To increase the resistance to liquid and substance flow through a lumen of a tubular organ, a flexible apparatus is appended to an exterior wall of the tubular organ such that it only partially surrounds the tubular organ. Ultimately, tissue ingrowth through the apparatus integrates the apparatus into the wall of the tubular organ. When the apparatus is applied to tubular tissue, flow though the tubular tissue is restricted. The apparatus may also support an optional expandable component, e.g., a balloon-like element that can be selectively inflated and/or deflated to restrict the lumen of the tubular tissue by a desired degree.