Functional Data Analysis of the Geometrical Features of the Internal Carotid Artery

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1 Milano, 11 Marzo 2008 Functional Data Analysis of the Geometrical Features of the Internal Carotid Artery Laura Maria SANGALLI Piercesare SECCHI Simone VANTINI Alessandro VENEZIANI

2 The ANEURISK Project 2 We don t know what to say to people with cerebral aneurysms. Choosing between treating or not can be a matter of choosing between Life and Death. Dr E. Boccardi Statistics Computer fluid dynamics Structure Mechanics Neuroradiology Medical Imaging Neurosurgery

3 Cerebral Aneurysms 3 Cerebral aneurysms: malformations of cerebral arteries, in particular of arteries belonging to or connected to the Circle of Willis. EPIDEMIOLOGICAL STATISTICS Incidence rate of cerebral aneurysms: 1/20 people Incidence rate of ruptured cerebral aneurysms per year: 1/10000 people per year Mortality due to a ruptured aneurysm: > 50%: Out of 9 patients with a ruptured aneurysm: 3 are expected to die before arriving at the hospital 2 to die after having arrived at the hospital 2 to survive with permanent cerebral damages 2 to survive without permanent cerebral damages

4 Cerebral Aneurysms 4 A CONJECTURE Onset, development and rupture of cerebral aneurysms are conditioned by the geometry of the vessel through its effects on blood fluid dynamics Is there a relationship between the geometrical features of the distal part of the ICA and the onset of cerebral aneurysms in the arterial district downstream of the ICA?

5 3D Angiographies 5 Observational Study conducted at Ospedale Ca Granda Niguarda Milano relative to 65 patients hospitalized from September 2002 to October With Cerebral Aneurysm Without Cerebral Aneurysm On Willis 33 On ICA 25 7

6 3D Angiographies 6 Observational Study conducted relative to 65 patients hospitalized at Ospedale Niguarda Ca Granda Milano from September 2002 to October With Cerebral Aneurysm Without Cerebral Aneurysm On Willis 33 On ICA 25 7

7 From 3D Angiographies to Discrete Data 7 IMAGE RECONSTRUCTION by L. Antiga, M. Piccinelli, Mario Negri-Bergamo 3D-arrays of gray scaled pixels Vessel Lumen (volume occupied by blood flow) identified by Centerlines and MISR s ELICITATION OF: 1) Centerline coordinates (centers of MIS) 2) Vessel width (Radius of MIS) x ij y ij z ij R ij t ij Right-Left Top-Down Front-Back MISR Abscissa

8 From Discrete Data to Functional Data Free Knot Regression Splines 8 Discrete Points of the Centerline x ij y ij z ij Left-Right Top-Down Front-Back For each patient Functional Expression of the Centerline x i (s) y i (s) z i (s) Left-Right Top-Down Front-Back Why? To Match Data To Estimate Derivatives To Reduce Dimensionality Data Registration Curvature Profiles

9 From Discrete Data to Functional Data Free Knot Regression Splines 9 Discrete Points of the Centerline x ij y ij z ij Left-Right Top-Down Front-Back For each patient Functional Expression of the Centerline x i (s) y i (s) z i (s) Left-Right Top-Down Front-Back Fixed: Spline Order by minimizing through an iterative algorithm: Regularity of Functional Data Not Fixed: Number and Position of Knots Spatial Variability Sum of Squared Residuals Complexity

10 From Discrete Data to Functional Data Free Knot Regression Splines 10 Row Data

11 From Discrete Data to Functional Data Free Knot Regression Splines 11 Fitted Data

12 From Discrete Data to Functional Data Free Knot Regression Splines 12 Curvature Profile

13 13 Local Polynomial Regression From Discrete Data to Functional Data Free Knot Regression Splines Free Knot Regression Splines

14 Data Registration 14 Unregistered first derivatives Registered first derivatives Registration Algorithm Goal: Decoupling Phase Variability from Amplitude Variability ( x i ( h 1 i ( t)), y i ( h 1 i Find optimal warping functions h 1,, h 65 such that ( t)), z i ( h 1 i ( t))) and ( x j ( h 1 j ( t)), y ( h 1 j ( t)), z are comparable with respect to amplitude j j ( h 1 j ( t))) for i, j = 1,...,65

