Hemomath pp 1-77 | Cite as

Hemorheology and Hemodynamics

  • Antonio Fasano
  • Adélia Sequeira
Part of the MS&A book series (MS&A, volume 18)


In this first approach to the subject we start describing a few basic facts about blood composition and about the circulatory system. Such notions will be enriched in the rest of the book, when needed. Concerning modeling, this chapter is devoted to the debatable question of the rheological properties of blood and to the various ways the circulatory system has been described in a mathematical way, both at the scale of main vessels and at the level of microcirculation. The peculiar phenomenon of vessels oscillation (vasomotion) will be considered briefly, and the literature on the mathematical modeling of diseases like atherosclerosis, affecting blood flow through arteries, will be illustrated.


  1. 1.
    C. Aalkjær, H. Nilsson, Vasomotion: cellular background for the oscillator and for the synchronization of smooth muscle cells. Br. J. Pharmacol. 144, 605–616 (2005)CrossRefGoogle Scholar
  2. 2.
    W.C. Aird, Discovery of the cardiovascular system: from Galen to William Harvey. J. Thromb. Haemost. 9(Suppl. 1), 118–129 (2011)CrossRefGoogle Scholar
  3. 3.
    J. Alastruey, K.H. Parker, J. Peiró, S.M. Byrd, S.J. Sherwin, Modelling the circle of Willis to assess the effects of anatomic variations and occlusions on cerebral flows. J. Biomech. 40(8), 1794–1805 (2007)CrossRefGoogle Scholar
  4. 4.
    D. Amadori, S. Ferrari, L. Formaggia, Derivation and analysis of a fluid-dynamical model in thin and long elastic vessels. Netw. Heterog. Media 2(1), 99–125 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    M. Anand, K.R. Rajagopal, A shear-thinning viscoelastic fluid model for describing the flow of blood. Int. J. Cardiovasc. Med. Sci. 4(2), 59–68 (2004)Google Scholar
  6. 6.
    M. Anand, J. Kwack, A. Masud, A new Oldroyd-B model for blood in complex geometries. Int. J. Eng. Sci. 72, 78–88 (2013)CrossRefGoogle Scholar
  7. 7.
    G. Astarita, G. Marrucci, Principles of Non-Newtonian Fluid Mechanics (McGraw Hill, New York, 1974)zbMATHGoogle Scholar
  8. 8.
    A.P. Avolio, Multi-branched model of the human arterial system. Med. Biol. Eng. Comp. 18, 709–718 (1980)CrossRefGoogle Scholar
  9. 9.
    A.C.L. Barnard, W.A. Hunt, W.P. Timlake, E. Varley, A theory of fluid flow in compliant tubes. Biophys. J. 6, 717–724 (1966)CrossRefGoogle Scholar
  10. 10.
    H.A. Barnes, Thixotropy - a review. J. Non-Newtonian Fluid Mech. 70, 1–33 (1997)CrossRefGoogle Scholar
  11. 11.
    P. Barter, The role of HDL–cholesterol in preventing atherosclerotic disease. Eur. Heart J. Suppl. 7, 4–8 (2005)CrossRefGoogle Scholar
  12. 12.
    S. Basting, A. Quaini, S. Čanić, R. Glowinski, Extended ALE Method for fluid-structure interaction problems with large structural displacements. J. Comput. Phys. 331, 312–336 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    S. Basting, A. Quaini, S. Čanić, R. Glowinski, On the implementation and benchmarking of an extended ALE method for FSI problems, in Fluid–Structure Interaction: Modeling, Adaptive Discretizations and Solvers. RICAM Publication, De Gruyter (Johann Radon Inst. for Comput. and Applied Mathematics, Austria) (2017, to appear)Google Scholar
  14. 14.
    W.H. Bauer, E.A. Collins, Thixotropy and dilatancy, in Rheology, Theory and Applications, ed. by F.R. Eirich, vol. 4 (Academic, New York, 1967)Google Scholar
  15. 15.
    E. Bazigou, T. Makine, Flow control in our vessels: vascular valves make sure there is no way back. Cell. Mol. Life Sci. 70, 1055–1066 (2013)CrossRefGoogle Scholar
  16. 16.
    L.S. Beale, The Microscope in Medicine (J. & A. Churchill, London, 1877)Google Scholar
  17. 17.
    H. Beirão da Veiga, On the existence of strong solutions to a coupled fluid–structure evolution problem. J. Math. Fluid Mech. 6(1), 21–52 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    A. Bertram, Elasticity and Plasticity of Large Deformations, An Introduction, chap. 7 Hyperelasticity (Springer, Berlin, 2012), pp. 209–229Google Scholar
  19. 19.
    E.C. Bingham, An investigation of the laws of plastic flow. U.S. Bur. Stand. Bull. 13, 309–353 (1916)Google Scholar
  20. 20.
    T. Bodnár, K.R. Rajagopal, A. Sequeira, Simulation of the three-dimensional flow of blood using a shear-thinning viscoelastic fluid model. Math. Model. Nat. Phenom. 6(5), 1–24 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    J.H. Breasted, The Edwin Smith Surgical Papyrus Published in Facsimile and Hieroglyphic Transliteration with Translation and Commentary in Two Volumes (The University of Chicago Press, Chicago, 1930)Google Scholar
  22. 22.
    C.P. Bryan, The Papyrus Ebers (Geoffrey Bles, London, 1930)Google Scholar
  23. 23.
    M. Bukač, S. Čanić, B. Muha, A nonlinear fluid–structure interaction problem in compliant arteries treated with vascular stents. Appl. Math. Optim. 73, 433–473 (2016)zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    M. Bulelzai, J. Dubbeldam, Long time evolution of atherosclerotic plaques. J. Theor. Biol. 297, 1–10 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    E. Burman, M. Fernández, Stabilization of explicit coupling in fluid—structure interaction involving fluid incompressibility. Comput. Methods Appl. Mech. Eng. 198(5–8), 766–784 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    A. Caggiati, The venous valves in the lower limbs. Phlebolymphology 20, 87–95 (2013)Google Scholar
  27. 27.
    A. Caggiati, P. Bertocchi, Regarding fact and fiction surrounding the discovery of the venous valves. J. Vasc. Surg. 33, 1317 (2001)Google Scholar
  28. 28.
    A. Caggiati, M. Phillips, A. Lametschwandtner, C. Allegra, Valves in small veins and venules. Eur. J. Vasc. Endovasc. Surg. 32, 447–452 (2006)CrossRefGoogle Scholar
  29. 29.
    V. Calvez, A. Ebde, N. Meunier, A. Raoult, Mathematical and numerical modeling of the atherosclerotic plaque formation. ESAIM Proc. 28, 1–12 (2009)zbMATHCrossRefGoogle Scholar
  30. 30.
    V. Calvez, J. Houot, N. Meunier, A. Raoult, G. Rusnakova, Mathematical and numerical modeling of early atherosclerotic lesions. ESAIM Proc. 30, 1–14 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    S. Čanić, E.H. Kim, Mathematical analysis of the quasilinear effects in a hyperbolic model of blood flow through compliant axi-symmetric vessels. Math. Methods Appl. Sci. 26(14), 1161–1186 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    S. Čanić, J. Tambača, G. Guidoboni, A. Mikelić, C.J. Hartley, D. Rosenstrauch, Modeling viscoelastic behavior of arterial walls and their interaction with pulsatile blood flow. SIAM J. Appl. Math. 67(1), 164–193 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    S. Čanić, B. Muha, M. Bukač, Fluid-structure interaction in hemodynamics: modeling, analysis, and numerical simulation, in Fluid-Structure Interaction and Biomedical Applications, ed. by T. Bodnár et al. Advances in Mathematical Fluid Mechanics (Birkhäuser, Basel, 2014), pp. 79–195Google Scholar
  34. 34.
