Patent Publication Number: US-11393593-B2

Title: Biological simulation apparatus and biological simulation apparatus control method

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application is a continuation application of International Application PCT/JP2014/067832 filed on Jul. 3, 2014 which designated the U.S., the entire contents of which are incorporated herein by reference. 
    
    
     FIELD 
     The embodiments discussed herein relate to a biological simulation apparatus and a biological simulation apparatus control method. 
     BACKGROUND 
     Advances in computational mechanics using computers are being applied not only to the industrial field but also to biomechanics. For example, a simulation of the blood flow in a cerebral aneurysm and a simulation of the blood flow after the Fontan procedure on a heart have recently been performed. As more advanced applications, fluid-structure interaction simulations such as an interaction simulation of heart pulsation and blood flow and an interaction simulation of heart valves and blood flow have been performed. A fluid-structure interaction simulation is a numerical simulation in which a technique of analyzing fluid and a technique of analyzing a structure are combined with each other. This numerical simulation is performed to analyze a dynamic problem in which the fluid and the structure motion while interacting with each other. 
     Examples of a technique for the interaction simulation of heart pulsation and blood flow include the following two methods. In one method, an ALE (Arbitrary Lagrangian Eulerian) method is applied to the fluid analysis, and interaction of the fluid and the structure is performed. In the other method, the interaction of the fluid and the structure is performed by using the Lagrange multiplier method. 
     The fluid-structure interaction analysis technique based on the ALE method tracks the interface between the blood as the fluid and the myocardial wall as the structure, namely, between the blood and the myocardial wall. In addition, this technique solves a motion equation of the blood described on the ALE fluid mesh that arbitrarily motions and deforms inside the fluid domain while maintaining consistency with the structure mesh that motions and deforms with material points based on the Lagrange description method on the interface. This technique is a highly accurate and stable technique. However, in the technique, there may be a limit of deformation which ensures the soundness of the mesh. Thus, it is difficult to track a structure such as a heart valve that undergoes very large deformation. The interaction analysis technique based on the ALE method belongs to a technique generally called an interface tracking method. The Lagrange multiplier method is a calculation technique for obtaining extreme values of a function with a constraint condition. 
     The fluid-structure interaction analysis method using the Lagrange multiplier method is an approximate solution method in which motion equations of the structure and the fluid are combined by using incompressibility of the fluid domain near a structural interface defined suitably by a delta function as a constraint condition. While the accuracy of the fluid-structure interaction analysis method using the Lagrange multiplier method is less than that of the ALE method used in the interaction with the heart wall, since the fluid mesh does not need to be matched with the structural interface, there is no risk of mesh failure. The fluid-structure interaction analysis method using the Lagrange multiplier method belongs to a technique generally called an interface capturing method. In addition, a mesh in the spatially-fixed Euler description method has conventionally been used as a fluid mesh in the interface capturing method. 
     See, for example, the following documents:
     Japanese Laid-open Patent Publication No. 2006-204463;   Japanese Laid-open Patent Publication No. 2011-248878;   M. A. Castro, C. M. Putman, J. R. Cebral, “Computational fluid dynamics modeling of intracranial aneurysms: effects of parent artery segmentation on intra-aneurysmal hemodynamics”, American Journal of Neuroradiology September 2006 27: 1703-1709;   Elaine Tang, Maria Restrepo, Christopher M. Haggerty, Lucia Mirabella, James Bethel, Kevin K. Whitehead, Mark A. Fogel, Ajit P. Yoganathan, “Geometric Characterization of Patient-Specific Total Cavopulmonary Connections and its Relationship to Hemodynamics”, JACC Volume 7, Issue 3, March 2014, pp. 215-224;   H. Watanabe, S. Sugiura, H. Kafuku, T. Hisada, “Multi-physics simulation of left ventricular filling dynamics using fluid-structure interaction finite element method”, Biophysical Journal, Vol. 87, September 2004, pp. 2074-2085;   J. De Hart, G. W. M. Peters, P. J. G. Schreurs, F. P. T. Baaijens, “A three-dimensional computational analysis of fluid-structure interaction in the aortic valve”, Journal of biomechanics, Vol. 36, Issue 1, January 2003, pp. 103-112;   Qun Zhang, Toshiaki Hisada, “Analysis of fluid-structure interaction problems with structural buckling and large domain changes by ALE finite element method”, Computer Methods in Applied Mechanics and Engineering, Volume 190, Issue 48, 28 Sep. 2001, Pages 6341-6357; and   Toshiaki Hisada, Takumi Washio, “Mathematical Considerations for Fluid-structure Interaction Simulation of Heart Valves”, The Japan Society for Industrial and Applied Mathematics, Vol. 16, No. 2, 27 Jun. 2006, pp. 36-50.   

     A heart simulation generally includes an electrical excitation propagation simulation and a mechanical pulsation simulation. Associating these two simulations with each other is important in performing a more realistic heart simulation on a biological body. In addition, collectively handling the pulsation of the heart wall and the blood flow therein without a contradiction in the mechanical pulsation simulation is important in not only accurately evaluating mechanical responses of the two but also handling clinically necessary indices (blood pressure, blood flow, etc.). In addition, motions of heart valves are often problems in actual heart diseases. Thus, to perform a more realistic heart simulation, it is appropriate to collectively handle the pulsation of the heart wall, the luminal blood flow, and the heart valves without a contradiction in the pulsation simulation. In addition, if a simulation of the entire heart including its valves can be performed highly accurately, the simulation can be used more widely. 
     As a method of stably and highly accurately performing the fluid-structure interaction analysis between the heart wall and the luminal blood, there is an interface-tracking fluid-structure interaction simulation technique in which the ALE method is used for the blood domain inside the heart. With this technique, a fluid equation is described from an ALE mesh that motions and deforms. However, it is difficult to track the interface of a structure such as a heart valve that undergoes very large and complex deformation. Thus, the ALE method cannot be applied. On the other hand, while the interface-capturing fluid-structure interaction simulation using the Lagrange multiplier method is inferior in stability and analysis accuracy, interaction analysis between a structure that undergoes large and complex deformation and blood can be performed. However, the conventional interface-capturing analysis methods that have been developed are only for fluid described from the spatially fixed Euler mesh. Namely, no method has been developed yet for fluid described from a mesh that motions and deforms such as an ALE mesh. Thus, analysis of the pulsation of a heart including a valve structure has remained as a very difficult fluid-structure interaction problem. 
     SUMMARY 
     According to one aspect, there is provided a biological simulation apparatus including: a memory configured to hold a geometric model that represents a structure of a biological organ; and a processor configured to perform a procedure including: representing, among domains in the geometric model, a structure domain in which tissues of the biological organ exist by using a structure mesh model based on a Lagrange description, representing a fluid domain in which fluid inside the biological organ exists by using an ALE (Arbitrary Lagrangian Eulerian) fluid mesh model based on an ALE description method, and performing a fluid-structure interaction simulation that obtains ever-changing equilibrium conditions by deforming the structure mesh model in accordance with a motion of the biological organ along with a progress of the simulation, generating a deformed ALE fluid mesh model by deforming the ALE fluid mesh model in such a manner that no gap is formed on a first interface located between a domain in which a site other than a certain site of the biological organ in the structure domain exists and the fluid domain or no overlap is formed with the structure domain, tracking the first interface by using the deformed ALE fluid mesh model, capturing a position of a second interface located between a domain in which the certain site in the structure domain exists and the fluid domain by using the deformed ALE fluid mesh model as a reference, and simultaneously solving both motions of the biological organ and the fluid therein, including the interaction of the biological organ and the fluid. 
     The object and advantages of the invention will be realized and attained by means of the elements and combinations particularly pointed out in the claims. 
     It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory and are not restrictive of the invention. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG. 1  illustrates a functional configuration example of an apparatus according to a first embodiment; 
         FIG. 2  illustrates examples of fluid-structure interaction analysis based on an interface-tracking analysis method using an ALE mesh when the biological organ is a heart; 
         FIG. 3  illustrates the behavior of a heart and the motion of valves; 
         FIGS. 4A to 4C  illustrate an example of fluid-structure interaction analysis based on an interface-capturing analysis method using the Lagrange multiplier method; 
         FIGS. 5A to 5C  illustrate analysis techniques that are compared with each other; 
         FIGS. 6A and 6B  illustrate coordinate systems for interface tracking and capturing; 
         FIG. 7  illustrates an example of a system configuration according to a second embodiment; 
         FIG. 8  illustrates an example of a hardware configuration of a prognosis prediction system according to the second embodiment; 
         FIG. 9  is a block diagram illustrating functions of the prognosis prediction system; 
         FIG. 10  is a flowchart illustrating an example of a procedure of prognosis prediction processing; 
         FIG. 11  illustrates a scheme for obtaining results of virtual treatments; 
         FIG. 12  illustrates an example of a procedure of generating heart mesh models obtained before a virtual operation; 
         FIG. 13  illustrates simulation processing and parameter adjustment processing; 
         FIG. 14  is a flowchart illustrating an example of a procedure of the simulation; 
         FIG. 15  is a flowchart illustrating an example of a procedure of differentiation and integration calculation processing; 
         FIG. 16  illustrates an example of processing for generating postoperative finite element mesh models; 
         FIG. 17  illustrates an example of how a geometric model changes after large blood vessels are removed; 
         FIGS. 18A and 18B  illustrate examples of visualization of parts of simulation results obtained before and after an operation; 
         FIG. 19  illustrates an example of information stored in a storage unit; 
         FIG. 20  illustrates an example of visualization processing; 
         FIG. 21  illustrates an example of a procedure of the visualization processing; 
         FIG. 22  illustrates an example of displaying operative procedures to be compared; 
         FIG. 23  is a flowchart illustrating an example of a procedure of result comparison processing; 
         FIG. 24  is a sequence diagram illustrating the first half of a procedure of prognosis prediction processing using an interactive system; and 
         FIG. 25  is a sequence diagram illustrating the second half of the procedure of the prognosis prediction processing using the interactive system. 
     
    
    
     DESCRIPTION OF EMBODIMENTS 
     Hereinafter, embodiments will be described below with reference to the accompanying drawings, wherein like reference characters refer to like elements throughout. A plurality of embodiments may be combined with each other without a contradiction. 
     First Embodiment 
       FIG. 1  illustrates a functional configuration example of an apparatus according to a first embodiment. A biological simulation apparatus  10  includes a storage unit  11  and an operation unit  12 . 
     The storage unit  11  holds a geometric model  1  that represents a structure of a biological organ. For example, the biological organ is a human heart. 
     On the basis of the geometric model  1 , the operation unit  12  simultaneously solves both the behavior of the biological organ and the motion of the fluid therein, including the interaction of the biological organ and the fluid, and performs a fluid-structure interaction simulation that obtains ever-changing equilibrium conditions. When performing the fluid-structure interaction simulation, the operation unit  12  generates finite element mesh models on the basis of the geometric model  1 . For example, the operation unit  12  generates a structure mesh model  2  that represents a structure domain in which the tissues of the biological organ exist and an ALE fluid mesh model  3  that represents a fluid domain in which the fluid inside the biological organ exists. For example, among the domains in the geometric model  1 , the operation unit  12  represents the structure domain in which the tissues of the biological organ exist by using the structure mesh model  2  based on the Lagrange description and the fluid domain in which the fluid inside the biological organ exists by using the ALE fluid mesh model based on the ALE description method. Among the interfaces made by the structure domain and the fluid domain, the operation unit  12  determines an interface between a domain in which a site other than a certain site  2   a  of the biological organ exists and the fluid domain to be a first interface  4 . In addition, among the interfaces made by the structure domain and the fluid domain, the operation unit  12  determines an interface between a domain in which the certain site  2   a  of the biological organ exists and the fluid domain to be a second interface  5 . For example, the certain site  2   a  is a site protruding into the domain in which the ALE fluid mesh model  3  exists. For example, the certain site  2   a  is a heart valve. 
     Along with the progress of the fluid-structure interaction simulation, the operation unit  12  deforms the structure mesh model  2  in accordance with the motion of the biological organ. In addition, the operation unit  12  generates a deformed ALE fluid mesh model  3  by deforming the ALE fluid mesh model  3  by using an interface-tracking analysis method in such a manner that the first interface  4  is tracked. More specifically, the ALE fluid mesh model  3  is deformed in such a manner that no gap is formed on the first interface  4  or no overlap is formed with the structure mesh model  2  (the structure domain). Namely, the first interface  4  is tracked by using the deformed ALE fluid mesh model  3 . Movement of nodes inside the deformed ALE fluid mesh model  3  is artificially controlled separately from the motion of the fluid so that the soundness of each of the meshes is maintained. In addition, when performing the fluid-structure interaction simulation, the operation unit  12  captures the position of the second interface  5  using the deformed ALE fluid mesh model  3  as a reference by using an interface-capturing analysis method. For example, on the coordinate space for defining the shape of the ALE fluid mesh model  3 , the coordinates of the second interface  5  are calculated. 
     In this way, the biological simulation apparatus  10  performs a highly accurate fluid-structure interaction simulation on a biological organ having a deformable site that has an interface difficult to track. Namely, a highly accurate simulation is performed by deforming the ALE fluid mesh model  3  in accordance with the interface-tracking analysis method and by tracking the first interface  4  with the deformed the ALE fluid mesh model  3 . In addition, the certain site  2   a  such as a heart valve, which significantly motions and which has an interface that is difficult to track by using the interface-tracking analysis method, is captured on the ALE fluid mesh model  3  by using the interface-capturing analysis method. In this way, the biological simulation apparatus  10  performs a fluid-structure interaction simulation on a biological organ having a site that has an interface difficult to track. For example, the ALE method may be used as the interface-tracking analysis method. In addition, for example, a method based on the Lagrange multiplier method may be used as the interface-capturing analysis method (see, for example, T. Hisada et al., “Mathematical Considerations for Fluid-structure Interaction Simulation of Heart Valves”). 
     The term “virtual operation” used hereinafter indicates a virtual operation that is performed on a biological data in a computer. Namely, a virtual operation is different from an actual operation performed by a doctor. 
     The storage unit  11  may hold a plurality of postoperative geometric models which represent structures of the biological organ that are obtained by performing a plurality of virtual operations using different operative procedures, respectively, on the biological organ. In this case, the operation unit  12  performs a fluid-structure interaction simulation on each of the plurality of postoperative geometric models and displays simulation results of the plurality of virtual operations on a monitor or the like for comparison. In this way, an appropriate operative procedure can accurately be determined before a doctor performs an actual operation. 
     In addition, the operation unit  12  may display possible techniques usable in a virtual operation on a monitor or the like and generate a postoperative geometric model by deforming the geometric model  1  in accordance with a selected technique. For example, the operation unit may generate a postoperative geometric model by deforming the geometric model  1  in accordance with a selected technique. After generating a postoperative geometric model, the operation unit  12  stores the generated postoperative geometric model in the storage unit  11 . Examples of the possible techniques used in a virtual operation include techniques used in a medical virtual operation and techniques used in a surgical virtual operation. The medical virtual operation is an operation performed without cutting open the chest, such as an operation using a catheter. The surgical operation is an operation performed by cutting open the chest, such as coronary circulation bypass surgery. 
     When creating a virtual postoperative geometric model based on a medical virtual operation, for example, the operation unit  12  displays possible techniques usable in the medical virtual operation on a monitor or the like and deforms the geometric model  1  in accordance with a selected technique. In this way, a postoperative geometric model can be generated in accordance with the procedure of the medical virtual operation. Likewise, the operation unit  12  displays possible techniques usable in a surgical operation on a monitor or the like and deforms the geometric model  1  in accordance with a selected technique. In this way, a postoperative geometric model can be generated in accordance with the procedure of the surgical operation. By preparing possible techniques of medical and surgical virtual operations, a postoperative geometric model in accordance with an operation assumed by a doctor can be generated, whether the doctor is a physician or a surgeon. 
     After performing a fluid-structure interaction simulation on each of the geometric models obtained by a plurality of virtual operations, the operation unit  12  can evaluate a simulation result of each of the plurality of virtual operations in view of a predetermined reference and rearrange and display the plurality of operations in view of the evaluation results. In this way, the doctor can easily determine the effectiveness of an individual operative procedure. 
     For example, the operation unit  12  may be realized by a processor of the biological simulation apparatus  10 . In addition, for example, the storage unit may be realized by a memory of the biological simulation apparatus  10 . 
     Second Embodiment 
     A second embodiment provides a prognosis prediction system that predicts prognosis of a biological body (for example, a patient) on the basis of the content of an operation on its heart. This prognosis prediction system is an example of the biological simulation apparatus  10  according to the first embodiment. 
     In the prognosis prediction system according to the second embodiment, a pulsation simulation based on a fluid-structure interaction of the heart wall, blood flow, and heart valves is performed by combining an interface-tracking analysis method and an interface-capturing analysis method with each other without a contradiction. In addition, the pulsation simulation is linked with an electrical excitation propagation simulation. In this way, a heart simulation that can be used clinically sufficiently is performed. As a result, a result from a cardiovascular operation (and a treatment on a heart) can accurately be predicted per biological body. In addition, when there are a plurality of treatment options, the most appropriate treatment can be selected. In addition, the results can be used for explanation to others (doctors and patients) and for medical education. 
     Namely, the heart simulation generally includes an electrical excitation propagation simulation and a mechanical pulsation simulation. Associating these two simulations with each other is important in performing a more realistic heart simulation. In addition, collectively handling the pulsation of the heart wall and the blood flow therein without a contradiction in the mechanical pulsation simulation is important in not only accurately evaluating mechanical responses of the two but also handling clinically necessary indices (blood pressure, blood flow, etc.). In addition, the motions of the heart valves are often problems in actual heart diseases. Thus, to perform a more realistic heart simulation, it is appropriate to collectively handle the pulsation of the heart wall, the luminal blood flow, and the heart valves without a contradiction in the pulsation simulation. Thus, in the prognosis prediction system according to the second embodiment, two kinds of interaction analysis techniques, namely, an interface-tracking analysis method and an interface-capturing analysis method, are combined with each other without a contradiction so that the motion of the entire heart including the valves can suitably be simulated. 
     In addition, in the prognosis prediction system according to the second embodiment, pre-processing is performed in which processing suitable for a heart operation based on the fluid-structure interaction simulation interactively proceeds while receiving a doctor&#39;s approval. In addition, in the prognosis prediction system according to the second embodiment, processing (post-processing) in which simulation results are analyzed and ranked on the basis of evaluation values which a clinician considers important is performed. In this way, the pulsation of a heart having a topologically arbitrary three-dimensional (3D) shape can be comprehensively reproduced. In addition, by simulating a virtual operation on the reproduced heart, modeling, pulsation analysis, and visualization can be performed. 
     Hereinafter, the fluid-structure interaction simulation according to the second embodiment will be described in detail. The fluid-structure interaction simulation to be described below is also an example of the fluid-structure interaction simulation according to the first embodiment. In the first or second embodiment, a simulation using fluid-structure interaction analysis in which the ALE method and the Lagrange multiplier method are combined with each other is performed. 
       FIG. 2  illustrates examples of fluid-structure interaction analysis based on an interface-tracking analysis method using an ALE mesh when the biological organ is a heart.  FIG. 2  illustrates an ALE mesh used in the analysis and blood flow obtained as an analysis result. When an interface of the heart deforms, the ALE mesh also deforms with the deformation of the heart. In addition, the fluid velocity (fluid velocity vectors) and the pressure of the blood are analyzed on the coordinate system in which the deformable ALE mesh is defined. In this way, accurate analysis is performed. 
     However, there is a limit to the deformation of the ALE mesh. Thus, it is difficult to track an interface of a site that significantly deforms such as a heart valve by using a mesh model based on the ALE fluid mesh. 
       FIG. 3  illustrates the behavior of a heart and the motions of valves. The left portion in  FIG. 3  illustrates a heart when the ventricles contract, and the right portion in  FIG. 3  illustrates the heart when the ventricles relax. For example, when the heart muscle of a left ventricle  8  contracts, an aortic valve  7  opens, and the blood in the left ventricle  8  is discharged. In this state, a mitral valve  6  is closed. When the heart muscle of the left ventricle  8  relaxes, the mitral valve  6  opens, and blood flows into the left ventricle  8 . In this state, the aortic valve  7  is closed. In this way, an individual valve opens or closes in accordance with the pulsation of the heart. When the heart motions, an individual valve deforms with the opening and closing operations more significantly than the atria and ventricles deform with the contraction and relaxation of the heart muscles. It is difficult to track such valves that deform significantly by using a mesh model based on the ALE mesh. 
     In contrast, with the interface-capturing interaction analysis technique using the Lagrange multiplier method, since the fluid mesh does not track an interface, the fluid around the heart valves can also be analyzed. 
       FIGS. 4A to 4C  illustrate an example of fluid-structure interaction analysis based on an interface-capturing analysis method using the Lagrange multiplier method.  FIG. 4A  illustrates a structure mesh model of an aortic valve created independently from a fluid mesh model.  FIG. 4B  illustrates a spatially-fixed Euler fluid mesh model representing the fluid domain inside an aorta.  FIG. 4C  illustrates simulation results in which the fluid velocities of the blood are indicated by vectors. Assuming that the wall of the aorta is a rigid body and does not deform, the fluid mesh can be represented by a spatially-fixed Euler mesh. 
     In this way, in conventional fluid-structure interaction analysis based on the Lagrange multiplier method, analysis using a spatially-fixed Euler fluid mesh model has been performed. Thus, fluid-structure interaction analysis can be performed on a site that undergoes extreme deformation such as a valve, and change of the blood flow around the valve over time can be analyzed, for example. However, in the fluid-structure interaction analysis based on the Lagrange multiplier method, since an interface is not tracked, the stability and analysis accuracy of this analysis are less than those of the fluid-structure interaction analysis based on the interface tracking method using the ALE mesh. 
     Thus, in the second embodiment, an interface that can be tracked by the interface-tracking analysis method is tracked by using a deformable ALE mesh, and an interface that is difficult to track such as a valve is captured from the deformable ALE mesh. 
       FIGS. 5A to 5C  illustrate analysis techniques that are compared with each other.  FIG. 5A  illustrates an interface-tracking analysis method using the ALE method.  FIG. 5B  illustrates an analysis method in which an interface is captured from a spatially-fixed Euler mesh.  FIG. 5C  illustrates an analysis method in which an interface is captured from a deformable ALE mesh. 
     A structure mesh model  41  representing a structure domain in which the tissues of a biological organ exist is deformed as the simulation time progresses from t to t+Δt. When the interface-tracking analysis method is used, a ALE fluid mesh model  42  formed by the ALE mesh also deforms as the structure mesh model  41  deforms. However, since a fluid mesh model  43  based on the interface-capturing analysis method is spatially fixed, the fluid mesh model  43  does not deform even when the structure mesh model  41  deforms. 
     In the analysis method in which an interface is captured from a deformable mesh, an interface of the structure mesh model  41  in a trackable range is tracked by using the deformable ALE fluid mesh model  42 . Regarding an untrackable site (for example, a valve site  41   a ), the position of the interface of the corresponding site is calculated on the coordinate system in which the ALE fluid mesh model  42  is defined. In this way, an interface of a site that significantly deforms such as the valve site  41   a  can be captured. 
     Next, interface-tracking and -capturing methods will be described in detail. 
       FIGS. 6A and 6B  illustrate coordinate systems for interface tracking and capturing.  FIGS. 6A and 6B  illustrate an Euler (space) coordinate system (x 1 , x 2 ), a Lagrangian (material) coordinate system (X 1 , X 2 ), and an ALE coordinate system (χ 1 , χ 2 ) that arbitrarily motions and deforms. A myocardial domain Ω S  is represented by the Lagrangian coordinate system. A blood domain Ω f  is represented by the ALE coordinate system. A heart valve  44  is protruding into the blood domain Ω f . While 3D representation is used in reality, two dimensional (2D) representation is used herein for simplicity. 
     An absolute velocity v i  at a material point X, a velocity w i  at the material point X observed from the ALE coordinate system, and a velocity (v i  with “{circumflex over ( )}”) in the ALE coordinate system controlled by an analyst are expressed as equations (1) to (3), respectively. 
     
