volumetric comparisons made on n = 65 pre-selected sections (containing cerebral aneurysms and vessel bifurcations) that were segmented by an expert. Branch sections were decomposed (using WACD) and voxelized to compute branch volumes (Fig. 8). In order to obtain topological consistence, it is crucial that the decomposition algorithm not separates the volumes into more parts. Volume decomposition is achieved by separation of a branched volume into a hierarchy of subvolumes. Sections containing cerebral aneurysms were decomposed into head, neck, and inlets/outlets. For correct volumetric decomposition, it is essential to maintain the hierarchical relationship
Author's personal copy Int J CARS (2013) 8:207–219 Fig. 8 a–d, f–i Decomposition of vessel branch section and aneurysm section, for ranges of ε (Table 2) where b, g obtained optimal decomposition, e, j voxelization of the sections
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between parent and child (volume of parent equals sum of subvolumes). Volumetric comparisons were made by comparing the number of voxels in each decomposed section. The sections were then decomposed using WACD and voxelized. In our experimental setup, parameters θ1 = −0.78, θ2 = 0.04, θ3 = 0.83, and δ = 0.67 were used. Table 2 shows the optimal decomposition achieved (for a single branch section and aneurysm section) for various concavity resolution ranges. We identify ranges that obtained topological constancy for both number of subparts and volumes that matched closest to expert segmented sections. For volume comparisons, the sum of the largest decomposed subvolume from the left child branch, right child branch, and the parent was considered. For sections contacting aneurysms, the sum of the largest decomposed subvolume for aneurysm head and inlet/outlet was considered. We determined the range of 150 < ε < 200 that matches closest to expert segmented section both in volume and number of decomposed parts.
c
e
d
h
i
j
our methodology for automated diagnosis, validation is performed using longitudinal study data on four patients with a prior history of unruptured and untreated cerebral aneurysms. Vascular decomposition Validation of the goodness of decomposition is performed in two stages, first by comparing it with an existing ACD approach [34] (that was shown to be effective for decomposing non-vascular objects) and next by testing how well it clustered sections whose correct number of parts (ground truth) was independently determined by an expert. However, in determining the ground truth for shape decomposition, our initial concern was with non-clinical, geometric judgment of primitive shapes (i.e., if a bifurcation was well clustered into three parts, etc.). The algorithm was tested on sections of the neurovasculature (BA, ACoA, PCoA, MCA, ICA, and SCA) that contained cerebral aneurysms, vessel bifurcations, tortuous sections, linear sections, and sections fused with the cranial base (Table 3; Fig. 10).
Experimental setup and results Effectiveness of decomposition The proposed vascular decomposition and correspondence methodology was tested on sections of the neurovasculature (BA, ACoA, PCoA, MCA, ICA, and SCA) from clinical data sets. Data were achieved for 48 aneurysms in 37 patients and pre-segmented sections (n = 177 that included sections containing cerebral aneurysms and vessel bifurcations) of vasculature used for validation. The above sections were classified as curved, linear, aneurysm, and branch sections. The vasculature being highly complex in structure, there is no generally accepted set of rules for validation of the effectiveness of shape-based decomposition such as the one proposed here. Hence, we validate first for geometric decomposition. To specifically address the clinical benefits of
Visual inspection: Comparisons were made between ACD and WACD (illustrated in Fig. 9). In this comparative test, WACD formed minimal number of clusters while maintaining an effective (meaningful) decomposition, whereas ACD clustered most of the sections non-optimally (bifurcations typically had more than 3 clusters). WACD obtained closer consistency with ground truth than ACD. In Fig. 9b, e, it can be seen that the aneurysm heads were subclustered in ACD, whereas WACD (c, f) contained the aneurysm head within a single cluster. Also in Fig. 10g, h, the convex hulls of the decomposed vascular parts show encapsulation of an aneurysm within a single cluster; the same results can be seen
