Most organs are only useful while they are still alive.
The only major organ that can currently be used without the death of the donor is the kidney. All organs used for transplants are removed while the body still has its life blood flowing through it and is still functioning. To do this it is increasingly common to declare a person brain dead before the body has stopped functioning. The implication is that the soul has departed form the body, thus allowing the body to be killed by removing the organs. To respond to the increasing number of organ transplants, it is important to understand the Biblical doctrine of body and soul, as it is central to many organ donations.