Skip to main content

Conformal mapping in optical biosensor applications


Optical biosensors are devices used to investigate surface-volume reaction kinetics. Current mathematical models for reaction dynamics rely on the assumption of unidirectional flow within these devices. However, new devices, such as the Flexchip, include a geometry that introduces two-dimensional flow, complicating the depletion of the volume reactant. To account for this, a previous mathematical model is extended to include two-dimensional flow, and the Schwarz–Christoffel mapping is used to relate the physical device geometry to that for a device with unidirectional flow. Mappings for several Flexchip dimensions are considered, and the ligand depletion effect is investigated for one of these mappings. Estimated rate constants are produced for simulated data to quantify the inclusion of two-dimensional flow in the mathematical model.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8


  • Carrier GF, Krook M, Pearson CE (2005) Functions of a complex variable: theory and technique (classics in applied mathematics). Society for Industrial and Applied Mathematics, Philadelphia

  • Driscoll T, Trefethen L (2005) Schwarz–Christoffel mapping (Cambridge monographs on applied and computational mathematics). Cambridge University Press, Cambridge

  • Driscoll TA (2013) The Schwarz–Christoffel toolbox for MATLAB.

  • Edwards DA (1999) Estimating rate constants in a convection-diffusion system with a boundary reaction. IMA J Appl Math 63:89–112

    Article  MathSciNet  Google Scholar 

  • Edwards DA (2001) The effect of a receptor layer on the measurement of rate constants. Bull Math Bio 63:301–327

    Article  Google Scholar 

  • Edwards DA (2011) Transport effects on surface reaction arrays: biosensor applications. Math Biosci 230:12–22

    Article  MathSciNet  MATH  Google Scholar 

  • GE Healthcare (2006) Biacore Flexchip product information. Uppsala

  • GE Healthcare (2007) Label-free interaction analysis in real-time using surface plasmon resonance, (technology note 23)

  • Mason T, Pineda AR, Wofsy C, Goldstein B (1999) Effective rate models for the analysis of transport-dependent biosensor data. Math Biosci 159:123–144

    Article  MATH  Google Scholar 

  • Rich RL, Cannon MJ, Jenkins J, Pandian P, Sundaram S, Magyar R, Brockman J, Lambert J, Myszka DG (2008) Extracting kinetic rate constants from surface plasmon resonance array systems. Anal Biochem 373(1):112–120

    Article  Google Scholar 

  • Zumbrum M (2014) The effect of receptor nonuniformity for surface reactions within optical biosensors. IMA J Appl Math (submitted)

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Matthew E. Zumbrum.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Zumbrum, M.E., Edwards, D.A. Conformal mapping in optical biosensor applications. J. Math. Biol. 71, 533–550 (2015).

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI:


  • Flexchip
  • Optical biosensors
  • Perturbation methods
  • Schwarz–Christoffel mapping
  • Surface-volume reactions

Mathematics Subject Classfication

  • 35C20
  • 92C45