Skip to main content
SpringerLink
Log in

Data Assimilation in Cardiovascular Fluid–Structure Interaction Problems: An Introduction

  • Chapter
  • First Online:
Fluid-Structure Interaction and Biomedical Applications

Part of the book series: Advances in Mathematical Fluid Mechanics ((AMFM))

Abstract

Numerical methods for incompressible fluid dynamics have recently received a strong impulse from the applications to the cardiovascular system. In particular, fluid–structure interaction methods have been extensively investigated in view of an accurate and possibly fast simulation of blood flow in arteries and veins. This has been strongly motivated by the progressive interest in using numerical tools not only for understanding the general physiology and pathology of the vascular system. The opportunity offered by medical images properly preprocessed and elaborated to simulate blood flow in real patients highlighted the potential impact of scientific computing on the clinical practice. Therefore, in silico experiments are currently extensively used in bioengineering for completing (and sometimes driving) more traditional in vivo and in vitro investigations. Parallel to the development of numerical models, the need for quantitative analysis for diagnostic purposes has strongly stimulated the design of new methods and instruments for measurements and imaging. Thanks to these developments, a huge amount of data is nowadays available. Data Assimilation is the accurate merging of measures (including images) and numerical simulations for a mathematically sound integration of different sources of information. The outcome of this process includes both the patient-specific measures and the general principles underlying the development of mathematical models. In this way, simulations are adapted to the availability of individual data and are therefore supposed to be more reliable; measures are correspondingly filtered by the mathematical models assumed to describe the underlying phenomena, resulting in a (hopefully) significant reduction of the noise.

This chapter provides an introduction to methods for data assimilation, mostly developed in fields like meteorology, applied to computational hemodynamics. We focus mainly on two of them: methods based on stochastic arguments (Kalman filtering) and variational methods. We also address some examples that have been approached with different techniques, in particular the estimation of vascular compliance from displacement measures.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 119.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 159.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    We remind that the wall shear stress (WSS) is a quantity of great relevance in biomedical applications for its correlation with pathologies such as atherosclerosis—see, e.g., [14].

  2. 2.

    Precise definitions of average and variance of a Gaussian variable will be given later on.

  3. 3.

    The choice of Gaussian distribution for white noise is reasonable, but arbitrary. We could have considered other distributions for zero-mean, uncorrelated components.

  4. 4.

    The third problem addressed in the Introduction, the identification of the system will be considered later on.

  5. 5.

    A similar problem has been investigated as a simplified model of superconductivity in [69].

  6. 6.

    This could be done also with unilateral constraints \(\|\boldsymbol{\alpha }\|\leq \) max-cost-allowed.

  7. 7.

    We remind that we assumed b to be divergence free.

  8. 8.

    Here we used the Lagrange multiplier χ 2 to prescribe the Dirichlet homogeneous boundary condition. Often, such condition is prescribed without using Lagrange multipliers but requiring directly that u and χ 1 vanish on the boundary.

  9. 9.

    This can be problematic in a clinical context, where patient-specific geometries differ one from the other and the snapshot computation is not trivially recycled. Anatomical atlas mapping ideal to real geometries are required.

  10. 10.

    Notice that we use the word “sites” for the location of measurements, as opposed to the word “nodes” for points where velocities are computed. In general sites and nodes are different, but we do not exclude that the intersection of sites set and nodes set in non-empty.

  11. 11.

    We define the signal to noise ratio as the ratio between the maximum of the absolute value of the signal and the standard deviation of the noise

  12. 12.

    This assumption may be questionable for arteries close to the heart (like the aortic arch), however it is in general quite acceptable.

  13. 13.

    See (6.37), and note that here the adjoint variable is denoted with \(\boldsymbol{\chi }\) as in this context ρ is used for the density.