       
         
           
             
               
                 
                   
                     
                       
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     If an equation of the motion of a continuum body derived from the law of conservation of momentum and the law of conservation of mass are written from the ALE coordinate system, equations (4) are (5) are obtained. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
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     In the above equations, ρ with “{circumflex over ( )}” and Π with “{circumflex over ( )}” indicate the mass density and the 1st Piola-Kirchhiff stress tensor defined by using the reference configuration in the ALE coordinate system as a reference. The second term on the right-hand side of equation (4) represents arbitrary body force. 
     For example, the motion equation of the luminal blood of the heart is described from the ALE coordinate system different from the Langrangian coordinate system used for the myocardial domain as illustrated in  FIG. 1  (See, for example, M. A. Castro et al., “Computational fluid dynamics modeling of intracranial aneurysms: effects of parent artery segmentation on intra-aneurysmal hemodynamics” and E. Tang et al., “Geometric Characterization of Patient-Specific Total Cavopulmonary Connections and its Relationship to Hemodynamics”), and the valve motions in the space of the ALE coordinate system. Thus, to cause the blood and the valve to interact with each other by applying the interface-capturing analysis method based on the Lagrange multiplier method proposed in “Multi-physics simulation of left ventricular filling dynamics using fluid-structure interaction finite element method” by H. Watanabe et al., the ALE coordinate system is basically used as the common coordinate system. Thus, the law of conservation of mass in one-side blood domain Ω K  (K=1, 2) defined by a delta function near a point A on the valve is imposed as expressed by the equation (6). The match between a blood velocity v F  and the velocity components in a normal direction n C  regarding a valve velocity v s  is imposed as expressed by the equation (7), where Γ C  represents an interface. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
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                               ∂ 
                               
                                 x 
                                 i 
                               
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       d 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         Ω 
                         K 
                       
                     
                     = 
                     
                       
                         0 
                         ⁢ 
                         
                           
 
                         
                         ⁢ 
                         K 
                       
                       = 
                       1 
                     
                   
                   , 
                   2 
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       n 
                       C 
                     
                     · 
                     
                       ( 
                       
                         
                           
                             v 
                             F 
                           
                           ⁡ 
                           
                             ( 
                             χ 
                             ) 
                           
                         
                         - 
                         
                           
                             v 
                             S 
                           
                           ⁡ 
                           
                             ( 
                             χ 
                             ) 
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     0 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     on 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       Γ 
                       c 
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     However, since the formulation based on the concept of the reference configuration of the ALE coordinate system complicates the equations, the equations including the motion equation of the blood are converted to be expressed in the Euler coordinate system. Consequently, since equations (4) and (5) are rewritten as equations (8) and (9), equations (6) and (7) are rewritten as equations (10) and (11), respectively. Note that the following relationships in equation (12) are used for the conversion. In equation (8), T ji  represents the Cauchy stress tensor component. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           ρ 
                           ⁢ 
                           
                             
                               ∂ 
                               
                                 v 
                                 i 
                               
                             
                             
                               ∂ 
                               t 
                             
                           
                         
                          
                       
                       χ 
                     
                     + 
                     
                       ρ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         c 
                         j 
                       
                       ⁢ 
                       
                         
                           ∂ 
                           
                             v 
                             i 
                           
                         
                         
                           ∂ 
                           
                             x 
                             j 
                           
                         
                       
                     
                   
                   = 
                   
                     
                       
                         ∂ 
                         
                           T 
                           ji 
                         
                       
                       
                         ∂ 
                         
                           x 
                           j 
                         
                       
                     
                     + 
                     
                       ρ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         g 
                         j 
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             ∂ 
                             ρ 
                           
                           
                             ∂ 
                             t 
                           
                         
                          
                       
                       χ 
                     
                     + 
                     
                       
                         c 
                         i 
                       
                       ⁢ 
                       
                         
                           ∂ 
                           ρ 
                         
                         
                           ∂ 
                           
                             x 
                             i 
                           
                         
                       
                     
                     + 
                     
                       ρ 
                       ⁢ 
                       
                         
                           ∂ 
                           
                             v 
                             i 
                           
                         
                         
                           ∂ 
                           
                             x 
                             i 
                           
                         
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             
                               ∫ 
                               
                                 Ω 
                                 K 
                               
                             
                             ⁢ 
                             
                               
                                 δ 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     x 
                                     - 
                                     
                                       x 
                                       A 
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 
                                   ( 
                                   
                                     
                                       ∂ 
                                       ρ 
                                     
                                     
                                       ∂ 
                                       t 
                                     
                                   
                                    
                                 
                                 x 
                               
                             
                           
                           + 
                           
                             
                               c 
                               i 
                             
                             ⁢ 
                             
                               
                                 ∂ 
                                 ρ 
                               
                               
                                 ∂ 
                                 
                                   x 
                                   i 
                                 
                               
                             
                           
                           + 
                           
                             ρ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 ∂ 
                                 
                                   v 
                                   i 
                                 
                               
                               
                                 ∂ 
                                 
                                   x 
                                   
                                     i 
                                     ⁢ 
                                     
                                         
                                     
                                   
                                 
                               
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       d 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         Ω 
                         K 
                       
                     
                     = 
                     0 
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       K 
                       = 
                       1 
                     
                     , 
                     2 
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       n 
                       C 
                     
                     · 
                     
                       ( 
                       
                         
                           
                             v 
                             F 
                           
                           ⁡ 
                           
                             ( 
                             x 
                             ) 
                           
                         
                         - 
                         
                           
                             v 
                             S 
                           
                           ⁡ 
                           
                             ( 
                             
                               X 
                               ⁡ 
                               
                                 ( 
                                 x 
                                 ) 
                               
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     0 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     on 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       Γ 
                       c 
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
             
               
                 
                   
                     χ 
                     = 
                     x 
                   
                   , 
                   
                     
                       
                         v 
                         ^ 
                       
                       i 
                     
                     = 
                     0 
                   
                   , 
                   
                     
                       v 
                       i 
                     
                     = 
                     
                       w 
                       i 
                     
                   
                   , 
                   
                     
                       c 
                       i 
                     
                     = 
                     
                       
                         
                           v 
                           i 
                         
                         - 
                         
                           
                             v 
                             ^ 
                           
                           i 
                         
                       
                       = 
                       
                         
                           
                             w 
                             j 
                           
                           ⁢ 
                           
                             
                               ∂ 
                               
                                 x 
                                 i 
                               
                             
                             
                               ∂ 
                               
                                 χ 
                                 j 
                               
                             
                           
                         
                         = 
                         
                           w 
                           i 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     In contrast, the motion equation of a heart muscle is expressed as the following equation by using the Lagrangian coordinate system. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           ρ 
                           0 
                         
                         ⁢ 
                         
                           
                             ∂ 
                             v 
                           
                           
                             ∂ 
                             t 
                           
                         
                       
                        
                     
                     X 
                   
                   = 
                   
                     
                       
                         ∇ 
                         X 
                       
                       ⁢ 
                       
                         · 
                         Π 
                       
                     
                     + 
                     
                       
                         ρ 
                         0 
                       
                       ⁢ 
                       
                         g 
                         ~ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     Next, regarding the constitutive equation of the material, the blood is expressed by the following equation as incompressible Newton fluid, wherein μ represents a viscosity coefficient.
 
 T=−pI+ 2μ D   (14)
 
 D= ½(∇ x     v+v     ∇   x )  (15)
 
∇ x   ·v= 0  (16)
 
     Thus, the first and second terms of the law of conservation of mass are eliminated. In addition, if the tangent stiffness of the constitutive equation of the heart muscle described in the Lagrange coordinate system is represented by a fourth-order tensor C, the following equations are established. In the following equations, S represents the 2nd Piola-Kirchhoff stress, F represents the deformation gradient tensor, and E represents the Green-Lagrange strain tensor.
 
Π= S·F   T   (17)
 
 F=x     ∇   x   (18)
 
Π=(det  F ) F   −1   ·T   (19)
 
 S=C:E   (20)
 
 E= ½{∇ x     u+u     ∇   x +(∇ x     u )·( u     ∇   x )}  (21)
 
     Thus, first, by displaying a variational form equation of the Navier-Stokes equation of the blood observed based on the ALE coordinate system in the Euler coordinate system, equation (22) is derived. Next, a variational form equation of the motion equation of the heart wall is derived as equation (23). In addition, by applying the divergence theorem to constraint condition equations (24) and (25) between the blood and the valve, equation (26) is derived. Consequently, the stationary condition equation of the entire system is expressed as equation (27). 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               W 
                               f 
                               ALE 
                             
                           
                           ≡ 
                           
                             
                               ∫ 
                               
                                 Ω 
                                 f 
                               
                             
                             ⁢ 
                             
                               δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 v 
                                 · 
                                 
                                   ρ 
                                   f 
                                 
                               
                               ⁢ 
                               
                                 
                                   ∂ 
                                   v 
                                 
                                 
                                   ∂ 
                                   t 
                                 
                               
                             
                           
                         
                          
                       
                       x 
                     
                     ⁢ 
                     d 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       Ω 
                       f 
                     
                   
                   + 
                   
                     
                       ∫ 
                       
                         Ω 
                         f 
                       
                     
                     ⁢ 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         v 
                         · 
                         
                           
                             ρ 
                             f 
                           
                           ⁡ 
                           
                             ( 
                             
                               v 
                               ⊗ 
                               
                                 ∇ 
                                 x 
                               
                             
                             ) 
                           
                         
                         · 
                         Cd 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         Ω 
                         f 
                       
                     
                   
                   + 
                   
                     2 
                     ⁢ 
                     μ 
                     ⁢ 
                     
                       
                         ∫ 
                         
                           Ω 
                           f 
                         
                       
                       ⁢ 
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         D 
                         ⁢ 
                         
                           : 
                         
                         ⁢ 
                         Dd 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           Ω 
                           f 
                         
                       
                     
                   
                   - 
                   
                     
                       ∫ 
                       
                         Ω 
                         f 
                       
                     
                     ⁢ 
                     
                       
                         ρ 
                         f 
                       
                       ⁢ 
                       
                         
                           ∇ 
                           x 
                         
                         ⁢ 
                         
                           · 
                           δ 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       vd 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         Ω 
                         S 
                       
                     
                   
                   - 
                   
                     
                       ∫ 
                       
                         Ω 
                         f 
                       
                     
                     ⁢ 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         p 
                         ⁡ 
                         
                           ( 
                           
                             
                               ∇ 
                               x 
                             
                             ⁢ 
                             
                               · 
                               v 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       d 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         Ω 
                         f 
                       
                     
                   
                   - 
                   
                     
                       ∫ 
                       
                         Ω 
                         f 
                       
                     
                     ⁢ 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         v 
                         · 
                         
                           ρ 
                           f 
                         
                       
                       ⁢ 
                       gd 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         Ω 
                         f 
                       
                     
                   
                   - 
                   
                     
                       ∫ 
                       
                         Γ 
                         f 
                       
                     
                     ⁢ 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         v 
                         · 
                         
                           τ 
                           _ 
                         
                       
                       ⁢ 
                       d 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         Γ 
                         f 
                       
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
             
               
                 
                   
                     δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       W 
                       S 
                     
                   
                   = 
                   
                     
                       
                         ∫ 
                         
                           Ω 
                           s 
                         
                       
                       ⁢ 
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             u 
                             . 
                           
                           · 
                           
                             ρ 
                             S 
                           
                         
                         ⁢ 
                         
                           u 
                           ¨ 
                         
                         ⁢ 
                         d 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           Ω 
                           S 
                         
                       
                     
                     + 
                     
                       
                         ∫ 
                         
                           Ω 
                           s 
                         
                       
                       ⁢ 
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           E 
                           . 
                         
                         ⁢ 
                         
                           : 
                         
                         ⁢ 
                         Sd 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           Ω 
                           S 
                         
                       
                     
                     - 
                     
                       
                         ∫ 
                         
                           Ω 
                           s 
                         
                       
                       ⁢ 
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             u 
                             . 
                           
                           · 
                           
                             ρ 
                             S 
                           
                         
                         ⁢ 
                         gd 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           Ω 
                           S 
                         
                       
                     
                     - 
                     
                       
                         ∫ 
                         
                           Γ 
                           s 
                         
                       
                       ⁢ 
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             u 
                             . 
                           
                           · 
                           
                             
                               τ 
                               _ 
                             
                             ~ 
                           
                         
                         ⁢ 
                         d 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           Γ 
                           S 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         
                           ∫ 
                           
                             Ω 
                             K 
                           
                         
                         ⁢ 
                         
                           
                             δ 
                             ⁡ 
                             
                               ( 
                               
                                 x 
                                 - 
                                 
                                   x 
                                   A 
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               ∇ 
                               x 
                             
                             ⁢ 
                             
                               · 
                               vd 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             Ω 
                             K 
                           
                         
                       
                       = 
                       0 
                     
                     , 
                     
                       K 
                       = 
                       1 
                     
                     , 
                     2 
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         n 
                         C 
                       
                       · 
                       
                         ( 
                         
                           v 
                           - 
                           
                             u 
                             . 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       0 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       on 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         Γ 
                         C 
                       
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         ∑ 
                         
                           K 
                           = 
                           1 
                         
                         2 
                       
                       ⁢ 
                       
                         
                           
                             ( 
                             
                               - 
                               1 
                             
                             ) 
                           
                           K 
                         
                         ⁢ 
                         
                           1 
                           2 
                         
                         ⁢ 
                         
                           
                             ∫ 
                             
                               Ω 
                               f 
                             
                           
                           ⁢ 
                           
                             ∇ 
                             
                               ⊗ 
                               
                                 
                                   
                                     δ 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         x 
                                         - 
                                         
                                           x 
                                           A 
                                         
                                       
                                       ) 
                                     
                                   
                                   · 
                                   
                                     v 
                                     ⁡ 
                                     
                                       ( 
                                       x 
                                       ) 
                                     
                                   
                                 
                                 ⁢ 
                                 d 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   Ω 
                                   K 
                                 
                               
                             
                           
                         
                       
                     
                     + 
                     
                       
                         ∫ 
                         
                           Γ 
                           c 
                         
                       
                       ⁢ 
                       
                         
                           δ 
                           ⁡ 
                           
                             ( 
                             
                               x 
                               - 
                               
                                 x 
                                 A 
                               
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           
                             n 
                             C 
                           
                           · 
                           
                             
                               u 
                               . 
                             
                             ⁡ 
                             
                               ( 
                               
                                 x 
                                 A 
                               
                               ) 
                             
                           
                         
                         ⁢ 
                         d 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           Γ 
                           C 
                         
                       
                     
                   
                   ≡ 
                   
                     C 
                     ⁡ 
                     
                       ( 
                       
                         
                           x 
                           ⁢ 
                           
                             : 
                           
                           ⁢ 
                           v 
                         
                         , 
                         
                           u 
                           . 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         W 
                         total 
                       
                     
                     ≡ 
                     
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           W 
                           f 
                           ALE 
                         
                       
                       + 
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           W 
                           S 
                           1 
                         
                       
                       + 
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           W 
                           S 
                           2 
                         
                       
                       + 
                       
                         
                           ∫ 
                           
                             Γ 
                             c 
                           
                         
                         ⁢ 
                         
                           
                             λ 
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                           ⁢ 
                           
                             C 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   x 
                                   ⁢ 
                                   
                                     : 
                                   
                                   ⁢ 
                                   δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   v 
                                 
                                 , 
                                 
                                   δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     u 
                                     . 
                                   
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             Γ 
                             C 
                           
                         
                       
                       + 
                       
                         
                           ∫ 
                           
                             Γ 
                             c 
                           
                         
                         ⁢ 
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             λ 
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                           ⁢ 
                           
                             C 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   x 
                                   ⁢ 
                                   
                                     : 
                                   
                                   ⁢ 
                                   v 
                                 
                                 , 
                                 u 
                               
                               ) 
                             
                           
                           ⁢ 
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             Γ 
                             C 
                           
                         
                       
                       - 
                       
                         
                           ∫ 
                           
                             Γ 
                             c 
                           
                         
                         ⁢ 
                         
                           
                             ɛ 
                             λ 
                           
                           ⁢ 
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             λ 
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                           ⁢ 
                           
                             λ 
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                           ⁢ 
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             Γ 
                             C 
                           
                         
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
     In the middle side of equation (27), δW S   1  and δW S   2  represent the first variations of the entire potential regarding the heart muscle and an individual valve. The last term in the middle side is a stabilization term. 
     In addition, connection between a heart valve and a wall, interaction made by contact between heart valves or between a heart valve and a heart wall, and connection with the systemic circulation analogy circuit can also be performed by using the Lagrange multiplier method. In this case, the stationary condition equation of the entire system is expressed as equation (28). 
     
       
         
           
             
               
                 
                   
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         W 
                         total 
                       
                     
                     ≡ 
                     
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           W 
                           f 
                           ALE 
                         
                       
                       + 
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           W 
                           S 
                           1 
                         
                       
                       + 
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           W 
                           
                             S 
                             ⁢ 
                             
                                 
                             
                           
                           2 
                         
                       
                       + 
                       
                         
                           ∫ 
                           
                             Γ 
                             c 
                           
                         
                         ⁢ 
                         
                           
                             λ 
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                           ⁢ 
                           
                             C 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   x 
                                   ⁢ 
                                   
                                     : 
                                   
                                   ⁢ 
                                   δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   v 
                                 
                                 , 
                                 
                                   δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     u 
                                     . 
                                   
