CROSS-REFERENCE TO RELATED APPLICATIONS 
     The present application is a divisional of U.S. application Ser. No. 12/868,624, filed Aug. 25, 2010, entitled “RECONFIGURING TISSUE FEATURES OF A HEART ANNULUS,” now abandoned, which claims priority to U.S. Provisional Patent Application No. 61/376,614, filed on Aug. 24, 2010, entitled “RECONFIGURING HEART FEATURES,” which is incorporated here in its entirety by reference. 
    
    
     BACKGROUND OF THE INVENTION 
     Field of the Invention 
     This description relates to reconfiguring heart features. 
     Description of the Related Art 
     The annulus of a heart valve (a fibrous ring attached to the wall of the heart), for example, maintains the shape of the valve opening and supports the valve leaflets. In a healthy heart, the annulus is typically round and has a diameter that enables the leaflets to close the valve tightly, ensuring no blood regurgitation during contraction of the heart. Because the annuluses of atrioventricular valves, for example, are supported more stably by the heart tissue on one side of the annulus than on the other side, and for other reasons, the size and shape of an annulus may become distorted over time. The distortion may prevent the valve from closing properly, allowing blood to regurgitate backwards through the valve. The distortion can be corrected, for example, during open heart surgery, by attaching a ring or other support around the annulus to restore its shape and size. 
     SUMMARY OF THE INVENTION 
     In one aspect, in general, a tool to attach a support to a heart valve annulus, includes a stabilizing body that includes features to stabilize an axial position of the tool relative to the annulus, and an attachment device connected to the stabilizing body, the stabilizing body and the attachment device being movable relative to one another under control from a location remote from the tool, from one configuration in which the support is held in a pre-attachment contracted state to another configuration in which the support is held in an expanded state for attachment at multiple locations around the annulus by action from the remote location. 
     Implementations may include one or more of the following features. The tool may be configured to permit blood to flow through during attachment. The attachment device may be connected to the stabilizing body in a configuration to permit the attachment device to be withdrawn from the support after attachment by action from the remote location. The tool may be for use with an annular support that includes sharp elements for attachment of the support to the annulus and connection elements to connect the support temporarily to the attachment device until the support has been attached to the annulus. The stabilizing body and the attachment device may form a tracking mechanism and the attachment device is configured to track the tracking mechanism during relative motion of the attachment device and the stabilizing body. The stabilizing body may be configured to match topological features of the annulus to provide stabilization without applying more than small radial forces to the annulus. The stabilizing body may be configured to support leaflets of the valve of the heart in at least a nearly closed position when the tool is in place in the valve. The stabilizing body may comprise a set of preformed flexible wires arranged around a central axis. The attachment device may comprise a set of tubes arranged around a central axis and extending to the remote location. 
     In one aspect, in general, a tool to attach a support to a heart valve annulus, includes flexible tubes that together carry the support, and flexible wires passing through the flexible tubes, the flexible wires forming a basket that is contoured to contact a heart valve annulus at multiple points along its periphery. 
     Implementations may include one or more of the following features. The basket may be collapsible for delivery to the annulus. The tool may include a sheath over the collapsed basket. The sheath may include tubes sized to receive the flexible wires. The basket may be expandable and the sheath resists expansion of the basket. The flexible wires may provide a passage through which blood can flow. Each of the flexible tubes may have at least two lumens. Each of the flexible wires may have a free end that passes through one of the two lumens of the corresponding tube. The support may be attached to another of the two lumens of each of the corresponding tube. Each of the flexible tubes may be configured to receive a strut feature of the support. The basket may be shaped to allow heart valve leaflets to partially close when the basket is deployed in a heart valve. The basket may conform to a shape of the heart valve annulus. The tool may include a guide catheter configured to guide the flexible wires along a common direction of travel. The tool may include a collar attached to the flexible wires and sized to receive the guide catheter. 
     In one aspect, in general, an apparatus includes an annular structure to be attached to a heart valve annulus, the annular structure being expandable and contractible between a contracted pre-attachment configuration and an expanded post attachment configuration, the structure including holding elements that are configured (a) to be held by an attachment tool to enable the attachment tool to cause expansion and contraction of the annular structure in connection with attaching the annular structure to the annulus, and (b) to restrain the annular structure from being expanded, after the annular structure has been attached and the holding elements of the structure are no longer held by the attachment tool. 
     In one aspect, in general, an apparatus includes structural elements connected to form a ring, the structural elements being capable of expanding and contracting to make the ring bigger or smaller, gripping elements attached to the structural elements to penetrate and grip heart tissue, and configuration elements attached to the respective structural elements and each controllable to permit or prevent the ring from expanding. 
     Implementations may include one or more of the following features. The configuration elements may be each sized to be received by a corresponding lumen of a delivery tool. The ring may include polygonal elements. The configuration elements may be capable of a first position flush with the polygonal elements and capable of a second position angled away from the polygonal elements. The polygonal elements may include diamond-shaped elements. The configuration elements may resist contraction of the polygonal elements when the configuration elements are in the position flush with the polygonal elements. The configuration elements may resist horizontal contraction of the polygonal elements. The configuration elements may resist vertical contraction of the polygonal elements. The configuration elements may resist horizontal expansion of the polygonal elements. The configuration elements may resist vertical expansion of the polygonal elements. 
     In one aspect, in general, a method includes positioning a delivery head near to or in contact with a heart valve annulus, causing a heart valve support to expand by moving the heart valve support along a basket of the delivery head, and attaching the expanded heart valve support to the heart valve annulus. 
     Implementations may include one or more of the following features. Aligning the delivery head with a heart valve annulus may include filling the annulus with the delivery head. The method may include contracting the heart valve support after attachment. 
     These and other aspects and features, and combinations of them, may be expressed as apparatus, methods, systems, and in other ways. 
     Other features and advantages will be apparent from the description and the claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIGS. 1A through 1E  show delivery of a heart valve support where a delivery tool is pushed into the valve, the hooks of a support are embedded into valve tissue, and the delivery tool is pulled away. 
         FIGS. 1F through 1H  show contracting a support to its final size and shape and leaving the support permanently in place to maintain the annulus in the desired final configuration and size. 
         FIGS. 2A through 2D  are perspective views of a heart valve support. 
         FIG. 2E  is a plan view of a recurved hook. 
         FIG. 3  is a section side view of a heart valve support showing the support body can be rolled about a central annular axis. 
         FIGS. 4A through 4C  are side and detailed views of a delivery tool and heart valve support. 
         FIG. 5  is a side view of a delivery tool. 
         FIGS. 6A and 6B  are sectional side views of a catheter delivery tool. 
         FIGS. 7A through 7B  show delivery of a heart valve support. 
         FIGS. 8A through 8I  show delivery of a heart valve support where the delivery tool is fed percutaneously through blood vessels and into the right atrium. A sheath is then retracted, exposing the valve support, and allowing the projections, the delivery head, and the support to expand. 
         FIG. 9A  is a plan view of a heart tissue support where the support body is a torus in the form of a helical spring. 
         FIG. 9B  is a perspective view of a fragment of a heart tissue support with burr hooks attached to the outside surface. 
         FIGS. 9C through 9E  are side views of burr hooks. 
         FIG. 9F  is a schematic view of a heart tissue support attached to annular tissue. 
         FIG. 9G  is a side view of a burr hook that has one barbed end. 
         FIG. 9H  is a side view of a portion of a support body surface containing burr hooks that each have two barbed ends facing in a first direction and shorter burr hooks each having one barbed end facing in a second direction. 
         FIGS. 9I through 9M  is a close-up view of portions of heart tissue support surfaces with burrs. 
         FIG. 9N  is a view of a heart tissue support and a delivery tool. 
         FIG. 9O  is a close-up view of a portion of a heart tissue support surface with burrs. 
         FIG. 9P  is a perspective view of a helically formed support. 
         FIG. 9Q  is a view of a heart tissue support and a delivery tool where the support body is placed on the delivery head and the coils of the helical spring stretch outward as the body expands to fit on the tool. 
         FIG. 9R  is a plan view of a heart tissue support having a binding section having burr hooks and a non-binding section having no burr hooks. 
         FIG. 9S  is a perspective view of a fragment of a heart tissue support having radiopaque markers indicating the borders between a binding section having burr hooks and a non-binding section having no burr hooks. 
         FIG. 9T  is a plan view of a heart tissue support that can have multiple sections having no burr hooks. 
         FIG. 9U  is a plan view of a heart tissue support that has an open section. 
         FIGS. 10A and 10B  are side views of a delivery tool, and a cross-section of a sheath. 
         FIGS. 10C and 10D  are cross-sectional views of a delivery tool and sheath. 
         FIG. 11A  is a perspective view of a delivery tool in a heart annulus. 
         FIG. 11B  is a view of the operator end of a delivery tool. 
         FIG. 11C  is a close-up view of a heart tissue support attached to a delivery tool. 
         FIGS. 11D and 11E  are close-up views of a portion of a heart tissue support attached to annular tissue. 
         FIG. 11F  is a close-up view of a heart tissue support and a delivery tool where the delivery head has a blade attached to one of the two rigid fingers that keep the support body in place. 
         FIGS. 12A and 12B  are views of a core of a delivery tool. 
         FIG. 12C  is a perspective view of a core of a delivery tool. 
         FIGS. 13A through 13D  show delivery of a heart valve support and a delivery tool with a collapsed (closed) conical head-end basket. 
         FIGS. 14A through 14D  are perspective views of portions of supports, where the support is constructed from several pieces including an elastic multiple-loop circular coil of strip material. 
         FIG. 15  is a perspective view of an anchor with grippers formed on a length of wire that includes a closed ring. 
         FIG. 16  is a perspective view of a gripper. 
         FIG. 17  is a side view of a gripper. 
         FIG. 18  is a perspective view of a covering wound around the other parts of a support. 
         FIG. 19  is a cutaway perspective view of a support. 
         FIG. 20  is a perspective view of a support. 
         FIG. 21  is an enlarged perspective view of a portion of a support. 
         FIGS. 22 through 25  are top views of a gripper with a pointed end and on each side of the pointed end, a pair of barbs. 
         FIGS. 26 and 27  show the detailed configuration of a Nitinol strip that includes a point and barbs. 
         FIGS. 28, 29, 30, and 31  are a perspective view, a sectional perspective view, a perspective view, and a sectional perspective view, respectively, of a support including anchors in the form of loops. 
         FIG. 32  is a top view of a gripper that allows a reversible process for installing and removing the grippers from the annulus tissue for repositioning. 
         FIGS. 33 through 35  are a top view, a top view, and a perspective view of a support on a hypothetical insertion tool. 
         FIGS. 36 through 39  are side views of an insertion tool that includes a dilator formed of six arms arranged at equal intervals around an insertion axis. 
         FIG. 40  is a side view of an insertion tool where each arm is formed of a stiff limb connected at one end to the outer tube, and at another end to a broader limb. 
         FIG. 41  shows a support mounted on an insertion tool ready for insertion. 
         FIGS. 42 and 43  show a dilator that can include round wire arms that are evenly spaced around the insertion axis and have each been shape set to the expanded configuration. 
         FIG. 44  is a side view of an insertion tool with a central ridge. 
         FIGS. 45 and 46  are perspective and enlarged perspective views of a portion of a support made of crimped stainless steel. 
         FIG. 47  is a perspective view of a support formed of three pieces. 
         FIG. 48  is a perspective view of an anchors where the orientation of the points of the grippers have been rotated to face generally in the insertion direction. 
         FIG. 49  is a perspective view of a coil. 
         FIG. 50  is a perspective view of a resilient ring. 
         FIG. 51  is a perspective view of a ring and coil assembly. 
         FIG. 52  is a perspective view of a support contracted in diameter when the insertion tool is removed from the support. 
         FIG. 53  shows relaxed anchors, driving the grippers to rotate and force the points towards each other, to hold onto the tissue securely. 
         FIGS. 54 and 55  are a perspective and side view of an interlock with embedded mating elements in a resilient ring. 
         FIGS. 56 and 57  are perspective views of an interlock. 
         FIGS. 58 and 59  are perspective views of a support that has a ring of successive hexagonal sections. 
         FIGS. 60A and 60B  are views of a hexagonal section of a support when the support expands and contracts. 
         FIGS. 61A and 61B  are top views of a support when the support expands and contracts. 
         FIG. 62  shows a support that is a complete loop of round cross-section wire wrapped helically and with the helical winding looped in a torus in a configuration of successive windings. 
         FIG. 63  shows a support having a series of helically coiled segments joined by intervening anchoring elements. 
         FIGS. 64A through 64D  show a support having coiled segments joined in a ring formation by connecting elements. 
         FIG. 65  shows a support made of a single continuous coil of flat wire. 
         FIGS. 66A and 66B  show a support having coiled segments made of flat wire joined in a ring formation by connecting elements. 
         FIGS. 67A and 67B  show a relatively flat support having doubled flat sinusoidal segments joined in a ring formation by connecting elements. 
         FIG. 68  shows a support having sinusoidal segments joined in a ring formation by connecting elements. 
         FIGS. 69A and 69B  show a support having crimped segments joined in a ring formation by anchoring elements. 
         FIG. 70  shows a support having segments joined in a ring formation. 
         FIG. 71  shows a support having doubled segments joined at junctions in a ring formation. 
         FIG. 72  shows a support having a metal ribbon coiled into a ring. 
         FIGS. 73A and 73B  show a support having a c-shaped ring. 
         FIG. 74  shows a support having an elastic polymer flat ring. 
         FIGS. 75A through 75D  show a delivery tool having a continuous cone forming the portion of the tool for delivering a support. 
         FIGS. 76A through 76C  show a delivery tool having a cone-shaped wire cage enclosing a balloon. 
         FIGS. 77A and 77B  show a delivery tool that has splaying projections spanning an upper ring and a base ring arranged around a shaft. 
         FIG. 78  shows a support having a ring of successive diamond sections touching at side corners. 
         FIGS. 79A through 79C  show delivery of a heart valve support. 
         FIGS. 80A and 80B  show a heart valve support attached to a delivery head. 
         FIGS. 81A and 81B , show a delivery head basket and a heart valve annulus. 
         FIGS. 82A through 82F  show a heart valve support including diamond sections and anchors extending downward from bottom corners of the diamond sections. 
         FIGS. 83A and 83B  shows a basket structured to fill and conform to the annulus. 
         FIGS. 84A through 84C  show a collapsible delivery head basket. 
         FIGS. 85A and 85B  show a delivery head where each of the wires forming the basket has another bend near the junction that forms (with the other wires) a projection. 
         FIGS. 86A through 86J  show delivery of a heart valve support that uses a guide catheter that can be used to stabilize the delivery tool within the heart valve annulus. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     This application is related to U.S. patent application Ser. No. 12/794,235, filed on Jun. 4, 2010, International application PCT/US2010/027943, filed on Mar. 19, 2010, U.S. patent application Ser. No. 12/563,293, filed on Sep. 21, 2009, U.S. patent application Ser. No. 12/407,656, filed on Mar. 19, 2009, and U.S. patent application Ser. No. 11/620,955, filed on Jan. 8, 2007, all of which are incorporated here in their entirety by reference. 
     As shown in the examples of  FIGS. 1A through 1G  distortion of an annulus  18  of a heart valve  16  can be corrected simply and quickly by the following steps: 
     A. Push  201  ( FIG. 1  A) a conical head-end basket  220  of a delivery tool  200  into the valve to force the distorted annulus ( 203 ,  FIG. 1  F) to conform to a desired configuration (e.g., a circle  205 ,  FIG. 1  G) and to a size that is larger (e.g., in diameter  207 ) than a desired final diameter  209  of the annulus ( FIG. 1  H). (The tool including the basket are shown in side view and the valve and annulus are shown in sectional side view.) 
     B. Continue to push  201  the delivery tool to drive an expanded heart valve support  100  (which has the desired configuration and the larger size and is temporarily held in its expanded configuration on the basket of the tool) towards the annulus to seat multiple (for example, eight, as shown, or a larger or smaller number of) recurved hooks  120  located along the periphery of the support simultaneously into the valve tissue at multiple locations along the periphery  121  of the annulus ( FIG. 1  B). 
     C. After the hooks are seated, pull  204  ( FIG. 1  C) on and evert the tip  230  of the head end basket from the inside to cause the support to roll so that the tips  122  of the hooks rotate  211  and embed themselves more securely into the annulus tissue ( FIG. 1  C). 
     D. After the hooks are further embedded, continue to pull  204  ( FIG. 1  D) on the inside  213  of the tip of the head-end basket to break the tool away from the support ( FIG. 1  E), allowing the support to contract to its final size and shape  215  ( FIG. 1  H) and leaving the support permanently in place to maintain the annulus in the desired final configuration and size. 
