Alessandro Veneziani

A Framework for Geometric Analysis of Vascular Structures: Application to Cerebral Aneurysms

There is well-documented evidence that vascular geometry has a major impact in blood flow dynamics and consequently in the development of vascular diseases, like atherosclerosis and cerebral aneurysmal disease. The study of vascular geometry 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 technologies is providing increasing amounts of data on the three-dimensional morphology of the in vivo vasculature, robust and objective 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 characterization of the geometric relationships between cerebral aneurysms and their parent vasculature.

Factorization methods for the numerical approximation of Navier-Stokes equations

We investigate a general approach for the numerical approximation of incompressible Navier-Stokes equations based on splitting the original problem into successive subproblems which are cheaper to solve. The splitting is obtained through an algebraic approximate factorization of the matrix arising from space and time discretization of the original equations. Several schemes based on approximate factorization are investigated. For some of these methods a formal analogy with well known time advancing schemes, such as the projection Chorin-Temams, can be pointed out. Features and limits of this analogy (that was earlier introduced in B. Perot, J. Comp. Phys. 108 (1993) 51-8) are addressed. Other, new methods can also be formulated starting from this approach: in particular, we introduce the so called Yosida method, which can be investigated in the framework of quasi-compressibility schemes. Numerical results illustrating the different performances of the different methods addressed are presented for a couple of test cases

Analysis of a Geometrical Multiscale Model Based on the Coupling of ODE and PDE for Blood Flow Simulations

The purpose of the present article is to provide a theoretical analysis heterogeneous model coupling ordinary differential equations (ODEs) and partial differential equations (PDEs), providing a local-in-time existence result for the solution. The heterogeneous problem will actually be split into subproblems. The solution of the original problem will be regarded as the solution of a suitable fixed point problem, based on the successive solution of the subproblems. Moreover, the authors study the role of matching conditions between the two submodels for the numerical simulation.

A Case Study in Exploratory Functional Data Analysis: Geometrical Features of the Internal Carotid Artery

This pilot study 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 analyses, we explore the AneuRisk dataset to highlight the relations between the geometric features of the internal carotid artery, expressed by its radius profile and centerline curvature, and the aneurysm location. After introducing a new similarity index for functional data, we eliminate ancillary variability of vessel radius and curvature profiles through an iterative registration procedure. We then reduce data dimension by means of functional principal components analysis. Last, a quadratic discriminant analysis of functional principal components scores allows us to discriminate patients with aneurysms in different districts.

Comparative finite element model analysis of ascending aortic flow in bicuspid and tricuspid aortic valve.

In bicuspid aortic valve (BAV) disease, the role of genetic and hemodynamic factors influencing ascending aortic pathology is controversial. To test the effect of BAV geometry on ascending aortic flow, a finite element analysis was undertaken. A surface model of aortic root and ascending aorta was obtained from magnetic resonance images of patients with BAV and tricuspid aortic valve using segmentation facilities of the image processing code Vascular Modeling Toolkit (developed at the Mario Negri Institute). Analytical models of bicuspid (antero-posterior [AP], type 1 and latero-lateral, type 2 commissures) and tricuspid orifices were mathematically defined and turned into a volumetric mesh of linear tetrahedra for computational fluid dynamics simulations. Numerical simulations were performed with the finite element code LifeV. Flow velocity fields were assessed for four levels: aortic annulus, sinus of Valsalva, sinotubular junction, and ascending aorta. Comparison of finite element analysis of bicuspid and tricuspid aortic valve showed different blood flow velocity pattern. Flow in bicuspid configurations showed asymmetrical distribution of velocity field toward the convexity of mid-ascending aorta returning symmetrical in distal ascending aorta. On the contrary, tricuspid flow was symmetrical in each aortic segment. Comparing type 1 BAV with type 2 BAV, more pronounced recirculation zones were noticed in the latter. Finally, we found that in both BAV configurations, maximum wall shear stress is highly localized at the convex portion of the mid-ascending aorta level. Comparison between models showed asymmetrical and higher flow velocity in bicuspid models, in particular in the AP configuration. Asymmetry was more pronounced at the aortic level known to be more exposed to aneurysm formation in bicuspid patients. This supports the hypothesis that hemodynamic factors may contribute to ascending aortic pathology in this subset of patients.

A model-based block-triangular preconditioner for the Bidomain system in electrocardiology

We introduce a preconditioner for the solution of the Bidomain system governing the propagation of action potentials in the myocardial tissue. The Bidomain model is a degenerate parabolic set of nonlinear reaction-diffusion equations. The nonlinear term describes the ion flux at the cellular level. The degenerate nature of the problem results in a severe ill conditioning of its discretization. Our preconditioning strategy is based on a suitable adaptation of the Monodomain model, a simplified version of the Bidomain one, which is by far simpler to solve, nevertheless is unable to capture significant features of the action potential propagation. The Monodomain preconditioner application to a non-symmetric formulation of the Bidomain system results at the algebraic level in a lower block-triangular preconditioner. We prove optimality of the preconditioner with respect to the mesh size, and corroborate our theoretical results with 3D numerical simulations both on idealized and real ventricle geometries.

