Patent Publication Number: US-2021192766-A1

Title: System and method for obtaining profile of eye lens capsule

Description:
TECHNICAL FIELD 
     The disclosure relates to a system and method for obtaining a profile of a lens capsule of an eye. 
     BACKGROUND 
     Humans have five basic senses: sight, hearing, smell, taste, and touch. Sight gives us the ability to visualize the world around us and connects us to our surroundings. Many people worldwide have issues with quality of vision and require the use of ophthalmic lenses, such as for example, intraocular lenses. The intraocular lens may be implanted into the eye in a cataract procedure to replace a human lens that has become cloudy. Having a profile of the lens capsule of the eye, prior to the procedure, assists in the selection of the intraocular lens. 
     SUMMARY 
     Disclosed herein is a system having a controller with at least one processor and at least one non-transitory, tangible memory on which instructions are recorded for executing a method for obtaining a profile of a lens capsule of an eye. Execution of the instructions by the processor causes the controller to obtain imaging data for a portion of the lens capsule visible through a pupil of the eye. The imaging data includes posterior datapoints and anterior datapoints and is transformed to an adjusted frame of reference having a first axis (X) and a second axis (Y). Also disclosed is a corresponding method for obtaining a profile of a lens capsule of an eye 
     The profile is represented by respective central surfaces and respective equatorial surfaces separated by respective transition points. The controller is configured to fit the imaging data in the adjusted frame of reference to the respective central surfaces in a predefined central region of the lens capsule. The method includes obtaining a transition coordinate as a coordinate value of the respective transition points in a positive X-domain. The controller is configured to determine a set of fitting parameters for the respective central surfaces and the respective equatorial surfaces based on the transition coordinate and a plurality of constraints. The profile is obtained based on the set of fitting parameters for the respective central surfaces and the respective equatorial surfaces. 
     The controller may be configured to select an intraocular lens based at least partially on the profile of the lens capsule. Transforming the imaging data to the adjusted frame of reference includes fitting the posterior datapoints and the anterior datapoints to a first circle and a second circle, respectively, and determining intersection points of the first circle and the second circle. Transforming the imaging data to the adjusted frame of reference includes transforming the posterior datapoints and the anterior datapoints such that they are centered about a respective center of the intersection points and rotated such that a tilt angle of rotation is zero. 
     The controller is configured to fit the respective central surfaces in the adjusted frame of reference to respective conic equations for a conic surface. The controller is configured to determine a conic x-intercept as a coordinate on the first axis (X) where the respective conic equations intersect in a positive-x domain, the transition coordinate being a product of the conic x-intercept and a predefined constant, the predefined constant being less than 1. 
     The respective central surfaces include a central anterior surface. The controller may be configured to represent the central anterior surface as an elliptical cone characterized by a first plurality of variables (Ka, Qa, Ra), the first plurality of variables (Ka, Qa, Ra) being obtained by fitting the imaging data to the central anterior surface in the predefined central region in the adjusted frame of reference. The central anterior surface Ca(x) may be defined as: 
     
       
         
           
             
               C 
                
               
                 a 
                  
                 
                   ( 
                   x 
                   ) 
                 
               
             
             = 
             
               Ka 
               + 
               
                 
                   
                     
                       - 
                       
                         
                           ( 
                           
                             1 
                             + 
                             
                               Q 
                                
                               a 
                             
                           
                           ) 
                         
                          
                         
                           [ 
                           
                             
                               
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     
                                       Q 
                                        
                                       a 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                                
                               
                                 x 
                                 2 
                               
                             
                             - 
                             
                               R 
                                
                               
                                 a 
                                 2 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                   
                     ( 
                     
                       1 
                       + 
                       
                         Q 
                          
                         a 
                       
                     
                     ) 
                   
                 
                 . 
               
             
           
         
       
     
     The respective central surfaces include a central posterior surface. The controller may be configured to represent the central posterior surface as an elliptical cone characterized by a second plurality of variables (Kp, Qp, Rp), the second plurality of parameters (Kp, Qp, Rp) being obtained by fitting the imaging data to the central posterior surface in the predefined central region in the adjusted frame of reference. The central posterior surface Cp(x) may be defined as: 
     
       
         
           
             
               Cp 
                
               
                 ( 
                 x 
                 ) 
               
             
             = 
             
               Kp 
               + 
               
                 
                   
                     
                       - 
                       
                         
                           ( 
                           
                             1 
                             + 
                             
                               Q 
                                
                               p 
                             
                           
                           ) 
                         
                          
                         
                           [ 
                           
                             
                               
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     
                                       Q 
                                        
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                                 2 
                               
                                
                               
                                 x 
                                 2 
                               
                             
                             - 
                             
                               R 
                                
                               
                                 p 
                                 2 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                   
                     ( 
                     
                       1 
                       + 
                       
                         Q 
                          
                         p 
                       
                     
                     ) 
                   
                 
                 . 
               
