Applications of variational data assimilation in computational hemodynamics

The development of new technologies for acquiring measures and images in order to investigate cardiovascular diseases raises new challenges in scientific computing. These data can be in fact merged with the numerical simulations for improving the accuracy and reliability of the computational tools. Assimilation of measured data and numerical models is well established in meteorology, whilst it is relatively new in computational hemodynamics. Different approaches are possible for the mathematical setting of this problem. Among them, we follow here a variational formulation, based on the minimization of the mismatch between data and numerical results by acting on a suitable set of control variables. Several modeling and methodological problems related to this strategy are open, such as the analysis of the impact of the noise affecting the data, and the design of effective numerical solvers. In this chapter we present three examples where a mathematically sound (variational) assimilation of data can significantly improve the reliability of the numerical models. Accuracy and reliability of computational models are increasingly important features in view of the progressive adoption of numerical tools in the design of new therapies and, more in general, in the decision making process of medical doctors.

[1]  R. Löhner,et al.  Hemodynamic analysis of intracranial aneurysms with moving parent arteries: Basilar tip aneurysms , 2010, International journal for numerical methods in biomedical engineering.

[2]  D. Xiu,et al.  Modeling uncertainty in flow simulations via generalized polynomial chaos , 2003 .

[3]  Max A. Viergever,et al.  A survey of medical image registration , 1998, Medical Image Anal..

[4]  Alessandro Veneziani,et al.  Methods for assimilating blood velocity measures in hemodynamics simulations: Preliminary results , 2010, ICCS.

[5]  Kenichi Funamoto,et al.  Numerical Validation of MR-Measurement-Integrated Simulation of Blood Flow in a Cerebral Aneurysm , 2009, Annals of Biomedical Engineering.

[6]  Joan Alexis Glaunès,et al.  Surface Matching via Currents , 2005, IPMI.

[7]  Charles Taylor,et al.  EXPERIMENTAL AND COMPUTATIONAL METHODS IN CARDIOVASCULAR FLUID MECHANICS , 2004 .

[8]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[9]  J. Hoffman,et al.  The incidence of congenital heart disease. , 2002, Journal of the American College of Cardiology.

[10]  Arthur Vidard,et al.  Assimilation of image sequences in numerical models , 2010 .

[11]  C. Ward.  Clinical significance of the bicuspid aortic valve , 2000, Heart.

[12]  Armando Manduca,et al.  Visualization of Tissue Elasticity by Magnetic Resonance Elastography , 1996, VBC.

[13]  H. Engl,et al.  Regularization of Inverse Problems , 1996 .

[14]  Dominique Chapelle,et al.  Robust filtering for joint state parameter estimation for distributed mechanical systems , 2008 .

[15]  R. Dwight.  Bayesian inference for data assimilation using least-squares finite element methods , 2010 .

[16]  Christian Vergara,et al.  A New Approach to Numerical Solution of Defective Boundary Value Problems in Incompressible Fluid Dynamics , 2008, SIAM J. Numer. Anal..

[17]  D. Chapelle,et al.  Reduced-order Unscented Kalman Filtering with application to parameter identification in large-dimensional systems , 2011 .

[18]  S. Griffis.  EDITOR , 1997, Journal of Navigation.

[19]  Enrique Zuazua,et al.  CONTROLLABILITY OF PARTIAL DIFFERENTIAL EQUATIONS AND ITS SEMI-DISCRETE APPROXIMATIONS , 2002 .

[20]  Alessandro Veneziani,et al.  A Variational Data Assimilation Procedure for the Incompressible Navier-Stokes Equations in Hemodynamics , 2011, Journal of Scientific Computing.

[21]  Alfio Quarteroni,et al.  Complex Systems in Biomedicine , 2006 .

[22]  Setsuo Takatani,et al.  A noninvasive tissue reflectance oximeter , 2006, Annals of Biomedical Engineering.

[23]  Jacques-Louis Lions,et al.  Remarks on approximate controllability , 1992 .

[24]  David A. Steinman,et al.  From image data to computational domains , 2009 .

[25]  Jean-Frédéric Gerbeau,et al.  Algorithms for fluid-structure interaction problems , 2009 .

[26]  Philippe Courtier,et al.  Unified Notation for Data Assimilation : Operational, Sequential and Variational , 1997 .

[27]  Assad A Oberai,et al.  Elastic modulus imaging: some exact solutions of the compressible elastography inverse problem , 2007, Physics in medicine and biology.

[28]  Lisa M. Brown,et al.  A survey of image registration techniques , 1992, CSUR.

