Irreversible electroporation (IRE) employs microsecond scale, mega-volt/m electric field pulses to impair the cell membrane. IRE is emerging as a valuable new minimally invasive technique. Of central importance in using IRE is the existence of electric fields, which while impairing the cell membrane, do not cause thermal Joule heating induced damage to the tissue. Our recent studies suggest that IRE could become an important technique to ablate vascular smooth muscle cells of the arterial wall and attenuate restenosis following angioplasty. This study was done to support the use of IRE in treatment of restenosis and is a fundamental investigation on the electric field parameters that can produce non-thermal IRE ablation of cells on the arterial wall. The study combines time-dependant finite-element models of the electric field equation and of the bio-heat equation with Henriques and Moritz thermal damage integral to evaluate the range of non-thermal IRE fields for use in blood vessels. The theoretical analysis is supported by temperature measurements during intravascular IRE of rodent carotid arteries, showing no significant temperature rise.