Blood Flow Velocity Field Estimation Via Spatial Regression With PDE Penalization

Abstract

We propose an innovative method for the accurate estimation of surfaces and spatial fields when prior knowledge of the phenomenon under study is available. The prior knowledge included in the model derives from physics, physiology, or mechanics of the problem at hand, and is formalized in terms of a partial differential equation governing the phenomenon behavior, as well as conditions that the phenomenon has to satisfy at the boundary of the problem domain. The proposed models exploit advanced scientific computing techniques and specifically make use of the finite element method. The estimators have a penalized regression form and the usual inferential tools are derived. Both the pointwise and the areal data frameworks are considered. The driving application concerns the estimation of the blood flow velocity field in a section of a carotid artery, using data provided by echo-color Doppler. This applied problem arises within a research project that aims at studying atherosclerosis pathogenesis. Supplementary materials for this article are available online.

• Finite elements
• Functional data analysis
• Object-oriented data analysis
• Penalized regression
• Spatial data analysis

Additional information

Notes on contributors

Laura Azzimonti

Laura Azzimonti (E-mail: [email protected]) is Post-Doctoral Fellow, Laura M. Sangalli (E-mail: [email protected]) is Assistant Professor, and Piercesare Secchi (E-mail: [email protected]) is Professor, MOX - Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy. Maurizio Domanin is Professor, U.O. di Chirurgia Vascolare Fondazione I.R.C.C.S. Ca’ Granda Ospedale Maggiore Policlinico, Milano, and Università degli Studi di Milano (E-mail: [email protected]). Fabio Nobile is Professor, Mathematics Institute of Computational Science and Engineering, École Polytechnique Fédérale de Lausanne, Switzerland (E-mail: [email protected]). The authors are grateful to James Ramsay for his advice on this work, to Christian Vergara - P.I. of the MACAREN@MOX project, to Elena Faggiano for the reconstruction of carotid geometries from MRI data, and to Silvia Romagnoli for the acquisition of eco-doppler data. The authors also thank the associate editor and three anonymous reviewers for their constructive comments. This work was supported by the program Dote Ricercatore Politecnico di Milano – Regione Lombardia, research project “Functional data analysis for life sciences,” and by MIUR Ministero dell’Istruzione dell’Università e della Ricerca, starting grant project FIRB Futuro in Ricerca “Advanced statistical and numerical methods for the analysis of high-dimensional functional data in life sciences and engineering,” http://mox.polimi.it/users/sangalli/firbSNAPLE.html, P.I.: Laura M. Sangalli.

Laura M. Sangalli

Laura Azzimonti (E-mail: [email protected]) is Post-Doctoral Fellow, Laura M. Sangalli (E-mail: [email protected]) is Assistant Professor, and Piercesare Secchi (E-mail: [email protected]) is Professor, MOX - Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy. Maurizio Domanin is Professor, U.O. di Chirurgia Vascolare Fondazione I.R.C.C.S. Ca’ Granda Ospedale Maggiore Policlinico, Milano, and Università degli Studi di Milano (E-mail: [email protected]). Fabio Nobile is Professor, Mathematics Institute of Computational Science and Engineering, École Polytechnique Fédérale de Lausanne, Switzerland (E-mail: [email protected]). The authors are grateful to James Ramsay for his advice on this work, to Christian Vergara - P.I. of the MACAREN@MOX project, to Elena Faggiano for the reconstruction of carotid geometries from MRI data, and to Silvia Romagnoli for the acquisition of eco-doppler data. The authors also thank the associate editor and three anonymous reviewers for their constructive comments. This work was supported by the program Dote Ricercatore Politecnico di Milano – Regione Lombardia, research project “Functional data analysis for life sciences,” and by MIUR Ministero dell’Istruzione dell’Università e della Ricerca, starting grant project FIRB Futuro in Ricerca “Advanced statistical and numerical methods for the analysis of high-dimensional functional data in life sciences and engineering,” http://mox.polimi.it/users/sangalli/firbSNAPLE.html, P.I.: Laura M. Sangalli.

