A valve may include (a) a housing defining a valve lumen adapted for fluid connection with a lumen of a catheter; (b) a substantially conical valve member coupled to the housing and extending across the valve lumen to control flow therethrough; (c) a plurality of slits extending through the valve member and separated from one another circumferentially around the valve member; and (d) a ramp formed within a distal end of the valve lumen extending proximally from a distal tip of the valve member toward a ramp vertex separated from the distal tip by a predetermined distance.