Patients with CML will be tested for matching with a donor (family member) and will undergo a medical history, physical examination and several tests (e.g., breathing tests, X-rays, and others) to determine eligibility for the study. Participants will then undergo apheresis to collect lymphocytes (white blood cells important in the immune system). In apheresis, whole blood is drawn through a needle in the arm, similar to donating a unit of blood. The required component-in this case, lymphocytes-are separated and removed, and the rest of the blood is returned through a needle in the other arm.