The Staff of Asclepius in Myth
Asclepius is the god of Healing. He is the son of Apollo and the nymph, Coronis. While pregnant with Asclepius, Coronis secretly took a second, mortal lover. When Apollo found out, he sent Artemis to kill her. While burning on the funeral pyre, Apollo felt pity and rescued the unborn child from the corpse. Asclepius was taught about medicine and healing by the wise centaur, Cheiron, and became so skilled in it that he succeeded in bringing one of his patients back from the dead. Zeus felt that the immortality of the Gods was threatened and killed the healer with a thunderbolt. At Apollo's request, Asclepius was placed among the stars as Ophiuchus, the serpent-bearer.
The children of Asclepius included his daughters Meditrina, Hygeia and Panacea who were symbols of medicine, hygiene and healing (literally, "all healing") respectively. Two of the sons of Asclepius appeared in Homer's Illiad as physicians in the Greek army (Machaon and Podalirius).
Note that the classic Hippocratic Oath is sworn "by Apollo the physician, by Æsculapius, Hygeia, and Panacea,..."