References

  1. H. Abou-Kandil, G. Freiling, V. Ionescu, G. Jank, Matrix Riccati Equations: In Control and Systems Theory (Springer, Berlin, 2003)

    Book  Google Scholar 

  2. H.T. Banks, A Functional Analysis Framework for Modeling, Estimation and Control in Science and Engineering (Taylor & Francis, London, 2012)

    Book  Google Scholar 

  3. H.T. Banks, K. Kunisch, Estimation Techniques for Distributed Parameter Systems. (Birkhauser, Boston, 1989)

    Google Scholar 

  4. P.E. Barbone, A.A. Oberai, Elastic modulus imaging: some exact solutions of the compressible elastography inverse problem. Phys. Med. Biol. 52, 1577 (2007)

    Article  Google Scholar 

  5. P.E. Barbone, C.E. Rivas, I. Harari, U. Albocher, A.A. Oberai, Y. Zhang, Adjoint-weighted variational formulation for the direct solution of inverse problems of general linear elasticity with full interior data. Int. J. Numer. Methods Eng. 81(13), 1713–1736 (2010)

    MathSciNet  MATH  Google Scholar 

  6. C. Bertoglio, P. Moireau, J.-F. Gerbeau, Sequential parameter estimation for fluid–structure problems: application to hemodynamics. Int. J. Numer. Methods Biomed. Eng. 28(4), 434–455 (2012)

    Article  MathSciNet  Google Scholar 

  7. L. Biegler, G. Biros, O. Ghattas, M. Heinkenschloss, D. Keyes, B. Mallick, L. Tenorio, B. Waanders, K. Willcox, Y. Marzouk, Large-Scale Inverse Problems and Quantification of Uncertainty. Wiley Series in Computational Statistics (Wiley, Chichester, 2011)

    Google Scholar 

  8. J. Blum, F.-X. Le Dimet, I. Michael Navon, Data assimilation for geophysical fluids, in Handbook of Numerical Analysis, vol. 14, ed. by P.G. Ciarlet (Elsevier, Amsterdam, 2009), pp. 385–441

    Google Scholar 

  9. P.B. Bochev, Analysis of least-squares finite element methods for the navier-stokes equations. SIAM J. Numer. Anal. 34, 1817–1844 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  10. P.B. Bochev, M.D. Gunzburger, Least-Squares Finite Element Methods (Springer, Berlin, 2009)

    MATH  Google Scholar 

  11. P.T. Boggs, J.W. Tolle, Sequential quadratic programming. Acta Numer. 4, 1–51 (1995)

    Article  MathSciNet  Google Scholar 

  12. D. Calvetti, E. Somersalo, An Introduction to Bayesian Scientific Computing: Ten Lectures on Subjective Computing. Surveys and Tutorials in the Applied Mathematical Sciences (Springer Science+Business Media, New York, 2007)

    Google Scholar 

  13. I. Campbell, W. Robert Taylor, Flow and atherosclerosis, in Hemodynamics and Mechanobiology of Endothelium (World Scientific, Hackensack, 2010)

    Google Scholar 

  14. D. Chapelle, A. Gariah, J. Sainte-Marie, Galerkin approximation with proper orthogonal decomposition: new error estimates and illustrative examples. ESAIM: Math. Model. Numer. Anal. 46, 731–757 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. M. D’Elia, A. Veneziani, Uncertainty quantification for data assimilation in a steady incompressible navier-stokes problem. ESAIM: Math. Model. Numer. Anal. 47, 1037–1057 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. M. D’Elia, L. Mirabella, T. Passerini, M. Perego, M. Piccinelli, C. Vergara, A. Veneziani, Some applications of variational data assimilation in computational hemodynamics, in Modelling of Physiological Flows, ed. by D. Ambrosi, A. Quarteroni, G. Rozza. MS&A Series (Springer, Berlin, 2011), pp. 363–394

    Google Scholar 

  17. M. D’Elia, M. Perego, A. Veneziani, A variational data assimilation procedure for the incompressible navier stokes equations in hemodynamics. J. Sci. Comput. 52(2), 340–359 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. H. Delingette, M. Sermesant, R. Cabrera-Lozoya, C. Tobon-Gomez, P. Moireau, R.M. Figueras i Ventura, K. Lekadir, A. Hernandez, M. Garreau, E. Donal, C. Leclercq, S.G. Duckett, K. Rhode, C.A. Rinaldi, A.F. Frangi, R. Razavi, D. Chapelle, N. Ayache, S. Marchesseau, Personalization of a cardiac electromechanical model using reduced order unscented kalman filtering from regional volumes. Med. Image Anal. 17, 816–829 (2013)