     The entire procedure can be performed in less than a minute in many cases. By temporarily forcing the annulus of the valve to expand to the desired circular shape, it is possible to attach the support quickly, easily, and somewhat automatically by forcing multiple gripping elements into the tissue at one time. Hooks are used in this example, although other types of gripping elements may be used as well. The physician avoids the time consuming steps of having to attach individual sutures or clips one at a time along the periphery of a distorted annulus and then cinch them together to reform the supported annulus to a desired shape and size. Thus, the physician does not even need to be able to see the annulus clearly (or at all). Once attached, when the tool is removed, the support automatically springs back to its final shape and size. 
     As shown in  FIGS. 2A and 2D , in some implementations the support includes a circular ring body  110  that bears the hooks  120 . The body  110  can be expanded from (a) a minimal-diameter long-term configuration ( FIG. 2A ) to which it conforms after it has been attached to the annulus to (b) an expanded delivery configuration ( FIG. 2D ) to which it conforms when it is held on the head-end basket of the tool and while it is being attached in the steps shown in  FIGS. 1A, 1B, and 1C . The long-term configuration is normally circular and has the diameter of a healthy annulus for a particular patient. When attached, the support maintains the healthy configuration of the annulus so that the valve will work properly. 
     In some examples, the body  110  has the same (e.g., circular) shape but different diameters in the delivery configuration and the long-term configuration. The body is constructed of a material or in a manner that biases the body to contract to the long-term configuration. For example, all or portions of the body  110  may be formed as a helical spring  110   a  such as a continuous helical spring connected at opposite ends to form a circular body or one or more interconnected helical spring segments ( FIG. 2B ). In some examples, the support body  110   b  may be a band of shape memory material such as Nitinol or a biologically compatible elastomer (or other material) that will return to the long-term configuration after being expanded to the delivery configuration ( FIG. 2C ). 
     The hooks  120  may number as few as three or as many as ten or twenty or more and may be arranged at equal intervals along the body or at unequal intervals as needed to make the body easy and quick to deliver, permanent in its placement, and effective in correcting distortion of the valve annulus. The hooks are configured and together mounted along the circular outer periphery so that they can be inserted simultaneously into the tissue along the periphery of the annulus and then firmly embedded when the tool is pulled away and the basket is everted. 
     In some examples, a portion or portions of the support body may not have hooks attached if, for example, a segment of the valve annulus shares a boundary with sensitive or delicate tissue, such as the atrioventricular (AV) node of the heart. This tissue should not be pierced by the hooks. A support body configured to avoid interfering with the AV node could have a section having no hooks attached or otherwise covered or protected to prevent penetration by hooks into the AV node. The support body should be positioned so that this special section of the support body is adjacent the sensitive or delicate tissue as the support body is put into place. The support body may have more than one special section lacking hooks, so that the operator has more than one option when placing the support body near the sensitive tissue. In some examples, the support body could have a section removed entirely, and would be shaped somewhat like the letter “C” instead of a complete ring. In any of these examples, the procedure described above could have an additional step preceding step A, in which the operator rotates the delivery head to position the section having no hooks or to position the gap in the support body to be adjacent to the sensitive tissue at the moment when the hooks are to be embedded in the other tissue. The support body may have radiopaque marks to help the operator view the positioning. 
     For this reason, as shown in  FIG. 2E , for example, each of the hooks has two pointed features. One pointed feature is a sharp free end  122  pointing away from the valve leaflets during delivery. The other pointed feature is a barb  128  formed at a bend between the sharp free end  122  and an opposite connection end  124  where the hook is attached, e.g., welded or glued, to the body  110 . The barb points toward the valve leaflets during delivery. Thus, the barb is arranged to penetrate the tissue when the tool is pushed toward the valve, and the sharp free end is arranged to embed the hook into the tissue when the tool is pulled away from the valve. 
     Each hook  120  can be formed of biologically compatible materials such as platinum, gold, palladium, rhenium, tantalum, tungsten, molybdenum, nickel, cobalt, stainless steel, Nitinol, and alloys, polymers, or other materials. During delivery the barbs of the hooks are together (and more or less simultaneously) forced into the tissue at a series of locations around the outer periphery of the temporarily expanded annulus. In a later step, the sharp free ends are forced to rotate somewhat away from the leaflets for secure (e.g., permanent) attachment. 
     To cause the hooks to rotate during delivery, the hooks  120  are attached permanently to the support body  110  and the support body can be rolled  123  ( FIG. 3 ) about a central annular axis  112  of the support body, as indicated. One way to cause the rolling of the support body and the associated rotation of the hooks is to enable the body to change its configuration by rotation of the entire body about an axis represented by the central circular axis  123 , much as a rubber o-ring can be rolled about its central circular axis. The reconfiguration of the body to cause the rotation of the hooks can be achieved in other ways. 
     In some examples, applying an axial force (arrows  113 ) to the inner peripheral edge of the ring (we sometimes refer to the support broadly as a ring) will cause the ring to tend to roll and the hooks to embed themselves in the annulus as intended. By appropriately mounting the inner periphery of the ring on the outer periphery of the delivery tool, the axial force  113  can be applied by pulling the tool away from the leaflets of the valve, as explained earlier. 
     For delivery to the valve annulus, the valve support  100  is first expanded to its delivery configuration and temporarily mounted on a delivery head  220  of the tool  200  ( FIG. 4A ). The support could be expanded enough in its temporary mounting on the tool and mounted far enough away from the tip along the conical head-end basket so that when the head-end basket of the tool is pushed against the annulus to force it to expand to the size and shape of the expanded support, the annulus first has reached a circular, non-distorted shape before the support hook barbs begin to penetrate the tissue. The tapered profile of the head-end basket of the delivery tool allows the tool to accommodate supports of various sizes. In some implementations, different shapes and sizes of baskets could be used for supports of different sizes. 
     The heart valve support  100  is held in place on the delivery head  220  using one or more releasable connections  246 . The connections  246  are arranged to translate forces from the tool  200  to the support  100  in each of two opposite directions  248  and  250 , toward or away from the leaflets of the valve. When the support has been embedded in the annulus and the tool is pulled in the direction  250  to release it from the support, the force on the connections  246  exceeds a predetermined threshold, and the connections break, releasing the tool from the support at the end of the delivery process. The connections  246  may be, in some examples, breakable sutures  252  ( FIG. 4A ), or some other breakaway structure such as clips or adhesive or a structure that can be manipulated from the tool by unscrewing or other manipulation. 
     In some examples, the connections  246  include retainers that can take, e.g., the configurations shown as  254   a  or  254   b  ( FIGS. 4B &amp; 4C , respectively). In the example shown in  FIG. 4B , the retaining element  254   a  has one rigid finger  256  to translate forces from the tool  200  to the support  100  when the tool is moved in direction  248  while the support is attached to the tool and being pushed into the heart tissue. A second deformable finger  258  aids in maintaining the connection between the support  100  and the tool  200  when the tool is moved in direction  250  and is deformable (dashed lines) to release the valve support  100  from the tool  200  when the force in direction  250  relative to the embedded support exceeds a predetermined threshold. 
     In the example shown in  FIG. 4C , the retaining element  254   b  includes a finger  260  having a crook  262  to receive the support  100  and to translate forces from the tool  200  to the support  100  when the tool is moved in direction  248 . The finger has a resiliently deformable tip  264  that is biased towards the tapered body  222  and helps to maintain the connection between the support  100  and the tool  200  and is deformable (shown in hidden lines) to release the valve support  100  from the tool  200  when the tool is moved in the second axial direction  250  against an embedded support and the force exceeds a predetermined threshold. 
     As shown in  FIG. 5 , in an example of a tool  200  that can be used for delivery of the support during open heart surgery, a basket  220  is connected at its broad end to a set of stiff wires or other rigid projections  216  that are splayed from a long shaft  210  having a handle  212  at the operator&#39;s end  214 . Thus the projections  216  connect the shaft  210  to the basket  220  and transfer pulling or pushing force between the shaft and the basket (and in turn to the support). 
     The example of the basket shown in  FIG. 5  includes a tapered body  222  having a network of interconnected struts  224  defining an array of openings  226  together forming a tapered semi-rigid net. In this example, the basket (which we also sometimes refer to as a delivery head)  220  has a rounded tip  228 . The head  222  tapers radially outwardly with distance along a longitudinal axis  234  of the head  220  from the tip  228  towards the operator. The broad end  232  of the tapered body  222  is firmly attached to the projections  216 , which taper in the opposite direction from the taper of the basket. The net formed by the struts  224  is semi-rigid in the sense of having enough stiffness to permit the operator to force the valve support against the heart tissue to cause the barbs of the hooks of the support to penetrate the tissue, and enough flexibility to permit the head-end basket to be everted when the operator pulls on the handle to evert the basket and release the support from the basket. 
     In some implementations, the shaft  210  defines a lumen  236  extending between the heart valve end  218  of the shaft  210  and the handle  212 . A wire  238  is arranged to move freely back and forth within the lumen  236 . The wire  238  has one end  240  that extends from the handle  212  and an opposite end  242  that is connected to the inside of tip  228 . The wire  238  can be pulled (arrow  244 ) to cause the delivery head  220  to collapse (hidden lines) and evert radially inwardly starting at the tip  228  as mentioned earlier. 
     Returning to a more detailed discussion of  FIGS. 1  A through  1 E, the operator begins the delivery of the support by pushing the tapered end  230  of the head basket  220  into the valve  16  (e.g., the tricuspid valve) to cause the valve leaflets  14  to spread apart. The tip  230  is small and rounded which makes it relatively easy to insert into the valve without requiring very precise guidance. Because the head-end basket is tapered, by continuing to push, the operator can cause the annulus  18  of the tricuspid valve  16  to expand in size and to conform to a desired shape, typically circular. During insertion, because of its symmetrical taper, the head-end basket tends to be self-centering. The taper of the basket  220  translates the insertion force in direction  248  into a radial force that causes the annulus  18  to expand and temporarily assume a desired shape (and a larger than final diameter). 
     As the operator continues to push on the tool, the ring of barbs of the hooks touch and then enter (pierce) the heart tissue along a ring of insertion locations defined by the outer periphery of the annulus, and the sharp free ends of the hooks enter and seat themselves within the tissue, much like fish hooks. Depending on how the operator guides the tool, the basket can be oriented during insertion so that essentially all of the hooks enter the tissue at the same time. Or the tool could be tilted during insertion so that hooks on one side of the support enter the tissue first and then the tool delivery angle could be shifted to force other hooks into the tissue in sequence. 
     Generally, when the number of hooks is relatively small (say between 6 and 20, comparable to the number of sutures that the physician would use in conventional stitching of a ring onto an annulus), it is desirable to assure that all of the hooks penetrate the tissue and are seated properly. 
     Once the hooks are embedded in the tissue, the operator pulls on the near end  240  of wire  238  to cause the basket  220  to collapse, evert, and be drawn out of the valve  16 . Eventually, the everted portion of the basket reaches the valve support  100 . By further tugging, the operator causes the body  110  of the support  100  to roll about its central axis (as in the o-ring example mentioned earlier) which causes the hooks  120  to embed more firmly in the tissue of the annulus  18  of the valve  16 . 
     Using a final tug, the operator breaks the connections between the tool  200  and the valve support  100  and removes the tool  200 , leaving the valve support  100  in place. As the everting basket  220  passes the points of connection  246 , the retaining forces exerted by the embedded hooks  120  of the support body  110 , acting in direction  248 , exceed the forces exerted by the withdrawing basket  220  on the support body  110  (through the connections  246 ), acting in direction  250 , thereby causing the connections  246  to break or release, in turn releasing the support  100 . 
     The tool  200  is then withdrawn, allowing the valve support  100 , along with the annulus  18 , to contract to the long-run configuration. 
     In implementations useful for delivery of the support percutaneously, as shown in  FIG. 6A , the delivery head  220   a  can be made, for example, from a shape memory alloy, such as Nitinol, which will allow the body  222   a  to be collapsed radially toward the longitudinal axis  234   a  prior to and during delivery of the head from a percutaneous entry point (say the femoral vein) into the heart. The delivery head  220   a  is biased towards the expanded, tapered configuration shown in  FIG. 6A . Thus, the delivery head  220   a , in the form of a tapered semi-rigid net, is connected to a catheter shaft  210   a  through projections  216   a  that splay radially outwardly from the catheter shaft  210   a  and taper in a direction opposite the taper of the delivery head  220   a . (Here we refer to the delivery head as the head-end basket.) 
     The projections  216   a  are resiliently mounted to the catheter shaft  210   a  and are biased towards the expanded, tapered orientation shown, for example, by spring biased projections  216   b  shown in  FIG. 6B . The projections  216   a  include springs  278 , e.g., torsion springs (as shown), mounted to the catheter shaft  210   a  and forming a resilient connection. 
     A wire  238   a  slides within a lumen  236   a  of the shaft  210   a  in a manner similar to the one described earlier. 
     The tool  200   a  also includes a sheath  280  in which the catheter shaft  210   a  can slide during placement of the support. The sheath  280 , the catheter shaft  210   a , and the wire  238   a  are all flexible along their lengths to allow the tool  200   a  to be deflected and articulated along a blood vessel to reach the heart and to permit manipulation of the delivery head once inside the heart. 
     To deliver the support percutaneously, as shown in  FIG. 7A , when the delivery head is prepared for use, the sheath  280  is retracted beyond the projections  216   a , allowing the delivery head  220   a  to expand. The valve support  100  is then expanded to the delivery configuration (either by hand or using an expansion tool) and mounted on the tapered body  222   a . The valve support  100  is connected to the delivery head  220   a  using releasable connections, e.g., breakable sutures and/or retaining elements (as described earlier). 
     The sheath  280  is then moved along the catheter shaft  210   a  towards the delivery head  220 , causing the projections  216   a  and the delivery head  220   a  to contract radially inwardly to fit within the sheath  280 , as shown in  FIG. 7B . In the contracted configuration, the tip  228   a  of the delivery head  220   a  bears against the end  282  of the sheath  280 . The rounded tip  228   a  may, e.g., provide easier delivery and maneuverability in navigating the blood vessels to reach the heart. 
     To deliver the support to the valve annulus, the end  230  of the tool  200   a  is fed percutaneously through blood vessels and into the right atrium  24  ( FIG. 8A ). The sheath  280  is then retracted, exposing the valve support  100  and allowing the projections  216   a , the delivery head  220   a , and the support  100  to expand, as shown in  FIG. 8A . 
     In steps that are somewhat similar to the open heart placement of the support, the catheter shaft  210   a  is then advanced, e.g., under image guidance, in the direction  248   a  along an axis  30  of the annulus  18 . The operator forces the distal end  230   a  of the self-centering delivery head  220   a  into the valve  16  ( FIG. 8B ) using feel or image guidance, without actually seeing the valve  16 . 
     Once the tip is in the valve  16 , the operator pushes on the end  214   a  of the catheter shaft  210   a  to force the tool further into the valve  16 . This causes the tapered body  222   a  of the delivery head  220   a  to restore the shape of the annulus  18  to a circle or other desired shape (such as the distinctive “D” shape of a healthy mitral valve). The tool  200   a  tends to be self-centering because of its shape. The net-like construction of the delivery head  220   a  (and the head used in open heart surgery, also) allows blood to flow through the valve even while the delivery head  220   a  is inserted. 
     As tool  200   a  reaches the position at which the support hooks touch the annulus, by giving an additional push, the operator drives the hooks  120  of the valve support  100  together into all of the annular locations at which it is to be attached, as shown in  FIG. 8C . In some examples, it may be possible for the operator to tilt the delivery head deliberately to cause some of the hooks to penetrate the tissue before other hooks. The configuration of the valve support  100  and the tool  200   a  and the manner of temporary attachment of the support  100  to the tool  200   a  tend to assure that the hooks  120  will penetrate the valve  16  at the correct positions, just along the outer edge of the annulus  18 . 
     Once the valve support  100  has been attached to the valve  16 , the operator pulls on the proximal end  240   a  causing the delivery head  220   a  to evert (hidden dashed lines) and be drawn out of the valve  16  (shown in  FIG. 8D ). Eventually the everted portion of the tool  200   a  reaches the valve support  100 . By further tugging, the operator causes the torus of the support  100  to roll around its periphery which jams the free ends of the hooks  120  securely into the annulus  18  of the valve  16 , as illustrated in  FIG. 8E , seating the support permanently and permitting later growth of tissue around the support  100 . The depth and radial extent of each of the placed hooks  120  can be essentially the same as a conventional suture so that their placement is likely to be as effective and familiar to the operator and others as conventional sutures. 