[29]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[30]  D K Smith,et al.  Numerical Optimization , 2001, J. Oper. Res. Soc..

[31]  Wing Kam Liu,et al.  Lagrangian-Eulerian finite element formulation for incompressible viscous flows☆ , 1981 .

[32]  D. Xiu,et al.  Stochastic Modeling of Flow-Structure Interactions Using Generalized Polynomial Chaos , 2002 .

[33]  Christian Vergara,et al.  Flow rate boundary problems for an incompressible fluid in deformable domains: Formulations and solution methods , 2010 .

[34]  George Em Karniadakis,et al.  Simulation of the human intracranial arterial tree. , 2009, Philosophical transactions. Series A, Mathematical, physical, and engineering sciences.

[35]  J. L. Lions,et al.  Exact Controllability for Distributed Systems. Some Trends and Some Problems , 1991 .

[36]  Thomas A. Manteuffel,et al.  Weighted least-squares finite elements based on particle imaging velocimetry data , 2010, J. Comput. Phys..

[37]  J. Modersitzki,et al.  Ill-posed medicine—an introduction to image registration , 2008 .

[38]  R S Reneman,et al.  A numerical and experimental analysis of the flow field in a two-dimensional model of the human carotid artery bifurcation. , 1987, Journal of biomechanics.

[39]  E. Haber,et al.  Image-Based CFD Simulation of a Compliant Blood Vessel , 2010 .

[40]  A. Quarteroni.  Cardiovascular mathematics , 2000 .

[41]  P. Hansen.  Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion , 1987 .

[42]  Frederica Darema.  Dynamic Data Driven Applications Systems (DDDAS) - A Transformative Paradigm , 2008, ICCS.

[43]  Li Yuan-yuan.  A SURVEY OF MEDICAL IMAGE REGISTRATION , 2006 .

[44]  K Perktold,et al.  Numerical simulation of pulsatile flow in a carotid bifurcation model. , 1986, Journal of biomedical engineering.

[45]  Paul Suetens,et al.  Fast Soft Tissue Deformation with Tetrahedral Mass Spring Model for Maxillofacial Surgery Planning Systems , 2004, MICCAI.

[46]  C. Taylor,et al.  Predictive medicine: computational techniques in therapeutic decision-making. , 1999, Computer aided surgery : official journal of the International Society for Computer Aided Surgery.

[47]  R. Glowinski,et al.  Exact and approximate controllability for distributed parameter systems , 1995, Acta Numerica.

[48]  E. Somersalo,et al.  Statistical and computational inverse problems , 2004 .

[49]  Jan Flusser,et al.  Image registration methods: a survey , 2003, Image Vis. Comput..

[50]  F. NOBILE,et al.  An Effective Fluid-Structure Interaction Formulation for Vascular Dynamics by Generalized Robin Conditions , 2008, SIAM J. Sci. Comput..

[51]  Raul Tempone,et al.  Analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients , 2009 .

[52]  Paul Suetens,et al.  Tetrahedral Mass Spring Model for Fast Soft Tissue Deformation , 2003, IS4TH.

[53]  Jingfeng Jiang,et al.  Linear and nonlinear elasticity imaging of soft tissue in vivo: demonstration of feasibility , 2009, Physics in medicine and biology.

[54]  David P. Dobkin,et al.  The quickhull algorithm for convex hulls , 1996, TOMS.

[55]  D. Ku,et al.  Pulsatile Flow and Atherosclerosis in the Human Carotid Bifurcation: Positive Correlation between Plaque Location and Low and Oscillating Shear Stress , 1985, Arteriosclerosis.

[56]  Max Gunzburger,et al.  Perspectives in flow control and optimization , 1987 .

[57]  Luca Gerardo-Giorda,et al.  Analysis and Optimization of Robin-Robin Partitioned Procedures in Fluid-Structure Interaction Problems , 2010, SIAM J. Numer. Anal..

[58]  R. Glowinski,et al.  A numerical approach to the exact boundary controllability of the wave equation (I) Dirichlet controls: Description of the numerical methods , 1990 .

[59]  Vivekanand Allada,et al.  Frequency of aortic root dilation in children with a bicuspid aortic valve. , 2004, The American journal of cardiology.

[60]  Ionel M. Navon,et al.  Data Assimilation for Geophysical Fluids , 2009 .

[61]  Christian Vergara,et al.  A Variational Approach for Estimating the Compliance of the Cardiovascular Tissue: An Inverse Fluid-Structure Interaction Problem , 2011, SIAM J. Sci. Comput..

