Skip to main content
SpringerLink
Log in
Book cover

Encyclopedia of Continuum Mechanics pp 1–8Cite as

Computational Models for Hemodynamics

  • Living reference work entry
  • First Online:

Synonyms

Mathematical modeling of the cardiovascular system

Definitions

Hemodynamics means fluid dynamics of the blood. Here, mathematical foundation and numerical approximation of the cardiocirculatory system, with emphasis on hemodynamics, are addressed.

Background

Cardiovascular diseases represent the major cause of death in Western countries, leading to more than 17.3 million deaths per year worldwide (about half of all deaths in Europe).

The cardiovascular system is composed of the heart, the vascular circulation (both arteries and veins), and the microcirculation (capillaries); see Fig. 1. Here, the mathematical and numerical description of the first two components is considered.

Fig. 1
figure 1

Representative scheme of the cardiovascular system

Full size image

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

References

  • Astorino M, Hamers J, Shadden C, Gerbeau J (2012) A robust and efficient valve model based on resistive immersed surfaces. Int J Numer Methods Biomed Eng 28(9):937–959

    Article  MathSciNet  Google Scholar 

  • Badia S, Nobile F, Vergara C (2008) Fluid-structure partitioned procedures based on Robin transmission conditions. J Comput Phys 227:7027–7051

    Article  MathSciNet  Google Scholar 

  • Balzani D, Brands D, Klawonn A, Rheinbach O, Schroder J (2010) On the mechanical modeling of anisotropic biological soft tissue and iterative parallel solution strategies. Arch Appl Mech 80(5):479–488

    Article  Google Scholar 

  • Barker A, Cai X (2010) Scalable parallel methods for monolithic coupling in fluid-structure interaction with application to blood flow modeling. J Comput Phys 229:642–659

    Article  MathSciNet  Google Scholar 

  • Boffi D, Gastaldi L (2003) A finite element approach for the immersed boundary method. Compos Struct 81(8–11):491–501

    Article  MathSciNet  Google Scholar 

  • Causin P, Gerbeau J, Nobile F (2005) Added-mass effect in the design of partitioned algorithms for fluid-structure problems. Comput Methods Appl Mech Eng 194(42–44):4506–4527

    Article  MathSciNet  Google Scholar 

  • Chorin A (1968) Numerical solution of the Navier-Stokes equations. Math Comput 22:745–762

    Article  MathSciNet  Google Scholar 

  • Colli Franzone P, Pavarino L, Scacchi S (2014) Mathematical cardiac electrophysiology. Springer International Publishing Switzerland

    Book  Google Scholar 

  • Crosetto P, Deparis S, Fourestey G, Quarteroni A (2011) Parallel algorithms for fluid-structure interaction problems in haemodynamics. SIAM J Sci Comput 33:1598–1622

    Article  MathSciNet  Google Scholar 

  • Elman H, Silvester D, Wathen A (2005) Finite elements and fast iterative solvers. Oxford Science Publications

    Google Scholar 

  • Fedele M, Faggiano E, Dedè L, Quarteroni A (2016) A patient-specific aortic valve model based on moving resistive immersed implicit surfaces. MOX-Report n. 23-2016, Department of Mathematics, Politecnico di Milano, Italy

    Google Scholar 

  • Fernández M, Gerbeau J, Grandmont C (2007) A projection semi-implicit scheme for the coupling of an elastic structure with an incompressible fluid. Int J Numer Meth Eng 69(4):794–821

    Article  MathSciNet  Google Scholar 

  • FitzHugh R (1961) Impulses and physiological states in theoretical models of nerve membrane. Biophys J 1(6):445–466

    Article  Google Scholar 

  • Formaggia L, Quarteroni A, Veneziani A (eds) (2009) Cardiovascular mathematics – modeling and simulation of the circulatory system. Springer, Milan

    MATH  Google Scholar 

  • Fritz T, Wieners C, Seemann G, Steen H, Dossel O (2014) Simulation of the contraction of the ventricles in a human heart model including atria and pericardium. Biomech Model Mechanobiol 13(3):627–641

    Article  Google Scholar 

  • Gee M, Kuttler U, Wall W (2011) Truly monolithic algebraic multigrid for fluid-structure interaction. Int J Numer Meth Eng 85(8):987–1016

    Article  MathSciNet  Google Scholar 

  • Goktepe S, Kuhl E (2010) Electromechanics of the heart: a unified approach to the strongly coupled excitation contraction problem. Comput Mech 45(2):227–243

    Article  MathSciNet  Google Scholar 

  • Hart JD, Peters G, Schreurs P, Baaijens F (2000) A two-dimensional fluid structure interaction model of the aortic value. J Biomech 33(9):10791088

    Google Scholar 

  • Heil M (2004) An efficient solver for the fully coupled solution of large-displacement fluid-structure interaction problems. Comput Methods Appl Mech Eng 193:1–23

    Article  MathSciNet  Google Scholar 

  • Hirt C, Amsden A, Cook J (1974) An arbitrary Lagrangian Eulerian computing method for all flow speeds. J Comput Phys 69:277–324

    MATH  Google Scholar 

  • Hodgkin A, Hukley A (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 117(4):500–544

    Article  Google Scholar 

  • Holzapfel G, Ogden R (2009) Constitutive modelling of passive myocardium: a structurally based framework for material characterization. Phil Trans R Soc A 367:3445–3475

    Article  MathSciNet  Google Scholar 

  • Holzapfel G, Ogden R (2010) Constitutive modelling of arteries. Proc R Soc Lond Ser A Math Phys Eng Sci 466(2118):1551–1596. https://doi.org/10.1098/rspa.2010.0058

    Article  MathSciNet  MATH  Google Scholar 

  • Keener J, Bogar K (1998) A numerical method for the solution of the bidomain equations in cardiac tissue. Chaos 8:234–241

