Ariel Williams is the sister of Rory Williams.
Her mom's family has a history of repeated cancer, which manifested in her sister, Rory.
Rory had brain cancer at age ten which was fully cured. However, when Rory was sixteen, she was diagnosed with osteosarcoma and was also discovered to have a tumor in her chest.
Rory having three cancers by age sixteen and the family history of prominent cancer diagnoses prompted Miranda Bailey to look for a genetic reason. She discovered a mutation of the P53 gene and concluded that the family carried Li-Fraumeni Syndrome, which affects their bodies' ability to regulate cell growth.
Rory became unresponsive and wasn’t breathing properly, so she was rushed into surgery. Despite the doctors' best efforts, Rory bled out into her chest and died. When the doctors came to tell the Williams family, they were distraught and it prompted Ariel to ask the doctors if she also had Li-Fraumeni, which she did.
Ariel gets along well with her parents, though she was exasperated at their desire to set up a foundation for Li-Fraumeni in their family's name.
Ariel got along well with her sister and even joked with her that her cancer was getting them both out of finals and had previously gotten them a puppy. When Rory died, it gave Ariel the push she needed to know whether or not she had the same syndrome Rory had. When it turned out that she did, she said it was good because she'd always wanted to be just like Rory.