Skip to main content
SpringerLink
Log in

Fundamentals of Vascular Bio-fluid and Solid Mechanics

  • Chapter
  • First Online:
Vascular Engineering

Abstract

While considering the circulatory system, a mechanical view point provides the basis for understanding various physiological and pathophysiological phenomena. In this section, we introduce the fundamentals of fluid and solid mechanics in relation to the circulatory system. In the first part on fluid mechanics, we start by introducing the concepts of Newtonian and non-Newtonian fluids, viscosity, and apparent viscosity. Then, the rheological properties of blood are described, and the universal mechanical law for flow through cylindrical tubes is derived. Based on that law, we consider the characteristics of tube flow for Newtonian and non-Newtonian fluids and present several representative examples of mathematical models for blood flow through vessels. These models have a close relationship with important physiological phenomena. In the second part, we first discuss the concept of continuum mechanics for a large deformation of the vascular wall. Then, we introduce passive hyperelastic models, an active smooth muscle model, and incorporations of residual strain and smooth muscle contractions. We demonstrate typical axisymmetric solutions of arterial wall stress for a tube model under physiological loading conditions, i.e., longitudinal stretch and intraluminal pressure. We also show some approaches for arterial diseases such as atherosclerosis, aortic aneurysm, and aortic dissection.

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 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover 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

References

  • Azuma T, Hasegawa M (1971) A rheological approach to the architecture of arterial walls. Jpn J Physiol 21(1):27–47

    Article  CAS  PubMed  Google Scholar 

  • Berne RM, Levy MN (1993) Physiology, 3rd edn. Mosby, St. Louis, p 362

    Google Scholar 

  • Burton AC (1954) Relation of structure to function of the tissues of the wall of blood vessels. Physiol Rev 34(4):619–642

    CAS  PubMed  Google Scholar 

  • Carew TE, Vaishnav RN, Patel DJ (1968) Compressibility of the arterial wall. Circ Res 23(1):61–68

    Article  CAS  PubMed  Google Scholar 

  • Casson N (1959) A flow equation for pigment-oil suspensions of the printing ink type. In: Mill CC (ed) Rheology of disperse systems. Pergamon Press, Oxford, p 84

    Google Scholar 

  • Chien S (1970) Shear dependence of effective cell volume as a determinant of blood viscosity. Science 168:977–979

    Article  CAS  PubMed  Google Scholar 

  • Chuong CJ, Fung YC (1983) Three-dimensional stress distribution in arteries. J Biomech Eng 105(3):268–274

    Article  CAS  PubMed  Google Scholar 

  • Chuong CJ, Fung YC (1984) Compressibility and constitutive equation of arterial wall in radial compression experiments. J Biomech 17(1):35–40. doi:10.1016/0021-9290(84)90077-0

    Article  CAS  PubMed  Google Scholar 

  • Cokelet GR, Merrill EW, Gilliland ER, Shin H, Britten A, Wells RE Jr (1963) The rheology of human blood – measurement near and at zero shear rate. Trans Soc Rheol 7:303–317

    Article  Google Scholar 

  • Cox RH (1978) Regional variation of series elasticity in canine arterial smooth muscles. Am J Physiol 234(5):H542–H551

    CAS  PubMed  Google Scholar 

  • Elger DF, Blackketter DM, Budwig RS, Johansen KH (1996) The influence of shape on the stresses in model abdominal aortic aneurysms. J Biomech Eng 118(3):326–332. doi:10.1115/1.2796014

    Article  CAS  PubMed  Google Scholar 

  • Fahraeus R, Lindqvist T (1931) The viscosity of the blood in narrow capillary tubes. Am J Physiol 96:562–568

    CAS  Google Scholar 

  • Farber TE (1995) Fluid dynamics for physicists. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Fung YC (1993) Biomechanics: mechanical properties of living tissues, 2nd edn. Springer, New York

    Book  Google Scholar 

  • Fung YC (1996) Biomechanics: circulation, 2nd edn. Springer, New York

    Google Scholar 

  • Fung YC, Sobin SS (1972) Pulmonary alveolar blood flow. Circ Res 30:470–490

    Article  CAS  PubMed  Google Scholar 

  • Hayashi K (1993) Experimental approaches on measuring the mechanical properties and constitutive laws of arterial walls. J Biomech Eng 115(4):481–488. doi:10.1115/1.2895528