    Article  Google Scholar 

  • Korakianitis T, Shi Y (2006) A concentrated parameter model for the human cardiovascular system including heart valve dynamics and atrioventricular interaction. Med Eng Phys 28(7):613–628

    Article  Google Scholar 

  • Nash M, Panfilov A (2004) Electromechanical model of excitable tissue to study reentrant cardiac arrhythmias. Prog Biophys Mol Biol 85(2–3):501–522

    Article  Google Scholar 

  • Niederer S, Smith N (2007) A mathematical model of the slow force response to stretch in rat ventricular myocytes. Biophys J 92(11):4030–4044

    Article  Google Scholar 

  • Niederer S, Smith N (2008) An improved numerical method for strong coupling of excitation and contraction models in the heart. Prog Biophys Mol Biol 96(1–3):90–111

    Article  Google Scholar 

  • Niederer S, Hunter P, Smith N (2006) A quantitative analysis of cardiac myocyte relaxation: a simulation study. Biophys J 90(5):1697–1722

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Pavarino L, Scacchi S, Zampini S (2015) Newton-Krylov-BDDC solvers for nonlinear cardiac mechanics. Comput Appl Mech Eng 295:562–580

    Article  MathSciNet  Google Scholar 

  • Perktold K, Thurner E, Kenner T (1994) Flow and stress characteristics in rigid walled and compliant carotid artery bifurcation models. Med Biol Eng Comput 32(1):19–26

    Article  Google Scholar 

  • Peskin C (1972) Flow patterns around heart valves: a numerical method. J Comput Phys 10(2):252–271

    Article  MathSciNet  Google Scholar 

  • Peskin C (2002) The immersed boundary method. Acta Numer 11:479–517

    Article  MathSciNet  Google Scholar 

  • Qu Z, Garfinkel A (1998) An advanced algorithm for solving partial differential equation in cardiac conduction. IEEE Trans Biomed Eng 46(9):1166–1168

    Google Scholar 

  • Quarteroni A, Valli A (1999) Domain decomposition methods for partial differential equations. Oxford science publications, Oxford

    Google Scholar 

  • Quarteroni A, Veneziani A, Vergara C (2016) Geometric multiscale modeling of the cardiovascular system, between theory and practice. Comput Methods Appl Mech Eng 302:193–252

    Article  MathSciNet  Google Scholar 

  • Quarteroni A, Lassila T, Rossi S, Ruiz-Baier R (2017a) Integrated heart – coupling multiscale and multiphysics models for the simulation of the cardiac function. Comput Methods Appl Mech Eng 314:345–407

    Article  MathSciNet  Google Scholar 

  • Quarteroni A, Manzoni A, Vergara C (2017b) The cardiovascular system: mathematical modelling, numerical algorithms and clinical applications. Acta Numer 26:365–590

    Article  MathSciNet  Google Scholar 

  • Robertson A, Sequeira A, Owens R (2009) Chapter 6. Rheological models for blood. In: Formaggia L, Quarteroni A, Veneziani A (eds) Cardiovascular mathematics. Spronger-Verlag Italia, Milano, pp 211–241

    Chapter  Google Scholar 

  • Smith NP, Nickerson DP, Crampin EJ, Hunter PJ (2004) Multiscale computational modelling of the heart. Acta Numer 13:371–431

    Article  MathSciNet  Google Scholar 

  • Taylor C, Figueroa C (2009) Patient-specific modeling of cardiovascular mechanics. Annu Rev Biomed Eng 11:109–134

    Article  Google Scholar 

  • Temam R (1969) Sur l’approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionaires (I). Arch Ration Mech Anal 32:135–153

    Article  Google Scholar 

  • Vergara C, Lange M, Palamara S, Lassila T, Frangi A, Quarteroni A (2016) A coupled 3D-1D numerical monodomain solver for cardiac electrical activation in the myocardium with detailed Purkinje network. J Comput Phys 308:218–238

    Article  MathSciNet  Google Scholar 

  • Vigmond E, Clements C (2007) Construction of a computer model to investigate sawtooth effects in the Purkinje system. IEEE Trans Biomed Eng 54(3):389–399

    Article  Google Scholar 

  • Yu Y, Baek H, Karniadakis G (2013) Generalized fictitious methods for fluid structure interactions: analysis and simulations. J Comput Phys 245:317–346

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    Alfio Quarteroni

  2. MOX, Dipartimento di Matematica, Politecnico di Milano, Milan, Italy

    Alfio Quarteroni & Christian Vergara

Authors
  1. Alfio Quarteroni
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christian Vergara
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christian Vergara .

Editor information

Editors and Affiliations

  1. Institut für Mechanik, Otto-von-Guericke-Universität Institut für Mechanik, Magdeburg, Germany

    Holm Altenbach

  2. Griffith University (Gold Coast Campus), Griffith School of Engineering Griffith University (Gold Coast Campus), Southport, Queensland, Australia

    Andreas Öchsner

Section Editor information

  1. Chair of Continuum Mechanics, Ruhr-University-Bochum, Bochum, Germany

    Daniel Balzani

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer-Verlag GmbH Germany, part of Springer Nature

About this entry

Check for updates. Verify currency and authenticity via CrossMark

Cite this entry

Quarteroni, A., Vergara, C. (2018). Computational Models for Hemodynamics. In: Altenbach, H., Öchsner, A. (eds) Encyclopedia of Continuum Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53605-6_35-1

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-53605-6_35-1

  • Received:

  • Accepted:

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-53605-6

  • Online ISBN: 978-3-662-53605-6

  • eBook Packages: Springer Reference EngineeringReference Module Computer Science and Engineering

Publish with us

Policies and ethics

Search

Navigation