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             Γ 
                             C 
                           
                         
                       
                       + 
                       
                         
                           ∫ 
                           
                             Γ 
                             c 
                           
                         
                         ⁢ 
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             λ 
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                           ⁢ 
                           
                             C 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   x 
                                   ⁢ 
                                   
                                     : 
                                   
                                   ⁢ 
                                   v 
                                 
                                 , 
                                 u 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             Γ 
                             C 
                           
                         
                       
                       - 
                       
                         
                           ∫ 
                           
                             Γ 
                             c 
                           
                         
                         ⁢ 
                         
                           
                             ɛ 
                             λ 
                           
                           ⁢ 
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             λ 
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                           ⁢ 
                           
                             λ 
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                           ⁢ 
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             Γ 
                             C 
                           
                         
                       
                       + 
                       
                         
                           ∫ 
                           
                             Γ 
                             L 
                           
                         
                         ⁢ 
                         
                           
                             
                               τ 
                               ⁡ 
                               
                                 ( 
                                 x 
                                 ) 
                               
                             
                             · 
                             
                               ( 
                               
                                 
                                   δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     x 
                                     L 
                                   
                                 
                                 - 
                                 
                                   
                                     W 
                                     L 
                                     W 
                                   
                                   ⁢ 
                                   δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     x 
                                     W 
                                   
                                 
                               
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     Γ L  represents a connection interface between a heart valve and a myocardial wall, χ L , and χ W  represent node coordinates of a heart valve and a myocardial wall, W R   W  represents an interpolation coefficient from a myocardial wall node to a connection point, Γ T,i  represents a surface that could be brought into contact, and C t  is a contact condition equation. In addition, Γ f,k  represents a blood domain interface connected to the systemic circulation analogy circuit, F k  represents a flow volume to the systemic circulation model on a connection surface. In addition, P k  represents the blood pressure of the systemic circulation model on a connection surface, and v k  and Q k  represent the blood pressure variable and the capacity variable in the systemic circulation model, respectively. With these parameters, the equation is solved as a system including the next balance equation of the systemic circulation analogy circuit.
 
 G   k ( P   k   ,F   k   ,V   k   ,Q   k )=0  (29)
 
     The following description will be made on a prognosis prediction system that predicts prognosis by performing finite element discretization and a heart simulation on the basis of the above formulation. A calculation procedure of the simulation will be described in detail with reference to a flowchart ( FIG. 14 ). 
       FIG. 7  illustrates an example of a system configuration according to the second embodiment. A terminal device  31  operated by a doctor  32  is connected to a prognosis prediction system  100  via a network  20 . Information inputted by the doctor  32  to the terminal device  31  is transmitted from the terminal device  31  to the prognosis prediction system  100  via the network  20 . For example, the content of a treatment performed on a biological body is transmitted from the terminal device  31  to the prognosis prediction system  100 . The prognosis prediction system  100  performs pre-processing, a simulation, and post-processing on the basis of the information transmitted from the terminal device  31  or information inputted by an operator  33  that operates the prognosis prediction system  100 . 
     For example, the prognosis prediction system according to the second embodiment may be realized by using a computer having a processor and a memory. 
       FIG. 8  illustrates an example of a hardware configuration of the prognosis prediction system according to the second embodiment. The prognosis prediction system  100  is comprehensively controlled by a processor  101 . The processor  101  is connected to a memory  102  and a plurality of peripheral devices via a bus  109 . The processor  101  may be a multiprocessor. Examples of the processor  101  include a CPU (Central Processing Unit), an MPU (Micro Processing Unit), and a DSP (Digital Signal Processor). At least a part of the functions realized by causing the processor  101  to perform a program may be realized by using an electronic circuit such as an ASIC (Application Specific Integrated Circuit) or a PLD (Programmable Logic Device). 
     The memory  102  is used as a main storage device of the prognosis prediction system  100 . The memory  102  temporarily holds at least a part of a program of an OS (Operating System) or an application program executed by the processor  101 . In addition, the memory  102  holds various types of data needed in processing performed by the processor  101 . For example, a volatile semiconductor memory device such as a RAM (Random Access Memory) is used as the memory  102 . 
     Examples of the peripheral devices connected to the bus  109  include an HDD (Hard Disk Drive)  103 , a graphics processing device  104 , an input interface  105 , an optical drive device  106 , a device connection interface  107 , and a network interface  108 . 
     The HDD  103  magnetically writes and reads data on its internal disk. The HDD  103  is used as an auxiliary storage device of the prognosis prediction system  100 . The HDD  103  holds an OS program, an application program, and various kinds of data. A non-volatile semiconductor memory device such as a flash memory may be used as the auxiliary storage device. 
     The graphics processing device  104  is connected to a monitor  21 . The graphics processing device  104  displays an image on a screen of the monitor  21  in accordance with an instruction from the processor  101 . Examples of the monitor  21  include a CRT (Cathode Ray Tube) display device and a liquid crystal display device. 
     The input interface  105  is connected to a keyboard  22  and a mouse  23 . The input interface  105  transmits a signal transmitted from the keyboard  22  or the mouse  23  to the processor  101 . The mouse  23  is an example of a pointing device. A different pointing device such as a touch panel, a tablet, a touchpad, or a trackball may also be used. 
     The optical drive device  106  reads data recorded on an optical disc  24  by using laser light or the like. The optical disc  24  is a portable recording medium holding data that can be read by light reflection. Examples of the optical disc  24  include a DVD (Digital Versatile Disc), a DVD-RAM, a CD-ROM (Compact Disc Read Only Memory), and a CD-R (Recordable)/RW (ReWritable). 
     The device connection interface  107  is a communication interface for connecting peripheral devices to the prognosis prediction system  100 . For example, a memory device  25 , a memory reader and writer  26 , etc. can be connected to the device connection interface  107 . The memory device  25  is a recording medium having a function of communicating with the device connection interface  107 . The memory reader and writer  26  is a device that reads and writes data on a memory card  27 . The memory card  27  is a card-type recording medium. 
     The network interface  108  is connected to the network  20 . The network interface  108  exchanges data with other computers or communication devices via the network  20 . 
     The prognosis prediction system  100  according to the second embodiment can be realized by the above hardware configuration. The terminal device  31  may be realized by a hardware configuration equivalent to that of the prognosis prediction system  100 . In addition, the biological simulation apparatus  10  according to the first embodiment may be realized by a hardware configuration equivalent to that of the prognosis prediction system  100  illustrated in  FIG. 8 . 
     The prognosis prediction system  100  realizes a processing function according to the second embodiment by performing a program recorded on a computer-readable recording medium, for example. The program holding the processing content executed by the prognosis prediction system  100  may be recorded on any one of various kinds of recording medium. For example, the program executed by the prognosis prediction system  100  may be stored in the HDD  103 . The processor  101  loads at least a part of the program in the HDD  103  onto the memory  102  and executes the loaded program. In addition, the program executed by the prognosis prediction system  100  may be recorded on a portable recording medium such as the optical disc  24 , the memory device  25 , or the memory card  27 . For example, after the program stored in the portable recording medium is installed in the HDD  103 , the program can be executed in accordance with an instruction from the processor  101 . The processor  101  may directly read the program from the portable recording medium and execute the read program. 
     The prognosis prediction system  100  may be a parallel computer system in which a plurality of calculation devices having the hardware configuration illustrated in  FIG. 8  are connected to each other via a high-speed transfer network and a plurality of calculation devices are operated in a parallel manner. In addition, the prognosis prediction system  100  may be a shared-memory calculation device having a large-scale memory. In addition, the prognosis prediction system  100  may include a front-end server separately from a calculation device that performs simulation operations, etc. In this case, by causing the front-end server to input a job to the calculation device, the simulation processing, etc. are executed on the calculation device. A job can be inputted via the network  20 , for example. 
     Next, functions of the prognosis prediction system  100  that performs an accurate simulation on a heart including valves and that predicts postoperative conditions of the heart will be described. 
       FIG. 9  is a block diagram illustrating functions of the prognosis prediction system. The prognosis prediction system  100  includes a storage unit  110 , a pre-processing unit  120 , a simulator  130 , and a post-processing unit  140 . 
     The storage unit  110  holds a biological organ&#39;s geometric model and biological data. A geometric model is numerical data with which a 3D shape of a biological organ whose dynamics are to be predicted and a 3D shape of an immediate biological organ can be expressed through visualization or the like by using biological data. For example, when the shape of a biological organ is approximated by a biological model, a curved surface of the biological organ being approximated by a group of fine triangles, the 3D shape of the biological organ is expressed by the coordinates of the corners of the triangles. The biological data includes image data representing the 3D shapes of the target biological organ and the immediate biological organ. The image data is, for example, image data with which the shape of a biological organ can be identified two or three dimensionally, such as a CT (Computed Tomography) image, an MRI (Magnetic Resonance Imaging) image, or an echocardiogram of a biological organ or an immediate biological organ thereof. Information about temporal change is attached to part of the data. Other examples of the biological data include data about indices representing biological conditions, such as test data based on a cardiovascular catheterization test, an electrocardiogram, blood pressure, etc. and numerical data based on laboratory test values, other physiological tests, data based on an imaging test, a medical history of the patient including a history of operations and records of past operations, and knowledge of the doctor. 
     The storage unit  110  also holds finite element mesh models generated by the pre-processing unit  120 . These finite element mesh models represent a biological heart on which a treatment has been performed. For example, when the biological organ is a heart, the finite element mesh models are tetra mesh models of the heart and the luminal blood, triangle mesh models of the heart valves, voxel mesh models of the heart and the luminal blood, and a voxel mesh model of the torso. For example, at least a part of the storage area of the memory  102  or the HDD  103  is used as the storage unit  110 . The storage unit  110  may be arranged in an external storage device connected to the prognosis prediction system  100  via the network  20 . 
     To reproduce a biological heart through a simulation, the pre-processing unit  120  extracts the shapes of the heart, immediate organs, and bones from the corresponding biological data and generates finite element mesh models of the heart and a finite element mesh model of the trunk from the extracted data. In addition, on the basis of the finite element mesh models representing the biological heart, the pre-processing unit  120  generates finite element mesh models of a postoperative heart obtained after a treatment presumed by the doctor is applied to the biological heart. For example, the finite element mesh models are generated on the basis of information interactively inputted by the doctor using the terminal device  31  or the like. 
     The simulator  130  performs an electrical excitation propagation simulation and a mechanical pulsation simulation in coordination with each other. In the mechanical pulsation simulation, the simulator  130  performs an accurate fluid-structure interaction simulation of a heart, including its valves. When performing the mechanical pulsation simulation, the simulator  130  captures the cusp interface of a heart valve from an interface-tracking ALE mesh that motions and deforms in conformity with an interface of the heart wall, to analyze the blood inside the heart wall. The operation screen of the simulator  130  may be displayed on the monitor  21  or the terminal device  31 . In addition, information may be inputted to the simulator  130  by using an input device of the prognosis prediction system  100  such as the keyboard  22  or the terminal device  31 . 
     The post-processing unit  140  analyzes and ranks simulation results on the basis of the evaluation values that the clinician considers important. For example, the post-processing unit  140  quickly and elaborately displays 3D images or moving images of numerical data about preoperative and postoperative conditions or a plurality of postoperative conditions obtained by a simulation. The post-processing unit  140  also analyzes data that can be compared and evaluated. The operation screen of the post-processing unit  140  may be displayed on the monitor  21  or the terminal device  31 . In addition, information may be inputted to the post-processing unit  140  by using an input device of the prognosis prediction system  100  such as the keyboard  22  or the terminal device  31 . 
     Among the elements illustrated in  FIG. 9 , an interactive system  150  is provided by the pre-processing unit  120  and the post-processing unit  140 . Namely, the pre-processing unit  120  and the post-processing unit  140  perform processing in accordance with instructions made interactively by the doctor  32  and the operator  33 . 
     Next, a procedure of prognosis prediction processing will be described. 
       FIG. 10  is a flowchart illustrating an example of a procedure of prognosis prediction processing. 
     [Step S 101 ] The pre-processing unit  120  acquires a captured image from the storage unit  110 . For example, the pre-processing unit  120  acquires a CT image or an MRI image of a biological body. 
     [Step S 102 ] The pre-processing unit  120  performs pre-processing. In the pre-processing, the pre-processing unit  120  performs shape extraction from biological data, creation of a geometric model of a biological organ, and generation of finite element mesh models of a heart having biologically geometric features. The pre-processing will be described in detail below (see  FIGS. 11 to 13 ). 
     In this step, the pre-processing unit  120  can also create finite element mesh models representing a postoperative heart obtained by assuming that an operation has been performed by a doctor, in addition of the finite element mesh models representing the current conditions of the heart of the biological body. If there are a plurality of possible treatments, the pre-processing unit  120  can create finite element mesh models representing a postoperative heart per treatment. 
     [Step S 103 ] The pre-processing unit  120  adjusts parameters used in the simulation. For example, the pre-processing unit  120  adjusts parameters so that simulation results based on the finite element mesh models approximate the corresponding biological data (an electrocardiogram, blood pressure, echo/MRI data, catheter test data, etc.). For example, the parameters are adjusted on the basis of information inputted by the doctor in view of the past simulation results. 
     [Step S 104 ] The simulator  130  performs an accurate fluid-structure interaction simulation of the heart including its valves, by using the created finite element mesh models and the set parameters. When the finite element mesh models representing the preoperative and postoperative conditions are created, the simulator  130  performs a simulation for each of the created finite element mesh models. The simulation processing will be described in detail below (see  FIGS. 14 and 15 ). 
     [Step S 105 ] The post-processing unit  140  performs processing for checking the preoperative conditions. For example, the post-processing unit  140  compares a simulation result obtained by using the finite element mesh models representing the current conditions of the heart of the biological body with data about the current conditions of the biological body and displays the comparison result. If the simulation result does not match the current conditions of the biological body, resetting of parameters is performed, for example, in accordance with an instruction from the doctor. 
     [Step S 106 ] The post-processing unit  140  displays the behavior of the heart obtained as the simulation result. For example, the post-processing unit  140  reproduces the behavior of the heart as an animation on the basis of the simulation result. The visualization processing for displaying the behavior will be described in detail below (see  FIG. 21 ). 
     [Step S 107 ] The post-processing unit  140  compares results per operative procedure. For example, by using data about the biological conditions, the post-processing unit  140  evaluates postoperative conditions in view of predetermined criteria. The comparison processing will be described in detail below (see  FIG. 23 ). 
     [Step S 108 ] The post-processing unit  140  displays the comparison result. 
     In this way, the prognosis prediction processing is performed. Next, each of the steps illustrated in  FIG. 10  will be described in detail. 
     First, in accordance with an instruction inputted by a doctor who sufficiently understands the clinical conditions of the biological body, the pre-processing unit  120  segments the heart and large blood vessel domains on the basis of the biological data. In this step, the pre-processing unit  120  can reflect a medical history, a history of operations, records of past operations of the patient, including test data based on an echocardiogram, catheterization test, etc., and knowledge of the doctor on the segmentation. The pre-processing unit  120  generates a surface mesh, formed by triangles, from the segment data and segments the surface mesh into finer sites. For example, in the case of a left ventricle, the pre-processing unit  120  segments the surface mesh into domains corresponding to a left ventricle fluid domain, a myocardial domain, a papillary muscle, an aortic valve, and a mitral valve. Next, the pre-processing unit  120  generates two kinds of volume mesh (tetra and voxel) by using the surface meshes as boundaries. The tetra mesh model includes a triangle surface mesh that has a material number set per detailed site information and a tetra volume mesh that has a material number different per domain spatially segmented by the surfaces of the triangle surface mesh. The voxel mesh model includes structured grid data having a material number indicating domain information per voxel. 
     Next, the pre-processing unit  120  generates a torso (trunk of the body) voxel mesh model having domains segmented in accordance with the electric conductivity of the human body. Next, the pre-processing unit  120  refers to the biological data obtained from CT, MRI, etc. and segments organ and bone domains into domains in accordance with luminance values. In this operation, in the range from the shoulders to the lower back, the pre-processing unit  120  suitably compensates for domains lacking sufficient images. Next, the pre-processing unit  120  sets predetermined material numbers to the segmented domains and adjusts the position of the torso voxel mesh model with the position of a heart voxel mesh. Finally, the pre-processing unit  120  performs voxel resampling to match the heart voxel mesh. 
     By performing the above processing, finite element mesh models are generated. For example, finite element mesh models segmented per site are created. 
     The finite element mesh models created in this way are largely classified into two kinds, namely, torso and heart voxel mesh models and a heart tetra mesh model (including the tetra mesh models of the heart wall and the luminal blood). First, the pre-processing unit  120  creates a heart voxel mesh model by matching the torso voxel mesh with the heart tetra mesh model. In this way, tetra and voxel meshes are associated with each other. Next, the pre-processing unit  120  sets a fiber distribution and a sheet distribution to the heart voxel mesh model on the basis of literature values or the like. In addition, the pre-processing unit  120  sets a Purkinje fiber distribution or an equivalent endocardial conductance distribution. In addition, the pre-processing unit  120  sets a site of earliest activation to the endocardium. In addition, the pre-processing unit  120  distributes cell models having three kinds of APD (Action Potential Duration) distribution in the long axis direction and the short axis direction. For example, a method in “Computational fluid dynamics modeling of intracranial aneurysms: effects of parent artery segmentation on intra-aneurysmal hemodynamics” by M. A. Castro et al. is used for these settings. The simulator  130  performs an electrical excitation propagation simulation by combining the above heart voxel mesh model and a torso model having a body surface on which electrodes for a standard 12-lead electrocardiogram are set. The pre-processing unit  120  adjusts parameters so that the simulation result matches the result of the standard 12-lead electrocardiogram test on the biological body. Consequently, the time history of the calcium ion concentration (calcium concentration history) of an individual tetra element constituting the heart (an individual element of the heart tetra mesh model) is determined. 
     Next, the pre-processing unit  120  sets a fiber distribution and a sheet distribution to the heart tetra mesh model, as is the case with the voxel mesh model. In addition, the pre-processing unit  120  sets boundary conditions for performing a mechanical simulation. In addition, the pre-processing unit  120  performs duplexing of pressure nodes for mechanical analysis and duplexing of nodes at an interface between the blood and the heart muscle. In addition, the pre-processing unit  120  provides the heart tetra mesh model with site segmentation information. These finite element mesh models are created for a relaxation phase of the heart. Thus, to obtain a natural shape corresponding to a free-stress condition, the pre-processing unit  120  performs a suitable mechanical finite element method and contracts the heart. The pre-processing unit  120  connects a systemic circulation model matching the conditions of the biological body to the heart represented by the created finite element mesh model. This systemic circulation model is an electrical circuit analogy model formed by suitably combining discrete resistance and capacitance defined by assuming that the blood pressure is voltage and the blood flow is current. 
     The simulator  130  performs a pulsation simulation on the basis of the heart tetra mesh model to which the above systemic circulation model is connected and the contraction force obtained by using the calcium concentration history of an individual finite element for an excitation-contraction coupling model. After the simulation, the pre-processing unit  120  adjusts parameters so that the parameters match biological data such as a pressure-volume relationship, an MRI image, an echocardiographic image, etc. and mechanical indices extracted from these items of information. In addition, when the oxygen saturation has already been measured, the pre-processing unit  120  solves an advection diffusion equation using a fluid velocity distribution of the blood obtained through fluid-structure interaction analysis by using a finite element method using the same mesh as the mechanical mesh and checks the conformity. 
     After the simulation, the post-processing unit  140  performs visualization processing on the basis of files in which results of the electrical excitation propagation simulation and the mechanical pulsation simulation performed as described above are stored. For example, the post-processing unit  140  causes a visualization system to read more than 100 files including data of various phenomena such as excitation propagation and interaction between the heart muscle and the fluid per beat and to generate and visualize 3D shapes from numerical data. The post-processing unit  140  enables observation from various points of view by changing the viewpoint in accordance with input information. In addition, by generating cross sections, the post-processing unit  140  enables observation of any one of the cross sections. In addition, by quickly generating a moving image of the behavior of the atria and displaying the behavior as an animation, the post-processing unit  140  enables realistic observation. In addition, the post-processing unit  140  can also extract and display a part of the heart muscle so that change of the extracted part over time can be checked. 
     The doctor refers to the visualization result and gains a better understanding of the pump performance of the heart, the hemodynamics, the load on the heart and the lungs, etc. As a result, the doctor can make diagnostic or clinical decisions. For example, the pump performance of the heart represents the motion of the heart and the motions and functions of the heart valves. For example, the hemodynamics includes the pressure generated by the heart, the pressure at an individual cardiovascular site, the flow volume of the blood at an individual cardiovascular site, the velocity of the blood flow at an individual cardiovascular site, the oxygen saturation of the blood at an individual cardiovascular site, the dissolved gas partial pressure of the blood at an individual cardiovascular site, and the blood concentration of drug at an individual cardiovascular site. For example, the load on the heart and lungs includes the energy consumption amount of the heart, the energy conditions of the blood fluid, the pressure caused in the heart muscle, the conditions of the coronary artery circulation, the systemic vascular resistance, and the pulmonary vascular resistance. 
     While what has been described is a technique for reproducing and visualizing the heart of the current biological body, the prognosis prediction system  100  can also visualize a result obtained by virtually performing several treatments on a heart model. 
       FIG. 11  illustrates a scheme for obtaining results of virtual treatments. First, on the basis of a biological data  51  (a CT image, an MRI image, an echocardiogram, etc.), the pre-processing unit  120  performs pre-processing (step S 110 ) for generating heart mesh models obtained before a virtual operation is performed. In this pre-processing, the pre-processing unit  120  performs processing for segmentation (step S 111 ) and so forth to create a geometric model  52  and finite element mesh models  53  of the heart of a biological body. The finite element mesh models  53  are models of the preoperative heart and chest of the biological body. The pre-processing for generating heart mesh models obtained before a virtual operation is performed will be described below in detail (see  FIG. 12 ). 
     By using the finite element mesh models  53  created from the biological data  51 , the pre-processing unit  120  performs a simulation and parameter adjustment (step S 112 ). In the simulation and parameter adjustment, the pre-processing unit  120  repeats a simulation while changing parameters so that the finite element mesh models approximate the biological data  58  (an electrocardiogram, blood pressure, echo/MRI data, catheterization test data, etc.). In the simulation, the pre-processing unit  120  performs an electrical excitation propagation simulation first before a mechanical pulsation simulation. The pre-processing unit  120  performs the mechanical pulsation simulation in accordance with a simulation method of fluid-structure interaction in which the above heart tetra mesh model is represented as a structure mesh model representing the structure domain and the luminal blood tetra mesh model as an ALE fluid mesh model representing the fluid domain. The simulation processing and the parameter adjustment processing will be described in detail below (see  FIG. 13 ). Next, a set of parameters (a best parameter set) with which the simulation result approximates the biological data the most is determined (step S 113 ). 
     In addition, on the basis of the geometric model created in the pre-processing (step S 110 ), the pre-processing unit  120  performs pre-processing (step S 120 ) for generating heart mesh models in accordance with a virtual treatment. In this pre-processing (step S 120 ), the pre-processing unit  120  performs shape deformation per virtual treatment (steps S 121  and S 122 ). For example, the pre-processing unit  120  performs shape deformation in accordance with a “virtual treatment A” to create a geometric model  54  and a finite element mesh model  55 . In addition, the pre-processing unit  120  performs shape deformation in accordance with a “virtual treatment B” different from the “virtual treatment A” to create a geometric model  56  and a finite element mesh model  57 . In addition, the simulator  130  performs simulations by using the best parameter set and the finite element mesh models  55  and  57  representing the conditions after the respective virtual treatments (steps S 123  and S 124 ). 
     The post-processing unit  140  performs post-processing (steps S 114 , S 125 , and S 126 ) for converting each of the simulation results into data that can be visualized and generates results  59  to  61  that can be visualized. The results  59  to  61  are stored in a storage device such as a memory or an HDD. The result  59  obtained from a model representing the heart of the current biological body is used as a reference for diagnosis by the doctor, along with the biological data  51  and  58 . 
     The doctor refers to the visualization results of these possible virtual operations and compares the results in terms of the pump performance of the postoperative heart, the hemodynamics, the load on the heart and the lungs, etc. Thus, the doctor can use these results as information for determining the most suitable operative procedure. By performing virtual operations and visualizing the results, for example, the doctor can predict the following contents. 
     For example, the doctor can predict conditions of a heart that has undergone an operation for congenital heart disease, conditions of a circulatory system that has undergone percutaneous coronary intervention or an aortocoronary bypass operation, conditions of a circulatory system that has undergone a heart valve replacement operation, conditions of a circulatory system that has undergone cardiac valvuloplasty, conditions of a circulatory system that has undergone cardiac valve annuloplasty, conditions of a circulatory system that has undergone pacemaker treatment including cardiac resynchronization treatment, conditions of a circulatory system that has undergone treatment for aortic disease, conditions of a circulatory system that has undergone treatment for pulmonary arteriopathy, conditions of a circulatory system that has undergone implantation of a circulatory assist device, and conditions of a circulatory system that has undergone other cardiovascular treatment. Examples of the operation for congenital heart disease include an open-heart operation for congenital heart disease, a catheterization operation for congenital heart disease, and an extracardiac operation for congenital heart disease. Examples of the catheterization operation for congenital heart disease include an operation using a defect closure device and an operation using a balloon catheter. Examples of the extracardiac operation for congenital heart disease include the Blalock-Taussig shunt, pulmonary artery banding, the Glenn operation, and TCPC (total cavopulmonary connection). The heart valve replacement operation and the cardiac valvuloplasty include a catheterization operation. Examples of the circulatory assist device include an intra-aortic balloon pump, a percutaneous cardiopulmonary support device, a left ventricular assist device, and a right ventricular assist device. The cardiovascular treatment includes medical treatment. 
     Next, the pre-processing (step S 110 ) for generating heart mesh models obtained before a virtual operation will be described in more detail. 
       FIG. 12  illustrates an example of a procedure of generating heart mesh models obtained before a virtual operation. The pre-processing unit  120  performs segmentation of the heart domain by using CT or MRI data and biological data  63  (step S 131 ). Next, the pre-processing unit  120  generates a triangle surface mesh segmented per site (step S 132 ). Next, the pre-processing unit  120  generates voxel and tetra volume meshes by using the surface mesh as boundaries (step S 133 ). As a result, a heat tetra mesh model  64  and a heart voxel mesh model  65  are generated. 
     In addition, the pre-processing unit  120  performs segmentation of the torso domain to generate a voxel mesh model (step S 134 ). In the segmentation of the torso domain, the lung, bone, and organ domains are segmented on the basis of the CT or MRI data  62 . Next, the pre-processing unit  120  performs extrapolation processing for compensating for insufficient domains (step S 135 ). After the extrapolation processing, the pre-processing unit  120  adjusts the position of the heart voxel model and the position of the torso voxel model (step S 136 ). Next, the pre-processing unit  120  performs resampling for reestablishing the structured grid of the torso in conformity with the heart voxel mesh (step S 137 ). In this way, a torso voxel mesh model  66  is generated. 
     The finite element mesh models  53  illustrated in  FIG. 11  include the heart tetra mesh model  64 , the heart voxel mesh model  65 , and the torso voxel mesh model  66  illustrated in  FIG. 12 . On the basis of these finite element mesh models  53 , the pre-processing unit  120  performs the simulation processing and the parameter adjustment processing. 
       FIG. 13  illustrates the simulation processing and the parameter adjustment processing. In  FIG. 13 , each item of information outputted as a simulation result is surrounded by a dashed line, and biological data is surrounded by a dashed-dotted line. The parameter adjustment processing is surrounded by a thick line. 
     The simulation-purpose finite element mesh models  53  obtained in the pre-processing (step S 110 ) are largely classified into two kinds, namely, models for an electrical excitation propagation simulation and models for a mechanical pulsation simulation. The former models are created by voxel finite elements and include a torso voxel mesh model  202  and a heart voxel mesh model  203  representing the heart in a relaxation phase. The pre-processing unit  120  sets fiber and sheet distributions  220  to the heart voxel mesh model  203  on the basis of literature values or the like. In addition, the pre-processing unit  120  sets a Purkinje fiber distribution  221  or an equivalent endocardial conductance distribution to the heart voxel mesh model  203 . In addition, the pre-processing unit  120  sets a site of earliest activation  222  to the endocardium of the heart voxel mesh model  203 . In addition, the pre-processing unit  120  distributes cell models having three kinds of APD distribution in the long axis direction and the short axis direction (distribution of three kinds of cells  223 ). 
     The simulator  130  performs an electrical excitation propagation simulation by combining the above heart voxel mesh model  203  and the torso voxel mesh model  202  having a body surface on which electrodes for an electrocardiogram are set. As a result, a 12-lead electrocardiogram  204  is obtained. Next, the pre-processing unit  120  performs parameter adjustment  205  so that the 12-lead electrocardiogram  204  obtained as a simulation result matches a measured 12-lead electrocardiogram  207 . Consequently, a calcium concentration history  206  of an individual tetra element constituting the heart (an individual element of the heart tetra mesh model) is determined. 
     Next, the pre-processing unit  120  sets the fiber and sheet distributions  220  to a heart tetra mesh model  208 . In addition, the pre-processing unit  120  sets boundary conditions for performing a mechanical simulation. In addition, the pre-processing unit  120  performs duplexing of pressure nodes for mechanical analysis and duplexing of nodes at an interface between the blood and the heart muscle. In addition, the pre-processing unit  120  provides the heart tetra mesh model  208  with site segmentation information. 
     In addition, the pre-processing unit  120  creates a luminal blood tetra mesh model  210  in a relaxation phase. The luminal blood tetra mesh model  210  is a tetra mesh model representing the shape of the domain in which blood inside the heart flows. The heart tetra mesh model  208  and the luminal blood tetra mesh model  210  are created for a relaxation phase of the heart. The pre-processing unit  120  creates a natural shape model corresponding to a free-stress condition from each of the heart tetra mesh model  208  and the luminal blood tetra mesh model  210 . For example, the pre-processing unit  120  performs a suitable mechanical finite element method, contracts the heart, and creates a natural-condition heart model  209  and a natural-condition luminal blood model  211 . 
     The pre-processing unit  120  adds, as simulation information, a combination  212  of cusp models of membrane elements and chorda tendineae models of beam elements to the natural-condition heart model  209  and the natural-condition luminal blood model  211 . In addition, the pre-processing unit  120  connects a systemic circulation model  214  adjusted to the biological conditions. On the basis of the various models to which the systemic circulation model  214  has been connected and the contraction force obtained by using the calcium concentration history of an individual finite element for an excitation contraction coupling model  213 , the simulator  130  analyzes the behavior of the pulsation and performs a fluid-structure interaction simulation  215 . Next, the pre-processing unit  120  adjusts parameters so that the parameters match biological data  218  based on blood pressure, echocardiography, MRI, catheterization, etc. and mechanical indices extracted from the biological data  218 . When the oxygen saturation has already been measured, the pre-processing unit  120  solves an advection diffusion equation  216  using a fluid velocity distribution of the blood obtained through a fluid-structure interaction simulation  215  by using a finite element method using the same mesh as the mechanical mesh and checks the conformity. 
     In this way, the simulation and the parameter adjustment are performed. In the simulation, fluid-structure interaction analysis in which the ALE method and the Lagrange multiplier method are combined with each other is used. The procedure of the fluid-structure interaction simulation performed by the simulator  130  is as follows. 
       FIG. 14  is a flowchart illustrating an example of a procedure of the simulation. 
     [Step S 151 ] The simulator  130  updates time of the time step. For example, the simulator  130  adds a predetermined time increment Δt to the current time t. When the simulation is started, the time is set to t=0. 
     [Step S 152 ] The simulator  130  calculates a stiffness matrix, internal force integration, and various conditional integrations and the corresponding differentiations. The matrix, differentiations, and integrations reflect the ALE fluid-structure interaction and fluid-structure interaction based on the Lagrange multiplier method. 
     [Step S 153 ] The simulator  130  synthesizes a global matrix A and a global right-side vector b. 
     [Step S 154 ] The simulator  130  calculates updated amounts of variables. 
     [Step S 155 ] The simulator  130  updates the ALE mesh. 
     [Step S 156 ] The simulator  130  determines whether the calculation result has converged. If so, the processing proceeds to step S 157 . If not, the processing returns to step S 152  and the calculation processing is performed again. 
     [Step S 157 ] The simulator  130  determines whether the simulation time t has reached finish time t end . For example, the finish time t end  is when a simulation of a single beat of the heart is finished. When the finish time t end  has been reached, the simulator  130  ends the simulation. If the finish time t end  has not been reached, the processing returns to step S 151 . 
     The processing in steps S 152  to S 155  will hereinafter be described in detail. 
       FIG. 15  a flowchart illustrating an example of a procedure of the differentiation and integration calculation processing. 
     [Step S 161 ] The simulator  130  calculates an internal force integrated value f fs  and a stiffness matrix A fs  of equation (30).
 
