Optimal Control of Surface Acoustic Wave Actuated Sorting of Biological Cells

  • Thomas FrankeEmail author
  • Ronald H. W. Hoppe
  • Christopher Linsenmann
  • Lothar Schmid
  • Achim Wixforth
Part of the International Series of Numerical Mathematics book series (ISNM, volume 165)


The sorting of biological cells using biological micro-electro-mechanical systems (BioMEMS) is of utmost importance in various biomedical applications. Here, we consider a new type of devices featuring surface acoustic wave (SAW) actuated cell sorting in microfluidic separation channels. The SAWs are generated by an interdigital transducer (IDT) and manipulate the fluid flow such that cells of different type leave the channel through designated outflow boundaries. The operation of the device can be formulated as an optimal control problem where the objective functional is of tracking type, the state equations describe the fluid-structure interaction between the carrier fluid and the cells, and the control is the electric power applied to the IDT.


Optimal control Biological cell sorting Surface acoustic waves Finite element immersed boundary method 

Mathematics Subject Classification (2010)

Primary 65K10 Secondary 49M05 74F10 


  1. 1.
    D. Boffi, L. Gastaldi, A finite element approach for the immersed boundary method. Comput. Struct. 81, 491–501 (2003)CrossRefMathSciNetGoogle Scholar
  2. 2.
    D. Boffi, L. Gastaldi, L. Heltai, Numerical stability of the finite element immersed boundary method. Math. Models Methods Appl. Sci. 17, 1479–1505 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    J.L. Carey, J.P. McCoy, D.F. Keren, Flow Cytometry in Clinical Diagnostics, 4th edn. (ASCP Press, Chicago, 2007)Google Scholar
  4. 4.
    M. Eisenstein, Cell sorting: divide and conquer. Nature 441, 1179–1185 (2006)CrossRefGoogle Scholar
  5. 5.
    T. Franke, S. Braunmüller, L. Schmid, A. Wixforth, Surface acoustic wave actuated cell sorting (SAWACS). Lab Chip 10, 789–794 (2010)CrossRefGoogle Scholar
  6. 6.
    T. Franke, R.H.W. Hoppe, C. Linsenmann, L. Schmid, C. Willbold, Numerical simulation of the motion and deformation of red blood cells and vesicles in microfluidic flows. Comput. Vis. Sci. 14, 167–180 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    T. Franke, R.H.W. Hoppe, C. Linsenmann, K. Zeleke, Numerical simulation of surface acoustic wave actuated cell sorting. Cent. Eur. J. Math. 11, 760–778 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    C. Geiger, C. Kanzow, Theorie und Numerik restringierter Optimierungsaufgaben (Springer, Berlin, 2002)CrossRefzbMATHGoogle Scholar
  9. 9.
    T.S. Hawley, R.G. Hawley, Flow Cytometry Protocols, vol. 263, 2nd edn. (Humana Press, Totowa, 2004)Google Scholar
  10. 10.
    R.H.W. Hoppe, C. Linsenmann, An adaptive Newton continuation strategy for the fully implicit finite element immersed boundary method. J. Comput. Phys. 231, 4676–4693 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    C. Peskin, Numerical analysis of flood flow in the heart. J. Comput. Phys. 25, 220–252 (1977)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    C. Peskin, The immersed boundary method. Acta Numer. 11, 479–517 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    H.M. Shapiro, Practical Flow Cytometry (Wiley-Liss, New York, 2003)CrossRefGoogle Scholar
  14. 14.
    L.A. Sklar, Flow Cytometry for Biotechnology (Oxford University Press, New York, 2005)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Thomas Franke
    • 1
    Email author
  • Ronald H. W. Hoppe
    • 2
    • 3
  • Christopher Linsenmann
    • 2
  • Lothar Schmid
    • 1
  • Achim Wixforth
    • 1
  1. 1.Institute of PhysicsUniversität AugsburgAugsburgGermany
  2. 2.Institute of MathematicsUniversität AugsburgAugsburgGermany
  3. 3.Department of MathematicsUniversity of HoustonHoustonUSA

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