Skip to main content
SpringerLink
Log in

Hemodynamics

  • Chapter
Pan Vascular Medicine

  • 47 Accesses

Abstract

The role of the cardiovascular system is to provide oxygen and nutrients to, as well as remove carbon dioxide and wastes from, the cells. The arteries and veins in the body do not constitute simply a passive conduit system for blood transportation, but rather an active component of blood circulation. Hemodynamics plays a very important role in vascular function. It is related to the cause and development of vascular disease, and can provide invaluable diagnostic information about vascular pathology. This chapter will focus on the principles of blood flow in the large vessels of the body. The same laws and principles that govern flow in simple tubes are valid for blood flow in arteries, veins, and in the heart, although the conditions in vivo are more complex. This is due to interactions between many regulatory factors that affect vascular function, such as the pulsatility of blood flow, the elasticity of the arteries and their complex geometry, the peripheral resistance, and the non-Newtonian nature of blood.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Abbreviations

A::

cross-sectional area of vessel or valve area (m2)

C::

speed of sound (m/s)

d::

internal diameter of pipe or vessel (m)

De::

Dean number (dimensionless)

f 0 ::

transmitted frequency (Hz)

F::

force (N)

G(t)::

magnetic field gradient (Tesla/m)

g::

gravitational acceleration (m/s2)

h::

pressure head (cmH2O)

L::

pipe or vessel length (m)

p or P::

pressure (Pa)

Q::

volumetric flow rate (m3/s)

r::

radial coordinate (m)

r c ::

radius of curvature (m)

R::

radius of pipe or vessel (m)

Re::

Reynolds number (dimensionless)

t::

time (s)

TE::

echo time (s)

V::

velocity (m/s)

V̄::

time-averaged velocity (m/s)

V′::

fluctuating velocity (m/s)

V max ::

maximum (centerline) velocity of fluid in a pipe (m/s)

V rms ::

root mean square of V′ (m/s)

x::

spatial coordinate (m)

z::

height (m)

α::

Womersley number (dimensionless)

δ::

gyromagnetic ratio (TeslA-1 • s-1)

Δf::

Doppler frequency shift (Hz)

ΔP::

pressure drop (Pa)

θ::

angle between the ultrasound beam direction and the particle motion direction

μ::

dynamic viscosity of fluid (kg/m • s)

v::

kinematic viscosity of fluid (m2/s)

ϱ::

density of fluid (kg/m3)

τ::

shear stress (N/m2)

τ wall ::

wall shear stress (N/m2)

φ::

phase shift

ω::

frequency of blood flow pulsation or heart rate (s1)

References

  1. White FM (1999) Fluid mechanics. McGraw-Hill, New York

    Google Scholar 

  2. Bird RB, Stewart WE, Lightfoot EN (1960) Transport phenomena. Wiley, New York

    Google Scholar 

  3. Streeter VL, Wylie EB, Bedford KW (1998) Fluid mechanics. McGraw-Hill, New York

    Google Scholar 

  4. Schlichting H (1978) Boundary-layer theory. McGraw-Hill, New York

    Google Scholar 

  5. Caro CG, Pedley Ti, Schröter RC, Seed WA (1978) The mechanics of the circulation. Oxford University Press, Oxford

    Google Scholar 

  6. Fung Y-C (1984) Biodynamics — circulation. Springer-Verlag, Berlin Heidelberg New York

    Google Scholar 

  7. Friedman MU, Deters OJ, Mark FF, Bargeron CB, Hutchins GM (1983) Arterial geometry affects hemodynamics — potential risk factor for atherosclerosis. Atherosclerosis 46: 225–231

    Article  CAS  PubMed  Google Scholar 

  8. Bargeron BC, Hutchings GM, Moore GW, Deters OJ, Mark FF, Friedman MH (1986) Distribution of geometric parameters of human aortic bifurcations. Arteriosclerosis 6: 109–113

    Article  CAS  PubMed  Google Scholar 

  9. McDonald DA (1974) Blood flow in arteries. Arnold, London

    Google Scholar 

  10. Dean WP (1927/1928) The streamline motion of fluid in a curved pipe. Phil Mag 4: 208 and 5: 673

    Google Scholar 

  11. Feuerstein IA, El Masry OA, Round GF (1976) Arterial bifurcation flows — effects of flow rate and area ratio. Can J Physiol Pharmacol 54795–807

