constructed of a tubular material such as nitinol which has a temperature dependent shape memory or some other tubular material having a permanent bias toward a curved shape the heat transfer element tubes 34 can be essentially straight originally at room temperature but trained to take on the outwardly flexed 34 basket 34 shape shown in fig3 at a lower temperature this allows easier insertion of the catheter assembly 10 through the vascular system of the patient with the essentially straight but flexible tubes then when the heat transfer element 34 is at the desired location in the feeding artery such as the internal carotid artery refrigerant flow is commenced as the expanding refrigerant cools the heat transfer element 34 down the heat transfer element takes on the basket shape shown in fig3 this enhances the heat transfer capacity while limiting the length of the heat transfer element a fourth embodiment of the heat transfer element is shown in fig4 this embodiment can be constructed of a material such as nitinol the heat transfer element 36 can be originally shaped as a long loop extending from the distal end of the catheter body 18 at room temperature but trained to take on the coiled tubular shape shown in fig4 at a lower temperature with the heat transfer element 36 coiled around the capillary tube 22 this allows easier insertion of the catheter assembly 10 through the vascular system of the patient with the essentially straight but flexible tubular loop shape.