δ W   f   ALE   +δW   S   1   δW   S   2   (30)
 
     [Step S 162 ] The simulator  130  calculates an internal force integrated value f C  of equation (31) and an integrated value C C  and differentiation B C  of equation (32).
 
∫ Γ     C   λ( x ) C ( x:δv,δ{dot over (u)} ) dΓ   C   (31)
 
∫ Γ     C   δλ( x ) C ( x:v,u ) dΓ   C   (32)
 
     [Step S 163 ] The simulator  130  calculates an internal force integrated value f R  of equation (33) and an integrated value C R  and differentiation B R  of equation (34).
 
∫ Γ     R   τ( x )·(δ x   R   −W   R   W   δx   W ) dl   (33)
 
∫ Γ     R   δτ( x )·( x   R   −W   R   W   x   W ) dl   (34)
 
     [Step S 164 ] The simulator  130  calculates an internal force integrated value f T  of equation (35) and an integrated value C T  and differentiation B T  of equation (36). 
     
       
         
           
             
               
                 
                   
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     [Step S 165 ] The simulator  130  calculates an internal force integrated value f f  of equation (37) and an integrated value C f  and differentiation B f  of equation (38).
 
∫ Γ     f,k     P   k   n   k   ·δvds   k   (37)
 
∫ Γ     f,k     δP   k   n   k   ·vds   k =0  (38)
 
     [Step S 166 ] The simulator  130  calculates a value C G  and differentiation B G  of G k  (P k , F k , V k , Q k ). 
     From the calculation results such as the matrix and integrated values calculated in the processing illustrated in  FIG. 15 , the simulator  130  creates and synthesizes the global matrix A and the global right-side vector b (step S 153 ). Specifically, the synthesization of the global matrix A and the global right-side vector b is performed as follows. 
     
       
         
           
             
               
                 
