More recently, the team has focused on an even tougher problem: tumors in the digestive system, including the colon and pancreas, and in ovaries, breasts and other organs, which cause more than 80 percent of the 596,000 cancer deaths in the United States each year.

The researchers analyze tumors for mutations — genetic flaws that set the cancer cells apart from normal ones. They also study TILs, looking for immune cells that can recognize mutations and therefore attack cancerous cells but leave healthy ones alone.

Ms. Ryan, from Rochester Hills, Mich., had colon cancer that spread to her lungs despite surgery, chemotherapy and radiation. With few options, she began looking into research programs and came across the TILs research at the National Cancer Institute. In December 2014, she called the institute, hoping to join the study.

But she was told, based on her scans and records, that she did not have a tumor big enough to yield TILs. A research nurse suggested she send her next set of scans; maybe, in the interim, the tumors would grow. Ms. Ryan took that advice — and was devastated to be turned down again.

“I felt sure I’d get in,” Ms. Ryan said. “My heart sank.”

The rejection left her sobbing. But then she and her husband pulled up images of her scans on their home computer, took screen shots and measurements of a lung tumor that seemed to match the study criteria, and sent them to the cancer institute. She included a polite note asking that, if her tumor was not eligible, she be told why.

“I was trying not to sound like a desperate maniac, but I was a desperate maniac,” she said.

In March 2015, she got in. Whether the screen shots were a deciding factor is not clear. Dr. Rosenberg said the team had been watching her progress and brought her in as soon as they identified operable tumors.

A month later, the researchers performed surgery, removing several lung tumors to search for TILs.

Ms. Ryan’s tissue turned out to be a medical gold mine. She had a KRAS mutation and her TILs included killer T-cells that locked onto the mutation like guided missiles.