The BBC said allowing the actor's face to be used would violate its rules on impartiality, The Daily Telegraph reported.
Denise Yates, head of the National Association for Gifted Children, said Doctor Who seemed like a natural match -- the wandering Time Lord is "very bright, he finds his way out of tight situations and creatively looks for solutions where there doesn't seem to be a solution."
"We've been looking for a while for a role model that children can look up to. We were racking our brains and one of the people that kept coming back to me was Doctor Who, he's a genius but everyone looks up to him," she said. "We asked if the BBC would let us have him, but a letter did come back and said we couldn't because they want to remain impartial."
"Doctor Who" ran on the BBC from 1963 to 1989, going through five doctors. The series was revived in 2005, and Smith is the third doctor this time around.