In medical diagnostic tests, including pregnancy testing and tests for typed red blood cells, a small fluid sample is placed at one end of a capillary channel, which has been lined with a dried reagent. If the sample contains the analyte (the substance being tested for) then an agglutination reaction occurs between it and the reagent in the channel, and the agglutinated complexes progressively slow the flow and may even block the channel, partially or completely, so that the flow only reaches the far end very slowly, or not at all. The aim is that this should give a reliable detection of quite low concentrations of analyte in the sample. Platform Diagnostics asked the Study Group to construct a mathematical model of the process, so that, for known binding forces in the agglutination complexes, we can design the channel size and shape, and the fluid viscosity, to maximize the reliable detection of low concentrations. A key question is how the flow time depends on channel size, fluid surface tension and viscosity, (a) in the absence of agglutination, and (b) in the presence of agglutination.

Problem presented by
Bernard Rapson (Platform Diagnostics)

Study Group contributors
Philip Bond (University of Oxford)
Richard Booth (University of Oxford
Chris Breward (University of Oxford)
Richard Clarke (University of Nottingham)
Linda Cummings (University of Nottingham)
Paul Dellar (Imperial College)
Laurent Duchemin (University of Cambridge)
Rosemary Dyson (University of Oxford)
Geoff Evatt (University of Oxford)
David Gelder (Mathematics for Manufacturers)
Ian Griffiths (University of Oxford)
Matthias Heil (University of Manchester)
Anne Juel (University of Manchester)
John King (University of Nottingham)
Andrew Lacey (Heriot-Watt University)
Shailesh Naire (Heriot-Watt University)
Kazuaki Nakane (Osaka Institute of Technology)
John Ockendon (University of Oxford)
Colin Please (University of Southampton)
Giles Richardson (University of Nottingham)
Gordon White (University of Oxford)
Rachel Zammett (University of Oxford)