An implant for insertion into a punctum of a patient comprises a body. The body has a distal end, a proximal end, and an axis therebetween. The distal end of the body is insertable distally through the punctum into the canalicular lumen. The body comprises a therapeutic agent included within an agent matrix drug core. Exposure of the agent matrix to the tear fluid effects an effective therapeutic agent release into the tear fluid over a sustained period. The body has a sheath disposed over the agent matrix to inhibit release of the agent away from the proximal end. The body also has an outer surface configured to engage luminal wall tissues so as to inhibit expulsion when disposed therein. In specific embodiments, the agent matrix comprises a non-bioabsorbable polymer, for example silicone in a non-homogenous mixture with the agent.