smooth vasculature reconstruction from patient...

Workshop On Virtual Reality Interaction and Physical Simulation (2005)F. Ganovelli and C. Mendoza (Editors)

Smooth Vasculature Reconstruction from Patient Scan

X. Wu1, V. Luboz1, K. Krissian2

1 The Simulation Group, Massachusetts General Hospital, 65 Landsdowne Street, Cambridge, MA 02139, USAEmail: [email protected] Phone: (001) 617-768-8791; Fax: (001) 617-768-8915

2 Surgical Planning Laboratory, Brigham and Women’s Hospital, Boston, MA 02122, USA

AbstractVR based simulations are ideal for procedure training and planning of neuro-interventional therapy withoutputting patients at risk. To achieve interactive visualization, real-time and robust physics-based modeling; asmooth and efficient 3D patient-specific neuro-vasculature is essential. This paper presents and evaluates a stream-lined reconstruction process from patient scan to smooth vascular surface. Semi-automatic tools have been de-veloped to reduce noise in data set, to segment vascular network, to estimate vessel centerlines and radii, and toreconstruct the associated smooth surface. The proposed scheme handles more general vascular topology thanprevious approaches and is more robust to present various bifurcation configurations. The accuracy and consis-tency of our technique are evaluated on a vascular phantom scanned in 12 different orientations as well as areal clinical data set. Experiments show that the proposed technique reaches a good balance in terms of meshsmoothness, compactness, and accuracy.

Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Computational Geometryand Object Modeling—curve, surface, solid, and object representations; I.4.5 [Image Processing and ComputerVision]: Reconstruction—transform methods; I.4.6 [Image Processing and Computer Vision]: Segmentation; J.3[Computer Applications]: Life and Medical Sciences—health;

1. Introduction

1.1. Background

In the context of neuro-vascular therapy, our team is de-veloping a real-time neuro-interventional simulation system.Its aim is to allow residents and physicians to learn andpractice without putting patients at risk. The foundation ofsuch simulator is the semiautomatic generation of vascularmodel from patient’s preoperative CTA. This model will beused to achieve smooth interactive vasculature visualization[SCC∗04], structured physiology modeling (for blood flowcomputation) and efficient real-time collision detection (CD)/ collision response (CR) between interventional tools andvessel walls [DLNC05]. Based on a skeleton segmentationfollowed by a surface reconstruction, the method presentedin this paper generates the patient specific virtual vascularskeleton and smooth surface.

1.2. Previous Work

Current techniques of segmenting vascular structure fromCTA scans can be divided in two main approaches:

• techniques for centerline enhancement, including multi-scale approaches, usually based on the Hessian matrix;

• and techniques for contour extraction, including statis-tical approaches: Expectation Maximization [WGKJ96],random Markov fields, and geometrical approaches: re-gion growing, adaptive thresholding, and active contoursthat can be explicit, like snakes, or implicit, like level sets[Set99, SLS∗02].

Both approaches are efficient for computing a suitable topo-logical representation of the vascular network either by com-puting ridges or by applying a thinning technique like ho-motopic skeletonisation. Given that CTA scans are relativelynoisy (background, dental artifacts...), these techniques areusually preceded by a noise reduction step. Their main draw-back is the lack of accuracy towards small vessels, though

c© The Eurographics Association 2005.

X. Wu & V. Luboz & K. Krissian / Smooth Vasculature Reconstruction from Patient Scan

they are indeed very efficient for carotid and other big ves-sels.

Tubular structure reconstructions on both artery [ZJP94]and airways [FPBAG04] have been studied in the pastdecade. There are many vessel surface reconstruction meth-ods including voxel-level tessellation [CL99], deformablesurface, e.g. level-set method. Although these methods aregeneric to process different types of surface, they often cre-ate highly detailed surfaces consuming a lot of memoryto store and yet missing tree graph information. Vessel re-construction methods based on contour estimation, on thecontrary, represent major characteristics of vascular objectswith low element count along with the vessel connectivity.For comprehensive introduction to these techniques, pleaserefer to an up-to-date survey [BFC02]. To fulfill the real-time simulation characteristics presented above, an efficient,smooth, and robust vascular surface reconstruction processis developed over [FWB04] based on the given vessel con-tour. The reconstructed surface can be easily refined throughpatch subdivision. Although our proposition bases on thesame framework as [FWB04], we aims both to vastly im-prove the quality under wider range of joint configurationsand graph topology and to enable adaptive meshing based onsegmented vessel geometry.

