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Hemorheology and Hemodynamics

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Hemomath

Part of the book series: MS&A ((MS&A,volume 18))

Abstract

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.

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Notes

  1. 1.

    Perhaps the most emblematic case is the miracle of the “liquefaction” of Saint Januarius’ blood. It is less known that the blood of many other saints periodically exhibits the same phenomenon. Saint Januarius (San Gennaro), the patron of Naples, died as a martyr around 305 ad, beheaded at the Solfatara Crater. According to the legend, his blood was collected by a pious woman (Eusebia) and preserved till our days (after many vicissitudes) in a sealed transparent container. A thousand of years later the saint’s relics ended in Naples and in 1389 chronicles started reporting the miracle. In our days the blood is exposed three times a year and upon mild agitation (most of the times) the reddish dust turns into liquid. Many explanations have been attempted for this intriguing phenomenon (including some thixotropic mixture prepared in the middle ages [93]). A professor of molecular biology of the University Federico II of Naples (prof. G. Geraci) has performed experiments with a sample of old blood saved in a sealed vial and with his own blood, suitably aged, reproducing the same “miracle”, thus concluding that “liquefaction” may well be just a blood property [94].

  2. 2.

    For a history of ancient Egyptian medicine see e.g. [59].

  3. 3.

    Though all kinds of body fluids were thought to flow through the heart. For instance semen was believed to be provided to testicles by two dedicated vessels (Ebers Papyrus).

  4. 4.

    The translation [21] with splendid reproductions of the Smith Papyrus is available on line.

  5. 5.

    The English translation [22] (parallel to [21]) is available on line.

  6. 6.

    Dates largely uncertain.

  7. 7.

    One of the oldest surviving text is Charaka Samhita (Fig. 1.5), written in Sanskrit (fifth to third century bc).

  8. 8.

    Two classes of medical approaches can be adopted to recuperate the broken equilibrium: Samana (for light alterations), and Shodana aimed at expelling the corrupted doshas. Panchakarma belongs to the second class.

  9. 9.

    Physical manifestations of yin and yang are natural opposites like dark and light, male and female, life and death, moist and dry, sun and moon, etc.

  10. 10.

    One more proof of how humans have tried to interpret life and the physical world on the basic of recurrent principles. Striking similarities can be found also among the many myths explaining creation.

  11. 11.

    A legendary, semi-divine figure between myth and history, he was credited with the invention of almost anything which started Chinese civilization.

  12. 12.

    In this paper it is reported how extremely heavy bloodletting accompanied the last days of Charles II and of George Washington.

  13. 13.

    Erasistratus (304–250 bc), a renowned Greek physician active in Syria, came to the conclusion that heart is a pump. Though he made some remarkable progress in understanding the role of brain and nerves, he believed (as many others) that arteries carried the “spiritual substance” (pneuma). In other words, arteries were full of air and if by chance they were contaminated by blood it would have caused illness. It is amazing that at that time there was a controversy about the path of drunk liquids: somebody claimed that they went to the trachea (a name probably due to Erisistratus himself) eventually reaching the lungs. He stated instead that anything which is ingested goes through the esophagus to the stomach. He opposed the practice of bloodletting. Herophilus (335–280 bc) established that the brain, and not the heart, was in control of motion.

  14. 14.

    The practice was legal in Alexandria.

  15. 15.

    He was particularly adversed by his former teacher, Franciscus de la Boe (1478–1555), better known as Jacobus Sylvius or Jacques Duboi, who was an irreducible worshiper of Galen’s works, to the point that if he had to admit that something in the human body was different from what Galen had described, then the mistake was in the body, in the sense that it had modified since Galen’s time. Sylvius was nevertheless the author of important discoveries in brain anatomy.

  16. 16.

    The list of scientists condemned by the Catholic Church for heresy is impressive. The case of Galileo Galilei is emblematic (1632). Despite their reputation, non-catholic Christian churches were not more tolerant about heresy, which most of the times consisted just in interpretations of the Scriptures (or of real world) different from the ones officially adopted by this or that Confession. An emblematic case was the one of the Spanish born Michael Servetus (1509–1553), Vesalius contemporary, and also a physiologist, who dared oppose Galen’s authority, providing his own description of pulmonary circulation [218]. Serveto was burnt at the stake (alive and with sulfur on his head) in Geneva, victim of the fanatic hate of John Calvin because of his refusal of the concept of Trinity.

  17. 17.

    We must recall that about three centuries earlier Ala-al-Din Abu al-Hasan Ali Ibn Abi al-Hazm al-Qarshi al-Dimashqi (known as Ibn Al-Nafis, 1213–1288), among his many other discoveries, described pulmonary circulation [238].

  18. 18.