             
           
         
       
     
     The set of fitting parameters include a first anterior parameter (Ga), a second anterior parameter (Pa), a first posterior parameter (Gp), a second posterior parameter (Pp) and respective coordinates (Xe, Ye) of a vertex in the adjusted frame of reference. The respective equatorial surfaces include an equatorial anterior surface and an equatorial posterior surface meeting at the vertex. The equatorial anterior surface and the equatorial posterior surface are represented by respective skewed parabola functions. The equatorial anterior surface is based in part on the first anterior parameter (Ga), the second anterior parameter (Pa) and the respective coordinates (Xe, Ye) of the vertex. The equatorial anterior surface Ea(x) may be defined as: Ea(x)=−[(1−Ga(Xe−x))][2√{square root over (Pa(Xe−x))}−Ye]. The equatorial posterior surface is based in part on the first posterior parameter (Gp), the second posterior parameter (Pp) and the respective coordinates (Xe, Ye) of the vertex. The equatorial posterior surface Ep(x) may be defined as: Ep(x)=+[(1−Gp(Xe−x))][2√{square root over (Pp(Xe−x))}+Ye]. 
     The respective central surfaces include a central posterior surface and a central anterior surface. The respective equatorial surfaces include an equatorial anterior surface and an equatorial posterior surface. The plurality of constraints includes a first equation matching respective values of the central anterior surface and the equatorial anterior surface at the transition coordinate, and a second equation matching the respective values of the central posterior surface and the equatorial posterior surface at the transition coordinate. 
     The plurality of constraints includes a third equation matching respective first derivatives of the central anterior surface and the equatorial anterior surface at the transition coordinate, and a fourth equation matching the respective first derivatives of the central posterior surface and the equatorial posterior surface at the transition coordinate. The plurality of constraints includes a fifth equation matching respective second derivatives of the central anterior surface and the equatorial anterior surface at the transition coordinate, and a sixth equation matching the respective second derivatives of the central posterior surface and the equatorial posterior surface at the transition coordinate. 
     Disclosed herein is a system including a controller having at least one processor and at least one non-transitory, tangible memory on which instructions are recorded for executing a method for obtaining a profile of a lens capsule of an eye. The profile is represented by respective central surfaces and respective equatorial surfaces separated by respective transition points. Execution of the instructions by the processor causes the controller to obtain a lens diameter and at least two variables from a set of variables, the set of variables including a lens thickness, a central anterior apex and a central posterior apex. A transition coordinate is set as a product of the lens diameter and a predefined constant, the predefined constant being less than 0.5. 
     The controller is configured to obtain a first plurality of variables (Ka, Qa, Ra) and a pair of anterior parameters (Ga, Pa) by simultaneously solving a first group of constraints based in part on the transition coordinate. The controller is configured to obtain a second plurality of variables (Kp, Qp, Rp) and a pair of posterior parameters (Gp, Pp) by simultaneously solving a second group of constraints based in part on the transition coordinate. 
     The controller is configured to obtain the profile based on the first plurality of variables (Ka, Qa, Ra), the pair of anterior parameters (Ga, Pa), the second plurality of variables (Kp, Qp, Rp) and the pair of posterior parameter (Gp, Pp). An updated value of the lens diameter is obtained based on the profile, and an updated value of the transition coordinate is obtained based on the updated value of the lens diameter. 
     When a difference between the updated value of the transition coordinate and the transition coordinate is greater than a predefined threshold, the controller is configured to update the first plurality of variables (Ka, Qa, Ra) and the pair of anterior parameters (Ga, Pa) by simultaneously solving the first group of constraints based in part on the updated value of the transition coordinate. When the difference between the updated value of the transition coordinate and the transition coordinate is greater than the predefined threshold, the controller is configured to update the second plurality of variables (Kp, Qp, Rp) and the pair of posterior parameter (Gp, Pp) by simultaneously solving the second group of constraints based in part on the updated value of the transition coordinate. 
     The respective central surfaces include a central anterior surface and a central posterior surface, the respective equatorial surfaces including an equatorial anterior surface and an equatorial posterior surface. The controller is configured to represent the central anterior surface and the central posterior surface as respective elliptical cones characterized by the first plurality of variables (Ka, Qa, Ra) and the second plurality of variables (Kp, Qp, Rp), respectively. The controller is configured to represent the equatorial anterior surface and the equatorial posterior surface as respective skewed parabolas characterized by the pair of anterior parameters (Ga, Pa) and the pair of posterior parameters (Gp, Pp), respectively. 
     The first group of constraints includes: a first equation matching respective values of the central anterior surface and the equatorial anterior surface at the transition coordinate; a second equation matching respective first derivatives of the central anterior surface and the equatorial anterior surface at the transition coordinate; a third equation matching respective second derivatives of the central anterior surface and the equatorial anterior surface at the transition coordinate; and a fourth equation matching a respective coordinate of the central anterior surface to the central anterior apex. 
     The second group of constraints includes: a fifth equation matching the respective values of the central posterior surface and the equatorial posterior surface at the transition coordinate; a sixth equation matching respective first derivatives of the central posterior surface and the equatorial posterior surface at the transition coordinate; a seventh equation matching respective second derivatives of the central posterior surface and the equatorial posterior surface at the transition coordinate; and an eighth equation matching a respective coordinate of the central posterior surface to the central posterior apex. 
     The above features and advantages and other features and advantages of the present disclosure are readily apparent from the following detailed description of the best modes for carrying out the disclosure when taken in connection with the accompanying drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a schematic illustration of a system for obtaining a profile of a lens capsule of an eye, the system having a controller; 
         FIG. 2  is a schematic flowchart for a method executable by the controller of  FIG. 1 , in accordance with a first embodiment; 
         FIG. 3  is a schematic fragmentary example of a sectional image of an eye, the image having posterior datapoints and anterior datapoints of a lens capsule; 
         FIG. 4  is a schematic illustration of the posterior datapoints and anterior datapoints of  FIG. 3 , in an original frame of reference; 
         FIG. 5 . is a schematic illustration of the posterior datapoints and anterior datapoints of  FIG. 4 , after transformation to an adjusted frame of reference; 
         FIG. 6  is a schematic diagram of a profile of the lens capsule obtained by the system of  FIG. 1 , in the adjusted frame of reference; 
         FIG. 7  is a schematic diagram of the profile of the lens capsule shown in  FIG. 6 , in the original frame of reference; and 
         FIG. 8  is a schematic flowchart for a method executable by the controller of  FIG. 1 , in accordance with a second embodiment. 
     
