Patent Publication Number: US-11647958-B2

Title: Catheter frame pieces used as large single axis sensors

Description:
PRIORITY CLAIM AND COPYRIGHT NOTICE 
     This application is a divisional filed under 35 USC § 121 from U.S. patent application Ser. No. 14/575,678, now allowed. This application also claims the benefit of priority under 35 USC § 120 to U.S. patent application Ser. No. 14/575,678, which prior application is hereby incorporated by reference as if set forth in full. A portion of the disclosure of this patent document contains material that is subject to copyright protection. The copyright owner has no objection to the facsimile reproduction by anyone of the patent document or the patent disclosure, as it appears in the Patent and Trademark Office patent file or records, but otherwise reserves all copyright rights whatsoever. 
    
    
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     This invention relates to apparatus and processes for diagnostic and surgical purposes. More particularly, this invention relates to an intra-body probe having a sensor of electromagnetic fields. 
     2. Description of the Related Art 
     Electrophysiology catheters are commonly-used for mapping electrical activity in the heart. Various electrode designs are known for different purposes. In particular, catheters having basket-shaped electrode arrays are known and described, for example, in U.S. Pat. No. 5,772,590, the disclosure of which is incorporated herein by reference. Such catheters are typically introduced into a patient through a guiding sheath with the electrode array in a folded position within the sheath so that the electrode array does not damage the patient during introduction. Within the heart, the guiding sheath is removed and the electrode array is permitted to expand to be generally basket-shaped. Some basket catheters include an additional mechanism in the form of a wire or the like connected to an appropriate control hand to assist in the expansion and contraction of the electrode array. 
     Such catheters may incorporate magnetic location sensors as described, for example, in U.S. Pat. Nos. 5,558,091, 5,443,489, 5,480,422, 5,546,951, and 5,568,809, and International Publication Nos. WO 95/02995, WO 97/24983, and WO 98/29033, the disclosures of which are incorporated herein by reference. Such electromagnetic mapping sensors typically have a length of from about 3 mm to about 7 mm. 
     SUMMARY OF THE INVENTION 
     It is common for multi-electrode catheters to have a wire frame on which electrodes are mounted. Embodiments of the invention provide a framework comprising several loops of wire forming a cage-like structure. When the frame is subjected to an electromagnetic field, each of the loops functions as a single axis magnetic sensor. Moreover, by partitioning the loops into triangles bends in the structure can be reconstructed. A solution, accurate to about a millimeter, can be obtained for each loop&#39;s location. An overall solution for the position of the catheter can be derived from data obtained from the loops. 
     There is provided according to embodiments of the invention a probe adapted for insertion into a heart of a living subject. A framework disposed on the distal end is formed by a plurality of electrically conducting wire loops defining a chamber. The loops are independently connectable to a receiver. There may be six to seven wire loops. 
     According to an aspect of the apparatus, the wire loops form spirals about an axis. 
     According to one aspect of the apparatus, the wire loops are deformable for deployment through a catheter lumen. 
     According to a further aspect of the apparatus, one of the wire loops contacts at least another of the wire loops. 
     There is further provided according to embodiments of the invention a method which is carried out by inserting a probe into a heart of a living subject. A framework disposed on the distal end is formed by a plurality of electrically conducting wire loops defining a chamber. The loops are independently connectable to a receiver. The method is further carried out by modeling the wire loops as respective polygons, subdividing the polygons into a plurality of triangles, exposing the wire loops to magnetic fluxes at respective frequencies, reading signals from the wire loops responsively to the magnetic fluxes at the respective frequencies, computing the theoretical magnetic fluxes in the polygons as respective sums of theoretical magnetic fluxes in the triangles thereof, and determining a location and orientation of the framework by relating the computed theoretical magnetic fluxes to the signals. 
     According to an aspect of the method, the polygons are hexagons. 
     In one aspect of the method subdividing the polygons into a plurality of triangles includes identifying local coordinates of the triangles in a local coordinate system, and transforming the local coordinates of the triangles to coordinates of a magnetic position tracking system. 
     According to another aspect of the method, transforming the local coordinates is performed by optimizing a cost function. 
     According to a further aspect of the method, computing the theoretical magnetic fluxes is based on areas and centroids of the triangles. 
     According to yet another aspect of the method, modeling the wire loops also includes applying a first constraint, wherein segments of the triangles of adjacent polygons are required to intersect. 
     According to still another aspect of the method, modeling the wire loops also includes applying a second constraint, wherein a vertex of each triangle of one polygon coincides with a vertex of an adjacent triangle of the one polygon. 
     According to an additional aspect of the method, modeling the wire loops also includes applying a third constraint, wherein adjacent polygons contact one another at exactly two points. 
    