Section 2 describes our method to segment and recon-struct the vascular network. Section 2.1 deals with semi-automatic tools which reduce data set noise, segment usingactive contours, compute the skeleton, and estimate the ves-sel radii. Section 2.2 describes the reconstruction scheme ofthe associated surface and detailed comparison over previ-ous methods. In Section 3, we present different tests per-formed on the phantom and used to evaluate the robustnessand the accuracy of our method. It is also applied on a patientdata set as an example. Finally, discussion and conclusionare presented in Section 4.

2. Methodology

The goal of our vascular surface reconstruction is to generatea smooth topologically preserved 2-manifold surface. Thesurface representation can be easily refined to suit the needsof applications ranging from efficient CD/CR, physical-based blood flow simulation, to multi-scale anatomical vi-sualization. Our process consists of two main steps: segmen-tation and reconstruction.

2.1. Segmentation

The segmentation technique developed here can be summa-rized in four consecutive steps:

• Enhancement and cleaning the patient data withanisotropic diffusion and morphological operators;

• Segmentation of the vessels through a level set evolution;• Skeletonisation to get the centerlines of the vessels;• Estimation of the vessel radii.

2.1.1. Enhancement and Cleaning

The first step of the segmentation is to filter the CTA im-ages. We use an anisotropic filter [Kri02], which reduces thedata noise while preserving small vascular structures there-fore improving their detection. It is a critical step since themajority of vessels within the brain have < 2.0mm radii. Theparameters of this algorithm are the standard deviation ofthe filter, the attachment coefficient, and the number of pix-els taken into account to get the filtered value of the pixeltreated. This nonlinear filter allows the intensity of vesselborders to be increased while lowering the noise intensitysimultaneously.

Then the skull, the sinuses, and the skin are removed us-ing morphological operations, i.e. dilation and erosion, onaccount of their similar intensity with that of the vessels. Inmost cases, the process starts with several dilation steps inorder to fill the small cavities in the anatomical structurestudied. These holes are in the vessels or next to them. Alarger number of erosion iterations are then applied to erasethe skin and sinuses. The bones are segmented through a reg-ular thresholding followed by additional steps of erosion toremove the transition part between the bones and the brain.The transition part has similar intensity to the vessels whichcould confuse the segmentation due to their proximity insome locations.

2.1.2. Level Set Segmentation

Manual seed selection and region growing methods wouldbe time consuming and less robust since disconnected partscould be missing. We segment the vessel contours by themeans of a level set evolution [OS88, Set99] applied on theenhanced data set. Here the initialization of the active con-tour is done using a threshold on the image intensity for bet-ter efficiency. The level set evolves a surface according tothree different forces: an advection force that pushes the sur-face towards the edges of the image; a smoothing term, pro-portional to the minimal curvature of the surface [LFG∗01],that keeps the surface smooth; a balloon force of equation(1)

e−(I−m)2

σ2 − τ (1)

that allows the contours to expand within vascular structures.These forces rely on the intensity statistics to either expandor shrink the evolving contour. I is the intensity, m stands forthe mean intensity of the vessels, σ is their standard devia-tion, and τ is a threshold (0.2 by default) that allows shrink-ing the contour when the position is unlikely to belong to avessel. For this study, a 3D surface of the vessels is recon-structed using the Marching-Cubes [LC87] as the iso-surfaceof intensity zero from the result of the level set evolution.This surface will be used later in this article as the referencefor the comparison to the one reconstructed by our method.



2.1.3. Skeletonization

From the images segmented by the level set, skeletonizationis applied to obtain the vessel centerlines. This process al-lows our simulator to efficiently perform CD and blood flowcomputation by supplying an abstract topological represen-tation of the vascular network. It is based on homotopic thin-ning where voxels are removed in the order of the Euclideandistance to the segmented surface. Voxels are iteratively re-moved from the object if they are simple [MBA93] and ifthey are not end-points, such that they have more than oneneighbor in a 3×3×3 neighborhood.

