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Hemodynamics

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ABSTRACT

A review is presented of the physical principles governing the distribution of blood flow and blood pressure in the vascular system. The main factors involved are the pulsatile driving pressure generated by the heart, the flow characteristics of blood, and the geometric structure and mechanical properties of the vessels. The relationship between driving pressure and flow in a given vessel can be understood by considering the viscous and inertial forces acting on the blood. Depending on the vessel diameter and other physical parameters, a wide variety of flow phenomena can occur. In large arteries, the propagation of the pressure pulse depends on the elastic properties of the artery walls. In the microcirculation, the fact that blood is a suspension of cells strongly influences its flow properties and leads to a nonuniform distribution of hematocrit among microvessels. The forces acting on vessel walls include shear stress resulting from blood flow and circumferential stress resulting from blood pressure. Biological responses to these forces are important in the control of blood flow and the structural remodeling of vessels, and also play a role in major disease processes including hypertension and atherosclerosis. Consideration of hemodynamics is essential for a comprehensive understanding of the functioning of the circulatory system. © 2016 American Physiological Society. Compr Physiol 6:975‐1003, 2016.

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Figure 1. Figure 1. Schematic representation of the systemic circulation as a network of resistances. (A) Basic elements of the systemic circulation. The pressure gradient between arterial pressure PA generated by the left heart and venous pressure PV drives blood through a network of blood vessels, consisting of the arteries, the microcirculation and the veins. Vascular segments are indicated by zigzag symbols, as in electrical circuits. The pulmonary circulation (not shown) has the same overall structure. (B and C) Hemodynamic interactions within a network of resistances, with flow driven by a pressure difference P1P2. Arrays of dots signify additional levels of branching in the network. (B) Increased flow resistance in one segment (*) (e.g., due to constriction or occlusion) causes a decrease in flow along all flow pathways containing that segment (dashed lines). (C) Decreased flow resistance along one flow pathway (*) (e.g., due to formation of a shunt pathway) causes increased flow on that pathway (heavy black lines) but reduced flow on parallel pathways (dashed lines).
Figure 2. Figure 2. Illustration of concepts underlying the definition of the stress tensor in a material. (A) The stress vector or traction T is defined as the force per unit area acting on a small surface ΔS in the material. In general, this vector has components parallel to the surface (shear force) and normal to the surface (normal force). (B) The traction acting on an arbitrarily oriented surface can be fully described in terms of the stress tensor σ. Each component σij of the stress tensor represents the i component of the traction acting on a surface oriented perpendicular to the coordinate axis xj. (C) The net force on a small cuboid of material resulting from a stress in the material is zero if the stress is uniform, because the traction vectors acting on opposite faces of the cuboid are equal and opposite. However, if the stress distribution is not uniform, the traction vectors do not cancel and a net force is generated.
Figure 3. Figure 3. Definition of simple shear flow of a fluid.
Figure 4. Figure 4. Coordinate systems used to describe the deformation of a body in continuum mechanics.
Figure 5. Figure 5. (A) Definition of geometry and forces, used in derivation of Poiseuille's law. (B) Parabolic velocity profile in Poiseuille flow.
Figure 6. Figure 6. Dependence of the bulk viscosity of human blood on hematocrit, for indicated shear rates. Curves are derived from polynomial expressions given by Chien et al. (), based on measurements using a coaxial‐cylinder viscometer.
Figure 7. Figure 7. Dependence of the relative bulk viscosity on shear rate for three different types of red blood cell suspensions as described in the text. Vertical arrows indicate effect of aggregation to increase viscosity relative to nonaggregating cells at very low shear rates, and effect of deformation to decrease viscosity relative to rigid cells, an effect that increases with shear rate.
Figure 8. Figure 8. Dimensions and numbers of vessels of various classification in the canine vasculature, based on observations of the mesenteric vascular bed by Mall (). Also included is an estimate of flow velocity in each type of segment, assuming a cardiac output of 2 L/min. Dashed lines at lower right hand side of figure indicate diameters of arteries corresponding to the veins of each classification, to show the difference in diameters between arteries and veins.
Figure 9. Figure 9. Intravascular pressure as a function of vessel diameter in different tissues and species. The “present data” refers to data obtained from mathematical model calculations for six mesenteric networks (). The other data are from Zweifach (), Gore (), Richardson and Zweifach () and Fronek and Zweifach (). Figure reproduced with permission from Pries et al. ().
Figure 10. Figure 10. Analysis of stresses in a pressurized cylindrical tube. (A) Thin‐walled theory, for a segment of length L and radius r. Pressure forces acting on the wall (dashed arrows) must balance tension in the wall (solid arrows), implying the Law of Laplace, T = (pipo)r. (B) Thick‐walled theory. The balance of forces is applied to a thin cylindrical shell of radius r and thickness dr. See text for details.
Figure 11. Figure 11. Schematic illustration of the mechanics of pulse propagation in an artery. Graphs show spatial variation of pressure and flow rate. Large arrow shows direction of propagation. Gray area represents artery, and small arrows indicate local fluid velocities. (A) Short high‐pressure pulse propagating in positive x‐direction. At the leading edge of the pulse, fluid is accelerated by the negative pressure gradient. This produces a negative spatial gradient of flow rate. By conservation of mass, fluid accumulates in this region, and wall must move outward. At the trailing edge of the pulse, fluid is decelerated by the positive pressure gradient, producing a positive spatial gradient of flow rate, and inward wall movement. (B) Short high‐pressure pulse propagating in positive x‐direction. Mechanism is as in A, but with reversed velocities. Note that an arbitrary (positive or negative) x‐independent velocity can be superimposed on the indicated velocities without affecting the mechanism. The x‐scale is greatly compressed here for illustrative purposes. In reality, the systolic pulse wave is much longer than the diameter (and the length) of the artery.
Figure 12. Figure 12. Analysis of wave propagation at an arterial bifurcation. An incident wave in branch 0 gives rise to transmitted waves in branches 1 and 2 and a reflected wave in branch 0.
Figure 13. Figure 13. Sequences of velocity profiles in a tube with a sinusoidally varying pressure gradient, for indicated values of unsteadiness parameter α. Velocity profiles represent one half of a complete cycle of the oscillation. Bottom profile corresponds to moment of maximum pressure gradient.
Figure 14. Figure 14. Development of boundary layer (shaded area) in fluid entering a tube. Velocity profiles indicate approach to fully developed flow.
Figure 15. Figure 15. Sketch of flow phenomena occurring during steady flow in a human carotid artery bifurcation, based on observations in a transparent postmortem sample (). Dashed lines indicate fluid streamlines. Curves across vessel diameters indicate local velocity profiles. Shaded area indicates region of flow separation, with separation point at the upstream end and reattachment point at the downstream end.
Figure 16. Figure 16. Two‐phase model for blood flow in a microvessel, with radius a. A central core region containing red blood cells, with viscosity μc and radius λa, is surrounded by a cell‐free or cell‐depleted layer, with viscosity μp and width δ. A typical resulting velocity profile is shown.
Figure 17. Figure 17. Variation of apparent viscosity with tube diameter for hematocrit HD = 0.45. The lower solid curve represents an empirical fit to experimental in vitro data (). The upper solid curve represents the dependence deduced from in vivo experiments (). The dashed curve corresponds to a two‐phase model with cell‐free layer width 1.8 μm, as discussed in the text.
Figure 18. Figure 18. Red blood cell partition in diverging microvascular bifurcations. Curves giving red blood cell flux fraction in one branch as a function of overall flow fraction entering that branch are derived from empirically derived relationships as described in the text, assuming a discharge hematocrit of 0.4 in the parent vessel. Assumed diameters of parent vessel, DF, and branches, Dα and Dβ, are indicated on each plot. (A) Symmetric bifurcation. (B) Asymmetric bifurcation.