    
    
     DETAILED DESCRIPTION 
     Referring to the drawings, wherein like reference numbers refer to like components,  FIG. 1  schematically illustrates system  10  for obtaining a profile of a lens capsule of an eye. Referring to  FIG. 1 , the system  10  includes a controller C having at least one processor  12  and at least one memory  14  (or non-transitory, tangible computer readable storage medium) on which are recorded instructions for executing one or more methods. Method  100  and Method  300  are shown in and described below with reference to  FIGS. 2 and 8 , respectively. 
     Referring to  FIG. 1 , the system  10  may include a user interface  16  for collecting user data from one or more clinical facilities or electronic medical record units. The system  10  may include a data management unit  18  for storing and/or facilitating transfer of the user data and other functions. The various components of the system  10  may be configured to communicate via a short-range network  20  and/or a long-range network  22 . Referring to  FIG. 1 , the controller C may be in communication with a remote server  24  and/or a cloud unit  26 , which may include one or more servers hosted on the Internet to store, manage, and process data. The cloud unit  26  may be a private or public source of information maintained by an organization, such as for example, a research institute, a company, a university and/or a hospital. 
     Referring to  FIG. 1 , the short-range network  20  may be a bus implemented in various ways, such as for example, a serial communication bus in the form of a local area network. The local area network may include, but is not limited to, a Controller Area Network (CAN), a Controller Area Network with Flexible Data Rate (CAN-FD), Ethernet, blue tooth, WIFI and other forms of data connection. The long-range network  22  may be a Wireless Local Area Network (LAN) which links multiple devices using a wireless distribution method, a Wireless Metropolitan Area Networks (MAN) which connects several wireless LANs or a Wireless Wide Area Network (WAN) which covers large areas such as neighboring towns and cities. Other types of connections may be employed. 
     The controller C may be configured to receive and transmit wireless communication to the remote server  24  through a mobile application  28 , shown in  FIG. 1 . The mobile application  28  may in communication with the controller C via the short-range network  20  such that it has access to the data in the controller C. In one example, the mobile application  28  is physically connected (e.g. wired) to the controller C. In another example, the mobile application  28  is embedded in the controller C. The circuitry and components of a remote server  24  and mobile application  28  (“apps”) available to those skilled in the art may be employed. 
     Referring to  FIG. 1 , the user interface  16  and/or the controller C may be configured to communicate with an imaging device  30 , which may be an optical coherence tomography machine. The imaging device  30  may be an ultrasound machine, a magnetic resonance imaging machine or other imaging device available to those skilled in the art. Additionally, the user interface  16  and/or the controller C may be in communication with a profile output module  32  and a lens selection module  34  for selecting an intraocular lens  36 , as described below. 
     Referring to  FIG. 3 , an example image of an eye E is shown. As described below and referring to  FIG. 3 , the controller C is configured to obtain imaging data for a portion  140  of a lens capsule  142  visible through the pupil  144  of the eye E. The lens capsule  142  has a lens thickness  146  and a lens diameter  148 . Also shown in  FIG. 3  are the cornea  152  and iris  154 . It is understood that  FIG. 3  is not to scale. 
     Referring now to  FIG. 2 , a flowchart of method  100  is shown. Method  100  need not be applied in the specific order recited herein and some blocks may be omitted. Per block  102  of  FIG. 2 , the method  100  includes obtaining imaging data for a portion  140  of the lens capsule  142  visible through the pupil  144  of the eye E. Method  100  enables a prediction of the entire shape of the lens capsule  142 , including the portions of the shape that are obscured by the iris  154 , using the portions  140  of the surface which is visible through the open pupil  144 . The imaging data may be obtained via ultrasound bio-microscopy, optical coherence tomography, magnetic resonance imaging or any other imaging modality available to those skilled in the art. The imaging data may be derived from a single image or from multiple images. The imaging data may be obtained from imaging device  30 . 
     Referring to  FIG. 3 , the imaging data includes posterior datapoints  160  (at a posterior side P) and anterior datapoints  170  (at an anterior side A). The posterior datapoints  160  and anterior datapoints  170  are shown plotted in  FIG. 4  in pixel units (referred to herein as the original frame of reference  150 ), with a horizontal axis Q and a vertical axis R. The image shown in  FIG. 4  (with posterior side P being above the anterior side A) is flipped vertically compared to  FIG. 3  (with anterior side A being above the posterior side P) since the coordinate value in the vertical axis R advances going downwards in  FIG. 3 . 
     Per block  104  of  FIG. 2 , the method  100  includes transforming the posterior datapoints  160  and anterior datapoints  170  from the original frame of reference  150  (shown in  FIG. 4 ) to an adjusted frame of reference  200  (shown in  FIGS. 5 and 6 ). First, referring to  FIG. 4 , the posterior datapoints  160  are fitted to the equation for a first circle  162  at the posterior side P. The anterior datapoints  170  are fitted to the equation for a second circle  172  at the anterior side A. The first circle  162  and the second circle  172  intersect at intersection points  180 ,  182 , shown in  FIG. 4 . 
     Second, referring to  FIG. 4 , the controller C is configured to determine a center  184  between the intersection points  180 ,  182 . The coordinate-transformed points (in the adjusted frame of reference  200  of  FIG. 5 ) are obtained by centering the posterior datapoints  160  and the anterior datapoints  170  about the center  184  and rotating the centered coordinates by an angle exactly opposite to a tilt angle  188 , such that the resulting tilt angle  188  becomes zero. The tilt angle  188  may be obtained as an arctangent of a ratio of the increase in the respective vertical coordinate value divided by an increase in the respective horizontal coordinate value between the intersection points  180 ,  182 . The adjusted frame of reference  200  is shown in  FIGS. 5-6  and has an X axis and a Y axis. 
     Per block  106  of  FIG. 2  and referring to  FIG. 5 , the controller C is configured to fit the posterior datapoints  160  and the anterior datapoints  170  in the adjusted frame of reference  200  to respective central surfaces  210  (see  FIG. 5 ) in a predefined central region  205  of the lens capsule  142 . In one example, the predefined central region  205  is in a range 3-7 mm. In one example, the predefined central region  205  is about 5 mm, corresponding to an average-sized open pupil. The predefined central region  205  may be correlated to the opening diameter of the pupil  144  of a specific patient. 
     In the embodiment shown, the respective central surfaces  210  are elliptical cones. The respective central surfaces  210  may be other types of conic surfaces. It is understood that the form of the respective central surfaces  210  may be varied. Referring to  FIG. 5 , trace  202  and trace  212  show extrapolation of the respective central surfaces  210  at the posterior side P and the anterior side A, respectively. The respective central surfaces  210  are extrapolated to determine a positive conic intersection point  220  and a negative conic intersection point  222 . The controller is configured to determine a conic x-intercept  224  as a coordinate on the X axis where the respective conic equations intersect in a positive-x domain. In other words, the conic x-intercept  224  is the X-coordinate of the positive conic intersection point  220 . 
     Referring to  FIGS. 6-7 , the profile L is represented by respective central surfaces  210  and respective equatorial surfaces  240 . The respective equatorial surfaces  240  include equatorial posterior surfaces  252  and equatorial anterior surfaces  262 . The central anterior surface  260  is flanked by the equatorial anterior surface  262  on either side. The central posterior surface  250  is flanked by the equatorial posterior surface  252  on either side. Referring to  FIG. 6 , the profile L of the lens capsule  142  is symmetric about the X-axis at X=0. 
     Per block  108  of  FIG. 2 , the controller C is configured to obtain a transition coordinate Xt (see  FIG. 6 ) which is the coordinate value of the respective transition points  274  in a positive X-domain. Stated differently, the transition coordinate Xt is the positive coordinate on the X axis for the respective transition points  274 . Referring to  FIG. 6 , the respective transition points  274  in the positive X-domain and the negative X-domain (corresponding to coordinate −Xt) are an equal distance from the X=0 line. The transition coordinate Xt is set as a product of the conic x-intercept  224  (Xi) and a predefined constant F, such that Xt=F*Xi. The predefined constant F is less than 1. In one example, the predefined constant F is within a range of 0.5 to 0.9. In one example, the predefined constant F is set to 0.7. 
     Referring to  FIG. 6 , the central posterior surface  250  and the central anterior surface  260  may be generated from expressions for the conic equations over the range −Xt&lt;x&lt;Xt, where Xt is the transition coordinate. In the example shown, the central anterior surface  260  is represented as an elliptical cone characterized by a first plurality of variables (Ka, Qa, Ra). The central anterior surface  260  or Ca(x) is defined as: 
     