    Google Scholar 

  12. Siouffi M, Pelissier R, Farahifar D, Rieu R (1984) The effect of unsteadiness on the flow through stenoses and bifurcations. J Biomech 17: 299–315

    Article  CAS  PubMed  Google Scholar 

  13. Ku DN, Giddens RP (1983) Pulsatile flow in a model carotid bifurcation. Arteriosclerosis 3: 31–39

    Article  CAS  PubMed  Google Scholar 

  14. Lutz RJ, Hsu L, Mcnawat I, Zrubck I, Edwards K (1983) Comparison of steady and pulsatile flow in a double branching arterial model. J Biomech 16: 753–766

    Article  CAS  PubMed  Google Scholar 

  15. El Masry OA, Feuerstein IA, Pound GF (1978) Experimental evaluation of streamline patterns and separated flows in a series of branching vessels with implications for atherosclerosis and thrombosis. Circ Res 43: 608–617

    Article  CAS  PubMed  Google Scholar 

  16. Cho YI, Back LU, Crawford DW (1985) Experimental investigation of branch flow ration, angle, and Reynolds number effects on the pressure and flow fields in arterial branch models. J Biomed Eng 107: 257–267

    CAS  Google Scholar 

  17. Walburn FI, Stein PD (1980) Flow in a symmetrically branched tube simulating the aortic bifurcation: the effects of unevenly distributed flow. Ann Biomed Eng 8: 159–173

    Article  CAS  PubMed  Google Scholar 

  18. Perktold K, Peter RO (1990) Numerical 3D-simulation of pulsatile wall shear stress in an arterial T-bifurcation model. J Biomech Eng 12: 2–12

    Article  CAS  Google Scholar 

  19. Perktold K, Resch M (1991) Numerical flow studies in human carotid artery bifurcations: basic discussion of the geometric factor in atherogenesis. J Biomech Engng 12: 111–123

    Article  Google Scholar 

  20. Fabregues S, Baijens K, Rieu R, Bergeron P (1998) Hemodynamics of endovascular prostheses. J Biomech 31: 45–54

    Article  CAS  PubMed  Google Scholar 

  21. Gijsen FJH, van de Vosse FN, Janssen JD (1999) The influence of the non-Newtonian properties of blood on the flow in large arteries: steady flow in a carotid bifurcation model. J Biomech 32: 601–608

    Article  CAS  PubMed  Google Scholar 

  22. Gijsen FJH, Allanic E, van de Vosse FN, Janssen JD (1999) The influence of the non-Newtonian properties of blood on the flow in large arteries: unsteady flow in a 90 ° curved tube. J Biomech 32: 705–713

    Article  CAS  PubMed  Google Scholar 

  23. Bellhouse BJ, Talbot L (1969) The fluid mechanics of the aortic valve. J Fluid Mech 35: 721–735

    Article  Google Scholar 

  24. Kilner PJ, Yang GZ, Mohiaddin RH, Firmin DN, Longmore DB (1993) Helical and retrograde secondary flow patterns in the aortic arch studied by three-directional magnetic resonance velocity mapping. Circulation 88: 2235–2247

    Article  CAS  PubMed  Google Scholar 

  25. Bogren HG, Buonocore MH (1994) Blood flow measurements in the aorta and major arteries with MR velocity mapping. JMRI 4: 119–130

    Article  CAS  PubMed  Google Scholar 

  26. Klipstein RH, Firmin DN, Underwood SR, Rees RSO, Longmore DB (1987) Blood flow patterns in the human aorta studied by magnetic resonance. Br Heart J 58: 316–323

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  27. Möller HE, Klocke H-K, Bongartz GM, Peters PE (1996) MR flow quantification using RACE: clinical application to the carotid arteries. JMRI 6: 503–512

    Article  PubMed  Google Scholar 

  28. Zarins CK, Giddens DP, Bharadvaj BK, Sottiural VS, Mabon RF, Glagov S (1983) Carotid bifurcation atherosclerosis: quantitative correlation of plaque localization with flow velocity profiles and wall shear stress. Circ Res 53: 502–514

