Automated Landmarking and Geometric Characterization of ... · Keywords: Cerebral angiography, Internal carotid artery, Segmentation, Geometric quantification, Computational anatomy,

Automated Landmarking and Geometric Characterization of the Carotid Siphon

Hrvoje Bogunovica,b,∗, Jose Marıa Pozoa,b, Ruben Cardenesa,b, Marıa Cruz Villa-Uriola,b, Raphael Blancc,Michel Piotinc, Alejandro F. Frangia,b,d

aCenter for Computational Imaging & Simulation Technologies in Biomedicine (CISTIB), Universitat Pompeu Fabra (UPF), Barcelona, SpainbNetworking Biomedical Research Center on Bioengineering, Biomaterials and Nanomedicine (CIBER-BBN), Barcelona, Spain

cDepartment of Interventional and Functional Neuroradiology, Foundation Rothschild Hospital, Paris, FrancedDepartment of Mechanical Engineering, University of Sheffield, Sheffield, UK

Abstract

The geometry of the carotid siphon has a large variability between subjects, which has prompted its study as a poten-tial geometric risk factor for the onset of vascular pathologies on and off the internal carotid artery (ICA). In this work,we present a methodology for an objective and extensive geometric characterization of carotid siphon parameterizedby a set of anatomical landmarks. We introduce a complete and automated characterization pipeline. Starting fromthe segmentation of vasculature from angiographic image and its centerline extraction, we first identify ICA by char-acterizing vessel tree bifurcations and training a support vector machine classifier to detect ICA terminal bifurcation.On ICA centerline curve, we detect anatomical landmarks of carotid siphon by modeling it as a sequence of fourbends and selecting their centers and interfaces between them. Bends are detected from the trajectory of the curvaturevector expressed in the parallel transport frame of the curve. Finally, using the detected landmarks, we characterizethe geometry in two complementary ways. First, with a set of local and global geometric features, known to affecthemodynamics. Second, using large deformation diffeomorphic metric curve mapping (LDDMCM) to quantify pair-wise shape similarity. We processed 96 images acquired with 3D rotational angiography. ICA identification had across-validation success rate of 99%. Automated landmarking was validated by computing limits of agreement withthe reference taken to be the locations of the manually placed landmarks averaged across multiple observers. For allbut one landmark, either the bias was not statistically significant or the variability was within 50% of the inter-observerone. The subsequently computed values of geometric features and LDDMCM were commensurate to the ones ob-tained with manual landmarking. The characterization based on pair-wise LDDMCM proved better in classifying thecarotid siphon shape classes than the one based on geometric features. The proposed characterization provides a richdescription of geometry and is ready to be applied in the search for geometric risk factors of the carotid siphon.

Keywords: Cerebral angiography, Internal carotid artery, Segmentation, Geometric quantification, Computationalanatomy, LDDMM, Landmarks.

1. Introduction

The locations where vascular pathologies tend to oc-cur more frequently are in general non-uniformly dis-tributed across the vasculature. For example, cerebralaneurysms (pathological bulging of arteries) have strongpreference for occurring at specific locations, and arefrequently found at or near the regions of high vascu-

∗Corresponding author, c/Roc Boronat 138, E08018 Barcelona,Spain. Tel: +34 93-542-1448, Fax: +34 93-542-1445

Email addresses: [email protected](Hrvoje Bogunovic), [email protected](Alejandro F. Frangi)

lar curvature in arteries of the Circle of Willis (Bris-man et al., 2006). Similarly, atherosclerosis (thicken-ing of the arterial wall) often occurs at carotid bifur-cation (Thomas et al., 2005) or near the bifurcationsof coronary arteries (Halon et al., 1983; Frangos et al.,1999).

Since geometry varies among different locations inthe vasculature, it is believed that this geometric varia-tion contributes to a corresponding variation in predis-posing hemodynamic forces (Lee et al., 2008b). Theseforces, coming from blood motion, are speculated toplay an important role in the initiation and localizationof pathologies, which in turn could explain their nonuni-

Preprint submitted to Medical Image Analysis February 13, 2012

form distribution (Zhu et al., 2009). Taking this intoaccount, Friedman et al. (Friedman et al., 1983) intro-duced the concept of “geometric risk factors” as the geo-metric features that provoke high hemodynamic stresseson the vessel wall.

An important condition for the geometric risk factorconcept is the presence of sufficient individual variabil-ity in geometry to induce important variations in indi-vidual hemodynamics (Friedman, 2002). One vessel ofclinical interest that satisfies this requirement is the in-ternal carotid artery (ICA). Located on each side of theneck, ICA is the main vessel that feeds blood to the ar-teries forming the anterior circulation of the brain. Thegeometry of ICA varies widely across the population, inparticular the part known as the carotid siphon (Krayen-buehl et al., 1982). The carotid siphon (Fig. 1) is the tor-tuous segment of the ICA that extends from the carotidcanal to the terminal bifurcation (ICA-TB) at which theICA bifurcates into the anterior cerebral artery (ACA)and the middle cerebral artery (MCA).

ICA terminal

bifurcation

ACAMCA

Carotid canal

Ca

roti

d s

iph

on

ICA

Figure 1: Carotid siphon with an aneurysm on the bifurcation with theposterior communicating artery.

Geometry of ICA is of special interest as incidencerate of aneurysms on it is high, as one third of allintracranial aneurysms occur along the carotid siphonor its terminal bifurcation (Brisman et al., 2006). Acouple of studies already identified effects linking ge-ometry and aneurysmal pathology on ICA. Piccinelliet al. (2011) concluded that ICA bends hosting rup-tured aneurysms tend to be shorter, having smaller ra-dius, lower maximum curvature, and the aneurysms arelocated closer to the bend center. Kim and Kang (2007)found that a relatively shorter length of the supraclinoidICA may be a risk factor for the development of an ICA-posterior communicating artery aneurysm due to higherhemodynamic stress. The geometry of ICA was also ofinterest to evaluate endovascular accessibility of lesionsand for treatment planning, which involves choosing theoptimal path and selecting the appropriate type of mi-crocatheters, guidewires and stents (Subramanian et al.,

2004; Jiang et al., 2004; Pakbaz and Kerber, 2007; Toy-ota et al., 2009).

The aim of this work is to provide a methodology forextensive geometric characterization of carotid siphonin an objective, robust and automated way, starting froman angiographic image. The characterization should al-low the comparison of carotid siphons within and be-tween subjects and measure their similarity. Such amethod would facilitate cataloguing the normal valuesand the variability of carotid siphon geometry to guidefuture exploration and identification of specific geomet-ric risk factors.