Figure 1. Schematic representation of the systemic circulation as a network of resistances. (A) Basic elements of the systemic circulation. The pressure gradient between arterial pressure PA generated by the left heart and venous pressure PV drives blood through a network of blood vessels, consisting of the arteries, the microcirculation and the veins. Vascular segments are indicated by zigzag symbols, as in electrical circuits. The pulmonary circulation (not shown) has the same overall structure. (B and C) Hemodynamic interactions within a network of resistances, with flow driven by a pressure difference P1P2. Arrays of dots signify additional levels of branching in the network. (B) Increased flow resistance in one segment (*) (e.g., due to constriction or occlusion) causes a decrease in flow along all flow pathways containing that segment (dashed lines). (C) Decreased flow resistance along one flow pathway (*) (e.g., due to formation of a shunt pathway) causes increased flow on that pathway (heavy black lines) but reduced flow on parallel pathways (dashed lines).


Figure 2. Illustration of concepts underlying the definition of the stress tensor in a material. (A) The stress vector or traction T is defined as the force per unit area acting on a small surface ΔS in the material. In general, this vector has components parallel to the surface (shear force) and normal to the surface (normal force). (B) The traction acting on an arbitrarily oriented surface can be fully described in terms of the stress tensor σ. Each component σij of the stress tensor represents the i component of the traction acting on a surface oriented perpendicular to the coordinate axis xj. (C) The net force on a small cuboid of material resulting from a stress in the material is zero if the stress is uniform, because the traction vectors acting on opposite faces of the cuboid are equal and opposite. However, if the stress distribution is not uniform, the traction vectors do not cancel and a net force is generated.


