3d vascular decomposition and classification for computer-aided detection

3514 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 60, NO. 12, DECEMBER 2013

3D Vascular Decomposition and Classificationfor Computer-Aided Detection

Ashirwad Chowriappa∗, Sarthak Salunke, Maxim Mokin, Peter Kan, and Peter D. Scott

Abstract—In this study, we propose a weighted approximateconvex decomposition (WACD) and classification methodology forcomputer-aided detection (CADe) and analysis. We start by ad-dressing the problem of vascular decomposition as a cluster opti-mization problem and introduce a methodology for compact geo-metric decomposition. The classification of decomposed vessel sec-tions is performed using the most relevant eigenvalues obtainedthrough feature selection. The method was validated using preseg-mented sections of vasculature archived for 98 aneurysms in 112patients. We test first for vascular decomposition and next for clas-sification. The proposed method produced promising results withan estimated 81.5% of the vessel sections correctly decomposed. Re-cursive feature elimination was performed to find the most compactand informative set of features. We showed that the selected subsetof eigenvalues produces minimum error and improved classifierprecision. The method was also validated on a longitudinal studyof four cases having internal cerebral aneurysms. Volumetric andsurface area comparisons were made between expert-segmentedsections and WACD classified sections containing aneurysms. Re-sults suggest that the approach is able to classify and detect changesin aneurysm volumes and surface areas close to that segmented byan expert.

Index Terms—Aneurysm, classification, segmentation, spectral,vascular decomposition.

I. INTRODUCTION

SUBARACHNOID hemorrhage (SAH) caused by the rup-ture of an aneurysm is one of the common causes of stroke.

Each year, approximately 795 000 Americans experience a newor recurrent stroke [1]. To perform pre- or post-stroke assess-ment, some form of analysis of the vasculature has to be carriedout [2], [3], such as comparative assessment among and betweenpatient vasculatures. Also, in the case of population-based com-parisons, and in diagnosis of neuro-vascular diseases [4], [5]. In

Manuscript received September 15, 2012; revised April 4, 2013; acceptedJune 27, 2013. Date of publication July 11, 2013; date of current version Novem-ber 18, 2013. Asterisk indicates corresponding author.

∗A. Chowriappa is with the Department of Computer Science and Engi-neering, State University of New York, Buffalo, NY 14260 USA (e-mail:[email protected]).

S. Salunke is with the Department of Mechanical and Aerospace Engi-neering, State University of New York, Buffalo, NY 14260 USA (e-mail:[email protected]).

M. Mokin and P. Kan are with the Department of Neuroscience, BuffaloGeneral Medical Center, Buffalo, NY 14203 USA (e-mail: [email protected]; [email protected]).

P. Scott is with the Department of Computer Science and Engineering, StateUniversity of New York, Buffalo, NY 14260 USA (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TBME.2013.2272721

this paper, we focus on shape decomposition and characteriza-tion as a fundamental approach to understand the nature of com-plex vascular structures in computer-aided detection (CADe)and analysis. The weighted approximate convex decomposition(WACD) methodology for vessel decomposition was first pro-posed in [5]. We extend this prior work and address the problemof vascular classification. Our approach is twofold. First, we de-compose the cerebral vasculature into substructures. Next, thedecomposed structures are classified to identify aneurysm sacs.To our knowledge, this is the first time that this approach hasbeen used to address the problem of 3-D vascular decompositionand classification.

A. Previous Work on Decomposition and Segmentation

The literature shows several 2-D/3-D vascular decompositionand segmentation methods [6]–[11]. For extended reviews onvessel segmentation algorithms, we refer the reader to the fol-lowing surveys [12]–[15]. Methods adapted from generic seg-mentation algorithms generally use a model-based approach,whereas others use the notion of centerline for vessel tracking.Model-based techniques naturally capture the physics and ge-ometry of vascular structures (more generally, tube-like struc-tures) that vary spatially. A geometric deformable model forsegmenting tubular structures is presented in [16]. The mainadvantage of this technique is its ability to segment twisted,convoluted, and occluded structures without the need for userinteraction. Christian et al. [17] presents an approach for simul-taneously separating and segmenting multiple interwoven tubu-lar tree structures. Also in [18], the authors discus a deformablemodel for detecting bifurcations and providing structuralanalysis.

