Marina Piccinelli, Alessandro Veneziani, David A. Steinman, Andrea …user.engineering.uiowa.edu/~raghavan/images/chung/... · 2012-03-23 · Marina Piccinelli, Alessandro Veneziani,

IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 28, NO. 8, AUGUST 2009 1141

A Framework for Geometric Analysis of VascularStructures: Application to Cerebral Aneurysms

Marina Piccinelli, Alessandro Veneziani, David A. Steinman, Andrea Remuzzi, and Luca Antiga*

Abstract—There is well-documented evidence that vascular ge-ometry has a major impact in blood flow dynamics and conse-quently in the development of vascular diseases, like atheroscle-rosis and cerebral aneurysmal disease. The study of vascular ge-ometry and the identification of geometric features associated witha specific pathological condition can therefore shed light into themechanisms involved in the pathogenesis and progression of thedisease. Although the development of medical imaging technolo-gies is providing increasing amounts of data on the three-dimen-sional morphology of the in vivo vasculature, robust and objec-tive tools for quantitative analysis of vascular geometry are stilllacking. In this paper, we present a framework for the geometricanalysis of vascular structures, in particular for the quantificationof the geometric relationships between the elements of a vascularnetwork based on the definition of centerlines. The framework isfounded upon solid computational geometry criteria, which conferrobustness of the analysis with respect to the high variability ofin vivo vascular geometry. The techniques presented are readilyavailable as part of the VMTK, an open source framework forimage segmentation, geometric characterization, mesh generationand computational hemodynamics specifically developed for theanalysis of vascular structures. As part of the Aneurisk project, wepresent the application of the present framework to the character-ization of the geometric relationships between cerebral aneurysmsand their parent vasculature.

Index Terms—Cerebral aneurysms, geometric quantification,three-dimensional modeling, vascular geometry.

Manuscript received February 25, 2009; revised April 10, 2009. First pub-lished May 12, 2009; current version published July 29, 2009. The Aneuriskproject is a joint research project of the Center for Modeling and ScientificComputing (MOX) of the Politecnico di Milano, Milan (Italy), the Mario NegriInstitute for Pharmacological Research, Bergamo (Italy), and the Ca’ GrandaHospital, Milan (Italy), supported by Siemens Medical Italia and FondazionePolitecnico. Asterisk indicates corresponding author.

M. Piccinelli is with the Department of Mathematics and Computer Science,Emory University, Atlanta, GA 30332 USA and also with the BiomedicalEngineering Department, Mario Negri Institute for Pharmacological Research,24020 Ranica, Italy.

A. Veneziani is with the Department of Mathematics and Computer Science,Emory University, Atlanta, GA 30332 USA.

D. A. Steinman is with the Department of Mechanical and Industrial Engi-neering, University of Toronto, Toronto, ON, M5S 2E4, Canada.

A. Remuzzi are with the Biomedical Engineering Department, Mario NegriInstitute for Pharmacological Research, 24020 Ranica, Italy and also with thewith the Industrial Engineering Department, University of Bergamo, 24020Ranica, Italy.

*L. Antiga are with the Biomedical Engineering Department, Mario Negri In-stitute for Pharmacological Research, 24020 Ranica, Italy (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/TMI.2009.2021652

I. INTRODUCTION

T HERE is well-documented evidence that vascular geom-etry has a major impact in the blood dynamics and, in turn,

in the origin and development of vascular disease, through theaction of forces exerted by flowing blood on the vascular wall.[1]–[4]. Typical examples of these relationships are the devel-opment of atherosclerotic lesions [5], [6] and the formation ofintracranial aneurysms preferentially at bifurcations and sharpbends [7], [8].

The study of vascular geometry in relation to the developmentof specific pathologic conditions can, therefore, shed light on thehemodynamic triggers involved in the pathogenesis and in theprogression of the disease [9], [10]. Furthermore, the identifica-tion of geometric quantities that are associated with a specificpathological condition, or that have some predictive power withrespect to the severity of disease progression is a valuable en-deavor in itself. In fact, a geometric quantity which is a surro-gate of a specific flow condition is amenable for inclusion in alarge-scale clinical trial and, once validated, it is directly usableas a clinical criterion.

The recent development of medical imaging devices, such asrotational angiography (RA) [11], computed tomography (CT),and magnetic resonance (MR), has lead to the availability oflarge amounts of data on the 3-D morphology of the in vivovasculature, for the investigation of these aspects for diagnosticand prognostic purposes [12]–[16]. Still, a quantitative anal-ysis of the relationship between vascular geometry and arterialphysiopathology is made difficult by the large variability of realanatomies on one hand, and by the objective difficulty in re-trieving quantitative data from images in a robust, operator-in-dependent way on the other.

In this paper, we present a framework devised for 3-D mod-eling and geometric characterization of vascular structures,readily available in the Vascular Modeling Toolkit (VMTK)[17], and we show how it can be employed for the charac-terization of cerebral aneurysms in relation to their parentvasculature. In particular, after a quick glance to image seg-mentation (Section II), we focus our attention on centerlinecalculation (Section III) and bifurcation identification andquantitative characterization (Section IV). Centerline of avessel is in general a significant synthesis of basic features ofa vessel (in terms of curvature, torsion, tortuosity), however,its definition from a general 3-D surface is not trivial. Sometechniques for a robust computation of the centerline based onthe concept of Voronoi diagram are presented here. As alreadypointed out, bifurcations are in general an interesting part ofthe vascular tree, quite often preferential site of atherosclerotic

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1142 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 28, NO. 8, AUGUST 2009

Fig. 1. Level set segmentation principles. (a) Image data obtained from 3-Drotational angiography (RA) of an internal carotid artery bearing an aneurysm.(b) Gradient magnitude of image intensity. (c) Location of the level set modelwith respect to the ridges of the gradient magnitude. (d) The final 3-D surfacemodel.

plaque development or aneurysm growth. Robust geometriccharacterization of bifurcations potentially plays an importantrole in understanding pathogenesis and disease progression. InSection V we illustrate examples of application of the presentframework to the case of middle cerebral artery and internalcarotid artery aneurysms. Medical images have been obtainedin the context of the Aneurisk project, a collaborative researchproject aimed at assessing the role of geometry and hemody-namics in the development and rupture of cerebral aneurysms.

II. IMAGE SEGMENTATION

The first step in our framework consists in building a geo-metric representation of a vascular network from imaging data,operation referred to as image segmentation. The goal is to reli-ably identify the surface corresponding to the boundary betweenthe lumen and the vascular wall. Assuming that for the givenimaging modality blood has a different intensity compared tothe vessel wall and surrounding tissue (this is the case for con-trast-enhanced RA, CT, and MR), the lumen boundary can beassumed to be located at the locations of maximal intensity tran-sition from the lumen to the outside. Following this basic as-sumption, we locate the lumen boundary where the magnitudeof the gradient of image intensity , where is the vector ofspace coordinates, has a ridge [see Fig. 1(a) and (b)].

