WASHINGTON — The government is expanding its “mystery disease” program, funding a network at six universities to help diagnose patients with diseases so rare they’ve been told they’re undiagnosable.

The National Institutes of Health has evaluated hundreds of these cold-case patients as part of a pilot program since 2008. There’s a waiting list.

So on Tuesday, the agency announced the NIH Undiagnosed Diseases Network, a four-year, $43 million initiative to bring more doctor-detectives on board in the quest to at least put a name to more patients’ puzzling symptoms, and eventually find treatments.

The centers include: Baylor College of Medicine in Houston; Duke University; Stanford University; University of California, Los Angeles; Vanderbilt University and the Harvard University teaching hospitals Brigham and Women’s, Massachusetts General and Boston Children’s.

The network “will focus on the rarest of disorders, often those affecting fewer than 50 people in the entire world,” said Dr. Eric Green, director of NIH’s National Human Genome Research Institute.

More than 3,000 patients have applied to the pilot program, which so far has enrolled about 600, NIH said. Gahl put the “solve rate” at about 25 percent.

The NIH will continue evaluating about 130 to 150 patients a year. The universities will gradually add additional ones until, by mid-2017, each should be admitting about 50 patients a year.

Participating patients spend a week in the designated center for head-to-toe exams. NIH said no patient would be turned away because of lack of insurance although each center would decide how to handle coverage.