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Author's personal copy 214 Fig. 9 The comparison between WACD and ACD, a, d, g the reconstructed mesh from clinical data, b, e, h decomposition using ACD and c, f, i decomposition using WACD, where t is the number of triangle and k is the number of clusters. It can be seen that WACD cluster sections with fewer number of clusters than ACD that are more consistent with the shape characteristics of the vessel sections
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around the Circle of Willis in Fig. 10i, j. For vessel branches, WACD clustered bifurcations with minimal number of cluster while maintaining an optimal representation of the branched section (Figs. 9i, 10b), whereas ACD excessively clustered the bifurcation (Fig. 9h). In Fig. 10d, f, WACD was used to cluster vessel sections fused with the skull. Although we did not intend to use it for separation of vessels fused to other structures, preliminary results seemed to favor its use. Section decomposition: n = 177 pre-segmented sections of vasculature were classified into curved, linear, aneurysm, and branch sections (Table 3). For each section, the ideal number of clusters was identified by an expert (e.g., some bifurcations were identified to have 3 clusters and others to have 4 clusters). Using WACD, we were able to cluster approximately 84.7 % of the structures into the correct number of components. The results suggest that the method is capable of producing an effective decomposition (close to expert segmentation). Table 3 shows the number of correctly clustered samples by WACD for each section.
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Longitudinal study analysis The purpose of the follow-up study analysis was to specifically address the clinical benefits of our methodology for automated diagnosis. Follow-up analysis was performed using data on four patients with a prior history of unruptured and untreated aneurysms. These aneurysms were not treated because they were considered to be too small. We perform analysis on the following subjects: Case 1: 31-year-old male with known history of bilateral ICA dissection and EC-IC bypass due to a moyamoya type changes post-ICA dissection having tiny basilar apex aneurysm. CT stroke study showed enlargement of the known basilar aneurysm. Case 2: 44-yearold female with known history of aneurysms and had previously stent coiled posterior communicating artery aneurysm on the right. A 5 × 5 mm incidental R Pcomm aneurysm, unchanged in size, was followed over a period of 2 months. Case 3: 62-year-old female was followed for incidental rightsided middle cerebral artery (MCA) aneurysm, overall
Author's personal copy Int J CARS (2013) 8:207–219 Fig. 10 Decomposition using WACD on sections of vasculature, a, b decomposition of vessel tree with bifurcations, c–f decomposition used to separate vessel sections fused with sections of cranium, i, j decomposition of the Circle of Willis with aneurysm present at anterior section
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Fig. 11 Case 3: 62-year-old female was followed for incidental right-sided MCA aneurysm. a Maximum intensity projection (MIP) of the internal carotid artery and MCA shows a 4 mm aneurysm (arrow) with a 3 mm wide neck in the M1 segment of the right MCA, b, d follow-up after a 6-month period shows noticeable change in aneurysm size, c 3D reconstruction of vasculature
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Structures
Expert classified range of true clusters
Samples/structure
Correctly clustered samples by WACD
Curved
5–7
38
29
Linear
1–3
56
48
Aneurysm
1–3
16
12
Branch
3–4
67
61
a
b
c
appeared stable (shown in Fig. 11) Case 4: 54-year-old female with post-carotid sacrifice and clipping of right MCA aneurysm and clipping of the left Pcom aneurysm. Followup studies show a small A1 aneurysm that appeared stable in size (Fig. 13).