The segmentation process is carried out with the useof implicit deformable models, whose deformation underimage-based forces is described using the level set technique[18]–[20]. The surface of the vascular model is representedas the iso-surface of level of a time-dependent function

fulfilling an appropriate differential equation driven bythe image-based forces. In particular, we employ

(1)

where are user-defined parameters [20]. The first termon the right hand side has the role of curvature regularizationand the section is an advective term with the role of attracting the0-level set of to the ridges of the gradient magnitude (Fig. 1).Equation (1) is spatially discretized using finite differences onthe image structured grid, with an explicit Euler time marchingscheme. The evolution of is tracked using upwind finite differ-ences in a narrow band of voxels around the 0-level set using thesparse field approach [21]. The implementation of the level setsolver is provided by the Insight Segmentation and RegistrationToolkit [22], which is a dependency of the Vascular ModelingToolkit (together with the Visualization Toolkit (VTK) [23]).The authors have recently extended such level set implemen-tation to nonisotropically spaced images, and contributed thechanges to the Insight Toolkit. Equation (1) needs to be properlycompleted by boundary and initial conditions. In particular, themethod yields accurate results when the initial condition is suchthat the 0-level set of is close enough to the target surface. Inour case, we resort to the following initialization method, re-ferred to as Colliding Fronts algorithm.

1) The user selects two seed points and interactivelyon the 3-D image. Let us denote by thetravel time of a wave originating from and traveling withvelocity , i.e., faster where the intensity is brighter (asit typically is the case within the lumen). Assuming that

, the fields are constructed to satisfy theEikonal equation for

(2)

numerically discretized by means of upwind finite differ-ences and solved efficiently on the image grid using theFast Marching Method [18].

2) The quantity

is computed, which is negative where the two waves travelin opposite directions and positive where they travel inthe same direction. This property can, therefore, be usedfor identifying the region comprised between the two seedpoints. In addition, since drives the speed of propaga-tion of the waves, these are assumed to significantly slowdown in correspondence of the lumen boundary, so thatbecomes positive. Therefore, the isosurface is ingeneral a good approximation of the lumen boundary in theregion of interest between the seed points. Last, au-tomatically excludes side branches, because the two frontsseep into a side branch traveling in the same direction.

It is worth pointing out that the purpose of the CollidingFronts algorithm is to provide a flexible way to selectivelyinitialize a surface close enough to the gradient magnituderidges that the inflation stage, normally required for deformablemodel-based approaches, is not needed, since the surface is

PICCINELLI et al.: A FRAMEWORK FOR GEOMETRIC ANALYSIS OF VASCULAR STRUCTURES: APPLICATION TO CEREBRAL ANEURYSMS 1143

Fig. 2. Level set initialization using the Colliding Fronts algorithm. (a) Posi-tioning of two seeds on the 3-D image for the identification the vascular branchof interest. (b) Initial surface generated with the Colliding Fronts algorithm.(c) The final level set model of the selected vascular tract.

already within the capture region of the advection term. Thisconfers robustness to the localization of the position of thelumen boundary while retaining flexibility in the choice ofthe segmented regions. In Fig. 2(a)–(c), we illustrate the ini-tialization steps for a single vascular segment. Thanks to theimplicit representation of the surface provided by the level setformulation, different tracts can be merged in successive stepsso to obtain a complete vascular network.

Notice that although the results presented in the sequel arebased on RA images, this segmentation technique can be ap-plied to images obtained with different imaging modalities (CTor MR). Moreover, we point out that the geometric character-ization of the vessel discussed in the subsequent paragraphs isindependent of the specific segmentation technique adopted andcan be applied to any piecewise linear representation of a 3-Dsurface representing a vascular network.

III. COMPUTATION OF THE CENTERLINES

Geometric analysis and synthesis of a vascular tree is in gen-eral a challenging task, both for the extreme anatomical vari-ability of in vivo morphologies and for the availability of manypotential quantities of interest. By exploiting the tubular struc-ture of vessels, a reasonable approach is to resort to the con-cept of centerline. Although this is an intuitive geometric en-tity, its formal definition is challenging especially at sites wherevascular segments exhibit a nontubular morphology, such as atbifurcations.

We define centerline as the line between two sections of thelumen, whose minimal distance from the boundary is maximalin some sense. Consequently, centerline computation can be for-mally described as a functional minimization problem. If is thecurvilinear abscissa along a curve connecting the genericpoints and , corresponding to the values and , respec-tively, we look for the curve , such that the functional

(3)

is minimal, with , being a cost-density functionto be properly constructed.

A method for the selection of based on the concepts ofmedial axis, Voronoi diagram and wave propagation, was pre-sented in a previous work [20]. Given the importance of theseconcepts in the foundations of the present framework, we brieflyrecall the theory behind the definition of the functional in (3).

Let be a volume with boundary . For any point in, we indicate with the minimal distance of from

the boundary, i.e.,

being the Euclidean distance. In other words, isthe radius of the sphere centered in and tangential in at leasta point of . Moreover, we define a maximal inscribed spherea sphere inscribed in which is not entirely contained in anyother inscribed sphere. The Medial axis is a surfacegiven by the locus of the centers of maximal inscribed spheresin . This is actually a nonmanifold surface contained in anddual to . In principle, once is known, can beretrieved. In practice, this is a challenging task, and approxima-tion procedures are required. A classical approximation methodfor discrete surfaces relies on the concept of Voronoi diagram,defined as follows [24], [25].

Let be a set of points on and be a point in . TheVoronoi region associated with is defined as the regionin whose points are closer to than to any other point of ,i.e.,

(4)

Given a point set in , the union of the Voronoi regionsconstitutes a tessellation of all . The collection

of the boundaries of the Voronoi regions for any pointin is called Voronoi diagram.

If we define as the counterpartof when this is computed only with respect to the set ,then the Voronoi diagram coincides with the surface collectingthe ridges of . In this respect, the Voronoi diagram canbe considered a sort of finite dimensional approximation of themedial axis. In our framework, the Voronoi diagram is com-puted by resorting to its dual, the Delaunay tessellation of ,denoted by . The latter is a tetrahedral reticulation ofthe point set , where the sphere circumscribed to each tetrahe-dron does not strictly contain any other point of . It is possibleto prove that the vertices of the Voronoi diagram correspond tothe circumcenters of Delaunay tetrahedra, and that the connec-tivity of the tessellation allows to compute the connectivity ofthe Voronoi diagram, since Voronoi vertices are connected byan edge if the corresponding Delaunay tetrahedra share a face.However, since the Delaunay tessellation of a point set producesa strictly convex tetrahedralization, tetrahedra located outsidethe surface have to be removed before the Voronoi diagram canbe computed, as we are only interested in the approximation of

inside the domain of interest. The removal of the non-in-ternal tetrahedra from the Delaunay tessellation is carried outby removing tetrahedra having circumcenter laying outside ,more formally

where are the vertices a tetrahedron, its circumcenter, andis the outward normal vector to the surface defined at .

After removal, internal or embedded Voronoi diagram


Fig. 3. (a) Internal Voronoi diagram of the 3-D model of an internal carotidartery bearing an aneurysm. (b) and (c) Details of the Voronoi diagram as a non-manifold surface composed of convex polygons, whose vertices are the centersof maximal inscribed spheres; the larger spheres are associated with basic fea-tures of the geometry and the smaller spheres capture the small-scale details ofthe model surface.

is defined from the Voronoi polygons whose connectivity iscomplete.

Fig. 3(a) and (b) illustrates for a portion of cere-bral vasculature. The internal Voronoi diagram is a nonmani-fold surface composed by convex polygons whose vertices arethe centers of maximal inscribed spheres, which coincide withthe circumspheres of the associated Delaunay tetrahedra. Themaximal inscribed sphere radius (MISR), hereafter denoted by

, is associated with each vertex, providing informationon the distance from the boundary. Notice that the spheres fea-turing a large radius (whose centers are in the deeper portion ofthe Voronoi diagram) are associated to the basic features of thegeometry, while the small-scale details are typically captured bysmall spheres, with centers are located on the outer portions ofthe Voronoi diagram.

