In the summer of 2014, my family and I were on vacation in Italy. Each morning for almost two weeks, I woke up with a terrible headache. They felt a lot like the sinus headaches I’d had since I was a teenager, but none of the routine medications helped.

My wife and I decided to wait until we got home to Texas to see if the headache came from allergies or the hard Italian pillows. But when I continued to wake up with the same headache...