Her sons Nicholas and Nathaniel are 2 weeks old now, and doctors say their chances of survival are good, but only if they receive heart transplants within the next three to six months.
While the boys remained hospitalized at UCLA's Mattel Children's Hospital, Draper and her husband, Michael, pleaded Monday for organ donations that could save their lives.
"We want our boys to have a chance," said Michael Draper. "We want them to come home."
Home is Phoenix, where Michael Draper works in admissions for the University of Phoenix.
Soon after they were born on July 11, Nicholas and Nathaniel were flown separately to UCLA. Their parents have only been able to hold each of the boys once since their births.
"It's just hard," said Michael Draper, 33. "When you walk into that setting it's hard to say, 'Let's just spend some time with our boys and pretend everything's normal."'
The Drapers, who are both registered organ donors, said they don't want other families to suffer the death of a child. But in the event that a baby does die, they hope the parents would be willing to donate the organs to help them or other parents.
Nicholas is already on a list of prospective recipients. His brother, Nathaniel, will be added soon. They will receive hearts in the order that they become available.
The boys are behind other babies on the list, but there are not many children of their age and size who need hearts, said Dr. Mark Plunkett, surgical director of the hospital's pediatric heart transplant program. He is unsure of how many babies are ahead of the Draper twins.
Plunkett said he believes that unless the boys improve dramatically in the next few months they will need transplants. Without them, he said, other organs could fail, further jeopardizing their health,
The boys suffer from dilated cardiomyopathy. It's an extremely rare condition for even one baby to be born with and even more so for twins.
Since the babies' arrival at UCLA, Michael Draper has continued to work in Phoenix while his 32-year-old wife remains in Los Angeles to be near the newborns. Along with the newborns, the couple also have a 5-year-old daughter and 4-year-old fraternal twins, who have been drawing pictures to give their new siblings.
"They're very excited to have new brothers," Nicole Draper said. "They understand that they're sick. ... They just want them to come home."