At this point, we define the centerline as the result of theminimization of the functional in (3) taking ,i.e., the domain where is looked for is restricted fromto , i.e., and the function is defined as

This way, is guaranteed to be medial, since it sits on topof the Voronoi diagram, and minimizes the line integral of theinverse of the MISR, i.e., it tends to the deeper portions of theVoronoi diagram (for this reason, such a defined is referredto as centering potential).

In practice, this minimization problem is recast into a min-imal cost path problem solved by first finding a solution to theEikonal equation

where are the arrival times of a wave traveling on theVoronoi diagram with speed , with initial condition

. Algorithmically, the solution is obtained using theFast Marching Method [18] extended to nonmanifold surfaces.Fig. 4 shows the embedded Voronoi diagram and the Eikonalsolution plotted in , where is the point set of thevertices of the triangulated surface representing the lumen. Thecenterline is finally computed by tracing the path from back

Fig. 4. (a) Voronoi diagram color-coded with the solution of the Eikonal equa-tion. (b) Computation of the centerline path between the endpoints. (c) Finalcenterline: each point is associated with the radius of maximal inscribed spheres(MISR).

to on along the direction of steepest descent of.

One of the potential drawbacks of resorting to Voronoi dia-grams for obtaining synthetic shape descriptors is its sensitivityto surface noise. Several algorithms for the extraction of a stablesubset of the Voronoi diagram have been proposed in the litera-ture [26]. It is, however, worth noting that in our case noise af-fects principally peripheral Voronoi elements, while the deeperones, which are targeted by our application, are less sensitiveto small changes in the shape of the surface. For this reason, inour framework we work directly on the entire diagram. Asan example, in Fig. 5 we show two models for the same vas-cular network. The former is obtained using the segmentationapproach presented in this paper, the latter is obtained by ex-traction of an intensity isosurface from the original images. Theassociated Voronoi diagrams are shown together with the sur-faces. Notice that, even if the second surface presents a largeramount of noise, the two centerlines extracted do not exhibitsignificant differences between the two models. Besides robust-ness, another advantange of this approach is that centerlines arewell defined even in presence of nontubular shapes, such as bi-furcations and saccular aneurysms, as shown in Section V.

A. Tube Functions and Surfaces

The union of the maximal inscribed spheres whose centers lieon the centerline defines a scalar function , the tubefunction, as follows:

(5)

being a generic point on the centerline and the correspondingmaximal inscribed sphere radius. The zero isosurface ofis referred to as tube, or canal surface. The tube function is neg-ative inside its surface. As shown in Fig. 6, the tube surface fora vascular segment is by construction strictly contained into thelumen—a maximal inscribed tube, with slight abuse of termi-nology [27].

There is a connection between the local radius of a tube sur-face and the radiological convention for measuring the diameterof a vessel. In fact, the diameter of a vessel is evaluated as the


Fig. 5. Comparison of the internal Voronoi diagram and the centerlines of twomodels obtained for the same image data: model 1 is obtained using level setsegmentation, model 2 is obtained using iso-surface extraction on the original3-D image, presenting a higher degree of surface noise; details of the internalVoronoi diagrams and of the centerlines are shown in the lower portion of theimage.

Fig. 6. Tube surface of the internal carotid artery: the surface is strictly con-tained into the lumen; from the two projections it can be noted that there is al-ways a projection in which the tube surface is tangent to the model, connectingtube radius to the radiological convention for measuring vessel diameters.

minimum diameter measured over a range of lateral projections.Fig. 6(b) shows that there is always a projection in which thetube surface is tangential to the model, so that the vessel size,

or the degree of a stenosis, is reflected by the radius of the tubesurface.

Tube functions will play an important role in the remainderof this paper, as they constitute the fundamental building blocksof our geometric characterization framework.

B. Geometric Characterization of Centerlines

Since centerlines are lines in space it is possible to proceedwith their geometric characterization based on classical differ-ential geometry of curves. In particular, we consider curvature,torsion, tortuosity as relevant quantities for the characterizationof a vascular segment, since they have been demonstrated tobe potentially linked to specific hemodynamic patterns [1], [9],[28] and disease processes [10], [29]–[31]. Using similar differ-ential geometry concepts, we then face the problem of defininglocal reference frames along the centerlines, which will serve asbuilding blocks for comparative geometric analysis as presentedin the following sections.

Before proceeding, let us first introduce a parametrization ofthe line through the associated curvilinear abscissa. After thisoperation, the line is reparameterized by means of the variable

, where is the arc length of the centerline. Theabscissa ranges over the 3-D course of the centerline and relatesevery point to its Euclidean distance from a point assumed to bethe origin, denoted by .

1) Curvature: Curvature (or more precisely extrinsic curva-ture) of the line , is defined as

(6)

and measures the deviation of the curve from a straight line.Since centerlines are guaranteed to be only continuous,

differentiation, here and in the following, is performed using fi-nite differences on a smoothed representation of the centerline,obtained using a Laplacian filter to get rid of spurious high-fre-quency noise. Note that this procedure is only employed for thecomputation of derivatives with respect to the parametric co-ordinate. An alternative, recently presented approach, is to usefree-knots regression splines to generate a high-order contin-uous representation of the curve, which allows for analytic com-putation of derivatives [32].

At a given point , curvature is the inverse of the radius ofthe osculating circle. i.e., the circle that approximates the curve(i.e., has a second-order contact with the curve) in . Theplane identified by the osculating circle is commonly referredto as osculating plane. is always positive, and it vanisheson rectilinear portions of the line.

2) Torsion: As curvature measures the deviation of the curvefrom a straight line, so torsion measures its local deviation fromlying on the osculating plane, or, equivalently, how sharply theline is twisting in space.

Considering a parameterized curve with non vanishing cur-vature, the torsion reads

(7)

Torsion reflects the rate of rotation of the osculating planearound the curve in the neighborhood of a point. Sign of


Fig. 7. Curvature, torsion, and tortuosity of the internal carotid artery center-line.

distinguishes between clockwise or counter-clockwise rotationsof the osculating plane.

3) Tortuosity: Given the length of the centerline and theEuclidean distance between its endpoints , tortuosity is de-fined as

(8)

Tortuosity is the relative increment in the length of a curve de-viating from a rectilinear line.

Fig. 7 displays the evaluation of curvature, torsion, and tortu-osity applied to the centerline extracted from an internal carotidsiphon.

C. Centerline Framing: Frenet and Parallel Transport Frames

The Frenet frame is a orthonormal coordinate reference framecomputed along the line, carrying an intrinsic description of thelocal geometry. The unit vectors making up the reference framehave a strong connection with the concepts of curvature andtorsion introduced in the previous section. Assuming that isat least three times differentiable, the Frenet frame is identifiedby the unit vectors tangent , binormal , and normal ,respectively, defined as

(9)

Vectors and belong to the plane orthogonal to . At eachpoint on the curve, is directed toward the center of the os-culating circle, while is aligned with the normal to the os-culating plane. In Fig. 8(a), the Frenet frame computed alonga centerline is shown. The Frenet frame is not defined whenthe curvature vanishes and by definition abruptly changes ori-entation when the direction of concavity of the curve changes.Fig. 8(b) shows an example of local osculating planes twistingaround the curve due to a concavity change.

Fig. 8. (a) Frenet frame along the internal carotid artery. (b) Detail of image(a) showing local osculating planes twisting around the centerline; � , � , �are, respectively, the tangent, normal, and binormal unit vectors of the frame.

