Complicating the problem, the number of trials has exploded in recent years. There are more than 100 Alzheimer’s studies looking for a whopping 25,000 participants, Mr. Dwyer said.

To begin filling them all, 37.5 million patients in the right age group would first have to be informed. Ten percent would be referred to a trial site for screening.

Just 4 percent will move forward with an evaluation, and of these, just over 17 percent will drop out, given the current rate, leaving roughly 125,000 to be screened. And with an 80 percent screening failure rate, that leaves 25,000 participants of the 37.5 million who were first informed.

The numbers make it clear: There’s no way scientists are going to find 25,000 participants for all of the Alzheimer’s trials that have been approved.

“The irony is that the science has never been more promising,” Mr. Dwyer said. “How many promising drugs will be abandoned or their evaluation seriously delayed? Some good science is going to be left on cutting-room floor.”

These trials are not just expensive; so far, they have been expensive failures.

For the most part, researchers have focused on a target that seemed obvious and approachable: a protein, beta amyloid, that starts to accrue in patients’ brains years before their memories falter. It is believed to be the first sign of Alzheimer’s disease.

For more than a decade, companies tried again and again with anti-amyloid drugs to slow or halt the disease, spending billions of dollars in clinical trials. Lilly alone invested more than $3 billion. Pfizer, after a series of failures, announced in January that it was getting out of the Alzheimer’s race altogether.