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.