Model of a Saccular Aneurysm of the Bifurcation Node of an Artery

  • V. A. KozlovEmail author
  • S. A. Nazarov

Modified Kirchhoff transmission conditions in a simple one-dimensional model of a branching artery developed by the authors, allow one to describe an anomaly of its bifurcation node, congenital or acquired due to trauma or disease of a vessel wall. The pathology of the blood flow through the damaged node and the methods of determining the aneurysm parameters from the data measured at the peripheral parts of the circulatory system by solving inverse problems are discussed.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    G. R. Kirchhoff, “Ueber den Durchgang eines elektrischen Stromes durch eine Ebene, insbesondere durch eine kreisformige,” Annalen der Physik und Chemie, Bd. LXIV, No. 4, 497–514 (1845).Google Scholar
  2. 2.
    L. Pauling, “The diamagnetic anisotropy of aromatic molecules,” J. Chem. Phys., 4 (1936).Google Scholar
  3. 3.
    J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. I, Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag, New York-Heidelberg (1972).Google Scholar
  4. 4.
    A. Bressan, S. Canic, M. Garavello, M. Herty, and B. Piccoli, “Flows on networks: recent results and perspectives,” European Math. Soc. (EMS Surveys in Mathematical Sciences), 1, No. 1. 47–111 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    V. A. Kozlov and S. A. Nazarov, “Surface enthalpy and the elastic properties of blood vessels,” Dokl. Akad. Nauk, 441, No. 1, 38–43 (2011).MathSciNetGoogle Scholar
  6. 6.
    V. A. Kozlov and S. A. Nazarov, “Asymptotic models of blood flow in arteries and veins,” Zap. Nauchn. Semin. POMI, 409, 80–106 (2012).Google Scholar
  7. 7.
    V. A. Kozlov and S. A. Nazarov, “An elementary on-dimensional model of a false aneurysm in the large femoral artery,” Zap. Nauchn. Semin. POMI, 426, 64–86 (2014).Google Scholar
  8. 8.
    V. A. Kozlov and S. A. Nazarov, “Transmission conditions in a one-dimensional model of a bifurcating artery with elastic walls,” Zap. Nauchn. Semin. POMI, 438, 138–177 (2015).Google Scholar
  9. 9.
    V. A. Kozlov and S. A. Nazarov, “Asymptotic models of anisotropic heterogeneous elastic walls of blood vessels,” J. Math. Sci., 213, No. 4, (2016).Google Scholar
  10. 10.
    V. A. Kozlov and S. A. Nazarov, “Effective one-dimensional images of arterial trees in the cardiovascular system,” Doklady Physics, 62, No. 3, 158–163 (2017).CrossRefGoogle Scholar
  11. 11.
    V. A. Kozlov and S. A. Nazarov, “One-dimensional model of flow in a junction of thin channels, including arterial trees,” Mat. Sb., 208, No. 8, 56–105 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    H. Le Dret, “Modeling of the junction between two rods,” J. Math. Pures Appl., 68, 365–397 (1989).MathSciNetzbMATHGoogle Scholar
  13. 13.
    S. A. Nazarov and A. S. Slutskii, “Asymptotic analysis of an arbitrary spatial system of thin rods,” Trudy Peterburg Mat. Obshch., Vol. X, 59–107, Amer. Math. Soc. Transl., Ser. 2, 214 (2005).Google Scholar
  14. 14.
    P. Kuchment (Editor), “Quantum graphs and their applications,” Waves in Random Media, 14, No 1 (2004).Google Scholar
  15. 15.
    P. Exner and H. Kovařík, Quantum Waveguides, Springer, Hidelberg (2015).Google Scholar
  16. 16.
    P. Kuchment “Graph models for waves in thin structures,” Waves in Random Media, 12, No. 12, R1–R24 (2002).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    P. Kuchment and O. Post, “On the spectrum of carbon nano-structures,” Commun. Math. Phys., 275, No. 3, 805–826 (2007).CrossRefzbMATHGoogle Scholar
  18. 18.
    E. Korotyaev and I. Lobanov, “Schrödinger operators on zigzag nanotubes,” Annales Henri Poincare, 8, No. 6. 1151–1176 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    A. V. Badanin and E. L. Korotyaev, “A magnetic Schrödinger operator on a periodic graph,” Mat. Sb., 201, No. 10, 3–46 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    I. Y. Popov, A. N. Skorynina, and I. V. Blinova, “On the existence of point spectrum for branching strips quantum graph,” J. Math. Phys., 55, No. 3 (2014).Google Scholar
