A drug delivery device includes a drug-eluting element defined by a plurality of outer surfaces including a delivery surface and one or more other non-delivery surfaces. The delivery surface is positioned against tissue of an eye and shaped to define an area of targeted tissue to receive a drug. The drug-eluting element holds the drug when the delivery surface is not positioned against the tissue. Responsive to the delivery surface being positioned against the tissue, the drug-eluting element releases the drug to the area of targeted tissue through the delivery surface. The drug delivery device includes one or more barrier structures disposed along the one or more non-delivery surfaces of the drug-eluting element. The one or more barrier structures substantially inhibit release of the drug from the drug-eluting element through the one or more non-delivery surfaces.