15 Data Registration 15 Similarity Index between Curves Class of Warping Functions The similarity index is bounded Class W is a group with respect to composition An iterative algorithm for registration is feasible Similarity index considers similar what doctors consider similar Increments in similarity obtained through the registration procedure are not artificial

16 Data Registration 16 Unregistered Derivatives Estimate reference first derivatives x 0 (t), y 0 (t), z 0 (t) Loess with Gaussian Kernel and span = 0.15 on the 65 patients For each patient, find h i that maximizes similarity between reference derivatives x 0 (t), y 0 (t), z 0 (t) and patient derivatives x i (h i -1 (t)), y i (h i -1 (t)), z i (h i -1 (t)) For each patient look for the affine transformation of the abscissa maximizing Iterative Registration Algorithm Registered Derivatives and Warping Functions

17 Data Registration 17 Unregistered first derivatives Registered first derivatives Registration Algorithm

18 Data Registration (MC Simulations) 18 MC simulations for testing the efficiency of the novel index RS(2005) Novel Index RS(2005) Novel Index

19 Data Registration 19 Unregistered Radius and Curvature Profiles Registered Radius and Curvature Profiles

20 Data Classification 20 Other Patients Patients with Aneurysm on Willis Curvature Radius

21 Data Classification + Functional PCA 21 Sample Autocorrelation Function and Std. Dev. for Radius Profiles Sample Autocorrelation Function and Std. Dev. for Curvature Profiles

22 PCA and Functional PCA (Overview) 22

23 PCA and Functional PCA (Overview) 23 Functional PCA PCA

24 Data Classification + Functional PCA 24

25 Data Classification + Functional PCA (Bootstrap Sampling) 25 Radius Profiles Curvature Profiles

26 Data Classification + Functional PCA 26 Functional PCA Scores

27 Data Classification + Functional PCA 27 L1ER = 21.5% APER = 16.9% ICA Willis ICA Willis ICA Willis ICA Willis Predicted Predicted Predicted Predicted Predicted Predicted Predicted Predicted ICA 23 9 ICA 35.4% 13.8% ICA 23 9 ICA 35.4% 13.8% Willis 5 28 Willis 7.7% 43.1% Willis 2 31 Willis 3.1% 47.7%

28 Conclusions (Medical Findings) 28 Willis group (patients with aneurysm on the Willis circle after ICA bifurcation): Large vessels, with low width variability (between patients) Significant tapering in the last 15mm next to ICA bifurcation High curvature variability (along ICA), presence of straight tracts and elbows Low curvature variability (between patients) Other Patients (patients with aneurysm on the ICA or without aneurysm): Narrow vessels with great variability in widthness (between patients) Constant tapering Constant (along ICA) and pronounced curvature Highly variable curvature profiles (between patients) Auxiliary Findings Landmarks identification: bends and syphon delimiters, curvature peaks Distribution of registered location for aneurysms on ICA Correlation structure of ICA centerlines

29 Case Selection for Blood Flow Numerical Simulations 29 Synthetic Functional Data (Velocity, Pressure, Shear Stress) NEW STATISTICAL ANALYSES Estimation of ANEURYSM RUPTURE RISK