  21. 21.
    V. A. Kozlov, S. A. Nazarov, and A. Orlof, “Trapped modes supported by localized potentials in the zigzag graphene ribbon,” C. R. Acad. Sci. Paris., Sér. 1, 354, No. 1. 63–67 (2016).Google Scholar
  22. 22.
    S. Molchanov and B. Vainberg, “Scattering solutions in networks of thin fibers: small diameter asymptotics,” Comm. Math. Phys., 273, No 2., 533–559 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    D. Grieser, “Spectra of graph neighborhoods and scattering,” Proc. London Math. Soc., 97, No. 3, 718–752 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    S. A. Nazarov, K. Ruotsalainen, and P. Uusitalo, “Asymptotics of the spectrum of the Dirichlet Laplacian on a thin carbon nano-structure,” C. R. Mecanique, 343, 360–364 (2015).CrossRefGoogle Scholar
  25. 25.
    S. A. Nazarov, “The spectrum of rectangular lattices of quantum waveguides,” Izv. Ross. Akad. Nauk, Ser. Mat., 81, No. 1, 31–92 (2017).MathSciNetGoogle Scholar
  26. 26.
    F. L. Bakharev, S. G. Matveenko, and S. A. Nazarov, Discrete spectrum of a cross-shaped waveguide, Algebra Analiz, 28, No. 2, 58–71 (2016).MathSciNetGoogle Scholar
  27. 27.
    F. S. Rofe-Beketov, “Selfadjoint extensions of differential operators in a space of vectorvalued functions,” Dokl. Akad. Nauk SSSR, 184, 1034–1037 (1969).MathSciNetzbMATHGoogle Scholar
  28. 28.
    B. S. Pavlov, The theory of extensions, and explicitly solvable models, Usp. Mat. Nauk, 4, No. 6, 99–132 (1987).Google Scholar
  29. 29.
    K. Pankrashkin, “Eigenvalue inequalities and absence of threshold resonances for waveguide junctions,” J. Math. Anal. Appl., 449, 907–925 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    S. A. Nazarov and K. I. Piletskas, “The Reynolds flow of a fluid in a thin three-dimensional channel,” Litovsk. Mat. Sb., 30, No. 4, 772–783 (1990).MathSciNetGoogle Scholar
  31. 31.
    S. A. Nazarov and K. Piletskas, “Asymptotic conditions at infinity for the Stokes and Navier–Stokes problems in domains with cylindrical outlets to infinity,” Quaderni di matematica, 4, 141–243 (1999).MathSciNetzbMATHGoogle Scholar
  32. 32.
    G. Panasenko and K. Piletskas, “Asymptotic analysis of the nonsteady viscous flow with a given flow rate in a thin pipe,” Applicable Analysis, 91, No. 3. 559–574 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    G. Panasenko and K. Piletskas, “Flows in a tube structure: equation on the graph,” J. Math. Phys., 55 081505 (2014); doi: Scholar
  34. 34.
    S. A. Nazarov, “The Navier–Stokes problem in thin or long tubes with periodically varying cross-section,” ZAMM, 80, No. 9, 591–612 (2000).MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    V. A. Kozlov and S. A. Nazarov, “One-dimensional model of viscoelastic blood flow through a thin elastic vessel,” J. Math. Sci., 207, No. 2 (2015).Google Scholar
  36. 36.
    Y. C. Fung, Biomechanics. Mechanical Properties of Living Tissues, Springer, New York–Berlin (1993).Google Scholar
  37. 37.
    Y. C. Fung, Biomechanics. Circulation, Second ed., Springer, New York–Berlin (2011).Google Scholar
  38. 38.
    F. Berntsson, M. Karlsson, V. A. Kozlov, and S. A. Nazarov, “A one-dimensional model of viscous blood flow in an elastic vessel,” Appl. Math Comput., 274, 125–132 (2016).MathSciNetzbMATHGoogle Scholar
  39. 39.
    F. Berntsson, M. Karlsson, V. A. Kozlov, and S. A. Nazarov, “A one-dimensional model of a false aneurysm,” International J. Research Engineering Sci., 6, No. 5, 61–73 (2017).Google Scholar
  40. 40.
    J. P. Hornak, The Basics of MRI, Interactive Learning Software (2008).Google Scholar
  41. 41.
    V. A. Kozlov, S. A. Nazarov, and G. Zavorokhin, “A fractal graph model of capillary type systems,” Complex Variables and Elliptic Equations (2017). Published online 17.07.2017.Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Linköping UniversityLinköpingSweden
  2. 2.Institute of Applied Engineering, RASSt. PetersburgRussia

Personalised recommendations