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     In the above equation (40), Δ represents an updated amount. A time integration method such as the Newmark-β method is used for time evolution. 
     Next, the blood domain is extracted from the entire system updated in this way, and updating the ALE mesh (step S 155 ) is calculated. Specifically, an equation for mesh control such as for a hyperelastic body is solved by using an interface with a new heart muscle as a boundary condition, and a new ALE mesh is generated. 
     If the total correction amount in the above process is not sufficiently small, the processing returns to step S 152 , and a calculation for updating the solution is performed. If the correction amount reaches to a predetermined value or less, the simulator  130  determines that convergence in the time increment of Δt has been achieved. Thus, calculation of the next time step is performed. When the finish time t end  of the analysis target is reached, the simulator  130  ends the present processing. 
     As described above, by performing a fluid-structure interaction simulation in which the ALE method and the Lagrange multiplier method are combined with each other, an accurate simulation can be performed even when there is a site that significantly motions. Upon completion of the simulation, the simulator  130  stores the simulation results in files for post-processing. 
     In the second embodiment, other than a simulation by using finite element mesh models of a heart created based on biological data, the simulator  130  can perform a simulation using finite element mesh models representing a heart after an individual virtual treatment is performed, as illustrated in  FIG. 11 . 
     Next, the pre-processing (step S 120 ) for generating heart mesh models based on respective virtual treatments will be described in more detail. In this pre-processing, the finite element mesh models  55  and  57  based on the respective virtual treatments are created not on the basis of the biological data but of the geometric model  52  created in the pre-processing (step S 110 ). When the finite element mesh models  55  and  57  based on the respective virtual treatments are created, the procedures of the virtual treatments are given in accordance with information inputted by a doctor or by the operator  33  who has received an instruction from a doctor. 
       FIG. 16  illustrates an example of processing for generating postoperative finite element mesh models. For example, the pre-processing unit  120  creates a preoperative triangle surface mesh model  72  and a preoperative voxel mesh model  73  in advance on the basis of a preoperative heart tetra mesh model  71 . Next, the pre-processing unit  120  performs postoperative finite element mesh model creation processing  75  in accordance with a doctor&#39;s instruction  74 . For example, the pre-processing unit  120  creates the preoperative triangle surface mesh model and the preoperative voxel mesh model on the basis of the preoperative heart tetra mesh model and deforms the preoperative triangle surface mesh model  72  and the preoperative voxel mesh model  73  in accordance with the doctor&#39;s instruction  74  (the deformation includes topological deformation). In this way, a postoperative triangle surface mesh model  76  and a postoperative voxel mesh model  77  are generated. The pre-processing unit  120  generates a postoperative heart tetra mesh model  78  on the basis of the postoperative triangle surface mesh model  76  and the postoperative voxel mesh model  77 . 
     To obtain a virtual 3D shape of a postoperative organ of a biological body whose dynamics and/or functions are to be predicted, for example, a doctor or the like or a person who has received an instruction from a doctor or the like refers to biological data of the biological body and gives the pre-processing unit  120  an instruction about morphological change of the heart or immediate organs on the basis of an operative procedure to be evaluated on a screen on which a preoperative geometric model is displayed. The operator  33  who has received a doctor&#39;s instruction orally or by written document may input the instruction to the pre-processing unit  120 . On the basis of the doctor&#39;s instruction, the pre-processing unit  120  corrects segment data of the preoperative geometric model, obtains a postoperative geometric model, and performs regeneration of the corresponding surface mesh. In addition, on the basis of the doctor&#39;s instruction, the pre-processing unit  120  may obtain a postoperative triangle surface mesh model assumed by the doctor or the like by directly changing the preoperative triangle surface mesh model  72 . In this way, a postoperative surface mesh (the postoperative triangle surface mesh model  76 ) assumed by the doctor can be obtained. In addition, the pre-processing unit  120  generates the postoperative voxel mesh model  77  by deforming the preoperative voxel mesh model  73  in accordance with a doctor&#39;s instruction. 
     In addition, the pre-processing unit  120  may present the doctor a list of possible techniques that can be used in a virtual operation and allow the doctor to select a technique to be used. The possible techniques are techniques used in medical virtual operations and techniques used in surgical virtual operations, for example. In addition, examples of the possible techniques include any technique relating to change of a morphological or mechanical phenomenon of a biological organ. For example, replacement of large blood vessels, reestablishment of the septum, suturation, bypass creation, flow path formation, banding (strictureplasty), valve replacement, valvuloplasty, etc. are displayed on a screen of the terminal device  31  or the like as possible techniques used in a virtual operation. For example, replacement of large blood vessels is a technique of separating two large blood vessels, a pulmonary artery and an aorta, from the heart to replace the large blood vessels with artificial blood vessels whose length has been adjusted as needed. If replacement of large blood vessels is adopted, the doctor specifies the position at which the aorta needs to be separated, for example. Accordingly, the pre-processing unit  120  separates the aorta at the specified position and deforms a finite element mesh model. If banding is selected, the doctor specifies the position at which a large blood vessel needs to be narrowed and the eventual diameter. The pre-processing unit  120  deforms the shape of the blood vessel of a finite element mesh model in accordance with the doctor&#39;s instruction. 
     In this way, by selecting a technique adopted in a virtual operation and deforming the preoperative triangle surface mesh model  72  and the preoperative voxel mesh model  73  in accordance with the change made by the technique, the postoperative heart tetra mesh model  78  (that has undergone topological change) is generated finally. 
     By performing this operation on all the procedures assumed, a plurality of heart finite element mesh models are created. For example, the finite element mesh models  55  and  57  illustrated in  FIG. 11  are created. 
       FIG. 17  illustrates an example of how a geometric model changes after large blood vessels are removed. For example, when a doctor inputs an instruction for removal of large blood vessels to the pre-processing unit  120 , a geometric model  52   a  is created by removing large blood vessels from the preoperative geometric model  52 . Next, when the doctor inputs an instruction for addition of an aorta to the pre-processing unit  120 , a geometric model  52   b  is created by adding an aorta to the geometric model  52   a . Next, when the doctor inputs an instruction for addition of a pulmonary artery to the pre-processing unit  120 , a geometric model  54  is created by adding a pulmonary artery to the geometric model  52   b . In this way, the shape is deformed in accordance with a virtual treatment. 
     By performing the shape deformation in this way, a finite element mesh model is created per virtual treatment procedure. In addition, a fluid-structure interaction simulation is performed for an individual finite element mesh model representing a heart after a virtual operation per virtual treatment procedure. The method of the simulation is the same as that of the simulation using a finite element mesh model representing the heart of a biological body before a virtual operation. 
       FIGS. 18A and 18B  illustrate examples of visualization of parts of simulation results obtained before and after an operation.  FIG. 18A  illustrates visualization of a simulation result of a preoperative heart, and  FIG. 18B  illustrates visualization of a simulation result of a postoperative heart. In the example in  FIGS. 18A and 18B , since the preoperative heart has a hole in its atrial septum, blood in the right atrium and blood in the left atrium are mixed. Since a wall has been formed between the right atrium and the left atrium of the postoperative heart, the blood in the right atrium and the blood in the left atrium are not mixed. 
     These simulation results are stored in the storage unit  110 . 
       FIG. 19  illustrates an example of information stored in the storage unit. The storage unit  110  holds biological data  111  and simulation results  112  to  114  obtained by respective simulations performed. In the example in  FIG. 19 , the pretreatment simulation result  112 , the simulation result  113  obtained after the “virtual treatment A” is performed, and the simulation result obtained after the “virtual treatment B” is performed are stored in the storage unit  110 . 
     The biological data  111  is information about a treatment target biological body. The biological data  111  includes various kinds of data such as a CT image, an MRI image, a 12-lead electrocardiogram, and blood pressure. 
     The simulation results  112  to  114  include positions of elements and nodes and physical quantities on elements and nodes per simulation time step. The elements are tetra elements, voxel elements, blood vessel elements, etc. The position of an element is a predetermined position in the element such as the center of gravity of the element. A node is an apex of an element, for example. 
     These simulation results are visualized by the post-processing unit  140 . 
       FIG. 20  illustrates an example of visualization processing. The post-processing unit  140  includes a visualization parameter input unit  141 , a data acquisition unit  142 , a myocardium visualization unit  143   a , an excitation propagation visualization unit  143   b , a coronary circulation visualization unit  143   c , a valve visualization unit  143   d , a graph visualization unit  143   e , a medical-image visualization unit  143   f , a blood flow visualization unit  143   g , a blood vessel visualization unit  143   h , a visualization result image display unit  144   a , a 3D display unit  144   b , a graph display unit  144   c , and a medical-data superposition and display unit  144   d.    
     The visualization parameter input unit  141  receives parameters indicating visualization conditions from the terminal device  31  and inputs the received parameters to other elements performing visualization processing. The data acquisition unit  142  acquires data stored in the storage unit  110  and inputs the acquired data to other elements performing visualization or display processing. 
     The myocardium visualization unit  143   a  visualizes the behavior of the myocardium and myocardial physical quantities, for example. For example, the myocardium visualization unit  143   a  represents change of a value corresponding to a physical quantity such as the myocardial pressure as a color change. 
     The excitation propagation visualization unit  143   b  visualizes propagation conditions of excitation of the heart. For example, the excitation propagation visualization unit  143   b  represents change of a value corresponding to a myocardial voltage as a color change. 
     The coronary circulation visualization unit  143   c  visualizes coronary circulation conditions. For example, the coronary circulation visualization unit  143   c  represents change of a physical quantity such as the velocity or pressure of the blood flow in the coronary circulation system as a color change. 
     The valve visualization unit  143   d  visualizes the motions of the valves and the motion of the blood around the valves. For example, the valve visualization unit  143   d  represents the blood flow around the valves as fluid velocity vectors. 
     The graph visualization unit  143   e  statistically analyzes simulation results and represents the results in the form of a graph. For example, on the basis of a simulation result obtained per virtual operation, the graph visualization unit  143   e  evaluates conditions of the biological body (for example, the patient) after an individual virtual operation and generates graphs that indicate the evaluation results. 
     The medical-image visualization unit  143   f  visualizes a medical image such as a CT image or an MRI image. For example, the medical-image visualization unit  143   f  generates display images from image data that indicates medical images. 
     The blood flow visualization unit  143   g  visualizes the blood flow in a blood vessel. For example, the blood flow visualization unit  143   g  generates a fluid velocity vector that indicates a blood flow velocity. 
     The blood vessel visualization unit  143   h  generates display blood vessel object that indicates an actual blood vessel on the basis of a blood vessel element. For example, when the positions and diameters of ends of a blood vessel element are already set, the blood vessel visualization unit  143   h  generates a cylindrical blood vessel object that connects the ends that have the diameters. 
     The visualization result image display unit  144   a  displays an image generated through visualization on the terminal device  31  or the monitor  21 . For example, the 3D display unit  144   b  displays a 3D image of a heart on the terminal device  31  or the monitor  21 . The graph display unit  144   c  displays graphs created by the graph visualization unit  143   e  on the terminal device  31  or the monitor  21 . For example, the medical-data superposition and display unit  144   d  superposes biological data onto a 3D image of a heart and displays the superposed image. 
     With the post-processing unit  140  having the functions as described above, for example, the blood flow can be visualized and displayed on a moving image showing the behavior of a heart. 
       FIG. 21  illustrates an example of a procedure of visualization processing. The processing illustrated in  FIG. 21  illustrates visualization processing in which a system configured by at least one computer processes data corresponding to a single beat in the electrical excitation propagation simulation and the mechanical pulsation simulation. By quickly visualizing and reproducing simulation results as a moving image of a single beat of the heart, the simulation results can be observed as the motion of a natural heart. For this purpose, a plurality of threads for performing visualization processing are activated in a computer. Specifically, a thread group  82  is formed by a plurality of threads, and among m steps corresponding to a single beat (m is an integer of 1 or more), an individual thread performs processing corresponding to m/n steps (n is an integer of 1 or more indicating the number of threads activated). In addition, each thread in the thread group finally creates results of the electrical excitation propagation simulation and the mechanical pulsation simulation as a plurality of bitmap images. A main process displays these bitmap images at 1 to 60 fps in chronological order. 
     Specifically, the terminal device  31  transmits a moving image display instruction to the post-processing unit  140  of the prognosis prediction system  100  (step S 201 ). Next, the main process  81  of the post-processing unit  140  transmits visualization parameters to the thread group  82  (step S 202 ). Each thread in the thread group  82  reads simulation result data from the storage unit  110  (step S 203 ). Next, on the basis of the read data, each thread in the thread group  82  performs visualization processing and rendering to which the specified visualization parameters are applied (step S 204 ). Next, each thread in the thread group  82  transmits rendering results to a frame buffer  83  (step S 205 ). The rendering results are written in the frame buffer  83  (step S 206 ). 
     When a bitmap image of a single frame is created on the frame buffer  83 , the corresponding thread in the thread group  82  acquires the bitmap image (step S 207 ). Next, the corresponding thread transmits the acquired bitmap image to the main process  81  (step S 208 ). The main process  81  arranges the bitmap images of the respective frames transmitted from the thread group  82  in chronological order and transmits the arranged bitmap images to the terminal device  31  (step S 209 ). In this way, the terminal device  31  displays a moving image. For example, a moving image indicating the behavior of a heart including the motions of the heart valves and the blood flow around the valves is displayed on the terminal device  31 . 
     In the second embodiment, a plurality of simulation results may be visualized in a parallel way and displayed side by side on the same screen. For example, simulation results of a heart on which a plurality of virtual operations have been performed may be arranged and displayed so that these simulation results can be compared with each other. 
       FIG. 22  illustrates an example of displaying operative procedures to be compared. When an “operative procedure A” and an “operative procedure B” are compared with each other, for example, the post-processing unit  140  displays the heart muscle and the blood flow on a cross section of the heart. When the blood flow in a domain has large oxygen saturation, the blood flow is visualized in red. In this way, postoperative effectiveness can be checked. 
     When displaying and comparing operative procedures, the post-processing unit  140  may calculate evaluation values for the operative procedures on the basis of a predetermined reference, rank the operative procedures on the basis of the evaluation values, and rearrange the operative procedures in descending order of rank. For example, the post-processing unit  140  may sort the operative procedures in the ascending or descending order by using parameter values specified by a doctor (for example, values corresponding to physical quantities) as evaluation values. 
     In addition, the post-processing unit  140  may use a weighted parameter set  84 , which is a parameter set obtained by giving weight to parameters on the basis of a doctor&#39;s instructions, and rank the operative procedures. For example, in view of disease conditions of a biological body such as a patient, a doctor gives more weight to parameters indicating conditions that a postoperative biological body calls for. The doctor gives less weight to parameters that do not relate to the disease conditions of the biological body. The post-processing unit  140  sets values of the weight to the parameters in accordance with the doctor&#39;s instructions. 
       FIG. 23  is a flowchart illustrating an example of a procedure of result comparison processing. First, the post-processing unit  140  selects weighted parameters in accordance with a doctor&#39;s instructions and generates the weighted parameter set  84  (step S 221 ). For example, when operative procedures to be compared are displayed, the doctor may specify one or more parameters used for the ranking. For example, when the heaviest and the second heaviest parameters are specified, the post-processing unit  140  generates the weighted parameter set  84  by first selecting a parameter corresponding to the heaviest weight. 
     Next, the post-processing unit  140  extracts values of the parameters indicated by the weighted parameter set  84  from result data  91  to  94 , indicating a simulation result per operative procedure (steps S 222  to S 225 ). For example, the post-processing unit  140  automatically sets a parameter value extraction point on the basis of information about sites of the heart added by the pre-processing. Next, the post-processing unit  140  extracts parameter values at the extraction point from data corresponding to a single pulse. For example, the post-processing unit  140  stores a maximal value of the parameter values at the extraction point corresponding to a single pulse as an evaluation value of the corresponding operative procedure in a memory. When a plurality of parameters are selected, the post-processing unit  140  performs predetermined calculation on the basis of an evaluation value per parameter and calculates the evaluation value of an operative procedure. For example, the post-processing unit  140  may multiply an evaluation value of a parameter with the weight of the parameter and uses the sum of the multiplication results as the evaluation value of the operative procedure. The post-processing unit  140  may normalize the evaluation value of each parameter. For example, the post-processing unit  140  may set “1” as an ideal value or a target value for a parameter and normalize the evaluation value of the parameter to a value between 0 and 1 on the basis of the difference from the ideal value or the target value. Next, the post-processing unit  140  may calculate the evaluation value of the operative procedure on the basis of the normalized evaluation value. 
     After calculating an evaluation value per operative procedure, the post-processing unit  140  sorts the operative procedures by the respective evaluation values (step S 226 ). For example, the post-processing unit  140  compares the evaluation values of the operative procedures with each other and arranges 3D display domains and graphs of the operative procedures in descending order of evaluation value. In the example in  FIG. 22 , an operative procedure having a larger evaluation value is arranged on the left. The post-processing unit  140  displays graphs indicating the evaluation values of the operative procedures on the arranged 3D display domains along with the simulation results of the respective operative procedures (step S 227 ). When the post-processing unit  140  sorts the operative procedures by different parameters, the processing returns to step S 221 . Otherwise, the processing proceeds to END (step S 228 ). 
     For example, the doctor  32  viewing the visualization screen checks an operative procedure indicating a result in which blood with more oxygen flows into the aorta and makes a diagnostic or clinical decision. 
     Examples of the parameters are as follows: 
     Flow amount [L] into aorta per beat 
     Aortic systolic pressure [mmHg] 
     Aortic diastolic pressure [mmHg] 
     Mean aortic pressure [mmHg] 
     Aortic pressure [mmHg] (pressure in contraction phase−pressure in relaxation phase) 
     Left ventricular systolic pressure 
     Left ventricular end-diastolic pressure 
     Flow amount [L] into pulmonary artery per beat 
     Pulmonary artery systolic pressure [mmHg] 
     Pulmonary artery diastolic pressure [mmHg] 
     Pulmonary artery mean pressure [mmHg] 
     Pulmonary artery pressure [mmHg] (pressure in contraction phase−pressure in relaxation phase) 
     Right ventricular systolic pressure 
     Right ventricular end-diastolic pressure 
     Mean right atrial pressure 
     Mean left atrial pressure 
     Flow amount into pulmonary artery/flow amount into aortal per beat 
     Pulmonary artery systolic pressure/aortic systolic pressure per beat 
     Maximum blood pressure of coronary artery 
     Maximum blood pressure of coronary artery−myocardial pressure 
     Flow amount [L] into coronary artery per beat 
     Maximum flow velocity [m/s] in specified region of interest 
     Amount of energy lost by viscosity at specified site (per beat) 
     Amount of ATP consumed at specified site (per beat) 
     Heart pulsation work amount 
     Pulsation work amount and ATP consumption amount 
     Mean oxygen saturation [%] of blood at specified site per beat 
     As described above, prognosis per operative procedure is evaluated, and an appropriate operative procedure can easily be determined. Medical staff can use these visualization results for explanation to patients and their families. In addition, the visualization results can also be used as educational material. 
     By interactively communicating with the interactive system  150  including the pre-processing unit  120  and the post-processing unit  140 , the doctor  32  or the operator  33  can give instructions to the prognosis prediction system  100  and receive an evaluation result about prognosis per operative procedure from the prognosis prediction system  100 . 
       FIG. 24  is a sequence diagram illustrating the first half of a procedure of prognosis prediction processing using the interactive system. The doctor  32  transmits a simulation request via the terminal device  31  (step S 311 ). For example, by using phone or electronic mail, the doctor  32  requests the operator  33  to perform a simulation. In response to the request, the operator  33  activates the prognosis prediction system  100  and starts the operation of the interactive system  150  (step S 312 ). The pre-processing unit  120  in the interactive system  150  notifies the terminal device  31  used by the doctor  32  of the start of the system (step S 313 ). 
     When notified of the start of the system, the doctor  32  registers biological data in the prognosis prediction system  100  by operating the terminal device  31  (step S 314 ). For example, the biological data stored in the terminal device  31  is transmitted to the prognosis prediction system  100  and stored in the storage unit  110 . Upon completion of the registration of the biological data, the pre-processing unit  120  transmits a notification of the completion of the registration of the biological data to the terminal device  31  used by the doctor  32  (step S 315 ). In addition, the pre-processing unit  120  displays the notification of the completion of the registration of the biological data on the monitor  21  used by the operator  33  (step S 316 ). 
     Next, the pre-processing unit  120  starts generating preoperative finite element mesh models and transmits a notification of the start of the generation of the preoperative models to the terminal device  31  used by the doctor  32  (step S 317 ). In addition, the pre-processing unit  120  displays the notification of the start of the generation of the preoperative models on the monitor  21  used by the operator  33  (step S 318 ). 
     The operator  33  instructs the pre-processing unit  120  to check whether the preoperative models have been generated, by using the keyboard  22  or the mouse  23  (step S 319 ). After the preoperative models are generated, the operator  33  receives a response to such effect from the pre-processing unit  120 . After the preoperative models are generated, the operator  33  instructs the pre-processing unit  120  to transfer data by using the keyboard  22  or the mouse  23  (step S 320 ). 
     The pre-processing unit  120  transfers the data used in the simulation to the simulator  130  (step S 321 ). For large data, the pre-processing unit  120  may notify the simulator  130  of the corresponding storage area in the storage unit  110  so that the simulator  130  can use the large data. When receiving the data, the simulator  130  transmits a notification of the completion of the data transfer (step S 322 ). 
     Next, the operator  33  operates the keyboard  22  or the mouse  23  to instruct the simulator  130  to perform a simulation (step S 323 ). Next, the operator  33  operates the keyboard  22  or the mouse  23  to input information for parameter adjustment to the simulator  130  (step S 324 ). Next, the simulator  130  performs a fluid-structure interaction simulation. Upon completion of the simulation, the simulator  130  outputs a notification of the completion of the simulation (step S 325 ). The notification of the completion of the simulation is displayed on the monitor  21 , for example. 
     The operator  33  operates the keyboard  22  or the mouse  23  to instruct the simulator  130  to transfer data (step S 326 ). In accordance with the instruction, the simulator  130  transfers data indicating a simulation result to the post-processing unit  140  (step S 327 ). For large data, the simulator  130  may notify the post-processing unit  140  of the corresponding storage area in the storage unit  110  so that the post-processing unit  140  can use the large data. When receiving the data, the post-processing unit  140  transmits a notification of the completion of the data transfer to the simulator  130  (step S 328 ). The notification of the completion of the data transfer is displayed on the monitor  21  used by the operator  33 . 
     The operator  33  requests the doctor  32  to check the simulation data by phone, electronic mail, or the like (step S 329 ). The doctor  32  transmits a data transmission instruction to the post-processing unit  140  via the terminal device  31  (step S 330 ). The post-processing unit  140  transmits a 3D image indicating the simulation result to the terminal device  31  for the terminal device  31  to display the 3D image (step S 331 ). The doctor  32  observes the 3D image of the heart displayed on the terminal device  31  and checks the simulation result. Next, the doctor  32  notifies the post-processing unit  140  of the completion of the observation via the terminal device  31  (step S 332 ). 
     In this step, if values of parameters obtained as the simulation result differ from the biological data, the processing may be performed again from the generation of the preoperative models. For example, the doctor  32  instructs the pre-processing unit  120  to perform the processing again from the generation of the preoperative models via the terminal device  31 . In accordance with this instruction, the pre-processing unit  120  generates preoperative models, and the simulator  130  performs a simulation on the basis of the generated preoperative models. When the simulation is performed again, the parameters are adjusted again, for example. 
       FIG. 25  is a sequence diagram illustrating the second half of the procedure of the prognosis prediction processing using the interactive system. The doctor  32  notifies the operator  33  of a request for interaction by phone or electronic mail (step S 341 ). The operator  33  operates the keyboard  22  or the mouse  23  to input an instruction for reading and displaying preoperative data to the pre-processing unit  120  (step S 342 ). The pre processing unit  120  reads data about the geometric model of the preoperative heart or the like from the storage unit  110  and displays the read data on a screen of the terminal device  31  and the monitor  21  (step S 343 ). The doctor  32  gives instructions about a plurality of operative procedures to the pre-processing unit  120  via the terminal device  31  (step S 344 ). The operator  33  recognizes the instructions about the operative procedures from the doctor  32  on the monitor  21 . Next, the operator  33  operates the keyboard  22  or the mouse  23  to instruct the pre-processing unit  120  to reflect the specified operative procedures (step S 345 ). In addition, the operator  33  operates the keyboard  22  or the mouse  23  to instruct the pre-processing unit  120  to generate postoperative models (step S 346 ). The pre-processing unit  120  generates postoperative models in accordance with the instructions. Upon completion of the generation of the postoperative models, the pre-processing unit  120  displays a notification of the completion of the model generation on the monitor  21  (step S 347 ). 
     The operator  33  requests the doctor  32  to check the postoperative mesh models by phone or electronic mail (step S 348 ). The doctor  32  transmits a data read check instruction to the pre-processing unit  120  via the terminal device  31  (step S 349 ). The pre-processing unit  120  transmits 3D images indicating the postoperative models to the terminal device  31  for the terminal device  31  to display the 3D images (step S 350 ). After the doctor checks the 3D images of the heart displayed on the terminal device  31 , the doctor  32  notifies the operator  33  of completion of checking the postoperative models and of simulation conditions by phone or electronic mail (step S 351 ). Upon reception of the notification from the doctor  32 , the operator  33  operates the keyboard  22  or the mouse to instruct the pre-processing unit  120  to transfer data (step S 352 ). The pre-processing unit  120  transfers data used in simulations to the simulator  130  (step S 353 ). When receiving the data, the simulator  130  transmits a notification of the completion the data transfer to the pre-processing unit  120  (step S 354 ). In this step, the notification of the completion of the data transfer is displayed on the monitor  21  used by the operator  33 . 
     Next, the operator  33  operates the keyboard  22  or the mouse  23  to instruct the simulator  130  to perform simulations (step S 355 ). Next, the simulator  130  performs fluid-structure interaction simulations. Upon completion of the simulations, the simulator  130  outputs a notification of the completion of the simulations (step S 356 ). The notification of the completion of the simulations is displayed on the monitor  21 , for example. 
     The operator  33  operates the keyboard  22  or the mouse  23  to instruct the simulator  130  to transfer data (step S 357 ). In accordance with the instruction, the simulator  130  transfers the data indicating the simulation results to the post-processing unit  140  (step S 358 ). When receiving the data, the post-processing unit  140  transmits a notification of the completion of the data transfer to the simulator  130  (step S 359 ). In this step, the notification of the completion of the data transfer is displayed on the monitor  21  used by the operator  33 . 
     The operator  33  requests the doctor  32  to check the simulation data by phone, electronic mail, or the like (step S 360 ). The doctor  32  transmits a data read check instruction to the post-processing unit  140  via the terminal device  31  (step S 361 ). The post-processing unit  140  transmits 3D images indicating the simulation results to the terminal device  31  for the terminal device  31  to display the 3D images (step S 362 ). The doctor  32  observes the 3D images of the heart displayed on the terminal device  31  and checks the simulation results. Next, the doctor  32  instructs the post-processing unit  140  to rearrange the operative procedures according to the evaluation results via the terminal device  31  (step S 363 ). In this step, the doctor  32  can specify weight per parameter used for the rearrangement. The post-processing unit  140  transmits 3D images of the rearranged operative procedures to the terminal device  31  for the terminal device  31  to display the 3D images (step S 364 ). 
     As described with reference to  FIGS. 1 to 25 , in the first and second embodiments, interaction between the heart wall and the luminal blood on the basis of the ALE method and interaction between the luminal blood and the heart valves on the basis of the Lagrange multiplier method are simultaneously achieved. As a result, interaction between the heart wall and the heart valves is also achieved. Namely, a heart simulation in which interaction among the three is mechanically accurately introduced is achieved. 
     In addition, heart and trunk finite element mesh models having geometric characteristics sufficient for a heart simulator to reproduce the heart conditions of a biological body can be generated. In addition, a heart finite element mesh model that is needed for a simulation for predicting postoperative heart conditions after an operative procedure specified by a doctor is applied to a heart is obtained. In addition, effectiveness of a plurality of operative procedures can be checked visually and quantitatively, and information for helping clinicians to make decisions can be provided. 
     The biological simulation apparatus, the control method thereof, and the control program thereof according to the embodiments can be used for a method for accurately deriving at least a part of the dynamics and/or functions of a biological organ on the basis of corresponding biological data. The following derivation method is an example of the usage of the apparatus according to the embodiments. Thus, the following derivation method is not limited to the usage of the apparatus according to the embodiments. 
     Namely, there is provided a method for deriving at least a part of the dynamics and/or functions of a biological organ through calculation on the basis of biological data. The method includes:
         determining a geometric model of a biological organ;   creating a finite element mesh model from the geometric model;   performing a simulation by using the finite element mesh model on the basis of biological data; and   deriving dynamics and/or functions of the biological organ from a result of the simulation.       