    Article  CAS  PubMed  Google Scholar 

  29. Ku DN, Giddens DP, Zarins CK, Glagov S (1985) Pulsatile flow and atherosclerosis in the human carotid bifurcation: positive correlation between plaque location and low and oscillating shear stress. Arteriosclerosis 5: 293–302

    Article  CAS  PubMed  Google Scholar 

  30. Long Q, Xu XY, Ariff B, Thorn SA, Hughes AD, Stanton AV (2000) Reconstruction of blood flow patterns in a human carotid bifurcation: a combined CFD and MRI study. JMRI 11: 299–311

    Article  CAS  PubMed  Google Scholar 

  31. Moore JA, Steinman DA, Prakash S, Johnston KW, Ethier CR (1999) A numerical study of blood flow patterns in anatomically realistic and simplified end-to-side anastomoses. J Biomech Eng 121: 265–272

    Article  CAS  PubMed  Google Scholar 

  32. Sivanesan S, How TV, Black RA, Bakran A (1999) Flow patterns in the radiocephalic arteriovenous fistula: an in vitro study. J Biomech 32: 915–925

    Article  CAS  PubMed  Google Scholar 

  33. Ensley AE, Lynch P, Chatzimavroudis GP, Lucas C, Sharma S, Yoganathan AP (1999) Toward designing the optimal total cavopul-monary connection: an in vitro study. Ann Thor Surg 68: 1384–1390

    Article  CAS  Google Scholar 

  34. Sharma S, Goudy S, Walker P, Panchal S, Ensley A, Kanter K, Tarn V, Fyfe D, Yoganathan AP (1996) In vitro flow experiments for determination of optimal geometry of total cavopulmonary connection for surgical repair of children functional single ventricle. JACC 27: 1264–1269

    Article  CAS  PubMed  Google Scholar 

  35. Lundbrook J (1962) Functional aspects of the veins of the leg. Am Heart J 64: 706–713

    Article  Google Scholar 

  36. Gorlin R, Gorlin G (1951) Hydraulic formula for calculation of area of stenotic mitral valve, other cardiac valves and central circulatory shunts. Am Heart J 41: 1–29

    Article  CAS  PubMed  Google Scholar 

  37. Young DF (1979) Fluid mechanics of arterial stenosis. J Biomech Eng 101: 157–175

    Article  Google Scholar 

  38. Yoganathan AP (1988) Fluid mechanics of aortic stenosis. Eur Heart J 9 [Suppl E]: 13–17

    Article  PubMed  Google Scholar 

  39. Yoganathan AP, Cape EG, Sung HW, Williams FP, Jimoh A (1988) Review of hydrodynamic principles for the cardiologist: applications to the study of blood flow and jets by imaging techniques. JACC 12: 1344–1353

    Article  CAS  PubMed  Google Scholar 

  40. Levine RA, Jimoh A, Cape EG, McMillan S, Yoganathan AP, Weyman AE (1989) Pressure recovery distal to a stenosis: potential cause of gradient “overestimation” by Doppler echocardiography. JACC 13: 706–715

    Article  CAS  PubMed  Google Scholar 

  41. Voelker W, Reul H, Stelzer T, Schmidt A, Karsch KR (1992) Pressure recovery in aortic stenosis: an in vitro study in a pulsatile flow model. JACC 20: 1585–1593

    Article  CAS  PubMed  Google Scholar 

  42. Kilner PJ, Manzara CC, Mohiaddin RH, Pennell DJ, Sutton MGStJ, Firmin DN, Underwood SR, Longmore DB (1993) Magnetic resonance jet velocity mapping in mitral and aortic valve stenosis. Circulation 87: 1239–1248

    Article  CAS  PubMed  Google Scholar 

  43. Heinrich RS, Marcus RH, Ensley AE, Gibson DE, Yoganathan AP (1999) Valve orifice area alone is an insufficient index of aortic stenosis severity: effects of the proximal and distal geometry on transaortic energy loss. J Heart Valve Dis 8: 509–515