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Aneurysm data From the above-described cases studies, CTA data sets break obtained were used for vascular reconstruction and analysis. Measurements were performed on the sections containing
Author's personal copy Int J CARS (2013) 8:207–219 Table 4 Volumetric comparisons between WACD volume of aneurysm head section and expert segmented volume (ESV)
Table 5 Surface comparisons between WACD surface area of aneurysm head section and expert segmented surface area (ESA)
Fig. 12 a shows WACD of case 3, initial follow-up study c, d reconstruction of follow-up I and II b decomposition of aneurysm head e vessel section from follow-up I (gray) aligned with vessel section from follow-up II (blue)
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Case
Preliminary
Follow-up I
Follow-up II
WACD voxels
ESV
% error
WACD voxels
ESV
%error
1
133
116
14.65
195
216
2
529
563
6.03
511
554
3
686
652
5.21
863
4
462
419
10.26
449
Case
Preliminary WACD SA
WACD voxels
ESV
% error
9.72
–
–
–
7.76
–
–
–
906
4.74
841
873
3.66
405
10.86
474
451
5.09
ESA
% error
Follow-up I
Follow-up II
ESA
% error
WACD SA
ESA
% error
WACD SA
1
8.22
10.63
22.67
15.89
17.60
11.42
–
–
–
2
41.03
44.69
8.18
39.83
43.07
7.54
–
–
–
3
55.47
51.98
6.71
70.02
77.19
10.03
84.06
77.49
8.47
4
37.04
33.78
9.62
35.10
31.37
11.89
37.89
35.31
7.30
a
b
c
d
e aneurysms; aneurysm maximum diameters ranged from 1 to 5 mm (mean, 2 mm ± 1 [standard deviation]), and aneurysm neck sizes ranged from 1 to 3 mm (mean, 1.4 mm ± 1.2). Volume of aneurysm in terms of voxels ranged from 116 to 873, and aneurysm surface area ranged from 10.63 to 35.31 mm2 (Tables 4, 5). None of the aneurysms had previously been treated with coils. Decomposition analysis Volume analysis: Form the data acquired on cases having pretty much diffused vascular aneurysms, segmentation of
the neurovasculature and 3D surface reconstruction was performed (10 CTA data sets). Using WACD, the vasculature was decomposed into subparts (Fig. 12). Aneurysm head sections from identified parent vessel sections were extracted, and comparative analysis (volumetric and surface area computation) was performed between subsequent follow-up data (preliminary, follow-up I, follow-up II). Table 4 explains our results; for volumetric comparisons made between expert segmented volume and WACD volumes, we obtained a mean error rate of 7.78 %. Figure 11a, b, d illustrates Case 3: where MCA aneurysm on right-hand side followed over a period of 2 months, (d) MIP shows noticeable change in aneurysm
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Author's personal copy 218 Fig. 13 Case 4: a small A1 aneurysm that appeared stable in size. a Maximum intensity projection (MIP) of CTA data shows a 3.5 mm aneurysm (arrow) with a 2.8 mm wide neck. b, c Reconstruction and WACD of aneurysm section
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a
b
c
size. Using WACD, we were able to detect volume changes in aneurysm structures that were close to that of expert segmented sections. Surface area analysis: Table 4 explains our results for surface area longitudinal comparisons (preliminary, follow-up I, follow-up II). Comparisons were made between surface areas of expert segmented aneurysm head sections and WACD segmented aneurysm head sections. We obtain a mean error rate of 10.38 % in surface area comparisons. In Fig. 13, we illustrate Case 4: MIP of a small left A1 aneurysm that appeared stable in size; (b, c) shows reconstruction followed by WACD performed. In the follow-up analysis, we obtained a mean error rate of 9.6 % for surface area comparisons and 8.76 % for volumetric comparisons. For the longitudinal study analysis using WACD, we were able detect both surface changes in aneurysm surface structures and changes in segmented volumes that were close to expert segmented sections.
Conclusion Our primary objective was the decomposition of the vasculature into a geometrically consistent (meaningful) set of convex regions for computer-aided diagnosis. We have demonstrated that WACD is a promising method for such vascular decomposition and can produce effective representations of complex vessel structures, which include bifurcations and aneurysms. Form our initial longitudinal study, findings suggest that our methodology can provide a basis for vascular analysis. However, a longer follow-up study would be necessary to determine the efficacy of WACD on cerebrovascular
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aneurysms and also a comparative study with patients with ruptured and unruptured aneurysms. Conflict of interest None.