     The biological simulation apparatus, the control method thereof, and the control program thereof according to the embodiments can be used for a method for predicting at least a part of the dynamics and/or functions of a biological organ before an operation is performed. The following method is an example of the usage of the apparatus according to the embodiments. Thus, the usage of the apparatus according to the embodiments is not limited to the following method. 
     Namely, there is provided a method for predicting at least a part of the dynamics and/or functions of a biological organ before an operation is performed. The method includes:
         determining a preoperative geometric model of a biological organ;   creating a preoperative finite element mesh model from the preoperative geometric model;   creating a postoperative finite element mesh model from the preoperative geometric model and/or the preoperative finite element mesh model;   performing a preoperative simulation by using the preoperative finite element mesh model on the basis of biological data and determining a parameter to be adjusted;   performing a postoperative simulation by using the postoperative finite element mesh model on the basis of the adjusted parameter; and   deriving predicted postoperative dynamics and/or functions of the biological organ from a result of the postoperative simulation.       

     The following description will be made on determining a geometric model of a biological organ, creating a finite element mesh model from the geometric model, performing a simulation and parameter adjustment, deriving prediction of dynamics and/or functions of the biological organ from a result of a simulation performed after a parameter is adjusted, creating a postoperative geometric model/a postoperative finite element mesh model from the preoperative geometric model/the preoperative finite element mesh model, performing a postoperative simulation by using the postoperative finite element mesh model on the basis of the adjusted parameter, and deriving prediction of dynamics and/or functions of the biological organ from a result of the postoperative simulation. 
     A geometric model of a biological organ is determined as follows. 
     A geometric model is numerical data with which a 3D shape of a biological organ whose dynamics are to be predicted and a 3D shape of an immediate biological organ can be expressed through visualization or the like by using biological data. For example, when a curved surface of a biological organ is approximated by a group of fine triangles, the 3D shape of the biological organ is expressed by the coordinates of the corners of the triangles. The biological data includes image data representing the 3D shapes of the target biological organ and the immediate biological organ. The image data is, for example, image data with which the shape of a biological organ can be identified two or three dimensionally, such as a CT image, an MRI image, or an echocardiogram of a biological organ or an immediate biological organ thereof. Information about temporal change is attached to part of the data. Other examples of the biological data include data about indices representing biological conditions, such as test data based on a cardiovascular catheterization test, an electrocardiogram, blood pressure, etc. and numerical data based on laboratory test values, other physiological tests, data based on an imaging test, a medical history of the patient including a history of operations and records of past operations, and knowledge of the doctor. 
     A geometric model of a biological organ is determined by segmenting the biological organ whose dynamics are to be predicted and an immediate domain of the biological organ from image data (domain segmentation). In this operation, biological data may be reflected as needed. When the biological organ is a heart, segmentation of the heart domain, the large blood vessel domain, and so forth is performed by using image data. 
     A finite element mesh model is created from a geometric model as follows. 
     A finite element mesh model represents geometric characteristics of a biological organ. For example, a finite element mesh model represents the structure domain of a biological organ and the fluid domain inside the biological organ. 
     As a suitable mode, a finite element mesh model includes a structure mesh model ( 2 ) that represents a structure domain in which the tissues of a biological organ exist by using the Lagrange description and an ALE fluid mesh model ( 3 ) that represents a fluid domain in which the fluid inside the biological organ exists by using the ALE (Arbitrary Lagrangian Eulerian) description method ( FIG. 1 ). 
     As another suitable mode, a finite element mesh model includes tetra mesh models of a biological organ and a domain therein, voxel mesh models of a biological organ and a domain therein, a tetra mesh model of a torso, and a voxel mesh model of a torso. 
     A tetra mesh model includes a triangle surface mesh that has a material number set per detailed site information and a tetra volume mesh that has a material number different per domain spatially segmented by the surfaces of the triangle surface mesh. A voxel mesh model includes structured grid data having a material number indicating domain information per voxel. 
     A torso voxel mesh model is structured grid data having domains segmented in accordance with the electric conductivity of the human body. 
     Tetra mesh models and a voxel mesh model A are created as follows. First, from segment data of a geometric model of a biological organ, a surface mesh formed by triangles is generated, and the surface mesh is segmented into finer sites. For example, the ventricles and luminal blood are segmented into finer sites such as the left ventricular free wall, the left ventricular papillary muscle, the blood domain in the left ventricle, the septum, the right ventricular free wall, the right ventricular papillary muscle, and the blood domain in the right ventricle. Thus, the sites include not only the myocardial domain but also the blood domains. For example, in the case of a heart, the sites include not only the myocardial domain but also the blood domains. Next, a triangle surface mesh segmented per site is generated. Next, two kinds of volume meshes (voxel and tetra) are created by using the surface mesh as boundaries. As a result, tetra mesh models (including a triangle surface mesh and a tetra volume mesh) and a voxel mesh model A (a voxel volume mesh) are generated. 
     When tetra elements are used to create a model of a thin structure, a large number of segments could be needed, and a mechanical problem could occur. In such cases, triangle shell elements or membrane elements may be used. For example, since a heart valve is a thin-walled structure, the heart valve may be segmented by using triangle shell elements or membrane elements. In the case of a linear structure such as a string, beam elements or cable elements may be used. 
     A torso voxel mesh model is created as follows. First, a torso domain is segmented. A torso domain is segmented into lungs, bones, organ domains mainly in accordance with luminance values from image data or other biological data. Next, when image data does not completely cover the torso domain, extrapolation processing for compensating for insufficient domains is performed. After the extrapolation processing, the position of the voxel mesh model A and the position of the torso voxel model are adjusted. The structured grid of the torso is reestablished in conformity with the voxel mesh A. 
     The finite element mesh models created in the above processing include the voxel mesh of the torso, the voxel mesh model A of the biological organ and the luminal blood therein, the tetra mesh model of the biological organ and the luminal blood therein, and the triangle mesh model of the thin-walled site. To associate the tetra and voxel meshes with each other, a voxel volume mesh is created by matching the torso voxel mesh model with the tetra mesh model of the biological organ. This will be used as a voxel mesh model B of the biological organ. 
     For example, when the biological organ is a heart, the finite element mesh models are tetra mesh models of the heart and the luminal blood, triangle mesh models of the heart valves, voxel mesh models of the heart and the luminal blood (corresponding to the above voxel mesh model B), and a voxel mesh model of the torso. 
     The simulation and the parameter adjustment are performed as follows. 
     As a suitable mode, the simulation is a fluid-structure interaction simulation about the motion of a biological organ and the motion of the fluid inside the biological organ. 
     It is preferable that the finite element mesh model created on the basis of a geometric model  1  include a structure mesh model  2  that represents a structure domain in which the tissues of a biological organ exist by using the Lagrange description and an ALE fluid mesh model  3  that represents a fluid domain in which the fluid inside the biological organ exists by using the ALE description method ( FIG. 1 ). Among the interfaces made by the structure domain and the fluid domain, an interface between a domain in which a site other than a certain site  2   a  of the biological organ exists and the fluid domain is determined to be a first interface  4 . In addition, among the interfaces made by the structure domain and the fluid domain, an interface between a domain in which the certain site  2   a  of the biological organ exists and the fluid domain is determined to be a second interface  5 . For example, the certain site  2   a  is a site protruding into the domain in which the ALE fluid mesh model  3  exists. For example, when the biological organ is a heart, the certain site  2   a  is a heart valve. 
     In the simulation, both the behavior of the biological organ and the motion of the fluid therein, including the interaction of the biological organ and the fluid, are simultaneously solved. A fluid-structure interaction simulation that obtains ever-changing equilibrium conditions is performed. The structure mesh model  2  is deformed in accordance with the motion of the biological organ, along with the progress of the fluid-structure interaction simulation. The ALE fluid mesh model is deformed by using an interface-tracking analysis method in such a manner that the first interface  4  is tracked. More specifically, the ALE fluid mesh model  3  is deformed in such a manner that no gap is formed on the first interface  4  or no overlap is formed with the structure mesh model  2  (the structure domain). Movement of nodes inside the ALE fluid mesh model  3  is artificially controlled separately from the motion of the fluid so that the soundness of each of the meshes is maintained. In addition, when performing the fluid-structure interaction simulation, the position of the second interface  5  using the ALE fluid mesh model  3  as a reference is captured by using an interface-capturing analysis method. For example, on the coordinate space for defining the shape of the ALE fluid mesh model  3 , the coordinates of the second interface  5  are calculated. 
     The fluid-structure interaction simulation enables a highly accurate simulation by deforming the ALE fluid mesh model  3  in accordance with the interface-tracking analysis method and by tracking the first interface  4  with the deformed the ALE fluid mesh model  3 . In addition, the certain site  2   a  such as a heart valve, which significantly motions and which has an interface that is difficult to track by using the interface-tracking analysis method, is captured on the ALE fluid mesh model  3  by using the interface-capturing analysis method. In this way, a fluid-structure interaction simulation is performed on a biological organ having a site that has an interface difficult to track. For example, the ALE method may be used as the interface-tracking analysis method. In addition, for example, a method based on the Lagrange multiplier method may be used as the interface-capturing analysis method (see, for example, T. Hisada et al., “Mathematical Considerations for Fluid-structure Interaction Simulation of Heart Valves”). 
       FIG. 2  illustrates examples of fluid-structure interaction analysis based on an interface-tracking analysis method using an ALE mesh when the biological organ is a heart.  FIG. 2  illustrates an ALE mesh used in the analysis and blood flow obtained as an analysis result. When an interface of the heart deforms, the ALE mesh also deforms with the deformation of the heart. In addition, the fluid velocity (fluid velocity vectors) and the pressure of the blood are analyzed on the coordinate system in which the deformable ALE mesh is defined. In this way, accurate analysis is performed. However, there is a limit to the deformation of the ALE mesh. Thus, it is difficult to track an interface of a site that significantly deforms such as a heart valve by using the ALE fluid mesh. 
       FIG. 3  illustrates the behavior of a heart and the motions of valves. The left portion in  FIG. 3  illustrates a heart when the ventricles contract, and the right portion in  FIG. 3  illustrates the heart when the ventricles relax. For example, when the heart muscle of a left ventricle  8  contracts, an aortic valve  7  opens, and the blood in the left ventricle  8  is discharged. In this state, a mitral valve  6  is closed. When the heart muscle of the left ventricle  8  relaxes, the mitral valve  6  opens, and blood flows into the left ventricle  8 . In this state, the aortic valve  7  is closed. In this way, an individual valve opens or closes in accordance with the pulsation of the heart. When the heart motions, an individual valve deforms with the opening and closing operations more significantly than the atria and ventricles deform with the contraction and relaxation of the heart muscles. It is difficult to track such valves that deform significantly by using the ALE fluid mesh model. 
     In contrast, with the Lagrange multiplier method, since the fluid mesh does not track an interface, the fluid around the heart valves can also be analyzed. 
       FIG. 4A to 4C  illustrates an example of fluid-structure interaction analysis based on an interface-capturing analysis method using the Lagrange multiplier method.  FIG. 4A  illustrates a structure mesh model of an aortic valve created independently from a fluid mesh model.  FIG. 4B  illustrates a spatially-fixed Euler fluid mesh model representing the fluid domain inside an aorta.  FIG. 4C  illustrates simulation results in which the fluid velocities of the blood are indicated by vectors. Assuming that the wall of the aorta is a rigid body and does not deform, the fluid mesh can be represented by a spatially-fixed Euler mesh. 
     In this way, in conventional fluid-structure interaction analysis based on the Lagrange multiplier method, analysis using a spatially-fixed Euler fluid mesh model has been performed. Thus, fluid-structure interaction analysis can be performed on a site that undergoes extreme deformation such as a valve, and change of the blood flow around the valve over time can be analyzed, for example. However, in the fluid-structure interaction analysis based on the Lagrange multiplier method, since an interface is not tracked, the stability and analysis accuracy of this analysis are less than those of the fluid-structure interaction analysis based on the interface tracking method using the ALE mesh. 
     Thus, an interface that can be tracked by the interface-tracking analysis method is tracked by using a deformable ALE mesh, and an interface that is difficult to track such as a valve is captured from the deformable ALE mesh. 
       FIGS. 5A to 5C  illustrate analysis techniques that are compared with each other.  FIG. 5A  illustrates an interface-tracking analysis method using the ALE method.  FIG. 5B  illustrates an analysis method in which an interface is captured from a spatially-fixed Euler mesh.  FIG. 5C  illustrates an analysis method in which an interface is captured from a deformable ALE mesh. 
     A structure mesh model  41  representing a structure domain in which the tissues of a biological organ exist is deformed as the simulation time progresses from t to t+Δt. When the interface-tracking analysis method is used, a ALE fluid mesh model  42  formed by the ALE mesh also deforms as the structure mesh model  41  deforms. However, since a fluid mesh model  43  based on the interface-capturing analysis method is spatially fixed, the fluid mesh model  43  does not deform even when the structure mesh model  41  deforms. 
     In the analysis method in which an interface is captured from a deformable mesh, an interface of the structure mesh model  41  in a trackable range is tracked by using the deformable ALE fluid mesh model  42 . Regarding an untrackable site (for example, a valve site  41   a ), the position of the interface of the corresponding site is calculated on the coordinate system in which the ALE fluid mesh model  42  is defined. In this way, an interface of a site that significantly deforms such as the valve site  41   a  can be captured. 
     Next, interface-tracking and -capturing methods will be described in detail. 
       FIGS. 6A and 6B  illustrate coordinate systems for interface tracking and capturing.  FIGS. 6A and 6B  illustrate an Euler (space) coordinate system (x 1 , x 2 ), a Lagrange (material) coordinate system (X 1 , X 2 ), and an ALE coordinate system (χ 1 , χ 2 ) that arbitrarily motions and deforms. A myocardial domain Ω S  is represented by the Lagrangian coordinate system. A blood domain Ω f  is represented by the ALE coordinate system. A heart valve  44  is protruding into the blood domain Ω f . While 3D representation is used in reality, two dimensional (2D) representation is used herein for simplicity. 
     An absolute velocity v i  at a material point X, a velocity w i  at the material point X observed from the ALE coordinate system, and a velocity (v i  with “{circumflex over ( )}”) in the ALE coordinate system controlled by an analyst are expressed as equations (41) to (43), respectively. 
     
       
         
           
             
               
                 
                   
                     
                       
                         v 
                         i 
                       
                       = 
                       
                         
                           ∂ 
                           
                             
                               x 
                               i 
                             
                             ⁡ 
                             
                               ( 
                               
                                 X 
                                 , 
                                 t 
                               
                               ) 
                             
                           
                         
                         
                           ∂ 
                           t 
                         
                       
                     
                      
                   
                   X 
                 
               
               
                 
                   ( 
                   41 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         w 
                         i 
                       
                       = 
                       
                         
                           ∂ 
                           
                             
                               χ 
                               i 
                             
                             ⁡ 
                             
                               ( 
                               
                                 X 
                                 , 
                                 t 
                               
                               ) 
                             
                           
                         
                         
                           ∂ 
                           t 
                         
                       
                     
                      
                   
                   X 
                 
               
               
                 
                   ( 
                   42 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           v 
                           ^ 
                         
                         i 
                       
                       = 
                       
                         
                           ∂ 
                           
                             
                               x 
                               i 
                             
                             ⁡ 
                             
                               ( 
                               
                                 χ 
                                 , 
                                 t 
                               
                               ) 
                             
                           
                         
                         
                           ∂ 
                           t 
                         
                       
                     
                      
                   
                   χ 
                 
               
               
                 
                   ( 
                   43 
                   ) 
                 
               
             
           
         
       
     
     If an equation of the motion of a continuum body derived from the law of conservation of momentum and the law of conservation of mass are written from the ALE coordinate system, equations (44) are (45) are obtained. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             ρ 
                             ^ 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               ∂ 
                               
                                 v 
                                 i 
                               
                             
                             
                               ∂ 
                               t 
                             
                           
                         
                          
                       
                       χ 
                     
                     + 
                     
                       
                         ρ 
                         ^ 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         w 
                         j 
                       
                       ⁢ 
                       
                         
                           ∂ 
                           
                             v 
                             i 
                           
                         
                         
                           ∂ 
                           
                             x 
                             j 
                           
                         
                       
                     
                   
                   = 
                   
                     
                       
                         ∂ 
                         
                           
                             Π 
                             ^ 
                           
                           ji 
                         
                       
                       
                         ∂ 
                         
                           x 
                           j 
                         
                       
                     
                     + 
                     
                       
                         ρ 
                         ^ 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         g 
                         i 
                       
                     
                   
                 
               
               
                 
                   ( 
                   44 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             ∂ 
                             
                               ρ 
                               ^ 
                             
                           
                           
                             ∂ 
                             t 
                           
                         
                          
                       
                       χ 
                     
                     + 
                     
                       
                         
                           ∂ 
                           
                             ρ 
                             ^ 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           w 
                           i 
                         
                       
                       
                         ∂ 
                         
                           x 
                           i 
                         
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   45 
                   ) 
                 
               
             
           
         
       
     
     In the above equations, ρ with “{circumflex over ( )}” and Π with “{circumflex over ( )}” indicate the mass density and the 1st Piola-Kirchhiff stress tensor defined by using the reference configuration in the ALE coordinate system as a reference. The second term on the right-hand side of equation (44) represents arbitrary body force. 
     For example, the motion equation of the luminal blood of the heart is described from the ALE coordinate system different from the Lagrangian coordinate system used for the myocardial domain as illustrated in  FIG. 1  (see, for example, M. A. Castro et al., “Computational fluid dynamics modeling of intracranial aneurysms: effects of parent artery segmentation on intra-aneurysmal hemodynamics” and E. Tang et al., “Geometric Characterization of Patient-Specific Total Cavopulmonary Connections and its Relationship to Hemodynamics”), and the valve motions in the space of the ALE coordinate system. Thus, to cause the blood and the valve to interact with each other by applying the interface-capturing analysis method based on the Lagrange multiplier method proposed in “Multi-physics simulation of left ventricular filling dynamics using fluid-structure interaction finite element method” by H. Watanabe et al., the ALE coordinate system is basically used as the common coordinate system. Thus, the law of conservation of mass in one-side blood domain Ω K  (K=1, 2) defined by a delta function near a point A on the valve is imposed as expressed by the equation (46). The match between a blood velocity v F  and the velocity components in a normal direction n C  regarding a valve velocity v s  is imposed as expressed by the equation (47), where Γ C  represents an interface. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               ∫ 
                               
                                 Ω 
                                 K 
                               
                             
                             ⁢ 
                             
                               
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                                 ⁡ 
                                 
                                   ( 
                                   
                                     χ 
                                     - 
                                     
                                       χ 
                                       A 
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 
                                   ( 
                                   
                                     
                                       ∂ 
                                       
                                         ρ 
                                         ^ 
                                       
                                     
                                     
                                       ∂ 
                                       t 
                                     
                                   
                                    
                                 
                                 χ 
                               
                             
                           
                           + 
                           
                             
                               
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                                   ^ 
                                 
                               
                               ⁢ 
                               
                                   
                               
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                                 w 
                                 i 
                               
                             
                             
                               
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                               ⁢ 
                               
                                   
                               
                             
                           
                         
                         ) 
                       
                       ⁢ 
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                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         Ω 
                         K 
                       
                     
                     = 
                     
                       
                         0 
                         ⁢ 
                         
                           
 
                         
                         ⁢ 
                         K 
                       
                       = 
                       1 
                     
                   
                   , 
                   2 
                 
               
               
                 
                   ( 
                   46 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       n 
                       C 
                     
                     · 
                     
                       ( 
                       
                         
                           
                             v 
                             F 
                           
                           ⁡ 
                           
                             ( 
                             χ 
                             ) 
                           
                         
                         - 
                         
                           
                             v 
                             S 
                           
                           ⁡ 
                           
                             ( 
                             χ 
                             ) 
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     0 
                     ⁢ 
                     
                         
                     
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                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
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                       c 
                     
                   
                 
               
               
                 
                   ( 
                   47 
                   ) 
                 
               
             
           
         
       
     
     However, since the formulation based on the concept of the reference configuration of the ALE coordinate system complicates the equations, the equations including the motion equation of the blood are converted to be expressed in the Euler coordinate system. Consequently, since equations (44) and (45) are rewritten as equations (48) and (49), equations (46) and (47) are rewritten as equations (50) and (51), respectively. Note that the following relationships in equation (52) are used for the conversion. In equation (48), T ji  represents the Cauchy stress tensor component. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           ρ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               ∂ 
                               
                                 v 
                                 i 
                               
                             
                             
                               ∂ 
                               t 
                             
                           
                         
                          
                       
                       χ 
                     
                     + 
                     
                       ρ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         c 
                         j 
                       
                       ⁢ 
                       
                         
                           ∂ 
                           
                             v 
                             i 
                           
                         
                         
                           ∂ 
                           
                             x 
                             j 
                           
                         
                       
                     
                   
                   = 
                   
                     
                       
                         ∂ 
                         
                           T 
                           ji 
                         
                       
                       
                         ∂ 
                         
                           x 
                           j 
                         
                       
                     
                     + 
                     
                       ρ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         g 
                         j 
                       
                     
                   
                 
               
               
                 
                   ( 
                   48 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
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                             ρ 
                           
                           
                             ∂ 
                             t 
                           
                         
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                     + 
                     
                       
                         c 
                         i 
                       
                       ⁢ 
                       
                         
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                           ρ 
                         
                         
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                             x 
                             i 
                           
                         
                       
                     
                     + 
                     
                       ρ 
                       ⁢ 
                       
                         
                           ∂ 
                           
                             v 
                             i 
                           
                         
                         
