One question is how Sharapova, who lives mainly in Florida, has been getting the drug.

A spokeswoman for the Food and Drug Administration said it was illegal to import unapproved drugs into the United States. But the agency will allow imports for personal use in certain situations, such as if a drug is used for a serious condition for which effective treatment is unavailable in the United States. Generally, only a three-month supply is allowed in those cases.

Sharapova’s lawyer, John Haggerty, said she had always complied with the F.D.A.’s guidelines on the personal importation of drugs.

Sharapova said she had been prescribed the drug for conditions including a magnesium deficiency, irregular electrocardiograms and prediabetes. Some experts said they did not see how meldonium would necessarily help those conditions (although there are at least two studies in rats showing that the compound might help with diabetes). The F.D.A. has approved many drugs for heart conditions and diabetes.

Meldonium appears to work by inhibiting the synthesis of a substance called carnitine, which the cells in the body need to burn fat to produce energy. But when cells are not getting enough oxygen, they can switch to burning glucose instead of fat. Glucose produces more energy for a given amount of oxygen than fat.

“Glucose is more efficient when you have limited oxygen,” said Dr. William R. Hiatt, a cardiologist and professor at the University of Colorado School of Medicine.

That could explain why the drug would help people whose heart cells were not getting enough oxygen. One randomized controlled trial showed that the use of meldonium increased how long patients with angina, or chest pains from inadequate blood flow to the heart, could ride an exercise bike.

That might also, in theory, suggest why the drug would help athletes. At maximum exertion, their cells might not be getting enough oxygen. Use of the drug might help switch the cells to burning glucose instead.