B. Our Vascular Decomposition Approach

In this study, we propose a model-based approach that de-composes the vasculature into subcomponents. Our approachuses convex decomposition, a topic that has been investigatedin [19], [20]. However, exact convex decomposition is not wellsuited for complex shapes such as vascular structures [seeFig. 1(b)]. An exact decomposition of a complex shape suchas a vascular tree can lead to an unacceptably large numberof components which may not have meaningful relationshipsin a global sense. To overcome this problem, we propose aWACD that is well suited for vessel decomposition illustratedin Fig. 1(c). It can be seen that an approximate decompositionas shown in Fig. 1(c)produces a clearer and more parsimoniousrepresentation of the vessel structures than an exact decompo-sition shown in Fig. 1(b).

0018-9294 © 2013 IEEE

CHOWRIAPPA et al.: 3D VASCULAR DECOMPOSITION AND CLASSIFICATION FOR COMPUTER-AIDED DETECTION 3515

Fig. 1. (a) Reconstructed surface mesh of a tortuous vessel tree, (b) exactconvex decomposition of vessel tree section having 286 clusters, (c) decom-position using a weighted approximate convex decomposition strategy havingnine clusters for the vessel tree. Clusters are shown as different colored convexhulls of varying sizes.

In the next step of our approach, the decomposed vessel sec-tions are classified based on the information coded by the Lapla-cian spectrum, represented by relevant eigenvalues.

C. Related Work

Classification requires an efficient shape representation, anddepends on the shape descriptors [21]. The literature showsseveral schemes for vascular classification in computer-aideddiagnosis and detection [22], [23]. In [24], the authors proposea method for vascular geometry classification to detect cerebralaneurysms using a color indicator. They use curvatures as thebase of their vascular coloring scheme. An approach that usestexture analysis is performed in [25] for vascular classification.Lauric et al. [26] introduce a writhe number as a novel aneurysmsurface descriptor. Their approach was shown to be effective forboth detection and classification.

D. Our WACD Classification Approach

Vessel classification is addressed by analysis in the spectraldomain. We obtain a 3-D shape representation that is invari-ant under natural deformations, and at the same time, con-tains enough information to perform shape characterization.Although classification can be directly performed without theneed for vascular decomposition. When dealing with complexand highly tortuous structures seen in the neuro-vasculature,such classification would require high dimensional features andan exponentially large training dataset. The WACD methodol-ogy is used to simplify vascular structures into parsimoniousdescriptions with useful physiological significance.

Spectral approaches in general use the k smallest eigenval-ues and embed the manifold onto a k dimensional subspacespanned by their associated eigenvectors [27]. These eigenval-ues can be used as simple shape descriptor that works quitewell on vessel structures. The literature shows spectral graphtheory used extensively to provide adequate characterization ofgeometric shapes [28] and also for shape matching [27]–[29].In [30], the authors propose a deformation invariant shape rep-resentation. They use the eigenvalues and eigenvectors of thegeodesic distance matrix after application of some kernel. Oneuseful property of spectral methods is their ability to computebend invariant signatures for manifold structures [28].

The remainder of the paper is organized as follows. In Sec-tion II–A, we introduce surface mesh generation for clinical

data. In Section II-B we compute the duel graph and define cur-vature parameters that drive the decomposition cost function.Vessel shape representation and feature selection are discussedin Section II-C. Finally, we present our results in Section III andconclude in Section V.

II. METHODOLOGY

A. Data Preprocessing and Mesh Generation

The clinical data-set consisting of contrast enhanced CTAdata (slick thickness: 1.25 mm. stored using DICOM stan-dards) was used for mesh generation. Volumes of interest (VOI)around identified vascular structures are specified using maxi-mum intensity projections (MIP). More specifically the follow-ing vessel sections are reconstructed: basilar arteries (BA), an-terior communicant (ACoA), posterior communicant (PCoA),middle cerebral (MCA), internal carotid (ICA), and superiorcerebelar (SCA). Prior to segmentation and reconstruction ananisotropic diffusion filter is applied [31]. This filter reduces thenoise while preserving small vascular structures enabling bet-ter segmentation which is important in the region of the brain.Next, we remove the skull bones, the sinuses and the skin, hav-ing similar intensity as the vessels, using double thresholding(dt1 , dt2 , dt3 , dt4). A 3-D model of the vessel is obtained asan iso-surface of intensity zero, resulting from the level setevolution, implemented utilizing ITK [32]. For a detailed de-scription, we refer the reader to [33]. In order for small vesselsto be included isosurface Iiso = 118 value is used. Values for thelevel set threshold tl = 36, variance σ2

l = 0.3, and number ofiteration it = 120 in ITK were chosen. The parameter settingsthat were used in double thresholding were dt1 = 183 (bone),dt2 = 318 (bone boundary), dt3 = 714 (mean bone boundary)and dt4 = max(I) (maximum intensity of the dataset).