                           ∂ 
                           
                             x 
                             i 
                           
                         
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   49 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             
                               ∫ 
                               
                                 Ω 
                                 K 
                               
                             
                             ⁢ 
                             
                               
                                 δ 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     x 
                                     - 
                                     
                                       x 
                                       A 
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 
                                   ( 
                                   
                                     
                                       ∂ 
                                       ρ 
                                     
                                     
                                       ∂ 
                                       t 
                                     
                                   
                                    
                                 
                                 x 
                               
                             
                           
                           + 
                           
                             
                               c 
                               i 
                             
                             ⁢ 
                             
                               
                                 ∂ 
                                 ρ 
                               
                               
                                 ∂ 
                                 
                                   x 
                                   i 
                                 
                               
                             
                           
                           + 
                           
                             ρ 
                             ⁢ 
                             
                               
                                 ∂ 
                                 
                                   v 
                                   i 
                                 
                               
                               
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                                   x 
                                   i 
                                 
                               
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       d 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         Ω 
                         K 
                       
                     
                     = 
                     0 
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       K 
                       = 
                       1 
                     
                     , 
                     2 
                   
                 
               
               
                 
                   ( 
                   50 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       n 
                       C 
                     
                     · 
                     
                       ( 
                       
                         
                           
                             v 
                             F 
                           
                           ⁡ 
                           
                             ( 
                             x 
                             ) 
                           
                         
                         - 
                         
                           
                             v 
                             S 
                           
                           ⁡ 
                           
                             ( 
                             
                               X 
                               ⁡ 
                               
                                 ( 
                                 x 
                                 ) 
                               
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     0 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     on 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       Γ 
                       c 
                     
                   
                 
               
               
                 
                   ( 
                   51 
                   ) 
                 
               
             
             
               
                 
                   
                     χ 
                     = 
                     x 
                   
                   , 
                   
                     
                       
                         v 
                         ^ 
                       
                       i 
                     
                     = 
                     0 
                   
                   , 
                   
                     
                       v 
                       i 
                     
                     = 
                     
                       w 
                       i 
                     
                   
                   , 
                   
                     
                       c 
                       i 
                     
                     = 
                     
                       
                         
                           v 
                           i 
                         
                         - 
                         
                           
                             v 
                             ^ 
                           
                           i 
                         
                       
                       = 
                       
                         
                           
                             w 
                             j 
                           
                           ⁢ 
                           
                             
                               ∂ 
                               
                                 x 
                                 i 
                               
                             
                             
                               ∂ 
                               
                                 χ 
                                 j 
                               
                             
                           
                         
                         = 
                         
                           w 
                           i 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   52 
                   ) 
                 
               
             
           
         
       
     
     In contrast, the motion equation of a heart muscle is expressed as the following equation by using the Lagrangian coordinate system. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           ρ 
                           0 
                         
                         ⁢ 
                         
                           
                             ∂ 
                             v 
                           
                           
                             ∂ 
                             t 
                           
                         
                       
                        
                     
                     X 
                   
                   = 
                   
                     
                       
                         ∇ 
                         X 
                       
                       ⁢ 
                       
                         · 
                         Π 
                       
                     
                     + 
                     
                       
                         ρ 
                         0 
                       
                       ⁢ 
                       
                         g 
                         ~ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   53 
                   ) 
                 
               
             
           
         
       
     
     Next, regarding the constitutive equation of the material, the blood is expressed by the following equation as incompressible Newton fluid, wherein μ represents a viscosity coefficient.
 
 T=−pI+ 2μ D   (54)
 
 D= ½(∇ x     v+v     ∇   x )  (55)
 
∇ x   ·v= 0  (56)
 
     Thus, the first and second terms of the law of conservation of mass are eliminated. In addition, if the tangent stiffness of the constitutive equation of the heart muscle described in the Lagrangian coordinate system is represented by a fourth-order tensor C, the following equations are established. In the following equations, S represents the 2nd Piola-Kirchhoff stress, F represents the deformation gradient tensor, and E represents the Green-Lagrange strain tensor.
 
Π= S·F   T   (57)
 
 F=x     ∇   x   (58)
 
Π=(det  F ) F   −1   ·T   (59)
 
 S=C:E   (60)
 
 E= ½{∇ x     u+u     ∇   x +(∇ x     u )·( u     ∇   x )}  (61)
 
     Thus, first, by displaying a variational form equation of the Navier-Stokes equation of the blood observed based on the ALE coordinate system in the Euler coordinate system, equation (62) is derived. Next, a variational form equation of the motion equation of the heart wall is derived as equation (63). In addition, by applying the divergence theorem to constraint condition equations (64) and (65) between the blood and the valve, equation (66) is derived. Consequently, the stationary condition equation of the entire system is expressed as equation (67). 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               W 
                               f 
                               ALE 
                             
                           
                           ≡ 
                           
                             
                               ∫ 
                               
                                 Ω 
                                 f 
                               
                             
                             ⁢ 
                             
                               δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 v 
                                 · 
                                 
                                   ρ 
                                   f 
                                 
                               
                               ⁢ 
                               
                                 
                                   ∂ 
                                   v 
                                 
                                 
                                   ∂ 
                                   t 
                                 
                               
                             
                           
                         
                          
                       
                       x 
                     
                     ⁢ 
                     d 
                     ⁢ 
                     
                         
                     
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                       f 
                     
                   
                   + 
                   
                     
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                         Ω 
                         f 
                       
                     
                     ⁢ 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         v 
                         · 
                         
                           
                             ρ 
                             f 
                           
                           ⁡ 
                           
                             ( 
                             
                               v 
                               ⊗ 
                               
                                 ∇ 
                                 x 
                               
                             
                             ) 
                           
                         
                         · 
                         Cd 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         Ω 
                         f 
                       
                     
                   
                   + 
                   
                     2 
                     ⁢ 
                     μ 
                     ⁢ 
                     
                       
                         ∫ 
                         
                           Ω 
                           f 
                         
                       
                       ⁢ 
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         D 
                         ⁢ 
                         
                           : 
                         
                         ⁢ 
                         Dd 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           Ω 
                           f 
                         
                       
                     
                   
                   - 
                   
                     
                       ∫ 
                       
                         Ω 
                         f 
                       
                     
                     ⁢ 
                     
                       
                         ρ 
                         f 
                       
                       ⁢ 
                       
                         
                           ∇ 
                           x 
                         
                         ⁢ 
                         
                           · 
                           δ 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       vd 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         Ω 
                         S 
                       
                     
                   
                   - 
                   
                     
                       ∫ 
                       
                         Ω 
                         f 
                       
                     
                     ⁢ 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         p 
                         ⁡ 
                         
                           ( 
                           
                             
                               ∇ 
                               x 
                             
                             ⁢ 
                             
                               · 
                               v 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       d 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         Ω 
                         f 
                       
                     
                   
                   - 
                   
                     
                       ∫ 
                       
                         Ω 
                         f 
                       
                     
                     ⁢ 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         v 
                         · 
                         
                           ρ 
                           f 
                         
                       
                       ⁢ 
                       gd 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         Ω 
                         f 
                       
                     
                   
                   - 
                   
                     
                       ∫ 
                       
                         Γ 
                         f 
                       
                     
                     ⁢ 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         v 
                         · 
                         
                           
                             τ 
                             ~ 
                           
                           _ 
                         
                       
                       ⁢ 
                       d 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         Γ 
                         f 
                       
                     
                   
                 
               
               
                 
                   ( 
                   62 
                   ) 
                 
               
             
             
               
                 
                   
                     δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       W 
                       S 
                     
                   
                   = 
                   
                     
                       
                         ∫ 
                         
                           Ω 
                           s 
                         
                       
                       ⁢ 
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             u 
                             . 
                           
                           · 
                           
                             ρ 
                             S 
                           
                         
                         ⁢ 
                         
                           u 
                           ¨ 
                         
                         ⁢ 
                         d 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           Ω 
                           S 
                         
                       
                     
                     + 
                     
                       
                         ∫ 
                         
                           Ω 
                           s 
                         
                       
                       ⁢ 
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           E 
                           . 
                         
                         ⁢ 
                         
                           : 
                         
                         ⁢ 
                         Sd 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           Ω 
                           S 
                         
                       
                     
                     - 
                     
                       
                         ∫ 
                         
                           Ω 
                           s 
                         
                       
                       ⁢ 
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             u 
                             . 
                           
                           · 
                           
                             ρ 
                             S 
                           
                         
                         ⁢ 
                         gd 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           Ω 
                           S 
                         
                       
                     
                     - 
                     
                       
                         ∫ 
                         
                           Γ 
                           s 
                         
                       
                       ⁢ 
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             u 
                             . 
                           
                           · 
                           
                             
                               τ 
                               ~ 
                             
                             _ 
                           
                         
                         ⁢ 
                         d 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           Γ 
                           S 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   63 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         
                           ∫ 
                           
                             Ω 
                             K 
                           
                         
                         ⁢ 
                         
                           
                             δ 
                             ⁡ 
                             
                               ( 
                               
                                 x 
                                 - 
                                 
                                   x 
                                   A 
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               ∇ 
                               x 
                             
                             ⁢ 
                             
                               · 
                               vd 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             Ω 
                             K 
                           
                         
                       
                       = 
                       0 
                     
                     , 
                     
                       K 
                       = 
                       1 
                     
                     , 
                     2 
                   
                 
               
               
                 
                   ( 
                   64 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         n 
                         C 
                       
                       · 
                       
                         ( 
                         
                           v 
                           - 
                           
                             u 
                             . 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       0 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       on 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         Γ 
                         c 
                       
                     
                   
                 
               
               
                 
                   ( 
                   65 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         ∑ 
                         
                           K 
                           = 
                           1 
                         
                         2 
                       
                       ⁢ 
                       
                         
                           
                             ( 
                             
                               - 
                               1 
                             
                             ) 
                           
                           K 
                         
                         ⁢ 
                         
                           1 
                           2 
                         
                         ⁢ 
                         
                           
                             ∫ 
                             
                               Ω 
                               f 
                             
                           
                           ⁢ 
                           
                             ∇ 
                             
                               ⊗ 
                               
                                 
                                   
                                     δ 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         x 
                                         - 
                                         
                                           x 
                                           A 
                                         
                                       
                                       ) 
                                     
                                   
                                   · 
                                   
                                     v 
                                     ⁡ 
                                     
                                       ( 
                                       x 
                                       ) 
                                     
                                   
                                 
                                 ⁢ 
                                 d 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   Ω 
                                   K 
                                 
                               
                             
                           
                         
                       
                     
                     + 
                     
                       
                         ∫ 
                         
                           Γ 
                           c 
                         
                       
                       ⁢ 
                       
                         
                           δ 
                           ⁡ 
                           
                             ( 
                             
                               x 
                               - 
                               
                                 x 
                                 A 
                               
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           
                             n 
                             C 
                           
                           · 
                           
                             
                               u 
                               . 
                             
                             ⁡ 
                             
                               ( 
                               
                                 x 
                                 A 
                               
                               ) 
                             
                           
                         
                         ⁢ 
                         d 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           Γ 
                           C 
                         
                       
                     
                   
                   ≡ 
                   
                     C 
                     ⁡ 
                     
                       ( 
                       
                         
                           x 
                           ⁢ 
                           
                             : 
                           
                           ⁢ 
                           v 
                         
                         , 
                         
                           u 
                           . 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   66 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         W 
                         total 
                       
                     
                     ≡ 
                     
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           W 
                           f 
                           ALE 
                         
                       
                       + 
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           W 
                           S 
                           1 
                         
                       
                       + 
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           W 
                           S 
                           2 
                         
                       
                       + 
                       
                         
                           ∫ 
                           
                             Γ 
                             c 
                           
                         
                         ⁢ 
                         
                           
                             λ 
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                           ⁢ 
                           
                             C 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   x 
                                   ⁢ 
                                   
                                     : 
                                   
                                   ⁢ 
                                   δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   v 
                                 
                                 , 
                                 
                                   δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     u 
                                     . 
                                   
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             Γ 
                             C 
                           
                         
                       
                       + 
                       
                         
                           ∫ 
                           
                             Γ 
                             c 
                           
                         
                         ⁢ 
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             λ 
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                           ⁢ 
                           
                             C 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   x 
                                   ⁢ 
                                   
                                     : 
                                   
                                   ⁢ 
                                   v 
                                 
                                 , 
                                 u 
                               
                               ) 
                             
                           
                           ⁢ 
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             Γ 
                             C 
                           
                         
                       
                       - 
                       
                         
                           ∫ 
                           
                             Γ 
                             c 
                           
                         
                         ⁢ 
                         
                           
                             ɛ 
                             λ 
                           
                           ⁢ 
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             λ 
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                           ⁢ 
                           
                             λ 
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                           ⁢ 
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             Γ 
                             C 
                           
                         
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   67 
                   ) 
                 
               
             
           
         
       
     
     In the middle side of equation (67), δW S   1  and δW S   2  represent the first variations of the entire potential regarding the heart muscle and an individual valve. The last term in the middle side is a stabilization term. 
     In addition, connection between a heart valve and a wall, interaction made by contact between heart valves or between a heart valve and a heart wall, and connection with the systemic circulation analogy circuit can also be performed by using the Lagrange multiplier method. In this case, the stationary condition equation of the entire system is expressed as equation (68). 
     
       
         
           
             
               
                 
                   
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         W 
                         total 
                       
                     
                     ≡ 
                     
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           W 
                           f 
                           ALE 
                         
                       
                       + 
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           W 
                           S 
                           1 
                         
                       
                       + 
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           W 
                           
                             S 
                             ⁢ 
                             
                                 
                             
                           
                           2 
                         
                       
                       + 
                       
                         
                           ∫ 
                           
                             Γ 
                             c 
                           
                         
                         ⁢ 
                         
                           
                             λ 
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                           ⁢ 
                           
                             C 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   x 
                                   ⁢ 
                                   
                                     : 
                                   
                                   ⁢ 
                                   δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   v 
                                 
                                 , 
                                 
                                   δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     u 
                                     . 
                                   
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             Γ 
                             C 
                           
                         
                       
                       + 
                       
                         
                           ∫ 
                           
                             Γ 
                             c 
                           
                         
                         ⁢ 
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             λ 
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                           ⁢ 
                           
                             C 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   x 
                                   ⁢ 
                                   
                                     : 
                                   
                                   ⁢ 
                                   v 
                                 
                                 , 
                                 u 
                               
                               ) 
                             
                           
                           ⁢ 
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             Γ 
                             C 
                           
                         
                       
                       - 
                       
                         
                           ∫ 
                           
                             Γ 
                             c 
                           
                         
                         ⁢ 
                         
                           
                             ɛ 
                             λ 
                           
                           ⁢ 
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             λ 
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                           ⁢ 
                           
                             λ 
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                           ⁢ 
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             Γ 
                             C 
                           
                         
                       
                       + 
                       
                         
                           ∫ 
                           
                             Γ 
                             L 
                           
                         
                         ⁢ 
                         
                           
                             
                               τ 
                               ⁡ 
                               
                                 ( 
                                 x 
                                 ) 
                               
                             
                             · 
                             
                               ( 
                               
                                 
                                   δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     x 
                                     L 
                                   
                                 
                                 - 
                                 
                                   
                                     W 
                                     L 
                                     W 
                                   
                                   ⁢ 
                                   δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     x 
                                     W 
                                   
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           dl 
                         
                       
                       + 
                       
                         
                           ∫ 
                           
                             Γ 
                             L 
                           
                         
                         ⁢ 
                         
                           
                             
                               δτ 
                               ⁡ 
                               
                                 ( 
                                 x 
                                 ) 
                               
                             
                             · 
                             
                               ( 
                               
                                 
                                   x 
                                   L 
                                 
                                 - 
                                 
                                   
                                     W 
                                     L 
                                     W 
                                   
                                   ⁢ 
                                   
                                     x 
                                     W 
                                   
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           dl 
                         
                       
                       - 
                       
                         
                           ∫ 
                           
                             Γ 
                             R 
                           
                         
                         ⁢ 
                         
                           
                             ɛ 
                             τ 
                           
                           ⁢ 
                           
                             
                               δτ 
                               ⁡ 
                               
                                 ( 
                                 x 
                                 ) 
                               
                             
                             · 
                             
                               τ 
                               ⁡ 
                               
                                 ( 
                                 x 
                                 ) 
                               
                             
                           
                           ⁢ 
                           dl 
                         
                       
                       + 
                       
                         
                           ∑ 
                           
                             i 
                             , 
                             j 
                           
                         
                         ⁢ 
                         
                           
                             ∫ 
                             
                               Γ 
                               
                                 T 
                                 , 
                                 i 
                               
                             
                           
                           ⁢ 
                           
                             
                               
                                 η 
                                 ij 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   x 
                                   i 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 C 
                                 t 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     
                                       x 
                                       i 
                                     
                                     ⁢ 
                                     
                                       : 
                                     
                                     ⁢ 
                                     
                                       u 
                                       f 
                                     
                                   
                                   , 
                                   
                                     u 
                                     j 
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               ds 
                               i 
                             
                           
                         
                       
                       + 
                       
                         
                           ∑ 
                           
                             i 
                             , 
                             j 
                           
                         
                         ⁢ 
                         
                           
                             ∫ 
                             
                               Γ 
                               
                                 T 
                                 , 
                                 i 
                               
                             
                           
                           ⁢ 
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 η 
                                 ij 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   x 
                                   i 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               
                                 C 
                                 t 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     
                                       x 
                                       i 
                                     
                                     ⁢ 
                                     
                                       : 
                                     
                                     ⁢ 
                                     
                                       u 
                                       i 
                                     
                                   
                                   , 
                                   
                                     u 
                                     j 
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               ds 
                               i 
                             
                           
                         
                       
                       - 
                       
                         
                           ∑ 
                           ij 
                         
                         ⁢ 
                         
                           
                             ∫ 
                             
                               Γ 
                               
                                 T 
                                 , 
                                 i 
                               
                             
                           
                           ⁢ 
                           
                             
                               ɛ 
                               η 
                             
                             ⁢ 
                             
                               
                                 δη 
                                 ij 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   x 
                                   i 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               
                                 η 
                                 ij 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   x 
                                   i 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             ds 
                           
                         
                       
                       + 
                       
                         
                           ∫ 
                           
                             Γ 
                             
                               f 
                               , 
                               k 
                             
                           
                         
                         ⁢ 
                         
                           
                             P 
                             k 
                           
                           ⁢ 
                           
                             
                               n 
                               k 
                             
                             · 
                             δ 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             vds 
                             k 
                           
                         
                       
                       + 
                       
                         
                           ∫ 
                           
                             Γ 
                             
                               f 
                               , 
                               k 
                             
                           
                         
                         ⁢ 
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               P 
                               k 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     n 
                                     k 
                                   
                                   · 
                                   v 
                                 
                                 - 
                                 
                                   F 
                                   k 
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             ds 
                             k 
                           
                         
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   68 
                   ) 
                 
               
             
           
         
       
     
     Γ L  represents a connection interface between a heart valve and a myocardial wall, χ L , and χ W  represent node coordinates of a heart valve and a myocardial wall, W R   W  represents an interpolation coefficient from a myocardial wall node to a connection point, Γ T,i  represents a surface that could be brought into contact, and C t  is a contact condition equation. In addition, Γ f,k  represents a blood domain interface connected to the systemic circulation analogy circuit, F k  represents a flow volume to the systemic circulation model on a connection surface. In addition, P k  represents the blood pressure of the systemic circulation model on a connection surface, and v k  and Q k  represent the blood pressure variable and the capacity variable in the systemic circulation model, respectively. With these parameters, the equation is solved as a system including the next balance equation of the systemic circulation analogy circuit.
 
 G   k ( P   k   ,F   k   ,V   k   ,Q   k )=0  (69)
 
     On the basis of the above formulation, finite element discretization is performed, and a heart simulation is performed. A procedure of the fluid-structure interaction simulation by a simulator is as follows. 
       FIG. 14  is a flowchart illustrating an example of a procedure of the simulation. 
     [Step S 151 ] The simulator  130  updates time of the time step. For example, the simulator  130  adds a predetermined time increment Δt to the current time t. When the simulation is started, the time is set to t=0. 
     [Step S 152 ] The simulator  130  calculates a stiffness matrix, internal force integration, and various conditional integrations and the corresponding differentiations. The matrix, differentiations, and integrations reflect the ALE fluid-structure interaction and fluid-structure interaction based on the Lagrange multiplier method. 
     [Step S 153 ] The simulator  130  synthesizes a global matrix A and a global right-side vector b. 
     [Step S 154 ] The simulator  130  calculates updated amounts of variables. 
     [Step S 155 ] The simulator  130  updates the ALE mesh. 
     [Step S 156 ] The simulator  130  determines whether the calculation result has converged. If so, the processing proceeds to step S 157 . If not, the processing returns to step S 152  and the calculation processing is performed again. 
     [Step S 157 ] The simulator  130  determines whether the simulation time t has reached finish time t end . For example, the finish time t end  is when a simulation of a single beat of the heart is finished. When the finish time t end  has been reached, the simulator  130  ends the simulation. If the finish time t end  has not been reached, the processing returns to step S 151 . 
     The processing in steps S 152  to S 155  will hereinafter be described in detail. 
       FIG. 15  is a flowchart illustrating an example of a procedure of the differentiation and integration calculation processing. 
     [Step S 161 ] The simulator  130  calculates an internal force integrated value f fs  and a stiffness matrix A fs  of equation (70).
 