    
     
       BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS 
       For a better understanding of the present invention, reference is made to the detailed description of the invention, by way of example, which is to be read in conjunction with the following drawings, wherein like elements are given like reference numerals, and wherein: 
         FIG.  1    is a pictorial illustration of a system for evaluating electrical activity in a heart of a living subject in accordance with an embodiment of the invention; 
         FIG.  2    is an elevation of a multi-electrode catheter in accordance with an embodiment of the invention; 
         FIG.  3    illustrates a model of a framework in the catheter shown in  FIG.  2    in accordance with an embodiment of the invention; 
         FIG.  4   , which is a representation of the triangles in the model shown in  FIG.  3    in accordance with an embodiment of the invention; 
         FIG.  5    is a model similar to  FIG.  3    that illustrates the intersection of faces in accordance with an embodiment of the invention; 
         FIG.  6    is a flow chart of a procedure for determining a catheter frame location in accordance with an embodiment of the invention; 
         FIG.  7    is a reconstruction of a catheter framework in accordance with an embodiment of the invention; 
         FIG.  8   , which is a bar chart describing an aspect of the reconstruction of  FIG.  7   ; 
         FIG.  9    is a bar chart describing a simulated reconstruction in accordance with an embodiment of the invention; and 
         FIG.  10    shows a wire framework in accordance with an alternate embodiment of the invention. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     In the following description, numerous specific details are set forth in order to provide a thorough understanding of the various principles of the present invention. It will be apparent to one skilled in the art, however, that not all these details are necessarily needed for practicing the present invention. In this instance, well-known circuits, control logic, and the details of computer program instructions for conventional algorithms and processes have not been shown in detail in order not to obscure the general concepts unnecessarily. 
     Documents incorporated by reference herein are to be considered an integral part of the application except that, to the extent that any terms are defined in these incorporated documents in a manner that conflicts with definitions made explicitly or implicitly in the present specification, only the definitions in the present specification should be considered. 
     Overview 
     Turning now to the drawings, reference is initially made to  FIG.  1   , which is a pictorial illustration of a system  10  for performing ablative procedures on a heart  12  of a living subject, which is constructed and operative in accordance with a disclosed embodiment of the invention. The system comprises a catheter  14 , which is percutaneously inserted by an operator  16  through the patient&#39;s vascular system into a chamber or vascular structure of the heart  12 . The operator  16 , who is typically a physician, brings the catheter&#39;s distal tip  18  into contact with the heart wall, for example, at an ablation target site. Electrical activation maps may be prepared, according to the methods disclosed in U.S. Pat. Nos. 6,226,542, and 6,301,496, and in commonly assigned U.S. Pat. No. 6,892,091, whose disclosures are herein incorporated by reference. One commercial product embodying elements of the system  10  is available as the CARTO® 3 System, available from Biosense Webster, Inc., 3333 Diamond Canyon Road, Diamond Bar, Calif. 91765. This system may be modified by those skilled in the art to embody the principles of the invention described herein. 
     Areas determined to be abnormal, for example by evaluation of the electrical activation maps, can be ablated by application of thermal energy, e.g., by passage of radiofrequency electrical current through wires in the catheter to one or more electrodes at the distal tip  18 , which apply the radiofrequency energy to the myocardium. The energy is absorbed in the tissue, heating it to a point (typically about 50° C.) at which it permanently loses its electrical excitability. When successful, this procedure creates non-conducting lesions in the cardiac tissue, which disrupt the abnormal electrical pathway causing the arrhythmia. The principles of the invention can be applied to different heart chambers to diagnose and treat many different cardiac arrhythmias. 
     The catheter  14  typically comprises a handle  20 , having suitable controls on the handle to enable the operator  16  to steer, position and orient the distal end of the catheter as desired for the ablation. To aid the operator  16 , the distal portion of the catheter  14  contains position sensors (not shown) that provide signals to a processor  22 , located in a console  24 . The processor  22  may fulfill several processing functions as described below. 
     Ablation energy and electrical signals can be conveyed to and from the heart  12  through one or more ablation electrodes  32  located at or near the distal tip  18  via cable  34  to the console  24 . Pacing signals and other control signals may be conveyed from the console  24  through the cable  34  and the electrodes  32  to the heart  12 . Sensing electrodes  33 , also connected to the console  24  are disposed between the ablation electrodes  32  and have connections to the cable  34 . 
     Wire connections  35  link the console  24  with body surface electrodes  30  and other components of a positioning sub-system for measuring location and orientation coordinates of the catheter  14 . The processor  22  or another processor (not shown) may be an element of the positioning subsystem. The electrodes  32  and the body surface electrodes  30  may be used to measure tissue impedance at the ablation site as taught in U.S. Pat. No. 7,536,218, issued to Govari et al., which is herein incorporated by reference. A temperature sensor (not shown), typically a thermocouple or thermistor, may be mounted on or near each of the electrodes  32 . 
