A Case Study in Functional Data Analysis; Investigatingthe Geometry of the Internal Carotid Artery for Cerebral

Aneurysms Classification(⋆)

Un caso studio di analisi di dati funzionali; classificazione degli aneurismicerebrali per mezzo delle caratteristiche geometriche chedescrivono la

carotide interna

Laura Maria Sangalli, Piercesare Secchi, Simone VantiniMOX - Dipartimento di Matematica, Politecnico di Milano

e-mail: piercesare.secchi@polimi.it

Riassunto: Analizziamo ricostruzioni di arterie carotidi interne, ottenute da immaginiangiografiche tridimensionali, e studiamo i loro profili di raggio e curvatura al fine diclassificare la diversa posizione degli aneurismi nel distretto arterioso cerebrale.

Keywords: two-stage method for functional data analysis, data smoothing and curvealignment, dimension reduction within Hilbert spaces, classification of functional data

1. Introduction

Cerebral aneurysms are deformations of cerebral vessels characterized by a bulge of thevessel wall. This pathology is common in the adult population; between 1% and 6%of adults develop a cerebral aneurysm during their lives (Rinkel et al., 1998). Usuallycerebral aneurysms are asymptomatic and not disrupting; however the rupture of acerebral aneurysm, even if quite uncommon, is a tragic eventthat implies death in morethan 50% of cases.

The origin of the aneurysmal pathology is still unclear. Possible explanations,discussed in the medical literature, focus on the interactions between biomechanicalproperties of artery walls and hemodynamic factors, such aswall shear stress andpressure; the hemodynamics is in turn strictly dependent onvascular geometry. See forinstance Hassanet al. (2005) and Castroet al. (2006). The study of these interactionsis the main goal of the AneuRisk Project, a scientific endeavorwhich joins researchersof different scientific fields ranging from neurosurgery andneuroradiology to statistics,numerical analysis and bio-engineering.

Most of the statistical modeling within the AneuRisk Projecthas been aimed atrevealing dependencies between vessel geometries and onset of aneurysm. Investigations

(⋆) This study is a product of the AneuRisk Project, a joint research program involving MOX Laboratoryfor Modeling and Scientific Computing (Dip. di Matematica, Politecnico di Milano), Laboratory ofBiological Structures (Dip. di Ingegneria Strutturale, Politecnico di Milano), Istituto Mario Negri (Ranica),Ospedale Niguarda Ca’ Granda (Milano), and Ospedale Maggiore Policlinico (Milano). The Project issupported by Fondazione Politecnico di Milano and Siemens-Medical Solutions Italia. Special thanks toAlessandro Veneziani (MOX and Emory University), leader ofthe AneuRisk Project, to Edoardo Boccardi(Ospedale Niguarda Ca’ Granda), who provided the 3D-angiographies and motivated our reaserch byposing fascinating medical questions, and to Luca Antiga and Marina Piccinelli (Istituto Mario Negri),who performed the image reconstructions.

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on this theme have been pioneered by Ujjieet al. (2001), Nader-Sephaiet al. (2004) andMill an et al. (2007). However, while these studies focus on the geometricfeatures ofthe aneurysm itself, we proceeded along a different and novel approach that considers thegeometry of the cerebral arteries in the neighborhood of theaneurysm. In particular,we looked for possible dependencies between the geometry ofthe distal part of theInternal Carotid Artery (ICA), and the onset of cerebral aneurysms in the arterial districtdownstream of the ICA. Our main goal has been to validate a conjecture, grounded onpractical experience of neuroradiologists at Niguarda Ca’ Granda Hospital (E. Boccardi,personal communication), that cerebral arteries of patients with an aneurysm at theterminal bifurcation of the ICA, or after it, are geometrically different from those ofpatients with an aneurysm along the ICA itself or healthy. To find statistical support forthis conjecture, we explored the AneuRisk dataset with the tools provided by functionaldata analysis, seeking evidence that could discriminate patients having an aneurysm ator after the terminal bifurcation of the ICA (Upper Group), from patients having ananeurysm along the ICA or healthy (Lower Group).

In this work we present a short account of our statistical analyses and findings; detailsare in Sangalliet al. (2007a), Sangalliet al. (2007b) and Vantini (2008).