Figure 3. Definition of simple shear flow of a fluid.


Figure 4. Coordinate systems used to describe the deformation of a body in continuum mechanics.


Figure 5. (A) Definition of geometry and forces, used in derivation of Poiseuille's law. (B) Parabolic velocity profile in Poiseuille flow.


Figure 6. Dependence of the bulk viscosity of human blood on hematocrit, for indicated shear rates. Curves are derived from polynomial expressions given by Chien et al. (), based on measurements using a coaxial‐cylinder viscometer.


Figure 7. Dependence of the relative bulk viscosity on shear rate for three different types of red blood cell suspensions as described in the text. Vertical arrows indicate effect of aggregation to increase viscosity relative to nonaggregating cells at very low shear rates, and effect of deformation to decrease viscosity relative to rigid cells, an effect that increases with shear rate.


Figure 8. Dimensions and numbers of vessels of various classification in the canine vasculature, based on observations of the mesenteric vascular bed by Mall (). Also included is an estimate of flow velocity in each type of segment, assuming a cardiac output of 2 L/min. Dashed lines at lower right hand side of figure indicate diameters of arteries corresponding to the veins of each classification, to show the difference in diameters between arteries and veins.


Figure 9. Intravascular pressure as a function of vessel diameter in different tissues and species. The “present data” refers to data obtained from mathematical model calculations for six mesenteric networks (). The other data are from Zweifach (), Gore (), Richardson and Zweifach () and Fronek and Zweifach (). Figure reproduced with permission from Pries et al. ().


Figure 10. Analysis of stresses in a pressurized cylindrical tube. (A) Thin‐walled theory, for a segment of length L and radius r. Pressure forces acting on the wall (dashed arrows) must balance tension in the wall (solid arrows), implying the Law of Laplace, T = (pipo)r. (B) Thick‐walled theory. The balance of forces is applied to a thin cylindrical shell of radius r and thickness dr. See text for details.


Figure 11. Schematic illustration of the mechanics of pulse propagation in an artery. Graphs show spatial variation of pressure and flow rate. Large arrow shows direction of propagation. Gray area represents artery, and small arrows indicate local fluid velocities. (A) Short high‐pressure pulse propagating in positive x‐direction. At the leading edge of the pulse, fluid is accelerated by the negative pressure gradient. This produces a negative spatial gradient of flow rate. By conservation of mass, fluid accumulates in this region, and wall must move outward. At the trailing edge of the pulse, fluid is decelerated by the positive pressure gradient, producing a positive spatial gradient of flow rate, and inward wall movement. (B) Short high‐pressure pulse propagating in positive x‐direction. Mechanism is as in A, but with reversed velocities. Note that an arbitrary (positive or negative) x‐independent velocity can be superimposed on the indicated velocities without affecting the mechanism. The x‐scale is greatly compressed here for illustrative purposes. In reality, the systolic pulse wave is much longer than the diameter (and the length) of the artery.


Figure 12. Analysis of wave propagation at an arterial bifurcation. An incident wave in branch 0 gives rise to transmitted waves in branches 1 and 2 and a reflected wave in branch 0.


Figure 13. Sequences of velocity profiles in a tube with a sinusoidally varying pressure gradient, for indicated values of unsteadiness parameter α. Velocity profiles represent one half of a complete cycle of the oscillation. Bottom profile corresponds to moment of maximum pressure gradient.


Figure 14. Development of boundary layer (shaded area) in fluid entering a tube. Velocity profiles indicate approach to fully developed flow.


Figure 15. Sketch of flow phenomena occurring during steady flow in a human carotid artery bifurcation, based on observations in a transparent postmortem sample (). Dashed lines indicate fluid streamlines. Curves across vessel diameters indicate local velocity profiles. Shaded area indicates region of flow separation, with separation point at the upstream end and reattachment point at the downstream end.


Figure 16. Two‐phase model for blood flow in a microvessel, with radius a. A central core region containing red blood cells, with viscosity μc and radius λa, is surrounded by a cell‐free or cell‐depleted layer, with viscosity μp and width δ. A typical resulting velocity profile is shown.


Figure 17. Variation of apparent viscosity with tube diameter for hematocrit HD = 0.45. The lower solid curve represents an empirical fit to experimental in vitro data (). The upper solid curve represents the dependence deduced from in vivo experiments (). The dashed curve corresponds to a two‐phase model with cell‐free layer width 1.8 μm, as discussed in the text.


