Carlsen - Anand, game 6 WCC 2014 -> why Black didn't play 10...Ne4 to try to equalize? What is wrong with 10...Ne4 instead of 10...Nd7?

It seems to me that it was a better move -> Black exchanges one more piece which relieves his cramped position and he obtains the bishop pair.

Can someone explain why 10...Ne4 is inferior to 10...Nd7?

Here is the position I am talking about:

[White "Carlsen, Magnus"]
[Black "Anand, Vishwanathan"]
[Event "WCC 2014"]
[StartFlipped "0"]
[fen "r1bk3r/1p3ppp/p1p1pn2/4P3/1bP5/2N5/PP3PPP/R1B1KB1R b KQ - 0 1 "]

1...Ne4
10...Ne4 doesn't help Black preserve the bishop pair. After the surprising move 11.a3!, Black is forced to take the knight on c3 with the bishop and then we reach a position similar to the one in the game, except that Black's knight is on a more vulnerable e4 square than on a safe square like d7. The pawn on c3 cannot be captured without a significant positional loss to Black. 

   [FEN "r1bk3r/1p3ppp/p1p1pn2/8/1bP1P3/2N5/PP3PPP/R1B1KB1R w KQ - 0 10"]
   [White "Carlsen"]
   [Black "Anand"]



   1. e5 Ne4 2. a3! Bxc3+ (2... Nxc3?? 3. axb4 Ne4 4. f3! +-) (2... Ba5 3. b4 Nxc3
   4. bxa5+=) 3. bxc3 Kc7 (3... Nxc3? 4. a4! Ne4 5. Be3 f6 6. Bd3 Ng5 7. O-O
   Nf7 8. exf6 gxf6 9. f4 +=) 4. Be3 b6 5. Bd3 Nc5 6. Bxc5 bxc5 7. O-O-O+=