The summary of the paper is the following. In sec-tion 2, we overview the state-of-the-art in the geomet-ric characterization of vasculatures. In section 3, westart presenting the automated characterization pipeline(Fig. 2), with a focus on identifying ICA and detect-ing anatomical landmarks of carotid siphon. Based onthese landmarks of correspondence, we propose twoapproaches for the geometric characterization (Fig. 2).One is based on computing several geometrically intu-itive features (section 4), while the other on measuringthe pair-wise similarity between carotid siphons takingtheir entire shape into account (section 5). In section 7,we validate the automated ICA identification and land-mark detection and compare the two characterizationapproaches in their ability to separate carotid siphonshaving different shape classes, related to endovascu-lar accessibility. Finally, section 8 discusses the bene-fits and limitations of the proposed characterization andpresents our conclusions.

2. Related work

The geometric characterization of vasculature and inparticular of ICA, have already attracted a lot of atten-tion. The state-of-the-art can be divided into two mainapproaches. The most common one is to represent thevessel shape with a set of geometric indices, which areconsidered as candidates for being geometric risk fac-tors. The other approach is to consider each point of acenterline as a function of its arc length parameter andthen apply functional data analysis (FDA) to explore thevariability in a population.

Bullitt et al. (2003, 2005) focused on the measure oftortuosity of the intracerebral vasculature. Three dif-ferent tortuosity metrics were compared by their ef-fectiveness in detecting several types of abnormalities.In Chen et al. (2002), curvature, torsion and tortuosityand their change along the heart cycle were computedin a selected region of a coronary tree. In Gielecki et al.

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segmentation

Angiogra-

phic image

ICA

identification

Landmarks

detection

Geometric

quantification

Geometric

features

Centerline

computation

Shape

similarity

metric

Pair-wise

similarities

Figure 2: The image-based pipeline for geometric characterization of the carotid siphon.

(2008), tortuosity and deviation index as well as cur-vature angle were computed for describing the terminalpart of the basilar artery. O’Flynn et al. (2007) describedthe anatomy of normal human abdominal aorta and itsside renal arteries with tortuosity, non-planarity of bifur-cations, branching angles, curvature and torsion. In theworks by Meng et al. (2008a,b), carotid siphon is char-acterized by its spatial complexity defined as the sum ofthe curvature and torsion energy. The above methods,apart that they limit the characterization to a small setof isolated geometric indices, require user-interaction.

A framework for geometric analysis of vasculatureis presented by Piccinelli et al. (2009). Vascular struc-tures, are objectively characterized using computationalgeometry. Using such framework, Piccinelli et al.(2011) presented a geometric characterization of ICAand searched for patterns that can be associated to thepresence and rupture status of aneurysms. Vessel cen-terline is partitioned into a sequence of quasi-planarbends. Each bend is then characterized with severalgeometric indices: torsion peaks at proximal and dis-tal endpoints, mean and maximum curvature, length,radius, angular orientation of aneurysm, etc. How-ever, their method does not guarantee the correspon-dence of the bends, which is reflected by the discrep-ancy in the number of bends obtained from differentsubjects. Furthermore, the number of such obtainedbends strongly depends on the applied scale for vesselcenterline smoothing.

As opposed to computing geometric indices, Sangalliet al. (2009) applied FDA to characterize a set of center-line curves of ICA. An atlas of curves was created wherethe reference curve and a set of affine transformationsof arc length parameter are simultaneously estimatedby Procrustes fitting. From the set of aligned curves,functional principal components analysis (PCA) of theirlocal radius and curvature was performed. In Sangalliet al. (2010), they extended their atlas to allow for mul-tiple reference curves. A method called k-mean align-ment is proposed for simultaneous alignment and clus-tering of spatial curves. However, as the transforma-

tions are restricted to be affine, the alignment does notassure correspondences between points and bends ofcurves. These inter-subject anatomical correspondencesare essential for computing geometrical descriptors. Be-sides, we consider them an important requirement forthe correct shape comparison between ICAs.

3. ICA segmentation

3.1. Segmentation of the vasculatureSegmentations of the vasculature are performed in

an automated way with a geometric deformable modelcalled Geodesic Active Regions (GAR) (Hernandez andFrangi, 2007; Bogunovic et al., 2011). The method wasdemonstrated to be accurate for 3D rotational angiog-raphy (3DRA) and time-of-flight magnetic resonance(TOF-MRA) images. The result of the segmentation is atriangular mesh modeling the vascular lumen with sub-voxel precision.

3.2. Vascular tree centerlines computationThe shape of tubular objects, like vessels, can be

approximated by the shape of their centerline (medialaxis), which is a 3D spatial curve. We obtain the setof vessel centerlines in two steps. First, to obtain theestimate of the topology of the vascular tree, fast topo-logical thinning based on collapsing fronts followed bya fast marching computation to assure centerline con-nectivity (Cardenes et al., 2010) was applied to obtainthe skeleton of the segmentation (Fig. 3). The skele-ton, due to imaging resolution and segmentation inac-curacies producing touching vessels, might not have thetopology of a tree. However, its end-points do corre-spond to the root and the terminal leaves of the under-lying vascular tree. The root was taken to be the end-point with the maximal associated radius at the lowestaxial plane, which corresponded to the ICA entering theimaged field of view.

Second, the set of accurate centerlines is obtained bybacktracking along the minimal cost path from the end-points toward the root using (Antiga et al., 2003), imple-mented in the open-source library VMTK (Antiga and

3

Figure 3: Segmented vascular mesh and its skeleton.

Steinman, 2011). Every point of such centerline, cor-responds to the center of a maximally inscribed sphereand the set of centerlines topologically form a rootedtree with the edges directed away from the root in ac-cordance with the blood flow.

3.3. ICA identification

To identify ICA in the extracted vascular tree, we ap-plied the method for detecting ICA-TB, that we prelimi-narily presented in (Bogunovic et al., 2010). A machinelearning based approach is applied, where a classifier istrained on a set of labeled bifurcation feature vectors.Then, a breadth-first traversal of tree bifurcations is per-formed until the first positive detection. With this strat-egy we only needed to differentiate ICA-TB from otherbifurcations along the ICA as the potential error wouldappear either as a false positive along the ICA or a falsenegative at its terminal bifurcation.

Origins of the bifurcations forming the vessel tree andtheir bifurcation vectors (unit vectors denoting direc-tions of parent vessel and the two daughter branches),are defined using the objective and robust criteriaof Antiga and Steinman (2004); Piccinelli et al. (2009).The two daughter branches are differentiated by their ra-dius: the larger daughter branch and the smaller daugh-ter branch. Then, each bifurcation is geometrically char-acterized with the following 15 dimensional feature vec-tor (Fig. 4(a)): ratios of mean vessel radii between eachpair of vessels forming the bifurcation (3); sagittal, axialand coronal-components of the three bifurcation vectors(9); angles between each pair of the bifurcation vectors(3).

