“You are trying to fool the body’s immune system,” Dr. Humar said. “That is not easy to do.”

Most of the scientific research so far has focused on liver and kidney transplant patients for several reasons, said Dr. James Markmann, chief of the division of transplant surgery at Massachusetts General Hospital.

Those organs can be transplanted from living donors, and so cells from the donor are available to use in an attempt to train the transplant patient’s immune system.

Far more people need kidneys than need any other organ — there are about 19,500 kidney transplants a year, compared with 8,000 transplanted livers. And those transplanted kidneys rarely last a lifetime of battering with immunosuppressive drugs.

“If you are 30 or 40 and get a kidney transplant, that is not the only kidney you will need,” said Dr. Joseph R. Leventhal, who directs the kidney and pancreas transplant programs at Northwestern University.

Another reason to focus on kidneys: “If something goes wrong, it’s not the end of the world,” Dr. Markmann said. If an attempt to wean patients from immunosuppressive drugs fails, they can get dialysis to cleanse their blood. Rejection of other transplanted organs can mean death.

The liver intrigues researchers for different reasons. It is less prone to rejection by the body’s immune system. When rejection does occur, there is less immediate damage to the organ.

And sometimes, after people have lived with a transplanted liver for years, their bodies simply accept the organ. A few patients discovered this by chance when they decided on their own to discard their anti-rejection drugs, generally because of the expense and side effects.