An Ontario family is appealing to the public for help finding a second liver donor for one of their ailing twin daughters. The family is facing a heart-breaking dilemma: The girls’ father is a viable match, but can provide a portion of his liver to only one of the twins.

Michael Wagner’s three-year-old adopted daughters, Binh and Phuoc, have a genetic disorder called Alagille syndrome that is gradually causing their livers to fail. Without a liver transplant, physicians say both girls will die.

The Wagners, who were told this week that Michael is a viable liver match, have left it for the doctors at Hospital for Sick Children in Toronto to decide which daughter receives the surgery. It is scheduled to take place in two weeks. The decision will be made based on which girl is healthier that day.

Michael Wagner said he would not want to make the decision.

“I am throwing my fate in someone else’s hands,” Michael told CTV News in an interview at the Wagner’s home in Kingston, Ont. on Saturday.

Hepatologist Dr. Binita Kamath said it is an unusual case.

“There is a difficulty in making a decision like this,” says Kamath, who is part of the care team. “It's not something that we commonly encounter, but I think the medical aspects are fairly clear, that we can distinguish between the severity of liver disease between the girls.”

As the family prepares for the surgery, the Wagners are appealing to the public for help locating a second liver donor. On their Facebook page, the Wagners have provided a link to Gift of Life webpage, and a list requirements for any potential donors.

The Wagners adopted the twins from an orphanage in Vietnam two years ago, knowing that they were in ill health.

"They were 9 pounds at 18 months,” Johanne said. “We left the orphanage that day and went to buy two little containers with dragonflies on them and that's what we were going to lay their ashes in if they didn't make it."

Alagille syndrome is a genetic disorder in which inadequate bile ducts make the liver unable to eliminate waste from the bloodstream, allowing it to build up.

The girls must be fed through a tube in their stomach, and the disorder has left them developmentally delayed. The girls also live with constant itching, outbreaks and medication as they wait for new livers.

"There's blood in the bed in the morning from scratching,” says Johanne. “There's no way they can concentrate. It rules their life."

The girls are the youngest of nine children. Everyone in the family pitches in to help.

“Our older daughters know how to work on a g-tube site, they know how to work a medical pump to pump food in, they know how to prep formula,” Michael said.

Johanne is confident that a second donor will be found, “hopefully very soon.”

Anyone interested in seeing if they are a match can contact Toronto General Hospital at 416-340-4800, ext. 6581.

With reports by CTV News' Peter Akman and with files from The Canadian Press