A Variational Data Assimilation Procedure for the Incompressible Navier-Stokes Equations in Hemodynamics

We propose a data assimilation (DA) technique for including noisy measurements of the velocity field into the simulation of the Navier-Stokes equations (NSE) driven by hemodynamics applications. The technique is formulated as an inverse problem where we use a Discretize-then-Optimize approach to minimize the misfit between the recovered velocity field and the data, subject to the incompressible NSE. The DA procedure for this nonlinear problem is a combination of two approaches: the Newton method for the NSE and the DA procedure we designed and tested for the linearized problem. We discuss conditions on the location of velocity measurements that guarantee the well-posedness of the minimization process for the linearized problem. Numerical results, with both noise-free and noisy data, certify the theoretical analysis. Moreover, we consider 2D non-trivial geometries and 3D axisymmetric geometries. Also, we study the impact of noise on non-primitive variables of medical interest.

A Variational Approach for Estimating the Compliance of the Cardiovascular Tissue: An Inverse Fluid-Structure Interaction Problem

Estimation of the stiffness of a biological soft tissue is useful for the detection of pathologies such as tumors or atherosclerotic plaques. Elastography is a method based on the comparison between two images before and after a forced deformation of the tissue of interest. An inverse elasticity problem is then solved for Youngs modulus estimation. In the case of arteries, no forced deformation is required, since vessels naturally move under the action of blood. Youngs modulus can therefore be estimated by solving a coupled inverse fluid-structure interaction problem. In this paper we focus on the mathematical properties of this problem and its numerical solution. We give some well posedness analysis and some preliminary results based on a synthetic data set, i.e., test cases where the exact Youngs modulus is known and the displacement dataset is numerically generated by solving a forward fluid-structure interaction problem. We address the problem of the presence of the noise in the measured displacement and of the proper sampling frequency for obtaining reliable estimates.

Geometry of the internal carotid artery and recurrent patterns in location, orientation, and rupture status of lateral aneurysms: an image-based computational study

BACKGROUND:Intracranial aneurysm development and rupture may be associated to the morphology of the parent vessel. OBJECTIVE:To quantitatively characterize the geometry of the internal carotid artery (ICA) in relation to the location and orientation of lateral aneurysms and to identify recurrent patterns associated with their rupture status. METHODS:The geometry of 54 ICAs hosting lateral aneurysms was analyzed by means of computational geometry techniques. The ICA was split into individual bends, and the bend hosting the aneurysm was described in terms of curvature, torsion, length, and radius. Aneurysm position and orientation with respect to the parent vessel and specifically the hosting bend were characterized, as well as angles between the portions of the parent artery immediately upstream of and downstream from the aneurysm and the aneurysm ostium. Differences in geometric parameters with respect to rupture status and their performance as classifiers were evaluated. RESULTS:ICA bends hosting ruptured aneurysms were shorter with a smaller radius, lower maximum curvature, and lower proximal torsion compared with those hosting unruptured lesions. Ruptured aneurysms occurred in more distal portions of the ICA, along the outer wall of the vessel, and closer to the curvature peak within the hosting bend than unruptured ones. The proximal portions of ICAs hosting ruptured aneurysms approached the ostium region at a smaller angle. CONCLUSION:Geometric factors relative to the ICA were associated with the distribution of aneurysms and their rupture status. The present work has potential implications in the quest for hemodynamic factors contributing to the development, progression, and rupture of intracranial aneurysms.

Automatic Neck Plane Detection and 3D Geometric Characterization of Aneurysmal Sacs

Geometric indices defined on intracranial aneurysms have been widely used in rupture risk assessment and surgical planning. However, most indices employed in clinical settings are currently evaluated based on two-dimensional images that inevitably fail to capture the three-dimensional nature of complex aneurysmal shapes. In addition, since measurements are performed manually, they can suffer from poor inter and intra operator repeatability. The purpose of the current work is to introduce objective and robust techniques for the 3D characterization of intracranial aneurysms, while preserving a close connection to the way aneurysms are currently characterized in clinical settings. Techniques for automatically identifying the neck plane, key aneurysm dimensions, shape factors, and orientations relative to the parent vessel are demonstrated in a population of 15 sidewall and 15 terminal aneurysms whose surface has been obtained by two trained operators using both level-set segmentation and thresholding, the latter reflecting typical clinical practice. Automatically-identified neck planes are shown to be in concordance with those manually positioned by an expert neurosurgeon, and automatically-derived geometric indices are shown to be largely insensitive to segmentation method or operator. By capturing the 3D nature of aneurysmal sacs and by minimizing observer variability, our approach allows large retrospective and prospective studies on aneurysm geometric risk factors to be performed using routinely acquired clinical images.