B. Vascular Decomposition

In this section, we introduce our WACD methodology. Fromthe computed vessel surface representation, convex decompo-sition is employed to partition the mesh into a minimal setof convex subsurfaces, S = {s1 , s2 , ..sn}. The two-manifoldsurface S is defined in R3 as V = {v1 , v2 , v3 , .., vi}, T ={t1 , t2 , t3 , .., tj}, and E = {e1 , e2 , e3 , . . . , ek} (where V is thevertex set, T the set of triangles, and E is the edge set). Wedefine the dual graph D∗ associated with the surface mesh S.Edge weights for D∗ are computed from shape indices to favorcertain features over others. In mesh decimation vertices of D∗

are iteratively clustered by applying a decimation operator thatminimizes a weighted cost function.

1) Computing Shape Index and Edge Weights: The curva-ture of a point on the surface can be defined by its maximumand minimum curvatures (k1 , k2). Using these curvature mea-sures, we determine whether the given point lies on a concave,convex, ridge, or saddle region (see Fig. 2). Saddle regions arecharacterized by being concave on one plane and convex fromanother (e.g., horse saddle) and most frequently correspond tosurfaces at vessel bifurcations. Ridges on the other hand can befound on aneurysm heads and ostiums (neck).


Fig. 2. Primitive shape types for corresponding surface characteristics of ananeurysm sac.

TABLE ISHAPE TYPES FOR SI RANGE

For each node in the graph D∗, we define a set of rings aroundthe node as follows; the ith ring around node vj is defined asthe set of vertices v∗ ∈ V for which there exists a shortest pathfrom vj to v∗ containing i edges. The L− ring neighborhood ofnode vj is defined as the set of rings i < L about node vj . Janand van Doorn presented the shape index in [34]. To capture theshape of the L− ring neighborhood (in our implementation a3-ring neighborhood), we use the shape index introduced by[35], where the shape index SI derived from the principal cur-vatures is given as

SI =12− 1

πarctan

kmin + kmax

kmin − kmax. (1)

The shape index varies in the interval [0, 1] and provides acontinuous gradation between shapes, such as concave shapes,hyperboloid shapes, and convex shapes (see Table I).

Edge weights are then computed from the SI ranges as fol-lows. Two neighboring vertices connected by an edge in thedual graph D∗ are assigned an edge weight λ= {λ1 , λ2 , λ3}determined by the following criteria

λ =

⎧⎪⎨

⎪⎩

λ1 , 0 < SI ≤ 0.5 (umbilic points)

λ2 , 0.5 < SI ≤ 0.75 (hyperbolic points)

λ3 , 0.75 < SI ≤ 1.0.

(2)

One advantage of using shape indices as edge weights is thattransition from one shape type to another is continuous; hence,they can be used to describe subtle shape variations.

2) Mesh Decimation: Following the assignment of edgeweights to the dual graph D∗, convex decomposition (meshdecimation) is initiated by iteratively applying half-edge oper-ations on neighboring vertices. We define a half-edge collapseoperator hcol(u, v) such that when applied to two vertices (u, v)connected by an edge in D∗ merges v with u and all incidentedges on v are connected to u. To keep track of the unified(merged) vertices caused by mesh decimation, we define H (u)

Fig. 3. Decomposition of vessel branch section for ε values.