     The console  24  typically contains one or more ablation power generators  25 . The catheter  14  may be adapted to conduct ablative energy to the heart using any known ablation technique, e.g., radiofrequency energy, ultrasound energy, and laser-produced light energy. Such methods are disclosed in commonly assigned U.S. Pat. Nos. 6,814,733, 6,997,924, and 7,156,816, which are herein incorporated by reference. 
     In one embodiment, the positioning subsystem comprises a magnetic position tracking arrangement that determines the position and orientation of the catheter  14  by generating magnetic fields in a predefined working volume and sensing these fields at the catheter, using field generating coils  28 . The positioning subsystem U.S. Pat. No. 7,756,576, which is hereby incorporated by reference, and in the above-noted U.S. Pat. No. 7,536,218. 
     As noted above, the catheter  14  is coupled to the console  24 , which enables the operator  16  to observe and regulate the functions of the catheter  14 . Console  24  includes a processor, preferably a computer with appropriate signal processing circuits. The processor is coupled to drive a monitor  29 . The signal processing circuits typically receive, amplify, filter and digitize signals from the catheter  14 , including signals generated by the above-noted sensors and a plurality of location sensing electrodes (not shown) located distally in the catheter  14 . The digitized signals are received and used by the console  24  and the positioning system to compute the position and orientation of the catheter  14  and to analyze the electrical signals from the electrodes. 
     Typically, the system  10  includes other elements, which are not shown in the figures for the sake of simplicity. For example, the system  10  may include an electrocardiogram (ECG) monitor, coupled to receive signals from one or more body surface electrodes, in order to provide an ECG synchronization signal to the console  24 . As mentioned above, the system  10  typically also includes a reference position sensor, either on an externally-applied reference patch attached to the exterior of the subject&#39;s body, or on an internally-placed catheter, which is inserted into the heart  12  maintained in a fixed position relative to the heart  12 . Conventional pumps and lines for circulating liquids through the catheter  14  for cooling the ablation site are provided. The system  10  may receive image data from an external imaging modality, such as an MRI unit or the like and includes image processors that can be incorporated in or invoked by the processor  22  for generating and displaying images that are described below. 
     Reference is now made to  FIG.  2   , which is an elevation of a multi-electrode catheter  37 , in accordance with an embodiment of the invention. A framework  39  is deformable and deployable through a shaft  41 . The framework  39  comprises several closed electrically conducting resilient wire loops  43 , typically at least 6 or 7 loops, as shown in  FIG.  2   , defining a chamber  45 . Each of the loops  43  functions independently as a single-axis magnetic location sensor when subjected to the magnetic field produced by field generating coils  28  ( FIG.  1   ). The loops  43  are electrically insulated from one another. Adjacent loops  43  contact one another and intersect, e.g., at points  47 , or be tangent, e.g., at point  49 . As the framework  39  deforms, the springiness of the structure keeps the loops  43  in contact, and the points of contact can slide along the frame. A larger or smaller number of loops than shown in  FIG.  2    may be provided on the framework  39 , limited by mechanical requirements of size, flexibility and number of electrodes desired. Three to eight loops are practical. The loops  43  in a current embodiment have areas of about 300 mm 2  such that the area bounded by a loop is the same order of magnitude as the area of one of the magnetic sensor coils (number of turns times coil area) in conventional catheters, such as the Navistar® catheter. Areas as low as 50 mm 2  may be useful. 
     The loops  43  experience electromagnetic fields and function as single-axis magnetic sensors. When the loops are subjected to electromagnetic fields at respective frequencies it has been found that the location of each loop can be determined to within 1 mm by combining signals obtained from the loops using the positioning subsystem of the system  10  ( FIG.  1   ). Once the locations of the loops are known, the location of distal end  51  of the catheter can also be determined. 
     As noted above, the loops  43  are formed of wires. Any conducting material can be used. Suitable materials include copper, stainless steel, and nitinol. Materials having shape memory may be advantageous in maintaining contact between the electrodes  53  and the endocardial surface of the heart chamber. The inventors have found in simulations that the average field strength over a large loop is the same as the field at the centroid of the loop. 
     The requisite size of the loops relates inversely to the intensity of the magnetic fields produced by field generating coils  28  ( FIG.  1   ). If the field is too weak, then the size of the loops would become impractical as the framework could not be easily accommodated in a cardiac chamber. On the other hand, if the loops were reduced in size, the required magnetic field strength would increase, in which case the field generating coils  28  and generators  25  would become expensive and might require additional protection for the operator and other personnel involved in the procedure. Sensor sensitivity is a function of the total sensor area. It is desirable that the total sensor area be dimensioned such that the sensors operate in magnetic fields generated by the CARTO system and other magnetic localizing systems. For a coil sensor sensitivity is a function of the area per loop times the number of loops. The formula for sensitivity to flux is 
     