The skeletonization leads to a set of rough centerlineswhich can still have connectivity discrepancies especiallynear small branches. Consequently, pruning is applied to re-move small leaves (lines connected at only one extremity,or line with no connection) from the centerline tree. Giventhe resolution of the medical data set, some lines remain dis-connected when they should be part of the same vessel. Weconnect them by using a semi-automatic process which se-lects close lines with a corresponding direction. The direc-tion criterion helps to match lines within a small curvaturedifference. This step often requires manual adjustment sincea few lines might be too long to be deleted by pruning ortoo distant to be connected automatically. These areas needhuman judgement to delete or connect appropriate lines andto complete the vascular skeleton.

The only manual intervention in the whole streamlinedreconstruction is mainly due to "breaks" in some vessels.Those breaks are discontinuities produced by artifacts, suchas the lead used by dentists, or by limited data resolution thatmakes thin vessels look like dash lines. This manual inter-action is non-existent given good data sets, e.g. higher res-olution with vessels clearly separated from each other andlittle artifacts without vessels and ambient area having sim-ilar grey scale. This is demonstrated in the test of phantomdata shown in Section 3. The manual task can take from 2minutes to several days of labor, depending on the level ofdetails expected for including thin vessels.

2.1.4. Vessel Radii Estimation

Once the vessels are finally connected, the radii of the center-lines are extracted to describe the circular surface of the ves-sels. This process is based on an algorithm growing a circlein the orthogonal plan of the centerline points. It computes,along circles of increasing radii, the intensity gradient, i.e.the derivatives of a Gaussian kernel with a given standarddeviation, in the binary image obtained from the level setevolution. It stops when a relevant local maximum of theintensity gradient is found on the cross-section therefore es-timating the radii along the centerlines.

2.2. Surface Reconstruction

The technique reconstructs topologically preserving quadri-lateral surface patches of branching tubular structure given

the vascular skeleton, i.e. the discrete vascular centerlinesand the radii at those sampling points. To be self-complete,we briefly introduce the main procedure which is based on[FWB04]. Please refer to the article for further details.

The base mesh generation is done by calling TileTree(S)recursively, where S is reference of a branch. Each branchis discretized into segments. Each segment has two circularcross sections (CS) and one line segment connecting the two.In the base mesh, circular CS is approximated by describedsquares. Subsequent subdivision of the square converges toa circle. TileTree() derives TileJoint(), who patches seg-ments in the bifurcation regions, and TileTrivially(), whois charge of patching the area between 2 end segments of S.TileTree(S) consists of three tasks:

1. Tiles S from the second segment to the one preceding thelast segment by TileTrivially() assuming the first seg-ment has been tiled in previous TileTree() call;

2. Tile the joint by means of the TileJoint() who patchesthe end segment of S and the beginning segments of otherbranches who share this joint;

3. TileTree() finishes by recursive calls of itself to the par-ent and children branches of S.

Our algorithm improves over [FWB04] in the first three ofthe four reconstruction sub-problems, decomposed by Mey-ers et al. [MSS92] as following:

• The correspondence problem is solved by filtering the rawskeletonization result and distributing cross sections ac-cording to its geometric profile, i.e. radius and curvature;

• The tiling problem is handled intrinsically by the skele-tonization and preserved by the above filtering, since crosssections are centered and ordered on the centerline;

• The branching problem is resolved by a recursive patchingscheme to connect the patches of branching vessels to thatof the trunk vessels regardless of vessel orientations;

• Surface-fitting problem is answered by Catmull-Clarksubdivision algorithm supplied in SUBDIVIDE2.0 † onthe base mesh to improve surface smoothness.

2.2.1. Handling generic directed graphs

In humans, artery vessels can form loops, e.g. the cerebralarterial circle—Circle of Willis. Our algorithm handles moregeneric directed graph structure where one branch is allowedto have multiple parents as well as multiple children. Onebranch can also connect to another single branch forming1-furcation or mono-furcation. This is useful to construct aunified directed graph for both artery and venous sides. Also,multiple trees can be reconstructed at the same time. To re-alize more comprehensive vascular network configurations,the propagation of so-called up-vector ~up, which is the firstof the four vectors equally dissecting the described squarecross-section, follows a recursive scheme. When a branch

† http://mrl.nyu.edu/∼biermann/subdivision



joint has multiple parents, the end cross section’s ~upini of

each parent branch Bini is projected onto the plane defined

by the joint location and a child centerline Boutj first normal

as ~upouti, j by a minimal rotation from one parent’s end normal

nini to one child’s beginning normal nout

j .