Figure 18. Red blood cell partition in diverging microvascular bifurcations. Curves giving red blood cell flux fraction in one branch as a function of overall flow fraction entering that branch are derived from empirically derived relationships as described in the text, assuming a discharge hematocrit of 0.4 in the parent vessel. Assumed diameters of parent vessel, DF, and branches, Dα and Dβ, are indicated on each plot. (A) Symmetric bifurcation. (B) Asymmetric bifurcation.
References
 1.Alastruey J, Khir AW, Matthys KS, Segers P, Sherwin SJ, Verdonck PR, Parker KH, Peiro J. Pulse wave propagation in a model human arterial network: Assessment of 1‐D visco‐elastic simulations against in vitro measurements. J Biomech 44: 2250‐2258, 2011.
 2. Asmar R , Rudnichi A , Blacher J , London GM , Safar ME . Pulse pressure and aortic pulse wave are markers of cardiovascular risk in hypertensive populations. Am J Hypertens 14: 91‐97, 2001.
 3. Avolio AP , Van Bortel LM , Boutouyrie P , Cockcroft JR , McEniery CM , Protogerou AD , Roman MJ , Safar ME , Segers P , Smulyan H . Role of pulse pressure amplification in arterial hypertension: Experts' opinion and review of the data. Hypertension 54: 375‐383, 2009.
 4. Batchelor GK . An Introduction to Fluid Mechanics. Cambridge: Cambridge University Press, 1967.
 5. Bertram CD , Butcher KS . Possible sources of discrepancy between sphygmomanometer cuff pressure and blood pressure quantified in a collapsible‐tube analogue. J Biomech Eng 114: 68‐77, 1992.
 6. Bovendeerd PH , Borsje P , Arts T , van de Vosse FN . Dependence of intramyocardial pressure and coronary flow on ventricular loading and contractility: A model study. Ann Biomed Eng 34: 1833‐1845, 2006.
 7. Burton AC. Physiology and Biophysics of the Circulation. Chicago: Year Book Medical Publishers, 1972.
 8. Buschmann I , Pries A , Styp‐Rekowska B , Hillmeister P , Loufrani L , Henrion D , Shi Y , Duelsner A , Hoefer I , Gatzke N , Wang H , Lehmann K , Ulm L , Ritter Z , Hauff P , Hlushchuk R , Djonov V , van Veen T , le Noble F . Pulsatile shear and Gja5 modulate arterial identity and remodeling events during flow‐driven arteriogenesis. Development 137: 2187‐2196, 2010.
 9.Canic S, Tambaca J, Guidoboni G, Mikelic A, Hartley CJ, Rosenstrauch D. Modeling viscoelastic behavior of arterial walls and their interaction with pulsatile blood flow. SIAM Journal on Applied Mathematics 67: 164‐193, 2006.
 10. Cardamone L , Valentin A , Eberth JF , Humphrey JD . Origin of axial prestretch and residual stress in arteries. Biomech Model Mechanobiol 8: 431‐446, 2009.
 11. Carlson BE , Arciero JC , Secomb TW . Theoretical model of blood flow autoregulation: Roles of myogenic, shear‐dependent, and metabolic responses. Am J Physiol Heart Circ Physiol 295: H1572‐H1579, 2008.
 12. Caro CG , Pedley TJ , Schroter RC , Seed WA . The Mechanics of the Circulation. Oxford: Oxford University Press, 1978.
 13. Chien S. Shear dependence of effective cell volume as a determinant of blood viscosity. Science 168: 977‐979, 1970.
 14. Chien S , Usami S , Taylor HM , Lundberg JL , Gregersen MI . Effects of hematocrit and plasma proteins on human blood rheology at low shear rates. J Appl Physiol 21: 81‐87, 1966.
 15. Coppola G , Caro C . Arterial geometry, flow pattern, wall shear and mass transport: Potential physiological significance. J R Soc Interface 6: 519‐528, 2009.
 16. Davies JE , Alastruey J , Francis DP , Hadjiloizou N , Whinnett ZI , Manisty CH , Aguado‐Sierra J , Willson K , Foale RA , Malik IS , Hughes AD , Parker KH , Mayet J . Attenuation of wave reflection by wave entrapment creates a “horizon effect” in the human aorta. Hypertension 60: 778‐785, 2012.
 17. Dobrin PB . Vascular mechanics. Compr Physiol (Suppl 8): 65‐102, 2011.
 18. Ellwein LM , Pope SR , Xie A , Batzel JJ , Kelley CT , Olufsen MS . Patient‐specific modeling of cardiovascular and respiratory dynamics during hypercapnia. Math Biosci 241: 56‐74, 2013.
