In Justice Douglas E. McKeon's fluorescent-lighted chambers in the Bronx, a new way of handling medical malpractice suits was on full, and sometimes gruesome, display. Around a polished wood table, lawyers haggled over the price for a lost nose ($300,000) and the missing tip of a finger ($50,000). The discussion was like some kind of malpractice bazaar, with lawyers blurting out terrible facts and big numbers. Judge-directed negotiation is seen by the Obama administration as offering states a way to curb liability expenses that have sharply increased healthcare costs nationally. Getting judges involved earlier, more often and much more actively in pushing for settlements, is its crucial ingredient. New York officials say the program bypasses years of court battles, limiting legal costs while providing injured patients with compensation that is likely to be less than a jury would award but can be paid out years earlier, without lengthy appeals. Under a $3 million federal grant, the city courts are now expanding the program beyond the Bronx, where it started in cases against city hospitals, to courts in Brooklyn and Manhattan, as well as to cases against private hospitals. It is to begin in Buffalo courts in the fall.