~upoutj =

∑i ~upout

i, j|∑i ~upout

i, j ||∑i ~upout

i, j |> ε

v v⊥ noutj |∑i ~upout

i, j | ≤ ε

(2)

Then these projected up-vector are averaged. If the averagedvector is close to singular, then an arbitrary unit length vectorv perpendicular to nout

j is chosen as ~upoutj .

2.2.2. Trunk branch selection based on angle and radiivariance

To patch the surface at vessel joints, both surface reconstruc-tion algorithms define at first two, i.e. incoming and outgo-ing, trunk branches and then form polygons to connect thetrunk surface and other joint branches’ base mesh. Previ-ous approach classifies vessels into forward and backwardbranches. And only forward branches are used to computethe average forward normal, navg, to avoid singularity. Thevessel i, whose starting normal ni is the closest to navg, is la-beled as the outgoing trunk branch. The centerline curve tan-gent ni(x) at location x is approximated by differentiating ad-jacent sampling points. Illustrated in Figure 1, when the sam-

Figure 1: Trunk branch selection: using both vessel’s aver-age radii and branching angle to determine the continuationtrunk of current branch, Bin

0 . Although θ j < θi, our algo-rithm chooses Bin

1 as the trunk branch of Bin0 , due to the

similarity of their average radii.

pling density is high, the approximated local tangents nin0 and

nini (x) at the joint of Bin

0 and Bin1 , respectively, will be similar

to each other, or otherwise opposite in direction. When cen-terlines are under sampled, the approximated normals can bemisleading. For example, in Figure 1 trunk branch selectionbased only on branching angles chooses Bout

0 as the trunkbranch and thus introduces patching artifact.

Our trunk branch selection scheme is based on bothbranching angle and vessel radii to reduce under-samplingartifacts, because this improves the robustness and thesmoothness of surface reconstruction. At a joint, there canbe more than one incoming as well as multiple outgoingbranches. Firstly nin

i , (i>0) are reversed. Then, we computethe disparity Ωi defined in

Ωi ≡ λθi +(1−λ)|ri− rin0 | (3)

where λ ∈ [0,1] is the weight balancing the influence of thebranching angle and that of the averaged vessel radius. λ =0.5 is used. The algorithm picks the branch with minimal Ω

as the trunk branch. In Figure 1, although θ j < θi, formedby (nin

0 ,−nin1 ), Bin

1 is still chosen as the trunk continuation ofBin

0 , due to the similarity of their average radii.

2.2.3. Adaptive cross sections distribution

Each sampling point on a centerline curve is the center ofthe circular cross section. In the previous approach, thesesampling points are obtained from a down-sampling processfrom the segmentation result. Evenly distributed samplingvertices do not accurately reflect the vessels geometry, e.g.diameter, curvature. For instance, the right external carotidartery with average radius 1.7mm will have the same densityof sampling points as that of the left common carotid arterywith radius 4.8mm. This potentially causes regions with ex-cessive surface patches and areas with insufficient patches toconnect the vessel geometry.

In order to incorporate the geometry characteristics, ouralgorithm adaptively distributes the sampling points accord-ing to both vessel’s radii and centerline curvature profiles

xi+1− xi = α( ri+11+βκi+1

+ ri1+βκi

) i ∈ [0,Nsegment −1](4)

where xi is the curvilinear coordinate of the cross sectioncenter along the centerline. ri and κi are the correspondingradius and Gaussian curvature, respectively, obtained by lin-ear interpolation between the ends of a raw skeleton segmentwho embeds xi. α > 0 is the desired distribution scalar. κi isestimated according to [COS∗98]. β > 0 is the weight ofcurvature influence on the distribution. Equation (4) statesthat after skeleton filtering, the centers of two adjacent crosssections are placed closer if the vessel is thin or has sharpturns. When a thick branch is straight, there is no need toplace more cross sections than needed. This approach com-promises the centerline smoothness and sharp feature preser-vation. This is shown in Figure 2. Assembling (4) for alli yields a set of (Nsegment − 1) nonlinear algebraic equa-tions with (Nsegment − 1) unknowns, since x0 and xN areset to be the curvilinear coordinates of the vessel end nodes.A multidimensional secant method - Broydn’s method - isused to solve for all xi. If Broydn’s method cannot give ananswer within the prescribed number of iteration steps, thefinal iteration result is used as the answer due to the globalconvergence property of Broydn’s method.



