I beleive you should continue the Doctor Who story. Its started off great, you should make it so she's hesistant to have fun. The difficult part is finding the right Doctor to incorporate into your story.
Like the girl could be walking home and she hears the sound of the tardis coming from an alley way.
She isnt scared to leave where she currently is. But the only thing she worries about is the children from the foster home.
David Tennet is Silly and Serious
Matt Smith is Corny and Oblivious
Peter Capaldi is Fearless and Caring
