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The method of administration that provides a steady state of analgesia is patient-controlled analgesia (PCA). PCA allows patients to self-administer a predetermined dose of pain medication through an intravenous pump. This method enables patients to manage their own pain relief in a manner that can be adjusted according to their specific needs, delivering consistent, controlled doses of medication at regular intervals. The continual ability to self-administer medication helps maintain a more stable plasma concentration of the analgesic, preventing peaks and troughs associated with other administration methods.

Intravenous bolus, while effective for providing rapid pain relief, typically leads to a quick spike in medication levels followed by a rapid decline, which may not sustain consistent analgesia. Oral medication and topical application can also provide effective pain relief but may not achieve the same level of control and consistency in analgesic effect as PCA, especially in acute settings where immediate response is required. Therefore, PCA is the most favorable option for ensuring a steady state of analgesia based on the capability to provide continuous and adjustable pain management.