The little girl that Alex and Stark treated onGrey's Anatomy Season 7, Episode 8 had biliary atresia. (And her name was Lily, okay? Lily. Remember that. Studies have show that when doctors... okay, we'll stop.) As you might have surmised, this condition affects bile — the liquid produced by the liver, which helps digest fat and transport waste materials to the intestine — and keeps it pent up within the liver, harming the cells of the liver and leading to liver failure.
As Grey's medical guru Meg Marinis explains, it can take two forms: fetal and perinatal. Fetal biliary atresia can develop while the baby is still a fetus, while perinatal biliary atresia makes itself known weeks after the baby is born. A telltale sign of liver dysfunction is jaundice (a yellowing of the skin) that remains for more than a normal amount of time, i.e. more than a couple of weeks. The causes of the condition are not yet definitively known, but it does not seem to be inherited from the parents.
It can be treated two ways. First is a hepato-porto-enterostomy — but you can just call it the Kasai procedure. In this operation, the intestine is connected to the small bile ducts in the liver in order to bypass the larger, blocked ducts. The other option is a liver transplant. Before, only livers from similarly young donors would work, but now doctors can split livers, meaning an adult can donate a portion of his or her liver, and then the livers will regenerate in the two separate bodies. Ain't science neat?