    Article  CAS  PubMed  Google Scholar 

  • Hayashi K, Stergiopulos N, Meister J-J, Greenwald SE, Rachev A (2001) Techniques in the determination of the mechanical properties and constitutive laws of arterial walls. In: Leondes C (ed) Biomechanical systems: techniques and applications, vol. II, Cardiovascular techniques. CRC Press, Boca Raton

    Google Scholar 

  • Holzapfel GA (2000) Nonlinear solid mechanics: a continuum approach for engineering. Wiley, Chichester

    Google Scholar 

  • Holzapfel GA (2009) Arterial tissue in health and disease: experimental data, collagen-based modeling and simulation, including aortic dissection. In: Holzapfel G, Ogden R (eds) Biomechanical modelling at the molecular, cellular and tissue levels, vol 508. CISM International Centre for Mechanical Sciences. Springer, Vienna, pp 259–344. doi:10.1007/978-3-211-95875-9_4

    Chapter  Google Scholar 

  • Holzapfel GA, Gasser TC, Ogden RW (2000) A new constitutive framework for arterial wall mechanics and a comparative study of material models. J Elast 61(1–3):1–48. doi:10.1023/A:1010835316564

    Article  Google Scholar 

  • Humphrey JD (2002) Cardiovascular solid mechanics: cells, tissues, and organs. Springer, New York

    Book  Google Scholar 

  • Mashima H (1986) Physiology, 18th edn. Bunkodo, Tokyo, pp 396–397, in Jap

    Google Scholar 

  • Matsumoto T, Goto T, Sato M (2004) Microscopic residual stress caused by the mechanical heterogeneity in the lamellar unit of the porcine thoracic aortic wall. JSME Int J, Ser A 47(3):341–348. doi:10.1299/Jsmea.47.341

    Article  Google Scholar 

  • Merrill EW, Cokelet GC, Britten A, Wells RE Jr (1963) Non-Newtonian rheology of human blood -effect of fibrinogen deduced by “subtraction”. Circ Res 13:48–55

    Article  CAS  PubMed  Google Scholar 

  • Murphy RA (1976) Contractile system function in mammalian smooth muscle. Blood Vessels 13(1–2):1–23

    CAS  PubMed  Google Scholar 

  • Murphy RA (1980) Mechanics of vascular smooth muscle. In: Handbook of physiology. American Physiological Society, Bethesda, pp 325–351

    Google Scholar 

  • Nichols WW, O’Rourke MF, Vlachopoulos C (2011) McDonald’s blood flow in arteries: theoretical, experimental and clinical principles, 6th edn. Hodder Arnold, London, pp 411–446

    Google Scholar 

  • Oka S (1981) Cardiovascular hemorheology. Cambridge University Press, Cambridge

    Google Scholar 

  • Okamoto RJ, Xu HD, Kouchoukos NT, Moon MR, Sundt TM (2003) The influence of mechanical properties on wall stress and distensibility of the dilated ascending aorta. J Thorac Cardiovasc Surg 126(3):842–850. doi:10.1016/S0022-5223(03)00728-1

    Article  PubMed  Google Scholar 

  • Pasta S, Phillippi JA, Gleason TG, Vorp DA (2012) Effect of aneurysm on the mechanical dissection properties of the human ascending thoracic aorta. J Thorac Cardiovasc Surg 143(2):460–467. doi:10.1016/j.jtcvs.2011.07.058

    Article  PubMed  Google Scholar 

  • Pries AR, Secomb TW, Gaehtgens P (2000) The endothelial surface layer. Pflugers Arch - Eur J Physiol 440:653–660

    Article  CAS  Google Scholar 

  • Rhodin JAG (1979) Architecture of the vessel wall. In: Burne RM (ed) Handbook of physiology, sec 2, vol 2. American Physiological Society, Bethesda, pp 1–31

    Google Scholar 

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

    Article  Google Scholar 

  • Sugihara-Seki M (2006) Transport of spheres suspended in the fluid flowing between hexagonally arranged cylinders. J Fluid Mech 55:309–321

    Article  Google Scholar 

  • Sugihara-Seki M, Fu BM (2005) Blood flow and permeability in microvessels. Fluid Dyn Res 37:82–132

    Article  Google Scholar 

  • Tang DL, Teng ZZ, Canton G, Yang C, Ferguson M, Huang XY, Zheng J, Woodard PK, Yuan C (2009) Sites of rupture in human atherosclerotic carotid plaques are associated with high structural stresses – an in vivo MRI-based 3D fluid-structure interaction study. Stroke 40(10):3258–3263. doi:10.1161/Strokeaha.109.558676