    C.G. Caro, T.J. Pedley, R.C. Schroter, W.A. Seed, R.H. Parker, The Mechanics of the Circulation, 2nd edn. (Oxford University Press, Oxford, 2012)zbMATHGoogle Scholar
  35. 35.
    P. Causin, J.-F. Gerbeau, F. Nobile, Added-mass effect in the design of partitioned algorithms for fluid–structured problems. Comput. Methods Appl. Mech. Eng. 194, 4506–4527 (2005)zbMATHCrossRefGoogle Scholar
  36. 36.
    A. Chambolle, B. Desjardins, M.J. Esteban, C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate. J. Math. Fluid Mech. 7(3), 368–404 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  37. 37.
    S.E. Charm, G.S. Kurland, Blood Flow and Microcirculation (Wiley, New York, 1974)Google Scholar
  38. 38.
    C.H.A. Cheng, S. Shkoller, The interaction of the 3D Navier–Stokes equations with a moving nonlinear Koiter elastic shell. SIAM J. Math. Anal. 42(3), 1094–1155 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  39. 39.
    C. Cheng, R. van Haperen, M. de Waard, L.C. van Damme, D. Tempel, L. Hanemaaijer, G.W. van Cappellen, J. Bos, C.J. Slager, D.J. Dunker, A.F. van der Steen, R. de Crom, R. Crams, Shear stress affects the intra- cellular distribution of eNOS: direct demonstration by a novel in vivo technique. Blood 106, 3691–3698 (2005)CrossRefGoogle Scholar
  40. 40.
    C. Cheng, D. Tempel, R. van Haperen, A. van der Baan, F. Grosveld, Mat. J.A.P. Daemen, R. Krams, R. de Crom, Atherosclerotic lesion size and vulnerability are determined by patterns of fluid shear stress. Circulation 113, 2744–2753 (2006)Google Scholar
  41. 41.
    S. Chien, S. Usami, R.J. Dellenback, M.I. Gregersen, Shear dependence of effective cell volume as a determinant of blood viscosity. Science 168, 977–979 (1970)CrossRefGoogle Scholar
  42. 42.
    S. Chien, R.G. King, R. Skalak, S. Usami, A.L. Copley, Viscoelastic properties of human blood and red cell suspensions. Biorheology 12, 341–346 (1975)CrossRefGoogle Scholar
  43. 43.
    Y.I. Cho, K.R. Kensey, Effects of the non-Newtonian viscosity of blood on flows in a diseased arterial vessel. Part I: steady flows. Biorheology 28, 241–262 (1991)Google Scholar
  44. 44.
    K.Y. Chyu, P.K. Shah, The role of inflammation in plaque disruption and thrombosis. Rev. Cardiovasc. Med. 2, 82–91 (2001)Google Scholar
  45. 45.
    P. Ciarlet, Mathematical Elasticity. Three-Dimensional Elasticity, vol. 1, 2nd edn. (Elsevier, Amsterdam, 2004)Google Scholar
  46. 46.
    M. Cilla, E. Peña, M.A. Martínez, Mathematical modelling of atheroma plaque formation and development in coronary arteries. J. R. Soc. Interface 11, 20130866 (2014)CrossRefGoogle Scholar
  47. 47.
    M. Cilla, I. Borrás, E. Peña, M.A. Martínez, M. Malvé, A parametric model for analysing atherosclerotic arteries: on the FSI coupling. Int. Commun. Heat Mass Transfer 67, 29–38 (2015)CrossRefGoogle Scholar
  48. 48.
    M. Cilla, M.A. Martínez, E. Peña, Effect of transmural transport properties on atheroma plaque formation and development. Ann. Biomed. Eng. 43(7), 1516–1530 (2015)CrossRefGoogle Scholar
  49. 49.
    C. Cobbold, J. Sherratt, S. Mexwell, Lipoprotein oxidation and its significance for atherosclerosis: a mathematical approach. Bull. Math. Biol. 64, 65–95 (2002)zbMATHCrossRefGoogle Scholar
  50. 50.
    D. Coutand, S. Shkoller, The interaction between quasilinear elastodynamics and the Navier–Stokes equations. Arch. Ration. Mech. Anal. 179(3), 303–352 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  51. 51.
    P. Cullen, J. Rauterberg, S. Lorkowski, The pathogenesis of atherosclerosis. Handb. Exp. Pharmacol. 170, 3–70 (2005)CrossRefGoogle Scholar
  52. 52.
    M.J. Davis, M.A. Hill, L. Kuo, Local regulation of microvascular perfusion, in Suppl. 9: Handbook of Physiology. The Cardiovascular System, Microcirculation, chap. 6 (American Physiological Society, 2011), pp. 161–284Google Scholar
  53. 53.
    C. de Wit, Closing the gap at hot spots. Circ. Res. 100, 931–933 (2007)CrossRefGoogle Scholar
  54. 54.
    L. Dintenfass, Blood Microrheology - Viscosity Factors in Blood Flow. Ischaemia and Thrombosis (Butterworth, Oxford, 1971)Google Scholar
  55. 55.
    L. Dintenfass, Blood Viscosity, Hyperviscosity and Hyperviscosaemia (MTP, Lancaster, 1985)Google Scholar
  56. 56.
    J. Donea, S. Giuliani, J.P. Halleux, An Arbitrary-Eulerian method for transient dynamic fluid-structure interactions. Comput. Methods Appl. Mech. Eng. 33(1–3), 689–723 (1982)zbMATHCrossRefGoogle Scholar
  57. 57.
    R.M. Dongaonkar, C.M. Quick, J.C. Vo, J.K. Meisner, G.A. Laine, M.J. Davis, R.H. Stewart, Blood flow augmentation by intrinsic venular contraction in vivo. Am. J. Phys. Regul. Integr. Comp. Physiol. 302, R1436–R1442 (2012)CrossRefGoogle Scholar
  58. 58.
    W. Dzwinel, K. Boryczko, D.A. Yuen, A discrete-particle model of blood dynamics in capillary vessels. J. Colloid Interface Sci. 258, 163–173 (2003)zbMATHCrossRefGoogle Scholar
  59. 59.
    N.I. Ebeid, Egyptian Medicine in the Days of the Pharaohs (General Egyptian Book Organization, Cairo, 1999)Google Scholar
  60. 60.
    N. El Khatib, S. Génieys, V. Volpert, Atherosclerosis initiation modeled as an inflammatory process. Math. Model. Nat. Phenom. 2(2), 126–141 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  61. 61.
    N. El Khatib, S. Génieys, B. Kazmierczak, V. Volpert, Mathematical modeling of atherosclerosis as an inflammatory disease. Philos. Trans. R. Soc. A. 367, 4877–4886 (2009)zbMATHCrossRefGoogle Scholar
  62. 62.
    N. El Khatib, S. Génieys, B. Kazmierczak, V. Volpert, Reaction-diffusion model of atherosclerosis development. J. Math. Biol. 65, 349–374 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  63. 63.