    This is a good place to mention a remarkable scientist, the German physiologist Adolf Eugen Fick (1829–1901), mainly known for two things: (1) the Fick’s principle for the determination of the cardiac output on the basis of the ratio between oxygen consumption and the arterio-venous oxygen difference; (2) the formulation of Fick’s law for diffusion, paralleling Fourier’s law for heat conduction.

  19. 19.

    Cells were first observed at the microscope by the physicist Robert Hooke (1635–1703) in a thin sample of cork (1665). He did not know what “cells” were, but he called them that way because of their particular and regular arrangement in the sample, resembling the one of monks cells. Hooke was an extremely versatile man in many disciplines. He can be considered the founder of elasticity theory, but he was also an engineer, astronomer, a paleontologist, an architect (a friend of Christopher Wren), Unfortunately he was in contrast with the president of the Royal Society, Isaac Newton, and that obscured his reputation in life. Only two centuries later cells were identified as the basic constituents of all living beings (by Theodor Schwann (1810–1882), a German physiologist, and Matias Jakob Schleiden (1804–1881), a German botanist).

  20. 20.

    Malpighi first described RBC’s as fat corpuscles (1663). Malpighi was also the discoverer of capillaries (1661) and of the filtrating units in kidneys, bearing his name. It is worth mentioning here an extraordinary character, Athanasius Kircher (1602–1680), a German Jesuit scholar, who wrote a great number of books in different areas. Kircher is mostly remembered for having built a machine for automatic music composition (the Arca Musurgia, 1650). An expert microbiologist, he was rightly convinced that the plague was caused by a microorganism (today known as the Yersinia pestis bacterium) that he thought to have found in blood with the help of the microscope in 1658. Most probably he had instead observed RBCs, the same year as Swammerdam.

  21. 21.

    There is some controversy about the discovery of platelets, generally attributed to the French physician Alfred Donné (1801–1878) in 1842.

  22. 22.

    It seems however that even Anthony Leeuwenhoek had observed them as early as 1678. Many others described cells of that kind in pus and other physiological fluids (see [220]). The French anatomist Joseph Lieutaud (1703–1780) called them “globuli albicans” (1749) [135]. The lymphatic system was then described by William Hewson (1739–1774) [109].

  23. 23.

    He formerly identified groups A, B, C, but “C” was later changed to “0” (zero). He was also the discoverer of the polio virus. The story of the ABO groups is actually more complicated. It is today recognized that the Czech serologist Jan Janský (1873–1921) had provided the complete 4-group classification (including group AB) before Landsteiner, who had nevertheless worked independently.

  24. 24.

    From the name of the monkey species (Rhesus) used in their tests.

  25. 25.

    The isovolumic contraction is a short transient phase preceding actual ventricles contraction, just before the sudden aortic pressure raise, during which the mitral valve closes (first heart sound). The isovolumetric relaxation is an equally short phase, accompanying the rapid aortic pressure drop, during which the mitral valve closes (second heart sound).

  26. 26.

    The values presented in this table are very crude estimates. It is quite difficult to find a complete and coherent table of this type when we compare all data provided by the different authors (see e.g. [166, 226, 227]). We also quote the paper [240]. Systolic values are of course much larger (e.g. peak velocity in aorta 65 cm/s, in venae cavae 38 cm/s, or more, though there are differences between superior and inferior v.c.). A simple calculation of the peak velocity in aorta can be done by dividing the blood volume ejected by the left ventricle (heart stroke = 50 ml) by the ejection time (300 ms) and by the aortic valve area (2.5 cm2), which gives 67 cm/s. Papers referring specifically to these main vessels are [153, 240].

  27. 27.

    The SI derived unit for pressure is the Pascal. It is equivalent to one Newton per square meter. The unit is named after Blaise Pascal (1623–1662), the eminent French mathematician, physicist and philosopher. 1 mmHg = 133. 32 Pa. The mmHg is also named Torr after Evangelista Torricelli (1608–1647), Italian physicist and mathematician.

  28. 28.

    A particular role in flow through veins is played by valves, representing a help in vessels with moderate pressure gradients. Some veins and venules are equipped with valves, whose importance will be discussed in several instances [15]. A study of the dynamics of vein valves has been performed in [141].

  29. 29.

    Knowing the velocity and pressure fields it is possible to obtain stresses, in particular the wall shear stress (WSS) which is the force per unit area exerted by the fluid tangentially to the vessel wall. WSS may cause alterations in the endothelium and has a great influence in many inflammatory diseases, including atherosclerosis, the development of aneurysms and clotting.

  30. 30.

    Studies on 1D models of blood flow were first introduced by the Swiss mathematician Leonhard Euler (1707–1783) in his seminal work [64].

  31. 31.