     Using a final tug, the operator breaks the connections  246  between the tool  200   a  and the valve support  100  and retracts the catheter shaft  210 , leaving the support  100  in place. The catheter shaft  210  is retracted to a position beyond the valve annulus  18  and the wire is advanced in the first direction allowing the delivery head  220   a  to assume its original tapered shape ( FIG. 8F ). The catheter shaft  210   a  is then retracted into the sheath  280  ( FIG. 8G ), and the tool  200   a  is withdrawn. 
     In some examples, as shown in  FIGS. 8H and 8I , the tip  228   a  of the tool  200   a , when everted, has a compressed dimension that is smaller than an internal diameter  284  of the sheath  280 , permitting the catheter shaft  210   a  to be retracted directly into the sheath  280  after deployment, with the everted tip held within the collapsed delivery basket, as shown in  FIG. 8I . 
     With the tool  200   a  withdrawn, the valve support  100  contracts, reshaping the annulus  18  such that the valve leaflets  14  coapt to prevent a backflow of blood during systole. 
     Other implementations are within the scope of the claims. 
     For example, distortion of either the tricuspid valve or mitral valve can be corrected. For tricuspid valve repair, the hooks can be arranged around only about three-quarters of the support and therefore the annulus. During the placement procedure, the operator will rotate the support to position the portion of the support having hooks. For mitral valve repair, the hooks can cover the entire periphery of the annulus. In this scenario, the hooks are arranged around the full circumference of the support. Alternatively, the hooks can cover only the posterior section of the annulus of the mitral valve. In this scenario, the hooks can be arranged around two-thirds of the support. Similarly to the tricuspid valve example, the operator will position the portion of the support having hooks against the posterior section of the mitral valve annulus. Further, for mitral valve repair, a back-up valve can be provided as part of the delivery tool to maintain heart function during the delivery procedure. Materials other than shape memory materials may be used as the material for the support body, and other ways can be used to force the support back to a desired size following expansion, including, for example, cross-bars that span the opening of the support. 
     In addition, the left atrial appendage of the heart can be closed by a similar technique. For example, the tool can be pushed into an opening of an atrial appendage causing the opening to assume a predetermined shape. The tool can continue to be pushed in order to embed the hooks of the expanded support into the periphery of the opening of the appendage. The tool can then be withdrawn, releasing the support, and allowing the support to contract. The support can have a relatively small contracted diameter such that, when the tool is withdrawn, releasing the support, the support can contract to a relatively small size, effectively closing off the appendage. 
     In addition to the open heart and percutaneous deployment procedures, the valve support can also be deployed through the chest. 
     The head-end of the tool need not be a basket, but can take any form, mechanical arrangement, and strength that enables the valve annulus to be forced open to a shape that corresponds to the shape of the support. The basket can be made of a wide variety of materials. The basket can be held and pushed using a wide variety of structural mechanisms that permit both pushing and pulling on the support both to seat and embed the support in the annulus tissue and disconnect the support from the tool. 
     The tool need not be conical. 
     The support could take a wide variety of configurations, sizes, and shapes, and be made of a wide variety of materials. 
     The hooks could be replaced by other devices to seat and embed the support using the pushing force of the tool. 
     The hooks of the support need not be embedded directly in the annulus but might be embedded in adjacent tissue, for example. 
     The support could take other forms and be attached in other ways. 
     In  FIG. 9A , the support body  110   a  can be a torus in the form of a helical spring (as mentioned earlier). Such a support body can have a native circumference  116  on the order of ten centimeters in its contracted state, and a proportional native diameter  114 . The circumference can be selected based on the physical requirements of a particular patient. 
     A close-up view of a fragment of this support body,  FIG. 9B , shows that some implementations have a number (e.g., a large or very large number, for example, as few as say 15, or 100, and up to hundreds or even thousands) of burr hooks  120   a  attached to an outer surface  111  of the support body  110   a . In the example shown in  FIG. 9B , the helical support body is wound from a flat strip that has the outer surface  111  and an inner surface  117 . Although  FIG. 9B  shows the burr hooks attached only to the outside surface, burr hooks could also be attached to the inner surface for manufacturing reasons or for other purposes. 
     The burr hooks, which are small relative to the body, are each configured to partially or fully pierce annular tissue when the part of the body to which the burr hook is attached is pushed against the tissue. 
     As shown in  FIG. 9C , in some examples, each burr hook  120   a  has a sharp free end  122   a  for piercing tissue and at least one barbed end  128   a ,  128   b  (two are shown in  FIG. 9C ) for keeping the burr hooks embedded in tissue. Each burr hook also has an end  124   a  that is attached to the surface of the support body. Once the support (we sometimes refer to the support structure simply as the support) is in contact with heart tissue, the embedded burr hooks hold the body in a proper position and configuration on the annulus. Burr hooks can be attached to the surface of the support body using glue, cement, or another type of adhesive, or formed from the support body as part of an industrial process, such as molding, etching, die cutting, welding, or another process, or can be attached by a combination of these techniques. Different burr hooks on a given support can be attached by different mechanisms. 
     Each burr hook  120   a  can be structured and attached so that the free end  122   a  points in a direction  122   b  perpendicular (or some other selected effective direction, or deliberately in random directions) to the body surface  111 . In some cases, the burr hook can be curved. A barbed end  128   a  could be located on a concave edge  113  ( FIG. 9D ) or a convex edge  115  ( FIG. 9E ) of a curved burr hook. 
     The burr hooks bear a resemblance to burr hooks on natural plant burrs. A different kind of attachment device could be used by analogy to metal tipped hunting arrows in which a sharp point has two broad and sharp shoulders that cut the tissue as the point enters. The tips of the two shoulders serve a similar function to the barbs, keeping the arrow embedded once it enters the tissue. 
     In some implementations, the burr hooks on a support body have two or more (in some cases, many) different shapes, sizes, orientations, materials, and configurations. By varying these features, for example, the orientations of the burr hooks, it may be more likely that at least some of the burr hooks will become embedded in the tissue, no matter how the support body is oriented at the moment that it comes into contact with the annulus. Varying the number, orientation, and curvature of the hooks may make it more likely that the support body will remain in place. For example, in such a support, a force applied to the support body in a particular direction may unseat or partially unseat some of the burr hooks by disengaging the barbed ends from the tissue, but the same force may not affect other burr hooks that have barbed ends oriented in a different direction or in a different configuration than the unseated burr hooks. The force applied to seat the support may cause some burr hooks to embed more securely than other burr hooks. 
     In use, typically not all of (in some cases not even a large portion of) the burr hooks will embed themselves in the tissue when the support body is pushed against the tissue, or remain embedded after placement. As shown in  FIG. 9F , there are enough burr hooks arranged in an appropriate way so only a fraction of the total hooks need be embedded in annular tissue (and in some cases only in certain regions) to create a physical bond to keep the support body properly in place. The proportion of burr hooks on a support that need to embed securely in the tissue could range from 1% to 10% or 40% or more. The averaging spacing of the successfully embedded burr hooks could range from, say, one burr hook per millimeter of support body length to one burr hook per two or three or more millimeters (or more) to secure the support appropriately. When burr hooks are grouped rather than arranged evenly on the support, the percentages of and distances between successfully embedded hooks may differ. 
     When the burr hooks come into contact with the annular tissue during delivery, some  131 ,  133 , but not necessarily all, of the burr hooks pierce the tissue and (when a retracting force is applied to the delivery tool) their barbs grip the tissue. Of the remaining burr hooks, some  135 ,  137  may (because of the contours of the tissue, for example) not even come into contact with the tissue, and others  139 ,  141  may not come into contact with the tissue with sufficient force or in the right orientation to pierce the tissue and have their barbs seat securely in the tissue. Some of the burr hooks  143 ,  145  may penetrate the tissue but fail to grip the tissue. Some of the burr hooks  147 ,  149  may only penetrate the tissue at the barbed end  128   a , and not with respect to the free end  122   a , providing a physical bond that may be weaker than one in which the free end has been embedded in the tissue. For some or many or most of the burr hooks that enter the tissue, however, the barbed ends  128   a  seat properly and resist forces in the direction  151  that would otherwise unseat the burr hook. Even though a wrenching force applied to a particular burr hook in direction  151  could still be large enough to unseat the barbed end, overall the combination of many burr hooks embedded in tissue tends to keep the support body set in place and in the proper configuration. Over time, some of the burr hooks that were not embedded when the support was placed may become embedded, and some of the burr hooks that were embedded when the support was placed may become unseated. 
     The resistance provided by each of the barb or barbs to removal of a given burr hook from the tissue may be relatively small. However, the aggregate resistance of the burr hooks that successfully embed themselves will be higher and therefore can reliably keep the support body in place and the annulus of the valve in a desirable shape. In addition, because there are a number (potentially a very large number) of small burr hooks spread over a relatively large area, the stress on any part of the tissue of the annulus is quite small, which helps to keep the support body properly seated and the valve shape properly maintained along its entire periphery, all without damaging the tissue. The fact that a large number of burr hooks at close spacings may become embedded along the length of the support means that the support may become attached to the annulus more evenly and continuously than might be the case with the relatively smaller number of hooks described earlier, and therefore perform better. 
     With respect to the implementations described beginning with  FIG. 1  A, the implementations shown beginning at  FIG. 9A  tend to have more and smaller hooks not all of which need to become embedded successfully. A common concept between the two arrangements is that the hooks penetrate by being pushed into the tissue and have retaining elements that become securely embedded in the tissue when a pulling force is applied at the end of the placement process. The two concepts are not mutually exclusive. Supports like those shown in  FIG. 1A  could also have burr hooks and supports like those shown in  FIG. 9A  could also have hooks of the kind shown in  FIG. 1  A. Placement of the support could rely on a combination of both kinds of hooks. 
     Each burr hook can be formed of a biologically compatible material such as platinum, gold, palladium, rhenium, tantalum, tungsten, molybdenum, nickel, cobalt, stainless steel, Nitinol, and alloys, polymers, or another material. As for the hooks shown beginning with  FIG. 1  A, the hooks can also be formed of a combination of such materials. An individual support body may exhibit burr hooks having a range of compositions. Some of the burr hooks attached to a support body may be composed of one material or combination of materials, and some of the burr hooks may be composed another material or combination of materials. Each burr hook may be unique in composition. Further, some parts of a burr hook may be composed of one set of materials, and other parts may be composed of another set of materials. In some examples, the region of the burr hook at the barbed end is composed of one set of materials, alloys, polymers, or mixtures, and the region of the burr hook at the free end is composed of another set of materials, alloys, polymers, or mixtures, and the rest of the burr hook is composed of a further set of materials, alloys, polymers, or mixtures.  FIG. 9G  shows an example burr hook that only has one barbed end  128   a . The burr hook extends from an attached end  124   a  to a free end  122   a  along the path of a principal axis  920  that (in this case) is perpendicular to the support body surface  111 . The barbed end spans a length  904  from the burr hook&#39;s free end  122   a  to the barbed end&#39;s free end  906 . This free end  906  forms a point spanning an acute angle  910  and the barbed end  128   a  spans an acute angle  911  to grab the tissue in response to any force that would otherwise pull an embedded burr hook away from tissue. 
     The length  901  of each burr hook could be between about 1 and 12 millimeters, as measured from the attached end  124   a  to the free end  122   a  along the principal axis. Each barbed end could extend a distance  902  from the burr hook lesser or greater than a principal width or diameter  903  of the burr hook as measured at the attached end. The cross-section of the body of the burr hook could be flat or cylindrical or ovoid or any other of a wide variety of shapes. 
     Different burr hooks may be placed on the support body surface in different sizes and configurations. For example, different burr hooks may have different lengths and different numbers and placement of barbed ends. As shown in  FIG. 9H , for example, a portion of support body surface  111  contains burr hooks  120   a  that each have two barbed ends  128   a ,  128   b  facing in a first direction  950  and shorter burr hooks  120   b  each having one barbed end  128   a  facing in a second direction  951 . Also, the burr hooks may be arranged on the body surface in various densities and patterns of distribution. For example, as shown in  FIG. 9I , the burr hooks may be placed on the surface of the body in repeating rows  930 . As shown in  FIG. 9J , the burr hooks may be placed on the surface in rows of different lengths and densities  931 ,  932 . As shown in  FIG. 9K , the burr hooks may be placed on the surface along are formations  933 . As shown in  FIG. 9L , the burr hooks may be placed on the surface as cluster formations  934 . As shown in  FIG. 9M , the burr hooks may be distributed randomly  935 . Other patterns may also be used. 
     A single support body can include a wide variety of patterns of burr hooks on its surface, because the physical characteristics of a particular heart valve may mean that the valve tissue is either more receptive or less receptive to a particular pattern of burr hook distribution. Some patterns may be more effective on some types of tissue, and other patterns may be more effective on other types of tissue. 
     In addition, as shown in  FIG. 9N , the burr hooks need not be present at the points where the body  110   a  contacts the delivery tool  220 , including in the area near the rigid fingers  256 ,  258 . This tends to prevent the burr hooks from causing the support body to stick to the tool. 
     As shown in  FIG. 9O , any two burr hooks may be placed at a distance  905  from each other greater than or less than the length  901 ,  901   a  of either one. 
     As shown in  FIG. 9P , when a support is formed helically, the ring can be considered to have a front side  961  (which faces the valve when the support is delivered), and a back side  960  that faces away from the valve. In some examples, the support body  110   a  does not have burr hooks  120   a  on the back side  960 . In these implementations of the support body, the back side  960  is covered by a sleeve  963 . After the support body has been attached to the annulus, the sleeve assists in the long-term process of integration with valve tissue. Over a period of time, heart tissue will attach to the support body as part of the process of healing. The sleeve is made of a material that allows this process to occur faster than without the sleeve. For example, the sleeve may be composed of a porous material, which allows tissue to grow into the sleeve, thus securing the support to the tissue more effectively than without the sleeve. The sleeve material may be a thermoplastic polymer such as Dacron (polyethylene terephthalate). The sleeve material may alternatively be a metal or another type of material. The sleeve can be placed on the support body at a location other than the back side. For example, the sleeve could be placed on the inner side  965  of the body, with burr hooks remaining on the outer side  964 . 
     The sleeve is formed as a half-torus in this example, but could have a wide variety of other configurations. Such a sleeve may be used with any kind of support, including the one shown beginning in  FIG. 1  A, could cover all or only part of the support, and could cover portions of the support that include hooks or barb hooks or both. In the latter case, the hook may be arranged to penetrate the sleeve during setup and before the support is placed into the heart. The sleeve could also cover a portion of the support meant to contact delicate or sensitive tissue, such as the AV node. In this case, the sleeve is made of a material that is less likely to damage or interfere with the operation of the delicate or sensitive tissue, as compared to other materials that may be used in the support. 
     Using burr hooks may make attaching the support faster, simpler, more reliable, and easier than for the larger hooks described earlier. The delivery tool operator may not need to apply as much force as might be necessary to embed larger hooks in the annular tissue. In some cases, the barbs would not need to be rotated as described for the larger hooks in order to embed them securely. The operator need not be concerned whether all of the burr hooks have become embedded. Once the operator has determined that the support body has made contact with the tissue and by inference that many of the burr hooks have become attached, the operator can tug on the support to confirm that it has been seated and then release the support body from the delivery tool using one of the mechanisms described earlier. Because of the ease of positioning, the procedure could be performed easily in a non-surgical context, such as in a catheterization laboratory. 
     As shown in  FIGS. 13A-13D , in the catheterization context, for a burr-hook support or any other kind of support being placed, the catheter may include a balloon  228   b  at the tip of the delivery tool. The balloon remains deflated as the catheter is passed through the patient&#39;s blood vessels into the heart, as in  FIG. 13A . When the tip of the catheter reaches the heart, the balloon can be inflated, shown in  FIG. 13B . The inflated balloon floats in the blood being pumped through the heart and (along with the delivery tool) is carried easily and to some extent automatically toward and into the valve that is to be repaired. The balloon can continue to move beyond the valve annulus, and, when located as shown in  FIG. 13C , supports the distal end of the catheter while the operator supports the proximal end of the catheter. The shaft of the catheter then serves as a “rail” supported at both ends and along which operations involving the delivery tool and the support can be performed with confidence that the rail is being held generally on axis with the valve. 
     In some of the examples described earlier, the annulus of the heart valve is expanded to the desired shape by pushing a conical surface, such as the basket, along the axis of and into the heart valve. Whether the delivery is done in the context of open heart surgery or in a catheterization lab, or elsewhere, the pushing of the conical surface into the annulus can be supplemented by or replaced by a technique in which the expansion of the annulus is done after the delivery tool is inserted into the valve. 
       FIG. 9A  shows one diameter of the support body, the native (long-term configuration) diameter  114 . Recall that this diameter is different from the diameter in the delivery configuration. The former diameter  114  is, as shown in  FIG. 9Q , smaller than the latter diameter  202  of the delivery tool at the point of support body attachment  247 . When the support body is placed on the delivery head  220 , the coils of the helical spring stretch outward as the body expands to fit on the tool. 
