Registration of Functional Data: Aligning Inner Carotid Artery

Registration of Functional Data:
Aligning Inner Carotid Artery Centerlines (⋆)
Registrazione di dati funzionali:
allineamento delle linee assiali di arterie carotidi interne
Laura Maria Sangalli, Simone Vantini
MOX, Dipartimento di Matematica - Politecnico di Milano
e-mail: [email protected]
Keywords: functional data analysis, curve registration, amplitude and phase variability
1. Introduction
This work is a product of the AneuRisk Project, a scientific program that aims at
evaluating the role of vascular geometry and hemodynamics in the pathogenesis of
cerebral aneurysms. By means of functional data analysis, we explore the AneuRisk
dataset to highlight the relations between the geometric features of the internal carotid
artery (ICA), expressed by its radius profile and centerline curvature, and the aneurysm
location (Sangalli et al. 2007a, Sangalli et al. 2007b). The dataset is based on 3Dangiographies of 65 patients hospitalized at the Neuroradiology Department of Niguarda
Ca’ Granda Hospital, Milano, from September 2002 to October 2005: the i-th sample
unit is a regular curve in R3 describing, as function of an abscissa s, the three spatial
coordinates (xi (s), yi (s), zi (s)) of the ICA centerline of the i-th patient.
2. Methods and results
We focus here on the centerlines registration procedure conceived and performed within
the former functional data analysis. This procedure is the key to statistically meaningful
comparisons among the geometrical features, radius and curvature, observed in different
patients. Indeed, it separates two types of variability affecting functional data: the
phase variability, related to the abscissa s and strongly dependent on the dimensions
and proportions of patients skulls, and the amplitude variability, the residuals variability
related to the geometrical features (Ramsay and Silverman, 2005). In detail, the
phase variability is removed by finding 65 warping functions hi of the abscissa s,
leading from the original centerlines (xi (s), yi (s), zi (s)) to new registered centerlines
−1
−1
(x̃i (s), ỹi (s), z̃i (s)), where x̃i = xi ◦ h−1
i , ỹi = yi ◦ hi , and z̃i = zi ◦ hi . We look
for the optimal warping functions in the class W of increasing affine transformations,
maximizing the following similarity index between the i-th centerline (x̃i (s), ỹi (s), z̃i (s))
and a reference centerline (x0 (s), y0 (s), z0 (s)):
(⋆)
This research has been carried out within AneuRisk Project, a joint research program involving MOX
Laboratory for Modeling and Scientific Computing (Dip. di Matematica, Politecnico di Milano), Laboratory
of Biological Structures (Dip. di Ingegneria Strutturale, Politecnico di Milano), Istituto Mario Negri
(Ranica), Ospedale Niguarda Ca’ Granda (Milano), and Ospedale Maggiore Policlinico (Milano). The
Project is supported by Fondazione Politecnico di Milano and Siemens-Medical Solutions, Italia.


R
R
′ ′
′ ′
′ ′
ds
x̃
x
ds
ds
ỹ
y
z̃
z
1
Si i 0
Si i 0
Si i 0

qR
qR
qR
qR
+ qR
+ qR
3
′2
′2
′2
′2
′2
′2 ds
ds
ds
ds
ds
ds
x̃
x
ỹ
y
z̃
z
Si i
Si 0
Si i
Si 0
Si i
Si 0
R
(1)
where Si is the support of the i-th centerline. The index (1) is a suitable 3D-generalization
in L2 (R; R3 ) of the cosine of the angle between the first derivatives of two functions in
L2 (R; R). The class W and the similarity index (1), jointly provide useful properties in
terms of formal consistency (e.g., the similarity index between two curves does not change
if the two curves are warped along the same function of class W ), pertinence to the real
problem and algorithm implementation. A Procrustes fitting criterion is used to jointly
estimate both the 65 warping functions and the reference centerline. Alternate expectation
and maximization steps are performed: at each expectation step, the reference curve is
estimated using all the curves obtained at the previous iteration; at each maximization
step, all curves are warped in order to maximize theirs similarity with the estimated
reference curve. The optimal warping functions hi are then obtained by composition of
the maximizing warping functions found at each iteration step. The algorithm converges
in few steps.
1.0
x~i ′(s )
0.0 0.5
−1.0
−1.0
x i ′(s )
0.0 0.5
1.0
Figure 1: First derivatives of the 65 centerlines before (left) and after (right) registration,
together with first derivatives of the reference centerline (solid black line)
−100
−80
−60
−40
−20
0
−100
−80
−60
−40
−20
0
−40
−20
0
−40
−20
0
0.5
−1.0
−1.0
y~i ′(s )
0.0
y i ′(s )
0.0 0.5
1.0
s
1.0
s
−100
−80
−60
−40
−20
0
−100
−80
−60
z~i ′(s )
0.0 0.5
−1.0
−1.0
z i ′(s )
0.0 0.5
1.0
s
1.0
s
−100
−80
−60
−40
s
−20
0
−100
−80
−60
s
References
Ramsay J.O., Silverman B.W. (2005) Functional Data Analysis, II ed., Springer, New
York.
Sangalli L.M., Secchi P., Vantini S., Veneziani A. (2007a) Functional data analysis.
A case study: geometrical features of the internal carotid artery, Tech. Rep.
24/2007, MOX, Dipartimento di Matematica, Politecnico di Milano, available at
http://mox.polimi.it/it/progetti/pubblicazioni.
Sangalli L.M., Secchi P., Vantini S., Veneziani A. (2007b) Smoothing and dimension
reduction of 3D centerlines of inner carotid arteries by free-knots regression splines,
Tech. Rep. 23/2007, MOX, Dipartimento di Matematica, Politecnico di Milano,
available at http://mox.polimi.it/it/progetti/pubblicazioni.