       
         
           
             
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     Electrodes  53  are typically disposed on the loops  43 . While only one electrode is shown on each loop in  FIG.  2   , any number of electrodes may be placed on the loops  43  in order to increase contact between the framework  39  and target tissue, and thereby improve the resolution of the electroanatomic map. The electrodes  53  are linked to the console  24  ( FIG.  1   ) by separate conductors (not shown). 
     Calibration 
     Reference is now made to  FIG.  3   , which shows a model  55  of the framework  39  ( FIG.  2   ) in accordance with an embodiment of the invention, in which the loops of the framework are here represented as hexagons  57 . However, the loops can be approximated by polygons with any number of sides. The polygons need not be regular polygons or even identical, so long as their segments approximately conform to the shapes of the loops. Any such polygon can be partitioned into triangles using an interior point as described below. 
     The borders of the hexagons  57  define respective surfaces  59 . If one of the loops  43  ( FIG.  2   ) lies in a plane, then the integral of the magnetic flux over its surface  59 , denoted as A, divided by the area of the loop is equal to the magnetic field at the centroid of the loop. This implies that a large sensor behaves like a small sensor located at the centroid. Using the spherical harmonic expansion model of the CARTO magnetic field, we verified this connection both analytically and numerically. 
     
       
         
           
             
               
                 
                   
                     
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     An electrical prototype of several catheters was constructed, where the external structure comprised six hexagonal shaped loops as shown in  FIG.  3   . The loops were calibrated using calibration procedures as described in U.S. Pat. Nos. 6,370,411 and 6,266,551, both to Osadchy et al., and herein incorporated by reference. The prototype was then tested using a CARTO system with various position shifts controlled by a robot. The resulting locations and orientations matched the relative location of the coil centroids within the catheter structure and the motion of the catheter as performed by the robot. 
     Simulations 
     It is possible to obtain a more detailed representation of the catheter than provided by the set of locations and orientations of the centroids of the loops  43 . This is achieved by exploiting knowledge of the structure of the framework  39 : specifically the shape of the loops  43  and the nature of intersections between adjacent loops, whether they cross or are tangent. 
     In one configuration which has been simulated, the loops are modeled as hexagons with various relative dimensions. A useful feature of hexagons is that they can be subdivided into triangles. That means even when the hexagon is deformed, and its border is no longer planar, the theoretical magnetic flux can be modeled as the sum of theoretical magnetic fluxes over the triangular segments, which are, by definition, planar. 
       FIG.  3    models a catheter whose framework comprises six hexagons  57 . As shown at the right of the figure, Each hexagon  57  contains six triangles  61 ,  63 ,  65 ,  67 ,  69 ,  71 , which have a common vertex  73 . 
     Reference is now made to  FIG.  4   , which is a representation of the triangles  71  ( FIG.  3   ), in accordance with an embodiment of the invention. The coordinates of each triangle can be defined in its own local coordinate system, Tri ij   local , where each Tr i  is a list of three points. Giving each vertex a label ( 1 ,  2 ,  3 ), the x-axis is the line connecting points  1  and  2 . The origin is the midpoint. The triangle lies in the plane z=0 and point  3  has a positive y-coordinate. We denote the transformations between the local coordinate frames as Rot ij   loc→CARTO  and T ij   loc→CARTO  where Rot refers to a 3×3 rotation matrix and T to a translation vector. The index i runs from 1 to 6 and refers to a particular hexagon, and the index j refers to a triangle within the hexagon. In the CARTO coordinate system Tri ij   CARTO =Rot ij   loc→CARTO ·Tri ij   local +T ij   loc→CARTO . The theoretical magnetic flux through each can be computed using the relation 
                     Flux   i     =       ∑     j   =   1     6     ⁢           ⁢       ∫       (     x   ,   y   ,   z     )     ∈     Tri   ij   CARTO         ⁢           B   CARTO     ⁡     (     x   ,   y   ,   z     )       ·       n   ^       Tri   ij   CARTO         ⁢       dA   ⁡     (     x   ,   y   ,   z     )       .                   Eq   .           ⁢     (   2   )                 
where dA(x,y,z) represents the surface element over the triangle and
 
               n   ^       Tri   ij   CARTO           
is the normal to the triangle, and B CARTO (x,y,z) are the CARTO fields, a set of nine vectors. Each Flux i  is a nine element vector. We can avoid the laborious calculation of surface integrals by applying Equation 1.
 
     Reverting to  FIG.  3   , the flux through each hexagon  57  is now expressed as 
     
       
         
           
             
               
                 
                   
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     The value of the signal from the hexagon is 
     
       
         
           
             
               
                 
                   
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     Continuing to refer to  FIG.  3    and  FIG.  4   , in order to use the signals in the hexagonal loops of the model  55  to determine the location and orientation of the structure, we have to define the problem differently from tracking of a single sensor. The locations and orientations of the triangles  71  are known in their local coordinate systems. The task is to solve for the parameters of the transformations of each triangle to the CARTO coordinate system. For each triangle  71  there are six unknowns, three rotation parameters and three translation parameters. For each hexagonal loop, each of which contains six triangles, we have one flux measurement from each transmitting coil, nine in the particular case of CARTO. So for a catheter containing n conducting hexagonal loops we have 36×n unknowns to be found using n×9 measured input values. To reduce the effective number of unknowns we apply our knowledge of the mechanical structure of the catheter to define constraints on the triangles. 
     Consider the internal structure of each hexagonal loop. Points  1  of all 6 triangles within a hexagon intersect at the vertex  73 .
 
 Tri   i,j   CARTO (1)= Tri   i,k≠j   CARTO (1)  Eq. (5).
 
     In local coordinates Equation 5 becomes
 
 Rot   i,j   loc→CARTO   Tri   i,j   local (1)+ T   i,j   loc→CARTO   =Rot   i,k≠j   loc→CARTO   Tri   i,k≠j   local (1)+ T   i,k≠j   loc→CARTO   Eq. (6).
 