     During delivery, shown in  FIGS. 13A-13D , when the support body has been attached to the annulus  18 , the operator releases the support from the delivery tool.  FIG. 13D  shows that, in the absence of the outward force previously applied by the delivery tool, the coils of the helical spring contract inwardly  1308  so that the support body returns to a final diameter  1309  of approximately its native diameter. Referring again to  FIG. 1H , recall that because the annulus is attached to the support body, the support body will also pull the annulus inward, reforming the annulus to a desired smaller diameter  209 . 
     If the support body is made of a material or alloy that is appropriately plastic, the support body may not fully contract to its original native diameter. However, if the support body is made of a shape memory alloy such as Nitinol, the memory effect of the alloy will tend to cause the support body to contract to a diameter nearly identical or identical to its original diameter. 
     As shown in  FIG. 9R , the support body  110   a  may have other portions bearing no burr hooks. As mentioned earlier, sensitive or delicate tissue such as the AV node should not be punctured or bound to hooks. In some examples, the support body  110   a  can have a binding section  972  having burr hooks and a non-binding section  974  having no burr hooks. A non-binding section  974  of sufficient length to abut the AV node spans an angle  975  between about 40 and 60 degrees of the support body circumference. The binding section  972  will span an angle  973  of the remaining circumference. In some examples, a non-binding section  974  is covered in a sleeve made of a material suited to contact the AV node or other sensitive tissue. 
     As shown in  FIG. 9S , the two sections  972 ,  974  can have radiopaque markers  976 ,  977  indicating the borders between the two sections. The markers  976 ,  977  are each in the shape of an arrow pointing to the non-binding section. During delivery, an operator can use the radiopaque markers  976 ,  977  to view the boundary of the non-binding section  974  and position the non-binding section  974  against the AV node or other sensitive tissue. 
     As shown in  FIG. 9T , the support body  110   a  can have multiple sections  974 ,  978  having no burr hooks. In some situations, the operator may be limited in the degree to which the delivery head can be rotated. In this example, the operator has multiple options for positioning the support body in order to avoid puncturing the AV node, and the operator would not have to rotate the delivery head more than about 90 degrees in any direction. Two non-binding sections are shown, but the support body can also have three or more of these sections. The non-binding sections  974 ,  978  span angles  975 ,  979  between about 40 and 60 degrees of the total circumference. In the example of two non-binding sections, there will also be two binding sections  980 ,  982  spanning angles  981 ,  983  of the remaining two lengths of circumference. 
     As shown in  FIG. 9U , the feature of the support body  110   a  that should abut the AV node can take the form of an open section  990 . As with the non-binding section described above, the open section  990  may span an angle  995  between about 40 and 60 degrees of the circle defined by the support body  110   a , while the support body spans the remaining angle  993 . The open section  990  can also have radiopaque markers on the open ends  992 ,  994  of the support body  110   a  to assist an operator in positioning the open section  990  against the AV node or other sensitive tissue. 
     As shown in  FIGS. 10A-10D , the delivery head  220  can include a sheath  280   a  for covering the support body during insertion.  FIGS. 10A and 10B  show the sheath in a side section, and  FIGS. 10C-10D  show the sheath as well as the delivery head in a cross-section at A-A in  FIG. 10B . The sheath  280   a  wraps around the delivery head  220 , including the support body  110   a , so that the burr hooks do not accidentally puncture or attach to any other tissue or devices prior to reaching the annulus. The sheath is made of a flexible material, such as rubber, silicone rubber, latex, or another biologically compatible material or combination of materials. The sheath can also be made of the same material or materials as the catheter. Recall that one implementation of the sheath is shown in  FIGS. 6A-6B  and described in the corresponding text. Other implementations of the sheath are possible. 
     For example, the implementation of the sheath  280   a  shown in side section in  FIG. 10A  is kept in place by attachment to an elastic retainer ring  1000  and a crossbar  1010  permanently affixed through and extending outward from the catheter shaft  210  perpendicular to the longitudinal axis  234 . The retainer ring  1000  is positioned closer to the operator and farther from the distal end than is the support body  110   a , and the crossbar  1010  is positioned farther from the operator and closer to the distal end than is the support body. This sheath  280   a  is permanently attached  1002  to the retainer ring  1000 . The sheath  280   a  is also attached to the crossbar temporarily at holes  1030 ,  1032  (visible in  FIG. 10B ) sized to fit the projecting tips  1020 ,  1022  of the crossbar  1010 . 
     As shown in  FIGS. 10B-10D , after insertion of the catheter into the valve and when the delivery head  220  is expanded in preparation for attaching the support body  110   a , the combination of the retainer ring and crossbar allows the sheath to automatically detach from the crossbar and retract upward away from the support body as part of the expansion procedure. The process by which this happens is as follows. 
     Referring to  FIG. 10B , when the delivery head expands outward  1006 , the diameter  1008  of the delivery head at the original point of retainer ring attachment  1012  increases to a diameter greater than the diameter  1009  of the retainer ring  1000 . As a result, the retainer ring rolls upward  1004  from a point  1012  to a point  1005  on the delivery head of smaller diameter. As the retainer ring rolls, it pulls the distal end of the sheath in the same upward direction  1004  along the delivery head  220  and away from the support body  110   a . Part of the sheath  280   a  wraps around the ring as part of the rolling process; in a sense, the retainer ring is “rolling up” the sheath, in the fashion of a scroll wrapping around a roller. The retainer ring  1000  is rubber or another biologically-compatible material with sufficient elasticity to allow the ring to roll up the expanding delivery head. 
     When the delivery head  220  expands, the sheath  280   a  is also released from the crossbar. A cross-section of the delivery head  220  including the crossbar  1010  is shown in  FIG. 10C . When the delivery tool is in transit to a heart valve, the delivery head  220  is in the collapsed configuration. The sheath  280   a  has holes  1030 ,  1032  configured to allow the crossbar  1010  to pass through, holding the distal end of the sheath to the crossbar. Because the crossbar projects beyond the sheath, the ends  1020 ,  1022  of the crossbar are rounded and smooth to prevent the crossbar from piercing or tearing any tissue that it contacts before the delivery head reaches its destination. Once the delivery head is positioned near or inside a heart valve and begins expanding outward  1006  from the shaft  210 , the delivery head pushes the sheath  280   a  outward. 
     During the expansion process, as shown in  FIG. 10D , the crossbar remains in place and does not extend outward or change configuration, because the crossbar is permanently and securely attached to the shaft  210 . As a result, the delivery head pushes the sheath beyond the tips  1020 ,  1022  of the crossbar, releasing the sheath from the crossbar. Thus, the sheath can move freely when the retainer ring rolls upward along the delivery head, as described above. The crossbar  1010  may be made of any of the materials used in the delivery tool, or another biologically-compatible material, provided that the crossbar is sufficiently rigid to keep the sheath  280   a  in place, as described. 
       FIG. 11A  shows another version of the delivery head  220   b . This version differs slightly from the versions of the delivery head already shown. Specifically, in this version  220   b , the rigid projections  216   b  are composed of an outer sleeve  1140  that encloses an inner arm  1142  attached to the shaft  210   b  by a hinge  1144 . When this version of the delivery head expands, the sleeve  1140  extends from the inner portion  1142 , and when the delivery head contracts, the sleeve withdraws along the length of the inner arm. This version of the delivery head is used in  FIG. 11A  to demonstrate the use of a tightening wire  1100 , but this tightening wire can be used with other versions of the delivery head as well. 
     As shown in  FIG. 11  B, this tightening wire  1100  is threaded into and back out of a hole  1103  at the operator end  214   b  of the delivery tool  200   b . In doing so, the wire traverses the interior of the shaft  210   b  of the delivery tool  200   b . The ends of the wire exterior to the operator end  214   b  form a loop  1102  to be manipulated by an operator. This wire  1100  can be used to activate a mechanism to adjust the shape of the support body  110   a  to a small degree, with the goal of contracting the final diameter  1309 , an example of which is shown in  FIG. 13B . Referring back to  FIG. 11A , at the other end of the delivery tool  200   b , the wire exits the shaft  210   b  at a hole  1105  placed at a point above the delivery head  220   b . The wire extends down the side of the delivery head  220   b , guided by hoops  1120 ,  1122 . As shown in  FIG. 11C , the wire is threaded along the interior of the helical coil  1150 ,  1152  of the support. At the position  1164  where the wire has completed a circumference of the support body  110   a , the wire returns up the side of the delivery head and back into the shaft. 
       FIG. 11C  also shows hoops  1124 ,  1126  that are placed on the struts  224   b  of the delivery head at regular intervals to keep the wire properly positioned. At the position  1164  where the wire meets itself and returns up the side of the delivery head, spools  1130 ,  1132 ,  1134 ,  1136  attached to the strut  224   b  guide the wire and prevent the wire from scraping against  1160 ,  1162  the helical loops  1150 ,  1152  at the wire exit region. The end of the wire that re-enters the hole  1105  ( FIG. 11A ) continues back up the shaft alongside itself, and exits the delivery tool ( FIG. 11B ) to form the loop  1102  by connecting with the other end. 
     When the support body  110   a  is firmly seated at the heart valve annulus  18  (for example, in the scenario shown in  FIG. 13C ), an operator can pull  1104  the loop  1102  ( FIG. 11B ) to reduce the final diameter of the support. When pulled, the wire tightens; as shown in  FIG. 11C , this brings  1106  the coils  1150 ,  1152  of the support closer together. 
     The adjusted circumference becomes permanent as the burr hooks of the support embed themselves in the annular tissue. Although some burr hooks will already have been embedded, the tightening procedure will pull out some of those burr hooks and embed other burr hooks in the tissue. This “bunches” annular tissue closer together.  FIG. 11D  shows an example of a portion of the support body  110   a  attached to the periphery  121  of an annulus before the support body is tightened. As shown in  FIG. 11E , after tightening, the support body  110   a  pulls the tissue at the periphery  121  closer together. The final diameter of the annulus will be slightly smaller due to this bunching effect. Once the delivery head is removed, the support body, and thus the attached annulus, will contract to the desired size. 
     Referring to  FIG. 11F , to detach the wire from the support body  110   a , the delivery head  220   b  has a blade  1170  attached to one of the two rigid fingers  256   b ,  258   b  that keep the support body in place. When the rigid finger  256   b  pulls away from the support body  110   a  after the support body is in place, the cutting segment  1172  of the blade structure severs the wire. The operator may pull the external loop after the wire has been severed to keep the stray ends of the wire from moving freely outside of the delivery tool when the tool is being removed from the annulus. 
     As shown in  FIGS. 12A through 12C , a delivery tool  200   b  for use in (but not only in) a catheterization context shares elements in common with the delivery tools discussed earlier, including the shaft  210   b , collapsible conical head end basket  220   b , set of struts  224   b , and operator end  214   b . This delivery tool  200   b  allows the operator to expand or contract the collapsible conical head-end basket  220   b  radially from a collapsed (closed) configuration (shown in  FIG. 12A ) to an expanded (open) configuration (shown in  FIG. 12B ), much in the way that an umbrella can be opened. For this purpose the basket can include a set of spars  1210 ,  1212 ,  1214 ,  1216 ,  1218  arranged about the axis, as shown in  FIG. 12C . Referring back to  FIG. 12B , each spar has one hinged end  1220 ,  1222  connected to a central collar  1200  that can ride up  1202  and down  1204  along a central shaft  1250  of the basket. Its other hinged end  1230 ,  1232  is connected to the hinged  1240 ,  1242  struts  224   b  of the basket in such a way that when the opening and closing mechanism is manipulated  1208  by the user to cause the collar  1200  to move back and forth along the shaft  1250 , the spars  1210 ,  1220  force  1206  the basket open or closed, akin to the mechanism of an umbrella. The operator end  214   b  of the delivery tool has a twist or slide control  1150  that enables the operator to control the collar. In  FIG. 12B , the control is a slide control, and can be slid downward, for example. In this way, the annulus can be expanded to the desired shape by radial forces  1206  that are not imposed by moving the entire basket linearly along the valve axis. Instead the basket is moved into the desired position linearly along the valve axis and then the annulus is expanded to its desired shape. The radial forces could also be imposed by a combination or sequence of moving the entire basket axially and expanding the basket laterally. 
     As shown in  FIG. 13A , radiopaque measurement marks  1310 ,  1312  can be placed on the shaft or basket at regular spacings according to a standard measurement unit (e.g., one mark per centimeter). The marks can be used to determine the distance that the delivery tool has traversed inside the heart and the location of the basket as it is inserted into the valve, allowing the operator to place the basket at a good position along the axis of the valve. 
     The placement of the support from the basket onto the annulus can be done either as part of the operation of opening the basket or following the opening of the basket. In the former case, illustrated in  FIGS. 13A through 13D , the basket would be inserted into the valve to a point where the basket is adjacent to the valve annulus. Simultaneously with the opening of the basket, burr hooks on the outer periphery of the support would be forced radially into the annulus tissue. In this method of placing the support, the porous sleeve described earlier and shown in  FIG. 9P  would be positioned on the inner periphery  965 , away from the embedded hooks. 
     In the other approach, akin to the process shown in  FIGS. 1A through 1  D, the basket would be inserted into the valve so that the support on the basket was positioned slightly upstream of the location of the annulus. The basket would then be opened to force the annulus into the desired shape, then the tool and basket would be pushed slightly to force the support into place, embedding the hooks. 
     In either approach, once the support is placed, the basket would be at least partially closed, releasing the basket from the support, and the tool would be withdrawn from the valve. 
     Further, in some implementations, a combination of the approaches could be used. For example, the basket could be partially opened, inserted into the annulus, and then fully opened. 
     The approach of  FIGS. 13A through 13D  follows these steps: 
     A. Position  1301  ( FIG. 13A ) the collapsed (closed) conical head-end basket  220   b  of the delivery tool  200   b  at the medial axis  30  of the valve with the support adjacent the annulus. (The tool and basket are shown in side view and the valve and annulus are shown in sectional side view.) 
     B. Press a button  1302  on the operator end  214   b  to inflate a balloon  228   b  ( FIG. 13B ) on the distal end  230   b  of the delivery tool, allowing the delivery head  220   b  to float into the correct position in the heart valve  16 . If necessary, rotate the delivery head to align any section of the support body not bearing burr hooks, or any gap in the support body, or any portion that is sheathed, with any section of the annulus abutting delicate or sensitive tissue. 
     C. Slide  1208  or twist the control  1150  to expand  1306  the basket bringing the support body  110   a  into contact with the distorted annulus  18 . The support bears burr hooks that embed themselves in valve tissue at the periphery  121  of the annulus  18  upon contact, thus attaching the support to the tissue ( FIG. 13C ). 
     D. When the basket  220   b  has reached a desired diameter  1303 , the expanded heart valve support  110   a  forces the annulus  18  to conform to a desired configuration (e.g., a circle) and to a size that is larger (e.g., in diameter) than a desired final diameter of the annulus. Optionally, pull  1104  the wire loop  1102  to tighten the coils of the support body  110   a  to achieve a smaller final diameter. 
     E. When the heart valve support is in its final position, to break the tool away from the support attachments  246   b , pull  1304  ( FIG. 13D ), allowing the support to contract  1308  to its final size (including final diameter  1309 ) and shape and leaving the support permanently in place to maintain the annulus in the desired final configuration and size. Deflate  1311  the balloon  228   b  by pressing the button on the operator end. 
     In some implementations, as shown in  FIGS. 14A through 14D , the support is constructed from several pieces including an elastic multiple-loop circular coil  302  of strip material  304 . The coil is encased in a tubular toroidal sheath  306 . A large number of burrs or hooks  308  (the number could be, for example, between 20 and 60, but could also be much larger in number, even orders of magnitude larger, or in some cases smaller) are mounted at regular small intervals  310  around the circumference of the toroidal sheath. 
     In some implementations, the multiple-loop circular coil is made of Nitinol strip, approximately ⅛ inch wide and approximately 10/1000-15/1000 inch thick. During fabrication, the Nitinol strip is shape set into a coil with final desired implant diameter. For purposes of insertion, the Nitinol coil would be expanded, as explained later. During expansion the ends  312 ,  314  of the strap would move circumferentially around the coil (in the directions indicated by arrows  316  and  318 ) to accommodate the increase in diameter of the ring. In  FIGS. 14A through 14D , the ring is shown in its native, unstressed diameter corresponding to the final desired implant diameter. The numbers of loops can be varied depending on the material used, the thickness, and other considerations. In some implementations the number of loops can be 3.5, or 5 or 8, or other numbers ranging from 1 to 10 or more. 
     In some implementations, other materials and combinations of them can be used to form the resilient coil. These could include, for example, plastics, metals, and coils of these and other materials. 