Although effective in accounting for the local geometry of acurve, due to its abrupt changes in orientation the Frenet framecannot be employed as a reference frame for the evaluation orthe comparison of circumferential positions around the center-line. For this purpose, we resort to a different moving frame, theparallel transport frame [33]. The basic idea is to sweep an ini-tial orthonormal coordinate frame along the line without beingaffected by torsion: at each point, the unit vectors making upthe frame are rotated by the exact amount required by curva-ture, resulting in the least possible change in direction. In prac-tice, given an initial frame at a point with parametric coordinate

(e.g., the Frenet frame in that point), featuring , , and, the frame at is obtained by rotating the frame at

by an angle (i.e., the angle be-tween the two successive tangent vectors) around the directionof the vector . Such constructed parallel trans-port frame minimizes the twist of the moving frame as it is sweptalong the centerline, and as such it can be used for comparingangular positions around the centerline between different loca-tions along the curve. Operatively, this is performed as in Fig. 9,where the comparison between the orientation of two differentplanes identified by their normal is shown: the two normals arefirst transported along the centerline, and their orientation com-pared at a generic point (since the angle between the two paralleltransported sets of normals is constant).

IV. ANALYSIS OF BIFURCATIONS

In the previous section, we faced the problem of providingtools for geometric characterization of vessels represented bycenterlines, computed independently between endpoints. No in-formation has been extracted yet on the topological relation-ship between centerlines representing difference branches of avascular network and the related geometric characterization. In-deed, the availability of tools for the robust analysis of the ge-ometry of bifurcations is of paramount importance for the ap-plications addressed here, given the fact that bifurcations are lo-


Fig. 9. Calculation of the angle between plane 1, defined by the normal vector� , and plane 2, defined by normal vector � , performed by parallel transportof the normal vectors along the centerlines, and by computation of the angle be-tween the transported sets of unit vectors � , � evaluated at a commonlocation.

cations where complex flow has a chance to develop, and thatseveral vascular diseases exhibit focal effects precisely aroundbifurcations, e.g., atherosclerosis [34], [35] and aneurysmal dis-ease of the cerebral circulation [36]–[39]. In this section, weillustrate how it is possible to identify and characterize bifurca-tions from the current definition of centerlines and tubes.

A. Identification

In our framework, the bifurcation is geometrically identifiedas the region preceding the location at which a centerline inter-sects the tube surface of another centerline and vice-versa. Inother words, the location where two tubes graft into each other.

Let us consider, for the sake of concreteness, the bifurcatingvessel shown in Fig. 10(a). Four reference points are identifiedon the two centerlines [see Fig. 10(b) and (c)].

1) and are the points where each centerline intersectsthe surface of the inscribed tube associated to the othercenterline.

2) and are the centers of the upstream maximal in-scribed spheres touching and .

These four points delimit a segment on each centerline whichidentifies the bifurcation region [Fig. 10(d)]. The center of mass

of the maximum inscribed spheres centered at the four ref-erence points is taken as the bifurcation origin

(10)

where are the radii of the maximal inscribed spheres asso-ciated with the centerline locations . Note that the bifurcationorigin does not in general correspond to the location where cen-terlines depart from each other, as such location might stronglydepend on the local shape of the cross section of the parent

Fig. 10. Definition of the bifurcation reference system. (a) 3-D model of a bi-furcating vessel and relative centerlines. (b) Intersecting tube surfaces (shownas shadows) generated from the two centerlines. (c) Identification of four refer-ence points with the associated maximal inscribed spheres. (d) Centerline tractsrelative to the bifurcation region, represented by thick tracts. (e) Bifurcationreference system: � , origin, � , bifurcation plane normal, � downstreamnormal. (f) Example of bifurcation origin in case of bifurcating vessels of dif-ferent sizes.

vessel as it approaches the bifurcation. Instead, it is more ro-bustly determined from the centers and radii of the spheres in-scribed within the bifurcation region.

The bifurcation plane is then defined by the normal tothe polygon defined by the four reference points evaluated in

. Since the four points are in general nonplanar, the normalis evaluated as the bilinear interpolation in of the normalsto the corner triangles of the polygon. Last, the bifurcation ref-erence frame is completed by defining the downstream direc-tion for the bifurcation as the normalized average of thevectors defined by the two reference points on each centerline.In Fig. 10(e) all the components of the reference system areillustrated.

Notice that the approach based on the centerlines and tubefunctions yields a robust characterization of both Y- andT-shaped bifurcations. For instance, for two bifurcating vesselswith a marked difference in size [see Fig. 10(f)], the tubefunction of the smaller vessel slides along the outer wall of thebifurcation region, giving origin to a small intersection areaand to reference points shifted towards the side of the largervessel. On the other hand, the reference points located on thecenterline of the larger vessel are associated to larger spheres


Fig. 11. (a) Splitting of the bifurcating centerlines into branches. (b) Branchvectors. (c) Projection of the branch vectors on the bifurcation plane.

and are therefore located in the neighborhood of the center ofthe vessel. By defining with (10) using the center of massof the inscribed spheres as a weighting factor, the origin of thebifurcation shifts towards the center of the main vessel, andthe bifurcation normal will be less influenced by the referencepoint relative to the small branch.

B. Angle Quantification

Once a bifurcation has been identified, each centerline issplit into the tracts and , correspondingto the branches of the bifurcation, and , beingthe length of the line. Tracts of different centerlines referred tothe same branch are grouped based on whether they are mutuallycontained within their respective tube surfaces.

In order to characterize the directions of the branches relativeto the bifurcation, we associate a unit vector to each branch af-ferring or emanating from a bifurcation, capturing the directionof centerlines immediately upstream or downstream the bifurca-tion, as illustrated in Fig. 11(b). In order to estimate directionsrobustly, each branch vector is taken from (to) the relative refer-ence point to (from) a point at a maximal inscribed sphere radiusdistance from it.

Once branch vectors are available, it is possible to quantify bi-furcation angles. Each vector is split into its in-plane and out-of-plane components with respect to the bifurcation plane [seeFig. 11(c)]. The angles comprised by the in-plane and out-of-plane components can then be computed.

By detecting bifurcations in a vascular tree, the completetopology of the vascular network can be retrieved, as shown inFig. 12.

C. Comparative Geometric Analysis

When comparing the geometry of vascular segments from se-rial acquisitions or in patient populations, bifurcation originscan constitute important landmarks for comparative analysis ofall geometric features (e.g., the distance of a lesion from a bifur-cation, or the angular position of a lesion around the centerlinewith respect to the bifurcation normal).

For this purpose, centerlines are reparameterized setting thelocation of zero curvilinear abscissa at the locationwhere the centerline is closest to the bifurcation origin .

Fig. 12. Identification of the bifurcations (a) and splitting of the centerline intobranches (b) in the case of a complex vascular network.

Fig. 13. Selection of a bifurcation as a landmark: the abscissa is set to zeroin correspondence of the bifurcation point, while the bifurcation plane normal� is used to compute a parallel transport frame along the centerline.

Similarly, the parallel transport reference system for centerlinesis generated by choosing as the initial normal to be trans-ported the bifurcation normal . This way, every orientationevaluated around the centerline can be given relative to the ori-entation of the bifurcation plane.

Fig. 13(a) and (b) illustrates the choice of the internal carotidartery (ICA) bifurcation as a landmark for both the abscissa andthe parallel transport reference frame. These concepts will bethen employed in Section V.

D. Surface Splitting

Identification of individual branches on centerlines allows thedecomposition of the 3-D surface into equivalent branches. Asdetailed above, once centerlines have been split into branches,the tracts belonging to the same branch are grouped and de-noted by . For instance, in the case of the bifurcating vesselof Fig. 14(a), the tracts relative to the two parent branches aregrouped into and , respectively, while the terminal tractsare taken individually as , , and . The tracts relative to the


Fig. 14. (a) Centerline network split into branches. (b) and (c) Splitting of the3-D model surface.

bifurcation region are not considered. Each group is then asso-ciated with the group function which is the union of thetube functions of the tracts composing the group.

