A fluid pump for medical applications. The pump includes a flexible cassette containing two pump chambers of differing volume, each chamber having a dedicated piston. The first piston pushes fluid from the first chamber to the second, and the second piston pushes fluid from the second chamber and out of the pump. Three valves are coordinated with the action of the pistons to control fluid flow into and out of the pump chambers by applying and releasing pressure to and from specific points of the flexible cassette. A control system controls the operation of the pistons and valves and directs the pistons to be advanced according to a time-varying velocity profile. The first pump chamber passively fills and then rapidly expels the collected fluid into the second chamber at low pressure. The second chamber, once filled, expels the fluid downstream at high pressure, producing a pulsatile pressure profile.