Anderson suffered a severe concussion after a collision with goalkeeper Ellery Balcombe during Tuesday’s defeat at Fleetwood Town.
And boss Darren Moore revealed the centre half had commenced the first stage of the FA’s concussion protocol, which lasts a minimum of six days.
That will see Anderson miss Saturday’s trip to Sunderland plus the visit of Accrington Stanley to the Keepmoat on Tuesday.
And he could be a doubt for the home clash with Hull City next weekend, should be not pass smoothly through each stage of the protocol.
“Tom will follow the protocol now and we’re into the first stages of that, which is just a rest,” Moore said.
“He’s got to make sure he gets no reactions from the collision he had on Tuesday.
“He’s doing that now and I’m pleased to report he feels a lot, lot better than he did on Tuesday night.
“He’s making some good progress.
“We’d rule him out of next Tuesday because we wouldn’t want to rush him back for that one. His long term health is more important.
“There’s a possibility he could be a doubt for next weekend but we’ll see.
“We’ll have had eyes on him by then and spoken to him. The medical team will have checked over him and we’ll see if he’s been able to do any training before then.
“It’s too early to say over Tom but Saturday and Tuesday he will be missing.”
Winger Sims could be in line to make his first appearance in a matchday squad since mid-December having trained well this week.
The Southampton loanee is in the final stages of his recovery from a hamstring injury which ended his initial spell at the Keepmoat.
And Moore says he will monitor the 23-year-old over the next day or so before deciding on whether to include him in the trip to the Stadium of Light.
“He’s out there, fine and there’s no problems,” he said.
“He’s been one building up his fitness levels. Because of where he’s been, it’s been about getting the right amount of data from him and get the volume of training into him.
“We’ve just got to keep assessing him and seeing where he’s at.
“If it takes an extra game or two but we get the work into him so when he comes back he stays back, then we’ll do that.
“He’s trained well and we’ll look at him maybe for the weekend and see where he is at.”
James Coppinger will not make the trip to Sunderland this week as he continues to recover from a calf strain.
But the veteran is not too far away from a return to action.
“He’s trained again today and we’re pleased to have him back out there and get his voice around the place again,” Moore said.
“For Copps, he’ll just want to get some good training sessions in and he’ll be pleased to be back out there with the boys.
“It’ll maybe a bit too soon this weekend with him just coming back in.”