We analytically examine the time-dependent adsorption of analyte (solute) on a finite-sized adsorption region as a model for sensors utilizing patterned or heterogeneous surfaces. We account for both reversible adsorption (assuming first-order reaction) and saturation of the adsorption patch that may arise either from packing constraints (finite area) or because of a finite number of binding sites (ligands). Our main conclusions include the following: (1) Saturation effects, due to either finite patch size or finite number of binding sites, become significant at extremely short times. (2) Increasing the strength of binding between the analyte and the adsorption sites increases the adsorbed amount at short times, but, at long times, the mass adsorbed on a weakly binding patch is higher than that on a strongly binding one. (3) The sensitivity of detection, as defined by the adsorption of the minimal analyte mass required for signaling, over a fixed period of time, does not scale as 1/detection time. As a result, increasing the time over which adsorption occurs increases sensitivity, but not linearly. Sensitivity of detection also increases with increasing patch area and initial binding strength.