     In some implementations, the overall shape of the coil could be different from the one shown in  FIG. 14A , including non-circular and non-planar shapes. 
     The coil (or other resilient core ring) needs to have enough strength and durability to be expandable to fit on the delivery tool, to be forced onto the heart valve annulus, to contract to pull the annulus back into the desired shape, to tolerate the force incurred when the insertion tool is disconnected, and to form a long-lasting and strong support for the annulus. It also needs to have enough resiliency to be able to contract the support and the annulus to which it is attached to the desired shape and size after insertion and to retain the support in essentially that shape and size against forces in the heart that may act against the support. 
     In some implementations, if there is a chance of exposure of the materials of which the coil is made to the blood or tissue of a patient, biocompatible materials are used. 
     The coil is held within the sheath  306  in a way that permits the coil to slide within the inner lumen of the sheath, especially as the coil is expanding for insertion and contracting after insertion. The sheath has an elasticity that allows it to move radially with the coil during expansion and contraction. Because the burrs or hooks (we sometimes refer to burrs and hooks and a wide variety of other gripping devices as grippers) are mounted on the sheath, and not on the coil, the expansion and contraction of the coil can occur without disruption of the angular locations of the grippers relative to the central axis of the support. 
     In some implementations, the sheath can be formed of a simple tube. To embed the coil in such a tube the coil can be unwound and wrapped through the tube repeatedly until all turns of the coil have been embedded. Once the coil is completely embedded, in the tube, one end of the tube can be pulled over and glued to the other end to finish the assembly. 
     In some implementations, the sheath can be formed of a specially molded piece that has the toroidal shape formed during molding and includes a way to secure the two ends together. 
     In some implementations, the sheath is meant to be sealed to prevent fluids from passing into the chamber that contains the coil. In some cases, the sheath is not sealed and fluid can pass freely. In some implementations, a fluid is used to fill the space within the sheath to provide lubrication for the sliding of the coil within the sheath and to displace air which could cause problems when the support is used inside the heart. The fluid could be blood or saline solution, for example. 
     The sheath must be strong enough to enclose the coil without breaking even when the support is expanded and contracted prior to, during, and after placement in the valve. As the diameter of the support is expanded and contracted, the cross-sectional diameter will also tend to change, and the amount of that change must not be so great as to disrupt the attachment of the grippers to the valve tissue, to constrain the sliding of the coil within the sheath, or to allow the grippers to become dislodged or disoriented relative to the sheath, among other things. The sheath can be resilient so that when the support is contracted after being expanded, the sheath contracts along with the coil. 
     A wide variety of materials can be used for the sheath, including silicone, plastics, and fabrics, for example. Combinations of materials can also be used. 
     As shown in  FIG. 14D , an outer surface  322  of the sheath can bear grooves  323  that accommodate (and hold in place) portions of the grippers, as explained below. In some implementations, the grooves can be parallel and lie at equal small intervals around the perimeter of the sheath. 
     The cross-sectional diameter of the sheath can be large enough so that the inner lumen accommodates the coil and allows it to slide, and the outer surface supports the grippers, and small enough that the support does not obstruct adequate flow of blood through the heart valve after installation. 
     As shown in  FIG. 15 , in some implementations, each of the grippers can be formed on a length of wire that includes a closed ring  324  that has about the same diameter  326  as (or slightly smaller than) the diameter of the cross section of the sheath. A straight section  328  extends from the ring and has the gripper  330  formed on its free end. 
     We sometimes refer to the entire piece that includes the gripper, and a portion to attach the gripper to the support, as an anchor  332 . 
     In some implementations, the anchor is prefabricated with the ring in its final shape and the gripper projecting from the ring. In some examples, the anchor is formed of stainless steel or another biocompatible material. 
     A wide variety of materials and combinations of them can be used to fabricate each of the anchors or groups of them, including metals and plastics. The cross-sectional shape of the anchors can vary and be, for example, round, oval, flat, or bent, or a variety of other shapes. 
     In some implementations, the anchors can be made from tiny fishhooks with the hook end serving as the gripper and the other end being bent to fit onto the support. 
     The thinner the anchors in the direction along the circumference of the sheath, the more anchors that can be fit onto the support. In some implementations, a larger number of thinner anchors would be useful in making the support easy to install and effective. In some cases, the arrangement of the anchors along the sheath can be other than regular and closely spaced. The spacing can be varied along the sheath or the number of anchors can be varied along the sheath, for example. 
     To install an anchor, its ring portion can be pulled open and slipped over the sheath, then released. In examples in which the outer surface of the sheath is molded to have grooves, the ring portions of the anchors can be seated in the grooves. 
     In some examples, the anchors can all be mounted to cause their grippers to point at a common angle  336  from a central axis  338  of the support as shown in  FIG. 14D  (in which some of the anchors have not yet been mounted). In some examples, the grippers can be pointed at different angles relative to the central axis. 
     In some examples, the anchors can be mounted in such a way that they do not tend to slip or rotate around the outer surface of the sheath, but rather maintain their installed orientations. In some implementations, when the supported is expanded and contracted prior to, during, and following insertion into the heart valve, the stretching and relaxing of the sheath may cause a change in its cross-sectional diameter and therefore an opening and closing of the rings and a corresponding reorientation of the angles of attack of the points of the grippers. This effect can be useful in installing and providing secure attachment of the grippers in the valve tissue. 
     In some cases, if the angle of attack of the points is shared in common by all of the grippers, then it may not be desirable to have the successive anchors along the perimeter be spaced too closely  310  because the adjacent gripper points could interfere with each other during insertion, and be less effective in gripping the valve tissue. For this reason, in some implementations, the angles of attack of the points of the grippers can be varied slightly from anchor to anchor which would permit a closer spacing while still allowing some clearance between successive grippers. In some cases the orientations of successive grippers could alternate back and forth around a central line. Other arrangements are also possible. 
     In  FIGS. 14A through 14D and 15 , the anchors are shown as each having a single free end bearing a point  340 . In some implementations, each anchor could provide for an extension of the other end  342  of the wire (for example, a symmetrical extension), as implied in dashed line  344 . A wide variety of other arrangements are also possible. 
     In  FIG. 15 , the gripper has three barbs on each side of the free end of the wire. In some implementations, there could be more or fewer barbs, and the barbs could have a wide variety of other configurations on the gripper. 
     In some implementations, each of the grippers  350  can be formed of wire or other cylindrical material and can be formed, machined, or molded, for example, to have the configuration shown in  FIGS. 16 and 17 , including a point  352  having two symmetrical faces  354 ,  356  each at an angle  358  of, for example, 25 degrees relative to a central axis  360  of the gripper. Below the point are two barbs that are formed, by laser cutting, machining or otherwise imparting slots  362  and  364  at a common angle (15 degrees in this example) to the central axis. 
     Once the barbs are formed they can be bent away from the axis in the directions  366  and  368  to form the final barbs. 
     A wide variety of other configurations and forms of manufacture are possible for the barbs and the grippers. In the particular example shown in  FIGS. 16 and 17 , the grippers are formed of Nitinol wire that is 1.26 mm in diameter and the length of the gripper to the bottom edge of the slots is 22.87 mm. 
     As shown in  FIG. 14D , in some examples, when installed each of the grippers extends from about 2 to about 4 millimeters (dimension  339 ) from the bottom of the sheath surface. 
     In some implementations, the support—which includes the coil, the sheath and portions of the anchors—is wrapped in a cloth covering as are many existing rings that are hand-sutured to the valve annulus by a surgeon. The cloth allows the heart tissue to attach itself securely to the support over time, making for a secure repair. 
     As shown in  FIG. 18 , in some cases, the cloth covering can be a thin strip of material that is helically wound around the other parts of the support. The material may be attached to the support by suturing, gluing, or in other ways. The helical winding allows an inelastic material to be employed and still accommodate the circumferential expansion of the support. In some examples, the cloth covering may include a series of independent tubular cloth segments placed over the support. The segmented arrangement will allow inelastic cloth to be used without hindering circumferential expansion of the support. 
     As the cloth is placed on the support, it is pulled over the grippers, each of which penetrates the cloth and remains ready for insertion. A wide variety of covering materials or combinations of them could be used including metal, fabric, and plastic. The covering should be able to accommodate the expansion and contraction of the support without becoming distorted and should be biocompatible and porous enough to accept and encourage the growth of tissue through its structure, 
     A wide variety of other configurations of parts and materials, and ways to assemble the parts of a support are possible. Different numbers of pieces can be used, and the functions described can be combined in different ways into different pieces of the support. 
     In some examples, shown in  FIGS. 19, 20, and 21 , the sheath can be made of two molded pieces that interlock. An outer annular housing  402  (sometimes called the outer piece) has upper and lower flat rings  404 ,  406  joined by an outer flat cylindrical wall  408 . The coil  407  sits within the housing. The other, inner piece  410  of the sheath is a cylindrical wall that is captured between the upper and lower rings  404 ,  406  in a way that permits the inner end  408  of the coil to be tightened or loosened by sliding it circumferentially  409 , causing the support to be expanded or contracted. During the sliding, the inner piece of the sheath slides circumferentially also. 
     In this example, the anchors  412  are formed from flat pieces of metal that are bent and then attached to the outer piece of the sheath. Each anchor includes an upper finger  417  that grasps the upper portion of the outer piece of the sheath, a vertical arm  419  and a lower finger  414  that grasps the bottom of the outer piece of the sheath. The gripper  416  extends downward from the lower finger. The inner piece of the sheath has a tab  418  that can be manipulated to pull or release the end of the coil to expand or contract the support. An opposite end of the inner piece of the sheath is attached to the end of the coil for this purpose. As a result, the support can be expanded or contracted without the anchors moving relative to the outer piece of the sheath. The tab  418  can be manipulated in a wide variety of ways, including by direct finger manipulation, use of an insertion tool in open heart surgery, or manipulation at the end of a catheter from a distant position in a catheter laboratory. 
     In some implementations of a gripper, as shown in  FIGS. 22 through 27 , there is a pointed end  430  and on each side of the pointed end, a pair of barbs  432 ,  434 ,  436 ,  438 . In the example shown in  FIGS. 22 and 23 , the barbs  434  and  438  are smaller. In the example of  FIGS. 24 and 25 , the two barbs on each side of the point have a similar size and shape. 
     In some examples, as shown in  FIGS. 26 and 27 , the detailed configuration of a Nitinol strip includes the point and the barbs. As shown in  FIG. 21 , in some configurations, the barbs are bent out of the plane of the strip from which the gripper is formed in order to be more effective as barbs. 
     In general, in some examples, the support to be embedded in the valve tissue can be configured to achieve three related functions: (1) the ability to easily insert the grippers of the support into the tissue once the support has been correctly located at the annulus; (2) the ability to retain the support in the tissue securely in a way that maintains the correct shape for the annulus of the valve and is durable and long lasting, in part by providing a substantial resistance to forces that could cause detachment of all or part of the support after insertion; (3) the ability to deliberately withdraw all or a portion of the grippers during or after the insertion procedure in order to relocate or reorient the support relative to the valve annulus if doing so would be useful. These three functions require a careful and subtle design of the grippers, the anchors, and the other parts of the support, because some design factors that favor one of the functions can be a negative influence on another of the functions. These functions should also be implemented in a device that is simple, foolproof in its operation, and easy to use. 
     For example, easier insertion of the grippers into the tissue can be achieved by reducing the size and profile of barbs on the grippers and aiming the points of the grippers directly at the tissue. Removal of some or all of the grippers to reposition the support would also be aided. But those same features could reduce the stability and durability of the attachment of the support to the tissue. By giving the barbs a broader or more obstructive profile or aiming the points of the grippers off a direct path to the tissue, the gripping is made more secure, but inserting the grippers is more difficult as is repositioning. 
     Among the design features that can be adjusted and traded-off to achieve a desired mix of the needed functions are the number, shape, size, orientation, and method of mounting the anchors, the grippers, and the barbs, the shape, size, orientation and other configuration of the body of the support, the materials used for all of the parts of the support, and a wide variety of other factors. 
     In some cases, a mechanism or configuration can be provided that allows a deliberately reversible process for inserting and removing the grippers in the tissue for repositioning. 
     For example, as shown in  FIGS. 28 through 31 , a support  450  could include anchors in the form of, say, 30 loops  452  equally spaced around the body  454  of the support. A cross-section of the body  454  could include a circular segment  456  along the inner periphery of the body, and a flat or concave section  458  along the outer periphery of the body. Each of the loops could include two free ends  460 ,  462 , one of which  460  is un-pointed and the other of which  462  has a sharp point. The loop does not have any barbed features. 
     In some modes of operation, prior to insertion, the curved sharp ends  462  of all of the grippers can be held away from body and aimed in the general direction of the annulus tissue. A sheath or other mechanism could be used to move them into and hold them in this temporary insertion position. During insertion, the insertion tool could be applied to force the grippers into the tissue. Once the pointed ends of the grippers are in the tissue, the sheath or mechanism could be manipulated to allow the anchors to assume their final shape, after following curved paths  464  through the tissue  466  and exiting from the tissue to lie next to the support body, as shown in  FIG. 31 . 
     This configuration has the advantage that the process could be reversed using a similar sheath or mechanism to withdraw the grippers through the tissue and back to the configuration of  FIG. 30 . Because the gripping has been achieved by the curvature of the shafts of the anchors and not by barbs on the sharp tips, reversing the process is relatively easy. Gripping is also secure. However, insertion may be more difficult than in other implementations, and the reversibility requires an additional mechanism. 
     In some examples, the support could be provided with an adjustment and locking feature that would permit the size (e.g., the diameter) and possibly the shape of the support to be adjusted or locked or both, by the surgeon or operator at the time of insertion. In some cases, the support could be adjusted to different possible sizes at the time of insertion rather than requiring that it reach only a single non-selectable designed size. 
     For example, as shown in  FIG. 45 , a core structural piece  570  of the support could be made of crimped stainless steel that is plastically deformed by an insertion tool (not shown). The tool could engage the top of the structural piece and force the piece temporarily to have a larger diameter for insertion. After pushing the support into the annulus to cause the grippers to attach to the tissue, the tool could collapse and allow the structural piece to collapse in diameter to its final size. 
     As shown in  FIG. 46 , in some cases, individual expansion elements  573 ,  575  would bear holes  576 ,  578  that have locations and spacing to mate exactly with the locations and spacings of pins  582 ,  584  in rigid locking elements  580  once the structural piece has been expanded or contracted to exactly the desired dimension. The locking elements would be held at the proper places in an annular silicone support that has inner and outer peripheral walls  574 ,  576  joined by an upper annular wall  578 . Pushing down on the silicone support when the support is properly sized will force the pins of the locking elements into the holes. 
     Referring to  FIGS. 47 through 53 , in some implementations, the support  600  could be formed of three pieces. 
     One of the pieces, an annular resilient (e.g., silicone) ring  606  has a cross-section that includes four linear segments defining a trapezoid, which provide stability to the shape of the ring. There are four corresponding faces of the ring. Pace  632  would have a configuration designed to match surfaces of a face of a dilator part of an insertion tool. 
     A second of the pieces is a metal ring  604  formed from a strip of, e.g., stainless steel having a curved cross-section and two overlapping ends  620 , and  622 . The curvature of the cross-section maintains the axial stability of the ring. Near one end  622 , the ring has a series of slots that are meant to mate with corresponding tabs  623  formed near the other end  620 . During fabrication and assembly the tabbed end of the ring is on the inside of the overlapping section  627  so that no mating and locking can occur. When finally installed, however, the tabbed end is on the outside of the overlapping section to permit locking. During manufacture, the silicone ring is molded around the metal ring. When the silicone ring is stretched and relaxed, the metal ring can expand and contract because the two ends are free to move relative to one another at the overlapping section. The support is essentially spring loaded. 
     The third piece of this example support is a double-pointed anchor  602 , many copies of which are arranged around the ring (in this version, but not necessarily, at regular intervals). In some implementations, each of the anchors is made from a single loop  602  of wire that has a gripper (a barb or a fish hook, for example) at opposite free ends  616 ,  618 . Each of the anchors is resilient and has a relaxed state shown in  FIG. 53 , with a distance  619  between the two grippers, and the points of the two grippers pointing generally towards each other. The loops of the anchors are placed on the metal ring and potted in the molded silicone ring. 
     After assembly, the support is stretched to a larger diameter and mounted on an insertion tool, not shown. The stretching has two effects. One, shown in  FIG. 51 , is that the two ends of the metal ring are pulled apart sufficiently to eliminate the overlap. The ends of the ring are biased so that the tabbed end moves to the outside relative to the slotted end. So when the two ends again form the overlap upon the later contraction of the ring, the tabs are positioned to mate with the slots. The ends of the metal ring are beveled to assist in achieving this arrangement as the ring contracts. 