2. A two-stage analysis

The AneuRisk dataset consists of three-dimensional cerebral angiographies of 65patients hospitalized at the Neuroradiology Department ofNiguarda Ca’ Granda Hospital(Milano), from September 2002 to October 2005. Among these patients, 33 have ananeurysm at the terminal bifurcation of the ICA or after it; they constitute the UpperGroup. The remaining patients belong to the Lower Group: of these, 25 have an aneurysmalong the ICA while 7 are healthy. None of the patients has other severe diseases affectingthe cerebral vascular system, apart for the possible aneurysm. Percentages of females andmales do not differ significantly from 1/2; age appears to be Normally distributed witha mean equal to 55.85 yrs and a standard deviation equal to 13.45 yrs. Gender and ageare not included in the statistical analysis because they are supposed to be related to theaneurysmal pathology only through their effect on vessel geometries. Each angiographyconsists of a three-dimensional array of gray-scaled pixels representing the patient’sscanned volume and the relative vessels network; lighter pixels show presence of flowingblood while darker pixels show absence of flowing blood. An image reconstructionalgorithm, devised and implemented at the Istituto Mario Negri (Ranica), identifies thelumen of the ICA within this volume and describes it in terms ofthe three spatialcoordinates of the vessel centerline and the radius length of the corresponding lumensection; details are in Piccinelliet al. (2007). Figure 1 shows one such reconstruction ofan ICA with an aneurysm.

The statistical information about the geometry of the ICA is of functional nature,although the image reconstruction algorithm provides, foreach patient, only a finitesample of observations of the vessel centerline and radius profile. Indeed, for everypatienti in our dataset, we know the three spatial coordinatesxij, yij andzij of the vesselcenterline, and the vessel radiusRij, for each pointsij on a fine grid along a curvilinearabscissa, that goes from the terminal bifurcation of the ICA towards the heart. Within-patient sample size is high, ranging from a minimum of 350 points to a maximum of 1380

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Figure 1: Image reconstruction of an ICAwith aneurysm. The grey curve inside thevessel is the centerline (grey scale refers tothe length of the vessel radius)

Figure 2: 3D image of a fitted centerlinetogether with rough data (the little bulletsshow the positions of the spline knots andthe big squares are siphon delimiters)

points; moreover within-patient signal to noise ratio is also high. This justifies the two-stage method that we followed for the analysis of these data,along a paradigm brilliantlyadvocated by J. O. Ramsay (see Ramsay and Silverman (2005) for an exhaustive accounton this approach to functional data analysis). In the first stage, separately for each patient,we obtain smooth and differentiable estimates of ICA centerlines and radius profiles.These curves are thus aligned by means of a novel registration algorithm that aims atdecoupling between-patientphaseandamplitudevariabilities. In the second stage of theanalysis, the dimension of registered data is reduced aiming at a few uncorrelated modesof variability, apt to discriminate at best patients belonging to the Upper Group from thosein the Lower Group. Analogous two-stage methods have been favored, for instance, bySheehyet al. (1999) for the analysis of human growth curves.

3. First stage: data smoothing and curve alignment

Our statistical analysis of the ICA geometries provided by the AneuRisk dataset hasbeen focused on the curvature function of the ICA centerline and on its radius profile,since these two geometric features, together with blood density, viscosity and velocity,determine the local hemodynamics.

Due to measurement and reconstruction errors, the centerlines provided by the imagereconstruction algorithm may be quite wiggly and need to be smoothed in order toobtain sensible estimates of their curvature functions. Wedo so by means of free knotregression splines, i.e. regression splines where the number and position of knots arenot fixed in advance, but chosen in a way to minimize a penalized average squarederror criterion. Since our data are 3D, the idea is to fit simultaneously the three spatialcoordinates(x(s), y(s), z(s)) of the centerline versus the curvilinear abscissas, lookingfor the optimal spline knots along the curvilinear abscissa. See Figure 2. Optimal knotsare searched by an algorithm which is a modification of the algorithm developed by Zhouand Shen (2001) in the 1D case. The first and second derivatives of the fitted centerline

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Figure 3: Left: from top to bottom, estimated first derivativesx′i(s), y