30 Publications 30 Technical Reports SANGALLI, L.M., SECCHI, P., VANTINI, S., VENEZIANI, A. (2007): Smoothing and dimension reduction of 3D centerlines of inner carotid arteries by free-knots regression splines. Tech. Rep., MOX 23/07, Dip. di Matematica, Politecnico di Milano. SANGALLI, L.M., SECCHI, P., VANTINI, S., VENEZIANI, A. (2007): Explorative Functional Data Analysis. A Case Study: Geometrical Features of the Internal Carotid Artery. Tech. Rep. MOX 24/07, Dip. di Matematica, Politecnico di Milano. SECCHI, P., VANTINI, S. (2006): Analisi Statistica della Morfologia Geometrica del Sistema Arterioso Cerebrale. Tech. Rep. AneuRisk Project. Conference Proceedings SANGALLI, L.M., SECCHI, P., VANTINI, S. (2008): A case study in functional data analysis; investigating the geometry of the internal carotid artery for cerebral aneurysms classification XLIV Riunione Scientifica Società Italiana di Statistica, Università della Calabria. SANGALLI, L. M., VANTINI, S. (2008): Registration of Functional Data: Aligning Inner Carotid Artery Centerlines XLIV Riunione Scientifica Società Italiana di Statistica, Università della Calabria. SANGALLI, L. M., VANTINI, S. (2008): Free knot regression splines for 3-dimensional functional data, with applications to the analysis of Inner Carotid Artery centerlines XLIV Riunione Scientifica Società Italiana di Statistica, Università della Calabria. SANGALLI, L.M., SECCHI, P., VANTINI, S. (2008): Explorative functional data analysis for 3D-geometries of the Inner Carotid Artery First international Workshop on Functional and Operatorial Statistics IWFOS'2008, University Paul Sabatier, Toulouse, France. SANGALLI, L.M., SECCHI, P., VANTINI, S. (2007): Functional Data Analysis for 3D-Geometries of the Inner Carotid Artery, Book of Short Papers S.Co Complex Models and Computational Intensive Methods for Estimation and Prediction, pp a cura di P. Mantovan, A. Pastore, S. Tonellato CLEUP Editore.

31 Thesis and Talks 31 PhD Thesis VANTINI, S. (2008): Functional Data Analysis of the Geometrical Features of the Internal Carotid Artery PhD in Mathematical Engineering, Politecnico di Milano. Master Degree Thesis VELE, S. (2007): Metodi numerici e statistici per l analisi di aneurismi cerebrali Master Degree in Mathematical Engineering, Politecnico di Milano. DALLA ROSA, M. (2008): Principal Differential Analysis e applicazioni allo studio delle caratteristiche geometriche della carotide interna Master Degree in Mathematical Engineering, Politecnico di Milano. Talks 27th September 2007, CRiSM Centre for Research in Statistical Methodology, Department of Statistics, Warwick University, UK. Invited Talk: Functional Data Analysis for 3D- Geometries of the Internal Carotid Artery ; speaker SANGALLI, L. M. 30th October 2007, Dipartimento di Scienze Economiche e Metodi Quantitativi, Univerisità del Piemonte Orientale. Invited Talk: Analisi di dati funzionali: un applicazione allo studio degli aneurismi cerebrali ; speaker SECCHI, P. 8th September 2007, S.Co. Fifth Conference on Complex Models and Computational Intensive Methods for Estimation and Prediction, Venezia, Italy. Invited Talk: Functional Data Analysis for 3D-Geometries of the Inner Carotid Artery ; speaker SECCHI, P. 26th September 2006, III International Symposium on Modelling of Physiological Flows (MPF2006), Bergamo, Italy. Contributed Talk: Statistical Analysis of the Geometry and Fluidynamics of Cerebral Arteries ; speaker VANTINI, S. June 2008, XLIV Riunione Scientifica Società Italiana di Statistica, Università della Calabria. Invited Talk: A case study in functional data analysis; investigating the geometry of the internal carotid artery for cerebral aneurysms classification ; speaker SECCHI, P. June 2008, XLIV Riunione Scientifica Società Italiana di Statistica, Università della Calabria. Contributed Talk: Registration of Functional Data: Aligning Inner Carotid Artery Centerlines ; speaker VANTINI, S. June 2008, XLIV Riunione Scientifica Società Italiana di Statistica, Università della Calabria. Contributed Talk: Free knot regression splines for 3-dimensional functional data, with applications to the analysis of Inner Carotid Artery centerlines ; speaker SANGALLI, L. M. June 2008, First international Workshop on Functional and Operatorial Statistics IWFOS'2008, University Paul Sabatier, Toulouse, France. Contributed Talk: Explorative functional data analysis for 3D-geometries of the Inner Carotid Artery ; speaker SECCHI, P.

