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
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
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
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?
3D Angiographies 5 Observational Study conducted at Ospedale Ca Granda Niguarda Milano relative to 65 patients hospitalized from September 2002 to October 2005. With Cerebral Aneurysm Without Cerebral Aneurysm On Willis 33 On ICA 25 7
3D Angiographies 6 Observational Study conducted relative to 65 patients hospitalized at Ospedale Niguarda Ca Granda Milano from September 2002 to October 2005. With Cerebral Aneurysm Without Cerebral Aneurysm On Willis 33 On ICA 25 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
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
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
From Discrete Data to Functional Data Free Knot Regression Splines 10 Row Data
From Discrete Data to Functional Data Free Knot Regression Splines 11 Fitted Data
From Discrete Data to Functional Data Free Knot Regression Splines 12 Curvature Profile
13 Local Polynomial Regression From Discrete Data to Functional Data Free Knot Regression Splines Free Knot Regression Splines
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
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
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
Data Registration 17 Unregistered first derivatives Registered first derivatives Registration Algorithm
Data Registration (MC Simulations) 18 MC simulations for testing the efficiency of the novel index RS(2005) Novel Index RS(2005) Novel Index
Data Registration 19 Unregistered Radius and Curvature Profiles Registered Radius and Curvature Profiles
Data Classification 20 Other Patients Patients with Aneurysm on Willis Curvature Radius
Data Classification + Functional PCA 21 Sample Autocorrelation Function and Std. Dev. for Radius Profiles Sample Autocorrelation Function and Std. Dev. for Curvature Profiles
PCA and Functional PCA (Overview) 22
PCA and Functional PCA (Overview) 23 Functional PCA PCA
Data Classification + Functional PCA 24
Data Classification + Functional PCA (Bootstrap Sampling) 25 Radius Profiles Curvature Profiles
Data Classification + Functional PCA 26 Functional PCA Scores
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%
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
Case Selection for Blood Flow Numerical Simulations 29 Synthetic Functional Data (Velocity, Pressure, Shear Stress) NEW STATISTICAL ANALYSES Estimation of ANEURYSM RUPTURE RISK
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. 2007 Complex Models and Computational Intensive Methods for Estimation and Prediction, pp. 427-432 a cura di P. Mantovan, A. Pastore, S. Tonellato CLEUP Editore.
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.
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
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