    Google Scholar 

  19. J. Donea, S. Giuliani, J.P. Halleux, An arbitrary lagrangian-eulerian finite element method for transient dynamic fluid-structure interactions. Comput. Methods Appl. Mech. Eng. 33(1–3), 689–723 (1982)

    Article  MATH  Google Scholar 

  20. R.P. Dwight, Bayesian inference for data assimilation using least-squares finite element methods, in IOP Conference Series: Materials Science and Engineering, vol. 10 (IOP Publishing, Bristol, 2010), p. 012224

    Google Scholar 

  21. B. Einarsson, Accuracy and Reliability in Scientific Computing, vol. 18 (Society for Industrial Mathematics, Philadelphia, 2005)

    Book  MATH  Google Scholar 

  22. H.W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems. Mathematics and its Applications (Springer, Berlin, 1996)

    Google Scholar 

  23. L. Formaggia, A. Veneziani, C. Vergara, A new approach to numerical solution of defective boundary value problems in incompressible fluid dynamics. SIAM J. Numer. Anal. 46(6), 2769–2794 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. L. Formaggia, A. Quarteroni, A. Veneziani, Cardiovascular Mathematics: Modeling and Simulation of the Circulatory System, vol. 1 (Springer, Berlin, 2009)

    Book  Google Scholar 

  25. L. Formaggia, A. Veneziani, C. Vergara, Flow rate boundary problems for an incompressible fluid in deformable domains: formulations and solution methods. Comput. Methods Appl. Mech. Eng. 9(12), 677–688 (2010)

    Article  MathSciNet  Google Scholar 

  26. P.C. Franzone, L.F. Pavarino, A Parallel Solver for Reaction-Diffusion Systems in Computational Electrocardiology, Math. Model. Methods in Appl. Sci. 14(6), 883–911 (2004) doi:10.1142/s0218202504003489

    Article  MATH  Google Scholar 

  27. B. Fristedt, N. Jain, N.V. Krylov, Filtering and Prediction: A Primer, STML vol. 38, AMS, Providence, RI (2007)

    Google Scholar 

  28. K. Funamoto, T. Hayase, Reproduction of pressure field in ultrasonic-measurement-integrated simulation of blood flow. Int. J. Numer. Methods Biomed. Eng. 29(7), 726–740 (2013)

    Article  MathSciNet  Google Scholar 

  29. G.P. Galdi, A.M. Robertson, R. Rannacher, S. Turek, Hemodynamical Flows: Modeling, Analysis and Simulation. Oberwolfach Seminar Series, vol. 37, Birkhauser Verlag AG, Basel (2008)

    Google Scholar 

  30. J.F. Gerbeau, D. Lombardi, Reduced-order modeling based on approximated lax pairs. Technical Report RR 8137, INRIA. arXiv:1211.4153v1 (November 2012)

    Google Scholar 

  31. E. Gilboa, P.S. La Rosa, A. Nehorai, Estimating electrical conductivity tensors of biological tissues using microelectrode arrays, in Engineering in Medicine and Biology Society (EMBC), 2012 Annual International Conference of the IEEE (2012), pp. 1040–1044

    Google Scholar 

  32. R. Glowinski, J.L. Lions, Exact and approximate controllability for distributed parameter systems. Acta Numer. 3, 269–378 (1994)

    Article  MathSciNet  Google Scholar 

  33. R. Glowinski, J.L. Lions, Exact and approximate controllability for distributed parameter systems. Acta Numer. 4, 159–328 (1995)

    Article  MathSciNet  Google Scholar 

  34. R. Glowinski, J.-L. Lions, J. He, Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach (Encyclopedia of Mathematics and its Applications), 1st edn. (Cambridge University Press, New York, 2008)

    Book  Google Scholar 

  35. G.H. Golub, C.F. Van Loan, Matrix Computations, vol. 3 (Johns Hopkins University Press, Baltimore, 1996)

    MATH  Google Scholar 

  36. L.S. Graham, D. Kilpatrick, Estimation of the bidomain conductivity parameters of cardiac tissue from extracellular potential distributions initiated by point stimulation. Ann. Biomed. Eng. 38(12), 3630–3648 (2010)

    Article  Google Scholar 

  37. M.D. Gunzburger, Perspectives in Flow Control and Optimization, vol. 5 (Society for Industrial Mathematics, Philadelphia, 2003)

