Brown damaged ligaments during the 2-1 home defeat to Walsall last weekend and missed the goalless draw with MK Dons at the Keepmoat Stadium on Tuesday night.
Rovers boss Brian Flynn confirmed: "Chris will be out for a minimum of four weeks and a maximum of eight weeks.
"It's not good for him at this stage and it's definitely not good for us.
"He's been playing exceptionally well and has played the part of a real good centre forward. It's a big blow."