    System (1.4) was proposed for the first time by Claude-Louis Navier in 1822 and later by George Stokes in 1845.

  32. 32.

    Named after Osborne Reynolds (1842–1912).

  33. 33.

    Marfan’s syndrome is named after Antoine Marfan (1858–1942), a French pediatrician who first described the condition in 1896. It is a genetic disorder 75% of the times inherited from a parent and 25% a new mutation, equally affecting males and females. Marfan syndrome involves a mutation to the gene resulting in connective tissue where the most serious complications are related to the heart and aorta, with an increased risk of valvular heart disease (mitral valve prolapse, aortic regurgitation) and aortic aneurysm.

  34. 34.

    A hyperelastic material is characterized by a nonlinear stress-strain response [18, 45].

  35. 35.

    Among the papers in which a shell model has been adopted we quote [32], containing a large introduction to the subject.

  36. 36.

    As for the fluid, it is not clear from physical arguments which should be the appropriate boundary conditions to impose on the artificial sections Γ Σ, in 0 and Γ Σ, out 0. It is important to refer that the geometrical multiscale approach has provided satisfactory answers [80]. Moreover, since the structure equations will be coupled with the fluid equations, the initial velocity for the fluid and the displacement should be compatible, i.e. \(\frac{\partial \eta _{0}} {\partial t} = \mathbf{u_{0}}\) on Γ w 0. 

  37. 37.

    The ALE approach will be considered only at the discrete level. In fact, due to time discretization, the domain changes from one time step to another and the ALE method through the introduction of the ALE time derivative that we will define here, allows to follow the evolution of quantities associated to the mesh nodes.

  38. 38.

    The problem with slip boundary conditions would require the specification of the slip velocity in terms of the normal stress at the vessel wall. As far as we know this is still a debated question.

  39. 39.

    The term multiscale is often used whenever two or more time and/or scales are present, like in turbulence modeling or when molecular and cellular-level biological phenomena are combined with macro-scale fluid and tissue mechanics. The term geometrical has been added to identify the present multiscale strategy, avoiding ambiguities.

  40. 40.

    The theoretical basis of this approach developed by Naghdi and co-authors (see e.g. [97, 99]) is similar to that used for rods in solid mechanics [98].

  41. 41.

    We will see in various instances that flows in vessels whose length is much larger, that the radius satisfy these conditions up to corrections which are of the same order as the radius/length ratio.

  42. 42.

    More complex algebraic relations can be considered (e.g. [108]). Here we prefer to use definition (1.16) since it separates the effects of the wall mechanical properties (Young modulus) from the geometrical ones (A 0).

  43. 43.

    In this Eq. (1.20) the term \(\frac{\partial A_{0}} {\partial z}\) may account for a variation of the area at the rest state, as in the case of tappering or stenosis, and the term \(\frac{\partial \beta } {\partial z}\) is related to the longitudinal variation of the mechanical properties of the vessel.

  44. 44.

    Under physiological hemodynamic conditions the flow regime is assumed to be subcritical and the typical values for c 1 and Q∕A are such that c 1 ≫ Q∕A. Indeed, characteristic values for c 1 are of the order of 103 m/s [236] whereas | Q∕A | is of the order of 10 [165].

  45. 45.

    This model was introduced by the German physiologist Otto Frank (1865–1944) in the late 1800s who described the heart and the systemic arterial system as a closed hydraulic circuit. He called it Windkessel in analogy with the device made by a water pump connected to a chamber, filled with water except for a pocket of air. As it is pumped, the water compresses the air, which in turn pushes the water out of the chamber.

  46. 46.

    It means the property of a system of being in a stable equilibrium as a result of interacting processes.

  47. 47.

    Albumin was discovered in urine in 1500 by one of the most famous physician and philosopher of that time: Philippus Aureolus Theophrastus Bombastus von Hohenheim (1493–1541), better known as Paracelsus, a name indicating that he was at least as great as the Roman physician Celsus.

  48. 48.

    Proteins heavier that albumin, divided in four groups: α 1, α 2, β, and γ.

  49. 49.

    Also known as erythrocyte volume fraction. The name hematocrit was coined in 1891 by the Swedish physician Magnus Blix (1849–1904).

  50. 50.

    Hemoglobin was discovered in 1840 by Friedrich Ludwig Hünefeld (1799–1882), a German physician and chemist. Drying the blood of an earthworm between two glass slides he observed the formation of red crystalline structures. The role of hemoglobin, however, became clear much later. Felix Hoppe-Seyler (1825–1895), German physiologist and chemist, gave it its name (1864) and recognized that it combines with oxygen. Among the many scientists that gave fundamental contributions to the study of hemoglobin we mention the celebrated French physician Claude Bernard (1813–1878) and the Austrian born Nobel laureate molecular biologist Max Perutz (1914–2002), who established its structure consisting of four elements, each with an iron atom, able to bind with oxygen. CO 2 and CO bind to other sites of the molecule. Hemoglobin is known to be synthesized by numerous cells other than erythrocytes for various purposes. See the review paper [207].