′i(s) andz′i(s) before

registration. Right: estimated first derivativesx′i(s), y′

i(s) and z′i(s) after registration.Darker lines are the first derivatives of the estimated reference centerline

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are thus used to estimate its curvature.Visual inspection of the first derivatives of estimated centerlines(x′

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for the 65 patients, makes evident the presence of strong between-patient phase variabilitythat needs to be decoupled from the between-patient amplitude variability to enablesensible comparisons across patients. Indeed between-patient phase variability couldbe due to different dimensions and proportions for patient skulls. See Figure 3. Thisdecoupling can be achieved by means of a registration procedure that finds the 65 warpingfunctionshi of the abscissa, that capture the phase variability, leading to the new registeredcenterlines(xi(s), yi(s), zi(s)), wherexi = xi ◦h−1

i , yi = yi ◦h−1

i , andzi = zi ◦h−1

i . Welook for the optimal warping functions on the space of increasing affine transformations,maximizing the following similarity index between(xi(s), yi(s), zi(s)) and a referencecenterline(x0(s), y0(s), z0(s)) :

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whereSi is the support of thei-th centerline. This index is inspired by the criterionproposed in Ramsay and Silverman (2005), but has more interesting mathematicalproperties and is moreover suitable for managing curves defined on different supports,as the ones we deal with. A Procrustes fitting criterion is used to estimate both the 65warping functions and the reference centerline. The iterative procedure converges in fewsteps.

These methods for data smoothing and registration are also the objects of twocommunications at theXLIV Riunione Scientifica della SIS, summarized in Sangalli andVantini (2008a) and Sangalli and Vantini (2008b).

4. Second stage: dimension reduction and classification

Many interesting traits emerge from the analysis of registered radius and curvatureprofiles. Figure 4 shows that the vessel gets progressively narrower toward the terminalbifurcation of the ICA - the so-called tapering effect. Moreover, it shows that two peaks

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Figure 4: Left: registered radius and curvature profiles (top and bottom respectively),with superimposed density estimate of aneurysms location. Right: first two principalcomponents of registered radius and curvature (top and bottom respectively), withboxplots of corresponding scores (dark for Lower Group and light for Upper Group)

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of curvature are usually present at about -3.5 and -2.0 cm to the terminal bifurcation.The same figure also displays a gaussian kernel density estimate of the aneurysm locationalong the ICA, based on the data relative to the 27 patients belonging to the Lower Groupand with an aneurysm on the ICA. Note that most of the ICA aneurysms are clustered intwo groups, both located in the terminal part of the vessel, where tapering is evident, andone located just after the last peak of curvature. These results provide evidence of a linkbetween morphology and aneurysm onset, induced by hemodynamics.

The autocovariance structures of radius and curvature profiles have been separatelyexplored by means of Functional Principal Component Analysis (FPCA) - see, forinstance, Ramsay and Silverman (2005) - in order to estimate their main uncorrelatedmodes of variability that discriminate at best patients belonging to the Upper Group fromthose belonging to the Lower Group. Since the 65 curves are known on different abscissaintervals, these analyses focus on the interval where all curves are available, i.e. for valuesof the abscissa between -3.37 and -0.78cm. Figure 4 shows thefirst two eigenfunctionsof the autocovariance function for radius and for curvaturerespectively, together withthe percentage of total variance explained. The eigenfunctions are not directly plotted;instead, sample means of radius and curvature (solid lines)are plotted together with twocurves obtained by adding/subtracting, to the sample means, the estimated normalizedeigenfunctions multiplied by the estimated standard deviation of the corresponding scores.The distributions of FPCA scores for the two groups of patients have significantlydifferent means and/or variances in all but two cases; this is confirmed by the p-valuesof the appropriate t-tests and F-tests for equality of meansand variances (Normalityassumption for score distributions has been verified with Shapiro tests). According tothese differences, Upper Group patients tend to have wider,more tapered and less curvedICA’s with respect to patients belonging to the Lower Group. Moreover the variance ofthese geometrical features is significantly smaller in the Upper Group than in the LowerGroup.

