The 23-year-old sustained the injury during the second T20I against Zimbabwe in Bulawayo this month and has been ruled out for at least six weeks.
Rana has been a leading light for Bangladesh, having taken 44 wickets in international cricket this year across formats - the most by any bowler - at an average of 18.31.
It is a massive blow to lose the paceman ahead of what will be Bangladesh first tour to Australia in 23 years with the first of two Test matches set to get underway in Darwin on 13 August.
A BCB statement said on Rana said: "He is expected to require approximately four weeks for tissue healing, followed by a further two to four weeks of bowling workload progression."
It has also been confirmed that left-arm seamer Shoriful Islam and wicketkeeper-batter Litton Das will also miss the first Test against Australia due to injury, with both players hopeful of being fit for the second game of series in Mackay from 22 August.
Bangladesh stand-in captain Towhid Hridoy was pleased with his charges after they claimed a 2-1 T20I series victory in Zimbabwe following a four-wicket win in the deciding game in Bulawayo.
Bangladesh spinner Rishad Hossain took four wickets as the visitors set up a deciding game in the T20I series against Zimbabwe with a 34-run victory in Bulawayo on Friday.
Zimbabwe drew first blood in the three-match T20I series at home against Bangladesh after a 32-run victory in Bulawayo on Wednesday.
Bangladesh claimed a consolation victory in the ODI series against Zimbabwe in Harare after the tourists prevailed by seven wickets in the third and final game.