    P. Espaõl, Fluid particle model. Phys. Rev. E 57(3), 2930–2948 (1998)CrossRefGoogle Scholar
  64. 64.
    L. Euler, Principia pro motu sanguinis per arterias determinando, in Opera Omnia, ed. by L. Euler, vol. 16(2) (Birkhäuser, Basel, 1989), pp. 178–196Google Scholar
  65. 65.
    E.A. Evans, R.M. Hochmuth, Membrane viscoelasticity. Biophys. J. 16(1), 1–11 (1976)CrossRefGoogle Scholar
  66. 66.
    R. Fåhraeus, The suspension stability of blood. Physiol. Rev. 9, 241–274 (1929)Google Scholar
  67. 67.
    R. Fåhraeus, T. Lindqvist, The viscosity of blood in narrow capillary tubes. Am. J. Physiol. 96, 362–368 (1931)Google Scholar
  68. 68.
    A. Farina, A. Fasano, Incompressible flows though slender oscillating vessels provided with distributed valves. Adv. Math. Sci. Appl. 25, 33–42 (2016)MathSciNetGoogle Scholar
  69. 69.
    A. Farina, A. Fasano, L. Fusi, A. Ceretani, F. Rosso, Modeling peristaltic flow in vessels equipped with valves: implications for vasomotion in bat wing venules. Int. J. Eng. Sci. 107, 1–12 (2016)MathSciNetCrossRefGoogle Scholar
  70. 70.
    A. Fasano, A. Farina, J. Mizerski, A new model for blood flow in fenestrated capillaries with application to ultrafiltration in kidney glomeruli. Adv. Math. Sci. Appl. 23, 319–337 (2013)zbMATHMathSciNetGoogle Scholar
  71. 71.
    J.J. Feher, Quantitative Human Physiology: An Introduction (Elsevier, Academic, Amsterdam, 2012)Google Scholar
  72. 72.
    M.A. Fernandéz, J.-F. Gerbeau, Algorithms for fluid–structure interaction problems, in Cardiovascular Mathematics. Modeling and Simulation of the Circulatory System, ed. by L. Formaggia, A. Quarteroni, A. Veneziani, vol. 1 (Springer, Berlin, 2009), pp. 307–346Google Scholar
  73. 73.
    M. Fernández, J. Gerbeau, C. Grandmont, A projection semi-implicit scheme for the coupling of an elastic structure with an incompressible fluid. Int. J. Numer. Methods Eng. 69, 794–821 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  74. 74.
    N. Filipovic, D. Nikolic, I. Saveljic, Z. Milosevic, T. Exarchos, G. Pelosi, O. Parodi, Computer simulation of three-dimensional plaque formation and progression in the coronary artery. Comput. Fluids 88, 826–833 (2013)CrossRefGoogle Scholar
  75. 75.
    T.M. Fischer, M. Stöhr-Lissen, H. Schmid-Schönbein, The red cell as a fluid droplet: tankread-like motion of the human erythrocyte membrane in shear flow. Science 202, 894–896 (1978)CrossRefGoogle Scholar
  76. 76.
    G.A. Fishbein, M.C. Fishbein, Arteriosclerosis: rethinking the current classification. Arch. Pathol. Lab. Med. 133, 1309–1316 (2009)Google Scholar
  77. 77.
    L. Formaggia, J.F. Gerbeau, F. Nobile, A. Quarteroni, On the coupling of 3D and 1D Navier–Stokes equations for blood flow problems in compliant vessels. Comput. Methods Appl. Mech. Eng. 191, 561–582 (2001)zbMATHCrossRefGoogle Scholar
  78. 78.
    L. Formaggia, J.F. Gerbeau, F. Nobile, A. Quarteroni, Numerical treatment of defective boundary conditions for the Navier–Stokes equations. SIAM J. Numer. Anal. 40, 376–401 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  79. 79.
    L. Formaggia, D. Lamponi, M. Tuveri, A. Veneziani, Numerical modeling of 1D arterial networks coupled with a lumped parameters description of the heart. Comput. Methods Biomech. Biomed. Eng. 9, 273–288 (2006)CrossRefGoogle Scholar
  80. 80.
    L. Formaggia, A. Moura, F. Nobile, On the stability of the coupling of 3D and 1D fluid-structure interaction models for blood flow simulations. Math. Model. Numer. Anal. 41(4), 743–769 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  81. 81.
    L. Formaggia, K. Perktold, A. Quarteroni, Basic mathematical models and motivations, in Cardiovascular Mathematics. Modeling and Simulation of the Circulatory System, ed. by L. Formaggia, A. Quarteroni, A. Veneziani, vol. 1 (Springer, Berlin, 2009), pp. 46–75Google Scholar
  82. 82.
    L. Formaggia, A. Quarteroni, A. Veneziani (eds.), Cardiovascular Mathematics. Modeling and Simulation of the Circulatory System, vol. 1 (Springer, Berlin, 2009), pp. 46–75Google Scholar
  83. 83.
    L. Formaggia, A. Quarteroni, A. Veneziani, Multiscale models of the vascular system, in Cardiovascular Mathematics. Modeling and Simulation of the Circulatory System, ed. by L. Formaggia, A. Quarteroni, A. Veneziani, vol. 1 (Springer, Berlin, 2009), pp. 395–446Google Scholar
  84. 84.
    L. Formaggia, A. Quarteroni, C. Vergara, On the physical consistency between three-dimensional and one-dimensional models in hemodynamics. J. Comput. Phys. 244, 97–112 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  85. 85.
    K.J. Franklin, Valves in veins: an historical survey. Proc. R. Soc. Med. Sect. Hist. Med. 21, 1–33 (1927)Google Scholar
  86. 86.
    A. Friedman, W. Hao, A mathematical model of atherosclerosis with reverse cholesterol transport and associated risk factors. Bull. Math. Biol. 77(5), 758–81 (2015)zbMATHMathSciNetCrossRefGoogle Scholar
  87. 87.
    A. Friedman, W. Hao, B. Hu, A free boundary problem for steady small plaques in the artery and their stability. J. Differ. Equ. 259(4) (2015). doi:10.1016/j.jde.2015.02.002Google Scholar
  88. 88.
    Y.C. Fung, Biomechanics: Mechanical Properties of Living Tissues (Springer, Berlin, 1993)CrossRefGoogle Scholar
  89. 89.
    Y.C. Fung, Biomechanics: Circulation (Springer, Berlin, 1997)CrossRefGoogle Scholar
  90. 90.
    I.T. Gabe, J.H. Gault, J. Ross, D.T. Mason, C.J. Mills, J.P. Schillingford, E. Braunwald, Measurement of instantaneous blood flow velocity and pressure in conscious man with a catheter-tip velocity probe. Circulation 40, 603–614 (1969)CrossRefGoogle Scholar
  91. 91.
    G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Linearised Steady Problems. Springer Tracts in Natural Philosophy, vol. 38, 2nd Corrected edn. (Springer, Berlin, 1998)Google Scholar
  92. 92.
    G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Nonlinear Steady Problems. Springer Tracts in Natural Phylosophy, vol. 39, 2nd Corrected edn. (Springer, Berlin, 1998)Google Scholar
  93. 93.
    L. Garlaschelli, F. Ramaccini, S. Della Sala, Working bloody miracles. Nature 353(6344), 507 (1991)Google Scholar
  94. 94.