     Another set of constraints on the internal structure is that point  3  of each triangle coincides with point  2  of the adjacent triangle.
 
 Tri   i,{1,2,3,4,5,6}   CARTO (3)= Tri   i,{2,3,4,5,6,1}   CARTO (2)  Eq. (7)
 
     We can also define constraints based on the relative disposition of the hexagonal loops. In the present configuration adjacent frames touch each other at two points, best seen in  FIG.  3    as points  75 ,  77 . Each vertex of the hexagons  57  is labeled with two indices. The first indexes each of the triangles  61 ,  63 ,  65 ,  67 ,  69 ,  71 . The second indexes the vertices ( 1 ,  2 ,  3 ) within each triangle of the hexagons. 
     Reference is now made to  FIG.  5   , which is a diagram similar to  FIG.  3    that illustrates the intersection of faces of model  55  in accordance with an embodiment of the invention. Indices of the hexagons and triangles are shown. Consider the intersections in visible in the front of the diagram. Segment  79  (defined by points ( 3 , 2 ), ( 3 , 3 )) of hexagon  81  on the right intersects segment  83  (defined by points ( 2 , 2 ), ( 2 , 3 )) of hexagon  82 . In like manner, segment  86  (defined by points ( 5 . 2 ), ( 5 , 3 )) of hexagon  81  intersects segment  84  (defined by points ( 6 , 2 ),( 6 , 3 )) of hexagon  82 . 
     A fit was performed that included the following conditions: 
     A vertex of each triangle meets at the center of a hexagon; 
     The other vertices of the triangles meet at points defining the vertices of a hexagon; 
     The measured flux from each hexagon equals the sum of the estimated fluxes through each triangle; and 
     There are wire crossing constraints, i.e., the triangles intersect at points, e.g., the intersections of segments  79 ,  83  and segments  84 ,  86 . 
     There is more than one way to define this constraint. Distances between two skewed lines may be computed using the formula 
               D   =              (       x   3     -     x   1       )     ·     [       (       x   2     -     x   1       )     ×     (       x   4     -     x   3       )       ]                     (       x   2     -     x   1       )     ×     (       x   4     -     x   3       )                ,         
where the differences are all vectors. The vectors x 1  and x 2  are endpoints of one segment, and the vectors x 3  and x 4  are endpoints of the other. This involves the addition of no 1  new parameters. For computational reasons we found it more convenient to define two parameters, v 12  and v 34 , the distance along the segments, and define the constraint as follows
 
 x   1   +v   12 ( x   2   −x   1 )= x   3   +v   34 ( x   4   −x   3 )  Eq. (8).
 
     For calculation purposes each of the terms x i  is expressed in terms of the local triangles and the transformations to the CARTO coordinate system. 
     Given a set of measurements for an array of hexagons defined by vertices or points, either from experiment or simulations, The parameters are found by optimizing a first cost function, wherein. 
     meas i  is the i th  measured signal from the system; and 
     locMeas i  is the location of the i th  point. 
     The value locMeas i  is determined, mutatis mutandis, by the method, disclosed in detail in commonly assigned U.S. Pat. No. 8,818,486, which is herein incorporated by reference. Briefly, the method involves generating a magnetic field in a predefined volume. A reference model is defined, which models the magnetic field at multiple points in the volume using spherical harmonics. The magnetic field is measured by a field detector, which is coupled to an intra-body probe inserted into an organ of a living body located in the volume. A second cost function is defined by comparing the measured magnetic field with the reference magnetic field model within the volume. The cost function is minimized by a computation over dipole terms in a derivative over the cost function so as to find a position and orientation that matches the measured magnetic field. The found position and orientation is outputted as the position and orientation of a probe in the organ. 
     The following constraints are incorporated into the cost function: 
     Centroid Constraint, 
     
       
         
           
             
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                         CARTO 
                       
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     The location of the first point in triangle  1  of each hexagon matches the location found by using the measured signal from the hexagon. 
     Flux Sum Constraint: 
     The term B CARTO (x,y,z) is the estimated field value at a location using a mathematical model such as described in the above-noted U.S. Pat. No. 8,818,486. That implies that the value Signal i  is also a model based value. 
     
       
         
           
             
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                 . 
               
             
           
         
       
     
     Optimally, the estimated flux in each hexagon matches the measured flux. 
     Triangle Points  1  for each hexagon meet: 
     
       
         
           
             
               cost 
               
                 vertex 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 1 
               
             
             = 
             
               
                 ∑ 
                 
                   i 
                   = 
                   1 
                 
                 
                   6 
                   ⁢ 
                   
                     ( 
                     hexagons 
                     ) 
                   
                 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 
                   ∑ 
                   
                     j 
                     = 
                     2 
                   
                   
                     6 
                     ⁢ 
                     
                       ( 
                       triangles 
                       ) 
                     
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                      
                     
                       ( 
                       
                         
                           
                             Tri 
                             
                               i 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               1 
                             
                             CARTO 
                           
                           ⁡ 
                           
                             ( 
                             1 
                             ) 
                           