  51. 51.

    For a non-Newtonian fluid, independently of the specific rheological model, one can define an apparent viscosity as the quantity measured by a viscometer (which is somehow an average measure of the fluid resistance to flow) for shear rates in the expected natural range. The term relative viscosity is also used meaning the ratio of the suspension viscosity (apparent viscosity) to the viscosity of the suspending fluid (plasma). Commonly used viscosity units are: poise (P)-named after the French physician Jean Louis Marie Poiseuille (1799–1869), centipoise (cP), which is 0.01 P and pascal-second (Pa s), the SI unit of viscosity equivalent to newton-second per square metre (Ns/m2). One poise is exactly 0.1 Pa s.

  52. 52.

    Robin S. Fårhaeus (1888–1968) and Torsten Lindqvist (1906–2007) were Swedish hematologists who described the effect in 1931. An earlier paper by a German team (P. Martini, A. Pierach, E. Scheryer, 1930) on the same issue unfortunately went unnoticed (see [231]).

  53. 53.

    Blood viscoelasticity was mentioned for the first time in this paper.

  54. 54.

    There is a large variety of published definitions for thixotropy, in the fields of industrial or biological applications. The following definition can be found in [14]: “When a reduction in magnitude of rheological properties of a system, such as elastic modulus, yield stress, and viscosity, for example, occurs reversibly and isothermally with a distinct time dependence on application of shear strain, the system is described as thixotropic”. Fluids whose behavior is opposite to thixotropic fluid (i.e. thickening under stress) are called rheopectic.

  55. 55.

    On top of that, experimenting with blood out of the body can find many obstacles. The simple process of extracting blood may apply high stresses, altering the original rheological properties. Then partial coagulation, particularly in the absence of flow, can severely influence the value of yield stress.

  56. 56.

    For isochoric motions the first invariant I D = tr D(u) is identically zero.

  57. 57.

    The presence of ϕ 2 is necessary to match experimental results on “real” fluids. The dependence on the value of II D is often neglected [7].

  58. 58.

    When not differently stated the data we report refer to room temperature.

  59. 59.

    Formulated by the British mathematician James G. Oldroyd (1921–1982).

  60. 60.

    Fenestrations in renal glomeruli are impervious to molecules whose molecular weight exceeds ∼ 12 kDa. The presence of albumin in urine is linked to pathological conditions (e.g. diabetes nephropathy).

  61. 61.

    The consequent oscillatory behavior of blood flow is called flowmotion.

  62. 62.

    The regulation of microvessels perfusion is an immensely complicated topic. See for instance the massive review [52] with its 34 pages of references. Section 6.2.1 is devoted to the role of vasomotion in flow autoregulation.

  63. 63.

    The non-solid complement of the nucleus in an eukaryote cell is the cytoplasm, which consist of a fluid, the cytosol, and of the organelles, performing various functions, and usually possessing a membrane. Among the organelles of eukaryotic cells the endoplasmic reticula, characterized by the presence of cisternae (in the form of sacs and tubes) have different structure in different organs, and in myocytes (muscle cells) we find the specialized version of sarcoplasmic reticulum, storing and releasing calcium ions.

  64. 64.

    A messenger molecule diffusing in the sarcoplasmic reticulum and triggering the opening of Ca ++ channels in the cell membrane.

  65. 65.

    A protein playing a major role in cells motility and in particular in skeletal muscle contraction.

  66. 66.

    Formula (1.57) produces the simple form (1.59) of the dimensionless Navier-Stokes equations. The actual maximal velocity provided by the Hagen-Poiseuille formula is 1/4 of that.

  67. 67.

    There is some controversy about the discovery of vein valves. According to A. Caggiati (a renown scientist in the field) the first to describe them was the Catalan Ludovicus Vassaeus in his De Anatomen Corporis Humani tabulae quator (Venice, 1542). See [26–28]. There were more or less contemporary or even earlier observations. In the old review paper [85] reference is made to unpublished findings on the subject by Giambattista Canano (1515–1579), professor of anatomy in Ferrara, that he reported to Vesalius in 1545. In the same year veins valves were described by the French anatomist Charles Estienne (1504–1564), who however claimed to have discovered them in 1538.

  68. 68.

    Notation is different from the case of arterioles, since it is preferable to have the maximum radius as a reference length.

  69. 69.

    Arteriosclerosis is the hardening, thickening or loss of elasticity of the artery walls [76].

  70. 70.