       
         
           
             
               Ca 
                
               
                 ( 
                 x 
                 ) 
               
             
             = 
             
               Ka 
               - 
               
                 
                   
                     
                       - 
                       
                         
                           ( 
                           
                             1 
                             + 
                             Qa 
                           
                           ) 
                         
                          
                         
                           [ 
                           
                             
                               
                                 
                                   ( 
                                   
                                     1 
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                                     Qa 
                                   
                                   ) 
                                 
                                 2 
                               
                                
                               
                                 x 
                                 2 
                               
                             
                             - 
                             
                               Ra 
                               2 
                             
                           
                           ] 
                         
                       
                     
                   
                   
                     ( 
                     
                       1 
                       + 
                       Qa 
                     
                     ) 
                   
                 
                 . 
               
             
           
         
       
     
     The first plurality of variables (Ka, Qa, Ra) are obtained by fitting the anterior datapoints  170  (in the adjusted frame of reference  200 ) in the predefined central region  205 . 
     The central posterior surface  250  is represented as an elliptical cone characterized by a second plurality of variables (Kp, Qp, Rp). The central posterior surface  250  or Cp(x) is defined as: 
     
       
         
           
             
               Cp 
                
               
                 ( 
                 x 
                 ) 
               
             
             = 
             
               Kp 
               + 
               
                 
                   
                     
                       - 
                       
                         
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                             + 
                             
                               Q 
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                               p 
                             
                           
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                                 2 
                               
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                 . 
               