References 1. AHA Statistical Update (2012) Heart disease and stroke statistics2012 update. A Report From the American Heart Association 2. Welter P, Hocken C, Deserno T, Grouls C, Gunther R (2010) Workflow management of content-based image retrieval for CAD support in PACS environments based on IHE. Int J Comput Assist Radiol Surgery 5(4):393–400 3. Radaelli AG, Peiró J (2010) On the segmentation of vascular geometries from medical images. Int J Numer Methods Biomed Eng 26(1):3–34 4. Tizon X (2004) Algorithms for the analysis of 3D magnetic resonance angiography images. Acta Universitatis Agriculturae Sueciae, Uppsala, ISBN 91-576-670-4 5. Thompson P (2004) Mapping cortical change in alzheimer’s disease, brain development and schizophrenia. Neuroimage 23(Suppl 1):S2–S18 6. Christian SW, Lorenzo P (2007) Comparative morphology of the hemolymph vascular system in scorpions-A survey using corrosion casting, MicroCT and 3D reconstruction. J Morphol 268(5):401– 413 7. Camarlinghi N, Gori I, Retico A, Bellotti R, Bosco P, Cerello P, Gargano G, Lopez Torres E, Megna R, Peccarisi M, Fantacci ME (2012) Combination of computer-aided detection algorithms for automatic lung nodule identification. Int J Comput Assist Radiol Surg 7(3):455–464 8. Francesco C, Roberto F, Christian D, Laurent M, Francesco M, Alfonso M, Antonio M, Sergio C (2010) Hepatic vessel segmentation for 3D planning of liver surgery: experimental evaluation of a new fully automatic algorithm. Acad Radiol 18(4):461–470 9. Funkhouser T, Kazhdan M, Shilane P, Min P, Kiefer W, Tal A, Rusinkiewicz S, Dobkin D (2004) Modeling by example. ACM Trans Graph (TOG)—Proceedings of ACM SIGGRAPH, 23(3):652–663
Author's personal copy Int J CARS (2013) 8:207–219 10. Tung T, Schmitt F (2004) Augmented reeb graphs for content-based retrieval of 3d mesh models. Proceedings In Shape Modeling and Applications, pp. 157–166 11. David L, Elsa DA, Isabelle B, Gareth F-L (2009) A review of 3D vessel lumen segmentation techniques: models, features and extraction schemes. Med Image Anal 13(6):819–845 12. Xiaopeng Z, Jianfei L, Zili L, Marc J (2008) Volume Decomposition and Hierarchical Skeletonization. In: Proceedings of The 7th ACM SIGGRAPH international conference on virtual-reality continuum and its applications in industry, Article No. 17 13. Lien JM, Keyser J, Amato NM (2006) Simultaneous shape decomposition and skeletonization. In: Proceedings of ACM symposium on solid and physical modeling, pp 219–228 14. Weimin H, Jiayin Z, Jiang L, Jing Z, Tao Y, Yi S, Gim HL, Chee KC, Stephen C (2011) 3D shape decomposition and comparison for gallbladder modeling. In: Proceedings of the SPIE 7964, 79642K 15. Dekanic K, Loncari S (2007) 3D vascular tree segmentation using level set deformable model. In Proceeding of the 5th international symposium on image and signal processing and analysis 16. Cohen LD, Deschamps T (2007) Segmentation of 3D tubular objects with adaptive front propagation and minimal tree extraction for 3D medical imaging. Comput Methods in Biomech Biomed Eng 10(4):289–305 17. Gooya A, Liao H (2008) A variational method for geometric regularization of vascular segmentation in medical images. IEEE Trans Image Process 17(8):1295–1312 18. Chen B, Kitasaka T, Honma H, Takabatake H, Mori M, Natori H, Mori K (2012) Automatic segmentation of pulmonary blood vessels and nodules based on local intensity structure analysis and