IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 28, NO. 8, AUGUST 2009 1141 A Framework for Geometric Analysis of Vascular Structures: Application to Cerebral Aneurysms Marina Piccinelli, Alessandro Veneziani, David A. Steinman, Andrea Remuzzi, and Luca Antiga* Abstract—There is well-documented evidence that vascular ge- ometry has a major impact in blood flow dynamics and conse- quently in the development of vascular diseases, like atheroscle- rosis and cerebral aneurysmal disease. The study of vascular ge- ometry and the identification of geometric features associated with a specific pathological condition can therefore shed light into the mechanisms involved in the pathogenesis and progression of the disease. Although the development of medical imaging technolo- gies is providing increasing amounts of data on the three-dimen- sional morphology of the in vivo vasculature, robust and objec- tive tools for quantitative analysis of vascular geometry are still lacking. In this paper, we present a framework for the geometric analysis of vascular structures, in particular for the quantification of the geometric relationships between the elements of a vascular network based on the definition of centerlines. The framework is founded upon solid computational geometry criteria, which confer robustness of the analysis with respect to the high variability of in vivo vascular geometry. The techniques presented are readily available as part of the VMTK, an open source framework for image segmentation, geometric characterization, mesh generation and computational hemodynamics specifically developed for the analysis of vascular structures. As part of the Aneurisk project, we present the application of the present framework to the character- ization of the geometric relationships between cerebral aneurysms and their parent vasculature. Index Terms—Cerebral aneurysms, geometric quantification, three-dimensional modeling, vascular geometry. Manuscript received February 25, 2009; revised April 10, 2009. First pub- lished May 12, 2009; current version published July 29, 2009. The Aneurisk project is a joint research project of the Center for Modeling and Scientific Computing (MOX) of the Politecnico di Milano, Milan (Italy), the Mario Negri Institute for Pharmacological Research, Bergamo (Italy), and the Ca’ Granda Hospital, Milan (Italy), supported by Siemens Medical Italia and Fondazione Politecnico. Asterisk indicates corresponding author. M. Piccinelli is with the Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30332 USA and also with the Biomedical Engineering Department, Mario Negri Institute for Pharmacological Research, 24020 Ranica, Italy. A. Veneziani is with the Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30332 USA. D. A. Steinman is with the Department of Mechanical and Industrial Engi- neering, University of Toronto, Toronto, ON, M5S 2E4, Canada. A. Remuzzi are with the Biomedical Engineering Department, Mario Negri Institute for Pharmacological Research, 24020 Ranica, Italy and also with the with the Industrial Engineering Department, University of Bergamo, 24020 Ranica, Italy. *L. Antiga are with the Biomedical Engineering Department, Mario Negri In- stitute for Pharmacological Research, 24020 Ranica, Italy (e-mail: antiga@mar- ionegri.it). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMI.2009.2021652 I. INTRODUCTION T HERE is well-documented evidence that vascular geom- etry has a major impact in the blood dynamics and, in turn, in the origin and development of vascular disease, through the action of forces exerted by flowing blood on the vascular wall. [1]–[4]. Typical examples of these relationships are the devel- opment of atherosclerotic lesions [5], [6] and the formation of intracranial aneurysms preferentially at bifurcations and sharp bends [7], [8]. The study of vascular geometry in relation to the development of specific pathologic conditions can, therefore, shed light on the hemodynamic triggers involved in the pathogenesis and in the progression of the disease [9], [10]. Furthermore, the identifica- tion of geometric quantities that are associated with a specific pathological condition, or that have some predictive power with respect to the severity of disease progression is a valuable en- deavor in itself. In fact, a geometric quantity which is a surro- gate of a specific flow condition is amenable for inclusion in a large-scale clinical trial and, once validated, it is directly usable as a clinical criterion. The recent development of medical imaging devices, such as rotational angiography (RA) [11], computed tomography (CT), and magnetic resonance (MR), has lead to the availability of large amounts of data on the 3-D morphology of the in vivo vasculature, for the investigation of these aspects for diagnostic and prognostic purposes [12]–[16]. Still, a quantitative anal- ysis of the relationship between vascular geometry and arterial physiopathology is made difficult by the large variability of real anatomies on one hand, and by the objective difficulty in re- trieving quantitative data from images in a robust, operator-in- dependent way on the other. In this paper, we present a framework devised for 3-D mod- eling and geometric characterization of vascular structures, readily available in the Vascular Modeling Toolkit (VMTK) [17], and we show how it can be employed for the charac- terization of cerebral aneurysms in relation to their parent vasculature. In particular, after a quick glance to image seg- mentation (Section II), we focus our attention on centerline calculation (Section III) and bifurcation identification and quantitative characterization (Section IV). Centerline of a vessel is in general a significant synthesis of basic features of a vessel (in terms of curvature, torsion, tortuosity), however, its definition from a general 3-D surface is not trivial. Some techniques for a robust computation of the centerline based on the concept of Voronoi diagram are presented here. As already pointed out, bifurcations are in general an interesting part of the vascular tree, quite often preferential site of atherosclerotic 0278-0062/$26.00 © 2009 IEEE