             
           
         
       
     
     The second plurality of parameters (Kp, Qp, Rp) being obtained by fitting the posterior datapoints  160  (in the adjusted frame of reference  200 ) in the predefined central region  205 . 
     Referring to  FIGS. 6-7 , the equatorial posterior surfaces  252  and the equatorial anterior surfaces  262  may be represented by respective skewed parabola functions. The equatorial anterior surfaces  262  and the equatorial posterior surfaces  252  meet at a vertex  270  in the positive X-domain and another vertex  272  in the negative X-domain. 
     Per block  110  of  FIG. 2 , the method  100  includes determining a set of fitting parameters for the respective central surfaces  210  and the respective equatorial surfaces  240 . The set of fitting parameters include a first anterior parameter (Ga), a second anterior parameter (Pa), a first posterior parameter (Gp), a second posterior parameter (Pp) and respective coordinates (Xe, Ye) of the vertex  270  in the positive X-domain in the adjusted frame of reference  200 . 
     The equatorial posterior surface  252  (see  FIGS. 6 and 7 ) is based in part on the first posterior parameter (Gp), the second posterior parameter (Pp) and the respective coordinates (Xe, Ye) of the vertex  270  in the positive X-domain. The equatorial posterior surface  252  or Ep(x) is defined as: 
         Ep ( x )=+[(1− Gp ( Xe−x ))][2√{square root over ( Pp ( Xe−x ))}+ Ye ].
 
     The equatorial anterior surface  262  (see  FIGS. 6 and 7 ) based in part on the first anterior parameter (Ga), the second anterior parameter (Pa) and respective coordinates (Xe, Ye) of the vertex  270 . The equatorial anterior surface  262  or Ea(x) is defined as: 
         Ea ( x )=−[(1− Ga ( Xe−x ))][2√{square root over ( Pa ( Xe−x ))}− Ye ].
 
     The set of fitting parameters are based on the transition coordinate Xt and a plurality of constraints. The plurality of constraints includes first, second, third, fourth, fifth and sixth equations. In the example shown, there are six fitting parameters and six constraint equations. The six constraint equations may be solved numerically, for example, using the MATLAB function fsolve, employing the trust-region algorithm. Other numerical algorithms available to those skilled in the art may be employed. 
     The first equation matches respective values of the central anterior surface  260  and the equatorial anterior surface  262  at the transition coordinate Xt, as follows: Ca(Xt)=Ea(Xt). The second equation matches the respective values of the central posterior surface  250  and the equatorial posterior surface  252  at the transition coordinate Xt, as follows: Cp(Xt)=Ep(Xt). 
     The third equation matches respective first derivatives of the central anterior surface  260  and the equatorial anterior surface  262  at the transition coordinate Xt, as follows: 
     
       
         
           
             
               
                 d 
                  
                 C 
                  
                 
                   a 
                    
                   
                     ( 
                     x 
                     ) 
                   
                 
               
               
                 d 
                  
                 x 
               
             
              
             
               
                 
                   
                      
                     
                       
                           
                       
                        
                       
                         x 
                         = 
                         Xt 
                       
                     
                   
                    
                   
                     = 
                     
                       
                         d 
                          
                         E 
                          
                         
                           a 
                            
                           
                             ( 
                             x 
                             ) 
                           
                         
                       
                       
                         d 
                          
                         x 
                       
                     
                   
                    
                 
                 
                   x 
                   = 
                   
                     X 
                      
                     t 
                   
                 
               
               . 
             
           
         
       
     
     The fourth equation matches the respective first derivatives of the central posterior surface  250  and the equatorial posterior surface  252  at the transition coordinate Xt, as follows: 
     
       
         
           
             
               
                 dCp 
                  
                 
                   ( 
                   x 
                   ) 
                 
               
               
                 d 
                  
                 x 
               
             
              
             
               
                 
                   
                      
                     
                       
                           
                       
                        
                       
                         x 
                         = 
                         Xt 
                       
                     
                   
                    
                   
                     = 
                     
                       
                         dEp 
                          
                         
                           ( 
                           x 
                           ) 
                         
                       
                       
                         d 
                          
                         x 
                       
                     
                   
                    
                 
                 
                   x 
                   = 
                   
                     X 
                      
                     t 
                   
                 
               
               . 
             
           
         
       
     
     The fifth equation matches respective second derivatives of the central anterior surface  260  and the equatorial anterior surface  262  at the transition coordinate Xt, as follows: 
     
       
         
           
             
               
                 
                   d 
                   2 
                 
                  
                 
                   Ca 
                    
                   
                     ( 
                     x 
                     ) 
                   
                 
               
               
                 dx 
                 2 
               
             
              
             
               
                 
                   
                      
                     
                       
                           
                       
                        
                       
                         x 
                         = 
                         Xt 
                       
                     
                   
                    
                   
                     = 
                     
                       
                         
                           d 
                           2 
                         
                          
                         
                           Ea 
                            
                           
                             ( 
                             x 
                             ) 
                           
                         
                       
                       
                         dx 
                         2 
                       
                     
                   
                    
                 
                 
                   x 
                   = 
                   
                     X 
                      
                     t 
                   
                 
               
               . 
             