 19. Euler L . Principia pro motu sanguinis per arterias determinando. In: Fuss PH , Fuss N , editors. Opera posthuma mathematica et physica anno 1844 detecta, Vol. 2. Petropoli: Apund Eggers et Socios, 1862, pp. 814‐823.
 20. Fåhraeus R . Die Strömungsverhältnisse und die Verteilung der Blutzellen im Gefäßsystem. Zur Frage der Bedeutung der intravasculären Erythrocytenaggregation. Klin Wochenschr 7: 100‐106, 1928.
 21. Fåhraeus R , Lindqvist T . The viscosity of the blood in narrow capillary tubes. Am J Physiol 96: 562‐568, 1931.
 22. Fedosov DA , Caswell B , Popel AS , Karniadakis GE . Blood flow and cell‐free layer in microvessels. Microcirculation 17: 615‐628, 2010.
 23. Fischer TM , Stöhr‐Liesen M , Schmid‐Schönbein H . The red cell as a fluid droplet: Tank tread‐like motion of the human erythrocyte membrane in shear flow. Science 202: 894‐896, 1978.
 24. Fishman AP , Richards DW . Circulation of the Blood: Men and Ideas. New York: Oxford University Press, 1964.
 25. Frank O . Die Grundform des arteriellen Pulses. Erste Abhandlung. Mathematische Analyse. Zeitschrift für Biologie 37: 483‐526, 1899.
 26. Fronek K , Zweifach BW . Microvascular pressure distribution in skeletal muscle and the effect of vasodilation. Am J Physiol 228: 791‐796, 1975.
 27. Fung YC. Biomechanics. Mechanical Properties of Living Tissues. Second Edition. New York: Springer, 1993.
 28. Fung YC. Biomechanics: Circulation. Second edition. New York: Springer‐Verlag, 1997.
 29. Fung YC , Liu SQ . Strain distribution in small blood vessels with zero‐stress state taken into consideration. Am J Physiol 262: H544‐H552, 1992.
 30. Gore RW . Pressures in cat mesenteric arterioles and capillaries during changes in systemic arterial blood pressure. Circ Res 34: 581‐591, 1974.
 31. Grandchamp X , Coupier G , Srivastav A , Minetti C , Podgorski T . Lift and down‐gradient shear‐induced diffusion in red blood cell suspensions. Phys Rev Lett 110: 2013.
 32. Greenwald SE , Carter AC , Berry CL . Effect of age on the in vitro reflection coefficient of the aortoiliac bifurcation in humans. Circulation 82: 114‐123, 1990.
 33. Greve JM , Les AS , Tang BT , Draney Blomme MT , Wilson NM , Dalman RL , Pelc NJ , Taylor CA . Allometric scaling of wall shear stress from mice to humans: Quantification using cine phase‐contrast MRI and computational fluid dynamics. Am J Physiol Heart Circ Physiol 291: H1700‐H1708, 2006.
 34. Hagenbach E . Uber die Bestimmung der Zähigkeit einer Flüssigkeit durch den Ausfluss aus Röhren. Poggendorf's Annalen der Physik und Chemie 108: 385‐426, 1860.
 35. Hales S. Statical essays: containing haemosticks, reprinted 1964, No. 22, History of Medicine series, Library of New York Academy of Medicine. New York: Hafner, 1733.
 36. Halpern D , Secomb TW . The squeezing of red blood cells through capillaries with near‐minimal diameters. J Fluid Mech 203: 381‐400, 1989.
 37. Hariprasad DS , Secomb TW . Two‐dimensional simulation of red blood cell motion near a wall under a lateral force. Physical Review E 90: 053014, 2014.
 38. Harvey W. Exercitatio Anatomica de Motu Cordis et Sanguinis in Animalibus. 1628. English translation with annotations, C.D. Leake. 4th edition. Springfield, Ill.: Thomas, 1958.
 39. Holzapfel GA , Gasser TC , Ogden RW . A new constitutive framework for arterial wall mechanics and a comparative study of material models. J Elast 61: 1‐48, 2000.
 40. Humphrey JD , Delange SL . An Introduction to Biomechanics. Solids and Fluids, Analysis and Design. New York: Springer, 2004.
 41. Johnson PC . The myogenic response. In: Bohr DF , Somlyo AP , Sparks HV, Jr ., editors. Handbook of Physiology, Section 2, The Cardiovascular System, Vol. II: Vascular Smooth Muscle. Bethesda, MD: American Physiological Society, 1980, pp. 409‐442.
 42. Jones CJ , Parker KH , Hughes R , Sheridan DJ . Nonlinearity of human arterial pulse wave transmission. J Biomech Eng 114: 10‐14, 1992.
 43. Kilner PJ , Yang GZ , Firmin DN . Morphodynamics of flow through sinuous curvatures of the heart. Biorheology 39: 409‐417, 2002.