(initially {∅} ) to be a vector containing the history of vertex u.With each operation of hcol (u, v), the vector H (u) is updatedas follows

H (u) ← H (u) ∪H (v) ∪ {v} . (3)

This decimation process is governed by a cost function C(u, v)that is weighted on λ and minimizes for concavity Con and theaspect ratio ρ. The cost associated with each hcol(u, v) is givenby

C(u, v) = αρ(u, v) + λCon(S(u, v))

N(4)

where Con(S(u, v)) is the concavity [19] of surface S(u, v),formed by the L− ring neighborhoods of (u, v), and N is thenormalization factor which is set as the diagonal of the boundingbox of S(u, v). The parameters α and λ control contributions ofthe aspect ratio ρ and concavity Con(S(u, v)). Setting α ≥ 2favors the generation of compact disks in which case the cost isunity. The aspect ratio AR of the surface S(u, v) is defined by

ρ(u, v) =γ(S(u, v))2

2π ∗ σ(S(u, v))(5)

where γ is the perimeter and σ is the area of the surface S(u, v).After each edge collapse operation λ is locally recomputed forsurface S(u, v), and D∗ is updated by (1 − δ)λ. Where δ is aninfluence parameter used to minimize the influence of newlyformed surface features caused by the decimation process insuccessive iterations of hcol.

3) Mesh Partitioning: With each iteration of the hcol op-erator, the lowest mesh decimation cost C is applied, and thenew partition ϕ(n) = {ϕn

1 , ϕn2 , ϕn

3 , ϕn4 , . . . , ϕn

k } is computedas follows

∀k, ϕnk = pn

k ∪H(pnk ) (6)

where, pnk represents the kth cluster in D∗ after n edge col-

lapse operations. This procedure is iteratively performed untilall edges of D∗ are in clusters with concavity lower than a deter-mined concavity resolution value ε. In Fig. 3, we illustrate thedecomposition of a vascular branch section.

4) Parameter Estimation and Computation Time: The esti-mation of the concavity resolution ε was determined by volu-metric comparisons made on n = 65 preselected sections thatwere segmented by an expert (containing cerebral aneurysmsand vessel bifurcations). In order to obtain topological consis-tence, it is essential that the WACD algorithm does not separatethe volumes into more parts (see Fig. 3). Vessel sections weredecomposed using WACD and voxelized. In our experimental


Fig. 4. Overview of the spectral embedding process. (a) Manifold-2 surface of a vascular tree. (d) Graph Laplacian operator is derived from this manifold andrepresented by the affinity matrix [A]. (e) The matrix [A] is eigen decomposed into eigenvectors V and eigenvalues λ. (f) Vertices of the manifold with vertexcoordinates y are projected onto the subspace spanned by the first k eigenvectors. (b) Resulting manifold-2 structure ling in this k dimensional subspace. (below)Shows embedding of aneurysm using eigenvectors ranging from 200 to 15.

TABLE IIDECOMPOSITIONS OBTAINED FOR CONCAVITY RESOLUTION RANGES,

50 < ε < 250

Fig. 5. Computation time for clustering with expected decomposition timeindicated by blue line.

setup, parameters λ1 = −0.78, λ2 = 0.04, λ3 = 0.83, and δ =0.67 were used. Table II shows the decomposition obtained (fora single branch section and aneurysm section) for various con-cavity resolution ranges. We determined on optimal range of150 < ε < 200 that match closest to expert segmented sectionboth in volume and number of decomposed parts.

We measured the computation time of WACD on a PC runningWindows 7, 64-bits with an i-7 processor clock speed 2.66 GHzand 6 GB of RAM. On average the estimated decompositiontime for a triangulated mesh with 3000 vertices was 250 s (seeFig. 5).

Fig. 6. (a) Bifurcation section with deforming transformation, (b) applied tothe inlets. Plots of the first k = 30 eigenvalues for the deformed and normalvessel sections show (c) little or no change to the first eight eigenvalues andprovide invariant shape representations.

C. Vessel Shape Representation and Feature Selection

Classification requires an efficient shape representation. Weobtain a 3-D shape representation that is invariant under naturaldeformations, and at the same time, contains enough informationto perform shape characterization.

1) Shape Representation and Characterization: The Lapla-cian spectrum is computed for decomposed sections of vascula-ture. The intuition behind spectral shape descriptors as featuresis that the Laplacian eigenvalues are meaningful descriptors ofthe input surface due to their intrinsic definition and invariancewith respect to deformations [28] (see Fig. 6). From the defini-tion of the Laplacian of a graph L = D − W , the normalizedgraph Laplacian L = I − D−1/2WD−1/2 is considered (in ourcase L is geometric), where W is the weight matrix whoseelements are W (i, j) if (i, j) is an edge in D∗ and zero other-wise and I is the identity matrix. Dij =

∑Wij , is an n × n

matrix of the row sums of W. The eigen decomposition of Lresults in eigenvalues 0 = λ0 ≤ λ1 ≤ . . . ≤ λn−1 , with small-est eigenvalue corresponding to the constant eigenvector (whichis the smoothest mesh function). The corresponding eigenvec-tors Vi , i = 0 . . . n − 1, are orthogonal and span the eigenspaceRn×n . This Laplace spectrum provides a descriptive featurevector, which is used to characterize the input vessel shape. Ge-ometric information intrinsic to the decomposed vessel section


is captured using the weight matrix W [36], [37], where W isconstructed using principal curvatures.