As a classifier, we employed C−Support vector ma-chine (C−SVM) (Chang and Lin, 2011) with a radial ba-sis function kernel. Optimal classifier parameter valueswere obtained by a simple grid search through multiplecombinations. The ones giving the best cross-validation(CV) score were chosen.

Finally, once the ICA-TB is detected, ICA is ex-tracted as a sequence of 4D points (3 spatial center-line coordinates plus the vessel radius) along a curvi-linear abscissa starting from the ICA-TB and proceed-ing toward the heart until it reaches the root of the tree(Fig. 4(b)).

Parent

branch

Larger

branch

Smaller

branch

(a) (b)

Figure 4: (a) Bifurcation characterization: Origin (black cube) andassociated bifurcation vectors. (b) Example of identified ICA and itscenterline.

3.4. Carotid siphon landmarksTo compare carotids within and between subjects, we

identify a set of sparse landmark points of anatomicalcorrespondence. They are essential as they will serveas a base for geometric characterization. We model thecarotid siphon, the part of ICA from terminal bifurca-tion to carotid canal, as a sequence of four bends named(from ICA-TB towards the heart): superior, anterior,posterior and inferior bend, following their anatomi-cal position with respect to the siphon center (Fig. 5).The inferior, posterior and anterior bends have been ob-served to be highly planar.For the superior bend, the pla-nar approximation is not found to be valid as its shaperesembles more a helix (non-zero torsion).

The landmarks we selected corresponded to: ICA-TB, the centers of the bends and to the interfaces be-tween the bends of the model. However, for the helicalsuperior bend, the location of its center turned out to behighly ambiguous and that landmark has consequentlybeen omitted. The final set of chosen seven landmarksis shown in Fig. 5.

As the landmarks are associated with the four-bendmodel, we first identify the four bends on a centerlinespatial curve. For this, we will make use of two natu-ral frames defined on a spatial curve: The Frenet-Serret

4

Superior

bend

Anterior

bend

Posterior

bend

Inferior

bend

Figure 5: Carotid siphon with the four bends (in color) and the sevenlandmarks (in black). White area denotes: outside the region of inter-est.

and the parallel transport one (Bishop, 1975; Hansonand Ma, 1995). We start by giving an overview of thetwo frames and then present the method for bends andlandmarks detection.

3.4.1. Frames on a spatial curveGiven a regular parameterized differentiable space

curve Γ with a normalized arc length parameter s,

Γ = {x(s)|s ∈ [0, 1], x ∈ R3}, (1)

the Frenet-Serret frame is defined locally by a triad: tan-gent T (s), normal N(s) and binormal B(s),

T (s) = x ′(s)

N(s) =T ′(s)

∥ T ′(s) ∥=

x ′′(s)∥ x ′′(s) ∥

B(s) = N(s) × T (s) =x ′(s) × x ′′(s)∥ x ′(s) × x ′′(s) ∥ . (2)

Thus, (T (s), N(s), B(s)) forms an orthonormal basis. Nis a unit vector pointing towards the center of the lo-cally osculating circle, i.e. in the direction the curve iscurved. The vector T ′(s) = κ(s)N(s), with the magni-tude being the scalar curvature κ(s) =∥ x ′′(s) ∥, is thencalled the curvature vector. Example of Frenet-Serretframe on a spatial curve is given in Fig. 6(a). Frenet-Serret frame can change orientation abruptly and is notdefined when curve is locally straight (x ′′(s) = 0).

The parallel transport frame (Bishop, 1975), is theframe obtained by parallel transport in the normal bun-dle of the curve. It can be obtained from any orthonor-mal basis {E1(0), E2(0)} spanning the plane orthogonalto the tangent T (0) at the initial point x(0), by paralleltransporting it along the curve. The following equation

defines such a frame:T ′(s)E1′(s)

E2′(s)

= 0 k1(s) k2(s)−k1(s) 0 0−k2(s) 0 0

T (s)E1(s)E2(s)

.(3)

E1′

and E2′

depend only on T and are parallel to it,hence are well defined everywhere on a regular spa-tial curve, regardless of curvature. Example is givenin Fig. 6(b). Such frame is smoothly varying and notaffected by the underlying torsion.

We are interested in representing the curvature vectorin the parallel transport frame:

T ′(s) = κ(s)N(s) = k1(s)E1(s) + k2(s)E2(s). (4)

Thus, k1 and k2 are the components of the curvaturevector with respect to basis {E1, E2} and every spatialcurve is uniquely represented in the (k1, k2) space upto a rotation. Indeed, this introduces a natural exten-sion to 3D of the notion of the oriented or signed curva-ture restricted to 2D plane curves. Such a representationavoids the computation of torsion (which requires third-order derivatives), hence we only require curves to be ofclass C2, which makes it more stable and robust to thelevel of noise on the extracted curve.

3.4.2. Bends and landmarks detectionBends are curved parts of the centerline and are sepa-

rated by a local curvature minimum at their ends. How-ever, the total number of the curvature extremums variesacross population and also depends on the scale and onthe extent of ICA visible in the image. To avoid falsepositive detections due to consecutive curvature mini-mums forming the same anatomical bend, we will makeuse of curvature vector expressed in the parallel trans-port frame. The main idea is to use the property that thecurvature vector changes orientation at the bend transi-tions, while E1 and E2 of the parallel transport frameremain stable along the curve (Fig. 6(c)). To detect thebend transitions, we then use the curve representationin the above defined (k1, k2) space. Thus, the changeof bends between two centerline points correspondingto local curvature maximums is expected to produce awide angle (θ > 90◦) between their vectors. On theother hand, if the angle between them is small (θ < 45◦),the two curvature maximums are expected to belong tothe same bend (Fig. 8(a)).

We argue that this is a more robust approach to bendsubdivision than the one based on observing the tor-sion and curvature peaks along the centerline, presentedby Piccinelli et al. (2011). The sensitivity of the torsion

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(a) (b)

(c)

Figure 6: Centerline spatial curve with: (a) Frenet-Serret frame withnormal N (green) and binormal B (purple) vectors. (b) Parallel trans-port frame with E1 and E2 (red and blue). (c) Normal vector N in theregion of bend transition changes the orientation with respect to E1and E2.

profile to the amount of noise on the centerline makesit difficult to select the level of centerline smoothing, asthe level appropriate for one subject is not necessarilyappropriate for the others. The proposed representationin (k1, k2) space is a more stable approach as the an-gle between curvature vectors of two centerline pointscorresponds to the amount (integral) of torsion betweenthem. Detecting bend transitions with a set of anglethresholds was able to consistently identify the corre-sponding bends across subjects. This is demonstratedin Fig. 7, where the results of the two approaches arecompared.