Piercesare Secchi

Laura Azzimonti (E-mail: [email protected]) is Post-Doctoral Fellow, Laura M. Sangalli (E-mail: [email protected]) is Assistant Professor, and Piercesare Secchi (E-mail: [email protected]) is Professor, MOX - Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy. Maurizio Domanin is Professor, U.O. di Chirurgia Vascolare Fondazione I.R.C.C.S. Ca’ Granda Ospedale Maggiore Policlinico, Milano, and Università degli Studi di Milano (E-mail: [email protected]). Fabio Nobile is Professor, Mathematics Institute of Computational Science and Engineering, École Polytechnique Fédérale de Lausanne, Switzerland (E-mail: [email protected]). The authors are grateful to James Ramsay for his advice on this work, to Christian Vergara - P.I. of the MACAREN@MOX project, to Elena Faggiano for the reconstruction of carotid geometries from MRI data, and to Silvia Romagnoli for the acquisition of eco-doppler data. The authors also thank the associate editor and three anonymous reviewers for their constructive comments. This work was supported by the program Dote Ricercatore Politecnico di Milano – Regione Lombardia, research project “Functional data analysis for life sciences,” and by MIUR Ministero dell’Istruzione dell’Università e della Ricerca, starting grant project FIRB Futuro in Ricerca “Advanced statistical and numerical methods for the analysis of high-dimensional functional data in life sciences and engineering,” http://mox.polimi.it/users/sangalli/firbSNAPLE.html, P.I.: Laura M. Sangalli.

Maurizio Domanin

Laura Azzimonti (E-mail: [email protected]) is Post-Doctoral Fellow, Laura M. Sangalli (E-mail: [email protected]) is Assistant Professor, and Piercesare Secchi (E-mail: [email protected]) is Professor, MOX - Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy. Maurizio Domanin is Professor, U.O. di Chirurgia Vascolare Fondazione I.R.C.C.S. Ca’ Granda Ospedale Maggiore Policlinico, Milano, and Università degli Studi di Milano (E-mail: [email protected]). Fabio Nobile is Professor, Mathematics Institute of Computational Science and Engineering, École Polytechnique Fédérale de Lausanne, Switzerland (E-mail: [email protected]). The authors are grateful to James Ramsay for his advice on this work, to Christian Vergara - P.I. of the MACAREN@MOX project, to Elena Faggiano for the reconstruction of carotid geometries from MRI data, and to Silvia Romagnoli for the acquisition of eco-doppler data. The authors also thank the associate editor and three anonymous reviewers for their constructive comments. This work was supported by the program Dote Ricercatore Politecnico di Milano – Regione Lombardia, research project “Functional data analysis for life sciences,” and by MIUR Ministero dell’Istruzione dell’Università e della Ricerca, starting grant project FIRB Futuro in Ricerca “Advanced statistical and numerical methods for the analysis of high-dimensional functional data in life sciences and engineering,” http://mox.polimi.it/users/sangalli/firbSNAPLE.html, P.I.: Laura M. Sangalli.

Fabio Nobile

Laura Azzimonti (E-mail: [email protected]) is Post-Doctoral Fellow, Laura M. Sangalli (E-mail: [email protected]) is Assistant Professor, and Piercesare Secchi (E-mail: [email protected]) is Professor, MOX - Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy. Maurizio Domanin is Professor, U.O. di Chirurgia Vascolare Fondazione I.R.C.C.S. Ca’ Granda Ospedale Maggiore Policlinico, Milano, and Università degli Studi di Milano (E-mail: [email protected]). Fabio Nobile is Professor, Mathematics Institute of Computational Science and Engineering, École Polytechnique Fédérale de Lausanne, Switzerland (E-mail: [email protected]). The authors are grateful to James Ramsay for his advice on this work, to Christian Vergara - P.I. of the MACAREN@MOX project, to Elena Faggiano for the reconstruction of carotid geometries from MRI data, and to Silvia Romagnoli for the acquisition of eco-doppler data. The authors also thank the associate editor and three anonymous reviewers for their constructive comments. This work was supported by the program Dote Ricercatore Politecnico di Milano – Regione Lombardia, research project “Functional data analysis for life sciences,” and by MIUR Ministero dell’Istruzione dell’Università e della Ricerca, starting grant project FIRB Futuro in Ricerca “Advanced statistical and numerical methods for the analysis of high-dimensional functional data in life sciences and engineering,” http://mox.polimi.it/users/sangalli/firbSNAPLE.html, P.I.: Laura M. Sangalli.