Form the weighted dual graph representation D∗ each edge(connecting vertices (i, j) ) has an associated weight {Wij}. Foredge e = (i, j), the weight matrix Wij is constructed among allpairwise vertices given by

Wij ={

(|Ki | + |Kj |) · |〈n,e〉| · l, if Ki < 0 or Kj < 0

δ, otherwise(7)

where Ki and Kj are the minimal principal curvature of verticesi and j. When Ki ≥ 0 and Kj ≥ 0 implies convexity at e (seeTable I). To ensure vertex connectivity, we set δ = 0.01. TheLaplacian eigenvalues become invariant to shape scales by nor-malizing the spectrum of the input shape. We define e to be thenormalized average of the principal curvature directions for Ki

and Kj , n is the (normalized) direction of e, and l is the lengthof e normalized by the average length of all edges in the mesh. InFig. 4, we show the projection of a vascular 2-manifold surfaceSεRn×3 onto the subspace spanned by the first k eigenvectorsgiven by

S′ = V(1...k) VT(1...k)S (8)

where V(1...k) is a matrix containing the first k columns of V ,and S is a vector of mesh coordinates in the Euclidian space.

We use the ARPACK package [38] together with SuperLU [39] for eigen decomposition. Neumann boundary condi-tion [40] are applied due to the presence of small holes or miss-ing triangles in the reconstruction of S. This results in the firsteigenvalue λ0 to be zero and its corresponding eigenvector V0is the constant vector. We ignore it and start with λ1 . The eigen-values can be considered as mesh frequencies, whose corre-sponding low frequency eigenvectors are used to capture globalshape characteristics. One useful property of spectral methods istheir ability to compute invariant signatures for manifold struc-tures [28]. In Fig. 6, we demonstrate invariance to vessel defor-mation. Very little or no change is observed in the first k = 8eigenvalues, that correspond to low frequency mesh character-istics. This invariance to deformation property also makes themrobust to noise. Next, eigenvalues of vessel shapes that are morerelevant for shape characterization are selected through FeatureSelection and used for classification.

2) Validation and Test Data: Presegmented sections (con-taining cerebral aneurysms and vessel bifurcations) of vascu-lature archived for 98 aneurysms in 112 patients were utilizedfor validation and testing. This dataset consisted of n = 732vascular sections that included the same vessel shape and iso-metrically deformed variations of the vessel shape class (seeFig. 7). These sections were classified by a clinical radiologicalexpert into regular, aneurysm, and bifurcation sections yieldingthe dataset (ground truth).

3) Feature Selection: In general, feature selection (FS) ad-dresses the problem of finding the most compact and informativeset of features. Classification performance depends heavily onthe dimension of this feature vector [41]. As the dimension of thefeature vector increases, the number of samples required to trainthe system increase exponentially. By using the eigenvalues as

Fig. 7. Dataset used for bifurcation, aneurysm, and regular class

TABLE IIIEXPERT LABELED SAMPLES (ES)

vascular shape features, the dimension of the feature space canbe reduced without compromising much from the content.

We used the recursive feature elimination (RFE) method,which uses a support vector machines (SVM) [42]–[45]. Recur-sive feature elimination is a wrapper method which performsbackward feature elimination. The idea is to find the featureswhich lead to the largest margin of class separation. This com-binatorial problem is solved in a greedy fashion. The algorithmbegins with the set of all features and successively eliminates thefeature which induces the smallest change in the cost function.Finally, RFE computes a ranking of the selected features.

We used 65% of the data, n = 476 vascular sections forfeature and model parameter selection. The Bootstrap cross-validation methodology [46] was used to split this data-set intotraining and validation sets. The Bootstrap methodology in gen-eral is random resampling with replacement. The training setwas used to select relevant features, and the validation set wasused to determine the performance of the classifier on the se-lected features. Using RFE, the subset of eigenvalues that pro-duce best performances with respect to classification accuracy(i.e., the lower the classification error, the higher the accuracy)is selected.

