Now, 15 months after the operation, Mr. Beyene, 39, who is from Eritrea, is tumor-free and breathing normally. He is back in Iceland with his wife and two small children, including a 1-year-old boy whom he had thought he would never get to know. In Stockholm earlier this year for a follow-up visit, he showed the long vertical scar on his chest and spoke quietly in English, the raspiness of his voice a leftover from radiation therapy.

His strength was improving every day, he said, and he could even run a little.

“Things are good,” Mr. Beyene said. “Life is much better.”

Imitating Nature

To make an organ, it helps to know how nature does it.

That is why Philipp Jungebluth, a researcher in Dr. Macchiarini’s lab, had mounted a heart and a pair of lungs inside a glass jar on a workbench and connected them by tubing to another jar containing a detergent-like liquid. The organs, fresh from a sacrificed rat, had slowly turned pale as the detergent dripped through and out of them, carrying away their living cells. After three days the cells were gone, leaving a glistening mass that retained the basic shape of the organs.

These were the heart and lungs’ natural scaffolds, or extracellular matrix — intricate three-dimensional webs of fibrous proteins and other compounds that keep the various kinds of cells in their proper positions and help them communicate.

Labs around the world are now experimenting with scaffolds. In some cases the goal is to use the natural scaffolds themselves to build new organs — to take a donor lung, for example, strip all its cells and reseed it with a patient’s own cells. Why not use what nature has perfected, this line of thinking goes, rather than try to replicate it in a synthetic scaffold?

Dr. Macchiarini and his team tried this beginning in 2008, successfully implanting reseeded windpipes from cadavers in about a dozen patients, most of whom are now living normal lives. Because the donor’s own cells are removed, this approach all but eliminates a major problem of transplants: the risk that foreign tissue will be rejected by the recipient. But it does not solve several other problems that may be just as troublesome. A donated windpipe may not be the right size; it has to be stripped of its cells and reseeded while the recipient waits; and the procedure still requires donor organs, which are in short supply.