    Cytokines and chemokines are small proteins made by cells of the immune system, both important in regulating responses to infection or injury by using chemical signals [193].

  71. 71.

    Molecules able to produce oxidative damage to cells.

References

  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)

    Article  Google Scholar 

  2. W.C. Aird, Discovery of the cardiovascular system: from Galen to William Harvey. J. Thromb. Haemost. 9(Suppl. 1), 118–129 (2011)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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. M. Anand, J. Kwack, A. Masud, A new Oldroyd-B model for blood in complex geometries. Int. J. Eng. Sci. 72, 78–88 (2013)

    Article  Google Scholar 

  7. G. Astarita, G. Marrucci, Principles of Non-Newtonian Fluid Mechanics (McGraw Hill, New York, 1974)

    MATH  Google Scholar 

  8. A.P. Avolio, Multi-branched model of the human arterial system. Med. Biol. Eng. Comp. 18, 709–718 (1980)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  10. H.A. Barnes, Thixotropy - a review. J. Non-Newtonian Fluid Mech. 70, 1–33 (1997)

    Article  Google Scholar 

  11. P. Barter, The role of HDL–cholesterol in preventing atherosclerotic disease. Eur. Heart J. Suppl. 7, 4–8 (2005)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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. 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. 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)

    Article  Google Scholar 

  16. L.S. Beale, The Microscope in Medicine (J. & A. Churchill, London, 1877)

    Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  18. A. Bertram, Elasticity and Plasticity of Large Deformations, An Introduction, chap. 7 Hyperelasticity (Springer, Berlin, 2012), pp. 209–229

    Google Scholar 

  19. E.C. Bingham, An investigation of the laws of plastic flow. U.S. Bur. Stand. Bull. 13, 309–353 (1916)

    Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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. C.P. Bryan, The Papyrus Ebers (Geoffrey Bles, London, 1930)

    Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  24. M. Bulelzai, J. Dubbeldam, Long time evolution of atherosclerotic plaques. J. Theor. Biol. 297, 1–10 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  26. A. Caggiati, The venous valves in the lower limbs. Phlebolymphology 20, 87–95 (2013)

    Google Scholar 

  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. A. Caggiati, M. Phillips, A. Lametschwandtner, C. Allegra, Valves in small veins and venules. Eur. J. Vasc. Endovasc. Surg. 32, 447–452 (2006)

    Article  Google Scholar 

  29. V. Calvez, A. Ebde, N. Meunier, A. Raoult, Mathematical and numerical modeling of the atherosclerotic plaque formation. ESAIM Proc. 28, 1–12 (2009)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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. 79–195

    Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  37. S.E. Charm, G.S. Kurland, Blood Flow and Microcirculation (Wiley, New York, 1974)

    Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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. P. Ciarlet, Mathematical Elasticity. Three-Dimensional Elasticity, vol. 1, 2nd edn. (Elsevier, Amsterdam, 2004)

    Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  49. C. Cobbold, J. Sherratt, S. Mexwell, Lipoprotein oxidation and its significance for atherosclerosis: a mathematical approach. Bull. Math. Biol. 64, 65–95 (2002)

    Article  MATH  Google Scholar 

  50. D. Coutand, S. Shkoller, The interaction between quasilinear elastodynamics and the Navier–Stokes equations. Arch. Ration. Mech. Anal. 179(3), 303–352 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  51. P. Cullen, J. Rauterberg, S. Lorkowski, The pathogenesis of atherosclerosis. Handb. Exp. Pharmacol. 170, 3–70 (2005)

    Article  Google Scholar 

  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. 161–284

    Google Scholar 

  53. C. de Wit, Closing the gap at hot spots. Circ. Res. 100, 931–933 (2007)

    Article  Google Scholar 

  54. L. Dintenfass, Blood Microrheology - Viscosity Factors in Blood Flow. Ischaemia and Thrombosis (Butterworth, Oxford, 1971)

    Google Scholar 

  55. L. Dintenfass, Blood Viscosity, Hyperviscosity and Hyperviscosaemia (MTP, Lancaster, 1985)

    Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  59. N.I. Ebeid, Egyptian Medicine in the Days of the Pharaohs (General Egyptian Book Organization, Cairo, 1999)

    Google Scholar 

  60. N. El Khatib, S. Génieys, V. Volpert, Atherosclerosis initiation modeled as an inflammatory process. Math. Model. Nat. Phenom. 2(2), 126–141 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  62. N. El Khatib, S. Génieys, B. Kazmierczak, V. Volpert, Reaction-diffusion model of atherosclerosis development. J. Math. Biol. 65, 349–374 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  63. P. Espaõl, Fluid particle model. Phys. Rev. E 57(3), 2930–2948 (1998)