Decomposition of the surface is carried out on the basis of. In fact, we can operate a partition of , in which the th

region is made up from points for which is lower than forthe union of all the other groups, more formally

(11)

In other words, is the region of influence of the branchassociated with the group function . The decomposition ofthe surface is performed according to the partition in (11),where the boundaries of each region are the lines where

. In Fig. 14(b) and (c), the resultingdecomposition of the surface is shown. Due to the gap left in thebifurcation region, the boundaries meet within the bifurcationregion. The actual location of a splitting boundary dependsfrom the relative size of the vessels, as represented by the tubefunctions.

Once the vascular network is split into branches, each seg-ment is topologically equivalent to a cylinder. This simplifiesits geometric characterization. For example, it is possible to ro-bustly generate cross sections of the vessel along the center-line, in order to evaluate their cross-sectional area [35] or shapefactor (i.e., the ratio between the minimum and maximum widthof the section). The step between two consecutive sections canbe either set as a fixed distance along the centerline, or speci-fied in terms of maximum inscribed spheres, which is a meanof measuring distances along centerlines which takes into ac-count the local size of the vessel and results in shorter steps forsmaller vessels. In Fig. 15, sections spaced one and four max-imal spheres apart for all three branches are depicted.

V. APPLICATIONS

In the previous sections, we presented the fundamentalbuilding blocks of our geometric analysis framework. All of

Fig. 15. Sections along the branches extracted at different steps, defined interms of maximum inscribed spheres. Sections are spaced one (left) and four(right) maximal inscribed spheres apart.

them are available within the VMTK open-source project.In this section, we present the actions of these tools on twoconcrete examples for the study of the geometric relationshipof cerebral aneurysms with respect to their parent vessels, atopic that has been subject of interest in the literature [38],[40], [41]. Data has been drawn from the Aneurisk projectimage database. Cases has been selected to span a variety ofaneurysmal configurations for demonstration purposes. Thefirst application deals with aneurysms developing at terminalbifurcations, such as at the bifurcation of the middle cerebralartery. The second application deals with lateral aneurysmsdeveloping along the siphon of the internal carotid artery. Sinceaneurysm formation, growth and rupture are thought to belinked to the action of hemodynamic forces, the study of thegeometric relationship between aneurysms and parent vesselsis of primary importance for the identification of hemodynamictriggers of development and risk factors of rupture. The casespresented here only serve as a sample. A thorough investigationof these relationships on a wide patient population is part of theongoing effort within the Aneurisk project.

A. Middle Cerebral Artery (MCA) Terminal Aneurysms

The localization of middle cerebral artery (MCA) aneurysmson the distal wall of the terminal bifurcation has been considereda possible evidence of the role of hemodynamics in aneurysmdevelopment, thought to be somehow related to the impact ofblood on the apex wall. A quantitative analysis of the geometricconfigurations of these aneurysms with respect to the bifurca-tion where they develop could provide strong hints to under-standing of the role of hemodynamics, especially if extended tolarge populations. In Fig. 16(a), a typical case of MCA terminalaneurysm is shown.

As described in the previous sections, the angle between thedirection of branches can be computed by resorting to branchvectors. In order to generate a branch vector accounting for theorientation of the aneurysm neck, an additional centerline isgenerated with one endpoint on the distal wall of the aneurysm


Fig. 16. (a) 3-D model of a terminal MCA aneurysm. (b) Centerlines of theparent vessel, daughter arteries and aneurysm sac. (c) Bifurcation referencesystem, computed considering the parent branch and the daughter arteries alone.(d) Bifurcation vectors for all branches originating from the bifurcation, in-cluding the aneurysm centerline.

Fig. 17. Steps for the calculation of the deviation of the neck from the directionof the parent branch in the bifurcation plane. (a)� and� are the basepoints of the respective bifurcation vectors. (b) Projection on the bifurcationplane is shown. (c) Neck basepoint from the direction �� of the parentartery is then computed.

sac, as displayed in Fig. 16(b). Even if the aneurysm centerlinedoes not account for the complex shape of the sac, it can repre-sent the orientation of the neck with respect to the parent vessel.

The bifurcation region is then identified from the vesselcenterlines and the bifurcation reference system is computed[Fig. 16(c)]. Bifurcation vectors are then computed in the twoconfigurations, namely excluding and including the aneurysmcenterline. Fig. 16(d) shows branch vectors in the sectionconfiguration. Recalling the definition of reference pointsduring the identification of a bifurcation, the base point of theaneurysm vector accounts for the location where the aneurysmis grafted on the parent vessel.

Given the branch vectors, the following geometric quantitiesare computed.

• The relative orientations of all the branches (including theone relative to the aneurysm neck), obtained from the an-gles between the in-plane and out-of-plane branch vectors.

• The geometry of the sections of the parent branch, thedaughter arteries and the aneurysm branch, extracted onemaximum inscribed sphere away from the bifurcation andcharacterized using section-related parameters such asarea, minimum and maximum section diameters.

• The deviation of the aneurysm neck from the direction ofthe parent branch (see Fig. 17).

Fig. 18. Three-dimensional models of MCA terminal aneurysms used in theanalysis.

More in detail, the latter quantity is determined as follows.1) The base points of the aneurysm vector and the

parent branch vector [in Fig. 17(a)] and their pro-jection on the bifurcation plane[in Fig. 17(b)] are computed, as well as the projectionof the parent branch vector on the bifurcation plane,

, applied in .2) The deviation of the neck from the parent

vessel direction is obtained as the distance offrom the direction [see Fig. 17(c)]; the directionof the vector provides information on thedeviation of the neck towards the smaller or the larger ofthe daughter arteries.

Analogous operations can be performed to obtain the deviationof the neck perpendicularly to the bifurcation plane, the out-of-plane deviation [as in Fig. 17(b)].

The quantities detailed above have been determined foreight representative patient-specific models of MCA terminalaneurysms, extracted from the Aneurisk project database ofRA models and shown in Fig. 18. Results of the analysis arepresented in Tables I–V, where rows are reported in the sameorder as the models in Fig. 18.

To attest the robustness of our methods to the segmenta-tion strategy, four different models were generated from theRA imaging data of four MCA terminal aneurysms from theAneurisk database: the model obtained by means of the level


TABLE IGEOMETRIC DATA ON EIGHT MCA ANEURYSMS: SECTION PARAMETERS.NOTATION: �� , SECTION AREA RATIO BETWEEN LARGEST DAUGHTER

