Cerebral Aneurysms - Department of Math/CS

Transcript

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
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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
SANGALLI, SECCHI, VANTINI, VENEZIANI
Cerebral Aneurysms
• 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
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Cerebral Aneurysms
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?
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3D Angiographies
Observational Study conducted at Ospedale Ca’ Granda Niguarda – Milano
relative to 65 patients hospitalized from September 2002 to October 2005.
With Cerebral
Aneurysm
On Willis
Without
Cerebral
Aneurysm
On ICA
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25
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3D Angiographies
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Observational Study conducted relative to 65 patients hospitalized
at Ospedale Niguarda Ca’ Granda Milano from September 2002 to October 2005.
With Cerebral
Aneurysm
On Willis
Without
Cerebral
Aneurysm
On ICA
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From 3D Angiographies to Discrete Data
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IMAGE RECONSTRUCTION by L. Antiga, M. Piccinelli, Mario Negri-Bergamo
3D-arrays of
gray scaled pixels
ELICITATION OF:
1)
Centerline coordinates
(centers of MIS)
2)
Vessel width
(Radius of MIS)
Vessel Lumen (volume occupied by blood flow)
identified by Centerlines and MISR’s
xij
Right-Left
yij
Top-Down
zij
Front-Back
Rij MISR
tij
Abscissa
From Discrete Data to Functional Data
Free Knot Regression Splines
Discrete Points
of the Centerline
For each
patient
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Functional Expression
of the Centerline
xi(s)
Left-Right
Top-Down
yi(s)
Top-Down
Front-Back
zi(s)
Front-Back
xij
Left-Right
yij
zij
Why?
•
To Match Data
Data Registration
•
To Estimate Derivatives
Curvature Profiles
•
To Reduce Dimensionality
Discrete Points
of the Centerline
For each
patient
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Functional Expression
of the Centerline
xi(s)
Left-Right
Top-Down
yi(s)
Top-Down
Front-Back
zi(s)
Front-Back
xij
Left-Right
yij
zij
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
Row Data
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Fitted Data
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Curvature Profile
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Local Polynomial Regression
Data Registration
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Registered first derivatives
Unregistered first derivatives
Registration
Algorithm
Goal: Decoupling Phase Variability from Amplitude Variability
Find optimal warping functions h1,…, h65 such that
( xi (hi−1 (t )), yi (hi−1 (t )), zi (hi−1 (t ))) and ( x j (h −j 1 (t )), y j (h −j 1 (t )), z j (h −j 1 (t ))) for i, j = 1,...,65
are comparable with respect to amplitude
Data Registration
Similarity Index between Curves
The similarity index is bounded
Class W is a group with respect to composition
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Class of Warping Functions
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
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Unregistered Derivatives
Loess with Gaussian Kernel and span = 0.15 on the 65 patients
For each patient, find hi that maximizes similarity between
reference derivatives x0‘(t), y0‘(t), z0‘(t)
and patient derivatives xi‘(hi-1(t)), yi‘(hi-1(t)), zi‘(hi-1(t))
For each patient look for the affine transformation of the abscissa maximizing
Registered Derivatives and Warping Functions
Iterative Registration Algorithm
Estimate
reference first derivatives x0‘(t), y0‘(t), z0‘(t)
Data Registration
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Unregistered first derivatives
Registered first derivatives
Registration
Algorithm
Data Registration
(MC Simulations)
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MC simulations
for testing the efficiency of the novel index
RS(2005)
Novel Index
RS(2005)
Novel Index
Data Registration
Unregistered
Radius and Curvature Profiles
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Registered
Radius and Curvature Profiles
Data Classification
Curvature
Radius
Other Patients
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Patients with Aneurysm on Willis
Data Classification + Functional PCA
Sample Autocorrelation Function
and Std. Dev. for Radius Profiles
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Sample Autocorrelation Function
and Std. Dev. for Curvature Profiles
PCA and Functional PCA
(Overview)
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Functional PCA
PCA
PCA and Functional PCA
(Overview)
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(Bootstrap Sampling)
Radius Profiles
Curvature Profiles
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Functional PCA Scores
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L1ER = 21.5%
APER = 16.9%
ICA Willis
ICA Willis
ICA Willis
ICA Willis
Predicted Predicted
Predicted Predicted
Predicted Predicted
Predicted Predicted
ICA
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ICA 35.4% 13.8%
ICA
23
9
ICA 35.4% 13.8%
Willis
5
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Willis 7.7% 43.1%
Willis
2
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Willis 3.1% 47.7%
Conclusions
(Medical Findings)
Willis group (patients with aneurysm on the Willis circle after ICA bifurcation):
•
Large vessels, with low width variability (between patients)
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Significant tapering in the last 15mm next to ICA bifurcation
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High curvature variability (along ICA), presence of straight tracts and elbows
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Low curvature variability (between patients)
Other Patients (patients with aneurysm on the ICA or without aneurysm):
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Narrow vessels with great variability in widthness (between patients)
•
Constant tapering
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Constant (along ICA) and pronounced curvature
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Highly variable curvature profiles (between patients)
Auxiliary Findings
•
Landmarks identification: bends and syphon delimiters, curvature peaks
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Distribution of registered location for aneurysms on ICA
•
Correlation structure of ICA centerlines
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Case Selection for
Blood Flow Numerical Simulations
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Synthetic Functional Data
(Velocity, Pressure, Shear Stress)
NEW STATISTICAL ANALYSES
Estimation of
ANEURYSM RUPTURE RISK
Publications
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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.
“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”
“Free knot regression splines for 3-dimensional functional data, with applications to the analysis of Inner Carotid Artery centerlines”
“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.
“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
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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 3DGeometries 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
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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
References
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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), 674-684. 8.
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, 662-680.
RAMSAY, J.O., SILVERMAN, B.W. (2005):
Functional Data Analysis, II ed., Springer New York NY.
“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.
Explorative functional data analysis for 3D-geometries of the Inner Carotid Artery
First international Workshop on Functional and Operatorial Statistics IWFOS'2008 19-21 June 2008 University Paul Sabatier, Toulouse
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.
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.
“Registration of Functional Data: Aligning Inner Carotid Artery Centerlines”
XLIV Riunione Scientifica Società Italiana di Statistica 2008 Arcavacata di Rende (CS), 25-27 giugno 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), 25-27 giugno 2008.
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, 609-628.
ZHOU, S., SHEN, X. (2001):
Spatially adaptive regression splines and accurate knot selection schemes. J. Amer. Statist. Assoc., 96, 247-259.

Cerebral Aneurysms - Department of Math/CS

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