 44. Kilner PJ , Yang GZ , Mohiaddin RH , Firmin DN , Longmore DB . Helical and retrograde secondary flow patterns in the aortic arch studied by three‐directional magnetic resonance velocity mapping. Circulation 88: 2235‐2247, 1993.
 45. Kim S , Popel AS , Intaglietta M , Johnson PC . Effect of erythrocyte aggregation at normal human levels on functional capillary density in rat spinotrapezius muscle. Am J Physiol Heart Circ Physiol 290: H941‐H947, 2006.
 46. Koller A , Kaley G . Endothelial regulation of wall shear stress and blood flow in skeletal muscle microcirculation. Am J Physiol 260: H862‐H868, 1991.
 47. Lakatta EG , Levy D . Arterial and cardiac aging: major shareholders in cardiovascular disease enterprises: Part I: aging arteries: A “set up” for vascular disease. Circulation 107: 139‐146, 2003.
 48. Langille BL . Arterial remodeling: Relation to hemodynamics. Can J Physiol Pharmacol 74: 834‐841, 1996.
 49. Langille BL , O'Donnell F . Reductions in arterial diameter produced by chronic decreases in blood flow are endothelium‐dependent. Science 231: 405‐407, 1986.
 50. Laurent S , Cockcroft J , Van BL , Boutouyrie P , Giannattasio C , Hayoz D , Pannier B , Vlachopoulos C , Wilkinson I , Struijker‐Boudier H , European Network for Non‐Invasive Investigation of Large Arteries. Expert consensus document on arterial stiffness: Methodological issues and clinical applications. Eur Heart J 27: 2588‐2605, 2006.
 51. Levick JR. An Introduction to Cardiovascular Physiology. Fourth Edition. London: Hodder Arnold, 2003.
 52. Lindert J , Werner J , Redlin M , Kuppe H , Habazettl H , Pries AR . OPS imaging of human microcirculation: A short technical report. J Vasc Res 39: 368‐372, 2002.
 53. Lipowsky HH , Kovalcheck S , Zweifach BW . The distribution of blood rheological parameters in the mirovasculature of the cat mesentery. Circ Res 43: 738‐749, 1978.
 54. Lipowsky HH , Usami S , Chien S . In vivo measurements of “apparent viscosity” and microvessel hematocrit in the mesentery of the cat. Microvasc Res 19: 297‐319, 1980.
 55. Lipowsky HH , Zweifach BW . Network analysis of microcirculation of cat mesentery. Microvasc Res 7: 73‐83, 1974.
 56. London GM , Guerin AP . Influence of arterial pulse and reflected waves on blood pressure and cardiac function. Am Heart J 138: 220‐224, 1999.
 57. Love AEH. A Treatise on the Mathematical Theory of Elasticity. 4th ed. New York: Dover, 1944.
 58. Luchsinger PC , Snell RE , Patel DJ , Fry DL . Instantaneous pressure distribution along the human aorta. Circ Res 15: 503‐510, 1964.
 59. Malek AM , Alper SL , Izumo S . Hemodynamic shear stress and its role in atherosclerosis. JAMA 282: 2035‐2042, 1999.
 60. Mall FP . Die Blut und Lymphwege im Dünndarm des Hundes. Abhandlungen der Mathematisch‐Physischen Classe der Königlich Sachsischen Gessellschaft der Wissenschaften 14: 151‐200, 1888.
 61. Malpighi M . De Pulmonibus . Observations anatomicae bologna. 1661. Translated by J. Young. Proc R Soc Med 23: 1‐11, 1929.
 62. Martini P , Pierach A , Schreyer E . Die Strömung des Blutes in engen Gefäβen. Eine Abweichung vom Poiseuille'schen Gesetz. Dt Arch Klin Med 169: 212‐222, 1930.
 63. McDonald DA . The relation of pulsatile pressure to flow in arteries. J Physiol 127: 533‐552, 1955.
 64. McDonald DA. Blood Flow in Arteries. 2nd ed. London: Edward Arnold, 1974.
 65. Milkiewicz M , Brown MD , Egginton S , Hudlicka O . Association between shear stress, angiogenesis, and VEGF in skeletal muscles in vivo. Microcirculation 8: 229‐241, 2001.
 66. Milnor WR. Hemodynamics. 2nd ed. Baltimore: Williams and Wilkins, 1989.
 67. Mitchell GF , Guo CY , Benjamin EJ , Larson MG , Keyes MJ , Vita JA , Vasan RS , Levy D . Cross‐sectional correlates of increased aortic stiffness in the community: The Framingham Heart Study. Circulation 115: 2628‐2636, 2007.
 68. Motomiya M , Karino T . Flow patterns in the human carotid artery bifurcation. Stroke 15: 50‐56, 1984.
 69. Mulvany MJ . Small artery remodelling in hypertension. Basic Clin Pharmacol Toxicol 110: 49‐55, 2012.
