Skip to main content
SpringerLink
Log in

On the Dynamic Suction Pumping of Blood Cells in Tubular Hearts

  • Conference paper
  • First Online:
Women in Mathematical Biology

Part of the book series: Association for Women in Mathematics Series ((AWMS,volume 8))

Abstract

Around the third week after gestation in embryonic development, the human heart consists only of a valveless tube, unlike a fully developed adult heart, which is multi-chambered. At this stage in development, the heart valves have not formed and so net flow of blood through the heart must be driven by a different mechanism. It is hypothesized that there are two possible mechanisms that drive blood flow at this stage—Liebau pumping (dynamic suction pumping (DSP) or valveless pumping) and peristaltic pumping. We implement the immersed boundary method (IBM) with adaptive mesh refinement (IBAMR) to numerically study the effect of hematocrit on the circulation around a valveless tube. Both peristalsis and DSP are considered. In the case of DSP, the heart and circulatory system is simplified as a flexible tube attached to a relatively rigid racetrack. For some Womersley number (Wo) regimes, there is significant net flow around the racetrack. We find that the addition of flexible blood cells does not significantly affect flow rates within the tube forWo ≤ 10, except in the case forWo ≈ 1. 5 where we see a decrease in average flow with increasing volume fraction. On the other hand, peristalsis consistently drives blood around the racetrack for allWo and for all hematocrit considered.

The original version of this chapter was revised. An erratum to this chapter can be found at https://doi.org/10.1007/978-3-319-60304-9_13

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

  1. Al-Roubaie, S., Jahnsen, E.D., Mohammed, M., Henderson-Toth, C., Jones, E.A.: Rheology of embryonic avian blood. Am. J. Physiol. Heart Circ. Physiol.301(6919), 2473–2481 (2011)

    Article  Google Scholar 

  2. Auerbach, D., Moehring, W., Moser, M.: An analytic approach to the Liebau problem of valveless pumping. Cardiovasc. Eng. Int. J.4, 201–207 (2004)

    Article  Google Scholar 

  3. Avrahami, I., Gharib., M.: Computational studies of resonance wave pumping in compliant tubes. J. Fluid Mech.608, 139–160 (2008)

    Google Scholar 

  4. Babbs, C.: Behavior of a viscoelastic valveless pump: a simple theory with experimental validation. BioMed. Eng. Online9(42), 19832–19837 (2010)

    Google Scholar 

  5. Baird, A.J.: Modeling valveless pumping mechanisms (Ph.D. thesis). Univ. N. C. Chapel Hill628, 129–148 (2014)

    Google Scholar 

  6. Baird, A.J., King, T., Miller, L.A.: Numerical study of scaling effects in peristalsis and dynamic suction pumping. Biol. Fluid Dyn. Model. Comput. Appl.628, 129–148 (2014)

    MATH  MathSciNet  Google Scholar 

  7. Berger, M.J., Oliger, J.: Adaptive mesh refinement for hyperbolic partial-differential equations. J. Comput. Phys.53(3), 484–512 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  8. Berger, M.J., Colella, P.: Local adaptive mesh refinement for shock hydrodynamics. J. Comput. Phys.82(1), 64–84 (1989)

    Article  MATH  Google Scholar 

  9. Bringley, T., Childress, S., Vandenberghe, N., Zhang, J.: An experimental investigation and a simple model of a valveless pump. Phys. Fluids20, 033,602 (2008)

    Article  MATH  Google Scholar 

  10. Chang, H.T., Lee, C.Y., Wen, C.Y.: Design and modeling of electromagnetic actuator in MEMS-based valveless impedance pump. Microsyst. Technol. Micro Nanosyst. Inf. Storage Process. Syst.13, 1615–1622 (2007)

    Google Scholar 

  11. Cooley, J., Tukey, J.W.: An algorithm for the machine calculation of complex Fourier series. Math. Comput.19, 297–301 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  12. Crowl, L.M., Fogelson, A.L.: Computational model of whole blood exhibiting lateral platelet motion induced by red blood cells. Int. J. Numer. Methods Biomed. Eng.26, 471–487 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Fogelson, A.L., Guy, R.D.: Immersed-boundary-type models of intravascular platelet aggregation. Comput. Methods Appl. Mech. Eng.197, 2087–2104 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  14. Forouhar, A.S., Liebling, M., Hickerson, A., Nasiraei-Moghaddam, A., Tsai, H.J., Hove, J.R., Fraser, S.E., Dickinson, M.E., Gharib, M.: The embryonic vertebrate heart tube is a dynamic suction pump. Science312(5774), 751–753 (2006)

    Article  Google Scholar 

  15. Griffith, B.E.: Simulating the blood-muscle-vale mechanics of the heart by an adaptive and parallel version of the immersed boundary method (Ph.D. thesis). Courant Institute of Mathematics, New York University (2005)

    Google Scholar 

  16. Griffith, B.E.: An adaptive and distributed-memory parallel implementation of the immersed boundary (ib) method (2014). URLhttps://github.com/IBAMR/IBAMR