    Article  Google Scholar 

  64. L. Euler, Principia pro motu sanguinis per arterias determinando, in Opera Omnia, ed. by L. Euler, vol. 16(2) (Birkhäuser, Basel, 1989), pp. 178–196

    Google Scholar 

  65. E.A. Evans, R.M. Hochmuth, Membrane viscoelasticity. Biophys. J. 16(1), 1–11 (1976)

    Article  Google Scholar 

  66. R. Fåhraeus, The suspension stability of blood. Physiol. Rev. 9, 241–274 (1929)

    Google Scholar 

  67. R. Fåhraeus, T. Lindqvist, The viscosity of blood in narrow capillary tubes. Am. J. Physiol. 96, 362–368 (1931)

    Google Scholar 

  68. A. Farina, A. Fasano, Incompressible flows though slender oscillating vessels provided with distributed valves. Adv. Math. Sci. Appl. 25, 33–42 (2016)

    MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  71. J.J. Feher, Quantitative Human Physiology: An Introduction (Elsevier, Academic, Amsterdam, 2012)

    Google Scholar 

  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. 307–346

    Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  76. G.A. Fishbein, M.C. Fishbein, Arteriosclerosis: rethinking the current classification. Arch. Pathol. Lab. Med. 133, 1309–1316 (2009)

    Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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. 46–75

    Google Scholar 

  82. L. Formaggia, A. Quarteroni, A. Veneziani (eds.), Cardiovascular Mathematics. Modeling and Simulation of the Circulatory System, vol. 1 (Springer, Berlin, 2009), pp. 46–75

    Google Scholar 

  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. 395–446

    Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  85. K.J. Franklin, Valves in veins: an historical survey. Proc. R. Soc. Med. Sect. Hist. Med. 21, 1–33 (1927)

    Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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.2015.02.002

    Google Scholar 

  88. Y.C. Fung, Biomechanics: Mechanical Properties of Living Tissues (Springer, Berlin, 1993)

    Book  Google Scholar 

  89. Y.C. Fung, Biomechanics: Circulation (Springer, Berlin, 1997)

    Book  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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. L. Garlaschelli, F. Ramaccini, S. Della Sala, Working bloody miracles. Nature 353(6344), 507 (1991)

    Google Scholar 

  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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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)

    Article  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  100. G. Greenstone, The history of bloodletting. Br. Columbia Med. J. 52, 12–14 (2010)

    Google Scholar 

  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. A.C. Guyton, J.E. Hall, A Textbook of Medical Physiology, 10th edn. (W.B. Saunders, Philadelphia, 2000)

    Google Scholar 

  103. R.E. Haddock, C.E. Hill, Rhythmicity in arterial smooth muscle. J. Physiol. 566, 645–656 (2005)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  105. S.I. Hajdu, A note from history: the discovery of blood cells. Ann. Clin. Lab. Sci. 33, 237–238 (2003)

    Google Scholar 

  106. G. Hansson, P. Libby, The immune response in atherosclerosis: a double edged sword. Nat. Immunol. 6, 508–519 (2006)

    Article  Google Scholar 

  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. K. Hayashi, K. Handa, S. Nagasaka, A. Okumura, Stiffness and elastic behaviour of human intracranial and extracranial arteries. J. Biomech. 13, 175–184 (1980)

    Article  Google Scholar 

  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. 30

    Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  111. G.A. Holzapfel, R.W. Ogden, Mechanics of Biological Tissue (Springer, Berlin, 2006)

    Book  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  115. M. Intaglietta, Vasomotion and flowmotion: physiological mechanisms and clinical evidence. Vasc. Med. Rev. 2, 1101–112 (1990)

    Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  119. J. Keener, J. Sneyd, Mathematical Physiology. II, System Physiology. Interdisciplinary Applied Mathematics, vol. 8/II, 2nd edn. (Springer, Berlin, 2009)

    Google Scholar 

  120. S.R. Keller, R. Shalak, Motion of a tank-treading ellipsoidal particle in a shear flow. J. Fluid Mech. 120, 27–47 (1982)

    Article  MATH  Google Scholar 

  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. 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)

    Article  Google Scholar 

  123. M. Koenigsberger, R. Sauser, J.L. Bény, J.J. Meister, Effects of arterial wall stress on vasomotion. Biophys. J. 91, 1663–1674 (2006)

    Article  Google Scholar 

  124. A. Krogh, The Anatomy and Physiology of the Capillaries (Yale University Press, New Haven, CT, 1938)

    Google Scholar 

  125. I. Kukavica, A. Tuffaha, Well-posedness for the compressible Navier–Stokes–Lamé system with a free interface. Nonlinearity 25(11), 3111–3137 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  127. E.M. Landis, Poiseuille’s law and capillary circulation. Am. J. Physiol. 103, 432–443 (1939)

