Abatacept for Treating Adults With Giant Cell Arteritis and Takayasu's Arteritis

Inclusion Criteria: * Diagnosis of GCA or TAK (defined below) * History of active GCA or TAK within the past 2 months * Age of 15 years or older * Willing to use an effective means of birth control throughout the study Specific Inclusion Criteria for Participants with GCA: * Participants must meet three of the following five criteria, including either Criterion 4 or 5: 1. Age at disease onset was equal to or greater than 50 years 2. Disease onset was recent or experiencing a new type of localized pain in the head 3. Erythrocyte sedimentation rate greater than 40mm in the first hour, as determined using the Westergren method 4. Temporal artery abnormality (i.e., temporal artery tenderness to palpation or decreased pulsation, unrelated to arteriosclerosis of cervical arteries) 5. Temporal artery or large vessel biopsy showing vasculitis characterized by a predominance of mononuclear cell infiltration or granulomatous inflammation, usually with multinucleated giant cell or characteristic changes of large vessel stenosis or aneurysm by arteriography Specific Inclusion Criteria for Participants with TAK: * Presence of abnormalities that are consistent with TAK identified using arteriography, plus at least one of the following criteria: 1. Age at disease onset was less than 50 years 2. Pain in the legs or arms 3. Decreased brachial artery pulse (one or both arteries) 4. Difference of more than 10mm Hg in blood pressure between the arms 5. Bruit over subclavian arteries or aorta Exclusion Criteria: * Evidence of active infection (including chronic infection) * Pregnant or breastfeeding * HIV infected, hepatitis C infected, or a positive hepatitis B surface antigen * Inability to comply with study guidelines * Inability to provide informed consent * Cytopenia, as defined by a platelet count of less than 80,000/mm3, an absolute neutrophil count of less than 1,500/mm3, and hematocrit less than 20% * Insufficient kidney function, as defined by a serum creatinine of more than 3 mg/dL or creatinine clearance of 20 ml/min or less * Other uncontrolled disease that could prevent safe study completion * History of any malignant neoplasm except adequately treated basal or squamous cell carcinoma of the skin or solid tumors treated with curative therapy and disease-free for at least 5 years * Receipt of an investigational agent or device within 30 days prior to study entry * A live vaccination within 4 weeks prior to study entry * Presence of a positive tuberculin skin test with induration of at least 5mm * Radiographic evidence suggestive of tuberculosis * Poor tolerability of blood draws or lack of adequate access to veins for medication administration and blood draws * History of treatment with rituximab within 12 months prior to study entry or history of treatment with rituximab more than 12 months prior to study entry, where the B lymphocyte count has not returned to normal * History of treatment with infliximab within the past 49 days, adalimumab within the past 28 days, or etanercept within the past 21 days. * Presence of any of the following diseases or conditions: 1. Microscopic polyangiitis 2. Churg-Strauss syndrome 3. Polyarteritis nodosa 4. Cogan's syndrome 5. Behcet disease 6. Sarcoidosis 7. Kawasaki disease 8. Tuberculosis or atypical mycobacterial infection 9. Deep fungal infection 10. Lymphoma, lymphomatoid granulomatosis, or other type of malignancy that mimics vasculitis 11. Cryoglobulinemic vasculitis 12. Systemic lupus erythematosus 13. Rheumatoid arthritis 14. Mixed connective tissue disease or any overlap autoimmune syndrome