    CAS  PubMed  Google Scholar 

  44. Khalifa AMA, Giddens DP (1981) Characterization and evolution of poststenotic flow disturbances. J Biomech 14: 279–296

    Article  CAS  PubMed  Google Scholar 

  45. Talukder N, Fulenwider JT, Mabon RF, Giddens DP (1986) Poststenotic flow disturbance in the dog aorta as measured with pulsed Doppler ultrasound. J Biomech Eng 108: 259–265

    Article  CAS  PubMed  Google Scholar 

  46. Solzbach U, Wollschlager H, Zeiher A, Just H (1987) Effect of stenotic geometry on flow behavior across stenotic models. Med Biol Eng Comp 25: 543–550

    Article  CAS  Google Scholar 

  47. Young DP, Tsai FY (1973) Flow characteristics in models of arterial stenoses. I. Steady flow. J Biomech 6: 395–410

    Article  CAS  PubMed  Google Scholar 

  48. Yongchareon W, Young DF (1979) Initiation of turbulence in models of arterial stenosis. J Biomech 12: 185–196

    Article  CAS  PubMed  Google Scholar 

  49. Vonrudden WJ, Blaisdell FW, Hall AD, Thomas AN (1964) Multiple arterial stenoses: effect on blood flow. Arch Surg 89: 307–315

    Article  Google Scholar 

  50. Feldman RL, Nichols WW, Pepine CJ, Conetta DA, Conti CR (1979) The coronary hemodynamics of left main and branch coronary stenoses — the effects of reduction in stenosis diameter, stenosis length, and number of stenoses. J Thor Cardiovasc Surg 77: 377–388

    CAS  Google Scholar 

  51. Karayannacos PE, Talukder N, Nerem RM, Roshon S, Vasko JS (1977) The role of multiple noncritical arterial stenosis in the pathogenesis of ischemia. J Thor Cardiovasc Surg 73: 458–469

    CAS  Google Scholar 

  52. Kilpatrick D, Webber SD, Colle J-P (1990) The vascular resistance of arterial stenoses in series. Angiology 41: 278–285

    Article  CAS  PubMed  Google Scholar 

  53. Allard L, Cloutier G, Durand L-G (1995) Doppler velocity ratio measurements evaluated in a phantom of multiple arterial disease. Ultrasound Med Biol 21: 471–480

    Article  CAS  PubMed  Google Scholar 

  54. Guo Z, Durand L-G, Allard L, Cloutier G, Fenster A (1998) In vitro evaluation of multiple arterial stenoses using three-dimensional power Doppler angiography. J Vase Surg 27: 681–688

    Article  CAS  Google Scholar 

  55. De Bruyne B, Pijls NHJ, Heyndrickx GR, Hodeige D, Kirkeeide R, Gould KL (2000) Pressure-derived fractional flow reserve to assess serial epicardial stenoses — theoretical basis and animal validation. Circulation 101: 1840–1847

    Article  CAS  PubMed  Google Scholar 

  56. Gould KL, Nakagawa Y, Nakagawa K, Sdringola S, Hess MJ, Haynie M, Parker N, Mullani N, Kirkeeide R (2000) Frequency and clinical implications of fluid dynamically significant diffuse coronary artery disease manifest as graded, longitudinal, base-to-apex myocardial perfusion abnormalities by noninvasive positron emission tomography. Circulation 101: 1931–1939

    Article  CAS  PubMed  Google Scholar 

  57. Ahmed SA, Giddens DP (1983) Velocity measurements in steady flow through axisymmetric stenoses at moderate Reynolds numbers. J Biomech 16: 505–516

    Article  CAS  PubMed  Google Scholar 

  58. Evans DH, Barrie MJ, Bentley S, Bell PRF (1980) The relationship between ultrasonic pulsatility index and proximal arterial stenosis in a canine model. Circ Res 46: 470–475

    Article  CAS  PubMed  Google Scholar 

  59. Budwig R, Elger D, Hooper H, Slippy J (1993) Steady flow in abdominal aneurysm models. J Biomech Eng 115: 418–423

    Article  CAS  PubMed  Google Scholar 

  60. Taylor TW, Yamaguchi T (1994) Three-dimensional simulation of blood in an abdominal aortic aneurysm — steady and unsteady flow cases. J Biomech Eng 116: 89–97