    G. Geraci, Il miracolo di S. Gennaro: esperienze e considerazioni di un biologo molecolare. Rend. Acc. Sc. Fis. Mat. Napoli LXXVII, 141–152 (2010)Google Scholar
  95. 95.
    C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate. SIAM J. Math. Anal. 40(2), 716–737 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  96. 96.
    R.J. Gratton, R.E. Gandley, J.F. McCarthy, W.K. Michaluk, B.K. Slinker, M.K. McLaughlin, Contribution of vasomotion to vascular resistance: a comparison of arteries from virgin and pregnant rats. J. Appl. Physiol. 85, 2255–2260 (1998)Google Scholar
  97. 97.
    A.E. Green, P.M. Naghdi, A direct theory of viscous fluids in pipes. I. Basic general developments. Philos. Trans. R. Soc. Lond. A 342, 525–542 (1993)zbMATHCrossRefGoogle Scholar
  98. 98.
    A.E. Green, P.M. Naghdi, M.L. Wenner, On the theory of rods. II. Developments by direct approach. Philos. Trans. R. Soc. Lond. A 337, 485–507 (1974)zbMATHGoogle Scholar
  99. 99.
    A.E. Green, P.M. Naghdi, M.J. Stallard, A direct theory of viscous fluids in pipes. II. Flow of incompressible viscous fluids in curved pipes. Philos. Trans. R. Soc. Lond. A 342, 543–572 (1993)zbMATHCrossRefGoogle Scholar
  100. 100.
    G. Greenstone, The history of bloodletting. Br. Columbia Med. J. 52, 12–14 (2010)Google Scholar
  101. 101.
    D. Gregg, Dynamics of blood and lymph flow, in The Physiological Basis of Medical Practice, ed. by C. Best, N. Taylor, 8th edn. (Williams and Wilkins, New York, 1966)Google Scholar
  102. 102.
    A.C. Guyton, J.E. Hall, A Textbook of Medical Physiology, 10th edn. (W.B. Saunders, Philadelphia, 2000)Google Scholar
  103. 103.
    R.E. Haddock, C.E. Hill, Rhythmicity in arterial smooth muscle. J. Physiol. 566, 645–656 (2005)CrossRefGoogle Scholar
  104. 104.
    R.E. Haddock, G.D.S. Hirst, C.E. Hill, Voltage independence of vasomotion in isolated irideal arterioles of the rat. J. Physiol. 540, 219–229 (2002)CrossRefGoogle Scholar
  105. 105.
    S.I. Hajdu, A note from history: the discovery of blood cells. Ann. Clin. Lab. Sci. 33, 237–238 (2003)Google Scholar
  106. 106.
    G. Hansson, P. Libby, The immune response in atherosclerosis: a double edged sword. Nat. Immunol. 6, 508–519 (2006)CrossRefGoogle Scholar
  107. 107.
    W. Hao, A. Friedman, The LDL-HDL profile determines the risk of atherosclerosis - a mathematical model. PLoS One 9(3), e90497 (2014)Google Scholar
  108. 108.
    K. Hayashi, K. Handa, S. Nagasaka, A. Okumura, Stiffness and elastic behaviour of human intracranial and extracranial arteries. J. Biomech. 13, 175–184 (1980)CrossRefGoogle Scholar
  109. 109.
    W. Hewson, Experimental Inquiries. Part I. A Description of the Lymphatic System in the Human Subject and Other Animals (J. Johnson, London, 1774), p. 30Google Scholar
  110. 110.
    J. Heywood, R. Rannacher, S. Turek, Artificial boundaries and flux and pressure conditions for the Incompressible Navier–Stokes Equations. Int. J. Numer. Methods Fluids 22, 325–352 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  111. 111.
    G.A. Holzapfel, R.W. Ogden, Mechanics of Biological Tissue (Springer, Berlin, 2006)CrossRefGoogle Scholar
  112. 112.
    T.J.R. Hughes, J. Lubliner, On the one-dimensional theory of blood flow in the larger vessels. Math. Biosci. 18, 161–170 (1973)zbMATHCrossRefGoogle Scholar
  113. 113.
    T.J.R. Hughes, W. Liu, T.K. Zimmermann, Lagrangian-Eulerian finite element formulation for incompressible viscous flows. Comput. Methods Appl. Mech. Eng. 29(3), 329–349 (1981)zbMATHMathSciNetCrossRefGoogle Scholar
  114. 114.
    A. Hundertmark-Zausková, M. Lukácová-Medidová, G. Rusnáková, Kinematic splitting algorithm for fluid-structure interaction in hemodynamics. Comput. Methods Appl. Mech. Eng. 265, 83–106 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  115. 115.
    M. Intaglietta, Vasomotion and flowmotion: physiological mechanisms and clinical evidence. Vasc. Med. Rev. 2, 1101–112 (1990)Google Scholar
  116. 116.
    J. Janela, A. Moura, A. Sequeira, A 3D non-Newtonian fluid-structure interaction model for blood flow in arteries. J. Comput. Appl. Math. 234(9), 2783–2791 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  117. 117.
    J. Janela, A. Moura, A. Sequeira, Absorbing boundary conditions for a 3D non-Newtonian fluid-structure interaction model for blood flow in arteries. Int. J. Eng. Sci. 48(11), 1332–1349 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  118. 118.
    T.W. Jones, Discovery that veins of the bat’s wing (which are furnished with valves) are endowed with rhythmical contractility and that the onward flow of blood is accelerated by each contraction. Philos. Trans. R. Soc. Lond. 142, 131–136 (1852)CrossRefGoogle Scholar
  119. 119.
    J. Keener, J. Sneyd, Mathematical Physiology. II, System Physiology. Interdisciplinary Applied Mathematics, vol. 8/II, 2nd edn. (Springer, Berlin, 2009)Google Scholar
  120. 120.
    S.R. Keller, R. Shalak, Motion of a tank-treading ellipsoidal particle in a shear flow. J. Fluid Mech. 120, 27–47 (1982)zbMATHCrossRefGoogle Scholar
  121. 121.
    S. Kim, Y.I . Cho, A.H. Jeon, B. Hogenauer, K.R. Kensey, A new method for blood viscosity measurement. J. Non-Newtonian Fluid Mech. 94, 47–56 (2000)Google Scholar
  122. 122.
    H.J. Kim, I.E. Vignon-Clementel, C.A. Figueroa, J.F. LaDisa, K.E. Jansen, J.A. Feinstein, C.A. Taylor, On coupling a lumped parameter heart model and a three-dimensional finite element aorta model. Ann. Biomed. Eng. 37, 2153–2169 (2009)CrossRefGoogle Scholar
  123. 123.
    M. Koenigsberger, R. Sauser, J.L. Bény, J.J. Meister, Effects of arterial wall stress on vasomotion. Biophys. J. 91, 1663–1674 (2006)CrossRefGoogle Scholar
  124. 124.
    A. Krogh, The Anatomy and Physiology of the Capillaries (Yale University Press, New Haven, CT, 1938)Google Scholar
  125. 125.
    I. Kukavica, A. Tuffaha, Well-posedness for the compressible Navier–Stokes–Lamé system with a free interface. Nonlinearity 25(11), 3111–3137 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  126. 126.