The landmark that will serve as a reference point toidentify all four anatomical bends is the one in the mid-dle, marking the interface between the anterior and theposterior bend. It is identified by combining the cur-vature information with the coronal coordinate of thecenterline (Fig. 8(b)). As the anterior bend is anatomi-cally positioned at the front, starting from the positionof maximum of the coronal coordinate of the centerlineand moving against the blood flow we search for thetwo neighboring curvature maximum points that haveθ > αant-post, where αant-post is a threshold parameter.

(a) (b) (c)

(d) (e) (f)

Figure 7: Subdivision of carotid siphon into bends. (a-c) Bend subdi-vision using the method of Piccinelli et al. (2011) with the same levelof Laplacian smoothing of the centerlines. The scale appropriate forcase (a), is to small for case (b) and too large for case (c). (e-f) Thefour bends detected on the same subjects using the proposed method,with fixed, small level of centerline smoothing.

The point of curvature minimum between the two suchmaximums is then the interface landmark.

From the anterior-posterior interface landmark, andmoving along the blood flow, we identify the anteriorand the superior bends by searching for their interfacelandmark point as curvature minimum where the twosurrounding points of curvature maximums have θ >αsup-ant. Similarly, moving opposite to the blood flow weidentify the posterior-inferior interface landmark withαpost-inf. Subsequently, the end of the inferior bend andROI is found with αinf-end. The threshold parameterswere fixed to αant-post = 60◦, αsup-ant = αpost-inf = 45◦,and αinf-end = 110◦ after observing θ values appearingin a subset of our data.

Once the bends are detected, we estimate their centrallandmarks. We model the central landmark to corre-spond to the center of the curved segment of the bend ata scale where its centerline has only one curvature max-imum. Thus, if a bend is initially composed of multiplecurvature peaks we observe the bend at a larger scale.This is achieved by convolving the centerline curve ofthe bend with a Gaussian function, as the standard de-viation of the Gaussian increases. Such curve evolu-tion is repeated until only one curvature maximum re-

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(c) (d)

Figure 8: Bends and landmarks detection: (a) (k1, k2) space of the centerline. Numbers denote the curvature peaks sequentially starting fromICA-TB. Between points 3 and 4 (θ1 ≈ 120◦) there is a transition of bends. Points 4 and 5 (θ2 ≈ 20◦) belong to the same bend (posterior). (b)Coronal coordinate and the curvature of the centerline. The global coronal coordinate maximum is denoted with vertical blue line. The interfacelandmarks between the four bends are denoted with vertical black lines. (c) Estimation of central segment (bounded with two vertical magentalines) and central landmark (vertical black line) of the posterior bend, using scale space. (d) The four bends with their central segments (in moresaturated color: green, red, yellow) and central landmarks (black).

mains. Then, the central segment is defined as the re-gion around the curvature maximum, delimited on bothsides by the mean of the curvature values at the maxi-mum and at the corresponding end. The central land-mark is taken as the midpoint of the central segment(Fig. 8(c),(d)).

4. Geometric quantification

Having identified the bends and the landmarks of thecarotid siphon, we are now able to compute a set of lo-cal and global features that quantify its geometry. Asthe shape is defined by the object geometry that is in-variant under similarity transformation (translation, ro-tation, and uniform scaling) (Kendall et al., 1999), theproposed set of features is accordingly made invariantunder this transformation. We compute the followingset of geometric features.

4.1. Bend lengths and average vessel radius

Lengths of each of the four bends are presented aspercentages of the region of interest occupied by each ofthe bends, obtained from their normalized arc lengths.

Vessel cross-section area is more related to hemody-namic properties than the radius of the maximally in-scribed sphere (Boskamp et al., 2004). Thus, we de-fine as local vessel radius, the radius of a circle hav-ing the same cross-section area. Along the centerlineof the bend, perpendicular cutting planes are automati-cally positioned to obtain the vessel cross-sections fromthe segmented mesh, and the circle equivalent radius is

computed. However, if the aspect ratio of minimal overmaximal cross-section diameter is below certain thresh-old (the value of 0.75 is chosen after visual observation),the section is considered to cross an aneurysm or a ves-sel bifurcation, the cross-section is ignored and its di-ameter value is linearly interpolated from its neighbors.

4.2. Osculating planes

For the bends that are observed to be quasi-planar (in-ferior, posterior, anterior), their osculating planes are fit-ted to the points forming the central segment of eachbend using least squares fit. The plane normal vectordefined by its sagittal, axial and coronal-components isthen used as a feature.

4.3. Change of osculating planes

The bends forming the siphon are concatenated in anon-planar way and the osculating planes change. Ingeneral, it has been shown that non-planar connection ofdouble-bend geometries influences the hemodynamics,especially the mixing and swirling of blood flow (Leeet al., 2008a). Thus, we quantify this change of osculat-ing planes of the siphon with the following values:

• Angles between all pairs of osculating plane nor-mal vectors ni computed directly as: arccos(ni ·n j), i , j.

• Directed angles between osculating plane normalvectors of consecutive bends, computed after par-allel transporting one to another on the normal bun-dle of the curve.

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• Two directed angles describing the ouf-of-planerotation (OPR) of two consecutive bends (proxi-mal and distal): inferior and posterior; posteriorand anterior. One (OPR1) corresponds to the rota-tion around the axis lying in the plane of the prox-imal bend while being orthogonal to the centerlinetangent at the transition landmark between the twobends (Fig. 9(a)). The other (OPR2) correspondsto the rotation around the axis defined by the cen-terline tangent at the transition landmark betweenthe two bends (Fig. 9(b)).

(a) OPR1 (b) OPR2

Figure 9: Illustration of the measured change of osculating planesbetween the posterior and the anterior bend. Each angle (betweenblue and red vectors) measures the rotation around the correspondingaxis (magenta).

4.4. Bending radiiThe points of the central segment of each planar bend

are fitted with a circle using Gauss-Newton method fornon-linear least squares optimization (Fig. 10). Radiusof the circle divided by the average vessel diameter ofthe corresponding bend is then used.

Figure 10: Circles (in black) fitted to the central segments of the ante-rior and the posterior bends.

4.5. Global features

Global features are computed from the region of in-terest starting from the terminal bifurcation of ICA un-til the most proximal landmark (center of the inferiorbend). We consider five global features:

Tortuosity Defined as the relative increment in thelength of a curve deviating from a rectilinear line,tortuosity χ is computed as (Piccinelli et al., 2009)

χ =Ld− 1, (5)

where L is the total arc length of the centerlineunder analysis and d is the Euclidean distance be-tween its endpoints.