                         
                         - 
                         
                           
                             Tri 
                             ij 
                             CARTO 
                           
                           ⁡ 
                           
                             ( 
                             1 
                             ) 
                           
                         
                       
                       ) 
                     
                      
                   
                   2 
                 
               
             
           
         
       
     
     Point  3  of each triangle in a hexagon meets point  2  of the next triangle: 
     
       
         
           
             
               cost 
               triangles 
             
             = 
             
               
                 ∑ 
                 
                   i 
                   = 
                   1 
                 
                 6 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 ( 
                 
                   
                     
                       
                         
                           
                              
                             
                               
                                 
                                   Tri 
                                   
                                     i 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                   
                                   CARTO 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   3 
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   Tri 
                                   
                                     i 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                   
                                   CARTO 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   2 
                                   ) 
                                 
                               
                             
                              
                           
                           2 
                         
                         + 
                         
                           
                              
                             
                               
                                 
                                   Tri 
                                   
                                     i 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                   
                                   CARTO 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   3 
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   Tri 
                                   
                                     i 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     3 
                                   
                                   CARTO 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   2 
                                   ) 
                                 
                               
                             
                              
                           
                           2 
                         
                         + 
                       
                     
                   
                   
                     
                       
                         
                           
                              
                             
                               
                                 
                                   Tri 
                                   
                                     i 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     3 
                                   
                                   CARTO 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   3 
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   Tri 
                                   
                                     i 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     4 
                                   
                                   CARTO 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   2 
                                   ) 
                                 
                               
                             
                              
                           
                           2 
                         
                         + 
                         
                           
                              
                             
                               
                                 
                                   Tri 
                                   
                                     i 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     4 
                                   
                                   CARTO 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   3 
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   Tri 
                                   
                                     i 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     5 
                                   
                                   CARTO 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   2 
                                   ) 
                                 
                               
                             
                              
                           
                           2 
                         
                         + 
                       
                     
                   
                   
                     
                       
                         
                           
                             
                                
                               
                                 
                                   
                                     Tri 
                                     
                                       i 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       5 
                                     
                                     CARTO 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     3 
                                     ) 
                                   
                                 
                                 - 
                                 
                                   
                                     Tri 
                                     
                                       i 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       6 
                                     
                                     CARTO 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     2 
                                     ) 
                                   
                                 
                               
                                
                             
                             2 
                           
                           + 
                           
                             
                                
                               
                                 
                                   
                                     Tri 
                                     
                                       i 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       6 
                                     
                                     CARTO 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     3 
                                     ) 
                                   
                                 
                                 - 
                                 
                                   
                                     Tri 
                                     
                                       i 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                     CARTO 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     2 
                                     ) 
                                   
                                 
                               
                                
                             
                             2 
                           
                         
                         ⁢ 
                         
                             
                         
                       
                     
                   
                 
                 ) 
               
             
           
         
       
     
     Adjacent hexagons meet at a point. The following equations show the meeting of triangle  3  from one hexagon with triangle  4  of the adjacent hexagon, e.g., segments  79 ,  83 . 
     
       
         
           
             
               x 
               ⁢ 
               
                   
               
               ⁢ 
               
                 1 
                 i 
               
             
             = 
             
               
                 Tri 
                 
                   i 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   4 
                 
                 CARTO 
               
               ⁡ 
               
                 ( 
                 2 
                 ) 
               
             
           
         
       
       
         
           
             
               x 
               ⁢ 
               
                   
               
               ⁢ 
               
                 2 
                 i 
               
             
             = 
             
               
                 Tri 
                 
                   i 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   4 
                 
                 CARTO 
               
               ⁡ 
               
                 ( 
                 3 
                 ) 
               
             
           
         
       
       
         
           
             
               If 
               ⁢ 
               
                   
               
               ⁢ 
               i 
             
             &lt; 
             6 
           
         
       
       
         
           
             
               x 
               ⁢ 
               
                   
               
               ⁢ 
               
                 3 
                 i 
               
             
             = 
             
               
                 Tri 
                 
                   1 
                   , 
                   3 
                 
                 CARTO 
               
               ⁡ 
               
                 ( 
                 2 
                 ) 
               
             
           
         
       
       
         
           
             
               x 
               ⁢ 
               
                   
               
               ⁢ 
               
                 4 
                 i 
               
             
             = 
             
               
                 Tri 
                 
                   1 
                   , 
                   3 
                 
                 CARTO 
               
               ⁡ 
               
                 ( 
                 3 
                 ) 
               
             
           
         
       
       
         
           
             
               x 
               ⁢ 
               
                   
               
               ⁢ 
               
                 3 
                 6 
               
             
             = 
             
               
                 Tri 
                 
                   1 
                   , 
                   3 
                 
                 CARTO 
               
               ⁡ 
               
                 ( 
                 2 
                 ) 
               
             
           
         
       
       
         
           
             
               x 
               ⁢ 
               
                   
               
