The present invention relates to covered endoprosthetic devices. Covered endoprosthetic devices comprise an endoprosthesis and a sheath. The sheath comprises a central portion and outer portions, wherein the central portion preferentially restricts or causes a restriction of blood flow. Blood flow can be reduced by the central portion of the sheath by varying the permeability of the sheath or by having projections on the sheath that slow blood flow. Permeability may be provided by perforations or holes in the material of the sheath or by varying the polymer structure that makes up the sheath itself. The outer portions of the sheath do not substantially reduce blood flow. Methods of using sheath-covered endoprosthetic devices of the invention to treat aneurysms, especially aneurysms in proximity to small perforator vessels or arteries, are also encompassed.