The 30-year-old Smith fell heavily while contesting a high ball during the Highlanders' 24-15 loss to the Waikato Chiefs last week and lay motionless on the ground before receiving medical treatment.
He was able to walk off the pitch but failed concussion tests and did not return.
"Concussions are always week-by-week," Brown told reporters in Dunedin. "He hasn't even been able to come into training this week."
Smith, who will be an integral part of All Blacks coach Steve Hansen's plans for the British and Irish Lions tour in June, has suffered four concussions in five years.
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"He's okay, but obviously with concussion you never know how long it's going to take," Brown said.
"We're going to be pretty cautious with Ben and just bring him back when he's 100 per cent."
Richard Buckman, who was sidelined for the whole of last season by a shoulder injury, will return at fullback in place of Smith for the clash with the Canterbury Crusaders in Dunedin on Saturday.
"He obviously missed all last year and he was sorely missed," Brown said.
"Richard has the uncanny ability to make the right decision at the right time. And his physicality at the back is something we wanted to get into play against the Crusaders."
Smith's absence, and a back injury to co-captain Ash Dixon, has given All Blacks loose forward Elliot Dixon the opportunity to captain the side.
"When Elliot speaks, everyone listens," Brown said.
"His leadership style is just going to be one of action and we're hoping he will have a big game for us and lead in that way."
(Reporting by Greg Stutchbury in Wellington; Editing by Nick Mulvenney)
