Kanye West and Kim Kardashian
06/21/2013 at 12:05 PM EDT
and her "beautiful daughter"
have left the hospital where she gave birth nearly a week ago.
A source tells PEOPLE the new mom slipped away early Friday in the middle of the night. But why did she stay so long at Cedars-Sinai Medical Center in Los Angeles?
"The baby is doing great and has been gaining weight just like her doctor was hoping," a source tells PEOPLE of the infant, whose name was revealed on Thursday night
to be North West. "The baby has had no problems at all."
Kardashian, 32, was rushed to Cedars-Sinai
June 14 due to complications and went in to early labor, which could explain why she remained in her hospital room for a week.
"If the baby was a little bit pre-term then you want to watch the baby a little bit longer," says Dr. Shelena Lalji, resident OB/GYN expert for AestheticTV.com
. "You just want to make sure the baby is completely independent and doesn't need any kind of additional respiratory help or monitoring."
Doting father Kanye West
did not leave his girlfriend's side, and the trio received a constant flow of visitors bringing cupcakes, gifts and well wishes.
"The nurses have had nothing but great things to say about Kim," says the source. "They say Kim has been very sweet and very easy to deal with."
Kardashian may not have made any diva demands, but her suite potentially cost over $4,000 per day.
"Insurance companies unfortunately want patients to go home as soon as possible," Dr. Lalji says. "But in her case, I am sure that is not an issue."