Speaking with Dr. Troy with a Doctor skill of at least 75% will present a dialogue option to introduce yourself as a fellow physician curious about the much-vaunted Vault City medical methods, leading to the cost-free accrual of the Vault City Training perk. Returning with an active Poison or Radiation counter level and asking him to cure either will then enable the derivative Vault City Inoculations perk as well.
Troy will heal you if you have been made a citizen of Vault City. If you have been made Captain of the Guard, he will heal the entire party.
You can use this knowledge to blackmail him for $500–$1000 each month. However, if you ever go to him for medical treatment again, you will be hit for 6999 damage.
If you are a citizen (otherwise, you will be kicked out), you can also present this information to Joanne Lynette, and she will kick Dr. Troy out of Vault City. Phyllis will not speak to you again if you do this.