Bending and twisting energy The energy required tobend and twist a straight line into its curved shape.It corresponds to the average value of the squarecurvature κ and torsion τ, respectively, over the to-tal arc length L of the centerline under analysis.Bending energy (BE) and twisting energy (TE) aredefined as (Meng et al., 2008b):

BE = L2∫ 1

0κ2(w)dw; TE = L2

∫ 1

0τ2(w)dw,

(6)The L2 factor guarantees scale invariance.

Curvature ratio and torsion ratio Dimensionless ra-tios of vessel radius with curvature and torsionradii form part of the Dean and Germano numbersthat characterize flows in curved tubes (Formaggiaet al., 2009). Thus, we define mean squared cur-vature ratio (CR) and mean squared torsion ratio(TR) as:

CR =∫ 1

0R2(w)κ2(w)dw;

TR =∫ 1

0R2(w)τ2(w)dw, (7)

where, R, κ and τ are local vessel radius, curvatureand torsion, respectively.

5. Shape similarity metric

In addition to geometric quantification, we charac-terize the variability of carotid siphon shapes using theframework of computational anatomy (Grenander andMiller, 1998). There, shape variations are modeled bydiffeomorphisms (differentiable transformations with

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differentiable inverse). One of the proposed paradigmsfor diffeomorphic registration is the large deformationdiffeomorphic metric mapping (LDDMM) (Beg et al.,2005), which apart from providing correspondences be-tween shapes defines a metric in shape space.

To establish the shape similarity distance mea-sure between carotid siphons, we use the large de-formation diffeomorphic metric curve mapping (LD-DMCM) (Glaunes et al., 2008), between each pair oftheir centerline curves. The registration of two spatialcurves C and S is performed by searching for a diffeo-morphism φ, which matches the given curves: φ(C) =S , taken as the end point t = 1 of a flow of diffeomor-phisms ϕt modeled by a time-dependent velocity vectorfield vt : Rd → Rd as

∂ϕt

∂t= vt(ϕt); ϕ0(x) = x. (8)

The distance between the two curves in the shapespace, D(C,S), is then defined by the length of the short-est diffeomorphism flow matching them:

D(C, S ) = inf ρ(ϕt), when ϕ0(C) = C, ϕ1(C) = S .(9)

The length is defined as the deformation cost function

ρ(ϕt) =(∫ 1

0||vt ||Vdt

) 12

, (10)

where the space V is a reproducing kernel Hilbert space(RKHS) of the smooth velocity fields with reproducingkernel being the Gaussian function with standard devia-tion σV , which determines the smoothness of the defor-mation.

The optimal transformation φ = ϕ1 is then computedby minimizing the energy functional

JC,S (ϕt) = γρ(ϕt)2 + E(ϕ1(C), S ), (11)

where E is a curve matching term and γ is a parame-ter of regularization weight. The matching term definedin Glaunes et al. (2008) was

Ecr(ϕ1(C), S ) = ||µϕ1(C) − µS ||2W∗ , (12)

where µC is a linear functional that embeds the curveC in a RKHS W∗ of currents, enabling comparisonof curves without assuming point correspondences be-tween them. The reproducing kernel is defined by theGaussian function with standard deviation σW , repre-senting a spatial scale of currents, and determines thescale of geometric details of curves that are taken into

account. σW was set to a small value of 2 mm.We extend the matching term by landmark matching

of our previously (subsection 3.4.2) extracted N = 7landmarks (xn, yn), n = 1, . . . ,N. This assures anatom-ically valid matching as the anatomical landmarks arerequired to correspond. We apply inexact landmarkmatching (Joshi and Miller, 2000) with normalized Eu-clidean metric, assuming independency between thelandmarks

Elm(ϕ1(x), y) =N∑

n=1

||yn − ϕ1(xn)||2σ2

n, (13)

where each landmark n has σn associated, which repre-sents the expected inaccuracy in its localization.

Thus, the final energy term that we minimize is

JC,S ,x,y(ϕt) = γρ(ϕt)2 + γcrEcr(ϕ1(C), S )+ γlmElm(ϕ1(x), y), (14)

where γ, γcr, and γlm are weights of the regularization,curve matching and landmark matching terms, respec-tively. As the Ecr and Elm matching terms are not sym-metric to the choice of source and target curves, neitheris the resulting pair-wise distance. To symmetrize it, wetake the distance to be the mean value from minimizingJC,S and JS ,C:

D(C, S ) =12

(D(C, S ) + D(S ,C)). (15)

Before the start of the LDDMCM registration, thetwo curves are registered under the similarity transfor-mation as any variability described by this transforma-tion is not considered as difference in shape. The impor-tance of adding the landmark matching term is demon-strated in Fig. 11. Although the transformed centerlinesare similar, the registration without using the landmarksdoes not provide correct anatomical correspondences ofthe bends and underestimates the geodesic distance inshape space, compared to the one that does match thelandmarks.

6. Evaluation methodology

In this section, the methodology applied for eval-uating the elements of the geometric characterizationpipeline is presented.

6.1. ICA classification performance

To evaluate the success of ICA-TB identification, 5-fold cross-validation (CV) was repeated 10 times and

9

(a) D = ρ(ϕt) = 5.7 (b) D = ρ(ϕt) = 6.8

Figure 11: Diffeomorphic registration of source centerline (blue) totarget centerline (red), with the registered centerline (green) and esti-mated distance D in shape space: (a) without (E = Ecr) and (b) withlandmark matching term (E = Ecr + Elm).

the estimated accuracy of correct classification wastaken as the average CV success rate for all repetitions.To provide a better understanding of which are the mostdiscriminating bifurcation features, we performed fea-ture selection as a sequential forward selection (SFS).Starting with an empty set, at each forward (inclusion)step, the feature added to the feature subset is the onethat maximizes the cross-validation (CV) classificationrate.

6.2. Landmark detection

As the later geometric characterization is based on thelandmarks, their detection has been extensively evalu-ated both qualitatively and quantitatively.

Qualitative evaluation. First, the carotid siphons werevisually inspected to check whether they have been beenpartitioned into the four bends and that none of themwere merged or split. Second, to evaluate the stabil-ity of the thresholds chosen for the four bends of themodel, we displayed the probability densities of anglesθ in (k1, k2) space (Fig. 8(a)) between curvature maxi-mums that belonged to the true and false transitions.

