Hugo is confined to a wheelchair due to his congenital muscular dystrophy, and is only expected to live to the age of about 25. Della was pushing his wheelchair around a skate park when she collapsed.
Since Della engages in activities typical for a boy her brother's age, the team quickly determines that she is trying to live the life he can't. However, she is very protective of him and refuses a bone marrow transplant despite the minimal risk to him.
However, after Della has a lung transplant, the lung is rejected and the team realizes they will never get another donor lung. As such, once Della is diagnosed, the only treatment is to take a lung from Hugo. Gregory House realizes this will shorten Hugo's life significantly, and when he discusses the plan with Lisa Cuddy, she refuses to let him present the option to the father.
However, the father comes to House asking if there is any option left. He finally admits that they could use Hugo's lung, but this would shorten his life. The father and mother get into an argument about what they should do, and when Della overhears, she tries to kill herself to prevent them from using Hugo's lung.
Finally, with time being very short, the parents inform House that they have decided against the transplant. Surprisingly, House loses it, calls them idiots, and tells them it's a no brainer given that Della will most likely live for decades longer than Hugo would have lived anyway. As he argues the point with Cuddy, Hugo overhears them. His parents rush to him, but he rushes to Della's side and tells her he couldn't live with himself if she died and tells her he wants to give her a lung so that she can outlive him.