An apparatus for use with a prosthetic heart valve at a native heart valve comprises an adjustable-lumen prosthetic valve support. In a delivery state, the prosthetic valve support is transluminally deliverable to the heart. The prosthetic valve support is transitionable, within the heart, into an implantation state for implantation at the native valve. In the implantation state, the prosthetic valve support has an inner edge that defines a lumen through the prosthetic valve support, the lumen having a first diameter. After the prosthetic valve support is implanted at the native valve, the prosthetic valve support is transitionable into an enlarged state in which the lumen has a second diameter that is greater than the first diameter. In the enlarged state, the prosthetic valve support is configured to support, at the native valve, the prosthetic heart valve within the lumen.