Skip to main content
Log in

Dynamics of pulsatile flow in fractal models of vascular branching networks

  • Original Article
  • Published:
Medical & Biological Engineering & Computing Aims and scope Submit manuscript

Abstract

Efficient regulation of blood flow is critically important to the normal function of many organs, especially the brain. To investigate the circulation of blood in complex, multi-branching vascular networks, a computer model consisting of a virtual fractal model of the vasculature and a mathematical model describing the transport of blood has been developed. Although limited by some constraints, in particular, the use of simplistic, uniformly distributed model for cerebral vasculature and the omission of anastomosis, the proposed computer model was found to provide insights into blood circulation in the cerebral vascular branching network plus the physiological and pathological factors which may affect its functionality. The numerical study conducted on a model of the middle cerebral artery region signified the important effects of vessel compliance, blood viscosity variation as a function of the blood hematocrit, and flow velocity profile on the distributions of flow and pressure in the vascular network.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Alastruey J, Parker K, Peiró J, Byrd S, Sherwin S (2007) Modelling the circle of Willis to assess the effects of anatomical variations and occlusion on cerebral flows. J Biomech 40:1794–1805

    Article  Google Scholar 

  2. Bagchi P (2007) Mesoscale simulation of blood flow in small vessels. Biophys J 92:1858–1877

    Article  Google Scholar 

  3. Brozici M, van der Zwan A, Hillen B (2003) Anatomy and functionality of leptomeningeal anastomoses: a review. Stroke 34:2750–2762

    Article  Google Scholar 

  4. Cassot F, Lauwers F, Fouard C, Prohaska S, Lawers-Cances V (2006) A novel three-dimensional computer assisted method for a quantitative study of microvascular networks of the human cerebral cortex. Microcirculation 13:1–18

    Article  Google Scholar 

  5. Dankelman J, Cornelissen A, Lagro J, VanBavel E, Spaan J (2007) Relation between branching pattern and perfusion in stochastic generated coronary arterial trees. Med Bio Eng Comput 45:25–34

    Article  Google Scholar 

  6. Edvinsson L, Krause DN (eds) (2002) Cerebral blood flow and metabolism. Lippincott Williams & Wilkins, Philadelphia

  7. Gabryś E, Rybaczuk M, Kedzia A (2006) Blood flow simulation through fractal models of circulatory system. Chaos Soliton Fract 27:1–7

    Article  MATH  Google Scholar 

  8. Gafiychuk VV, Lubashevsky I (2001) On the principles of the vascular network branching. J Theor Biol 212:1–9

    Article  Google Scholar 

  9. Herman P, Kocsis L, Eke A (2001) Fractal branching pattern in the pial vasculature in the cat. J Cerebr Blood F Met 21:741–753

    Article  Google Scholar 

  10. Horton R (1945) Erosional development of stream and their drainage basins: hydrophysical approach to quantitative geormorphology. Geol Soc Am Bull 56:275–370

    Article  Google Scholar 

  11. Huo Y, Kassab G (2006) Pulsatile blood flow in the entire coronary arterial tree: theory and experiment. Am J Physiol Heart C 291:H1074–H1087

    Article  Google Scholar 

  12. Kaimovitz B, Huo Y, Lanir Y, Kassab GS (2008) Diameter asymmetry of porcine coronary arterial trees: structural and functional implications. Am J Physiol Heart C 294:H714–H723

    Article  Google Scholar 

  13. Karau KL, Krenz GS, Dawson CA (2001) Branching exponent heterogeneity and wall shear stress distribution in vascular trees. Am J Physiol Heart C 280:H1256–1263

    Google Scholar 

  14. Karch R, Neumann F, Neumann M, Schreiner W (1999) A three-dimensional model for arterial tree representation, generated by constrained constructive optimization. Comput Biol Med 29:19–38

    Article  Google Scholar 

  15. Lapi D, Marchiafava P, Colantuoni A (2008) Geometric characteristics of arterial network of rat pial microcirculation. J Vasc Res 45:69–77

    Article  Google Scholar 

  16. Li JKJ (2004) Dynamics of the vascular system. World Sci. Publ. Co., Singapore

    Google Scholar 

  17. Liu Y, Kassab GS (2007) Vascular metabolic dissipation in murray’s law. Am J Physiol Heart C 292:H1336–H1339

    Article  Google Scholar 

  18. Matthys KS, Alastruey J, Peiró J, Khir AW, Segers P, Verdonck PR, Parker KH, Sherwin SJ (2007) Pulse wave propagation in a model human arterial network: assessment of 1-D numerical simulations against in-vitro measurements. J Biomech 40:3476–3486