  17. Griffith, B.E., Peskin, C.S.: On the order of accuracy of the immersed boundary method: higher order convergence rates for sufficiently smooth problems. J. Comput. Phys.208, 75–105 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  18. Griffith, B.E., Hornung, R., McQueen, D., Peskin, C.S.: An adaptive, formally second order accurate version of the immersed boundary method. J. Comput. Phys.223, 10–49 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  19. Hickerson, A.I.: An experimental analysis of the characteristic behaviors of an impedance pump (Ph.D. thesis). Calif. Inst. Technol.608, 139–160 (2005)

    Google Scholar 

  20. Hickerson, A., Rinderknecht, D., Gharib, M.: Experimental study of the behavior of a valveless impedance pump. Exp. Fluids38, 534–540 (2005)

    Article  Google Scholar 

  21. Hieber, S., Koumoutsakos, P.: An immersed boundary method for smoothed particle hydrodynamics of self-propelled swimmers. J. Comput. Phys.227, 8636–8654 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  22. Hove, J.R., Koster, R.W., Forouhar, A.S., Acevedo-Bolton, G., Fraser, S.E., Gharib, M.: Intracardiac fluid forces are an essential epigenetic factor for embryonic cardiogenesis. Nature421(6919), 172–177 (2003)

    Article  Google Scholar 

  23. Jones, S.K., Laurenza, R., Hedrick, T.L., Griffith, B.E., Miller, L.A.: Lift- vs. drag-based for vertical force production in the smallest flying insects. J. Theor. Biol.384, 105–120 (2015)

    Google Scholar 

  24. Jung, E.: Two-dimensional simulations of valveless pumping using the immersed boundary method (Ph.D. thesis). Courant Inst. Math. N. Y. Univ.608, 139–160 (1999)

    Google Scholar 

  25. Jung, E., Peskin, C.: 2-d simulations of valveless pumping using immersed boundary methods. SIAM J. Sci. Comput.23, 19–45 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  26. Kenner, T., Moser, M., Tanev, I., Ono, K.: The Liebau-effect or on the optimal use of energy for the circulation of blood. Scr. Med.73, 9–14 (2000)

    Google Scholar 

  27. Kriebel, M.E.: Conduction velocity and intracellular action potentials of the tunicate heart. J. Gen. Physiol.50, 2097–2107 (1967)

    Article  Google Scholar 

  28. Lauga, E.: Propulsion in a viscoelastic fluid. Phys. Fluids19, 083,104 (2007)

    Article  MATH  Google Scholar 

  29. Lee, D.S., Yoon, H.C., Ko, J.S.: Fabrication and characterization of a bidirectional valveless peristaltic micropump and its application to a flow-type immunoanalysis. Sensors Actuators103, 409–415 (2004)

    Article  Google Scholar 

  30. Lee, C.Y., Chang, H.T., Wen, C.Y.: A MEMS-based valveless impedance pump utilizing electromagnetic actuation. J. Micromech. Microeng.18, 225–228 (2008)

    Google Scholar 

  31. Maes, F., Chaudhry, B., Ransbeeck, P.V., Verdonck, P.: Visualization and modeling of flow in the embryonic heart. IFMBE Proc.22(6919), 1875–1878 (2008)

    Google Scholar 

  32. Maes, F., Chaudhry, B., Ransbeeck, P.V., Verdonck, P.: The pumping mechanism of embryonic hearts. IFMBE Proc.37, 470–473 (2011)

    Article  Google Scholar 

  33. Malone, M., Sciaky, N., Stalheim, L., Klaus, H., Linney, E., Johnson, G.: Laser-scanning velocimetry: A confocal microscopy method for quantitative measurement of cardiovascular performance in zebrafish embryos and larvae. BMC Biotechnol.7, 40 (2007)

    Article  Google Scholar 

  34. Manner, J., Wessel, A., Yelbuz, T.M.: How does the tubular embryonic heart work? looking for the physical mechanism generating unidirectional blood flow in the valveless embryonic heart tube. Dev. Dyn.239, 1035–1046 (2010)

    Article  Google Scholar 

  35. Manopoulos, C.G., Mathioulakis, D.S., Tsangaris, S.G.: One-dimensional model of valveless pumping in a closed loop and a numerical solution. Phys. Fluids18, 201–207 (2006)

    Article  Google Scholar 

  36. Mathur, S.R., Sun, L., Das, S., Murthy, J.Y.: Application of the immersed boundary method to fluid, structure, and electrostatics interaction in MEMS. Numer. Heat Transfer Part B Fundam. Int. J. Comput. Methodol.62, 399–418 (2012)

    Article  Google Scholar 

  37. Meier, J.: A novel experimental study of a valveless impedance pump for applications at lab-on-chip, microfluidic, and biomedical device size scales (Ph.D. thesis). Calif. Inst. Technol. 8636–8654 (2011)

    Google Scholar 

  38. Miller, L.A., Peskin, C.S.: When vortices stick: an aerodynamic transition in tiny insect flight. J. Exp. Biol.207, 3073–3088 (2004)

    Article  Google Scholar 

  39. Miller, L.A., Peskin, C.S.: A computational fluid dynamics of clap and fling in the smallest insects. J. Exp. Biol.208, 3076–3090 (2009)