           
         
       
     
     The sixth equation matches the respective second derivatives of the central posterior surface  250  and the equatorial posterior surface  252  at the transition coordinate Xt, as follows: 
     
       
         
           
             
               
                 
                   d 
                   2 
                 
                  
                 
                   Cp 
                    
                   
                     ( 
                     x 
                     ) 
                   
                 
               
               
                 dx 
                 2 
               
             
              
             
               
                 
                   
                      
                     
                       
                           
                       
                        
                       
                         x 
                         = 
                         Xt 
                       
                     
                   
                    
                   
                     = 
                     
                       
                         
                           d 
                           2 
                         
                          
                         
                           Ep 
                            
                           
                             ( 
                             x 
                             ) 
                           
                         
                       
                       
                         dx 
                         2 
                       
                     
                   
                    
                 
                 
                   x 
                   = 
                   
                     X 
                      
                     t 
                   
                 
               
               . 
             
           
         
       
     
     Per block  112  of  FIG. 2 , the controller C is configured to obtain the profile L based on the set of fitting parameters (obtained in block  110 ) applied to the respective central surfaces  210  and the respective equatorial surfaces  240 .  FIG. 6  shows the profile L after the set of fitting parameters have been obtained. Additionally, per block  112 , the profile L may be re-adjusted to the original frame of reference  150  shown in  FIG. 4 .  FIG. 7  is a schematic diagram of the profile L after the opposite transformation from that of block  104 , including de-centering and re-tilting. In order to provide visualization for clinicians, the profile L in the original frame of reference  150  may be superimposed onto the image of the eye E in  FIG. 3 . 
     Also, per block  112 , the controller C may be configured to select an intraocular lens  36  based at least partially on the profile L of the lens capsule  142 . Obtaining an accurate shape of the lens capsule L optimizes selection of the power of the intraocular lens  36 . This effect is heightened where the intraocular lens  36  is an accommodative lens which may change its shape in response to external forces. In other words, the intraocular lens  36  may react differently to the same accommodative changes mediated by the ciliary muscles, depending on the geometric dimension and shape of the lens capsule  142 . 
     Referring now to  FIG. 8 , a flowchart of the method  300 , executable by the controller C of  FIG. 1 , is shown. Method  300  need not be applied in the specific order recited herein and some blocks may be omitted. Per block  302  of  FIG. 8 , the controller C is configured to obtain a lens diameter  148  (see  FIG. 3 ) and at least two variables from a set of variables. The set of variables includes a lens thickness  146  (see  FIG. 3 ), respective coordinates on the Y-axis of the central anterior apex  264  (see  FIG. 6 ) and the central posterior apex  254  (see  FIG. 6 ). The set of variables may be obtained from the imaging device  30  of  FIG. 1  or any other source. Method  300  enables a prediction of the entire shape of the lens capsule  142  using a handful of parameters. 
     Referring to  FIG. 7 , the method  300  includes representing the profile L with respective central surfaces  210  and respective equatorial surfaces  240  in the adjusted frame of reference  200 , as shown in  FIG. 6 . The respective central surfaces  210  include a central anterior surface  260  and a central posterior surface  250 , represented as elliptical cones characterized by a first plurality of variables (Ka, Qa, Ra) and a second plurality of variables (Kp, Qp, Rp), respectively. The respective equatorial surfaces  240  include an equatorial anterior surface  262  and an equatorial posterior surface  252 , represented as skewed parabolas characterized by a pair of anterior parameters (Ga, Pa) and a pair of posterior parameters (Gp, Pp), respectively. 
     Per block  304  of  FIG. 8 , the controller C is configured to set the transition coordinate Xt (see  FIG. 6 ) as a product of the lens diameter  148  (LD) and a predefined constant J, such that Xt=J*LD. The predefined constant J is less than 0.5. In one example, the predefined constant J is 0.35. The transition coordinate Xt may also be obtained as a product of the respective X-coordinate (Xe) of the vertex  270  in the positive X-domain and the predefined constant J, such that Xt=J*(2*Xe). The lens diameter  148  (LD) may be set as twice the value of the respective X-coordinate (Xe) of the vertex  270 , LD=2*Xe. 
     Per block  306  of  FIG. 8 , the controller C is configured to obtain the first plurality of variables (Ka, Qa, Ra) and the pair of anterior parameters (Ga, Pa) by simultaneously solving a first group of constraints based in part on the transition coordinate Xt. The first group of constraints includes four equations which may be solved numerically, for example, using the MATLAB function fsolve. Since there are five unknowns and four equations, the Levenberg-Marquardt method may be used. Other numerical algorithms available to those skilled in the art may be employed. 
     The first equation matches respective values of the central anterior surface  260  and the equatorial anterior surface  262  at the transition coordinate Xt, as follows: Ca(Xt)=Ea(Xt). The second equation matches respective first derivatives of the central anterior surface  260  and the equatorial anterior surface  262  at the transition coordinate Xt, as follows: 
     
       
         
           
             
               
                 d 
                  
                 C 
                  
                 
                   a 
                    
                   
                     ( 
                     x 
                     ) 
                   
                 
               
               
                 d 
                  
                 x 
               
             
              
             
               
                 
                   
                      
                     
                       
                           
                       
                        
                       
                         x 
                         = 
                         Xt 
                       
                     
                   
                    
                   
                     = 
                     
                       
                         d 
                          
                         E 
                          
                         
                           a 
                            
                           
                             ( 
                             x 
                             ) 
                           
                         
                       
                       
                         d 
                          
                         x 
                       
                     
                   
                    
                 
                 
                   x 
                   = 
                   
                     X 
                      
                     t 
                   
                 
               
               . 
             