     Also, as the silicone ring expands, the cross-sectional diameter of the silicone ring contracts; because the anchors are potted within the silicone ring, as the ring stretches in length and contracts in diameter, the matrix squeezes the loops  610  of the anchors and forces them into a temporary configuration shown in  FIG. 48 , in which the distance  619  has increased and the orientation of the points of the grippers has rotated to face generally in the insertion direction, ready for insertion. 
     As shown in  FIG. 52 , when the insertion tool is removed from the support, the support contracts in diameter, which reconfigures the annulus to the desired shape and size. And the silicone rings expands in cross-sectional diameter, which allows the anchors to relax ( FIG. 53 ), driving the grippers to rotate and force the points towards each other, to hold onto the tissue securely. As the metal ring contracts, the tabs and slots cooperate in a ratchet action which permits the support to contract to its final shape and size, while prevent a reverse expansion from occurring again. 
     In some cases, shown in  FIGS. 54 and 55 , the locking of the final diameter of the support can be achieved by embedding mating elements in a resilient ring  700 . One set of elements  704  can be embedded in one plane of the ring, and a corresponding set of elements  706  to be mated can be embedded in a second plane of the ring. The embedding is done in a way that permits the two different kinds of mating elements to slide relative to one another as the support is expanded and contracted prior to and during installation. When the proper diameter of the support has been reached, a tool can be used to press down on the silicone ring to cause the mating elements to occupy the same plane and be interlocked. 
     In some examples, two interlocking elements  722  and  724  can be formed at the ends of a resilient metal coil  720  that forms part of the support. Once installed and properly sized, the support can be locked by pushing down to cause the interlocking elements to mate. 
     In some cases, a support could have a central annular lumen filled with uncured polyurethane and arranged so that the diameter or shape or both of the support could be adjusted at the time of insertion. Once the desired diameter or shape or both have been reached, ultraviolet light, which could be delivered through a delivery tool or in other ways, would be used to cure and harden the polyurethane. Current curable materials and lighting can achieve curing in about 20 to 30 seconds. 
       FIGS. 32 through 35  show another example configuration that allows a reversible process for installing and removing the grippers from the annulus tissue for repositioning. Each of the anchors  470  incorporates a scissoring or pincering mechanism that has two pointed (but not barbed) grippers  472 ,  474  on opposite free ends of a 0.015 inch Nitinol wire loop. To form the each anchor, the wire is wound on a jig in the shape  476  shown in  FIG. 32 , which is the open configuration of the anchor. Then heat is used to memory set that open shape. The loop diameter  478  in this example could be about 0.20 inches for mounting on a toroidal resilient stretchable support body having a cross-sectional diameter  480  of about 0.25 inches. 
     When the loop of each anchor is opened up to force it onto the larger diameter  480  support body, the configuration of the anchor automatically causes the two pointed free ends to close up into a gripping configuration as shown in  FIG. 33 . Prior to installation and before the support has been loaded onto the insertion tool, the support body is in its contracted installed shape as shown in  FIG. 33 , with all of the pincers closed. In  FIGS. 34 and 35  the support has been stretched to its insertion configuration, in which the diameter  482  is larger to fit onto (here a simulated) insertion tool  484 . Because of the shape and configuration of the support body (for example, a silicone tube), when the body is stretched, its cross-sectional diameter is reduced allowing the anchors to relax to their native, open shape, ready for insertion. 
     Insertion proceeds by pushing the support towards the opened and properly shaped annulus causing the sharp points of the grippers to penetrate the tissue. As the insertion tool is removed from the support, the support body contracts to the final desired shape and diameter of the valve annulus. As it contracts, the pincers are forced to grasp the tissue of the annulus and hold the support securely in place. Thus, the support is relatively easy to insert and can be removed and repositioned by reversing the process, that is by expanding the support body, which releases the pincers. 
     A wide variety of insertion tools (which we also sometimes call dilators) can be used to attach a support to the heart valve annulus tissue. Some have been described earlier and we describe others below. 
     An important principle of the configuration and operation of at least some examples of insertion tools is that they enable a surgeon or catheter operator to install the support reliably and easily in a wide range of patients having heart valves that are in a wide variety of conditions and have a wide variety of shapes and sizes. In other words, insertion can be achieved routinely and simply. This can be done by an insertion tool that automatically and easily temporarily expands and reconfigures any heart valve annulus to adopt a common expanded shape or size or both so that a support that has been pre-expanded to the common shape or size or both can be attached without concern for the upstretched context and configuration of the patient&#39;s valve annulus. The support is configured so that after insertion the support can be reconfigured automatically or by manipulation to a final secure stable desired shape and size, with the insertion tool removed. 
       FIGS. 36 through 39  illustrate an example of an insertion tool  500  that includes a dilator  502  formed of six arms  504  arranged at equal intervals around an insertion axis  506 . Each of the arms is formed of a 0.125″ wide spring steel metal strip that is bent at two places  508  and  510 . Ends  512  of the arms are gathered together and held by a segment of plastic tubing  513  on the end of an aluminum inner tube  514  (0.28″ outside diameter, 0.24″ inside diameter). The opposite ends  516  of the arms are gathered together and held by a segment of tubing and a shaft collar  518  to an aluminum outer tube  520  (0.37″ outer diameter, 0.30″ inner diameter). The outer tube is connected to a handle  522 . The inner tube, which slides within the outer tube along the insertion axis, is manipulated by a second handle  524 . 
     By pushing or pulling  526  on the second handle relative to the first handle, the inner tube is moved back and forth relative to the outer tube, which causes the arms to dilate as in  FIG. 38  or contract as in  FIG. 37 . A thin molded sleeve of, e.g., silicone,  530  protects the mechanism and protects the heart tissue and the support from damage. Prior to installation of the support in the heart valve, the support is stretched and mounted on the dilator at the central ridge  532 . It can be held in place by force and friction or can be lashed with sutures that are cut after installation, or the central ridge can be provided with a concavity in which the support is seated. Another view of the central ridge  532  is shown in  FIG. 44 . 
     As shown in  FIGS. 42 and 43 , in some examples, a dilator can include round wire arms  550  that are evenly spaced around the insertion axis and have each been shape set to the expanded configuration shown in  FIG. 42 . The ends  552 ,  554  of each wire are secured respectively to two circular hubs  556   558 . The upper hub  556  has a central hole (not shown) that is threaded to receive a threaded rod  560  to which a handle  562  is clamped. The other end  559  of the threaded rod is fixed to the hub  558 . Using the handle to turn  564  the threaded rod advances it or withdraws it (depending on the direction of rotation) through the upper hub, toward or away from the lower hub. The rod pushes or pulls on the lower hub, thereby increasing or decreasing the distance  566  between the two hubs and forcing the arms to contract or allowing them to expand to the shape set expanded configuration. 
     As shown in  FIG. 40 , in some implementations each arm  538  of an insertion tool  540  is formed of a stiff limb  544  connected at one end  546  to the outer tube  548 , and at another end  549  to a broader limb  550 . The other end  551  of the second limb is connected to the inner tube  554  at a tip  556 . The limbs are joined by a hinged element that allows them to pivot relative to each other. On each of the arms, a clip  560  has a recess to capture the support at one location along its perimeter. 
       FIG. 41  shows a support mounted on an insertion tool ready for insertion. 
       FIGS. 58 and 59  show a version  730  of the support. This version  730  has a ring of successive hexagonal sections  732 ,  734  touching at short edges  736 ,  738 . At the junction of longer edges  740 ,  742 ,  744 ,  746  of the hexagonal sections are sharp free ends  748 ,  750 , pointing in opposite directions. Further, on each hexagonal section, one sharp free end  750  is longer than the other sharp free end  748  and has barbs  752 ,  754 ,  756  for gripping tissue  757  that the barbed sharp free end  750  has pierced. All of the barbed sharp free ends  750  point in the same direction  751  on all of the hexagonal sections  732 ,  734 . The other set of free ends  748  have no barbs and can further stabilize the support by piercing other adjacent tissue if any is present, lodging themselves inside and further securing the support to the tissue. All of the other free ends  748  point in the same direction  753  which is opposite the direction  751  that the barbed sharp free ends  750  point to. 
     This version  730  of the support is resilient and can be expanded to a delivery configuration and later will contract to a final configuration. As shown in  FIGS. 60A and 61A , when the support is expanded  760  to a larger diameter  762  in a delivery configuration, e.g. by a delivery tool, each hexagonal section  732  increases in width  770  and decreases in height  772 . As shown in  FIGS. 60B and 61B , when the support contracts  764  to a smaller diameter  766  in a final configuration, each hexagonal section  732  decreases in width  770  and increases in height  772 . In some implementations, this version  730  of the support can be made of a flexible shape memory material such as Nitinol or a biologically compatible elastomer (or other material) that is configured to contract  764  the support to the final configuration after insertion into tissue. For example, the support may be configured to contract upon a period of exposure to the temperature of the human body. In some implementations, this version  730  of the support can expand to 38.2 millimeters in diameter or more and contract to 6.5 millimeters in diameter or less. 
       FIG. 62  shows a support  800 . Support  800  is a complete loop of round cross-section wire wrapped helically and with the helical winding looped in a torus in a configuration of successive windings  802 ,  804 . The loop includes anchors  806 ,  808  each of which is bonded to a respective one of the windings  802 ,  804 . The anchors  806 ,  808  are bonded at points of attachment  810 ,  812  such that sharp free ends  814 ,  816  of the anchors  806 ,  808  all point in the same direction  818  for piercing heart tissue and anchoring the support. 
       FIG. 63  shows a support  820  having a series of helically coiled segments  822 ,  824  joined by intervening anchoring elements  826 ,  828 . The coiled segments  822 ,  824  and the anchoring elements  826 ,  828  alternate within the ring formation in such a way that every coiled segment joins with an anchoring element. The coiled segments  822 ,  824  are expandable and contractible and are made up of successive windings  827 ,  829  such that a single segment could have anywhere from one winding to a dozen windings or more. The anchoring elements  826 ,  828  can be rigid or semi-rigid relative to the coiled segments  822 ,  824 . The ends  830 ,  832  of the coiled segments  822 ,  824  tightly fit through holes  834 ,  836  in the anchoring elements  826 ,  828  to form a secure connection between the coiled segments and the anchoring elements. The anchoring elements  826 ,  828  have anchors  838 ,  840  with sharp free ends  842 ,  844  all pointing in the same direction  846  for piercing heart tissue and anchoring the support. The anchors  838 ,  840  have two pairs of barbs  839 ,  841  for gripping pierced tissue. Each anchoring element  826 ,  828  could have as few as one anchor or as many as several dozen. The anchoring elements  826 ,  828  could be flat, round, or another shape, and are made of a biologically-compatible material such as a metal, a flexible or semi-flexible material such as Nitinol, or another material. Generally, a support may be easier and cheaper to manufacture if it uses dedicated anchoring elements as a platform to bear the anchors, rather than attaching anchors directly to other elements of the support such as the flexible coiled segments. For example, the anchors may be easier to attach to anchoring elements, or the anchoring elements could be manufactured separately from other elements like the coiled segments. 
       FIGS. 64A through 64D  show a support  848  having coiled segments  850 ,  852  joined in a ring formation by connecting elements  854 ,  856 . Both ends of each of the coiled segments  850 ,  852  terminate in sharp free ends  862 ,  864  all pointing in the same direction  866  for piercing heart tissue and anchoring the support. The free ends  862 ,  864  of the coiled segments fit tightly through holes  868 ,  870  in the connecting elements  854 ,  856  to form a secure connection between the coiled segments and the connecting elements. The coiled segments  850 ,  852  and the connecting elements  854 ,  856  alternate within the ring formation in such a way that every coiled segment joins with a connecting element. In some implementations, as shown in  FIGS. 64A and 64B , each of the connecting elements  854 ,  856  joins a free end  864 , of one of the coils. oriented at the outer edge  858  of the ring to a free end  862 , of the next one of the coils, oriented at the inner edge  860  of the ring. 
     As shown in  FIGS. 64C and 64D , in some implementations, some connecting elements  872  are arranged to join ends  874 ,  876  both oriented at the outer edge  858  of the ring and some connecting elements  878  arranged to join ends  880 ,  882  both oriented at the inner edge  860  of the ring. A combination of the arrangements of  FIGS. 64A and 64C  would also be possible. 
       FIG. 65  shows a support  1400  made of a single continuous coil of flat wire  1402 . Flat wire  1402  can be used in applications where other types of wire are not desirable or less desirable. For example, flat wire  1402  may provide advantages in manufacturing the support or attaching anchors or hooks.  FIGS. 66A and 66B  show a support  1404  having coiled segments  1406 ,  1408  made of flat wire joined in a ring formation by connecting elements  1410 ,  1412 . The coiled segments  1406 ,  1408  terminate in sharp free ends  1414 ,  1416  all pointing in the same direction  1418  for piercing heart tissue and anchoring the support. The free ends  1414 ,  1416  have barbs  1420 ,  1422  for gripping pierced heart tissue. The barbs are in the form of multiple pairs that line the free ends  1414 ,  1416  from the tip  1415  to the point of attachment  1417  with the respective connecting element. The free ends  1414 ,  1416  of the coiled segments  1406 ,  1408  fit tightly through holes  1424 ,  1426  in the connecting elements  1410 ,  1412  to form a secure connection between the coiled segments and the connecting elements. In some implementations, the coiled segments  1406 ,  1408  and the connecting elements  1410 ,  1412  alternate within the ring formation in such a way that every coiled segment joins with a connecting element. For example, the connecting elements  1410 ,  1412  can be arranged to join a free end  1414  oriented at the outer edge  1428  of the ring to a free end  1416  oriented at the inner edge  1430  of the ring. Other arrangements of the coiled segments  1406 ,  1408  and connecting elements  1410 ,  1412  are possible. 
       FIGS. 67A and 67B  show a relatively flat support  1432  having doubled flat sinusoidal segments  1434 ,  1436  joined in a ring formation by connecting elements  1438 ,  1440 . In use, this support  1432  sits flat against heart tissue. The doubled sinusoidal segments  1434 ,  1436  and the connecting elements  1438 ,  1440  alternate within the ring formation in such a way that every doubled sinusoidal segment joins with a connecting element. The connecting elements  1438 ,  1440  can be rigid or semi-rigid relative to the doubled sinusoidal segments  1434 ,  1436 . The doubled sinusoidal segments  1434 ,  1436  are expandable and contractible and are each made of two sinusoidal wires  1442 ,  1444 . 
     The peaks and valleys of the sinusoid of the first sinusoidal wire  1442  are inverted relative to the peaks and valleys for the second sinusoidal wire  1444  such that a peak  1446  of the first sinusoidal wire  1442  oriented toward the outer edge  1448  of the ring formation is positioned opposite a peak  1450  of the second sinusoidal wire  1444  oriented toward the inner edge  1452  of the ring formation. One sinusoidal wire  1442  in each double sinusoidal segment  1432  terminates in sharp free ends  1454 ,  1456  all pointing in the same direction  1462  for piercing heart tissue and anchoring the support. The sharp free ends  1454 ,  1456  have barbs  1464 ,  1466  for gripping pierced heart tissue. One sinusoidal wire  1444  in each double sinusoidal segment  1434  terminates in flat free ends  1458 ,  1460 , which do not aid in piercing the heart tissue. In some configurations, both sinusoidal wires  1442 ,  1444  terminate in sharp free ends. The sharp free ends  1454 ,  1456  and flat free ends  1458 ,  1460  of the sinusoidal wires  1442 ,  1444  fit tightly through holes  1468 ,  1470 ,  1472 ,  1474  in the connecting elements  1438 ,  1440  to form a secure connection between the double sinusoidal segments  1434 ,  1436  and the connecting elements. 
       FIG. 68  shows a support  1476  having sinusoidal segments  1478 ,  1480  joined in a ring formation by connecting elements  1482 ,  1484 . The sinusoidal segments  1478 ,  1480  and the connecting elements  1482 ,  1484  alternate within the ring formation in such a way that every pair of sinusoidal segments are joined by a connecting element. The connecting elements  1482 ,  1484  can be rigid or semi-rigid relative to the double sinusoidal segments  1478 ,  1480 . The sinusoidal segments  1478 ,  1480  are expandable and contractible and terminate in sharp free ends  1482 ,  1484  for piercing heart tissue and anchoring the support. One sharp free end  1482  on each sinusoidal segment  1478 ,  1480  points in one direction  1486 , and the other sharp free end  1484  points in another direction  1488 . The sharp free ends  1482 ,  1484  fit tightly through holes  1490 ,  1492  in the connecting elements  1482 ,  1484  to form a secure connection  1491  between the sinusoidal segments and the connecting elements. 
