Doctor Tideman is dying of cancer. You can promise to find the cure, which has allegedly been developed at the secret base in Darwin. Talk to Thrasher near the museum at the Citadel to learn about the whereabouts of Darwin and head towards there. (Darwin is in a highly irradiated area, make sure that you, at least, have bought level 2 Rad Suits before you get there. You can obtain them, e.g. at Melson's).
Darwin is infested with pestilence that was developed by doctor Finister (one of the main villains of the Wasteland 1, years ago. You will have to make your way through a group of infected mutants. A single mutant is not a threat, but their strength is in numbers so, you may have some problems, if you are short of ammo. Head towards the underground Darwin facility (you can open the door, e.g. with Computers skill). There, inside a small room, in a white locker, you will find bottle with the medicine, which you now need to take to Tideman.
Once you walk into the Darwin underground complex, you will be immediately infected and you will have to find a way to treat the infection. To learn the details, see the chapter about Darwin.