               ⁢ 
               
                 4 
                 6 
               
             
             = 
             
               
                 Tri 
                 
                   1 
                   , 
                   3 
                 
                 CARTO 
               
               ⁡ 
               
                 ( 
                 3 
                 ) 
               
             
           
         
       
       
         
           
             
               cost 
               
                 cross 
                 ⁢ 
                 _ 
                 ⁢ 
                 top 
               
             
             = 
             
               
                 
                   ( 
                   
                     
                       ( 
                       
                         
                           x 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             3 
                             i 
                           
                         
                         - 
                         
                           x 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             1 
                             i 
                           
                         
                       
                       ) 
                     
                     · 
                     
                       ( 
                       
                         
                           ( 
                           
                             
                               x 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 2 
                                 i 
                               
                             
                             - 
                             
                               x 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 1 
                                 i 
                               
                             
                           
                           ) 
                         
                         × 
                         
                           ( 
                           
                             
                               x 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 4 
                                 i 
                               
                             
                             - 
                             
                               x 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 3 
                                 i 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                   
                   ) 
                 
                 2 
               
               
                 
                    
                   
                     ( 
                     
                       
                         ( 
                         
                           
                             x 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               2 
                               i 
                             
                           
                           - 
                           
                             x 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               1 
                               i 
                             
                           
                         
                         ) 
                       
                       × 
                       
                         ( 
                         
                           
                             x 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               4 
                               i 
                             
                           
                           - 
                           
                             x 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               3 
                               i 
                             
                           
                         
                         ) 
                       
                     
                     ) 
                   
                    
                 
                 2 
               
             
           
         
       
     
     The following equations show the meeting of triangle  6  from one hexagon with triangle  1  of adjacent hexagon, e.g., segments  84 ,  86  ( 84  and  85  of  FIG.  5   ) 
     
       
         
           
             
               x 
               ⁢ 
               
                   
               
               ⁢ 
               
                 1 
                 i 
               
             
             = 
             
               
                 Tri 
                 
                   i 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   6 
                 
                 CARTO 
               
               ⁡ 
               
                 ( 
                 2 
                 ) 
               
             
           
         
       
       
         
           
             
               x 
               ⁢ 
               
                   
               
               ⁢ 
               
                 2 
                 i 
               
             
             = 
             
               
                 Tri 
                 
                   i 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   6 
                 
                 CARTO 
               
               ⁡ 
               
                 ( 
                 3 
                 ) 
               
             
           
         
       
       
         
           
             
               If 
               ⁢ 
               
                   
               
               ⁢ 
               i 
             
             &lt; 
             6 
           
         
       
       
         
           
             
               x 
               ⁢ 
               
                   
               
               ⁢ 
               
                 3 
                 i 
               
             
             = 
             
               
                 Tri 
                 
                   1 
                   , 
                   1 
                 
                 CARTO 
               
               ⁡ 
               
                 ( 
                 2 
                 ) 
               
             
           
         
       
       
         
           
             
               x 
               ⁢ 
               
                   
               
               ⁢ 
               
                 4 
                 i 
               
             
             = 
             
               
                 Tri 
                 
                   1 
                   , 
                   1 
                 
                 CARTO 
               
               ⁡ 
               
                 ( 
                 3 
                 ) 
               
             
           
         
       
       
         
           
             
               x 
               ⁢ 
               
                   
               
               ⁢ 
               
                 3 
                 6 
               
             
             = 
             
               
                 Tri 
                 
                   1 
                   , 
                   1 
                 
                 CARTO 
               
               ⁡ 
               
                 ( 
                 2 
                 ) 
               
             
           
         
       
       
         
           
             
               x 
               ⁢ 
               
                   
               
               ⁢ 
               
                 4 
                 6 
               
             
             = 
             
               
                 Tri 
                 
                   1 
                   , 
                   1 
                 
                 CARTO 
               
               ⁡ 
               
                 ( 
                 3 
                 ) 
               
             
           
         
       
       
         
           
             
               cost 
               
                 cross 
                 ⁢ 
                 _ 
                 ⁢ 
                 bottom 
               
             
             = 
             
               
                 
                   ( 
                   
                     
                       ( 
                       
                         
                           x 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             3 
                             i 
                           
                         
                         - 
                         
                           x 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             1 
                             i 
                           
                         
                       
                       ) 
                     
                     · 
                     
                       ( 
                       
                         
                           ( 
                           
                             
                               x 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 2 
                                 i 
                               
                             
                             - 
                             
                               x 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 1 
                                 i 
                               
                             
                           
                           ) 
                         
                         × 
                         
                           ( 
                           
                             
                               x 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 4 
                                 i 
                               
                             
                             - 
                             
                               x 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 3 
                                 i 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                   
                   ) 
                 
                 2 
               
               
                 
                    
                   
                     ( 
                     
                       
                         ( 
                         
                           
                             x 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               2 
                               i 
                             
                           
                           - 
                           
                             x 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               1 
                               i 
                             
                           
                         
                         ) 
                       
                       × 
                       
                         ( 
                         
                           
                             x 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               4 
                               i 
                             
                           
                           - 
                           
                             x 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               3 
                               i 
                             
                           
                         