Figure 2: Left: Cross section distribution is denser at thin-ner regions of a vessel. Cross sections are farther apart asthe vessel diameter increases. Right: the distribution densityis higher where a vessel turns or twists. Few cross sectionsare placed where the centerline curve is flatter.

Figure 3: Depending on the outgoing vessel’s orientation,Felkel’s method connects Child( j) to Seg(0) and Child(i)to Seg(N− 1). Our method connects Child(i) and Child( j)both Seg(N − 1) and Seg(0). The bottle-neck effect is re-duced and skeleton symmetry is preserved.

2.2.4. Robust bifurcation tiling

Because the base polygon is always a tetragon as in allother surface patches, the recursive joint tiling algorithm,in previous publication [FWB04], generates only quadran-gle tiles. Our algorithm differs vastly from previous methodby introducing two techniques, i.e. end-segment-groupingand adjacent-quadrant-grouping, use neighbor quadrilateralpatches to form base polygon before tiling joint patches. Themotivation behind these ideas is to improve the smoothnessof the reconstructed vascular surface with less patching arti-fact and preserving branching symmetry.

Instead of using different schemes to connect the maintrunk with the backward branches or the forward branches,a single joint tiling scheme has been developed. We joinbranches and the main trunk in the same way regardless thebranching angles. Therefore a single recursive joint tilingprocess is needed for a branch joint. End-segment-groupingunifies all the outgoing branches together such that the con-necting patches connect the bottom of the outgoing branch’s

Figure 4: Child(i) centerline lies close to the boundary ofQ0 and Q3. Using only one quadrant Q3 induces unwantedtwisting artifact. Adjacent-quadrant-grouping uses both Q0and Q3 to connect Child(i)’s base mesh to the trunk surface.

base mesh with both end segments of the trunk branches, i.e.Seg(N−1) and Seg(0), demonstrated in the left of Figure 3.

When the outgoing centerline forms a small angle with thetrunk centerline, the previous approach produces bottle-neckeffect which can not be eliminated by surface subdivision.In Figure 3, the bottle-neck effect is reduced when both endsegments are deployed for the joint tiling. When the outgo-ing centerline lies near or close to the bisection plane of twotrunk centerlines, using a single end segment cannot presentthe symmetry. The symmetry of this bisection situation isnicely preserved by connecting the base mesh of Child(i)with the side of both Seg(N− 1) and Seg(0). End-segment-grouping not only reduces the patching artifacts in both ex-treme cases, but yields a smoother transition from the trunkto the branches under all branching angles.

We improve the bifurcation tiling not just along thetrunk centerline direction. Adjacent-quadrant-grouping isdesigned to use both adjacent sides of the end hexahedronsegments. In Figure 4, when a child centerline lies close tothe boundary of a quadrant, e.g. Child(i) centerline lies inquadrant Q3, but close to the boundary of Q0 and Q3, theformer algorithm still uses only one quadrant, Q3. The in-duced artifact is apparent. This situation is resolved in ourapproach by adding the neighboring quadrant into the tiling.In this case, the adjacent Q0 and Q3 are grouped together as awhole when connecting with the base mesh of Child(i) to thetrunk mesh. Grouping 2 adjacent quadrants is sufficient topreserve skeleton symmetry. When Child(i) centerline linesbisects a quadrant, our approach uses only the current quad-rant for the tiling as in [FWB04].

Because the joint tiling involves more than one trunkpatches, the base polygon can have up to 12 edges. Therecursive joint tiling algorithm has to exam the branchingcenterline’s orientation and tile a minimally twisted polygonsurface. The pseudo-code of the recursive joint tiling is pre-sented.



Tile_Bifurcation(Base_Polygon,Segment, Branch)

Inverse Segment direction due to graphconnectivity.

if (Segment intersects Base_Polygon)if (Intersection close to the

edge of Base_Polygon)Base_Polygon = Expand(Base_Polygon);

//Form a new polygon without//overlap edges.Base_Polygon = Form_Polygon(Base_Polygon,

Segment_Tetragon);

Branch = Current Segment’shosting branch.

