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Mathematical modelling of a magnetic immunoassay

Roberts, Lewis; Griffith, Thom; Champneys, Alan; Piano, Martina; Kiely, Janice; Luxton, Richard W


Lewis Roberts

Thom Griffith

Alan Champneys

Martina Piano
Research Fellow Biosensor reproducibility testing

Janice Kiely
Professor in Bio-electronics/Res In CoDi

Richard Luxton
Research Centre Director-IBST/Professor


© The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. A mathematical model is developed to describe the action of a novel form of fluidic biosensor that uses paramagnetic particles (PMPs) that have been pre-coated with target-specific antibodies. In an initial phase the particles are introduced to a sample solution containing the target which then binds to the particles via antigen-antibody reactions. During the test phase a magnet is used to draw the PMPs to the sensor surface which is similarly coated with specific antibodies. During this process, cross-links are formed by the antigens thereby binding the PMPs to the sensor surface. After the magnetic field is removed, a voltage change across an inductor below the sensor surface is recorded, which is deemed to depend on the number of magnetic particles that have been bound to the sensor surface. The fundamental question addressed is to explain the range of experimentally observed dose-response curves, and how this depends on the various parameters of the problem. In particular, observations have shown both rising and falling dose-response curves, as well as 'hooked' dose-response curves possessing local maxima. Initially a particle-dynamics computational model is produced to determine the time scales of the key processes involved, but is shown to be unable to produce differently shaped dose-response curves. The computational model suggests spatio-temporal effects are unimportant, therefore a homogenized rateequation model is developed for each of the key phases of the immunoassay process. Binding rates are shown to depend on various geometric factors related to the diameter of the PMPs and the size of the sensor surface. The dose-response is shown to depend crucially on various saturation effects during each phase, and conditions can be derived, in some cases analytically, for each of the three qualitatively different curve types. Furthermore, non-dimensionalization reveals 5 key dimensionless parameters and the dependence of these curve shapes on each is revealed. The results point to future quantitative approaches to sensor design and calibration.


Luxton, R., Roberts, L., Griffith, T., Champneys, A., Piano, M., Kiely, J., & Luxton, R. W. (2017). Mathematical modelling of a magnetic immunoassay. IMA Journal of Applied Mathematics, 82(6), 1253-1282.

Journal Article Type Article
Acceptance Date Oct 24, 2017
Online Publication Date Nov 17, 2017
Publication Date Dec 1, 2017
Journal IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications)
Print ISSN 0272-4960
Electronic ISSN 1464-3634
Publisher Oxford University Press (OUP)
Peer Reviewed Peer Reviewed
Volume 82
Issue 6
Pages 1253-1282
Public URL
Publisher URL


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