    Google Scholar 

  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. J. Lee, N.P. Smith, Theoretical modeling in hemodynamics of microcirculation. Microcirculation 15, 699–718 (2008)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  131. J. Lequeurre, Existence of strong solutions to a fluid-structure system. SIAM J. Math. Anal. 43(1), 389–410 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  133. P. Libby, P. Ridker, A. Maseri, Inflammation and atherosclerosis. Circulation 105(9), 1135–1143 (2002)

    Article  Google Scholar 

  134. D. Liepsch, St. Moravec, Pulsatile flow of non-Newtonian fluid in distensible models of human arteries. Biorheology 21, 571–586 (1984)

    Article  Google Scholar 

  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. 11–12

    Google Scholar 

  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. H.H. Lipowsky, Microvascular rheology and hemodynamics. Microcirculation 12, 5–15 (2005)

    Article  Google Scholar 

  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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  140. G.D.O. Lowe (ed.), Clinical Blood Rheology, vols. I and II (CRC, Boca Raton, FL, 1998)

    Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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 Na∕K pump and Na∕Ca 2 exchanger modulates intercellular communication. Circ. Res. 100, 1026–1035 (2007)

    Article  Google Scholar 

  145. H. McCann, Pricking the Vessels: Bloodletting Therapy in Chinese Medicine (Singing Dragons, London, 2014)

    Google Scholar 

  146. D.A. McDonald, Blood Flow in Arteries, 2nd edn. (The Camelot, Southampton, 1974)

    Google Scholar 

  147. G. Mchedlishvili, Basic factors determining the hemorheological disorders in the microcirculation. Clin. Hemorheol. Microcirc. 30, 179–180 (2004)

    Google Scholar 

  148. C. McKay, S. McKee, N. Mottram, T. Mulholand, S. Wilson, Towards a model of atherosclerosis. Strathclyde Mathematics Research Report (2005)

    Google Scholar 

  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. 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. 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. 135–143

    Google Scholar 

  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. 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)

    Article  Google Scholar 

  154. M.E. Mitchell, A.N. Sidawy, The pathophysiology of atherosclerosis. Semin. Vasc. Surg. 11(3), 134–141 (1998)

    Google Scholar 

  155. S. Mitrovska, S. Jovanova, I. Matthiesen, C. Libermans, Atherosclerosis: Understanding Pathogenesis and Challenge for Treatment (Nova Science, New York, 2009)

    Google Scholar 

  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)

    Article  Google Scholar 

  157. C. Morris, H. Lecar, Voltage oscillations in the barnacle giant muscle fiber. Biophys. J. 35, 193–123 (1981)

    Article  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  159. B. Muha, S. Čanić, Existence of a solution to a fluid-multi-layered-structure interaction problem. J. Differ. Equ. 256, 658–706 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  162. Q.D. Nguyen, D.V. Boger, Measuring the flow properties of yield stress fluids. Annu. Rev. 24, 47–88 (1992)

    MATH  Google Scholar 

  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.2007.10.008

    Article  Google Scholar 

  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. 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)

    Article  Google Scholar 

  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. R.G. Owens, A new microstructure-based constitutive model for human blood. J. Non-Newtonian Fluid Mech. 14, 57–70 (2006)

    Article  MATH  Google Scholar 

  168. L.A. Parapia, History of bloodletting by phlebotomy. Br. J. Hematol. 143, 490–495 (2008)

    Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  171. L. Pater, J.Wvd. Berg, An electrical analogue of the entire human circulatory system. Med. Electron. Biol. Eng. 2, 161–166 (1964)

    Article  Google Scholar 

  172. K. Perktold, D. Hilbert, Numerical solution of pulsatile flow in a carotid bifurcation. J. Biomed. Eng. 8, 193–199 (1986)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  176. C. Peskin, Flow patterns around heart valves. Ph.D. thesis, Albert Einstein College of Medicine, New York, 1972

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  179. C. Peskin, D. McQueen, A three-dimensional computational method for blood flow in the heart. J. Comput. Phys. 81, 372–405 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  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. A.S. Popel, P.C. Johnson, Microcirculation and hemorheology. Annu. Rev. Fluid Mech. 37, 43–69 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  183. C. Pozrikidis, Axisymmetric motion of a file of red blood cells through capillaries. Phys. Fluids 17 (2005). doi:10.1063/1.1830484

    Google Scholar 

  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. 7–127

    Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  186. A. Quarteroni, M. Tuveri, A. Veneziani, Computational vascular fluid dynamics: problems, models and method. Survey article. Comput. Vis. Sci. 2, 163–197 (2000)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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.2016.05.031