Tile_Bifurcation(Base_Polygon,Next_Segment, Branch);

else //No intersectionif (None of the connected Segments

intersects Base_Polygon)Form_Min_Twisted_Patch(Base_Polygon,

Segment_Tetragon);

Tile_Bifurcation(Base_Polygon,Next_Segment, Branch);

end

With these improvements, the proposed reconstructionscheme is able to handle more general directed graph. It isless prone to artifacts due to initial data sampling. It is alsomore robust to present full range of bifurcation configura-tion. The reconstructed smooth vascular surface is suitablefor CD/CR, flow computation and visualization.

3. Tests and evaluation

The evaluation of our method has been performed on a phan-tom simulating a simplified vascular network. This phantom,shown in Figure 5(a), is composed of a Plexiglas box filledwith silicon gel and nylon tubing. Vessels are modeled bytubing with radii ranging from 2.34mm (simulating the mid-dle cerebral artery) to 0.78mm (simulating small brain ves-sels). This phantom is simple compared to standard patientdata: there is only one material around the tubing (the sili-cone gel), and the contrast agent in the vessels makes themeasy to segment. It will not be as easy in a real data set,where there are different materials with heterogenous in-tensities, and the contrast agent doesn’t highlight the ves-sels that clearly. We decided to use the phantom as a firsttest because of its simplicity on those two problems, know-ing that we would have to deal with them in a regular dataset. In another hand, the phantom is already presenting awide variety of geometries, vessel diameters, vessel inter-sections, cycle, and therefore presents a network complexenough for a first evaluation of our method before movingto a real data set. Moreover, no grand truth in the vascular

segmentation/reconstruction domain is available to compareour method with others. The method presented in this paperhas been applied to CTA scans of the phantom (Figure 5(b)).The segmentation generates a skeleton (Figure 5(c)) whichis then reconstructed into a smooth surface by our tiling al-gorithm as shown in Figure 5(d).

Figure 5: (a) The silicon phantom with nylon tubing mim-icking a vascular structure, (b) an image of the CTA wherethe tubes are filled with contrast agent, (c) result of the skele-tonization after pruning and smoothing, and (d) reconstruc-tion of the 3D surface.

We evaluated the rotational invariance and robustness ofour method on CTA scans of the phantom in 12 different ori-entations. The phantom orientations are obtained via a 45

or 90 rotation on one or more axes in the scanner. The CTAscan resolution is 0.6x0.6x1.25mm.

3.1. Evaluation of the robustness of the vessel lengthsand radii

The segmentation method, described in 2.1, was applied tothe 12 data sets with following parameters:



Figure 6: Brand-Altman plots for (a) the tube lengths and(b) the average radii. The upper and the lower limits repre-sent 2σ. Very few length and radius values deviate from theiraverages.

• 4 steps of anisotropic filtering with a standard deviation(SD) of 0.8, an attachment coefficient of 0.05, and takinginto account the intensity value of the 8 pixels around theone computed;

• a 19 pixel dilation step follow by a 26 erosion step to re-move the box walls. Each step is multiplied by samplingsize of the erosion and dilation required by the user. Forthis case, a sample size of 0.25 has been chosen. Thesemorphological operations are required for the real datawhile for the phantom it is not really useful since the box(which has an intensity closed to the one of the contrastagent) is well separated from the tubing;

• 1000 level set iterations; using intensity threshold of 2300with a SD of 750, and a 4 pixel distance search from apoint already in the computed surface;

• 5mm pruning;• the connection of centerlines with smaller than 2mm gaps

and the same direction (based on the tangent direction attheir ends), and smoothing of those lines;

• radius estimation with gradient computed from the deriv-atives of a Gaussian kernel with SD 0.4mm, and 10 itera-tions in the radius range 0.01mm and 15mm.

No manual work was required to improve the line junctions.However, manual interaction was needed to correct the ori-entation of a few lines in 7 of 12 cases. It is impossible to

deal with this problem automatically because of a few loopsexisted in the human vascular network, and here in the phan-tom. The reorientation task takes < 2 minutes on each dataset. From the corrected lines and the radii, the 3D surfacecan be reconstructed as in Figure 5(d).