                         ) 
                       
                     
                     ) 
                   
                    
                 
                 2 
               
             
           
         
       
     
     The total cost function to be minimized is
 
cost full   =w 1 cost centroid   +w 2 cost Flux_Hex   +w 3 cost vertex1   +w 4 cost triangles   +w 5 cost cross_top   +w 6 cost cross_bottom  
 
     The variables w1-w6 are relative weighting terms. In the results presented here they are all set to 1. 
     The parameters are found by minimizing the above cost function. In simulations, the optimization is begun with the entire structure displaced and deformed by bending at some of the hexagon joints. The reconstruction accuracy was at the sub-millimeter level. 
     Reference is now made to  FIG.  6   , which is a flow chart summarizing the procedure for determining the catheter location in accordance with an embodiment of the invention. 
     At initial step  85  a catheter framework is modeled as a series of hexagons subdivided into triangles. 
     Next, at step  87  local coordinates of each triangle are defined. 
     Next, at step  89  the local coordinates of the triangles are transformed into CARTO coordinates. The initial parameters of the transform can be a priori values or can be obtained from a previous solution as shown in step  90 . 
     Next, at step  101  theoretical fluxes and signals for the triangles and hexagons are obtained as described above. 
     Step  103  applies measured signals to the solution in step  103  for the parameters of the transformations of each triangle to the CARTO coordinate system by optimizing the cost function described above—applying Equation 4 and Equation 5 to the triangles using their areas and centroids (in CARTO coordinates) to compute the flux in each of the hexagons and then to compute the signal obtained from each of the hexagons. 
     At final step  107  the location of the catheter is reported, using the best values of the transformation parameters and Carto coordinates. 
     Example 
     A fit was performed that included the following conditions:
         a vertex of each triangle meets at the center of a hexagon;   the other vertices of the triangles meet at points defining the vertices of a hexagon;   the measured flux from each hexagon equals the sum of the estimated fluxes through each triangle;   the hexagons are not deformed; and   the wire crossing constraint is met using the formula.       

     
       
         
           
             D 
             = 
             
               
                 
                    
                   
                     
                       ( 
                       
                         
                           x 
                           3 
                         
                         - 
                         
                           x 
                           1 
                         
                       
                       ) 
                     
                     · 
                     
                       [ 
                       
                         
                           ( 
                           
                             
                               x 
                               2 
                             
                             - 
                             
                               x 
                               1 
                             
                           
                           ) 
                         
                         × 
                         
                           ( 
                           
                             
                               x 
                               4 
                             
                             - 
                             
                               x 
                               3 
                             
                           
                           ) 
                         
                       
                       ] 
                     
                   
                    
                 
                 
                    
                   
                     
                       ( 
                       
                         
                           x 
                           2 
                         
                         - 
                         
                           x 
                           1 
                         
                       
                       ) 
                     
                     × 
                     
                       ( 
                       
                         
                           x 
                           4 
                         
                         - 
                         
                           x 
                           3 
                         
                       
                       ) 
                     
                   
                    
                 
               
               . 
             
           
         
       
     
     Reference is now made to  FIG.  7   , which is an exemplary reconstruction of a catheter framework similar to the framework  39  ( FIG.  2   ) using data obtained using a robot, in accordance with an embodiment of the invention and complying with the above-noted conditions. The figure shows an exemplary robot position. The procedure described with respect to  FIG.  6    may be used to compute the signals obtained from hexagons  93 . More generally, the procedure of  FIG.  6    is applicable to models of frameworks containing any number of loops. Reconstructed hexagons  93  are shown in solid lines and hexagons  95  in broken lines. The hexagons  95  represent actual positions of the loops based on knowledge of the robot location and engineered geometry of the catheter 
     Reference is now made to  FIG.  8   , which is a bar chart showing the result of the fit. The chart shows the distribution of errors in the locations of each point on the triangles. This result is for one of 27 robot locations. The other locations give similar results. 
     When the hexagons are allowed to deform by bending at the vertices, which is the case in an actual flexible catheter, simulations show a slight loss of accuracy. Reference is now made to  FIG.  9   , which illustrates a simulated fit in which the hexagons were allowed to deform in accordance with an embodiment of the invention. The vertex location errors tend to be larger than those shown in  FIG.  8   . 
     Alternate Embodiment 
     The construction of a wire framework is not limited to the embodiment shown in  FIG.  2   . The principles of the invention can be applied to other frame shapes and arrangement. Reference is now made to  FIG.  10   , which shows a wire framework  97  deployable through a catheter  99 , in accordance with an alternate embodiment of the invention. Individual wires spiral about an axis, contact one another and form closed loops as in  FIG.  2   . The pitch of the spirals may be identical, but are not necessarily so. Other constructions will occur to those skilled in the art and may be applied as described above. 
     It will be appreciated by persons skilled in the art that the present invention is not limited to what has been particularly shown and described hereinabove. Rather, the scope of the present invention includes both combinations and sub-combinations of the various features described hereinabove, as well as variations and modifications thereof that are not in the prior art, which would occur to persons skilled in the art upon reading the foregoing description.