    Article  MathSciNet  Google Scholar 

  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. K.R. Rajagopal, A.R. Srinivasa, A thermodynamic frame work for rate type fluid models. J. Non-Newtonian Fluid Mech. 80, 207–227 (2000)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  193. T.J. Reape, P. Groot, Chemokines and atherosclerosis. Atherosclerosis 147, 213C225 (1999)

    Google Scholar 

  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)

    Article  Google Scholar 

  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. 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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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. 63–120

    Google Scholar 

  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. 211–242

    Google Scholar 

  201. M.C. Roco (ed.), Particulate Two-Phase Flow. Series in Chemical Engineering (Butterworth-Heinemann, Boston, 1993)

    Google Scholar 

  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)

    Article  Google Scholar 

  203. R. Ross, Atherosclerosis - an inflammatory disease. Mass. Med. Soc. 340(2), 115–126 (1999)

    Google Scholar 

  204. S. Rossi, Anisotropic modeling of cardiac mechanical activation. Ph.D. in Mathematics, IST, Lisbon, and EPFL, 2014

    Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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. 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. G.W. Scott-Blair, An equation for the flow of blood, plasma and serum through glass capillaries. Nature 183, 613–614 (1959)

    Article  Google Scholar 

  210. T.W. Secomb, Mechanics and computational simulation of blood flow in microvessels. Med. Eng. Phys. 33, 800–804 (2010)

    Article  Google Scholar 

  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. 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)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  214. T. Silva, Mathematical modeling of the atherosclerosis physiopathology. Ph.D. thesis, University of Lisbon, 2016

    Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  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 978-4-431-56455-3

    Google Scholar 

  217. D.U. Silverthorn, Human Physiology. An Integrated Approach, 7th edn. (Prentice Hall, Upper Saddle River, NJ, 2015)

    Google Scholar 

  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. 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. T.P. Stossel, The early history of phagocytosis, in Phagocytosis: The Host, ed. by S. Gordon (JAI, Stamford, CT, 1999)

    Google Scholar 

  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)

    Article  Google Scholar 

  222. I. Surovtsova, Effects of compliance mismatch on blood flow in an artery with endovascular prosthesis. J. Biomech. 38, 2078–2086 (2005)

    Article  Google Scholar 

  223. J.M. Tarbell, Mass transport in arteries and the localisation of atherosclerosis. Annu. Rev. Biomed. Eng. 5, 79–118 (2003)

    Article  Google Scholar 

  224. M.G. Taylor, The input impedance of an assembly of randomly branching elastic tubes. Biophys. J. 6, 29–51 (1966)

    Article  Google Scholar 

  225. M. Thiriet, Biology and Mechanics of Blood Flows. Part I: Biology. CRM Series in Mathematical Physics (Springer, Berlin, 2008)

    Google Scholar 

  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. 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. 1–46

    Google Scholar 

  228. G.B. Thurston, Viscoelasticity of human blood. Biophys. J. 12, 1205–1217 (1972)

    Article  Google Scholar 

  229. G.B. Thurston, Non-Newtonian viscosity of human blood: flow induced changes in microstructure. Biorheology 31(2), 179–192 (1994)

    Article  Google Scholar 

  230. G.B. Thurston, Viscoelastic properties of blood and blood analogs. Adv. Hemodyn. Hemorheol. 1, 1–30 (1996)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. A.C. van der Wal, A.E. Becker, Atherosclerotic plaque rupture-pathologic basis of plaque stability and instability. Cardiovasc. Res. 41, 334–344 (1999)

    Article  Google Scholar 

  235. F.J. Walburn, D.J. Schneck, A constitutive equation for whole human blood. Biorheology 13, 201–210 (1976)

    Article  Google Scholar 

  236. J.J. Wang, K.H. Parker, Wave propagation in a model of the arterial circulation. J. Biomech. 37, 457–470 (2004)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  238. J.B. West, Ibn al-Nafis, The pulmonary circulation, and the Islamic Golden Age. J. Appl. Physiol. 105(6), 1877–1880 (2008)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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. T. Wick, Fluid-structure interactions using different mesh motion techniques. Comput. Struct. 89(13–14), 1456–1467 (2011)

    Article  Google Scholar 

  243. R.P. Wideman, Effect of venous flow on frequency of venous vasomotion in the bat wing. Circ. Res. 5, 641–644 (1957)

    Article  Google Scholar 

  244. Y. Yang, Mathematical modeling and simulation of the evolution of plaques in blood vessels. Ph.D. thesis, Heidelberg University, 2014

    Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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 

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    Antonio Fasano

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    Adélia Sequeira

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Fasano, A., Sequeira, A. (2017). Hemorheology and Hemodynamics. In: Hemomath. MS&A, vol 18. Springer, Cham. https://doi.org/10.1007/978-3-319-60513-5_1

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