    Article  Google Scholar 

  19. Mayer S (1996) On the pressure and flow-rate distributions in tree-like and arterial-venous networks. B Math Biol 58:753–785

    Article  MATH  Google Scholar 

  20. Moore S, David T, Chase J, Arnold J, Fink J (2006) 3D models of blood flow in the cerebral vasculature. J Biomech 39:1454–1463

    Article  Google Scholar 

  21. Murray C (1926) The physiological principle of minimum work. I. The vascular system and the cost of blood volume. P Natl Acad Sci USA 12:207–214

    Article  Google Scholar 

  22. Olufsen MS (1999) Structured tree outflow condition for blood flow in larger systemic arteries. Am J Physiol 276:H257–268

    Google Scholar 

  23. Olufsen MS, Peskin CS, Kim WY, Pedersen EM, Nadim A, Larsen J (2000) Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions. Ann Biomed Eng 28:1281–1299

    Article  Google Scholar 

  24. Popel AS, Johnson PC (2005) Microcirculation and hemorheology. Annu Rev Fluid Mech 37:43–69

    Article  MathSciNet  Google Scholar 

  25. Pries A, Secomb T, Gaehtgens P (1996) Review—biophysical aspects of blood flow in the microvasculature. Cardiovasc Res 32:654–667

    Google Scholar 

  26. Pries A, Secomb T, Gessner T, Sperandio M, Gross J, Gaehtgens P (1994) Resistance to blood flow in microvessels in vivo. Circ Res 75:904–915

    Google Scholar 

  27. Reneman RS, Hoeks AP (2008) Wall shear stress as measured in vivo: consequences for the design of the arterial system. Med Biol Eng Comput 46:499–507

    Article  Google Scholar 

  28. Rossitti S, Lofgren J (1993) Vascular dimensions of the cerebral arteries follow the principle of minimum work. Stroke 24:371–377

    Google Scholar 

  29. Schreiner W, Karch R, Neumann M, Neumann F, paul Szawlowski, Roedler S (2006) Optimized arterial trees supplying hollow organs. Med Eng Phys 28:416–429

    Article  Google Scholar 

  30. Schreiner W, Neumann F, Neumann M, End A, Muller M (1996) Structural quantification and bifurcation symmetry in arterial tree models generated by constrained constructive optimization. J Theor Biol 180:161–174

    Article  Google Scholar 

  31. Schreiner W, Neumann F, Neumann M, End A, Roedler SM (1997) Anatomical variability and funtional ability of vascular trees modeled by constrained constructive optimization. J Theor Biol 187:147–158

    Article  Google Scholar 

  32. Sherwin SJ, Formaggia L, Peiró J, Franke V (2003) Computational modeling of 1D blood flow with variable mechanical properties and its application to the simulation of wave propagation in the human arterial system. Int J Numer Meth Fluids 43:673–700

    Article  MATH  Google Scholar 

  33. Taber L (1998) An optimization principle for vascular radius including the effects of smooth muscle tone. Biophys J 74:109–114

    Article  Google Scholar 

  34. Ursino M (1988) A mathematical study of human intracranial hydrodynamics. i. the cerebrospinal fluid pulse pressure. Ann Biomed Eng 16:379–402

    Article  Google Scholar 

  35. Ursino M, Lodi CA (1997) A simple mathematical model of the interaction between intracranial pressure and cerebral hemodynamics. J Appl Physiol 82:1256–1269

    Google Scholar 

  36. West GB, Brown JH, Enquist BJ (1997) A general model for the origin of allometric scaling laws in biology. Science 276:122–126

    Article  Google Scholar 

  37. Westerhof N, Stergiopulos N, Noble MI (2005) Snapshots of hemodynamics—an aid for clinical research and graduate education. Springer, Boston

    Google Scholar 

Download references

Acknowledgment

The authors acknowledge the constructive comments made by the reviewers.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Anh Bui.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bui, A., Šutalo, I.D., Manasseh, R. et al. Dynamics of pulsatile flow in fractal models of vascular branching networks. Med Biol Eng Comput 47, 763–772 (2009). https://doi.org/10.1007/s11517-009-0492-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11517-009-0492-6

Keywords

Navigation