    K. Laganà, R. Balossino, F. Migliavacca, G. Pennati, G. Dubini, T.Y. Hsiab, M.R. de Leval, E.L. Bove, Multiscale modelling in biofluid-dynamics: application to reconstructive pediatric cardiac surgery. J. Biomech. 39(6), 1010–1020 (2006)CrossRefGoogle Scholar
  127. 127.
    E.M. Landis, Poiseuille’s law and capillary circulation. Am. J. Physiol. 103, 432–443 (1939)Google Scholar
  128. 128.
    A.T. Layton, A. Edwards, Mathematical Modeling in Renal Physiology. Lecture Notes on Mathematical Modeling in the Life Sciences (Springer, Berlin, 2013)Google Scholar
  129. 129.
    J. Lee, N.P. Smith, Theoretical modeling in hemodynamics of microcirculation. Microcirculation 15, 699–718 (2008)CrossRefGoogle Scholar
  130. 130.
    H. Lei, D.A. Fedosov, B. Caswell, G. Karniadakis, Blood flow in small tubes: quantifying the transition to the non-Newtonian regime. J. Fluid Mech. 722, 214–239 (2013)zbMATHCrossRefGoogle Scholar
  131. 131.
    J. Lequeurre, Existence of strong solutions to a fluid-structure system. SIAM J. Math. Anal. 43(1), 389–410 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  132. 132.
    J. Lequeurre, Existence of strong solutions for a system coupling the Navier–Stokes equations and a damped wave equation. J. Math. Fluid Mech. 15(2), 249–271 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  133. 133.
    P. Libby, P. Ridker, A. Maseri, Inflammation and atherosclerosis. Circulation 105(9), 1135–1143 (2002)CrossRefGoogle Scholar
  134. 134.
    D. Liepsch, St. Moravec, Pulsatile flow of non-Newtonian fluid in distensible models of human arteries. Biorheology 21, 571–586 (1984)CrossRefGoogle Scholar
  135. 135.
    J. Lieutaud, Elementa Physiologiae (Grune & Stratton, Amsterdam, 1749), pp. 82–84. Translated in C. Dreyfus, Milestones in the History of Haematology, New York, 1957, pp. 11–12Google Scholar
  136. 136.
    H.H. Lipowsky, Shear stress in the circulation, in Flow-Dependent Regulation of Vascular Function, ed. by J.A. Bevan, C. Kaley, G.M. Rubanyi (Oxford University Press, New York, 1995)Google Scholar
  137. 137.
    H.H. Lipowsky, Microvascular rheology and hemodynamics. Microcirculation 12, 5–15 (2005)CrossRefGoogle Scholar
  138. 138.
    B. Liu, D. Tang, Computer simulations of atherosclerosis plaque growth in coronary arteries. Mol. Cell. Biomech. 7(4), 193–202 (2010)Google Scholar
  139. 139.
    R. Loubere, P.-H. Maire, M. Shashkov, J. Breil, S. Galera. ReALE: a reconnection-based arbitrary-Lagrangian –Eulerian method. J. Comput. Phys. 229(12), 4724–4761 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  140. 140.
    G.D.O. Lowe (ed.), Clinical Blood Rheology, vols. I and II (CRC, Boca Raton, FL, 1998)Google Scholar
  141. 141.
    F. Lurie, R.L. Kistner, B. Eklof, D. Kessler, Mechanism of venous valve closure and role of the valve in circulation: a new concept. J. Vasc. Surg. 38, 955–961 (2003)CrossRefGoogle Scholar
  142. 142.
    A.M. Malek, S.L. Alper, S. Izumo, Hemodynamic shear stress and its role in atherosclerosis. J. Am. Med. Assoc. 282(21), 2035–2042 (1999)CrossRefGoogle Scholar
  143. 143.
    S.S Mao, N. Ahmadi, B. Shah, D. Beckmann, A. Chen, L. Ngo, F.R. Flores, Y.I. Gao, M.J. Budoff, Normal thoracic aorta diameter on cardiac computed tomography in healthy asymptomatic adult; Impact of age and gender. Acad. Radiol. 15, 827–834 (2008)Google Scholar
  144. 144.
    V.V. Matchkov, H. Gustafsson, A. Rahman, D.M. Boedtkjer, S. Gorintin, A.K. Hansen, E.V. Bouzinova, H.A. Praetoriu, C. Aalkjaer, H. Nilsson, Interaction between NaK pump and NaCa 2 exchanger modulates intercellular communication. Circ. Res. 100, 1026–1035 (2007)CrossRefGoogle Scholar
  145. 145.
    H. McCann, Pricking the Vessels: Bloodletting Therapy in Chinese Medicine (Singing Dragons, London, 2014)Google Scholar
  146. 146.
    D.A. McDonald, Blood Flow in Arteries, 2nd edn. (The Camelot, Southampton, 1974)Google Scholar
  147. 147.
    G. Mchedlishvili, Basic factors determining the hemorheological disorders in the microcirculation. Clin. Hemorheol. Microcirc. 30, 179–180 (2004)Google Scholar
  148. 148.
    C. McKay, S. McKee, N. Mottram, T. Mulholand, S. Wilson, Towards a model of atherosclerosis. Strathclyde Mathematics Research Report (2005)Google Scholar
  149. 149.
    E.W. Merrill, G.R. Cokelet, A. Britten, R.E. Wells, Non-Newtonian rheology of human blood. Effect of fibrinogen deduced by subtraction. Circ. Res. 13, 48–55 (1963)Google Scholar
  150. 150.
    E.W. Merrill, E.R. Gilliland, G.R. Cokelet, H. Shin, A. Britten, R.E. Wells Jr., Rheology of human blood, near and at zero flow. Effects of temperature and hematocrit level. Biophys. J. 3, 199–213 (1963)Google Scholar
  151. 151.
    E.W. Merrill, W.G. Margetts, G.C. Cokelet, E.R. Gilliland, The Casson equation and rheology of the blood near shear zero, in Proceedings Fourth International Congress on Rheology, Part 4, ed. by A.L. Copley (Interscience, New York, 1965), pp. 135–143Google Scholar
  152. 152.
    C. Meyer, G. de Vries, S.T. Davidge, D.C. Mayes, Reassessing the mathematical modeling of the contribution of vasomotion to vascular resistance. J. Appl. Physiol. 92, 888–889 (2002)Google Scholar
  153. 153.
    G.S. Mintz, M.N. Kotler, W.R. Parry, A.S. Iskandrian, S.A. Kane, Real-time inferior vena caval ultrasonography: normal and abnormal findings and its use in assessing right-heart function. Circulation 64, 1018–1025 (1981)CrossRefGoogle Scholar
  154. 154.
    M.E. Mitchell, A.N. Sidawy, The pathophysiology of atherosclerosis. Semin. Vasc. Surg. 11(3), 134–141 (1998)Google Scholar
  155. 155.
    S. Mitrovska, S. Jovanova, I. Matthiesen, C. Libermans, Atherosclerosis: Understanding Pathogenesis and Challenge for Treatment (Nova Science, New York, 2009)Google Scholar
  156. 156.
    P.C.F. Moller, J. Mewis, D. Bonn, Yield stress and thixotropy: on the difficulty of measuring yield stress in practice. Soft Matter 2, 274–288 (2006)CrossRefGoogle Scholar
  157. 157.
    C. Morris, H. Lecar, Voltage oscillations in the barnacle giant muscle fiber. Biophys. J. 35, 193–123 (1981)CrossRefGoogle Scholar
  158. 158.
