When a pair of conjoined twin girls arrived at a New England hospital last year, doctors were presented with a gut-wrenching case.

In an extremely rare occurrence, the twins were connected by the abdomen and pelvis. They had three legs, one of which had a misshapen foot. Imaging tests showed they shared a single liver, while each child had one complete kidney, which drained into a single bladder, and one undersized kidney. The gastrointestinal tracts were fused together. There was one anus and one vagina.

With the help of a non-profit organization, the girls' parents brought them to the United States to seek treatment. They were 22 months old at the time.

One twin was larger than the other and was alert and interacting with caregivers, while the smaller twin was less active and less able to engage.

Requests were sent to around 20 hospitals across the country. Doctors at Mass General Hospital for Children agreed to take on the case, which they detail in a new report published in the New England Journal of Medicine.

The doctors were quickly hit with the realization that the smaller child, referred to as "Twin A," was dying – and not only that, she was slowly killing her sister, "Twin B."

"Twin B is normal and living relatively healthy and we know that she can live without her sister, whereas Twin A relies completely on her sister. Her sister is her life support," lead study author Dr. Allan Goldstein, M.D., surgeon-in-chief at Mass General Hospital for Children, told CBS News.

Testing showed Twin A's heart was undersized and had significant congenital defects. She had underdeveloped blood vessels between her heart and lungs and relied on the blood supply from her sister.

Twin A became progressively sicker, with several episodes of respiratory distress that required admission to the pediatric intensive care unit.

It became clear to the doctors that Twin A would not survive surgery for separation. But her declining health made it likely she would die even if the twins remained conjoined. Her death would mean the inevitable death of her twin. There was also the chance that if separation surgery were performed, both twins would die.