    Article  Google Scholar 

  40. Miller, L.A., Santhanakrishnan, A., Jones, S.K., Hamlet, C., Mertens, K., Zhu, L.: Reconfiguration and the reduction of vortex-induced vibrations in broad leaves. J. Exp. Biol.215, 2716–2727 (2012)

    Article  Google Scholar 

  41. Mittal, R., Iaccarino, C.: Immersed boundary methods. Annu. Rev. Fluid Mech.37, 239–261 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  42. Mohammed, M., Roubaie, S., Jahnsen, E., Jones, E.: Drawing first blood: Measuring avian embryonic blood viscosity. SURE Poster Presentation61, 33–45 (2011)

    Google Scholar 

  43. Ottesen, J.: Valveless pumping in a fluid-filled closed elastic tube-system: one-dimensional theory with experimental validation. J. Math. Biol.46, 309–332 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  44. Peskin, C.: Numerical analysis of blood flow in the heart. J. Comput. Phys.25, 220–252 (1977)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  46. Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T.: Fast Fourier transform. Ch. 12 in Numerical Recipes in FORTRAN: The Art of Scientific Computing2, 490–529 (1992)

    Google Scholar 

  47. Randall, D.J., Davie, P.S.: The hearts of urochordates and cephalochordates. Comp. Anat. Dev.1, 41–59 (1980)

    Article  Google Scholar 

  48. Reckova, M., Rosengarten, C., deAlmeida, A., Stanley, C.P., Wessels, A., Gourdie, R.G., Thompson, R.P., Sedmera, D.: Hemodynamics is a key epigenetic factor in development of the cardiac conduction system. Circ. Res.93, 77 (2003)

    Google Scholar 

  49. Samson, O.: A review of valveless pumping: History, applications, and recent developments (2007). URLhttp://www.researchgate.net/publication/267300626_A_Review_of_Valveless_Pumping_History_Applications_and_Recent_Developments

  50. Santhanakrishnan, A., Miller, L.A.: Fluid dynamics of heart development. Cell Biochem. Biophys.61, 1–22 (2011)

    Article  Google Scholar 

  51. Tucker, D.C., Snider, C., Woods Jr., W.T.: Pacemaker development in embryonic rat heart culturedin oculo. Pediatr. Res.23, 637–642 (1988)

    Google Scholar 

  52. Tytell, E., Hsu, C., Williams, T., Cohen, A., Fauci, L.: Interactions between internal forces, body stiffness, and fluid environment in a neuromechanical model of lamprey swimming. Proc. Natl. Acad. Sci. U. S. A.107, 19832–19837 (2010)

    Article  Google Scholar 

  53. Waldrop, L.D., Miller, L.A.: Large-amplitude, short-wave peristalsis and its implications for transport. Biomech. Model. Mechanobiol.15, 629–642 (2016)

    Article  Google Scholar 

  54. Weinbaum, S., Zhang, X., Han, Y., Vink, H., Cowin, S.: Mechanotransduction and flow across the endothelial glycoalyx. Proc. Natl. Acad. Sci. U. S. A.100, 7988–7995 (2003)

    Article  Google Scholar 

  55. Zhu, L., He, G., Wang, S., Miller, L.A., Zhang, X., You, Q., Fang, S.: An immersed boundary method by the lattice Boltzmann approach in three dimensions. Comput. Math. Appl.61, 3506–3518 (2011)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Steven Vogel for conversations on scaling in various hearts. We would also like to thank Lindsay Waldrop, Austin Baird, Jiandong Liu, Leigh Ann Samsa, and William Kier for discussions on embryonic hearts. This project was funded by NSF DMS CAREER #1151478 awarded to L.A.M. Funding for N.A.B. was provided from an National Institutes of Health T32 grant [HL069768-14; PI, Christopher Mack].

Author information

Authors and Affiliations

  1. Departments of Mathematics and Biology, University of North Carolina, Chapel Hill, NC, 27599, USA

    Nicholas A. Battista & Laura A. Miller

  2. Departments of Biostatistics and Mathematics, UNC Gillings School of Global Public Health, Chapel Hill, NC, 27599, USA

    Andrea N. Lane

Authors
  1. Nicholas A. Battista
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andrea N. Lane
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Laura A. Miller
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nicholas A. Battista .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Duke University, Durham, North Carolina, USA

    Anita T. Layton

  2. Departments of Mathematics and Biology, University of North Carolina, Chapel Hill, North Carolina, USA

    Laura A. Miller

Rights and permissions

Reprints and permissions

Copyright information

© 2017 The Author(s) and the Association for Women in Mathematics

About this paper

Cite this paper

Battista, N.A., Lane, A.N., Miller, L.A. (2017). On the Dynamic Suction Pumping of Blood Cells in Tubular Hearts. In: Layton, A., Miller, L. (eds) Women in Mathematical Biology. Association for Women in Mathematics Series, vol 8. Springer, Cham. https://doi.org/10.1007/978-3-319-60304-9_11

Download citation

Publish with us

Policies and ethics

Search

Navigation