    B. Muha, S. Čanić, Existence of a weak solution to a nonlinear fluid-structure interaction problem modeling the flow of an incompressible, viscous fluid in a cylinder with deformable walls. Arch. Ration. Mech. Anal. 207(3), 919–968 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  159. 159.
    B. Muha, S. Čanić, Existence of a solution to a fluid-multi-layered-structure interaction problem. J. Differ. Equ. 256, 658–706 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  160. 160.
    B. Muha, S. Čanić, Existence of a weak solution to a fluid-structure interaction problem with the Navier slip boundary condition. J. Differ. Equ. 260(12), 8550–8589 (2016)zbMATHMathSciNetCrossRefGoogle Scholar
  161. 161.
    C.D. Murray, The physiological principle of minimum work. I. The vascular system and the cost of blood volume. Proc. Natl. Acad. Sci. USA 12, 207–214 (1926)CrossRefGoogle Scholar
  162. 162.
    Q.D. Nguyen, D.V. Boger, Measuring the flow properties of yield stress fluids. Annu. Rev. 24, 47–88 (1992)zbMATHGoogle Scholar
  163. 163.
    M.K. O’Connell, S. Murthya, S. Phanb, C. Xuc, J. Buchanand, R. Spilker, R.L. Dalman, C.K. Zarins, W. Denk, C.A. Taylor, The three-dimensional micro- and nanostructure of the aortic medial lamellar unit measured using 3D Confocal & Electron Microscopy Imaging. Matrix Biol. 27(3), 171–181 (2008). doi:10.1016/j.matbio.2007.10.008CrossRefGoogle Scholar
  164. 164.
    M.S. Olufsen, Structured tree outflow condition for blood flow in larger systemic arteries. Am. J. Physiol. 276, H257–H268 (1999)Google Scholar
  165. 165.
    M.S. Olufsen, C.S. Peskin, W.Y. Kim, E.M. Pedersen, A. Nadim, J. Larsen, Numerical simulation and experimental validation of blood flow in arteries with structured tree outflow conditions. Ann. Biomed. Eng. 28, 1281–1299 (2000)CrossRefGoogle Scholar
  166. 166.
    J.T. Ottensen, M.S. Olufsen, J.K. Larsen, Applied Mathematical Models in Human Physiology. SIAM Monographs on Mathematical Modeling and Computation (SIAM, Philadelphia, 2004)Google Scholar
  167. 167.
    R.G. Owens, A new microstructure-based constitutive model for human blood. J. Non-Newtonian Fluid Mech. 14, 57–70 (2006)zbMATHCrossRefGoogle Scholar
  168. 168.
    L.A. Parapia, History of bloodletting by phlebotomy. Br. J. Hematol. 143, 490–495 (2008)Google Scholar
  169. 169.
    K.H. Parker, C.J. Jones, Forward and backward running waves in the arteries: analysis using the method of characteristics. J. Biomech. Eng. 112, 322–326 (1990)CrossRefGoogle Scholar
  170. 170.
    D. Parthimos, R.E. Haddock, C.E. Hill, T.M. Griffith, Dynamics of a three-variable nonlinear model of vasomotion: comparison of theory and experiment. Biophys. J. 93, 1534–1556 (2007)CrossRefGoogle Scholar
  171. 171.
    L. Pater, J.Wvd. Berg, An electrical analogue of the entire human circulatory system. Med. Electron. Biol. Eng. 2, 161–166 (1964)CrossRefGoogle Scholar
  172. 172.
    K. Perktold, D. Hilbert, Numerical solution of pulsatile flow in a carotid bifurcation. J. Biomed. Eng. 8, 193–199 (1986)CrossRefGoogle Scholar
  173. 173.
    K. Perktold, R. Peter, Numerical 3D-simulation of pulsatile wall shear stress in an arterial T-bifurcation model. J. Biomed. Eng. 12, 2–12 (1990)CrossRefGoogle Scholar
  174. 174.
    K. Perktold, G. Rappitsch, Computer simulation of local blood flow and vessel mechanics in a compliant carotid artery bifurcation model. J. Biomech. 28, 845–856 (1995)CrossRefGoogle Scholar
  175. 175.
    K. Perktold, M. Resh, R.O. Peter, Three-dimensional numerical analysis of pulsatile blood flow and wall shear stress in the carotid artery bifurcation. J. Biomech. 24, 409–420 (1991)CrossRefGoogle Scholar
  176. 176.
    C. Peskin, Flow patterns around heart valves. Ph.D. thesis, Albert Einstein College of Medicine, New York, 1972Google Scholar
  177. 177.
    C. Peskin, Flow patterns around heart valves: a numerical method. J. Comput. Phys. 10, 252–271 (1972)zbMATHMathSciNetCrossRefGoogle Scholar
  178. 178.
    C. Peskin, The immersed boundary method. Acta Numer. 11, 479–517 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  179. 179.
    C. Peskin, D. McQueen, A three-dimensional computational method for blood flow in the heart. J. Comput. Phys. 81, 372–405 (1989)zbMATHMathSciNetCrossRefGoogle Scholar
  180. 180.
    H.M. Poiseuille, Recherches expérimentales sur le movement des liquids dans les tubes de très petites diamètres. C. R. Acad. Sci. 11, 961–967 (1840)Google Scholar
  181. 181.
    A.S. Popel, P.C. Johnson, Microcirculation and hemorheology. Annu. Rev. Fluid Mech. 37, 43–69 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  182. 182.
    G. Porenta, D.F. Young, T.R. Rogge, A finite-element model of blood flow in arteries including taper, branches, and obstructions. J. Biomech. Eng. 108, 161–167 (1986)CrossRefGoogle Scholar
  183. 183.
    C. Pozrikidis, Axisymmetric motion of a file of red blood cells through capillaries. Phys. Fluids 17 (2005). doi:10.1063/1.1830484Google Scholar
  184. 184.
    A. Quarteroni, L. Formaggia, Mathematical modelling and numerical simulation of the cardiovascular system, in Computational Models for the Human Body, Special volume, ed. by N. Ayache (Guest editor). Handbook of Numerical Analysis, vol. XII, ed. by P.G. Ciarlet (Elsevier, North Holland, New York, 2004), pp. 7–127Google Scholar
  185. 185.
    A. Quarteroni, A. Veneziani, Analysis of a geometrical multiscale model based on the coupling of ODE’s and PDE’s for blood flow simulations. SIAM J. Multiscale Model. Simul. 1(2), 173–195 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  186. 186.
    A. Quarteroni, M. Tuveri, A. Veneziani, Computational vascular fluid dynamics: problems, models and method. Survey article. Comput. Vis. Sci. 2, 163–197 (2000)zbMATHCrossRefGoogle Scholar
  187. 187.
    A. Quarteroni, A. Veneziani, P. Zunino, Mathematical and numerical modeling of the solute dynamics in blood flow and arterial walls. SIAM J. Numer. Anal. 39, 1488–1511 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  188. 188.
    A. Quarteroni, A. Veneziani, C. Vergara, Geometric multiscale modeling of the cardiovascular system, between theory and practice. Comput. Methods Appl. Mech. Eng. 302, 193–252 (2016)MathSciNetCrossRefGoogle Scholar
  189. 189.