           
         
       
     
     The third equation matches respective second derivatives of the central anterior surface  260  and the equatorial anterior surface  262  at the transition coordinate Xt, as follows: 
     
       
         
           
             
               
                 
                   d 
                   2 
                 
                  
                 
                   Ca 
                    
                   
                     ( 
                     x 
                     ) 
                   
                 
               
               
                 dx 
                 2 
               
             
              
             
               
                 
                   
                      
                     
                       
                           
                       
                        
                       
                         x 
                         = 
                         Xt 
                       
                     
                   
                    
                   
                     = 
                     
                       
                         
                           d 
                           2 
                         
                          
                         
                           Ea 
                            
                           
                             ( 
                             x 
                             ) 
                           
                         
                       
                       
                         dx 
                         2 
                       
                     
                   
                    
                 
                 
                   x 
                   = 
                   
                     X 
                      
                     t 
                   
                 
               
               . 
             
           
         
       
     
     The fourth equation matches the respective coordinates (Y coordinate) of the central anterior surface  260  and the central anterior apex  264  (Yac) when the X coordinate is zero, such that Ca(0)=Yac. 
     Per block  308  of  FIG. 8 , the controller C is configured to obtain the second plurality of variables (Kp, Qp, Rp) and the pair of posterior parameter (Gp, Pp) by simultaneously solving a second group of constraints based in part on the transition coordinate Xt. The second group of constraints includes four equations (fifth through eighth equations) which may be solved numerically, for example, using the MATLAB function fsolve. Since there are five unknowns and four equations, the Levenberg-Marquardt method may be used. Other numerical algorithms available to those skilled in the art may be employed. 
     The fifth equation matches the respective values of the central posterior surface  250  and the equatorial posterior surface  252  at the transition coordinate Xt, as follows: Cp(Xt)=Ep(Xt). The sixth equation matches the respective first derivatives of the central posterior surface  250  and the equatorial posterior surface  252  at the transition coordinate Xt, as follows: 
     
       
         
           
             
               
                 dCp 
                  
                 
                   ( 
                   x 
                   ) 
                 
               
               
                 d 
                  
                 x 
               
             
              
             
               
                 
                   
                      
                     
                       
                           
                       
                        
                       
                         x 
                         = 
                         Xt 
                       
                     
                   
                    
                   
                     = 
                     
                       
                         dEp 
                          
                         
                           ( 
                           x 
                           ) 
                         
                       
                       
                         d 
                          
                         x 
                       
                     
                   
                    
                 
                 
                   x 
                   = 
                   
                     X 
                      
                     t 
                   
                 
               
               . 
             
           
         
       
     
     The seventh equation matches the respective second derivatives of the central posterior surface  250  and the equatorial posterior surface  252  at the transition coordinate Xt, as follows: 
     
       
         
           
             
               
                 
                   d 
                   2 
                 
                  
                 
                   Cp 
                    
                   
                     ( 
                     x 
                     ) 
                   
                 
               
               
                 dx 
                 2 
               
             
              
             
               
                 
                   
                      
                     
                       
                           
                       
                        
                       
                         x 
                         = 
                         Xt 
                       
                     
                   
                    
                   
                     = 
                     
                       
                         
                           d 
                           2 
                         
                          
                         
                           Ep 
                            
                           
                             ( 
                             x 
                             ) 
                           
                         
                       
                       
                         dx 
                         2 
                       
                     
                   
                    
                 
                 
                   x 
                   = 
                   
                     X 
                      
                     t 
                   
                 
               
               . 
             
           
         
       
     
     The eighth equation matches the respective coordinates (Y coordinate) of the central posterior surface  250  and the central posterior apex  254  (Ypc) when the X coordinate is zero, such that Cp(0)=Ypc. The central anterior apex  264  (Yac), central posterior apex  254  (Ypc) and the lens thickness  146  are related as follows: [Ypc=Yac+Lens Thickness]. 
     The output of block  306  may be used to obtain the central anterior surface  260  and the equatorial anterior surface  262 . The central anterior surface  260  or Ca(x) is defined as: 
     
       
         
           
             
               Ca 
                
               
                 ( 
                 x 
                 ) 
               
             
             = 
             
               Ka 
               - 
               
                 
                   
                     
                       - 
                       
                         
                           ( 
                           
                             1 
                             + 
                             Qa 
                           
                           ) 
                         
                          
                         
                           [ 
                           
                             
                               
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     Qa 
                                   
                                   ) 
                                 
                                 2 
                               
                                
                               
                                 x 
                                 2 
                               
                             
                             - 
                             
                               Ra 
                               2 
                             
                           
                           ] 
                         
                       
                     
                   
                   
                     ( 
                     
                       1 
                       + 
                       Qa 
                     
                     ) 
                   
                 
                 . 
               