       FIGS. 69A and 69B  show a support  1500  having crimped segments  1502 ,  1504  joined in a ring formation by anchoring elements  1506 ,  1508 . The accordion-crimped flat-metal segments  1502 ,  1504  and the anchoring elements  1506 ,  1508  alternate within the ring formation in such a way that successive crimped segments are joined by an anchoring element. The crimped segments  1502 ,  1504  and the anchoring elements  1506 ,  1508  can be joined by welding or bonding, for example, or the entire support could be formed from a single piece of material. The crimped segments  1502 ,  1504  can be made of a metal, e.g. stainless steel or another biologically compatible material, and can expand and collapse and the anchoring elements  1506 ,  1508  can be rigid or semi-rigid relative to the crimped segments  1502 ,  1504 . The anchoring elements  1506 ,  1508  have two parallel rows of evenly spaced anchors  1510 ,  1512  with arrow-shaped free ends  1514 ,  1516  all pointing in the same direction  1518  for piercing heart tissue and anchoring the support. The anchors  1510 ,  1512  have barbs  1520 ,  1522  for gripping pierced heart tissue. Each anchoring element  1506 ,  1508  could have as few as one anchor or as many as several dozen. The anchors  1510 ,  1512  can be arranged in one or more rows  1524 ,  1526 , for example, one row  1524  lined up along the outer edge  1528  of the ring formation and one row  1526  lined up along the inner edge  1530  of the ring formation. 
       FIG. 70  shows a support  1532  having arc segments  1534 ,  1536  joined in a ring formation. The arc segments  1534 ,  1536  are welded or bonded at junctions  1538 ,  1540  bearing anchors  1542 ,  1544  with sharp free ends  1546 ,  1548  all pointing in the same direction  1550  for piercing heart tissue and anchoring the support. Further, the angle  1552  of the junctions  1538 ,  1540  between the arc segments  1534 ,  1536  is variable, allowing the support to expand and contract. For example, when the angle  1552  is reduced, the support contracts (e.g. by a delivery tool for a delivery configuration), and when the angle  1552  is increased, the support expands. The arc segments  1534 ,  1536  could be made of wire or cut from coils of a spring, for example. 
       FIG. 71  shows a support  1554  having doubled arc segments  1556 ,  1558  joined at junctions  1560 ,  1562  in a ring formation. The doubled arc segments  1556 ,  1558  have a pair of joined single are segments  1564 ,  1566  each terminating in anchors  1568 ,  1570  with sharp free ends  1576 ,  1578  all pointing in the same direction  1584  for piercing heart tissue and anchoring the support. Further, the separation distance  1586  of the single arc segments  1564 ,  1566  is variable, allowing the support to expand and contract. For example, when the separation distance  1586  is reduced, the support contracts (e.g. by a delivery tool for a delivery configuration), and when the separation distance  1586  is increased, the support expands. The single arc segments  1564 ,  1566  could be made of wire or cut from coils of a spring, for example. 
       FIG. 72  shows a support  1588  having a metal ribbon  1590  coiled into a ring. The metal ribbon  1590  can be wrapped onto itself to form multiple overlapping layers  1592 ,  1594 . When the support expands, the layers  1592 ,  1594  slide  1596  apart relative to each other, and when the support contracts, the overlaps  1592 ,  1594  slide  1598  together relative to each other. One edge  1600  of the metal ribbon  1590  bears anchors  1602 ,  1604  with sharp free ends  1606 ,  1608  all pointing in the same direction  1610  for piercing heart tissue and anchoring the support. The anchors  1602 ,  1604  also have barbs  1612 ,  1614  for gripping heart tissue. The anchors  1602 ,  1604  can be attached to the metal ribbon  1590  using one of several methods such as welding or bonding, for example, or they could be formed or cut directly from the metal ribbon  1590 , for example. 
       FIGS. 73A and 73B  show a support  1616  having a c-shaped ring  1618 . The c-shaped coil  1618  has a gap  1620  that allows the support to expand and contract. When the support expands, the gap  1620  increases in width  1622 , and when the support contracts, the gap  1620  decreases in width  1622 . The c-shaped coil  1618  is supported by an attached secondary ring  1624 , which also has a gap  1626  positioned across the diameter  1628  from the gap  1620  of the c-shaped coil  1618 . The secondary ring  1624  assists in maintaining the ring shape of the support by attenuating any physical distortion when the support expands and contracts. The c-shaped coil  1618  bears anchors  1632 ,  1634  all pointing in the same direction  1640  for piercing heart tissue and anchoring the support with sharp free ends  1636 ,  1638  curved slightly inward relative to the c-shaped coil  1618 . The anchors  1632 ,  1634  can be attached to the c-shaped coil  1618  using one of several methods such as welding or bonding, for example, or they could be formed or cut directly from the c-shaped coil  1618 , for example. 
     The slight curve of the free ends  1636 ,  1638  resists forces that pull on the support when the anchors  1632 ,  1634  are embedded in annular tissue. Some or all of the anchors  1632 ,  1634  could also have barbs, just as the barbed anchors shown on some of the other supports herein (e.g. the supports in  FIGS. 62-72 ) could also have curved ends. If desired, any straight anchor could be bent to form a curve. Although the free ends  1636 ,  1638  shown in  FIGS. 73A and 73B  all curve inward, some or all of the free ends could also curve outward, to the side, have multiple curves, or have any combination of these curve configurations. 
       FIG. 74  shows a support  1642  having an elastic polymer flat ring  1644 . In use, this support  1642  sits flat against heart tissue. The elastic polymer flat ring  1644  is elastic enough to allow expansion during insertion (e.g. by an insertion tool) and is stiff enough to support a heart valve annulus after implantation. If desired, the support  1642  can also be folded during delivery, e.g., folded in half along the diameter  1646  of the support. The elastic polymer flat ring  1644  bears anchors  1648 ,  1650  with sharp free ends  1652 ,  1654  all pointing in the same direction  1656  for piercing heart tissue and anchoring the support. The anchors  1648 ,  1650  also have barbs  1658 ,  1660  for gripping heart tissue. 
     The supports shown in  FIGS. 62-74  could be used with any of the implementations of the delivery tool shown throughout this description, including the delivery tool  200  shown in  FIG. 1A , the delivery tool  200   a  shown in  FIG. 6A , the delivery tool  200   b  shown in  FIG. 11A , and the insertion tools shown in  FIGS. 36-44 , as well as other implementations of the delivery tool, for example. In general, the support chosen does not necessarily limit the choice of delivery tool. The variations of the support insertion process, such as the variations shown in  FIGS. 1A-1D ,  FIGS. 8A-8I , and  FIGS. 13A-13D , are not necessarily limited to any combination of support and delivery tool. 
       FIGS. 75A through 75D  show a delivery tool  1662  having a continuous cone  1664  forming the portion of the tool for delivering a support  1665 . The cone  1664  is made of a material such as rubber or a flexible polymer that allows it to expand and contract and slide smoothly against a heart valve annulus. The cone  1664  has an upper flange  1666  providing a shelf  1668  against which the support  1665  can securely rest. When the support  1665  is being delivered, the upward force  1670  upon the support by the annulus (not shown) is countered by the shelf  1668  of the upper flange  1666 . This delivery tool  1662  also has a shaft  1672  that connects to the cone  1664  by several splaying projections  1674 ,  1676  that spread apart away from the shaft  1672  when the delivery tool expands and pull together toward the shaft  1672  when the delivery tool contracts. The head  1678  of this delivery tool  1662  has one or more openings  1680 ,  1682  allowing blood to flow past the delivery tool so as to not impede blood flow through the annulus. In some implementations of the delivery tool  1662 , as shown in  FIG. 75D , the upper flange  1666  is divided into angled or shaped segments  1684 ,  1686 . The angled or shaped segments  1684 ,  1686  form a jagged shelf  1668   a . The jagged configuration of the shelf  1668   a  allows portions of the support  1665  to shift slightly during delivery, which allows anchors, hooks, or grippers of the support to attach to heart tissue at slightly different angles relative to each other. 
       FIGS. 76A through 76C  show a delivery tool  1688  having a cone-shaped wire cage  1690  enclosing a balloon  1692 . The wire cage  1690  is expandable and contractible. When the balloon  1692  inflates with air, the force of the balloon against the wire cage  1690  causes the wire cage to expand. Air flows through a shaft  1691 , which is surrounded by the balloon  1692 . The wire cage  1690  has splaying projections  1694 ,  1696  extending from attachment points  1695 ,  1697  at a base ring  1698  up to attachment points  1699 ,  1701  at a top sinusoidal ring  1702 . The splaying projections  1694 ,  1696  spread apart away from the balloon  1692  when the balloon expands and pull together toward the balloon when the balloon contracts. The splaying projections  1694 ,  1696  also attach to an intermediate sinusoidal ring  1704  located on the wire cage  1690  halfway between the base ring  1698  and the top sinusoidal ring  1702 . Because the splaying projections  1694 ,  1696  attach at different points  1695 ,  1697  on the sinusoidal rings, some of the splaying projections  1694  are positioned to contact the balloon  1692 , while the other splaying projections  1696  are positioned away from the balloon  1692  and are instead positioned to contact annular tissue (not shown) during a support ring delivery procedure. The other, outer splaying projections  1696  form an outer edge  1706  of the delivery tool. The configuration provides a gap  1708  between the balloon  1692  and the outer edge  1706 , and during a delivery procedure, blood can flow through the gap  1708  unimpeded by the balloon  1692 . For example, in some implementations of the delivery tool  1668 , the maximum diameter  1710  of the balloon  1692  is 28 millimeters, and the maximum diameter  1712  of the outer edge  1706  of the delivery tool is 35 millimeters. In this example, blood can flow through the gap  1708  at a rate similar to the rate of blood flow through a heart valve having a 21 millimeter flow area. 
       FIGS. 77A and 77B  show another delivery tool  1714 . This delivery tool  1714  has splaying projections  1722 ,  1724  spanning an upper ring  1716  and a base ring  1718  arranged around a shaft  1720 . An annular support ring (not shown) can be placed over the splaying projections  1722 ,  1724  for delivery. The splaying projections  1722 ,  1724  each have a point of attachment  1726  at the upper ring  1716  and another point of attachment  1728  at the base ring  1718 . The splaying projections  1722 ,  1724  spread apart away from the shaft  1720  in an expanded configuration and pull together toward the shaft  1720  in a contracted configuration. The upper ring  1716  and base ring  1718  have slots  1717 ,  1719  allowing the splaying projections  1722 ,  1724  to articulate at the points of attachment  1726 ,  1728 . In a collapsed configuration, as shown in  FIG. 77A , the splaying projections  1722 ,  1724  lie flat against the shaft  1720 . In an expanded configuration for delivering an annular support ring, as shown in  FIG. 77B , the upper ring  1716  slides  1730  down along the shaft  1720  toward the base ring  1718 , causing the splaying projections  1722 ,  1724  to bend at an angle  1732 . The angle  1732  begins at 180 degrees in the collapsed configuration and can decrease to less than 90 degrees in the expanded configuration. For example, in  FIG. 77B , the angle  1732  is about 60 degrees. 
       FIG. 78  shows a support  1760  having a ring of successive diamond sections  1736 ,  1738  touching at side corners  1740 ,  1742 . The bottom corners  1744 ,  1746  of the diamonds bear anchors  1748 ,  1750  all pointing in the same direction  1752  for piercing heart tissue and anchoring the support. The anchors  1748 ,  1750  have sharp free ends  1754 ,  1756  that curve slightly toward the geometric center  1758  of the ring formation. The slight curve of the free ends  1754 ,  1756  resists forces that pull on the support when the anchors  1748 ,  1750  are embedded in annular tissue. In some implementations, the anchors  1748 ,  1750  may have barbs for lodging in tissue, and in some implementations, the anchors  1748 ,  1750  may be replaced by hooks. The anchors  1748 ,  1750  can be attached to the diamond sections  1736 ,  1738  using one of several methods such as welding or bonding, for example, or they could be formed or cut directly from the same material from which the diamond sections  1736 ,  1738  are formed or cut, for example. The diamond sections  1736 ,  1738  and anchors  1748 ,  1750  could all be cut (for example, laser cut) as a single piece from tubing. The support  1760  could be used with any one of several implementations of the delivery tool, for example, the implementations shown in this description. 
     Generally, this support  1760  is similar in structure to a stent. The diamond sections  1736 ,  1738  could be different sizes, and other kinds of polygonal sections could be substituted for the diamond sections  1736 ,  1738 . For example, hexagonal sections or zig-zag-shaped wire sections could be used, or a combination of different shapes and sizes could be used. While diamond sections  1736 ,  1738  may touch at side corners  1740 ,  1742 , other types of polygons may touch at points other than corners. 
     The support  1760  is resilient and can be expanded to a delivery configuration and later will contract to a final configuration. The support can be made of a flexible shape memory material such as Nitinol or a biologically compatible elastomer (or other material) that is configured to contract the support to the final configuration after insertion into tissue. For example, the support may be configured to contract upon a period of exposure to the temperature of the human body. 
       FIGS. 79A through 79C  show one example of a delivery procedure for the support  1760 . As shown in  FIG. 79A , the support  1760  is placed in a collapsed configuration on the delivery head  1762  of a delivery tool  1764 . The support  1760  and delivery head  1762  are covered in a sheath  1766  that can be removed when the delivery head  1762  arrives at a heart valve annulus  1768 . In the collapsed configuration, the diamond sections  1736 ,  1738  are stretched vertically, reducing the diameter of the support  1760 . As shown in  FIG. 79B , splaying projections  1770 ,  1772  attached to the delivery head  1762  push  1774  outward on the support  1760 , expanding the support to a diameter  1776  greater than the diameter  1769  of the heart valve annulus  1768  ( FIG. 79A ). As shown in  FIG. 79C , the support  1760  is lowered onto the heart valve annulus  1768  and the anchors  1748 ,  1750  lodge inside the annular tissue. The delivery head  1762  is collapsed and pulled  1778  away from the support  1760 , upon which the support  1760  contracts  1780 , pulling the heart valve annulus  1768  to a smaller diameter  1782  than its original larger diameter  1769  ( FIG. 79A ). 
     In general, the delivery tool  1764  expands both the support  1760  and the heart valve annulus  1768  to the same diameter and brings the support anchors  1748 ,  1750  into radial alignment with the circumference of the annulus, thereby allowing attachment of the support to the annulus. Release or removal of the delivery tool  1764  allows the support  1760  to collapse to its preferred and predetermined size and retain the heart valve annulus at that size. 
       FIG. 80A  shows an example of a delivery head  1800  that includes of a cluster of pre-formed flexible wires  1802 ,  1804  arranged at equal intervals around a central axis  1801 . In an upper portion  1805  of the delivery head, the upper ends  1803  of the wires  1802 ,  1804  are parallel and arranged around an imaginary cylinder. In a lower portion  1806 , the wires  1802 ,  1804  each have an upper outward bend  1808  and a lower inward bend  1810 , and meet at a junction  1812 . This portion  1806  of the wires defines a pre-formed but flexible basket  1814 . The wires  1802 ,  1804  can be made of any flexible material such as Nitinol or a biologically compatible elastomer (or other material). In some examples, opposite (with respect to the axis  1807 ) pairs of wires are coupled or joined continuously at the junction  1812 , and in some examples the wires all end at the junction and a coupling is used to hold the ends in place. 
     Each of the upper ends of wires  1802 ,  1804  passes through one lumen of a corresponding tube  1816 ,  1818  that has multiple lumens. All of the multiple-lumen tubes  1816 ,  1818  attach to a heart valve support  1820  and hold the heart valve support during delivery to a heart valve annulus of a patient. The multiple-lumen tubes  1816 ,  1818  can be made of any flexible material such as Nitinol or a biologically compatible elastomer (or other material). Because the wires  1802 ,  1804  are held in position around the axis by the junction at the bottom of the basket, the upper free ends of the wires maintain their cylindrical arrangement and keep the ends of the multiple-lumen tubes in which they are held also in the preformed but flexible cylindrical arrangement. The wires  1802  can also be held in position during delivery by a support sheath or other support structure (not shown). 
     When initially deployed into the heart, the support can be located at the upper end of the basket as shown in  FIG. 80A . As shown in  FIG. 80B , for purposes of installing the support onto the annulus, the multiple-lumen tubes  1816 ,  1818  then can be pushed simultaneously  1822  downward which causes the one lumen of each of the multiple-lumen tubes to slip along the corresponding one of the wires  1802 ,  1804  to carry the heart valve support  1820  along the portion of the wires forming the basket  1814 . Because of the shapes of the wires, this motion of the tubes also expands the heart valve support radially with respect to the axis, as the tubes progress to wider portions of the basket. The motion of the tubes and the shape of the basket wires also cause the shape of the support to become reconfigured as shown in  FIG. 80B  relative to  FIG. 80A . If the basket is near to or in contact with a heart valve annulus, the motion of the multiple-lumen tubes  1816 ,  1818  puts the heart valve support  1820  in contact with annular tissue. The pushing action, as well as other actions of the delivery head, can be carried out by an operator using delivery controls (e.g. controls such as a loop  1102  or twist or slide control  1150  as shown in  FIG. 13A ). 