    Article  PubMed  PubMed Central  Google Scholar 

  • Thubrikar MJ (2007) Vascular mechanics and pathology. Springer, New York

    Book  Google Scholar 

  • Vito RP, Dixon SA (2003) Blood vessel constitutive models-1995-2002. Annu Rev Biomed Eng 5:413–439. doi:10.1146/annurev.bioeng.5.011303.120719

    Article  CAS  PubMed  Google Scholar 

  • Vito RP, Hickey J (1980) The mechanical properties of soft tissues.2. The elastic response of arterial segments. J Biomech 13(11):951–957. doi:10.1016/0021-9290(80)90166-9

    Article  CAS  PubMed  Google Scholar 

  • von Maltzahn WW, Besdo D, Wiemer W (1981) Elastic properties of arteries – a nonlinear two-layer cylindrical model. J Biomech 14(6):389–397. doi:10.1016/0021-9290(81)90056-7

    Article  Google Scholar 

  • Weinbaum S, Zhang X, Han Y, Vink H, Cowin SC (2003) Mechanotransduction and flow across the endothelial glycocalyx. PNAS 100:7988–7995

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Weinbaum S, Tarbell JM, Damiano ER (2007) The structure and function of the endothelial glycocalyx layer. Annu Rev Biomed Eng 9:121–167

    Article  CAS  PubMed  Google Scholar 

  • Whitmore RL (1968) Rheology of the circulation. Pergamon, Oxford

    Google Scholar 

  • Yamada H (1999) A mathematical model of arteries in the active state (Incorporation of active stress and activation parameter). JSME Int J, Ser C 42(3):545–551

    Article  Google Scholar 

  • Yamada H (2012) Fundamental mechanics and biomechanics. Corona Publishing, Tokyo, in Jap

    Google Scholar 

  • Yamada H, Hasegawa Y (2007) A simple method of estimating the stress acting on a bilaterally symmetric abdominal aortic aneurysm. Comput Methods Biomech Biomed Engin 10(1):53–61. doi:10.1080/10255840601086531

    Article  CAS  PubMed  Google Scholar 

  • Yamada H, Sakata N (2013) Low pressure condition of a lipid core in an eccentrically developed carotid atheromatous plaque: a static finite element analysis. J Biorheol 27(1):9–17. doi:10.1007/s12573-012-0051-x

    Article  Google Scholar 

  • Yamada H, Shinoda T, Tanaka E, Yamamoto S (1999) Finite element modeling and numerical simulation of the artery in active state. JSME Int J, Ser C 42(3):501–507

    Article  Google Scholar 

  • Yamada H, Yoshitake Y, Iwata N (2007) Comparisons of the finite element analysis solutions and the analytical ones for various opening-angled arterial walls. In: Proc Mech Eng Congress, 2007 Japan, No. 07–1, vol. 5, pp 163–164, in Jap

    Google Scholar 

  • Yamada H, Yuri K, Sakata N (2010) Correlation between stress/strain and the retention of lipoproteins and rupture in atheromatous plaque of the human carotid artery: a finite element study. J Biomech Sci Eng 5(4):291–302

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Pure and Applied Physics, Kansai University, Suita, Japan

    Masako Sugihara-Seki

  2. Department of Biological Functions Engineering, Graduate School of Life Science and Systems Engineering, Kyushu Institute of Technology, Kitakyushu, Japan

    Hiroshi Yamada

Authors
  1. Masako Sugihara-Seki
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hiroshi Yamada
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Masako Sugihara-Seki .

Editor information

Editors and Affiliations

  1. Waseda University, Tokyo, Tokyo, Japan

    Kazuo Tanishita

  2. The University of Tokyo, Tokyo, Tokyo, Japan

    Kimiko Yamamoto

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer Japan

About this chapter

Cite this chapter

Sugihara-Seki, M., Yamada, H. (2016). Fundamentals of Vascular Bio-fluid and Solid Mechanics. In: Tanishita, K., Yamamoto, K. (eds) Vascular Engineering. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54801-0_2

Download citation

  • DOI: https://doi.org/10.1007/978-4-431-54801-0_2

  • Published:

  • Publisher Name: Springer, Tokyo

  • Print ISBN: 978-4-431-54800-3

  • Online ISBN: 978-4-431-54801-0

  • eBook Packages: MedicineMedicine (R0)

Publish with us

Policies and ethics

Search

Navigation