The Brand-Altman method[BA86] has been used to eval-uate the results of the segmentation step and more speciallythe lengths and the radii of the phantom lines. In this graph-ical statistical method, the differences (or alternatively theratios) between the two measurements are plotted againstthe averages of the two techniques consequently showingthe variations of the data in comparison of the standarddeviation, σ. Figure 6(a) shows that length variation stayswithin 1.0mm, while 2σlength = 3.5mm. In only 6 out of204 cases (17tubes x 12scans), the radius variation is outof [−2σradius,2σradius], where 2σradius = 0.2mm, as shownin Figure 6(b).

3.2. Evaluation of the mesh accuracy and smoothness

In this section, we measure surface smoothness and thedistance between two surfaces. The distance between twosurfaces is computed using MESH software (available athttp://mesh.berlios.de). It uses the Hausdorff distance tocompute the maximum, mean and root-mean-square errorsbetween two given surfaces as described in [Aspert2002].In this paper, the Hausdorff distance is computed from theprocessed surface to the reference iso-surface. For evaluatingthe smoothness of a given surface, we compute Root MeanSquare (RMS) of both the minimal and the maximal curva-tures of the surface, κ1 and κ2 respectively. The principalcurvatures are computed by fitting a second order polyno-mial to each vertex and its direct neighbors: using this smallsize neighborhood allows taking into account the noise of thesurface in the smoothness estimation. The lower the value,the smoother the surface is.

Our reconstruction has been applied to the phantom withthree levels of subdivision, L0, L1, L2. The base level cor-responds to square cross sections, while the level 1 to octa-gon, and the level 2 to 16-edge polygons. Figure 7 depictsthe RMS and maximal distance error from our reconstructedsurface at different subdivision levels, L0,1,2, to the surfaceS0, by applying Marching Cubes to the level set segmenta-tion result. The RMS error in the upper plot is always lessthan one voxel (i.e. less than 0.6mm). The maximal errorin the lower plot happens at the bifurcation regions whosecross sections are far from circular. The maximal errors areunder 4mm throughout the data sets. Under both measures,the error monotonically decays as the surface is refined.

The results of our method has been compared, in termof smoothness and distance error, to the ones obtainedusing the Visual Tool Kit library (VTK, available athttp://www.vtk.org ). For the comparison, we used an arbi-trarily chosen data set, named M7. This data set has been



Figure 7: RMS and maximal distance error between the re-constructed surface and the iso-surface S0 for 12 data sets at3 subdivisions. Base Mesh is the reconstructed surface withzero subdivision.

Figure 8: RMS distance error on M7 in function of the num-ber of triangles, after different decimation levels of the orig-inal iso-surface.

rotated of an angle of 45 on the x axis and on the z axis.Figure 8 shows the distance between the original iso-surfaceS0 and L0, L1, L2 compared to the distance between S0 andits decimation levels Sd

0 , d ∈ [0,9] using the vtkDecimateProalgorithm and derived from [SL92]. We plot the root meansquare distance versus the logarithm of the number of trian-gles. The RMS obtained after decimation is always smaller,because most errors occur at vessel extremities or close tojunctions. However, our model allows simpler meshes, withreasonable error (RMS<0.6mm) and good smoothness asdisplayed in Figure 9. This figure shows the RMS of theminimal curvature, κ1, on our models L0, L1, L2 and onthe decimated iso-surface, Sd

0 . It is almost constant (0.03)for all subdivision levels and much better than any Sd

0 (be-

Figure 9: RMS of min curvature vs. triangle count.

Figure 10: Smoothness evolution for different smoothinglevels compared to the smoothness our model at level 1.

tween 0.3 and 0.7). For completeness, we also depict theRMS of κ1 and κ2 for L1 and S8

0 with approximately thesame similar number of triangles. Figure 10 plots the evolu-tion of these smoothness measures according to the numberof smoothing iterations. S8

0 has been smoothed by applyingthe vtkSmoothPolyDataFilter. It shows that the RMS of themaximal curvature, κ2, always increases with the smoothingdue to vessel shrinkage. At the same time, the RMS of mini-mal curvature, κ1, decreases to 0.25 while our correspondingreconstructed surface, L1, have a value of 0.03. This showsour model smoothness superiority over VTK smoothing.

