MEDICAL EMER DECLARED; ACR MD-82 PERFORMS LNDG PROC DOWNWIND LNDG OVERSHOT WITH ACFT GROSS WT OVER THAT PERMITTED FOR NORMAL LNDG. BLOWS TIRES ON ROLLOUT.

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DEPARTED BWI ON RWY 28 AT XX51Z. WITH THE FO FLYING AND DURING CLBOUT BTWN FL190 AND FL200; I WAS INFORMED OF A POSSIBLE MEDICAL EMER BY THE #1 FLT ATTENDANT. A PAX HAD LOST CONSCIOUSNESS AND OXYGEN WAS BEING GIVEN TO HIM. I ASKED THE FLT ATTENDANT TO CONFIRM THE SERIOUSNESS OF THE PAX. THE FLT ATTENDANT INFORMED ME THAT THE PAX WAS VERY ILL AND NEEDED IMMEDIATE MEDICAL CARE AND THIS OBSERVATION WAS CONFIRMED BY 2 MEDICAL DOCTORS WHO WERE PAX ABOARD THE FLT. AT FL230; I TOLD CTR THAT WE HAD TO RETURN TO BWI; BECAUSE OF AN ILL PAX IN NEED OF MEDICAL ATTN. A MEDICAL EMER WAS DECLARED AND I REQUESTED MEDICAL CREWS TO BE STANDING BY IN BWI. THE FLT WAS GIVEN PRIORITY BY ATC AND WE LANDED DOWNWIND ON RWY 10 IN BWI AT XY12Z. THE WIND AT TOUCHDOWN WAS 240/3 AND THE AIRCRAFT LNDG WT WAS 129400 LBS. THE TOUCHDOWN WAS SMOOTH AND DURING LNDG ROLLOUT BRAKING WAS APPLIED. ABOUT 1/2 WAY DOWN THE RWY; I FELT 1 TIRE ON THE L MAIN GO FLAT AND ABOUT 2/3 WAY DOWN THE RWY; I FELT ANOTHER TIRE ON THE R SIDE OF THE ACFT GO FLAT. SMOKE WAS RPTED BY THE BWI TWR COMING FROM THE TIRES DURING THE LNDG AND A B-767 AT THE END OF RWY 28 CONFIRMED THAT WE HAD A FLAT ON THE L OUTBOARD AND THE R INBOARD MAIN TIRES. DUE TO THE FACT THAT WE HAD A MEDICAL EMER AND THAT WE HAD 1 MAIN TIRE ON EACH SIDE STILL INFLATED AND I COULD SEE THE ARPT EMER FIRE TRUCK AND THE MEDICAL EMER CREWS AT THE GATE; I PROCEEDED TO THE GATE. THE ACFT'S TAXI TO THE GATE WAS NORMAL AND BLOCKED IN AT THE GATE AT XY15Z. ABOUT 5 MINS AFTER GATE ARR THE OTHER MAIN TIRE ON THE L SIDE DEFLATED FOLLOWED BY ABOUT 10 MINS LATER THE REMAINING MAIN TIRE ON THE OTHER SIDE DEFLATING. MEDICAL ASSISTANCE WAS GIVEN TO THE ILL PAX ON THE ACFT FOR SEVERAL MINS AND THEN HE WAS TRANSPORTED TO THE LCL HOSPITAL. SUPPLEMENTAL INFO FROM ACN 263539: WITH THE FIELD IN SIGHT WE WERE CLRED A VISUAL APCH TO RWY 10 FROM 30 NM W OF BWI AT 10000 FT. WE INTERCEPTED THE GS TO RWY 10 AND LANDED ON RWY 10. THERE WAS A SLIGHT TAILWIND AND I TOUCHED DOWN 2500-3000 FT DOWN THE RWY. I LOWERED THE NOSEWHEEL TO THE RWY; DEPLOYED THE THRUST REVERSERS AND INITIATED BRAKING. AFTER APPROX 1500 FT OF LNDG ROLL; A L MAIN TIRE DEFLATED AND JUST PRIOR TO CLRING; BRAKING WITH 1000 FT REMAINING; A R MAIN TIRE DEFLATED. TWR RPTED THERE WAS SMOKE UNDER THE ACFT WHICH WE ASSUMED WAS FROM THE HVY BRAKING AND DEFLATED TIRES. TAILWINDS DURING THE DSCNT WERE STRONGER THAN WE THOUGHT THEY WERE; MAKING BOTH DSNDING AND SLOWING DOWN AT THE SAME TIME DIFFICULT.