An SVM with a Gaussian kernel was used as the classifier;the kernel width and the soft-margin parameter were selectedthrough 10 Bootstrap iterations using all 200 eigenvalues. Wetrain a different classifier for each class of vessel sections (bifur-cation, aneurysm and regular, shown in Fig. 7), where relevanteigenvalues are selected during RFE. These eigenvalues repre-sent the shape features that are shared by the class members(see Table III) and maximize the discrimination with respectto the nonmember shapes. If the selected eigenvalues changeat each Bootstrap iteration, this indicates that the selectionbased on eigenvalues strongly depends on the training set; such


Fig. 8. (a, d, g) the reconstructed mesh from clinical data, and (b, e, h) thedecomposition using WACD, where t is the number of triangle and k is thenumber of clusters. Classification of decomposed sections labeled as vesselclass and aneurysm class (AN) in green are shown in (c, f, i).

selections are less representative for the class of vessel shapes.The eigenvalues that are selected several times within all theBootstrap iterations are selected.

Using RFE, the number of eigenvalues selected for each ves-sel class ranged from 8 to 11 (Table III). This tells us that a verysmall fraction of eigenvalues (about the 6% of the input spec-trum) characterize the vessel shape features shared by a givenclass. We observed that the classifier uses almost the same se-lected features for all iterations of Bootstrap cross validation.This demonstrates the stability of the eigenvalues selected bythe classifier.

III. RESULTS

The proposed vascular decomposition and classificationmethodology were tested on labeled sections of the neuro-vasculature: BA, AcoA, PcoA, MCA, ICA, and SCA extractedfrom clinical data sets. We test first for vascular decompositionand next for classification. To demonstrate the clinical benefitsof this approach for computer-assisted detection (CADe), wealso test the method using longitudinal study data on four pa-tients with a prior history of unruptured and untreated cerebralaneurysms.

A. Experimental Study 1: WACD Decomposition

Table III documents our results; using WACD approximately81.5% of the structures were clustered into the same number ofcomponents identified as the ground truth. The results suggestedthat the proposed WACD method is capable of producing effec-tive decomposition, a parsimonious description basis on whichvessel classification can be performed. In Figs. 8 and 9, we

Fig. 9. Decomposition using WACD on sections of vasculature, (a) and(b) the decomposition of vessel tree with bifurcations, (c, d, e, f) decompo-sition used to separate vessel sections fused with sections of cranium.

Fig. 10. Shows decomposed and labeled vessel section, where the green sec-tions are labeled as aneurysm class, and yellow sections are labeled as vesselclass (VE).

illustrate the decomposition using WACD, where close confor-mity with ground truth was obtained. In Fig. 8(b) and (e), itcan be seen that WACD contained the aneurysm sac within asingle cluster, the same results can be seen around the circleof Willis in Fig. 11(b). For vessel branches, WACD clusteredbifurcations with minimal number of clusters, while maintain-ing close structural representation of the branched section [seeFig.8(h)]. In Fig. 9(d) and (f), WACD was used to cluster vesselsections fused with the skull. Although we did not intend to useit for separation of vessels fused to other structures, preliminaryresults seemed to favor its use.

B. Experimental Study 2: Vessel Classification

The performance of the classifier was tested on expert labeleddata. Using the ground truth dataset, 35% (n = 256) of vesselsections that remained after REF, was used to test the SVM clas-sifier. Comparisons were made between classifier performanceusing the first k (10, 20, 40 and 80) eigenvalues and the selected


Fig. 11. (a) Circle of Willis is reconstructed for a CTA dataset, where t isthe mesh triangles. (b) WACD is performed generating k sub-sections on thereconstructed vasculature. In (c), the labeled aneurysm section is highlighted ingreen and nonaneurysm sections are highlighted in yellow (VE).