[62]  P. Tallec,et al.  Filtering for distributed mechanical systems using position measurements: perspectives in medical imaging , 2009 .

[63]  Vincent J. Ervin,et al.  Numerical Approximation of a Quasi-Newtonian Stokes Flow Problem with Defective Boundary Conditions , 2007, SIAM J. Numer. Anal..

[64]  Fabio Nobile,et al.  Added-mass effect in the design of partitioned algorithms for fluid-structure problems , 2005 .

[65]  M. Thubrikar,et al.  The congenitally bicuspid aortic valve: how does it function? Why does it fail? , 2004, The Annals of thoracic surgery.

[66]  D. Xiu,et al.  Modeling Uncertainty in Steady State Diffusion Problems via Generalized Polynomial Chaos , 2002 .

[67]  Allan R. Robinson,et al.  OVERVIEW OF DATA ASSIMILATION , 2007 .

[68]  A. Quarteroni,et al.  Numerical Approximation of Partial Differential Equations , 2008 .

[69]  David L. Hayes,et al.  Mayo Clinic Practice of Cardiology , 1996 .

[70]  P. Tallec,et al.  Joint state and parameter estimation for distributed mechanical systems , 2008 .

[71]  Trevor Q. Robbie,et al.  Hemodynamic predictors of aortic dilatation in bicuspid aortic valve by velocity-encoded cardiovascular magnetic resonance , 2010, Journal of cardiovascular magnetic resonance : official journal of the Society for Cardiovascular Magnetic Resonance.

[72]  Guang-Zhong Yang,et al.  MR Image-Based Geometric and Hemodynamic Investigation of the Right Coronary Artery with Dynamic Vessel Motion , 2010, Annals of Biomedical Engineering.

[73]  R. Wulandana,et al.  An inelastic multi-mechanism constitutive equation for cerebral arterial tissue , 2005, Biomechanics and modeling in mechanobiology.

[74]  J L Lions,et al.  On the controllability of distributed systems. , 1997, Proceedings of the National Academy of Sciences of the United States of America.

[75]  Howard C. Elman,et al.  ASSESSMENT OF COLLOCATION AND GALERKIN APPROACHES TO LINEAR DIFFUSION EQUATIONS WITH RANDOM DATA , 2011 .

[76]  Mario Bertero,et al.  Inverse problems in biomedical imaging: modeling and methods of solution , 2006 .

[77]  Maxime Sermesant,et al.  Cardiac Function Estimation from MRI Using a Heart Model and Data Assimilation: Advances and Difficulties , 2005, FIMH.

[78]  E. Zuazua,et al.  Exact controllability for semilinear distributed systems , 1988, Proceedings of the 27th IEEE Conference on Decision and Control.

[79]  Yusheng Feng,et al.  Theory and methodology for estimation and control of errors due to modeling, approximation, and uncertainty , 2005 .

[80]  L. Antiga,et al.  Comparative finite element model analysis of ascending aortic flow in bicuspid and tricuspid aortic valve. , 2010, Artificial organs.

[81]  E. Topol.  Textbook of Cardiovascular Medicine , 1997 .

[82]  Sevan Goenezen,et al.  Inverse Problems , 2008 .

[83]  Robert W. Walters,et al.  Uncertainty analysis for fluid mechanics with applications , 2002 .

[84]  Greg Welch,et al.  Welch & Bishop , An Introduction to the Kalman Filter 2 1 The Discrete Kalman Filter In 1960 , 1994 .

[85]  J. Lions.  Optimal Control of Systems Governed by Partial Differential Equations , 1971 .

[86]  Daniel Rueckert,et al.  Fast Spatio-temporal Free-Form Registration of Cardiac MR Image Sequences , 2004, FIMH.

[87]  A. Peirce.  Computer Methods in Applied Mechanics and Engineering , 2010 .

[88]  Thomas Bonesky.  Morozov's discrepancy principle and Tikhonov-type functionals , 2008 .

[89]  Ahmet Erdemir,et al.  Considerations for reporting finite element analysis studies in biomechanics. , 2012, Journal of biomechanics.

[90]  R. Ogden,et al.  A New Constitutive Framework for Arterial Wall Mechanics and a Comparative Study of Material Models , 2000 .

[91]  Tiziano Passerini,et al.  Shear Stress and Angiotensin II in the Development and Localization of Abdominal Aortic Aneurysms , 2009 .

[92]  Frederica Darema.  Dynamic Data Driven Applications Systems (DDDAS) , 2012 .