    Article  CAS  PubMed  Google Scholar 

  61. Viswanath N, Rodkiewicz CM, Zajac S (1997) On the abdominal aortic aneurysms: pulsatile state considerations. Med Eng Phys 19: 343–351

    Article  CAS  PubMed  Google Scholar 

  62. Aenis M, Stancampiano AP, Wakhloo AK, Lieber BB (1997) Modeling of flow in a straight stented and non-stented side wall aneurysm model. J Biomech Eng 119: 206–212

    Article  CAS  PubMed  Google Scholar 

  63. Yu SCM, Zhao JB (2000) A steady flow analysis on the stented and non-stented sidewall aneurysm models. Med Eng Phys 21: 133–141

    Article  Google Scholar 

  64. Asakura T, Karino T (1990) Flow patterns and spatial distribution of atherosclerotic lesions in human coronary arteries. Circ Res 66: 1045–1066

    Article  CAS  PubMed  Google Scholar 

  65. Friedman MU, Bargcron CB, Hutchins GM, Mark FF, Deters OJ (1980) Hemodynamic measurements in human arterial casts, and their correlation with histology and luminal area. J Biomech Eng 102: 247–251

    Article  CAS  PubMed  Google Scholar 

  66. Fox IA, Hugh AE (1966) Localization of atheroma: a theory based on boundary layer separation. Br Heart J 28: 388–399

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  67. Moore JE, Xu C, Glagov S, Zarins CK, Ku DN (1994) Fluid wall shear stress measurements in a model of the human abdominal aorta: oscillatory behavior and relationship to atherosclerosis. Atherosclerosis 110: 225–240

    Article  CAS  PubMed  Google Scholar 

  68. Kisslo J, Adams DB, Belkin RN (1988) Doppler color flow imaging. Churchill Livingstone, New York

    Google Scholar 

  69. Perry GJ, Helmcke F, Nanda NC, Byard C, Soto B (1987) Evaluation of aortic insufficiency by Doppler color flow mapping. JACC 9: 952–959

    Article  CAS  PubMed  Google Scholar 

  70. Baumgartner H, Kratzer H, Helmreich G, Kuhn P (1988) Quantitation of aortic regurgitation by colour coded cross-sectional Doppler echocardiography. Eur Heart J 9: 380–387

    CAS  PubMed  Google Scholar 

  71. Taylor AL, Eichhorn EJ, Brickner ME, Eberhart RC, Grayburn PA (1990) Aortic valve morphology: an important in vitro determinant of the proximal regurgitant jet width by Doppler color flow mapping. JACC 16: 405–412

    Article  CAS  PubMed  Google Scholar 

  72. Reimold SC, Atkinson CM, Luna FB, Lee RT (1993) Influence of jet impingement on color Doppler parameters of aortic regurgitation. Echocardiography 10: 113–119

    Article  CAS  PubMed  Google Scholar 

  73. Recusani F, Bargiggia GS, Yoganathan AP, Raisaro A, Valdes-Cruz LM, Sung H-W, Bertucci C, Gallati M, Moises VA, Simpson IA, Tronconi L, Sahn DJ (1991) A new method for quantification of regurgitant flow rate using color Doppler flow imaging of the flow convergence region proximal to a discrete orifice. Circulation 83: 594–604

    Article  CAS  PubMed  Google Scholar 

  74. Simpson IA, Sahn DJ, Valdes-Cruz LM, Chung KJ, Sherman FS, Swensson RE (1988) Color Doppler flow mapping in patients with coarctation of the aorta: new observations and improved evaluation with color flow diameter and proximal acceleration as predictors of severity. Circulation 77: 736–744

    Article  CAS  PubMed  Google Scholar 

  75. Deeg KH, Hofbeck M, Singer H (1993) Diagnosis of subclavian steal in infants with coarctation of the aorta and interruption of the aortic arch by color-coded Doppler sonography. J Ultrasound Med 12: 713–718