     Although the example shown uses multiple-lumen tubes arranged to surround wires, other structures could be used to bear the heart valve support. For example, the heart valve support could be affixed to a structure that travels along a delivery head along tracks, grooves, rails, or another kind of structure that guides the heart valve support along the delivery head to a heart valve annulus. 
       FIG. 81A  shows a cross-section of the basket  1814  deployed within a heart valve annulus  1824 . The wires  1802 ,  1804  are in contact with the inner peripheral edge of the annular tissue  1826 , holding the delivery head  1800  in place axially and radially within the annulus. The forces applied by the wires against the inner peripheral edge of the annular tissue are small, sufficient to hold the head in place, but typically not enough to significantly deform the existing shape of the peripheral edge. The wires  1802 ,  1804 , because of their flexibility, have slight bends  1825 ,  1827  at the points of contact  1831 ,  1833  with the annular tissue  1826 . In some implementations, the wires may be pre-formed to have bends that correspond to and mate with the topography of the annulus. In other implementations, the wires have the form shown in  FIG. 80A  prior to deployment, and the wires become bent during insertion to a shape such as the one shown in  FIG. 81A . 
     The wires  1802 ,  1804  pass through inner lumens  1834 ,  1836  of the multiple-lumen tubes  1816 ,  1818 . Vertical struts  1838 ,  1840  of the heart valve support  1820 , which are bent at common angles relative to the body of the support, are inserted in outer lumen  1842 ,  1844  of the multiple-lumen tubes  1816 ,  1818 , keeping the heart valve support attached to the tubes as the basket is deployed. 
     In some implementations, the inner lumens  1834 ,  1836  can all slide along the corresponding wires in unison, and the outer lumens  1842 ,  1844  that carry the heart valve support therefore all move in unison. 
     In the view shown in  FIG. 81A , the multiple-lumen tubes  1816 ,  1818  have traveled down along the wires  1802 ,  1804 , and carried the heart valve support  1820 . The motion  1828  of the multiple-lumen tubes  1816 ,  1818  and the heart valve support  1820  has caused anchors  1830 ,  1832  attached to the support to pierce the annular tissue  1826 . The anchors  1830 ,  1832 , are splayed outward in  FIG. 81A  relative to their axial direction in  FIG. 80A , prior to deployment, because of the shapes of the wires. In some implementations, the curvature of the basket  1814  in the area where the anchors  1830 ,  1832  contact the annular tissue  1826  (for example, the inward bend  1810  as shown in  FIG. 80A ) causes curvature of the path of the anchors as the anchors approach the annular tissue. This may cause the anchors  1830 ,  1832  to pierce the annular tissue  1826  along a curved path  1835 , which may provide further resistance to forces upon the anchors after the anchors are seated. 
       FIG. 81B  shows a close-up view of one of the multiple-lumen tubes  1816  near annular tissue  1826 . One of the wires  1802 , at the location of its inward bend  1810 , is shown in contact with the annular tissue  1826 . The outer lumen  1842  for carrying a heart valve support (not shown) travels along the same path  1829  as the inner lumen  1834  through which the wire  1802  passes. As the inner lumen  1834  travels along the path  1829  toward the inward bend  1810  of the wire  1802 , the outer lumen  1842  also travels along the same path  1829  directly to the annular tissue  1826 . When the outer lumen  1842  carries a heart valve support, the heart valve support is also led directly to the annular tissue, in a direction that has a substantial radial component. The radial component can be, for example, more prominent than the axial component of motion. 
       FIG. 82A  shows an example heart valve support  1820 . The heart valve support  1820  has substantially the same annular structure as the heart valve support  1760  as shown in  FIG. 78  including diamond sections  1846 ,  1848  and anchors  1830 ,  1832  extending downward from bottom corners  1850 ,  1852  of the diamond sections. The anchors  1830 ,  1832  also have barbs  1854 ,  1856  for lodging in tissue and resisting removal. The vertical struts  1838 ,  1840  attached to the heart valve support  1820  extend upward from the bottom corners  1850 ,  1852  of the diamond sections  1846 ,  1848 . Also, each of the vertical struts  1838 ,  1840  can be tilted at an angle  1858  toward (or, in some implementations, away) from the imaginary surface defined by the diamond section to which it is attached. The tilting can be done on a temporary basis, for example, by manipulating the vertical struts  1838 ,  1840  with a tool or, if the vertical struts are made of a shape memory material such as Nitinol, applying a stimulus such as heat or electricity. This tilted configuration provides adequate clearance for each of the vertical struts to be inserted into a corresponding lumen of the multiple-lumen tubes while the other lumen can slide along the corresponding wire. This arrangement also allows the support to be temporarily expandable. In some implementations, once the vertical struts  1838 ,  1840  are released from the multi-lumen tubes, the vertical struts spring back to become flush with the diamond sections  1846 ,  1848 . 
       FIG. 82B  shows the vertical struts  1838 ,  1840  positioned flush with the diamond sections  1846 ,  1848 . In this configuration, the free ends of the vertical struts are positioned so if the support has forces on it tending to expand it radially or contract it axially, the forces are resisted by the vertical struts pushing against the rigid corners  1847 ,  1849  of the diamonds. In that way, the struts enable the support to resist contraction. Thus, the vertical struts act as support braces that resist vertical contraction and horizontal expansion of the diamond sections. For example, this configuration of the vertical struts can be used when the heart valve support  1820  has been placed in a static long-term configuration within a heart valve annulus. If the vertical struts  1838 ,  1840  are made of a shape memory material such as Nitinol, the vertical struts can be positioned by an appropriate shape memory stimulus such as heat or electricity. In some examples, the vertical struts  1838 ,  1840  can be manually positioned, for example, by a delivery head. 
       FIG. 82C  shows a configuration of the heart valve support  1820  having horizontal struts  1839 ,  1841 . The horizontal struts  1839 ,  1841 , like the vertical struts, can be oriented flush with the diamond sections  1846 ,  1848  to prevent the diamond sections from horizontally contracting, although this arrangement may not be useful for an annuloplasty support in which contraction of the support may be useful. 
     Other arrangements of struts and other mechanical elements can be provided to achieve similar resistance to radial expansion of the support when in some configurations while permitting radial expansion while in other configurations. 
       FIG. 82D  shows the heart valve support  1820  attached to annular tissue  1826 . The anchors  1830 ,  1832  are embedded within the tissue, and the vertical struts  1838 ,  1840  are flush with the diamond sections  1846 ,  1848  to limit the flexibility of the diamond sections and thereby prevent or reduce expansion of the support and heart valve annulus. 
       FIG. 82E  shows the heart valve support  1820  attached to the multiple-lumen tubes  1816 ,  1818 . The vertical struts  1838 ,  1840  are angled inward from the diamond sections  1846 ,  1848  and inserted inside the respective outer lumens  1842 ,  1844  of the multiple-lumen tubes  1816 ,  1818 . In this example, the ends  1862 ,  1864  of the lumens  1842 ,  1844  containing the vertical struts  1838 ,  1840  extend beyond the ends  1866 ,  1868  of the lumens  1834 ,  1836  into which the flexible wires are inserted. 
       FIG. 82F  is a close-up view of the heart valve support  1820  attached to the multiple-lumen tubes  1816 ,  1818 , showing the vertical struts  1838 ,  1840  inserted inside lumens  1842 ,  1844  of the multiple-lumen tubes  1816 ,  1818 . 
     As shown in  FIG. 83A , the basket  1814  is structured to fill and conform to the annulus  1824 . When the basket is inserted  1882  into the annulus  1824  by pushing, the basket may apply slight pressure  1884  to the annular tissue  1826 , allowing the basket to fit tightly against the annular tissue.  FIG. 83B  shows the basket advanced  1886  further into the annulus  1824 . In reaction to the pressure  1884  applied to the annular tissue  1826 , the annular tissue  1826  also applies pressure  1892  upon the wires  1802 ,  1804 , causing a slight bend  1825 ,  1827  in the wires. In some configurations, the wires  1802 ,  1804  are pre-formed with one or more bends, allowing the basket  1814  to conform to the shape of the annulus and exert minimal pressure  1884 , or none at all. In some configurations, the basket  1814  will take on an irregular (non-cylindrical) shape, and when the basket is used to deliver a heart valve support to the annulus (for example, as seen in  FIG. 80B ), the heart valve support will also take on an irregular shape as it tracks the shape of the wires forming the basket. 
       FIGS. 84A-84C  show the removal of the delivery head  1800  after the heart valve support  1820  has been attached to annular tissue. As shown in  FIG. 84A , a sheath  1874  can be used to cover the basket  1814  and protect the delivery head  1800 . For example, if the delivery head  1800  travels through a patient as part of a catheterization procedure, then the sheath  1874  protects the delivery head during the delivery and removal of the delivery head from the patient. In some implementations, the sheath  1874  is made of a stiff material and the basket  1814  can be pulled  1870  up into the sheath to collapse  1872  the basket. In some implementations, the sheath  1874  can be advanced over the basket  1814  to collapse the basket, or the basket can be collapsed independent of the action of the sheath. As the basket collapses  1872 , the heart valve support  1820  contracts. 
     As shown in  FIG. 84B , the multiple-lumen tubes  1816 ,  1818  are pulled  1876  upward along the wires  1802 ,  1804  to detach them from the vertical struts  1838 ,  1840  of the heart valve support  1820 . In some implementations, the vertical struts  1838 ,  1840  spring into place to line up with the diamond sections  1846 ,  1848  of the heart valve support  1820 . Because the heart valve support  1820  has contracted, the diamond sections have increased in height, allowing the vertical struts to clear the upper corners  1847 ,  1849  of the diamond sections when the vertical struts move into a position flush with the diamond sections, as shown. In some implementations, the sheath  1874  is advanced  1878  over the basket  1814  to cover the basket in preparation for travel away from the annulus for removal from the patient. 
     In some configurations, the heart valve support  1820  is attached to annular tissue having an irregular shape (a shape other than a shape resembling a ring), for example, caused by disease or distortion. The heart valve support  1820  can be shaped to mimic the shape of the annular tissue (for example, during delivery as shown in  FIG. 80B ). When the heart valve support  1820  contracts, the heart valve support re-forms into the shape of a ring, also reconfiguring the distorted or diseased annulus into the shape of a ring. 
       FIG. 84C  shows the basket  1814  fully collapsed and withdrawn into the sheath  1874 . In this view, the delivery head  1800  is prepared for travel, for example, through the body of a patient in configurations in which the delivery head is part of a catheter. 
       FIG. 85A  shows a version of the delivery head  1800   a  in which each of the wires  1802 ,  1804  forming the basket  1814  has another bend  1894  near the junction  1812  that forms (with the other wires) a projection  1896 . In some implementations, the projection  1896  extends into the heart valve farther than a basket  1814  would extend absent the projection. The effect of the projection  1896  extending into the heart valve may allow the basket  1814  to be more firmly seated within the heart valve.  FIG. 85B  shows a cross section of the basket  1814  of the delivery head  1800   a  within a heart valve annulus  1824  (similar to the cross section view shown in  FIG. 81A ). The basket  1814  allows leaflets  1897 ,  1899  of a heart valve (e.g. a mitral valve) to close and the shape of the basket supports the leaflets during closure. 
       FIGS. 86A through 86J  show the heart valve support being delivered by another version of the delivery head  1800   c . As shown in  FIG. 86A , this delivery process uses a guide catheter  1900 , which is a narrow rod that can be used to stabilize the delivery tool within the heart valve annulus  1824 . The guide catheter  1900  is inserted  1902  into the heart  1904  with enough depth to keep the guide catheter centered in place within the annulus. In some implementations, the guide catheter  1900  terminates in a j-wire  1906  which has a round and smooth shape that does not puncture heart tissue. The j-wire may touch the far end of the inner wall of the heart chamber that is downstream of the heart valve, which helps to stabilize the position and orientation of the delivery tool within the valve. 
     As shown in  FIG. 86B , the collapsed delivery head  1800   c  is inserted into the annulus  1824 . The delivery head  1800   c  surrounds the guide catheter  1900  and the guide catheter acts as a rail for centering the delivery head within the annulus. In some implementations, the delivery head  1800   c  has a collar  1908  at the junction  1812  of the wires of the delivery head. In some implementations, the wires are bonded to the collar  1908 . The collar  1908  is a termination point of the wires and surrounds and contacts the guide catheter  1900  with a level of friction that allows the collar to slide up and down the guide catheter. 
     The delivery head  1800   c  is protected by a multiple-tube sheath  1875  that covers the wires. The multiple-tube sheath  1875  also holds the wires in a cylindrical formation until the multiple-tube sheath is retracted and removed. 
       FIG. 86C  shows a close-up of the multiple-tube sheath  1875 , which has integrated tubes  1877 ,  1879 . The individual wires  1802 ,  1804  pass through the tubes  1877 ,  1879  arranged in a ring formation within the multiple-tube sheath  1875 . The multiple-tube sheath  1875  covers the wires  1802 ,  1804  up to the point at which the wires  1802 ,  1804  are bonded to the collar  1908 . 
       FIG. 86D  shows the delivery head  1800   c  unsheathed and the basket  1814  expanded so that the wires  1802 ,  1804  contact annular tissue  1826 , further securing the delivery head within the annulus  1824  and preparing the basket to guide the delivery of the heart valve support. In some implementations, the basket  1814  automatically springs open when the sheath is removed. 
       FIG. 86E  shows the multiple-lumen tubes  1816 ,  1818  pulling  1910  the heart valve support  1820  along the wires  1802 ,  1804  in a direction toward the annulus  1824 . The heart valve support  1820  remains contracted as it approaches the basket  1814 . 
     As shown in  FIG. 86F , as the multiple-lumen tubes  1816 ,  1818  pull the heart valve support  1820  along the wires forming the basket  1814 , the heart valve support expands  1912  and the anchors  1830 ,  1832  splay outward. 
     As shown in  FIG. 86G , the multiple-lumen tubes  1816 ,  1818  continue to pull  1910  the heart valve support  1820  until the anchors  1830 ,  1832  contact the annular tissue  1826  at locations near the points of contact  1831 ,  1833  of the basket  1814  and the annular tissue. 
     As shown in  FIG. 86H , a sheath  1874  is advanced  1914  down the delivery head  1800   c  as the basket  1814  contracts  1916 . In some implementations, the sheath  1874  is made of a sufficiently stiff material to cause the basket  1814  to contract  1916  by applying inward pressure to the wires  1802 ,  1804  as the sheath advances  1914 . The contraction of the basket  1814  causes the heart valve support  1820  (still attached to the delivery head) to contract, pulling the annular tissue  1826  inward. (For comparison, the outline of the un-contracted annular tissue  1826   a  is shown.) 
     As shown in  FIG. 86I  the multiple-lumen tubes  1816 ,  1818  retract  1922  upward away from the basket  1814  to under the sheath  1874 . As the multiple-lumen tubes  1816 ,  1818  retract, they detach from the vertical struts  1838 ,  1840  of the heart valve support  1820 . In some implementations, in response, the vertical struts  1838 ,  1840  automatically spring into alignment with the diamond sections  1846 ,  1848  of the heart valve support  1820  in a position that braces the diamond sections and resists horizontal expansion of the diamond sections. At this point, the heart valve support  1820  is in a configuration that it will remain in for the long term to keep the annulus  1824  in a contracted, repaired state. 
     As shown in  FIG. 86J , after the multiple-lumen tubes have retracted up along the wires into the sheath  1874 , the basket  1814  fully collapses. The entire collapsed basket  1814  can then be pulled  1924  up into the sheath  1874  so that the delivery head  1800   c  is protected by the sheath as it is removed (for example, removed from a patient). The heart valve support  1820  remains attached to the annular tissue  1826  in a long-term configuration. 
     Other implementations are within the scope of the following claims.
Among other things, a tool to attach a support to a heart valve annulus includes a stabilizing body that includes features to stabilize an axial position of the tool relative to the annulus, and an attachment device connected to the stabilizing body, the stabilizing body and the attachment device being movable relative to one another under control from a location remote from the tool. The support may have an expandable tubular body having a plurality of struts, a plurality of tissue anchors extending from distally facing apexes in a distal direction post-deployment, wherein axial distal advance of the implantable annulus support causes the plurality of tissue anchors to axially engage tissue, and the implantable annulus support is self-contractible from a radially enlarged engagement configuration for engaging tissue of the mitral valve annulus, to a reduced, deployed configuration for modifying mitral valve annulus geometry.