3.3. Results on a clinical data set

To evaluate the robustness of our process on a real patientdata set, we select a region of the neuro-vasculature in Fig-ure 11(a). This anatomical part contains the end of the verte-bral arteries, joining into the basilar artery which then split-ting into the posterior cerebral arteries. This testing data isrepresentative of the whole network for several reasons. Itpresents several junctions, transits sharply from big to rela-tively small vessels, bifurcates in wide angles, and has junc-tions close to each other. To get the reference mesh of it,



Figure 11: (a) anatomy of the circle of Willis; (b) segmentediso-surface and color code showing the difference with thereconstructed surface shown in (c). The color code in (b)ranges from blue (0.0mm) to red (3.0mm), (d) reconstructedarterial side.

we generate an iso-surface from the local volume shown inFigure 11(b). After skeletonization and surface reconstruc-tion, we obtained the surface in Figure 11(c). Figure 11(b)shows the color code of the Hausdorff distance between thetwo surfaces. The result shows excellent fitting (RMS error< 0.4mm) with only 5% of iso-surface triangles. There is nomanual interaction for this model since the vessels are bigenough to be segmented perfectly by the level set. Using ourstreamlined process on a full CTA data set gives the arterialside of the neuro-vasculature depicted in Figure 11(d).

In our case, we are aiming a physics based simulation re-quiring a complete and directed vascular graph rather than amere geometric model. For this patient, it took several daysof manual labor to connect and reorient the vessel center-lines. Moreover, recovering tiny vessels (R < 0.5mm) in thismodel is very difficult because the segmentation can confusethem with noise due to CTA resolution (0.6mm). As it canbe seen in Figure 12(a), there are many disconnected vesselsegments after the automatic phase. This is why, in order toget a physiologically complete model, the skeleton has to bemanually corrected as in Figure 12(b).

The resulting network has been integrated in our neuro-

Figure 12: (a) Skeleton in the center of the brain after thepruning and the automatic connection. Some centerlines stillneed to be added to the complete network. (b) Skeleton afterthe manual correction (reorientation and connection).

vascular simulator. Figure 13(a) shows the fluoroscopy ofthe skull augmented by the reconstructed smooth vascularsurface. In Figure 13(b), a zoom-in view of Figure 13(a) de-picts the smooth surface of a joint.

Figure 13: (a) Reconstructed vascular surface along withthe fluoroscopic view of the same patient skull. (b) Zoom-inview on a bifurcation surface.

4. Conclusion

This article presents a streamlined process for segmentingand reconstructing a structured, smooth, and efficient neuro-vascular network from a patient CTA scan. The evaluation ofour method has been performed both on a vascular phantomand on a real clinical data set. Despite the 12 differently ori-ented phantom data sets, our method consistently produceshomogeneous skeletons and radii. The length variation stayswithin 0.6σlength, while the radius estimation is also accu-rate. Moreover, the RMS of Hausdorff distance between thereconstructed and the reference surfaces is always less thanone voxel. Similar result is also obtained on real patient data.These results demonstrates the accuracy and robustness of



our method. Our reconstructed surfaces is very efficient be-cause the excellent fitting is achieved by using only 5% ofiso-surface triangles. At the mean time, reconstructed sur-faces are more than 10 times smoother than the reference.

The main drawbacks of our method are that it is not fullyautomatic to preserve vessel orientation and intense manuallabor is required to recover tiny vessels who appear to bedisconnected line segments scattered in segmentation result.Also, estimating CS using more comprehensive representa-tion should be devised rather than circles used in this article.

5. Future Work

In the future, the main effort will focus on reducing theamount of manual work by improving centerline and smallvessel detection via re-orienting the lines and separating tan-gent vessels, which are currently merged under the imagingresolution. Both tasks would benefit from an a priori knowl-edge based on anatomical atlas or templates. Indeed, approx-imating vascular surface by general cylinder is really simpleand might raise accuracy problem in modeling vessel lumenwith stenosis or aneurysm. We are developing the estimationof the cross sections by ellipses which will vastly improvereconstruction quality at stenosis, aneurysm, and junction ar-eas without sacrificing mesh smoothness and efficiency. Anautomatic correction of centerline orientation is also underinvestigation based on graph theory.

We will further validate the presented method on a largeset of patient data while integrating it into the neuro-vascularintervention training system.

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