TABLE IVFALSE POSITIVE RATE AND FALSE NEGATIVE RATE (FNR) OF CLASSIFYING

ANEURYSM SECTIONS

set of eigenvalues obtained by REF. For the two class problem(i.e., aneurysm or not aneurysm, bifurcation or not bifurcation,regular or not regular), we obtained a classification error of 21%for aneurysm sections, 12% for regular vessel sections and 19%for bifurcation sections (see Table II). However, in classifyingvascular sections our primary concern lies with the classifiererror in detection of cerebral aneurysms, given by the false neg-ative (FN) error (aneurysm sections that were classified as vesselsections) and the false positive (FP) error (vessel sections thatwere classified as aneurysm sections) in classifying aneurysmsacs. We evaluate the classifier for precision PRE (positive pre-dictive value) and error rate (ER) on classification of cerebralaneurysms. Table IV summarizes the classification error rateand precision rate with respect to the optimal set of eigenvalues(using RFE) and the first k (10, 20, 40, 80) eigenvalues. Our re-

Fig. 12. Case 3: 62 year old female was followed for incidental right-sidedMCA aneurysm. (a) MIP of the internal carotid artery and MCA shows a 4 mmaneurysm (arrow) with a 3-mm-wide neck in the M1 segment of the right MCA.(b) 3-D reconstruction of the patient vasculature.

sults showed the selected set of eigenvalues using RFE producesminimum error and improved classifier precision.

True Positive (TP): Aneurysm section classified as aneurysm.False Positive (FP): Vessel section (bifurcation and regular) clas-sified as aneurysm.False Negative (FN): Aneurysm section classified as vesselsection.False Positive Rate = FP/(FP+TN).False Negative Rate = FN/(TP/FN).Error Rate (ER): (FP + FN)/(TP+FN+TN+FP).Precision Rate (PRE): TP/(TP+FP).

In Fig. 8(a), the reconstruction of a large anterior communi-cating artery (ACOM) aneurysm is shown. Using WACD [seeFig. 8 (b)], the aneurysm section is decomposed, and aneurysmsac classified [see Fig. 8 (c)] shown in green. Also, in Fig. 8 (d), amedium sized ACOM aneurysm of nearly half the mesh densitythan that shown in Fig. 8(a) is decomposed using WACD andclassified Fig. 8(e) and (f). In decomposing arterial branch struc-tures, a section of the anterior cerebral artery (ACA) is shown inFig. 8(h). However, it is misclassified for an aneurysms section[shown in green Fig. 8(i)].

1) WACD-Based Classification of the Circle of Willis: It hasbeen suggested that the geometric characteristics of the cere-bral vessels of the circle of Willis may provide meaningful


TABLE VVOLUMETRIC AND SURFACE AREA COMPARISONS BETWEEN WACD VOLUME (VOX), SURFACE AREA (SA) OF EXTRACTED ANEURYSM HEAD SECTION AND

VOLUME (EV), SURFACE AREA (EA) OF EXPERT SEGMENTED ANEURYSM HEAD SECTION

association with vascular disease formation and progression[47]. We test if the methodology is capable of effectively decom-posing the circle of Willis and detecting cerebral aneurysms. Inparticular, cerebral aneurysms (pathological dilation of the ves-sel) are frequently located at or near arterial bifurcations andafter regions of high vascular curvature in arteries of the cir-cle of Willis [48]. In Fig.11, we illustrate the decompositionand classification of the circle of Willis using the WACD clas-sification approach. The decomposed sections are labeled (c)and classified as aneurysms and vessel sections. Anterior com-municating artery (ACoA) aneurysms are the most commonaneurysms [48]. Fig.10 shows the classification and labeling ofACoA aneurysm sacs.

C. Experimental Study 3: Longitudinal Study Analysis

We next address the clinical benefits of this proposed method-ology for automated vascular decomposition and detection ofcerebral aneurysms. Follow up analysis was performed usingdata on four patients with a prior history of unruptured and un-treated aneurysms. These aneurysms were not treated becausethey were determined to be small and nonthreatening. We per-form analysis on the following subjects; Case 1: 31 year oldmale with known history of bilateral ICA dissection and EC-ICbypass due to a moyamoya type changes post-ICA dissectionhaving tiny basilar apex aneurysm. CT stroke study showed en-largement of the known basilar aneurysm. Case 2: 44 year oldfemale with known history of aneurysms and had previouslystent coiled posterior communicating artery aneurysm on theright. A 5 × 5 mm incidental R Pcomm aneurysm, unchangedin size was followed over a period of two months. Case 3: 62year old female was followed for incidental right-sided mid-dle cerebral artery (MCA) aneurysm, overall appeared stable(shown in Fig.12). Case 4: 54 year old female with post carotidsacrifice and clipping of right MCA aneurysm and clipping ofthe left Pcom aneurysm. Follow up studies shows a small A1aneurysm that appeared stable in size.