 70. Mynard JP , Smolich JJ . Wave potential and the one‐dimensional windkessel as a wave‐based paradigm of diastolic arterial hemodynamics. Am J Physiol Heart Circ Physiol 307: H307‐H318, 2014.
 71. Nichols WW , O'Rourke MF . McDonald's Blood Flow in Arteries. Theoretical, experimental and clinical principles. Fourth Edition. London: Arnold, 1998.
 72. Parker KH . A brief history of arterial wave mechanics. Med Biol Eng Comput 47: 111‐118, 2009.
 73. Pedley TJ. The Fluid Mechanics of Large Blood Vessels. Cambridge: Cambridge University Press, 1980.
 74. Poiseuille JLM . Recherches expérimentales sur le mouvement des liquides dans les tubes de très‐petits diamêtres. Mémoires presentés par divers savants à l'Académie Royale des Sciences de l'Institut de France IX: 433‐544, 1846.
 75. Pranay P , Henriquez‐Rivera RG , Graham MD . Depletion layer formation in suspensions of elastic capsules in Newtonian and viscoelastic fluids. Physics of Fluids 24: 2012.
 76. Pries AR , Ley K , Claassen M , Gaehtgens P . Red cell distribution at microvascular bifurcations. Microvasc Res 38: 81‐101, 1989.
 77. Pries AR , Neuhaus D , Gaehtgens P . Blood viscosity in tube flow: Dependence on diameter and hematocrit. Am J Physiol 263: H1770‐H1778, 1992.
 78. Pries AR , Reglin B , Secomb TW . Remodeling of blood vessels: Responses of diameter and wall thickness to hemodynamic and metabolic stimuli. Hypertension 46: 725‐731, 2005.
 79. Pries AR , Secomb TW . Microvascular blood viscosity in vivo and the endothelial surface layer. Am J Physiol Heart Circ Physiol 289: H2657‐H2664, 2005.
 80. Pries AR , Secomb TW . Blood flow in microvascular networks. In: Tuma RF , Duran WN , Ley K , editors. Handbook of Physiology: Microcirculation. 2nd ed. San Diego: Academic Press, 2008, pp. 3‐36.
 81. Pries AR , Secomb TW . Origins of heterogeneity in tissue perfusion and metabolism. Cardiovasc Res 81: 328‐335, 2009.
 82. Pries AR , Secomb TW , Gaehtgens P . Design principles of vascular beds. Circ Res 77: 1017‐1023, 1995.
 83. Pries AR , Secomb TW , Gaehtgens P . Structure and hemodynamics of microvascular networks: Heterogeneity and correlations. Am J Physiol 269: H1713‐H1722, 1995.
 84. Pries AR , Secomb TW , Gaehtgens P . Structural autoregulation of terminal vascular beds: Vascular adaptation and development of hypertension. Hypertension 33: 153‐161, 1999.
 85. Pries AR , Secomb TW , Gaehtgens P . The endothelial surface layer. Pflugers Arch 440: 653‐666, 2000.
 86. Pries AR , Secomb TW , Gaehtgens P , Gross JF . Blood flow in microvascular networks. Experiments and simulation. Circ Res 67: 826‐834, 1990.
 87. Pries AR , Secomb TW , Gessner T , Sperandio MB , Gross JF , Gaehtgens P . Resistance to blood flow in microvessels in vivo. Circ Res 75: 904‐915, 1994.
 88. Recek C. Conception of the venous hemodynamics in the lower extremity. Angiology 57: 556‐563, 2006.
 89. Rhodin JAG . Architecture of the vessel wall. Compr Physiol (Suppl 7): 1‐31, 2011.
 90. Richardson DR , Zweifach BW . Pressure relationships in the macro‐ and microcirculation of the mesentery. Microvasc Res 2: 474‐488, 1970.
 91. Rodbard S . Vascular caliber. Cardiology 60: 4‐49, 1975.
 92. Safar ME , Levy BI , Struijker‐Boudier H . Current perspectives on arterial stiffness and pulse pressure in hypertension and cardiovascular diseases. Circulation 107: 2864‐2869, 2003.
 93. Sagawa K . Baroreflex control of systemic arterial pressure and vascular bed. Compr Physiol (Suppl 8): 453‐496, 2011.
 94. Sato M , Hayashi K , Niimi H , Moritake K , Okumura A , Handa H . Axial mechanical properties of arterial walls and their anisotropy. Med Biol Eng Comput 17: 170‐176, 1979.
 95. Schiffrin EL . Vascular remodeling in hypertension: Mechanisms and treatment. Hypertension 59: 367‐374, 2012.
 96. Secomb TW . Mechanics of blood flow in the microcirculation. Symp Soc Exp Biol 49: 305‐321, 1995.
 97. Secomb TW . Mechanics of red blood cells and blood flow in narrow tubes. In: Pozrikidis C , editors. Modeling and Simulation of Capsules and Biological Cells. Boca Raton, Florida, Chapman & Hall/CRC, 2003, pp. 163‐196.
