An apparatus is provided for quantitatively determining amounts and identities of substances whithin patterns of magnetic particles, such as within lateral flow membranes, microfludic systems, and the like. Magnetic particles are complexed with substances to be determined and are excited in an oscillating magnetic field. The magnetizations of the magnetic particles are thereby caused to oscillate at the excitation frequency to create their own fields. These fields are inductively coupled to at least one sensor such as sensing coils fabricated in a gradiometer configuration. The output signals from the sensing coils are appropriately amplified and processed to provide useful output indications.