1) Aneurysm Data: Data were acquired on the above casesthat had pretty much diffused vascular aneurysms. CTA datasetsobtained were used for vascular reconstruction and analy-sis. Measurements were performed on the sections containinganeurysms; aneurysm maximum diameters ranged from 1 to5 mm (mean, 2 mm ± 1[standard deviation]), and aneurysmneck sizes ranged from 1 to 3 mm (mean, 1.4 mm ± 1.2). Thevolume of aneurysm in terms of voxels ranged from 116 to 873,and aneurysm surface area ranged from 10.63 to 35.31 mm2 (see

Fig. 13. Reconstruction of aneurysm sections from preliminary data sets isshown (b). WACD is applied to case 3 and in (a) the decomposition of aneurysmsection is shown. In (c) decomposed aneurysm section is classified and aneurysmsac identified.

Table V). None of the aneurysms had previously been treatedwith coils.

2) Vascular Decomposition and Classification: Segmenta-tion of the neuro-vasculature and 3-D surface reconstruction wasperformed (10 CTA data sets). Using WACD, the reconstructedvasculature was decomposed resulting in the subparts of ves-sel sections (see Fig. 13). Next, decomposed vessel sectionswere classified as aneurysm and nonaneurysm (vessel) sections.Follow-up analysis (volumetric and surface area computation)was performed on the extracted (classified) aneurysm sacs fromlongitudinal data (preliminary, follow-up I, follow-up II). Expertsegmented aneurysm sections are used as ground truth. In ourstudy, eight of the ten aneurysms were correctly classified. How-ever, by specifying a volume of interest (VOI) around aneurysmsections (in the experimental setup a 8 cm3 VOI was used) allaneurysms were correctly classified and extracted. Table V doc-uments our results; volumetric and surface area comparisonswere made between expert segmented aneurysm sections andWACD classified aneurysm sections. We obtained a mean error


rate of 7.78% for volumetric comparison and 10.38% for surfacearea comparisons between expert segmented aneurysm sections.Fig. 12(a) and (b) illustrates Case 3: where MCA aneurysm onthe right-hand side followed over a period of 2 months, MIPindicated noticeable change in aneurysm size (see Table V).Using WACD and classification [see Fig. 13(a) and (c)], the de-composed aneurysm sac was extracted. We were able to detectchanges in aneurysm volumes (4.5% error) and surface areas(8.4% error) that were close to that segmented by an expert.

IV. DISCUSSION

The quantitative results aforementioned validate the feasibil-ity of the proposed vascular decomposition and classificationmethodology. Results obtained for vascular decomposition sug-gest that we attain near optimal decomposition close to expertsegmented sections, with approximately 81.5% of the vessel sec-tions decomposed into the correct number of components. Thisis an essential step for vessel classification. Since classificationof large sections of vasculature relies on vessel decompositionto simplify the problem.

Vessel classification is performed in the spectral domain. Weobtain a 3-D shape representation that is invariant under naturaldeformations, and at the same time, contains enough informa-tion to perform effective shape characterization. RFE was per-formed to find the most compact and informative set of features.We showed that the selected subset of eigenvalues producesminimum error and improved classifier precision.

The method was also validated for CADe on a longitudinalstudy of four cases having internal cerebral aneurysms. The mainreason for the follow-up study analysis was to demonstrate theeffectiveness of this approach in automated detection of cerebralaneurysms. Results show we were able classify aneurysms anddetect changes in aneurysm sac volumes and surface areas thatwere close to expert segmented sections.

V. CONCLUSION

Our primary objective was the decomposition of the vascu-lature into a geometrically consistent (meaningful) set of con-vex regions, and to characterize these decomposed sections forcomputer-aided detection. We have demonstrated that the pro-posed approach is a promising method for vascular analysiscapable of producing robust, effective representations, and theclassification of complex vessel structures which include bi-furcations and aneurysms. Form our initial longitudinal studyfindings suggest that our methodology can provide a basis forvascular analysis. However, a much larger follow-up study, andalso a comparative study with patients with ruptured and unrup-tured aneurysms, would be necessary to determine the clinicalefficacy of this methodology on identifying and labeling cere-brovascular aneurysms.

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3d vascular decomposition and classification for computer-aided detection

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