ARTERY AND PARENT ARTERY; �� , SECTION AREA RATIO BETWEEN

SMALLEST DAUGHTER ARTERY AND PARENT ARTERY; ��, BIFURCATION

AREA RATIO BETWEEN SUM OF DAUGHTER ARTERY SECTION AREAS

AND PARENT ARTERY; � , SECTION AREA AT ANEURYSM BASE

POINT � , IN ��

TABLE IIGEOMETRIC DATA ON EIGHT MCA ANEURYSMS: IN-PLANE ANEURYSM

DEVIATION. NOTATION: �� , DEVIATION OF THE ANEURYSM NECK FROM THE

PARENT BRANCH DIRECTION, IN MM; �� DEVIATION OF THE ANEURYSM

NECK RELATIVE TO THE RADIUS OF THE PARENT ARTERY; �� , SIDE OF

DEVIATION OF THE ANEURYSM NECK, �, LARGER DAUGHTER,�, SMALLER

DAUGHTER BRANCH

TABLE IIIGEOMETRIC DATA ON EIGHT MCA ANEURYSMS: OUT-PLANE ANEURYSM

DEVIATION. NOTATION IS ANALOGOUS TO FOR OUT-OF-PLANE QUANTITIES

TABLE IVGEOMETRIC DATA ON EIGHT MCA ANEURYSMS: IN-PLANE ANGLES. NOTATION

(FOR CONCISENESS, ALL VECTORS ARE MENTIONED IN LIEU OF THEIR

IN-PLANE COMPONENTS): � , ANGLE BETWEEN PARENT BRANCH VECTOR AND

THE DOWNSTREAM UNIT VECTOR � OF THE BIFURCATION REFERENCE

SYSTEM; � AND � , ANGLE BETWEEN THE PARENT BRANCH VECTOR

AND THE LARGEST AND SMALLEST DAUGHTER ARTERY, RESPECTIVELY;�, ANGLE BETWEEN THE PARENT BRANCH AND THE ANEURYSM NECK

DIRECTION; � � , ANGLE BETWEEN DAUGHTER ARTERIES, OR BIFURCATION

ANGLE; � � AND � �, ANGLE BETWEEN THE ANEURYSM NECK DIRECTION

AND THE LARGEST AND SMALLEST DAUGHTER ARTERY, RESPECTIVELY.ANGLES ARE EXPRESSED IN DEGREES

set segmentation and two additional models obtained throughsimple iso-surface extraction of the RA data at two user-definedthresholds, after the application of a Gaussian filter on the

TABLE VGEOMETRIC DATA ON EIGHT MCA ANEURYSMS: OUT-PLANE ANGLES.

NOTATION IS ANALOGOUS TO TABLE IV, WHERE ANGLES ARE EVALUATED

BETWEEN OUT-OF-PLANE COMPONENTS OF THE MENTIONED VECTORS

TABLE IVCOMPARISON BETWEEN IN-PLANE BRANCH ANGLES OF FOUR MCA

ANEURYSM MODELS OBTAINED WITH LEVEL SET SEGMENTATION AND

ISO-SURFACE EXTRACTION AT TWO USER-DEFINED THRESHOLDS. NOTATION:��, LEVEL SETS MODEL, �, HIGHER THRESHOLD MODEL, ��, LOWER

THRESHOLD MODEL. NOTATION FOR ANGLES IS SAME AS TABLE IV

image (with standard deviation 2 pixels) for the suppression ofnoise. For each model, centerlines and the bifurcation vectorswere extracted and in-plane and out-of-plane angles betweenthe branches were computed as described. Results are reportedin Tables VI and VII, and show excellent robustness of theparameters to the segmentation method. However, in somecases, iso-surface extraction with user-specified thresholdscan produce changes in the geometry of the surface with theinclusion of spurious structures potentially leading to changesin the geometric parameters (see case ). For this reason, weargue for the adoption a robust segmentation method in orderto obtain reliable geometric parameters.

B. ICA Lateral Aneurysms

In this section, we present the extension of the treatment ofterminal aneurysms to the analysis of lateral aneurysms, whichare not associated to a major bifurcation. These aneurysms typi-cally arise in correspondence of sharp bends, as is the case alongthe carotid siphon.

Fig. 19 shows the cases that have been retrieved from theAneurisk project data base, and that will be employed in this sec-tion. The ICA and the aneurysm centerlines are also displayed.Regarding the latter, it is worth noting that, even if the positionof the centerline endpoint on the aneurysm sac obviously influ-ences the aneurysm centerline trajectory, the orientation of thecenterline through the neck is fairly stable with respect to theselection of the endpoints [Fig. 19(d)], which ensures that thedirection of the aneurysm neck is robust to this choice.

Once the two centerlines are extracted, we consider the loca-tion of the aneurysm neck as a bifurcation, which we refer to as


TABLE VIICOMPARISON BETWEEN OUT-OF-PLANE BRANCH ANGLES OF FOUR MCAANEURYSM MODELS OBTAINED WITH LEVEL SET SEGMENTATION AND

ISO-SURFACE EXTRACTION AT TWO USER-DEFINED THRESHOLDS. NOTATION:��, LEVEL SETS MODEL,��, HIGHER THRESHOLD MODEL,��, LOWER

THRESHOLD MODEL. NOTATION FOR ANGLES IS SAME AS TABLE V

Fig. 19. 3-D models of ICA lateral aneurysms. 3-D robustness of the orienta-tion of the centerline through the neck with respect to the selection of the distalcenterline endpoint on the aneurysm.

aneurysm bifurcation. We can then characterize the aneurysmbifurcation using the geometric quantities described in the pre-vious section, namely the bifurcation area, the aneurysm bifur-cation reference system and the bifurcation vectors. By meansof the reference system, the position and orientation of the neckwith respect to the ICA are quantified. The origin of the refer-ence system accounts for the location of the aneurysm neckalong the carotid siphon, the normal to the aneurysm bifurca-tion plane represents the orientation of the plane on which

Fig. 20. Extraction of the position of the neck along the siphon centerline asthe origin of the aneurysm bifurcation � ; the bifurcation plane normal �and the direction of the aneurysm branch vector � are determined.

Fig. 21. Eight cases of ICA lateral aneurysms, shown with the respectiveaneurysm bifurcation planes and vectors.

the aneurysm develops in the neck region, and the branch vectorrelative to the aneurysm centerline identifies the main direc-tion of the sac immediately distal to the neck.

In Fig. 20 we depict the details of the aneurysm neck regionfor one of the models together with the relative geometric fea-tures. Assuming the ICA terminal bifurcation as a landmark,curvilinear abscissa of the point allows us to quantify theposition of the aneurym along the siphon, while the aneurysmbifurcation plane locally identifies the aneurysm centerline di-rection in the neck region.

Fig. 21 shows the geometric characterization of eight lateralaneurysms. The description of the local geometry of the carotidsiphon, i.e., the local plane of curvature of the parent vessel, isperformed by means of the Frenet frame defined along the ICAcenterline. At each point, the binormal vector identifiesthe osculating plane where the ICA locally lies [see Fig. 10(a)and (b)]; consequently the osculating plane calculated at the


TABLE VIIIPOSITION AND ORIENTATION OF THE NECK OF THE ICA ANEURYSM

MODELS SHOWN IN FIG. 19. NOTATION: � , ANEURYSM NECK

ABSCISSA WITH RESPECT TO THE ICA TERMINAL BIFURCATION

(NEGATIVE SINCE THE ICA BIFURCATION IS DOWNSTREAM THE ICA)IN MM; � , ANEURYSM NECK ORIENTATION, IN DEGREES,

WITH RESPECT TO THE LOCAL ICA OSCULATING PLANE

Fig. 22. Definition of the orientation of the aneurysm neck with respect to thelocal parent vessel geometry: (a) Osculating plane of the parent artery at theaneurysm site. (b) Aneurysm bifurcation normal � parallel transported alongthe siphon centerline. (c) Comparison between the aneurysm bifurcation planeand the siphon osculating plane at the aneurysm site. (d) Computation of theangle between the two planes.

point defines the orientation of the parent vessel in the prox-imity of the aneurysm neck.