    MATH  Google Scholar 

  38. P.C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems. SIAM Monographs on Mathematical Modeling and Computation (Society for Industrial and Applied Mathematics, Philadelphia, 1998)

    Google Scholar 

  39. J.J. Heys, T.A. Manteuffel, S.F. McCormick, M. Milano, J. Westerdale, M. Belohlavek, Weighted least-squares finite elements based on particle imaging velocimetry data. J. Comput. Phys. 229(1), 107–118 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  40. K. Hinsch, 3-Dimensional particle velocimetry. Meas. Sci. Technol. 6, 742–753 (1995)

    Article  Google Scholar 

  41. T.J.R. Hughes, W.K. Liu, T.K. Zimmermann, Lagrangian-eulerian finite element formulation for incompressible viscous flows. Comput. Methods Appl. Mech. Eng. 29(3), 329–349 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  42. J. Humpherys, P. Redd, J. West, A fresh look at the kalman filter. SIAM Rev. 54(4), 801–823 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  43. S.J. Julier, J.K. Uhlmann, A new extension of the kalman filter to nonlinear systems, in Proceedings of SPIE 3068, Signal Processing, Sensor Fusion, and Target Recognition VI, 182 (1997), pp. 182–193

    Google Scholar 

  44. S.J. Julier, J.K. Uhlmann, Unscented filtering and nonlinear estimation. Proc. IEEE 92(3), 401–422 (2004)

    Article  Google Scholar 

  45. T. Kailath, Lectures Notes on Wiener and Kalman Filtering (Springer, Berlin, 1981)

    Book  Google Scholar 

  46. J. Kaipio, E. Somersalo, Statistical and Computational Inverse Problems (Applied Mathematical Sciences), vol. 160, 1st edn. (Springer, Berlin, 2004)

    Google Scholar 

  47. R.E. Kalman, A new approach to linear filtering and prediction problems. Trans. ASME J. Basic Eng. 82, 35–45 (1960)

    Article  Google Scholar 

  48. K. Kunisch, M. Wagner, Optimal control of the bidomain system (iii): existence of minimizers and first-order optimality conditions. ESAIM: Math. Model. Numer. Anal. 47, 1077–1106 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  49. K. Kunisch, M. Wagner, Optimal control of the bidomain system (ii): uniqueness and regularity theorems for weak solutions. Annali di Matematica Pura ed Applicata 192, 1–36 (2012)

    Google Scholar 

  50. P. Lancaster, L. Rodman, Algebraic Riccati Equations (Oxford Science Publications, New York, 1995)

    MATH  Google Scholar 

  51. T. Lassila, A. Manzoni, A. Quarteroni, G. Rozza, A reduced computational and geometrical framework for inverse problems in hemodynamics. Int. J. Numer. Methods Biomed. Eng. 29(7), 741–776 (2013)

    Article  MathSciNet  Google Scholar 

  52. J. Modersitzki, FAIR: Flexible Algorithms for Image Registration. Fundamentals of Algorithms (Society for Industrial and Applied Mathematics, Philadelphia, 2009)

    Google Scholar 

  53. P. Moireau, D. Chapelle, Reduced-order unscented kalman filtering with application to parameter identification in large-dimensional systems. ESAIM: Control Optim. Calc. Var. 17(02), 380–405 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  54. A.M. Mood, F.A. Graybill, D.C. Boes, Introduction to the Theory of Statistics (McGraw-Hill, New York, 1974)

    MATH  Google Scholar 

  55. C. Nagaiah, K. Kunisch, G. Plank, Numerical solutions for optimal control of monodomain equations. PAMM 9(1), 609–610 (2009)

    Article  Google Scholar 

  56. F. Nobile, C. Vergara, An effective fluid-structure interaction formulation for vascular dynamics by generalized Robin conditions. SIAM J. Sci. Comput. 30(2), 731–763 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  57. J. Nocedal, S. Wright, Numerical Optimization (Springer, Berlin, 2000)

    Google Scholar 

  58. M. Perego, A. Veneziani, C. Vergara, A variational approach for estimating the compliance of the cardiovascular tissue: an inverse fluid-structure interaction problem. SIAM J. Sci. Comput. 33(3), 1181–1211 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  59. K.B. Petersen, M.S. Pedersen, The matrix cookbook. Technical report, http://matrixcookbook.com (2008)