    CAS  PubMed  Google Scholar 

  76. Steinke W, Kloetzsch C, Hennerici M (1990) Carotid artery disease assessed by color Doppler flow imaging: correlation with standard Doppler sonography and angiography. AJNR 11: 259–266

    CAS  PubMed  Google Scholar 

  77. Bluth EI, Merritt CR (1992) Doppler color imaging. Carotid and vertebral arteries. Clin Diagn Ultrasound 27: 61–96

    CAS  PubMed  Google Scholar 

  78. Moran PR (1982) A flow velocity zeugmatographic interlace for NMR imaging in humans. MRI 1: 197–203

    Article  Google Scholar 

  79. Bryant DJ, Payne JA, Firmin DN, Longmore DB (1984) Measurement of flow with NMR imaging using a gradient pulse and phase difference technique. JCAT 8: 588–593

    CAS  Google Scholar 

  80. Meier D, Maier S, Bosiger P (1988) Quantitative flow measurements on phantoms and on blood vessels with MR. Magn Res Med 8: 25–34

    Article  CAS  Google Scholar 

  81. Bendei P, Buonocore E, Bockisch A, Besozzi MC (1989) Blood flow in the carotid arteries: quantification by using phase-sensitive MR imaging. AJR Am J Roentgenol 152: 1307–1310

    Article  Google Scholar 

  82. Chatzimavroudis GP, Walker PG, Oshinski JN, Franch RH, Pettigrew RI, Yoganathan AP (1997) The importance of slice location on the accuracy of aortic regurgitation measurements with magnetic resonance phase velocity mapping: an in vitro investigation. Ann Biomed Eng 25: 644–652

    Article  CAS  PubMed  Google Scholar 

  83. Underwood SR, Firmin DN, Klipstein RH, Rees RSO and Longmore DB (1987) Magnetic resonance velocity mapping: clinical application of a new technique. Br Heart J 57: 404–412

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  84. Mohiaddin RH, Wann SL, Underwood R, Firmin DN, Rees S, Long-more DB (1990) Vena caval flow: assessment with cine MR velocity mapping. Radiology 177: 537–541

    CAS  PubMed  Google Scholar 

  85. Kondo C, Caputo GR, Semelka R, Foster E, Shimakawa A, Higgins CB (1991) Right and left ventricular stroke volume measurements with velocity-encoded cine MRI: In-vitro and in-vivo validation. AJR Am J Roentgenol 157: 9–16

    Article  CAS  PubMed  Google Scholar 

  86. Pelc LR, Pele NJ, Rayhill SC, Castro LJ, Glover GH, Herfkens RJ, Miller DC, Jeffrey RB (1992) Arterial and venous blood flow: noninvasive quantification with MR imaging. Radiology 185: 809–812

    CAS  PubMed  Google Scholar 

  87. Chatzimavroudis GP, Walker PG, Oshinski JN, Franch RH, Pettigrew RI, Yoganathan AP (1997) Slice location dependence of aortic regurgitation measurements with MR phase velocity mapping. Magn Res Med 37: 545–551

    Article  CAS  Google Scholar 

  88. Chatzimavroudis GP, Oshinski JN, Franch RH, Pettigrew RI, Walker PG, Yoganathan AP (1998) Quantification of aortic regurgitation with magnetic resonance phase velocity mapping: a clinical investigation of the importance of slice location. J Heart Valve Dis 7: 94–101

    CAS  PubMed  Google Scholar 

Download references

Authors
  1. Ajit P. Yoganathan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. George P. Chatzimavroudis
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Herz-Zentrum Coswig, Klinik für Kardiologie und Angiologie, Lerchenfeld 1, 06869, Coswig (Anhalt), Germany

    Peter Lanzer MD

  2. The Cleveland Clinic Foundation, One Clinic Center, 9500 Euclid Avenue, Cleveland, 44195, OH, USA

    Eric J. Topol MD

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Yoganathan, A.P., Chatzimavroudis, G.P. (2002). Hemodynamics. In: Lanzer, P., Topol, E.J. (eds) Pan Vascular Medicine. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56225-9_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-56225-9_7

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-62565-7

  • Online ISBN: 978-3-642-56225-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation