LOS ANGELES — As baby boomers enter their golden years, the number of people afflicted with Alzheimer’s disease is expected to reach 13.8 million by 2050 — millions more than previously anticipated, according to a new study in the journal Neurology.

If researchers can’t find a way to reduce the prevalence of the brain disease, the cost to care for all of these patients could top $1 trillion a year, experts say.

Alzheimer’s is a progressive brain disease that damages patients’ memory and cognitive skills, ultimately leaving them unable to care for themselves. Scientists aren’t sure how it starts, but they believe it causes plaques and tangles to form in the brain, slowly killing neurons and causing the entire brain to shrink. Between 60 percent and 80 percent of dementia cases are believed to be a consequence of Alzheimer’s, according to the Alzheimer’s Association.

The risk of developing the disease rises with age. So while deaths from breast and prostate cancer, heart disease, stroke and HIV all fell between 2000 and 2008, the number of Alzheimer’s-related deaths grew by two-thirds.