Apparatus for use in a heart of a subject, the apparatus comprising: a frame assembly transluminally advanceable to the heart, and comprising: an inner frame that defines: a tubular portion that defines a longitudinal lumen therethrough, and an upstream support portion , coupled to the tubular portion ; and an outer frame that: is coupled to the inner frame , circumscribes the tubular portion , and defines a plurality of flanges that are coupled to the tubular portion ; and a plurality of prosthetic valve leaflets , coupled to the tubular portion , and disposed within the lumen , wherein: the frame assembly : has a compressed state for transluminal delivery to the heart, and has an expanded state, in which: the upstream support portion extends radially outward from the tubular portion , the flanges extend radially outward from the tubular portion and toward the upstream support portion , the tubular portion has a transverse cross-sectional area, and the frame assembly defines a toroidal space between the flanges , the upstream support portion , and the tubular portion , the toroidal space circumscribing the tubular portion and having a cross-sectional area that is 5-10 percent of the transverse cross-sectional area of the tubular portion . a frame assembly transluminally advanceable to the heart, and comprising: an inner frame that defines: a tubular portion that defines a longitudinal lumen therethrough, and an upstream support portion , coupled to the tubular portion ; and an outer frame that: is coupled to the inner frame , circumscribes the tubular portion , and defines a plurality of flanges that are coupled to the tubular portion ; and an inner frame that defines: a tubular portion that defines a longitudinal lumen therethrough, and an upstream support portion , coupled to the tubular portion ; and a tubular portion that defines a longitudinal lumen therethrough, and an upstream support portion , coupled to the tubular portion ; and an outer frame that: is coupled to the inner frame , circumscribes the tubular portion , and defines a plurality of flanges that are coupled to the tubular portion ; and is coupled to the inner frame , circumscribes the tubular portion , and defines a plurality of flanges that are coupled to the tubular portion ; and a plurality of prosthetic valve leaflets , coupled to the tubular portion , and disposed within the lumen , has a compressed state for transluminal delivery to the heart, and has an expanded state, in which: the upstream support portion extends radially outward from the tubular portion , the flanges extend radially outward from the tubular portion and toward the upstream support portion , the tubular portion has a transverse cross-sectional area, and the frame assembly defines a toroidal space between the flanges , the upstream support portion , and the tubular portion , the toroidal space circumscribing the tubular portion and having a cross-sectional area that is 5-10 percent of the transverse cross-sectional area of the tubular portion . the upstream support portion extends radially outward from the tubular portion , the flanges extend radially outward from the tubular portion and toward the upstream support portion , the tubular portion has a transverse cross-sectional area, and the frame assembly defines a toroidal space between the flanges , the upstream support portion , and the tubular portion , the toroidal space circumscribing the tubular portion and having a cross-sectional area that is 5-10 percent of the transverse cross-sectional area of the tubular portion . <EOS>