    A. Quarteroni, T. Lassila, S. Rossi, R. Ruiz-Baier, Integrated Heart - Coupling multiscale and multiphysics models for the simulation of the cardiac function. Comput. Methods Appl. Mech. Eng. 314, 345–407 (2017). doi:10.1016/j.cma.2016.05.031MathSciNetCrossRefGoogle Scholar
  190. 190.
    A. Quarteroni, A. Manzoni, C. Vergara, The cardiovascular system: mathematical modelling, numerical algorithms and clinical applications. Acta Numer. 1–225 (2017)Google Scholar
  191. 191.
    K.R. Rajagopal, A.R. Srinivasa, A thermodynamic frame work for rate type fluid models. J. Non-Newtonian Fluid Mech. 80, 207–227 (2000)zbMATHCrossRefGoogle Scholar
  192. 192.
    J.-P. Raymond, M. Vanninathan, A fluid-structure model coupling the Navier-Stokes equations and the Lamé system. J. Mat. Pura Appl. 102(3), 546–596 (2014)zbMATHCrossRefGoogle Scholar
  193. 193.
    T.J. Reape, P. Groot, Chemokines and atherosclerosis. Atherosclerosis 147, 213C225 (1999)Google Scholar
  194. 194.
    J.J. Reho, X. Zheng, S.A. Fisher, Smooth muscle contractile diversity in the control of regional circulations. Am. J. Physiol. Heart Circ. Physiol. 306, H163–H172 (2014)CrossRefGoogle Scholar
  195. 195.
    A. Remuzzi, B.M. Brenner, V. Pata, G. Tebaldi, R. Mariano, A. Belloro, G. Remuzzi, Three-dimensional reconstructed glomerular capillary network: blood flow distribution and local filtration. Am. J. Physiol. 263, F562–F572 (1992)Google Scholar
  196. 196.
    P. Reymond, F. Merenda, F. Perren, D. Rufenacht, N. Stergiopulos, Validation of a one-dimensional model of the systemic arterial tree. Am. J. Physiol. Heart Circ. Physiol. 297, H208–222 (2009)CrossRefGoogle Scholar
  197. 197.
    C. Rivadulla, C. de Labra, K.L. Grieve, J. Cudeiro, Vasomotion and neurovascular coupling in the visual thalamus in vivo. PLOS One 6, e28746 (2011)CrossRefGoogle Scholar
  198. 198.
    A.M. Robertson, A. Sequeira, A director theory approach for modeling blood flow in the arterial system: an alternative to classical 1D models. Math. Models Methods Appl. Sci. 15(6), 871–906 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  199. 199.
    A.M. Robertson, A. Sequeira, M.V. Kameneva, Hemorheology, in Hemodynamical Flows: Modeling, Analysis and Simulation, ed. by G.P. Galdi, R. Rannacher, A.M. Robertson, S. Turek. Oberwolfach Seminars, vol. 37 (Birkhäuser, Basel, 2008), pp. 63–120Google Scholar
  200. 200.
    A.M. Robertson, A. Sequeira, R.G. Owens, Hemorheology, in Cardiovascular Mathematics. Modeling and Simulation of the Circulatory System, ed. by L. Formaggia, A. Quarteroni, A. Veneziani, vol. 1 (Springer, Berlin, 2009), pp. 211–242Google Scholar
  201. 201.
    M.C. Roco (ed.), Particulate Two-Phase Flow. Series in Chemical Engineering (Butterworth-Heinemann, Boston, 1993)Google Scholar
  202. 202.
    E. Rooz, D.F. Young, T.R. Rogge, A finite-element simulation of pulsatile flow in flexible obstructed tubes. J. Biomech. Eng. 104, 119–124 (1982)CrossRefGoogle Scholar
  203. 203.
    R. Ross, Atherosclerosis - an inflammatory disease. Mass. Med. Soc. 340(2), 115–126 (1999)Google Scholar
  204. 204.
    S. Rossi, Anisotropic modeling of cardiac mechanical activation. Ph.D. in Mathematics, IST, Lisbon, and EPFL, 2014Google Scholar
  205. 205.
    S. Rossi, R. Ruiz-Baier, L.F. Pavarino, A. Quarteroni, Orthotropic active strain models for the numerical simulation of cardiac biomechanics. Int. J. Numer. Methods Biomed. Eng. 28, 761–788 (2012)MathSciNetCrossRefGoogle Scholar
  206. 206.
    S. Rossi, T. Lassila, R. Ruiz-Baier, A. Sequeira, A. Quarteroni, Thermodynamically consistent orthotropic activation model capturing ventricular systolic wall thickening in cardiac electromechanics. Eur. J. Mech. A. Solids 48, 129–142 (2014)MathSciNetCrossRefGoogle Scholar
  207. 207.
    D. Saha, M. Patgaonkar, A. Shroff, K. Ayyar, T. Bashir, K.V.R. Reddy, Hemoglobin expression in nonerythroid cells: novel or ubiquitous? Int. J. Inflamm. 2014, 8 pp. (2014)Google Scholar
  208. 208.
    H. Schmid-Schönbein, R.E. Wells, Rheological properties of human erythrocytes and their influence upon anomalous viscosity of blood. Physiol. Rev. 63, 147–219 (1971)Google Scholar
  209. 209.
    G.W. Scott-Blair, An equation for the flow of blood, plasma and serum through glass capillaries. Nature 183, 613–614 (1959)CrossRefGoogle Scholar
  210. 210.
    T.W. Secomb, Mechanics and computational simulation of blood flow in microvessels. Med. Eng. Phys. 33, 800–804 (2010)CrossRefGoogle Scholar
  211. 211.
    J. Serrin, Mathematical principles of classical fluid mechanics, in Handbuch der Physik, vol. VIII/I, ed. by S. Flugge, C. Truesdell (Springer, Berlin, 1959)Google Scholar
  212. 212.
    S.J. Sherwin, L. Formaggia, J. Peiró, V. Frank, Computational modeling of 1D blood flow with variable mechanical properties and application to the simulation of wave propagation in the human arterial system. Int. J. Numer. Methods Fluids 43, 673–700 (2003)zbMATHCrossRefGoogle Scholar
  213. 213.
    S.J. Sherwin, V. Frank, J. Peiró, One-dimensional modelling of a vascular network in space-time variables. J. Eng. Math. 47(3–4), 217–250 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  214. 214.
    T. Silva, Mathematical modeling of the atherosclerosis physiopathology. Ph.D. thesis, University of Lisbon, 2016Google Scholar
  215. 215.
    T. Silva, A. Sequeira, R.F. Santos, J. Tiago, Existence, uniqueness, stability and asymptotic behavior of solutions for a mathematical model of atherosclerosis. Discrete Cont. Dyn. Syst. Ser. S 9(1), 343–362 (2016)zbMATHMathSciNetGoogle Scholar
  216. 216.
    T. Silva, J. Tiago, A. Sequeira, Mathematical analysis and numerical simulations for a model of atherosclerosis, in Mathematical Fluid Dynamics, Present and Future, ed. by Y. Shibata, Y. Suzuki. Springer Proceedings in Mathematics & Statistics (Springer, Berlin, 2016), pp. 577–595. ISBN 978-4-431-56455-3Google Scholar
  217. 217.
    D.U. Silverthorn, Human Physiology. An Integrated Approach, 7th edn. (Prentice Hall, Upper Saddle River, NJ, 2015)Google Scholar
  218. 218.
