FLC OF LGT ACR ACFT DEVIATED FROM ASSIGNED TRACK DUE WX ENCOUNTER.

Narrative

WE WERE BEING VECTORED FOR AN ILS RWY 11R APCH AT MSP. WE HAD WORKED OUR WAY THROUGH A CORRIDOR FROM THE W WITH OUR WX RADAR AND THE HELP OF ATC. THE ARPT WAS CLR OF TSTMS. OUR FLT CONDITIONS AT THE TIME WERE CLR WITH PATCHY CLOUDS ALL AROUND AND THESE LARGE WALL OF CLOUDS TO OUR L AND STRAIGHT AHEAD. IT STARTED WITH AN ASSIGNED HDG OF 020 DEGS. WE WERE ABOUT TO INTERCEPT 11R LOC 10-11 MI OUT. ATC ADVISED US HE WAS GOING TO TAKE US THROUGH THE LOC; WE SUSPECTED FOR SPACING BEHIND ANOTHER ACFT. THERE WAS A LARGE CELL OFF THE L SIDE OF OUR ACFT WITH A MASSIVE LINE 5 MI OFF THE NOSE. THE CTLR WANTED US TO TURN L TO A 220 DEG HDG. WE TOLD HIM WE WERE UNABLE DUE TO THE TSTM OFF OUR L SIDE. HE THEN GAVE US A HDG OF 360 DEGS. WE FLEW ANOTHER MI OR 2. WE REQUESTED A TURN TO THE R TO KEEP US OUT OF THE LARGE LINE AHEAD. THE CTLR DID NOT ANSWER. WE MADE 2 MORE CALLS WITH NO ANSWER. THIS IS WHEN I MADE THE DECISION TO MAKE THE TURN. IF I DID NOT TURN IMMEDIATELY; WE WOULD HAVE FLOWN INTO THE LARGE LINE. I TURNED TO A 090 DEG HDG WHILE THE COPLT ANNOUNCED ON THE RADIO WE WERE MAKING AN EMER TURN FOR WX AVOIDANCE. WE MONITORED TCASII WHILE NOT IN CONTACT WITH ATC. WE MADE ANOTHER CALL TO ATC AT WHICH TIME ANOTHER CTLR ANSWERED GIVING US A R 220 DEG HDG. HE THEN CLRED US FOR THE APCH. THE REMAINDER OF THE FLT WAS UNEVENTFUL. I TRIED CALLING ATC BY PHONE BUT GOT NO ANSWER.