A catheter includes a sealed multi-lumen heat transfer extension designed to internally circulate a coolant, and thereby cool tissue or fluid surrounding the catheter. The heat transfer extension includes a tube having a distally positioned region that coils about the tube's longitudinal axis. The tube houses multiple lumens running longitudinally along the tube. These lumens include one or more supply lumens and one or more return lumens. A distal fluid exchange reservoir resides at the tube's tip, for the purpose of redirecting fluid from the supply lumen(s) to the return lumen(s). The heat transfer extension may include a shape memory structure causing the heat transfer extension to vary its shape according to temperature. Namely, the extension assumes a coiled shape under predetermined shape-active temperatures, and it assumes a non-coiled shape under other predetermined shape-relaxed temperatures. The catheter also includes an interface having supply and return lines to provide coolant to the heat transfer extension, and remove coolant returning therefrom. To connect the supply/return lines with the supply/return lumens, the invention may include a fluid transfer housing. Optionally, one or more of the lumens may be configured as flow-through lumens to exchange fluids with the patient.