Configurational Statistics of Magnetic Bead Detection with Magnetoresistive Sensors

Magnetic biosensors detect magnetic beads that, mediated by a target, have bound to a functionalized area. This area is often larger than the area of the sensor. Both the sign and magnitude of the average magnetic field experienced by the sensor from a magnetic bead depends on the location of the bead relative to the sensor. Consequently, the signal from multiple beads also depends on their locations. Thus, a given coverage of the functionalized area with magnetic beads does not result in a given detector response, except on the average, over many realizations of the same coverage. We present a systematic theoretical analysis of how this location-dependence affects the sensor response. The analysis is done for beads magnetized by a homogeneous in-plane magnetic field. We determine the expected value and standard deviation of the sensor response for a given coverage, as well as the accuracy and precision with which the coverage can be determined from a single sensor measurement. We show that statistical fluctuations between samples may reduce the sensitivity and dynamic range of a sensor significantly when the functionalized area is larger than the sensor area. Hence, the statistics of sampling is essential to sensor design. For illustration, we analyze three important published cases for which statistical fluctuations are dominant, significant, and insignificant, respectively.