Mee sustained the injury in Monday's 4-2 Premier League loss at West Ham United, joining fellow defenders Rico Henry, Ethan Pinnock and Aaron Hickey on a long list of injured players.
"It’s our entire back four from last season now out,” Frank told reporters ahead of Saturday's derby at home to Chelsea.
"If you look at key players, I have potentially seven starters out. The back four, Bryan (Mbeumo), Kevin (Schade) and Josh Dasilva.
"We have a good squad, we have good players, but that makes it a little bit more tough," he said.
Brentford, 16th with 25 points, have lost their last three games as they struggle to stay above the relegation zone.
"I'm very optimistic. I'm a massive fighter and my players are massive fighters," Frank said.
"When you are in a situation when things are going against you, you can only do one thing. Get tighter together... and fight. I can't wait for tomorrow."