surface propagation in 3D chest CT images. Int J Comput Assist Radiol Surg 7(3):465–482 19. Bing-yin R, Yong-bo Z, Daniel XBC (2010) A novel method for 3D-segmentation of vascular images. The 3rd international conference of bionic engineering, Advances in Natural Science 3(2) 20. Kirbas C, Quek FKH (2004) A review of vessel extraction techniques and algorithms. ACM Comput Surv (CSUR) 36(2):81–121 21. Attene M, Biasotti S, Mortara M, Patane G, Spagnuolo M, Falcidieno B (2006) Computational methods for understanding 3d shapes. Comput Graph Special Issue Comput Graph 30(3):323–333 22. Carrillo J, Orkisz M, Hoyos M (2005) Extraction of 3D vascular tree skeletons based on the analysis of connected components evolution. Lect Notes Comput Sci 3691:604–611 23. Florez-Valencia L, Azencot J, Vincent F, Orkisz M, Magnin IE (2006) Segmentation and quantification of blood vessels in 3D images using a right generalized cylinder state model. IEEE international conference on image processing, pp 2441–2444
219 24. Piccinelli M, Veneziani A, Steinman DA, Remuzzi A, Antiga L (2009) A framework for geometric analysis of vascular structures: application to cerebral aneurysms. IEEE Trans Med Imaging 28(8):1141–1155 25. Marcela HH, Maciej O, Philippe CD, Isabelle EM (2006) Assessment of carotid artery stenoses in 3D contrast-enhanced magnetic resonance angiography, based on improved generation of the centerline. Mach Graph Vis 14(4):349–378 26. Antiga L (2004) Robust and objective decomposition and mapping of bifurcating vessels. IEEE Trans Med Imaging 23(6):704–713 27. Franchini E, Morigi S, Sgallari F (2010) Segmentation of 3D tubular structures by a PDE-based anisotropic diffusion model. MMCS, volume 5862 of lecture notes in computer science, pp 224–241 28. McIntosh C, Hamarneh G (2006) Vessel Crawlers: 3D physicallybased deformable organisms for vasculature segmentation and analysis. IEEE computer society conference on computer vision and pattern recognition (CVPR’06), vol. 1, pp 1084–1091 29. Xiaopeng Z, Bo X, Wujun C, Marc J (2009) Volume decomposition and hierarchical skeletonization for shape analysis. Pattern Recognit D:47–72 30. Wei Ma, Bo X, Xiaopeng Z, Hongbin Z (2008) Decomposition of branching volume data by tip detection. IEEE international conference on image processing, pp 1948–1951 31. Lien JM, Amato NM (2006) Approximate convex decomposition. Comput Geom 35(1–2):100–123 32. Lien JM, Amato NM (2004) Approximate convex decomposition of polygons. In: Proceedings of the 20th annual ACM symposium computational geometry (SoCG), pp 17–26 33. Katz S, Tal A (2003) Hierarchical mesh decomposition using fuzzy clustering and cuts. ACM Trans Graph (TOG)—Proceedings of ACM SIGGRAPH 22(3):954–961 34. Lien JM, Amato NM (2008) Approximate convex decomposition of polyhedra and its applications. Comput Aided Geom Des (CAGD) 25(7):503–522 35. Newman T, Hong Y (2006) A survey on marching cubes algorithm. Comput Graph 30(5):854–879 36. Kawata Y, Niki N, Ohmatsu H, Kakinuma R, Eguchi K, Kaneko M, Moriyama N (1998) Quantitative surface characterization of pulmonary nodules based on thin-section CT images. IEEE Trans Nucl Sci 45(4):2132–2138 37. Cees W, Vincent FR, Frans MV, Lucas JV (2010) Detection and segmentation of colonic polyps on implicit isosurfaces by second principal curvature flow. IEEE Trans Med Imaging 29(3):688–689
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