    C. Stefanadis, M. Karamanou, G. Androutsos, Michael Servetus (1511–1553) and the discovery of pulmonary circulation. Hell. J. Cardiol. 50, 373–378 (2009)Google Scholar
  219. 219.
    N. Stergiopulos, B.E. Westerhof, N. Westerhof, Total arterial inertance as the fourth element of the Windkessel model. Am. J. Physiol. 276, H81–H88 (1999)Google Scholar
  220. 220.
    T.P. Stossel, The early history of phagocytosis, in Phagocytosis: The Host, ed. by S. Gordon (JAI, Stamford, CT, 1999)Google Scholar
  221. 221.
    M. Stücker, J. Steinbrügge, C. Ihrig, K. Hoffmann, D. Ihrig, A. Röchling, D.W. Lübbers, H. Jungmann, P. Altmeyer, Rhythmical variations of haemoglobin oxygenation in cutaneous capillaries. Acta Derm. Venereol. 78, 408–411 (1998)CrossRefGoogle Scholar
  222. 222.
    I. Surovtsova, Effects of compliance mismatch on blood flow in an artery with endovascular prosthesis. J. Biomech. 38, 2078–2086 (2005)CrossRefGoogle Scholar
  223. 223.
    J.M. Tarbell, Mass transport in arteries and the localisation of atherosclerosis. Annu. Rev. Biomed. Eng. 5, 79–118 (2003)CrossRefGoogle Scholar
  224. 224.
    M.G. Taylor, The input impedance of an assembly of randomly branching elastic tubes. Biophys. J. 6, 29–51 (1966)CrossRefGoogle Scholar
  225. 225.
    M. Thiriet, Biology and Mechanics of Blood Flows. Part I: Biology. CRM Series in Mathematical Physics (Springer, Berlin, 2008)Google Scholar
  226. 226.
    M. Thiriet, Biology and Mechanics of Blood Flows. Part II: Mechanics and Medical Aspects. CRM Series in Mathematical Physics (Springer, Berlin, 2008)Google Scholar
  227. 227.
    M. Thiriet, K.H. Parker, Physiology and pathology of the cardiovascular system: a physical perspective, in Cardiovascular Mathematics. Modeling and Simulation of the Circulatory System, ed. by L. Formaggia, A. Quarteroni, A. Veneziani, vol. 1 (Springer, Berlin, 2009), pp. 1–46Google Scholar
  228. 228.
    G.B. Thurston, Viscoelasticity of human blood. Biophys. J. 12, 1205–1217 (1972)CrossRefGoogle Scholar
  229. 229.
    G.B. Thurston, Non-Newtonian viscosity of human blood: flow induced changes in microstructure. Biorheology 31(2), 179–192 (1994)CrossRefGoogle Scholar
  230. 230.
    G.B. Thurston, Viscoelastic properties of blood and blood analogs. Adv. Hemodyn. Hemorheol. 1, 1–30 (1996)CrossRefGoogle Scholar
  231. 231.
    L.N. Toksvang, R.M.G. Berg, Using a classic paper by Robin Fåhraeus and Torsten Lindqvist to teach basic hemorheology. Adv. Physiol. Educ. 37(2), 129–133 (2013)CrossRefGoogle Scholar
  232. 232.
    E.F. Toro, L.O. Müller, M. Cristini, E. Menegatti, P. Zamboni, Impact of jugular vein valve function on cerebral venous haemodynamics. Curr. Neurovasc. Res. 12(4), 384–397 (2015)CrossRefGoogle Scholar
  233. 233.
    M. Ursino, G. Fabbri, E. Belardinelli, A mathematical analysis of vasomotion in the peripheral vascular bed. Cardioscience 3, 13–25 (1992)Google Scholar
  234. 234.
    A.C. van der Wal, A.E. Becker, Atherosclerotic plaque rupture-pathologic basis of plaque stability and instability. Cardiovasc. Res. 41, 334–344 (1999)CrossRefGoogle Scholar
  235. 235.
    F.J. Walburn, D.J. Schneck, A constitutive equation for whole human blood. Biorheology 13, 201–210 (1976)CrossRefGoogle Scholar
  236. 236.
    J.J. Wang, K.H. Parker, Wave propagation in a model of the arterial circulation. J. Biomech. 37, 457–470 (2004)CrossRefGoogle Scholar
  237. 237.
    S.L. Waters, J. Alastruey, D.A. Beard, P.H.M. Bovendeerd, P.F. Davies, G. Jayaraman, O.E. Jensen, J. Lee, K.H. Parker, A.S. Pople, T.W. Secomb, S.J. Sherwin, R.J. Shipley, N.P. Smith, F. van de Vosse, Theoretical models for coronary vascular biomechanics: progress & challenges. Prog. Biophys. Mol. Biol. 104(1–3), 49–76 (2011)CrossRefGoogle Scholar
  238. 238.
    J.B. West, Ibn al-Nafis, The pulmonary circulation, and the Islamic Golden Age. J. Appl. Physiol. 105(6), 1877–1880 (2008)CrossRefGoogle Scholar
  239. 239.
    N. Westerhof, F. Bosman, C.J. De Vries, A. Noordergraaf, Analog studies of the human systemic arterial tree. J. Biomech. 2: 121–143 (1969)CrossRefGoogle Scholar
  240. 240.
    L. Wexler, D.H. Bergel, I.T Gabe, G.S. Makin, C.J. Mills, Velocity of blood flow in normal human venae cavae. Circ. Res. 23, 349–359 (1968)Google Scholar
  241. 241.
    I.S. Whitaker, J. Rao, D. Izadi, P.E Butler, Historical article: hirudo medicinalis: ancient origins of, and trends in the use of medicinal leeches throughout history. Br. J. Oral Maxillofac. Surg. 42, 133–137 (2004)Google Scholar
  242. 242.
    T. Wick, Fluid-structure interactions using different mesh motion techniques. Comput. Struct. 89(13–14), 1456–1467 (2011)CrossRefGoogle Scholar
  243. 243.
    R.P. Wideman, Effect of venous flow on frequency of venous vasomotion in the bat wing. Circ. Res. 5, 641–644 (1957)CrossRefGoogle Scholar
  244. 244.
    Y. Yang, Mathematical modeling and simulation of the evolution of plaques in blood vessels. Ph.D. thesis, Heidelberg University, 2014Google Scholar
  245. 245.
    Y. Yang, W. Jäger, M. Neuss-Radu, T. Richter, Mathematical modeling and simulation of the evolution of plaques in blood vessels. J. Math. Biol. 72(4), 973–996 (2016)zbMATHMathSciNetCrossRefGoogle Scholar
  246. 246.
    K.K. Yeleswarapu, M.V. Kameneva, K.R. Rajagopal, J.F. Antaki, The flow of blood in tubes: theory and experiment. Mech. Res. Commun. 25(3), 257–262 (1998)zbMATHCrossRefGoogle Scholar
  247. 247.
    B.W. Zweifach, Quantitative studies of microcirculatory structure and function. I: analysis of pressure distribution in the terminal vascular bed in cat mesentery. Circ. Res. 34, 843–857 (1974)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Antonio Fasano
    • 1
  • Adélia Sequeira
    • 2
  1. 1.Fabbrica Italiana Apparecchi Biomedicali (FIAB)Università degli Studi di FirenzeFirenzeItaly
  2. 2.Instituto Superior TécnicoUniversidade de LisboaLisboaPortugal

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