Quantitative evaluation. The landmarks from the four-bend model attempt to reproduce the human intuition ofwhere are the corresponding points representing bendtransitions and their centers. As the model is just anapproximation of objects having much larger anatom-ical variability, these points do not necessarily coin-cide with the curvature properties like the maximum

and minimum. Thus, we considered expert observeras the best reference for identifying these correspond-ing points along the siphon. The reference landmarkpositions were obtained as the average across multipleobservers of the manually placed ones. To evaluate theaccuracy of the automatically determined landmarks wethen computed:

• The limits of agreement of the automatically de-termined landmarks with the reference, which rep-resent the 95% confidence interval of the differ-ences (Bland and Altman, 1986) and are expressedas bias and standard deviation σloa.

• The standard deviation of the inter-observer vari-ability σo, computed using one-way analysis ofvariance (ANOVA) (Carstensen et al., 2008).

• The variability index I, defined as the ratio of theabove two standard deviations:

I =σloa

σo. (16)

If the value of this index I < 1, the landmarks fromthe automated method deviate from the referenceless than the manually placed landmarks vary be-tween observers.

6.3. Geometric quantification

The values of the computed geometric features (sec-tion 4) depend on the estimated landmark positions.Thus, we compared the values of features computedfrom automatically determined landmarks to the onesobtained with manually placed landmarks. Reference istaken to be the average of the values obtained from in-dividual landmarkings by each observer. We then com-puted for each feature the variability index I (Eq. 16)and the normalized mean error computed as absolute er-ror divided by the population range of values.

6.4. Shape similarity metric

The pairwise shape similarities also depend on theestimated landmark positions. We take the referenceto be the shape distances obtained using the referencelandmark positions and enforcing the exact landmarkmatching during the registration process. Exact land-mark matching is enforced by setting a high value to theweight γlm in Eq. 14. The distances based on automatedlandmarking were then computed with three differentoptions for the landmark matching terms: no landmarkmatching, exact landmark matching, and the proposedinexact landmark matching. The limits of agreement

10

with the reference for these three variants are computedand compared. In addition, we evaluated how differentchoices of the parameter σV , which defines the smooth-ness of the deformation field, affect the obtained dis-tances and the matching residuals (Eq. 11).

6.5. Carotid siphon shape classesTo evaluate and compare the characterizations based

on the proposed geometric features and LDDMCM, twoclinicians labeled the class of siphon shape followingthe classification proposed by Krayenbuehl et al. (1982).Such classification has been used in evaluating the vas-cular accessibility between the guide-catheter and thelesion (Kim et al., 2008). Essentially, there are fourshape classes: U, C, V and S (Fig. 12), with variationbeing mainly in the part of the anterior and the poste-rior bend. We then expect that the siphons characterizedby the two proposed schemes will cluster by classes i.e.the siphons belonging to the same class will have simi-lar geometric features and small LDDMCM distance toeach other.

(a) U-type (b) C-type

(c) V-type (d) S-type

Figure 12: Examples of carotid siphon shape classes.

The following steps were then performed. First,we applied dimensionality reduction to map all thecarotid siphons to an Euclidean submanifold. Princi-pal Geodesic Analysis (PGA) (Fletcher et al., 2004) isapplied on the geometric features as they are composedof a mixture of features in R+ (radii and lengths), SO(2)(angles), and S2 (normal vectors). Each normal vector istreated as a single feature, having two degrees of free-dom. Similarly, for LDDMCM, classical multidimen-sional scaling (CMDS) (Pekalska and Duin, 2005) is

applied. We performed two dimensionality reductions.One to 2D, for the purpose of visualizing and qualita-tively evaluating the achieved clustering. The other, to asmallest dimension still preserving the 99% of the totaldata variance, for the purpose of quantitative evaluationas a classification success rate of a linear classifier. Inthe obtained submanifold, we trained a classifier usinglinear discriminant analysis (LDA) and leave-one-outcross-validation classification rate is reported as a mea-sure of how well separated the four classes are. Finally,we looked at the LDDMCM classification performancefor different choices of σV while for the characteriza-tion based on geometric features we performed SFS offeatures to identify the most relevant ones.

7. Results

The geometric characterization pipeline was retro-spectively applied to 96 images acquired with 3DRA,from 86 patients (age range: 33 − 76, mean age: 53years, 74% women). Contrast was injected to enhancethe vessels comprising anterior cerebral circulation ofeither left (43) or right (53) hemisphere (10 patientshad both sides imaged). Acquisitions were performedwith an angiographic unit: Allura Xper FD20 (PhilipsHealthcare, Best, The Netherlands). On a dedicatedworkstation, 3D images were reconstructed with a 2563

matrix having a voxel size of 0.29 × 0.29 × 0.29 (mm).All images were successfully segmented and had theirvascular tree centerlines and topology extracted.

7.1. ICA classification performance

From all 96 vascular trees, the feature vectors of 297bifurcations along ICA were manually labeled as “ter-minal” (96) or “non-terminal” (201), and supplied toSVM classifier for training and cross-validation (CV)(Fig. 13). Feature selection revealed that the two fea-tures that contributed the most to the ICA-TB discrim-ination were: ratio of mean radii between the smallerand the larger daughter branches and axial componentof the smaller daughter branch vector. At peak CVrate (6 selected features) ICA-TB was misclassified inonly one case (99% success) rate, producing one falsepositive and zero false negatives (99.5% specificity and100% sensitivity). The false positive sample was theonly example available of a bifurcation of ICA with atentorial marginal branch, hence during its testing nonewere present in the training data. The results show thatthe chosen features describe adequately the bifurcation.

11

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Figure 13: Identification of ICA-TB: Cross-validation (CV) classifi-cation rates for sequential forward selection of features.

7.2. Landmark detection

Qualitative evaluation. Detection of landmarks for par-titioning the carotid siphon into bends failed in threecases (97% success rate) (Fig. 14). Failure occurredmostly when the transition between the posterior andthe inferior bend was missed due to small angle betweenvectors in (k1, k2) space (θ < αpost-inf = 45◦, Fig. 14(d)).

(a) (b)

(c) (d)

Figure 14: Automated landmarking:(a-c) Examples of successfulcases. (d) Example of a case where the posterior and the inferior bendwere incorrectly detected as one (θpost-inf = 40◦).

Evaluation of threshold stability is shown in Fig. 15.Of the four thresholds, the αant-post, αinf-end are the morestable ones, as they separate two distributions tightlygrouped around their means. We can observe that anychoice of αant-post in the range of [40o − 80o] would pro-

duce the same results. Choice of αpost-inf and αant-supis more critical but given that they were evaluated ona large number of cases, a good generalization is ex-pected.

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Figure 15: Distribution of angles in (k1, k2) space that present bendtransition (BT) and non-bend transition (No-BT), in a form of a box-plot and histogram. (a) anterior-superior, (b) anterior-posterior, (c)posterior-inferior, (d) inferior-end.

