“You can’t use the remainder for the patient the next time she comes in or use it on another patient, so it has to be discarded as waste,” Ms. Traylor said.

Safety standards permit nurses to use drug leftovers in other patients only if used within six hours and only in specialized pharmacies.

Told that she was using only about half of the drug that was purchased, Ms. Haddad said she was shocked.

“No wonder my premiums keep going up,” she said.

Medicare and many private insurers charge patients drug co-payments of as much as 20 percent, which can add up to tens of thousands of dollars annually for the latest drugs; much is spent on cancer medicines that patients never receive, according to the study.

Dr. Dixie-Lee Esseltine, vice president for oncology clinical research at Takeda, wrote in an email that the pharmaceutical firm “worked closely with the F.D.A. to establish the Velcade vial size of 3.5 mg to ensure that one vial of Velcade would provide an adequate amount of the drug for a patient of almost any size.”

Velcade is sold in Britain in both 1-milligram and 3.5-milligram vials.

Takeda is expected to earn $309 million this year on supplies of Velcade that are discarded, an amount that represents 30 percent of the drug’s overall sales in the United States, the cancer researchers estimated. If Takeda provided an additional vial size of 0.25 milligram, waste would be slashed by 84 percent, also reducing Velcade’s sales in the United States by $261 million annually, the researchers calculated.