The 25-year-old, the Dons' player of the year last season, was taken off on a stretcher during the first half of the 3-1 win at Dundee on Saturday.
"It was very upsetting to find that Andy had a broken leg after the game," Brown told RedTV.
"It looked very serious at the time of the accident, and it was an accident. Sometimes the tackle has been violent or malicious, I'm most certain it wasn't that.
"Andy was very unfortunate and he went to Ninewells Hospital. Archie (Knox, the assistant manager) and I went in last night, but unfortunately he hadn't come back from the operating theatre."
Brown spoke to Considine on the phone on Sunday night and the operation was a success.
"Ryan Jack (who broke his leg in October) this morning was starting to run and train with us," Brown added.
"It will be a delight when I see that starting to happen in a few months with Andy Considine."
Considine is expected to miss most of the rest of the season.