Quantitative evaluation. Three observers manuallyplaced the landmarks on sequentially chosen subset of50 cases. The results are shown in Fig. 16. The tran-sition between the superior and the anterior bend (L1)has the largest localization error in both bias and stan-dard deviation. However, this is the location with thelargest inter-observer variability as well. For other land-marks, either the bias is not statistically significant orthe variability is within 50% of the inter-observer one.The central landmarks (L2, L4, L6) are particularly welldetected with no significant bias and the deviation fromthe reference being below the inter-observer one for theposterior and the inferior bends.

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Figure 16: (a) Limits of agreement between automated landmark-ing and the reference. Bias is denoted with a marker and 95% con-fidence interval, while the bars correspond to 95% limits of agree-ment (±2σloa). Agreements are compared with inter-observer lim-its of agreement for the manual measurements (b) Variability indexfor each landmark. Landmarks are ordered from distal to proximal:superior-anterior (L1), central anterior (L2), anterior-posterior (L3),central posterior (L4), posterior-inferior (L5) and central inferior (L6).

7.3. Geometric quantification

The evaluation results for the total set of geometricfeatures is shown in Fig. 17. We can observe that allthe features have I < 1.8 and normalized mean abso-lute error below 8% with more than half of them havingI < 1.3 and error less than 3%. Thus, we can assumethat landmark localization is sufficiently accurate not toaffect the computed geometric features.

7.4. Shape similarity metric

For the registrations using inexact landmark match-ing, previously evaluated σloa (Fig. 16(a)) of each land-mark’s limits of agreement are used to normalize cor-responding Euclidean distances (σn = σ

loan ) in Eq. 13.

Thus, the larger the landmark’s discrepancy from the

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Figure 17: Boxplot for geometric features, showing the distributionof: (a) variability index and (b) normalized mean error. Boxes spanthe lower (higher) quartiles and whiskers extend up to 1.5 the inter-quartile range.

reference, the smaller is its influence during the regis-tration.

The parameters γcr and γ were experimentally set toγcr = 1 and γ = 0.1, for all registrations. Observing theeffect of the parameter σV on the registration results,for large σV (> 6 mm), due to strong smoothness con-straint on the deformation, the final matching precisiondeteriorates. At such scales, the deformation is not ac-counting for the details that we consider to be part ofthe difference in geometry as opposed to noise. SmallσV (< 1 mm), allows highly irregular speed fields andnearby regions start to move independently. The ob-tained distance is not representative of the one in theshape space and the registration optimization is likely toend in a local minimum. The analysis (Fig. 18) confirmsthis observation. The values of σV from the range [1, 6]produce small matching error and the optimal choice de-pends on a priori assumptions and the final application.We chose σV = 4.5 mm, which is the scale of the aver-age vessel diameter. This choice is later reevaluated forthe application to siphon shape classification.

For the inexact landmark matching, to find γlm wetook a small sample of the first 10 cases and used itas a training set. Then, the sweep search with γlm =

{2−3, . . . , 23} was performed and the value that pro-duced the best agreement with the reference was used(γlm = 1).

To compare the results with the three different land-mark matching terms, the distances based on auto-mated landmarking were then computed with: no land-mark matching (γlm = 0), exact landmark matching(γlm = 100), and the proposed inexact landmark match-ing (γlm = 1). The limits of agreement with the ref-

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E(φ1(C),S)

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Figure 18: Sensitivity to the parameter σV of: The distance D,the matching residual E after the registration with inexact landmarkmatching (average curves across pair-wise registrations), and thecross-validation (CV) siphon shape classification rate.

erence for these three variants are shown in Fig. 19.We can observe: First, that using landmarks is impor-tant as otherwise the obtained similarity distances areunderestimated. Second, using the proposed inexactlandmark matching showed improvements in terms ofsmaller bias and standard deviation compared to enforc-ing exact landmark matching, since any inaccuraciesin landmark localization influence less the registrationprocess. The distances obtained with inexact landmarkmatching had the variability index I = 1.1, which isclose to the variability obtained with the manual land-marking.

No Landmarks Exact Landmark Match Inexact Landmark Match

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Figure 19: Limits of agreements of distances based on automatedlandmarking with the reference distances. Results for LDDMCM withthe three types of landmark matching terms are shown: None, exactand inexact.

7.5. Carotid siphon shape classesAs many siphons had ambiguous shape, only the

ones where both clinicians agreed on the shape classwere used (43 cases). The first step in the evalua-tion is the dimensionality reduction and we look at theresidual variances as a measure of the obtained statis-tical fit (Fig. 20(a)). Residual variance is defined as

1−R2, where R is the correlation coefficient between thepairwise point distances in a subspace and the originalspace. We can observe that the items with characteriza-tion based on the LDDMCM can be better representedin a low dimension (< 5D) than the ones characterizedby the geometric features. For other dimensionality re-ductions, both characterizations produce similar resid-ual variances and the more dimensions used, the betterthe statistical fit.

Observing, the items mapped to a 2D Euclidean sub-manifold (Fig. 20(b)&(c)), LDDMCM appears to pro-duce more discriminating clusters than the geometricfeatures. This is confirmed by the classification of la-beled items mapped to a 12-dimensional submanifold(where residual variance for both approaches is negli-gible), as there CVLDDMCM = 84% while CVFeatures =

63%. To further evaluate the importance of using land-mark matching in the registration process, we computedthe CV rate for the results obtained without the land-mark matching term: CVLDDMCM no landmarks = 77%,which produced worse class separability than the pro-posed method.

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LDDMCM classification for different choice of σV

parameter (Fig. 18), shows that the performance is verystable in the range [3, 5] mm, and still quite stable (77%-86%) in the entire evaluated range [0.5, 10] mm. The re-sults of geometric feature selection are shown in Fig. 21.

14

Using the feature selection improves the CV rate and thepeak CVFeatures = 77% is reached already with four fea-tures. The four features in the order of importance were:tortuosity, normal vector of the anterior bend’s osculat-ing plane, length of the anterior bend and the bendingradius of the anterior bend. This is in good agreementwith the visually observed variability of shape types,which is mostly due to the change in the geometry ofthe anterior and the posterior bends.

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8. Discussion and conclusions

We presented a pipeline for extensive geometric char-acterization of carotid siphon. Starting from angio-graphic image the pipeline identifies and extracts thecenterline and radius of the ICA. On the extracted cen-terline we automatically detect anatomical landmarks ofthe region of interest corresponding to carotid siphon,which are prerequisites for the proposed geometric char-acterization. Landmarks as points of anatomical corre-spondence are used for computing both: geometric fea-tures and LDDMCM shape similarity.