  60. M. Piccinelli, L. Mirabella, T. Passerini, E. Haber, A. Veneziani, 4d image-based cfd simulation of a compliant blood vessel. Technical report, Technical Report TR-2010-27, Department of Mathematics & CS, Emory University, www.mathcs.emory.edu (2010)

  61. A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics. Texts in Applied Mathematics Series (Springer GmbH, Berlin, 2000)

    Google Scholar 

  62. A. Quarteroni, L. Formaggia, A. Veneziani, Complex Systems in Biomedicine (Springer, Berlin, 2007)

    Google Scholar 

  63. G. Rozza, K. Veroy, On the stability of the reduced basis method for stokes equations in parametrized domains. Comput. Methods Appl. Mech. Eng. 196(7), 1244–1260 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  64. G. Rozza, D.B.P. Huynh, A.T. Patera, Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Methods Eng. 15(3), 229–275 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  65. S. Salsa, Partial Differential Equations in Action: From Modelling to Theory (Springer, Berlin, 2008)

    Google Scholar 

  66. O. Scherzer, The use of morozov’s discrepancy principle for tikhonov regularization for solving nonlinear ill-posed problems. Computing 51(1), 45–60 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  67. R. Todling, Estimation theory and foundations of atmospheric data assimilation. DAO Office Note 1:1999 (1999)

    Google Scholar 

  68. F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods, and Applications, vol. 112 (American Mathematical Society, Providence, 2010)

    Google Scholar 

  69. K. Urban, A.T. Patera, A new error bound for reduced basis approximation of parabolic partial differential equations. C. R. Math. 350(3–4), 203–207 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  70. A. Veneziani, C. Vergara, Inverse problems in cardiovascular mathematics: toward patient-specific data assimilation and optimization. Int. J. Numer. Methods Biomed. Eng. 29(7), 723/725 (2013). Editorial of the special issue “Inverse Problems in Cardiovascular Mathematics”

    Google Scholar 

  71. C.R. Vogel, Computational Methods for Inverse Problems. Frontiers in Applied Mathematics (Society for Industrial and Applied Mathematics, Philadelphia, 2002)

    Google Scholar 

  72. E.A. Wan, R. Van der Merwe, The unscented kalman filter for nonlinear estimation, in Adaptive Systems for Signal Processing, Communications, and Control Symposium 2000. AS-SPCC. The IEEE 2000 (2000), pp 153–158

    Google Scholar 

  73. H. Yang, A. Veneziani, Variational estimation of cardiac conductivities by a data assimilation procedure. Technical Report TR-2013-007, Math&CS, Emory University (July 2013)

    Google Scholar 

Download references

Acknowledgements

The authors wish to thank Tiziano Passerini (Siemens, Princeton, NJ, USA) and Marina Piccinelli (Emory University, Department of Radiology) for several contributions in the development of methods and codes used for the topics considered in the chapter.

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Emory University, Atlanta, GA, USA

    Luca Bertagna & Alessandro Veneziani

  2. Computer Science Research Institute, Sandia National Laboratories, Albuquerque, NM, USA

    Marta D’Elia & Mauro Perego

Authors
  1. Luca Bertagna
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marta D’Elia
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mauro Perego
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Alessandro Veneziani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alessandro Veneziani .

Editor information

Editors and Affiliations

  1. Department of Technical Mathematics, Czech Technical University in Prague, Prague, Czech Republic

    Tomáš Bodnár

  2. Swanson School of Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania, USA

    Giovanni P. Galdi

  3. Institute of Mathematics, Academy of Sciences of the Czech Republic, Prague 1, Czech Republic

    Šárka Nečasová

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Basel

About this chapter

Cite this chapter

Bertagna, L., D’Elia, M., Perego, M., Veneziani, A. (2014). Data Assimilation in Cardiovascular Fluid–Structure Interaction Problems: An Introduction. In: Bodnár, T., Galdi, G., Nečasová, Š. (eds) Fluid-Structure Interaction and Biomedical Applications. Advances in Mathematical Fluid Mechanics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0822-4_6

Download citation

Publish with us

Policies and ethics

Search

Navigation