A group of patients waiting for liver transplants and hospitals have filed suit to block new rules they contend will reduce their access to the life-sustaining organs by transferring hundreds to medical centers in large cities, where the demand is higher.

The plaintiffs — which include transplant centers in Georgia, Michigan, Kansas, Missouri and elsewhere, as well as people on the waiting list for livers — say the rules will leave patients in those places with about 20 percent fewer organs than the current policy, resulting in an increase in deaths.

They are seeking a temporary restraining order to prevent the rules from taking effect. The Department of Health and Human Services, named as a defendant in the case, agreed to delay the rule change from April 30 to May 14 to allow a court hearing to be held May 7.

The lawsuit is just the latest step in decades of wrangling over how to allocate scarce, valuable livers and other organs in a nation where there are far too few to meet the need. For people with end-stage liver disease, a transplant is the only option.

There are 13,546 people on the list for livers; in 2018, 8,250 of the organs were transplanted, according to the United Network for Organ Sharing (UNOS), a nonprofit organization that runs the transplant system. At least three people die each day waiting for livers.

Overall, there are nearly 114,000 people on the list for kidneys, hearts, livers, lungs and other organs.

For years, transplant hospitals generally had first shot at organs donated in their areas. The new policy, adopted last December, offers livers to the sickest patients as far as 500 nautical miles from the donor. It was approved after a lawsuit was filed in New York by a group of patients who said they were waiting longer for livers than people in other parts of the country who were less ill. The government, concerned that the policy in place was illegal, ordered transplant authorities to consider a better approach.

Critics, including many of the hospitals that filed suit Monday, say the policy change that resulted would allow transplant hospitals in New York, San Francisco, Chicago and elsewhere to reach far into other parts of the country where the shortage is not as severe.

The medical centers in the lawsuit say their patients will have access to 256 fewer livers.

In a statement, UNOS, the other defendant in the case, said the new plan would provide “a fairer, more equitable system for all liver patients — no matter where they live.” The organization said the new policy would “reduce waitlist mortality by roughly 100 fewer deaths each year” and “correct an inequity” that “led to unfair advantages and disadvantages based on where liver transplant recipients live.”

Ryutaro Hirose and Sander Florman, transplant surgeons in San Francisco and New York, argued in a statement that the system must be changed because regional disparities have become too sharp.

“A moderately ill patient in Kansas has a better than 60 percent chance of getting a liver transplant within 30 days, while a similarly ill patient in Minnesota has only a 6 percent chance,” they wrote. In California, the odds are 1 percent, they said.

“Why the stark disparity? Our current distribution system is based on arbitrary geographic boundaries drawn more than 30 years ago, when transplantation was in its infancy,” they said.