A method and 
system for performing 
anastomosis uses an anvil to control and support a tissue site during an 
anastomosis procedure involving tissue bonding techniques such as 
tissue welding and 
adhesive tissue bonding. The anvil is particularly useful for supporting a wall of a coronary 
artery during attachment of a graft vessel in a coronary 
artery bypass 
graft procedure. The anvil is inserted into a pressurized or unpressurized 
target vessel and is pulled against an inner wall of the 
target vessel causing tenting of the thin tissue of the vessel wall. A graft vessel is then advanced to the 
anastomosis site and an end of the graft vessel is positioned adjacent an exterior of the 
target vessel. When 
tissue welding is used, a graft vessel fixture is positioned over the tissue surfaces to be welded in order to clamp the graft and target vessel tissue together. The tissue contacting surfaces of the anvil and / or graft vessel fixture are provided with one or more energy applying surfaces. Energy in the form of RF power, 
laser energy or ultrasonic energy is then applied to the compressed graft and target vessel tissue to weld the vessels together. When 
adhesive bonding is used, the 
adhesive may be applied to 
mating surfaces of the graft and / or target vessels either before or after the vessels are brought into contact. After tissue bonding is complete, an incision is formed in the wall of the target vessel to allow 
blood flow between the target vessel and the graft vessel. The incision may be made with an electro-cautery 
cutting device.