             
           
         
       
     
     The equatorial anterior surface  262  or Ea(x) is defined as: Ea(x)=−[(1−Ga(Xe−x))][2√{square root over (Pa(Xe−x))}−Ye]. 
     The output of block  308  may be used to obtain the central posterior surface  250  and the equatorial posterior surface  252 . The central posterior surface  250  or Cp(x) is defined as: 
     
       
         
           
             
               Cp 
                
               
                 ( 
                 x 
                 ) 
               
             
             = 
             
               Kp 
               + 
               
                 
                   
                     
                       - 
                       
                         
                           ( 
                           
                             1 
                             + 
                             
                               Q 
                                
                               p 
                             
                           
                           ) 
                         
                          
                         
                           [ 
                           
                             
                               
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     
                                       Q 
                                        
                                       p 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                                
                               
                                 x 
                                 2 
                               
                             
                             - 
                             
                               R 
                                
                               
                                 p 
                                 2 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                   
                     ( 
                     
                       1 
                       + 
                       
                         Q 
                          
                         p 
                       
                     
                     ) 
                   
                 
                 . 
               
             
           
         
       
     
     The equatorial posterior surface  252  or Ep(x) is defined as: Ep(x)=+[(1−Gp(Xe−x))][2√{square root over (Pp(Xe−x))}+Ye]. 
     Per block  310  of  FIG. 8 , the controller C is configured to obtain the profile L based on the first plurality of variables (Ka, Qa, Ra), the pair of anterior parameters (Ga, Pa), the second plurality of variables (Kp, Qp, Rp) and the pair of posterior parameter (Gp, Pp). The controller C is configured to obtain an updated value of the lens diameter  148  based on the profile L, and an updated value of the transition coordinate Xt based on the updated value of the lens diameter  148 . 
     Per block  312  of  FIG. 8 , the controller C is configured to determine if a difference between the updated value of the transition coordinate and the transition coordinate is less than a predefined threshold, in other words, if the updated value of the transition coordinate and the transition coordinate converge to within the predefined threshold. If so, the method  300  is ended. The controller C may be configured to select an intraocular lens  36  based at least partially on the profile L of the lens capsule  142 . 
     If not, as shown by line  313 , the method  300  loops back to block  306  and the controller C is configured to update the first plurality of variables (Ka, Qa, Ra) and the pair of anterior parameters (Ga, Pa) by simultaneously solving the first group of constraints based in part on the updated value of the transition coordinate. Additionally, the controller C is configured to update the second plurality of variables (Kp, Qp, Rp) and the pair of posterior parameter (Gp, Pp) by simultaneously solving the second group of constraints based in part on the updated value of the transition coordinate. 
     In summary, the system  10  (via execution of the method  100  and/or method  300 ) enables the prediction of a profile L of the lens capsule  142  with relatively high accuracy while requiring a relatively small number of parameters. The system  10  uses separate parameter values for the anterior side A and the posterior side P of the lens capsule  142 , thus capturing the physiologic asymmetric nature of the shape of the lens capsule  142 , which can be flatter on one side compared to the other. 
     The controller C of  FIG. 1  includes a computer-readable medium (also referred to as a processor-readable medium), including a non-transitory (e.g., tangible) medium that participates in providing data (e.g., instructions) that may be read by a computer (e.g., by a processor of a computer). Such a medium may take many forms, including, but not limited to, non-volatile media and volatile media. Non-volatile media may include, for example, optical or magnetic disks and other persistent memory. Volatile media may include, for example, dynamic random-access memory (DRAM), which may constitute a main memory. Such instructions may be transmitted by one or more transmission media, including coaxial cables, copper wire and fiber optics, including the wires that comprise a system bus coupled to a processor of a computer. Some forms of computer-readable media include, for example, a floppy disk, a flexible disk, hard disk, magnetic tape, other magnetic medium, a CD-ROM, DVD, other optical medium, punch cards, paper tape, other physical medium with patterns of holes, a RAM, a PROM, an EPROM, a FLASH-EEPROM, other memory chip or cartridge, or other medium from which a computer can read. 
     Look-up tables, databases, data repositories or other data stores described herein may include various kinds of mechanisms for storing, accessing, and retrieving various kinds of data, including a hierarchical database, a set of files in a file system, an application database in a proprietary format, a relational database management system (RDBMS), etc. Each such data store may be included within a computing device employing a computer operating system such as one of those mentioned above and may be accessed via a network in one or more of a variety of manners. A file system may be accessible from a computer operating system and may include files stored in various formats. An RDBMS may employ the Structured Query Language (SQL) in addition to a language for creating, storing, editing, and executing stored procedures, such as the PL/SQL language mentioned above. 
     The detailed description and the drawings or FIGS. are supportive and descriptive of the disclosure, but the scope of the disclosure is defined solely by the claims. While some of the best modes and other embodiments for carrying out the claimed disclosure have been described in detail, various alternative designs and embodiments exist for practicing the disclosure defined in the appended claims. Furthermore, the embodiments shown in the drawings or the characteristics of various embodiments mentioned in the present description are not necessarily to be understood as embodiments independent of each other. Rather, it is possible that each of the characteristics described in one of the examples of an embodiment can be combined with one or a plurality of other desired characteristics from other embodiments, resulting in other embodiments not described in words or by reference to the drawings. Accordingly, such other embodiments fall within the adjusted framework of the scope of the appended claims.