The first two eigenfunctions of radius and curvature autocovariances are in fact theoptimal set of eigenfunctions to discriminate the two groups of patients, when the

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Table 1: Relative and conditional Confusion Matrices, estimated according to L1ER (left)and to APER (right). Row labels refer to true classes, columnlabels refer to predictedclasses. Total L1ER and APER are reported on top

L1ER = 21.54% APER= 16.92%

Rel. Lower UpperLower 35.38% 13.85%Upper 7.69% 43.08%

Rel. Lower UpperLower 35.38% 13.85%Upper 3.08% 47.69%

Cond. Lower UpperLower 71.88% 28.12%Upper 15.15% 84.85%

Cond. Lower UpperLower 71.88% 28.12%Upper 6.06% 93.94%

Figure 5: L1ER (left) and APER (right), as functions ofkR and kC . The broken linecorresponds to points such thatkR = kC

classification is performed by means of Quadratic Discriminant Analysis (QDA) ofFPCA scores. Indeed, let the eigenfunctions of the autocovariance be ordered accordingto the decreasing proportion of explained total variance. The set of the firstkR andkC eigenfunctions - respectively of radius and curvature autocovariance - span finitedimensional spaces where each patient’s radius profile and centerline curvature functionare represented by score vectors of dimensionkR andkC , respectively. For each possiblecombination ofkR andkC , we performed a QDA for classifying patients in the two groupsusing as predictor the concatenation of the two score vectors; the effectiveness of thediscriminant rule generated by each QDA has been quantified by the Leave-One-Out ErrorRate (L1ER) as shown in Figure 5. The minimum of L1ER is reached for kR = 2 andkC = 2. The same optimal choice forkR andkC results from inspection of the ApparentError Rate (APER) - the fraction of patients in the dataset thatare misclassified by thediscriminant rule - due to the presence of a marked elbow whenkR = 2 andkC = 2. Thisoptimal choice corresponds to using, for QDA, only the scores relative to the first andsecond eigenfunctions of the radius and curvature autocovariances. According to L1ER, ifkR = kC = 2 are used, 21.54% of new patients would be misclassified; hence, the number

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of misclassified patients, using the prediction rule induced by this QDA, is estimated to beless than half the number of patients that would be misclassified by randomly assessingpatients to the Lower or Upper group without taking into account FPCA scores.

Estimated membership probabilities for the 65 patients usingkR = kC = 2 are reportedin Figure 6. Patients are sorted according to group membership probabilities. Differentcolors refer to the real membership of the patients. Some facts have to be noticed:

Figure 6: Estimated Lower/Uppergroup membership probabilities forthe 65 patients

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1. Estimated Lower Group membership probabi-lities, for the Lower Group patients, range from1 to almost 0. This means that a patient inthe Lower Group may have geometrical featuressimilar to those characterizing the Upper Group.

2. Estimated Upper Group membership proba-bilities for the Upper Group patients are all, buttwo, greater than 0.5. This means that (nearly)no Upper Group patient has geometrical featuressimilar to those characterizing the Lower Group.

3. Many Lower Group patients have anestimated Lower Group membership probabilityapproximately equal to 1, while no UpperGroup patient has such a high Upper Groupmembership probability.

In terms of geometrical characterization of the two groups,these facts show that theLower Group patients are more spread in the scores space, whereas the patients in theUpper Group are concentrated in a smaller region, nested within the region covered bythe Lower Group. These conclusions are confirmed by inspection of the conditional errorrates reported in Table 1. According to L1ER, predicting correctly a Lower Group patientis more difficult than predicting correctly an Upper Group patient; in fact the estimatedprobability of misclassifying a patient belonging to the Lower Group (28.12%) is nearlytwice as big as the probability of misclassifying a patient belonging to the Upper Group(15.15%).

5. Conclusions

Our two-stage functional data analysis of ICA radius and curvature profiles, forpatients included in the AneuRisk dataset, validates the conjecture formulated by theNeuroradiology Group of the AneuRisk Project concerning thegeometrical differenceof cerebral arteries of patients with an aneurysm at the terminal bifurcation of the ICA,or after it, from those of patients with an aneurysm along theICA itself or healthy. Theanalysis generated an efficient classification of patients belonging to the two groups -Upper and Lower - by means of a quadratic discriminant analysis applied to suitablerepresentations of ICA radius and curvature profiles in spaces of low finite dimension.These results, if confirmed on a larger dataset, could support clinical practice.

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