The comparison between the orientation of the local ICA os-culating plane and the aneurysm bifurcation plane, i.e., between

and , ultimately represents the angular orientation of theaneurysm with respect to its parent vessel. This is a significantquantity, as it assesses whether the aneurysm developed on theouter wall of a bend and with which orientation with respectto the bend itself. The quantity bears a direct connection withhemodynamic forces originating from flow in curved vessels.In Table VIII, quantitative information on the location and ori-entation of the lateral aneurysms depicted in Fig. 21 is reported.

The sequence in Fig. 22 illustrates the steps needed to obtainthe angle between the ICA osculating plane at any point of theICA and the aneurysm bifurcation plane, performed using theparallel transport technique, as presented in Section III. This al-lows the evaluation of the angular orientation of the aneurysmwith respect to any osculating plane, e.g., upstream the neck.This is again a hemodynamically relevant quantity, as it ac-counts for the role of upstream geometry in determining hemo-dynamic forces at the neck region.

Fig. 23. Subset of digital phantoms used for validation. Left panel: sample ide-alized bifurcation models, bifurcation angle of 30 and 60 , respectively. Rightpanel: sample idealized models of bend bearing a sidewall aneurysm at differentorientations, from bottom to top, aneurysm orientation 0 , 45 , and 90 withrespect to the parent vessel osculating plane, respectively.

More complete quantitative analyses of the geometry of ICAand its possible relations with the presence of aneurysms willbe presented in future work [42], [43].

VI. CONCLUSION

In this paper, we presented a complete framework for robustcharacterization of vascular geometry, and we provided a de-scription of the application of the framework to the case of cere-bral aneurysms.

The present framework is implemented as part of the VascularModeling Toolkit open-source project. As such, it is available tothe modeling community for use and further development. De-spite thorough testing is performed as part of the developmentprocess, the open nature of the project lends itself to additionaltesting by third parties, which is a recommended aspect for theadoption of the project in a scientific setting. The setup of a re-gression testing framework as part of the development processis essential in this respect, and will be part of future activity.

Thanks to the objective criteria introduced in our framework,and to the availability of the tools as part of the Vascular Mod-eling Toolkit, quantitative geometric features are readily avail-able from imaging data, and can be easily employed at a largescale for explorative studies, clinical trials and clinical routine,in order to elucidate and exploit the role of geometry and hemo-dynamics in vascular disease.

APPENDIX

Evaluation of the basic quantities presented in this manuscripthas been performed on digital phantoms generated by compo-sition of analytic implicit functions. In particular, as shown inFig. 23, surface models of Y-shaped bifurcations at imposed bi-furcation angles and sidewall aneurysms at imposed orientationswith respect to the local osculating plane were generated bysampling appropriately defined tube functions on a regular gridand extracting the zero isosurface of the obtained scalar fields. Inorder to avoid degeneration of the Voronoi diagram into a line inregions where analytic models were perfectly cylindrical, noisewas added to the surface by displacing surface points in randomdirections by an amount equal to 1% the radius of the tubes.Centerlines were then extracted and geometric quantities werecomputed following the criteria described in the manuscript. Bi-furcation angles were computed for bifurcation models, and theorientation of the aneurysm with respect to the local osculatingplane were computed, in addition to the mean radius, curvature


TABLE IXBIFURCATION ANGLE � � EVALUATED ON DIGITAL PHANTOMS OF Y-SHAPED

BIFURCATIONS AT DIFFERENT IMPOSED BIFURCATION ANGLES � �

TABLE XANEURYSM ORIENTATION ANGLE � EVALUATED ON DIGITAL PHANTOMS

OF SIDEWALL ANEURYSMS AT DIFFERENT IMPOSED ORIENTATIONS WITH

RESPECT TO THE PARENT VESSEL OSCULATING PLANE � . IN ADDITION,MEAN RADIUS �, MEAN CURVATURE � AND MEAN TORSION � OF THE

PARENT ARTERY ARE REPORTED. TRUE VALUES FOR �, � AND �

WERE 1, 0.2 AND 0, RESPECTIVELY

and torsion of the parent artery (nominally 1, 0.2, and 0, respec-tively). As shown in Tables IX and X, the proposed definitionsaccurately capture the geometric features of the analytic models.

REFERENCES

[1] C. Caro, D. J. Doorly, M. Tarnawski, K. Scott, Q. Long, and C.Dumoulin, “Non-planar curvature and branching of arteries andnon-planar-type flow,” Proc. R. Soc., vol. 452, no. 1944, pp. 185–197,1996.

[2] K. Perktold, R. Peter, M. Resch, and G. Langs, “Pulsatile non-newto-nian blood flow in three-dimensional carotid bifurcation models: A nu-merical study of flow phenomena under different bifurcation angles,”J. Biomed. Eng., vol. 13, no. 6, pp. 507–1, 1991.

[3] S. Sherwin, O. Shah, D. Doorly, M. McLean, N. Watkins, C. Caro, J.Peiro, M. Tarnawski, and C. Dumoulin, “Visualization and computa-tional study of flow at model planar and non-planar end-to-side arterialbypass graft,” J. Physiol.-London, vol. 504P, p. P44, 1997.

[4] S. Sherwin, O. Shah, D. Doorly, J. Peiro, Y. Papaharilaou, N. Watkins,C. Caro, and C. Dumoulin, “The influence of out-of-plane geometryon the flow within a distal end-to-side anastomosis,” J. Biomech. Eng.,vol. 122, no. 1, pp. 86–95, 2000.

[5] U. Schulz and P. Rothwell, “Major variation in carotid bifurcationanatomy: A possible risk factor for plaque development?,” Stroke, vol.32, no. 11, pp. 2522–2529, 2001.

[6] A. Malek, S. Alper, and S. Izumo, “Hemodynamic shear stress and itsrole in atherosclerosis,” JAMA, vol. 282, no. 21, pp. 2035–2042, 1999.

[7] Y. Hoi, H. Meng, S. Woodward, B. Bendok, R. Hanel, L. Guterman,and L. Hopkins, “Effects of arterial geometry on aneurysm growth:Three-dimensional computational fluid dynamics study,” J. Neuro-surg., vol. 101, no. 4, pp. 676–681, 2004.

[8] G. Foutrakis, H. Yonas, and R. Sclabassi, “Saccular aneurysm forma-tion in curved and bifurcating arteries,” Am. J. Neuroradiol., vol. 20,no. 7, pp. 1309–1317, 1999.

[9] M. Friedman, “Variability of 3-D arterial geometry and dynamics andits pathologic implications,” Biorheology, pp. 513–517, 2002.

[10] M. Friedman, O. Deters, F. Mark, C. Bargeron, and G. Hutchins, “Ar-terial geometry affects hemodynamics. A potential risk factor for ath-erosclerosis,” Atherosclerosis, vol. 46, pp. 225–231, 1983.

[11] J. A. Garcia, S. Y. Chen, J. C. Messenger, I. P. Casserly, A. Hansgen,O. Wink, B. Movassaghi, A. J. Klein, and J. D. Carroll, “Initial clinicalexperience of selective coronary angiography using one prolonged in-jection and a 180 degrees rotational trajectory,” Catheter Cardiovasc.Interv., vol. 70, no. 2, pp. 190–196, 2007.

[12] P. O’Flynn, G. O’Sullivan, and A. Pandit, “Methods for three-dimen-sional geometric characterization of the arterial vasculature,” Ann.Biomed. Eng., vol. 35, no. 8, pp. 1368–1381, 2007.

[13] C. Taylor, M. Draney, J. Ku, D. Parker, B. Steele, K. Wang, and C.Zarins, “Predictive medicine: Computational techniques in therapeuticdecision making,” Comput. Aided Surg., vol. 4, no. 5, pp. 231–247,1999.

