It's not :) It is an app version of the same game. You're the first person who has made that connection in months of beta testing and showing people the app. I'm sure other people will recognize it as well, but I don't think it's as recognizable as was, say, Words With Friends was an app version of Scrabble.
Quarto is a great game, and I think it's well suited for an app like this where you can take your time thinking about your turn. However, it would be nice to give the game designer some credit.
I like your design, but I think you need some copy on privacy and data security. For physician access, how do you handle HIPAA compliance? It would be good to mention this on the front page, at least under the "For Physicians" section.
Your description of research as "pebbles" reminds me of this representation of PhD research (posted on HN a couple years ago). I like how it captures the idea of pushing out the boundaries of what we know just a tiny bit at a time.
This book provides a great introduction to game theory. What I like best about it is the way that it introduces simple examples in the first chapter, and then expands upon these examples in the following chapters, adding new complications each time.