 98. Secomb TW , Pries AR . Blood viscosity in microvessels: Experiment and theory. Comptes Rendus Physique 14: 470‐478, 2013.
 99. Segers P , Mynard J , Taelman L , Vermeersch S , Swillens A . Wave reflection: Myth or reality? Artery Research 6: 7‐11, 2012.
 100. Sharman JE , Davies JE , Jenkins C , Marwick TH . Augmentation index, left ventricular contractility, and wave reflection. Hypertension 54: 1099‐1105, 2009.
 101. Skalak R . Wave propagation in blood flow. In: Fung YC , editors. Biomechanics Symposium. New York: American Society of Mechanical Engineers, 1966, pp. 20‐46.
 102. Skalak R , Keller SR , Secomb TW . Mechanics of blood flow. J Biomech Eng 103: 102‐115, 1981.
 103. Stranden E . Edema in venous insufficiency. Phlebolymphology 18: 3‐15, 2011.
 104. Strony J , Beaudoin A , Brands D , Adelman B . Analysis of shear stress and hemodynamic factors in a model of coronary artery stenosis and thrombosis. Am J Physiol 265: H1787‐H1796, 1993.
 105. Sutera SP , Seshadri V , Croce PA , Hochmuth RM . Capillary blood flow. II. Deformable model cells in tube flow. Microvasc Res 2: 420‐433, 1970.
 106. Sutera SP , Skalak R . The history of Poiseuille's law. Ann Rev Fluid Mech 25: 1‐19, 1993.
 107. Tanaka TT , Fung YC . Elastic and inelastic properties of the canine aorta and their variation along the aortic tree. J Biomech 7: 357‐370, 1974.
 108. Trendelenburg F. Über die Unterverbindung der Vena saphena magna bei Unterschenkelvarizen. Beitr Klin Chir 7: 195‐210, 1891.
 109. Tyberg JV , Davies JE , Wang Z , Whitelaw WA , Flewitt JA , Shrive NG , Francis DP , Hughes AD , Parker KH , Wang JJ . Wave intensity analysis and the development of the reservoir‐wave approach. Med Biol Eng Comput 47: 221‐232, 2009.
 110. Ursino M . Interaction between carotid baroregulation and the pulsating heart: A mathematical model. Am J Physiol 275: H1733‐H1747, 1998.
 111. Vaishnav RN , Vossoughi J . Residual stress and strain in aortic segments. J Biomech 20: 235‐239, 1987.
 112. van de Vosse FN , Stergiopulos N . Pulse wave propagation in the arterial tree. Ann Rev Fluid Mech 43: 467‐499, 2011.
 113. Vand V. Viscosity of solutions and suspensions. I. Theory. J Phys Colloid Chem 52: 277‐299, 1948.
 114. Weizsacker HW , Pinto JG . Isotropy and anisotropy of the arterial wall. J Biomech 21: 477‐487, 1988.
 115. West JB. Respiratory Physiology ‐ The Essentials. Baltimore: Williams and Wilkins, 1974.
 116. Westerhof N , Boer C , Lamberts RR , Sipkema P . Cross‐talk between cardiac muscle and coronary vasculature. Physiol Rev 86: 1263‐1308, 2006.
 117. Westerhof N , Lankhaar JW , Westerhof BE . The arterial Windkessel. Med Biol Eng Comput 47: 131‐141, 2009.
 118. Wexler L , Bergel DH , Gabe IT , Makin GS , Mills CJ . Velocity of blood flow in normal human venae cavae. Circ Res 23: 349‐359, 1968.
 119. Womersley JR . Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known. J Physiol 127: 553‐563, 1955.
 120. Wootton DM , Ku DN . Fluid mechanics of vascular systems, diseases, and thrombosis. Annu Rev Biomed Eng 1: 299‐329, 1999.
 121. Young T . Hydraulic investigations, subservient to an intended Croonian lecture on the motion of the blood. Phil Trans Roy Soc 98: 164‐186, 1808.
 122. Young T . On the functions of the heart and arteries. The Croonian lecture. Phil Trans Roy Soc 99: 1‐31, 1809.
 123. Zarins CK , Giddens DP , Bharadvaj BK , Sottiurai VS , Mabon RF , Glagov S . Carotid bifurcation atherosclerosis. Quantitative correlation of plaque localization with flow velocity profiles and wall shear stress. Circ Res 53: 502‐514, 1983.
 124. Zweifach BW . Quantitative studies of microcirculatory structure and function. II. Direct measurement of capillary pressure in splanchnic mesenteric vessels. Circ Res 34: 858‐866, 1974.

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Timothy W. Secomb. Hemodynamics. Compr Physiol 2016, 6: 975-1003. doi: 10.1002/cphy.c150038