32 People 32 Group Coordinator Piercesare SECCHI Postdoc Laura Maria SANGALLI PhD Student Simone VANTINI PhD Scholarship supported by Fondazione Politecnico di Milano and Siemens Medical Solutions, Italia Master Degree Students Matilde DALLA ROSA Stefania VELE Valeria VITELLI

33 References 33 ADAMS, R. A. (1975): Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, Pure and Applied Mathematics, Vol. 65. ANTIGA, L., ENE-IORDACHE, B., REMUZZI, A. (2003): Computational geometry for patient-specific reconstruction and meshing of blood vessels from MR and CT angiography. IEEE Trans Med Imaging, 22(5), BACIGALUPPI, S., VENEZIANI, A.,M. COLLICE, M., BOCCARDI, E., GALLI, C., TORRESIN, A., SECCHI, P., FORMAGGIA, L., PASSERINI, T., VANTINI, S., DUBINI, G., MIGLIAVACCA, F., SOCCI, L., REMUZZI, A., ANTIGA, L., ENE-IORDACHE, B., PICCINELLI, M., GAINI, S. M. (2005): Progetto Aneurisk. Strumenti numerici e statistici di indagine per il rischio di rottura negli aneurismi cerebrali, Atti del 54 Congresso Nazionale, Società Italiana di Neurochirurgia. HASSAN, T., TIMOFEEV, EV., SAITO, T., SHIMIZU, H., EZURA, M., MATSUMOTO, Y., TAKAYAMA, K., TOMINAGA, T., TAKAHASHI, A. (2005): A proposed parent vessel geometry-based categorization of saccular intracranial aneurysms:computational flow dynamics analysis of the risk factors for lesion rupture. J. Neurosug., 103, RAMSAY, J.O., SILVERMAN, B.W. (2005): Functional Data Analysis, II ed., Springer New York NY. SANGALLI, L.M., SECCHI, P., VANTINI, S. (2007): Functional Data Analysis for 3D-Geometries of the Inner Carotid Artery, Book of Short Papers S.Co Complex Models and Computational Intensive Methods for Estimation and Prediction, pp a cura di P. Mantovan, A. Pastore, S. Tonellato CLEUP Editore. SANGALLI, L.M., SECCHI, P., VANTINI, S. (2008): Explorative functional data analysis for 3D-geometries of the Inner Carotid Artery First international Workshop on Functional and Operatorial Statistics IWFOS' June 2008 University Paul Sabatier, Toulouse SANGALLI, L.M., SECCHI, P., VANTINI, S., VENEZIANI, A. (2007): Smoothing and dimension reduction of 3D centerlines of inner carotid arteries by free-knots regression splines. Tech. Rep., MOX 23/07, Dip. di Matematica, Politecnico di Milano. SANGALLI, L.M., SECCHI, P., VANTINI, S., VENEZIANI, A. (2007): Explorative Functional Data Analysis. A Case Study: Geometrical Features of the Internal Carotid Artery. Tech. Rep. MOX 24/07, Dip. di Matematica, Politecnico di Milano. SANGALLI, L. M., VANTINI, S. (2008): Registration of Functional Data: Aligning Inner Carotid Artery Centerlines XLIV Riunione Scientifica Società Italiana di Statistica 2008 Arcavacata di Rende (CS), giugno SANGALLI, L. M., VANTINI, S. (2008): Free knot regression splines for 3-dimensional functional data, with applications to the analysis of Inner Carotid Artery centerlines XLIV Riunione Scientifica Società Italiana di Statistica 2008 Arcavacata di Rende (CS), giugno SECCHI, P., VANTINI, S. (2006): Analisi Statistica della Morfologia Geometrica del Sistema Arterioso Cerebrale. Tech. Rep. AneuRisk Project. ZHANG, C. (2003): Calibrating the Degrees of Freedom for Automatic Data Smoothing and Effective Curve Checking. J. Amer. Statist. Assoc., 98, ZHOU, S., SHEN, X. (2001): Spatially adaptive regression splines and accurate knot selection schemes. J. Amer. Statist. Assoc., 96,

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