The statement adds:
Bresnan will begin a recovery and rehabilitation programme and a date for his return to cricket will be determined in due course.
Bresnan, 28, has played in three of this summer's Ashes tests, claiming 10 wickets at an average of 29.6 and with an economy rate of 3.25 (ESPNcricinfo).
He was in the England side that drew at Old Trafford to ensure they retained the Ashes and was also part of the victorious teams at Trent Bridge and Chester-le-Street.
With the bat, Bresnan scored 103 runs in those Tests, averaging 23.75, and with a highest score of 43 last time out.
Here's what Bresnan had to say on Twitter about his injury:
His absence means an opportunity will likely arise for another of England's bowling options. Chris Tremlett and Steve Finn are both options for the selectors, as England prepare for the fifth test at The Oval.