The method to identify ICA from a vascular tree bydetecting ICA-TB had a high classification success rate(99%). Looking at the two most discriminative fea-tures: ratio of mean radii between the smaller and thelarger daughter branches and axial component of thesmaller branch vector, the classifier learned that ICA-TB branches into two similarly big vessels as opposedto a narrow side vessel and also that the smaller branchvector (corresponding to ACA) is pointing towards thetop of the head.

One of the main elements of the paper was automatedlandmark detection as it is essential for later character-ization. Landmarks were associated to the four bendmodel of the carotid siphon. Apart from the superiorbend, the other three bends were modeled as planar. Analternative would be to split the superior bend into twosmaller quasi-planar bends, but we observed that those

two bends are not consistently present between subjects.Direct application of the bend detection method by Pic-cinelli et al. (2011), which uses curvature and torsioncenterline profiles, was not suitable as it does not assurebend correspondences and the results are sensitive to theapplied centerline smoothing scale. Thus, a more robustmethod, based on the curvature vector expressed in theparallel transport frame is proposed.

Landmarks are essential for geometric features asthey define, in an automated and consistent manner, thecorresponding regions of interest for local and globalfeatures. They are crucial for computing LDDMCMas they assure anatomically correct registrations. Thevalidation study showed that the automatically obtainedlandmarks are in agreement with the ones selected man-ually and that they do not affect the obtained geomet-ric quantities or LDDMCM distances with respect tomanual landmarking. Using automated approach avoidsobserver variability and enables high reproducibilityamong a set of individuals, which is imperative in per-forming robust population studies.

The presented set of geometric features were selectedfrom typical geometric quantities used in the litera-ture, known to affect hemodynamics. In general, theyshould be defined and selected in accordance to the hy-pothesis one is testing. Characterization based on ge-ometric features and LDDMCM are two complemen-tary approaches. The former has the advantage thatany observed associations or variabilities of features arestraightforward to interpret. The later quantifies shapedifferences directly hence is more effective in capturingsubtle changes in geometry but its interpretation is moreabstract. Both approaches avoid the need for any pointcorrespondences, apart from the sparse set of anatomi-cal landmarks.

The explanatory power of the characterization wasevaluated by observing the discrimination of siphonshape classes in the submanifold coordinates, seen asmodes of anatomical variation of population. The char-acterization based on LDDMCM proved better in clas-sifying the carotid siphon shape classes than the onebased on geometric features. These shape classes al-ready have direct clinical relevance in selecting the en-dovascular treatment strategy, but they could be re-placed by any other clinical condition of interest, e.g.presence of aneurysm or its rupture status. Then, in-sights about the relation of the geometry of carotidsiphon with its clinical condition could be obtained andautomatically quantified in a large population.

The proposed methodology has limitations. As weassume all carotid siphons can be modeled with thefour bends, the landmarking fails when two neighbor-

15

ing bends are perceived as one large bend. In ourdatabase, this assumption was rarely violated (3%). Ingeneral, the method could be extended to learn the ge-ometric properties of the bends, to detect unrealistic(unprobable) solutions. Regarding the computation ofLDDMCM similarities, the drawback of the registra-tion metric is that it is not symmetric with respect tothe source and target curves. Thus, the resulting shapedistance was taken as the mean value from the two reg-istrations. In addition, adding a new case requires reg-istering it with all the cases from the database. For alimited size database, like ours, this is not a problem, asthe registration of spatial curves is generally much fasterthan the registration of volumetric images. In the caseof larger datasets, a low-dimensional manifold could befirst learned from a set of curves used as a training set.Then, each new centerline curve would be projected di-rectly on the manifold, as similarly presented for brainimages in Gerber et al. (2010).

8.1. Conclusions and OutlookGeometry of carotid siphon has a large variability

across subjects, which makes it a good candidate to bea potential risk factor for the onset of vascular patholo-gies on and off the ICA. We have presented a completeand automated pipeline for geometric characterizationof the carotid siphon. The proposed approach, basedon anatomical landmarks, enables the analysis througha set of geometric features and LDDMCM shape simi-larities.

Some elements of the proposed pipeline are new andrepresent contributions in themselves. In this sense, themain contributions of the paper are the following:

• Algorithm for the automated identification of ICA-TB and its discriminating features.

• Algorithm for the automated detection of vesselbends based on the curvature vector expressed inthe parallel transport frame and its application toanatomical landmarking of carotid siphon.

• Definition and computation of geometric quantitieslike angles of rotation between osculating planesof consecutive bends, having a known influence onhemodynamics.

• The use of LDDMCM similarity metric for ves-sels and the importance of using the inexact land-mark matching to obtain anatomically valid defor-mations.

Although the methodology is tuned to carotidsiphons, it is applicable to other vessels. The classifier

used for the identification of ICA can be extended todetect more bifurcations for the purpose of anatomicallabeling of the vascular tree and this is currently understudy. The bend identification algorithm and the geo-metric quantities defined on them are generic and can beapplied to any vessel (e.g. aorta, coronaries or periph-eral arteries) or tubular structure. However, the numberand the type of bends might not be as consistent alongthe population as they are for the carotid siphon. In thatcase, classical LDDMCM, without the landmarks, canbe used as a shape similarity metric and the number ofbends can then become a geometric feature in itself.

The presented characterization is the first step in thepursuit of geometric risk factors of carotid siphon. Iden-tifying these factors was not the aim of this paper, butwill form part of the future work. In addition, the tech-niques used here will be extended to a more distal level,in an effort to characterize the complete Circle of Willis.

Acknowledgments

This work was partially supported by the CENIT pro-gramme of CDTI, the Industrial and Technological De-velopment Centre of Spain, under the research projectcvREMOD (CEN-20091044), and by Philips Health-care B.V. (Best, The Netherlands). H. Bogunovic issupported by the FI-DGR 2009 fellowship of AGAURand R. Cardenes by the Beatriu de Pinos programmeof AGAUR. A.F. Frangi holds an ICREA-AcademiaAward by the Institutcio Catalana de Recerca i EstudisAvancats (ICREA).

Antiga, L., Ene-Iordache, B., Remuzzi, A., 2003. Computational ge-ometry for patient-specific reconstruction and meshing of bloodvessels from MR and CT angiography. IEEE Trans. Med. Imag.22, 674–684.

Antiga, L., Steinman, D.A., 2004. Robust and objective decomposi-tion and mapping of bifurcating vessels. IEEE Trans. Med. Imag.23, 704–713.

Antiga, L., Steinman, D.A., 2011. VMTK: the Vascular ModelingToolkit. http://www.vmtk.org.

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