You can learn a lot about scientific advances from going on a tour of a medical facility. Check to see if Santo Spirito Hospital is open to visitors while you're in town.
You'll find Santo Spirito Hospital in the commercial district of Rione XIV Borgo. It's free to enter, so you can leave your wallet tucked away in your pocket. While you're in these parts, why not cross off a few other great sights, such as St. Peters Basilica, Roman Forum and Piazza Navona?