[14] J. Cebral, M. Castro, S. Appanaboyina, C. Putman, D. Millan, and A.Frangi, “Efficient pipeline for image-based patient-specific analysis ofcerebral aneurysm hemodynamics: Technique and sensitivity,” IEEETrans. Med. Imag., vol. 24, no. 4, pp. 457–467, Apr. 2005.

[15] J. Cebral, M. Castro, J. Burgess, R. Pergolizzi, M. Sheridan, and C.Putman, “Characterization of cerebral aneurysms for assessing riskof rupture by using patient-specific computational hemodynamicsmodels,” AJNR, vol. 26, no. 10, pp. 2550–2559, 2005.

[16] M. Castro, C. Putman, and J. Cebral, “Computational fluid dynamicsmodeling of intracranial aneurysms: Effects of parent artery segmen-tation on intra-aneurysmal hemodynamics,” AJNR, vol. 27, no. 8, pp.1703–1709, 2006.

[17] L. Antiga and D. A. Steinman, Vascular Modeling Toolkit. Jan. 26,2009 [Online]. Available: http://www.vmtk.org,

[18] , J. Sethian, Ed., Level Set Methods and Fast Marching Methods, 2nded. Cambridge, U.K.: Cambridge Univ. Press, 1999.

[19] J. Sethian, “A fast marching level set method for monotonically ad-vancing fronts,” Proc. Nat. Acad. Sci., vol. 93, no. 4, pp. 1591–1595,1996.

[20] L. Antiga, B. Ene-Iordache, and A. Remuzzi, “Computational geom-etry for patient-specific reconstruction and meshing of blood vesselsfrom MR and CT angiography,” IEEE Trans. Med. Imag., vol. 22, no.5, pp. 674–684, May 2003.

[21] R. T. Whitaker, “A level-set approach to 3-D reconstruction from rangedata,” Int. J. Comput. Vis., vol. 29, no. 3, pp. 203–231, 1998.

[22] L. Ibanez, W. Schroeder, L. Ng, and J. Cates, The ITK Software Guide,2nd ed. Albany, NY: Kitware, 2005.

[23] W. Schroeder, K. Martin, and W. E. Lorensen, The VisualizationToolkit, An Object Oriented Approach to 3-D Graphics, 4th ed. Al-bany, NY: Kitware, 2005.

[24] T. Dey and W. Zhao, “Approximate medial axis as a voronoi subcom-plex,” in Proc. 7th ACM Symp. Solid Model. Appl., 2002, pp. 356–366.

[25] N. Amenta, S. Choi, and R. Kolluri, “The power crust, unions of ballsand the medial axis transform,” Computational Geometry: TheoryAppl, vol. 19, no. 2, pp. 127–153, 2001.

[26] T. Dey and W. Zhao, “Approximating the medial axis from the voronoidiagram with a convergence guarantee,” Algorithmica, vol. 38, no. 1,pp. 179–200, 2003.

[27] L. Antiga and D. Steinman, “Robust and objective decomposition andmapping of bifurcating vessels,” IEEE Trans Med. Imag., vol. 23, no.6, pp. 704–713, Jun. 2004.

[28] K. Lee, K. H. Parker, C. Caro, and S. Sherwin, “The spectral/hp elementmodeling of steady flow in non-planar double bends,” Int. J. Numer.Meth. Fluids, pp. 1–16, 2006.

[29] E. Bullitt, G. Gerig, S. Pizer, W. Lin, and S. Aylward, “Measuring tortu-osity of the intracerebral vasculature from MRA images,” IEEE Trans.Med. Imag., vol. 22, no. 9, pp. 1163–1171, Sep. 2003.

[30] M. Poullis, R. Warwick, A. Oo, and R. Poole, “Ascending aortic curva-ture as an independent risk factor for type a dissection and ascendingaortic aneurysm formation: A mathematical model,” Eur. J. Cardio-thorac. Surg., vol. 33, no. 6, pp. 995–1001, 2008.

[31] H. Zhu, Z. Ding, R. Piana, T. Gehrig, and M. Friedman, “Cataloguingthe geometry of the human coronary arteries: A potential tool for pre-dicting risk of coronary artery disease,” Int. J. Cardiol., 2008.

[32] S. V. L. M. Sangalli, P. Secchi, and A. Veneziani, Efficient estimationof 3-Dimensional centerlines of inner carotid arteries and their curva-ture functions by free knot regression splines 2008 [Online]. Available:http://www.mathcs.emory.edu

[33] R. Bishop, “There is more than one way to frame a curve,” Amer. Math.Monthly, vol. 82, no. 3, pp. 246–251, 1975.

[34] J. Thomas, L. Antiga, S. Che, J. Milner, D. Steinman, J. Spence, B.Rutt, and D. Steinman, “Variation in the carotid bifurcation geometryof young versus older adults: Implications for geometric risk of athero-sclerosis,” Stroke, vol. 36, no. 11, pp. 2450–2456, 2005.

[35] S. Lee, L. Antiga, and D. S. J. D. Spence, “Geometry of the carotidbifurcation predicts its exposure to disturbed flow,” Stroke, vol. 39, no.8, pp. 2341–2347, 2008.

[36] T. Ingebrigtsen, M. Morgan, K. Faulder, L. Ingebrigtsen, T. Sparr, andH. Schirmer, “Bifurcation geometry and the presence of cerebral arteryaneurysms,” J. Neurosurg., vol. 101, no. 1, pp. 108–113, 2004.


[37] H. Kasuya, T. Shimizu, K. Nakaya, A. Sasahara, T. Hori, and K.Takakura, “Angles between a1 and a2 segments of the anterior cerebralartery visualized by three dimensional computed tomographic angiog-raphy and association of anterior communicating artery aneurysms,”Neurosurgery, vol. 45, no. 1, pp. 89–93, 1999.

[38] T. Sadatomo, K. Yuki, K. Migita, E. Taniguchi, Y. Kodama, and K.Kurisu, “The characteristics of the anterior communicating arteryaneurysm complex by three-dimensional digital subtraction angiog-raphy,” Neurosurg. Rev., vol. 29, no. 3, pp. 201–207, 2006.

[39] W. v. Rooij, M. Sluzewski, and G. Beute, “Internal carotid bifurcationaneurysms: Frequency, angiographic anatomy and results of coiling in50 aneurysms,” Neuroradiology, vol. 50, no. 7, pp. 583–587, 2008.

[40] T. Sadatomo, K. Yuki, K. Migita, E. Taniguchi, Y. Kodama, and K.Kurisu, “Evaluation of relation among aneurysmal neck, parent arteryand daughter arteries in middle cerebral artery aneurysms, by three-dimensional digital subtraction angiography,” Neurosurg. Rev., vol. 28,no. 3, pp. 196–200, 2005.

[41] T. Sadatomo, K. Yuki, K. Migita, E. Taniguchi, Y. Kodama, and K.Kurisu, “Morphological differences between ruptured and unrupturedcases in middle cerebral artery aneurysms,” Neurosurgery, vol. 62, no.3, pp. 602–609, 2008.

[42] M. Piccinelli, S. Bacigaluppi, E. Boccardi, B. Ene-Iordache, A. Re-muzzi, A. Veneziani, and L. Antiga, Influence of internal carotid arterygeometry on aneurysm location and orientation: A computational ge-ometry study 2008 [Online]. Available: http://www.mathcs.emory.edu

[43] L. Sangalli, P. Secchi, S. Vantini, and A. Veneziani, A case study in ex-plorative functional data analysis: Geometrical features of the internalcarotid artery 2008 [Online]. Available: http://www.mathcs.emory.edu

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