FLT CREW LOSS OF ACFT CONTROL AT THE START OF TKOF ROLL CAUSED A TKOF ABORT THAT RESULTED IN RWY EXCURSION.

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START AND TAXI; NORMAL. F/O WAS TO EXECUTE TKOF; LINEUP ON CENTERLINE OF RWY; GOOD LINEUP ON CENTER. F/O INCREASED PWR TO 80% N2; CALLED 80%. F/O CALLED ROLLING AND ADVANCED PWR TOWARD TKOF EPR. AS PWR WAS ADVANCED BEYOND 80% N2; I REALIZED THE ACFT HAD STARTED A YAW TO THE L; AND HAD STARTED PROGRESSING TOWARDS THE EDGE OF THE RWY. AS SOON AS IT BECAME APPARENT THAT WE WERE HEADED OFF THE RWY; I IMMEDIATELY ATTEMPTED TO STEER BACK TO CENTERLINE; BUT HAD NO STEERING WITH NOSEWHEEL STEERING; OR RUDDER PEDALS. AT THIS TIME I CALLED FOR AN ABORT; IMMEDIATELY WENT TO IDLE WITH THE THROTTLES; AND ALSO ATTEMPTED TO BRAKE THE ACFT TO A STOP. THERE WAS NO BRAKING ABILITY AT THIS POINT. AS I REDUCED PWR; THE ACFT STARTED TO YAW TO THE R. THIS WAS APPARENTLY CAUSED BY MY NOT HAVING CLOSED THE THROTTLE ON THE #1 ENG. BY THE TIME I REALIZED THAT THE #1 THROTTLE WAS STILL FORWARD OF 2; 3; AND 4; THE ACFT WAS EXITING THE RWY ON THE R SIDE. AS I CLOSED THE THROTTLE; THE ACFT LEFT THE RWY; THEN STARTED A SKID IN THE MUD; FINALLY COMING TO REST FACING THE RWY. AFTER THE ACFT HAD COME TO A STOP IN THE INFIELD BETWEEN RWYS 16L AND 16R AT SEA; I WAS ABLE TO PWR THE ACFT OUT OF THE MUD AND BACK ONTO THE RWY W/O ANY DAMAGE. I THEN TAXIED THE ACFT BACK TO PARKING; DID A WALK AROUND TO INSPECT FOR DAMAGE; AND THE ONLY DAMAGE THAT WAS FOUND ON THE ACFT WAS A BROKEN CABLE GUIDE BRACKET FOR THE NOSEWHEEL STEERING CABLES. NONE OF THE AIRFRAME OR LNDG GEAR WAS DAMAGED. I ALSO MIGHT ADD THAT THERE WAS SOME OBVIOUS MUD INGESTION ON THE #3 ENG; AND POSSIBLY SOME IN THE #2. OUR COMPANY MECHANICS PERFORMED AN EXTENSIVE INSPECTION OF THE ACFT SUBSEQUENT TO THE INCIDENT; AND NO FURTHER DAMAGE WAS FOUND. THEY DID HAVE TO REMOVE AND CLEAN ALL WHEELS AND BRAKES; AND REPLACE THE #5 TIRE DUE TO SOME CUTS. THE ENGS WERE ALL FLUSHED; AND BORE-SCOPED TO CHK FOR INTERNAL DAMAGE. AGAIN; NO DAMAGE APPARENT. LATER ON THAT EVENING; ANOTHER CREW WAS FLOWN TO SEA TO FERRY THE ACFT TO WILMINGTON; OH (ILN) FOR FURTHER INSPECTION AND TESTING. I WAS INFORMED THAT AT ILN; SOME TAXI AND ENG SPOOL TESTS WERE RUN; AND IT WAS DETERMINED AT THAT TIME THAT THERE WAS A DEFINITE SPOOL TIME DIFFERENCE BETWEEN THE NUMBERS 1 AND 4 ENGS. THE #1 ENG WAS CONSISTENTLY SLOWER TO ACCELERATE THAN THE #4; ESPECIALLY WITH ENG ANTI-ICE SELECTED ON. (PLEASE NOTE; THOUGH NOT INCLUDED IN MY RPT; THE ENG ANTI-ICE HAD BEEN SELECTED ON FOR TKOF AT SEA). THIS; I BELIEVE; COULD HAVE CONTRIBUTED TO THE DIRECTIONAL CONTROL PROBS THAT I EXPERIENCED AT SEA. SUPPLEMENTAL INFO FROM ACN 81935: THE ACFT WAS LOADED WITH AN AFT C.G; BUT STILL WITHIN LIMITS.