δ W   f   ALE   +δW   S   1   +δW   S   2   (70)
 
     [Step S 162 ] The simulator  130  calculates an internal force integrated value f C  of equation (71) and an integrated value C C  and differentiation B C  of equation (72).
 
∫ Γ     C   λ( x ) C ( x:δv,δ{dot over (u)} ) dΓ   C   (71)
 
∫ Γ     C   δλ( x ) C ( x:v,u ) dΓ   C   (72)
 
     [Step S 163 ] The simulator  130  calculates an internal force integrated value f R  of equation (73) and an integrated value C R  and differentiation B R  of equation (74).
 
∫ Γ     R   τ( x )·(δ x   R   −W   R   W   δx   W ) dl   (73)
 
∫ Γ     R   δτ( x )·( x   R   −W   R   W   x   W ) dl   (74)
 
     [Step S 164 ] The simulator  130  calculates an internal force integrated value f T  of equation (75) and an integrated value C T  and differentiation B T  of equation (76). 
     
       
         
           
             
               
                 
                   
                     ∑ 
                     
                       i 
                       , 
                       j 
                     
                   
                   ⁢ 
                   
                     
                       ∫ 
                       
                         Γ 
                         
                           T 
                           , 
                           i 
                         
                       
                     
                     ⁢ 
                     
                       
                         
                           η 
                           ij 
                         
                         ⁡ 
                         
                           ( 
                           
                             x 
                             i 
                           
                           ) 
                         
                       
                       ⁢ 
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           C 
                           t 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 x 
                                 i 
                               
                               ⁢ 
                               
                                 : 
                               
                               ⁢ 
                               
                                 u 
                                 i 
                               
                             
                             , 
                             
                               u 
                               j 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         ds 
                         i 
                       
                     
                   
                 
               
               
                 
                   ( 
                   75 
                   ) 
                 
               
             
             
               
                 
                   
                     ∑ 
                     
                       i 
                       , 
                       j 
                     
                   
                   ⁢ 
                   
                     
                       ∫ 
                       
                         Γ 
                         
                           T 
                           , 
                           i 
                         
                       
                     
                     ⁢ 
                     
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                       ⁢ 
                       
                           
                       
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     [Step S 165 ] The simulator  130  calculates an internal force integrated value f f  of equation (77) and an integrated value C f  and differentiation B f  of equation (78).
 
∫ Γ     f,k     P   k   n   k   ·δvds   k   (77)
 
∫ Γ     f,k     δP   k   n   k   ·vds   k =0  (78)
 
     [Step S 166 ] The simulator  130  calculates a value C G  and differentiation B G  of G k  (P k , F k , V k , Q k ). 
     From the calculation results such as the matrix and integrated values calculated in the processing illustrated in  FIG. 15 , the simulator  130  creates and synthesizes the global matrix A and the global right-side vector b (step S 153 ). Specifically, the synthesis of the global matrix A and the global right-side vector b is performed as follows. 
     
       
         
           
             
               
                 
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     After the synthesis of the global matrix A and the global right-side vector b, processing for calculating updated amounts of variables is performed (step S 154 ). Specifically, the following equation is solved. 
     
       
         
           
             
               
                 
                   
                     
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     In the above equation (80), Δ represents an updated amount. A time integration method such as the Newmark-β method is used for time evolution. 
     Next, the blood domain is extracted from the entire system updated in this way, and updating the ALE mesh (step S 155 ) is calculated. Specifically, an equation for mesh control such as for a hyperelastic body is solved by using an interface with a new heart muscle as a boundary condition, and a new ALE mesh is generated. 
     If the total correction amount in the above process is not sufficiently small, the processing returns to step S 152 , and a calculation for updating the solution is performed. If the correction amount reaches to a predetermined value or less, the simulator  130  determines that convergence in the time increment of Δt has been achieved. Thus, calculation of the next time step is performed. When the finish time t end  of the analysis target is reached, the simulator  130  ends the present processing. 
     As described above, by performing a fluid-structure interaction simulation in which the ALE method and the Lagrange multiplier method are combined with each other, an accurate simulation can be performed even when there is a site that significantly motions. 
     In another suitable embodiment, the biological organ is a heart, and the simulation includes an electrical excitation propagation simulation and a mechanical pulsation simulation. The mechanical pulsation simulation is performed in accordance with a simulation method of fluid-structure interaction in which the heart tetra mesh model is a structure mesh model that represents a structure domain and the luminal blood tetra mesh model is a fluid mesh model that represents a fluid domain ( FIG. 13 ). 
     The electrical excitation propagation simulation and parameter adjustment can be performed as follows. First, fiber and sheet distributions are set to a heart voxel mesh model on the basis of literature values or the like. In addition, a Purkinje fiber distribution or an equivalent endocardial conductance distribution is set. In addition, a site of earliest activation is set to the endocardium. In addition, cell models having three kinds of APD distribution are distributed in the long axis direction and the short axis direction. For example, a method in “Computational fluid dynamics modeling of intracranial aneurysms: effects of parent artery segmentation on intra-aneurysmal hemodynamics” by M. A. Castro et al. is used for these settings. Next, a simulator performs an electrical excitation propagation simulation by combining the heart voxel mesh model set as described above and a torso voxel mesh model having a body surface on which electrodes for a standard 12-lead electrocardiogram are set. Next, parameters are adjusted so that the simulation result matches the result of the standard 12-lead electrocardiogram test on the biological body whose dynamics are to be predicted. Examples of the parameters include a site of earliest activation of the endocardium and three kinds of APD distribution. 
     By the electrical excitation propagation simulation and the parameter adjustment, for example, the time history of the calcium ion concentration (calcium concentration history) of an individual element of the heart tetra mesh model constituting the heart can be determined. 
     The mechanical pulsation simulation and parameter adjustment can be performed as follows. 
     First, as is the case with the heart voxel mesh model, fiber and sheet distributions are set to the heart tetra mesh model. In addition, boundary conditions for performing a mechanical simulation are set, and duplexing of pressure nodes for mechanical analysis and duplexing of nodes at an interface between the blood and the heart muscle are performed, for example. In addition, information about an individual site of the heart is also given. Next, a natural-condition heart model (tetra element) is created from the heart tetra mesh model. Namely, while the heart tetra mesh model is created for a relaxation phase of the heart, to obtain a natural shape corresponding to a free-stress condition, a natural-condition heart model (tetra element) is created by performing suitable mechanical finite element analysis and reducing the heart. Along with this reduction operation, the tetra mesh of the luminal blood domain is also deformed simultaneously. When the mesh of the luminal blood domain is deformed significantly, a tetra mesh is newly created. 
     As a simulation finite element model, a combination of cusp models of shell or membrane elements and chorda tendineae models of beam elements or cable elements is added to the natural-condition heart model (tetra model) and luminal blood model (tetra model), as needed. In addition, a systemic circulation model adjusted to the biological conditions is connected. The systemic circulation model is an electrical circuit analogy model formed by suitably combining discrete resistance and capacitance defined by assuming that the blood pressure is voltage and the blood flow is current. On the basis of the contraction force obtained by using the calcium concentration history of an individual tetra element of the natural-condition heart model to which the systemic circulation model is connected for an excitation contraction coupling model, a mechanical pulsation simulation is performed. 
     Parameters are adjusted so that the simulation result match the biological data of the biological body whose dynamics and/or functions are to be derived, such as a result of a cardiovascular catheterization test, a blood pressure value, a pressure-volume relationship, and image data about an MRI image, a CT image, an echocardiogram, and mechanical indices extracted from the biological data. When the oxygen saturation or concentration of the blood of the biological body whose dynamics are to be predicted has already been measured, an advection diffusion equation using a fluid velocity distribution of the blood obtained through fluid-structure interaction analysis is solved by using a finite element method using the same ALE fluid mesh as that used in the fluid-structure interaction analysis, and the conformity is checked. Examples of the parameters adjusted include the transition rate that determines the stochastic state transition of myosin molecules in the excitation-contraction coupling model, coordination parameters, and the resistance and capacitance of the systemic circulation model. 
     The parameter adjustment is performed on the biological body whose dynamics and/or functions are to be derived, so as to match the corresponding biological data. Likewise, the parameter adjustment is also performed on virtual condition change that does not involve morphological change of the biological organ, namely, on virtual biological data set to virtual change of a function of the biological organ or virtual change of a motion or a physical property of fluid. When the biological organ is a heart, for example, the virtual change of a function signifies performing parameter adjustment for virtually reducing the contraction of the heart muscle of the heart or changing an APD distribution in the electrical excitation propagation simulation. The virtual change of a motion or a physical property of fluid signifies performing parameter adjustment for virtually decreasing or increasing the amount of blood that circulates in the body. 
     In accordance with the following method, dynamics and/or functions of the biological organ are derived from the simulation result after the parameter adjustment. 
     The simulation result after the parameter adjustment includes the positions of element or nodes and physical quantities on elements or nodes per simulation time step. The elements are heart tetra elements, heart voxel elements, luminal blood tetra elements, etc. The position of an element is a predetermined position in the element such as the center of gravity of the element. A node is an apex of an element, for example. For example, such a simulation result is visualized as graphs or the like, and dynamics and/or functions of the biological organ can be derived. 
     The dynamics and/or functions of the biological organ derived from the simulation result after the parameter adjustment include information relating to the motion of the biological organ having temporal and spatial resolution that cannot be observed by the measurement techniques practically used. The measurement techniques practically used can only observe the motion of the biological organ in a limited spatial range with limited temporal or spatial resolution. In the derivation of the simulation result, the motion of the entire biological organ or the entire site to be observed that cannot be observed by the measurement techniques practically used or physical quantities that cannot be clinically observed can be observed. Consequently, detailed dynamics and/or functions of the observation target can be obtained. 
     In addition, the dynamics and/or functions of the biological organ derived from the simulation result after the parameter adjustment on virtual biological data can be used for prediction of dynamics or functions of the cardiovascular system after medication treatment or fluid or blood infusion treatment is performed on the biological body. 
     When the biological organ is a heart, the dynamics and/or functions obtained by visualization or other derivation methods can be used for prediction of the pump performance of the heart, the hemodynamics, the load on the heart and the lungs, etc. For example, the pump performance of the heart represents the motion of the heart and the motions and functions of the heart valves. For example, the hemodynamics includes the pressure generated by the heart, the pressure at an individual cardiovascular site, the flow volume of the blood at an individual cardiovascular site, the velocity of the blood flow at an individual cardiovascular site, the oxygen saturation of the blood at an individual cardiovascular site, the dissolved gas partial pressure of the blood at an individual cardiovascular site, and the blood concentration of drug at an individual cardiovascular site. For example, the load on the heart and lungs includes the energy consumption amount or efficiency of the heart, the energy conditions of the blood fluid, the pressure caused on the heart muscle, the conditions of the coronary circulation, the systemic vascular resistance, and the pulmonary vascular resistance. Among these derivation results, those for which clinical measurement techniques exist accurately match their respective measured values to such an extent that the derivation results can be applied for clinical purposes. The dynamics and/or functions of the biological organ derived from the simulation result after the parameter adjustment indicate the conditions of the biological body to such an extent that the dynamics and/or functions can practically be used for clinical purposes. Thus, for example, a doctor can gain a better understanding of the pump performance of the heart, the hemodynamics, the load on the heart and the lungs, etc. and can make diagnostic or clinical decisions by using the dynamics and/or functions as a reference. 
     Upon completion of the simulation, the visualization processing for predicting the dynamics and/or functions of the biological organ can be performed by displaying a space-time distribution of physical variables corresponding to a result of the simulation. For example, numerical data of and temporal change of an individual phenomenon, such as the membrane potential or stress by the myocardial excitation propagation or the velocity or the pressure of the blood flow per beat, is visualized by 2D or 3D image processing. In this processing, observation from various points of view is enabled by changing the viewpoint in accordance with input information. In addition, by generating cross sections, observation of any one of the cross sections is enabled. In addition, by quickly generating a moving image of the behavior of the heart and displaying the behavior as an animation, observation suitable for examination, diagnosis, or treatment is enabled. In addition, a part of the heart muscle can be extracted and displayed so that change of the extracted part over time can be checked. 
       FIG. 20  illustrates an example of visualization processing. Namely, the post-processing unit  140  includes a visualization parameter input unit  141 , a data acquisition unit  142 , a myocardium visualization unit  143   a , an excitation propagation visualization unit  143   b , a coronary circulation visualization unit  143   c , a valve visualization unit  154   d , a graph visualization unit  143   e , a medical-image visualization unit  143   f , a blood flow visualization unit  143   g , a blood vessel visualization unit  143   h , a visualization result image display unit  144   a , a 3D display unit  144   b , a graph display unit  144   c , and a medical-data superposition and display unit  144   d.    
     The visualization parameter input unit  141  receives parameters indicating visualization conditions from the terminal device  31  and inputs the received parameters to other elements performing visualization processing. The data acquisition unit  142  acquires data stored in the storage unit  110  and inputs the acquired data to other elements performing visualization or display processing. 
     The myocardium visualization unit  143   a  visualizes the behavior of the myocardium and myocardial physical quantities, for example. For example, the myocardium visualization unit  143   a  represents change of a value corresponding to a physical quantity such as the myocardial pressure as a color change. 
     The excitation propagation visualization unit  143   b  visualizes propagation conditions of excitation of the heart. For example, the excitation propagation visualization unit  143   b  represents change of a value corresponding to a myocardial voltage as a color change. 
     The coronary circulation visualization unit  143   c  visualizes coronary circulation conditions. For example, the coronary circulation visualization unit  143   c  represents change of a physical quantity such as the velocity or pressure of the blood flow in the coronary circulation system as a color change. 
     The valve visualization unit  143   d  visualizes the motions of the valves and the motion of the blood around the valves. For example, the valve visualization unit  143   d  represents the blood flow around the valves as fluid velocity vectors. 
     The graph visualization unit  143   e  statistically analyzes simulation results and represents the results in the form of a graph. For example, on the basis of a simulation result obtained per virtual operation, the graph visualization unit  143   e  evaluates conditions of the patient after an individual virtual operation and generates graphs that indicate the evaluation results. 
     The medical-image visualization unit  143   f  visualizes a medical image such as a CT image or an MRI image. For example, the medical-image visualization unit  143   f  generates display images from image data that indicates medical images. 
     The blood flow visualization unit  143   g  visualizes the blood flow in a blood vessel. For example, the blood flow visualization unit  143   g  generates a fluid velocity vector that indicates a blood flow velocity. 
     The blood vessel visualization unit  143   h  generates display blood vessel object that indicates an actual blood vessel on the basis of a blood vessel element. For example, when the positions and diameters of ends of a blood vessel element are already set, the blood vessel visualization unit  143   h  generates a cylindrical blood vessel object that connects the ends that have the diameters. 
     The visualization result image display unit  144   a  displays an image generated through visualization on the terminal device  31  or the monitor  21 . For example, the 3D display unit  144   b  displays a 3D model of a heart on the terminal device  31  or the monitor  21 . The graph display unit  144   c  displays graphs created by the graph visualization unit  143   e  on the terminal device  31  or the monitor  21 . For example, the medical-data superposition and display unit  144   d  superposes biological data onto a 3D model of a heart and displays the superposed image. 
     In accordance with the following method, a postoperative geometric model/postoperative finite element mesh model is created from a preoperative geometric model/preoperative finite element mesh model. 
     A postoperative finite element mesh model is created by deforming a preoperative geometric model and creating a postoperative geometric model or by directly deforming a preoperative finite element mesh model. 
     A postoperative geometric model is a virtual 3D shape of a postoperative biological organ whose dynamics and/or functions are to be predicted or an immediate biological organ of the biological organ. The postoperative geometric model can be created by changing segment data of the corresponding preoperative geometric model. A postoperative finite element mesh model can be created from the postoperative geometric model. A postoperative finite element mesh model can also be created by deforming a triangle surface mesh model and a voxel mesh model of the corresponding preoperative finite element mesh model (the deformation includes topological deformation). In addition, a postoperative geometric model can be created by deforming a preoperative finite element mesh model through mechanical analysis and newly generating a triangle surface mesh model on the basis of the deformed preoperative finite element mesh model. If a postoperative geometric model is created in this way, a postoperative finite element mesh can be generated by using this postoperative geometric model. 
       FIG. 16  illustrates an example of processing for generating postoperative finite element mesh models. For example, a preoperative triangle surface mesh model and a preoperative voxel mesh model are created on the basis of a preoperative heart tetra mesh model. Next, the preoperative triangle surface mesh model and the preoperative voxel mesh model are deformed in accordance with an operative procedure assumed (the deformation includes topological deformation). In this way, a postoperative triangle surface mesh model and a postoperative voxel mesh model are generated. A postoperative heart tetra mesh model is generated on the basis of the postoperative triangle surface mesh model and the postoperative voxel mesh model. 
     To obtain a virtual 3D shape of a postoperative organ of a biological body whose dynamics and/or functions are to be predicted, for example, a doctor or the like or a person who has received an instruction from a doctor or the like can refer to biological data of the biological body and indicate change of the morphology or shape of the heart or immediate organs on the basis of an operative procedure to be evaluated on a screen on which a preoperative geometric model is displayed. On the basis of the indication about the postoperative organ of the biological body, a postoperative geometric model can be obtained by correcting segment data of the preoperative geometric model, and regeneration of the corresponding surface mesh can be performed. In addition, on the basis of the indication, a postoperative triangle surface mesh model assumed by the doctor or the like can be obtained by directly changing the preoperative triangle surface mesh model. In addition, on the basis of the indication, the postoperative voxel mesh model and the postoperative heart tetra mesh model can be regenerated by deforming the preoperative voxel mesh model (the deformation includes topological deformation). 
     A virtual 3D shape of a postoperative organ of a biological body whose dynamics and/or functions are to be predicted (or a postoperative geometric model or a postoperative finite element mesh model based on the virtual 3D shape) can be created for techniques used in the following virtual operations. Examples of the techniques include techniques used in medical operations and techniques used in surgical operations. Specifically, replacement of large blood vessels, reestablishment of the septum, suturation, bypass creation, flow path formation, banding (strictureplasty), valve replacement, and valvuloplasty. Namely, examples of the techniques include any technique relating to change of a morphological or mechanical phenomenon of a biological organ. For example, replacement of large blood vessels is a technique of separating two large blood vessels, a pulmonary artery and an aorta, from the heart to replace the large blood vessels with artificial blood vessels whose length has been adjusted as needed. If a doctor or the like indicates the position at which the aorta needs to be separated, the preoperative heart tetra mesh model is deformed so that the aorta at the indicated position is separated. In the case of banding, if a doctor or the like indicates or instructs the position at which a blood vessel needs to be narrowed and the eventual diameter, the shape of the blood vessel of the preoperative heart tetra mesh model is deformed. 
       FIG. 17  illustrates an example of how a geometric model changes after large blood vessels are removed. For example, when a doctor or the like inputs an instruction for or indicates removal of large blood vessels, a geometric model  52   a  is created by removing large blood vessels from a preoperative geometric model  52 . Next, when the doctor or the like inputs an instruction for or indicates addition of an aorta, a geometric model  52   b  is created by adding an aorta to the geometric model  52   a . Next, when the doctor or the like inputs an instruction for or indicates addition of a pulmonary artery, a geometric model  54  is created by adding a pulmonary artery to the geometric model  52   b . In this way, the shape is deformed by the operation. 
     A postoperative finite element mesh model may be created for each of a plurality of different techniques (for example  FIG. 11 ), and postoperative simulations may be performed by using the created postoperative finite element mesh models. In this way, regarding each of the techniques, prediction of postoperative dynamics and/or functions can be derived, and the obtained results can be compared with each other. In addition, a doctor or the like can use the results when selecting a most suitable operative procedure. 
     A postoperative simulation is performed by using a postoperative finite element mesh model on the basis of adjusted parameters as follows. 
     A postoperative simulation is performed on a postoperative finite element mesh model on the basis of adjusted parameters in the same way as the above simulation (hereinafter, a preoperative simulation) is performed. As the adjusted parameters, a combination of parameters (a best parameter set) that approximates the biological data the most, which is obtained by the preoperative simulation and parameter adjustment can be used. The adjusted parameters may be a combination of parameters that approximate the virtual biological data the most, which is set in view of virtual condition change of the biological organ, such as virtual change of a function of the biological organ or virtual change of a physical property or motion of fluid of the biological organ. The adjusted parameters may be obtained by performing a simulation and parameter adjustment on the postoperative finite element mesh model. For example, the optimized parameters are parameters included in the electrical conductance of the heart muscle, the site of earliest activation, the APD spatial distribution, the excitation-contraction coupling model, the systemic circulation model, and the like. 
     When a postoperative finite element mesh model is created for each of a plurality of different techniques (for example,  FIG. 11 ), a simulation is performed on each of the postoperative finite element mesh models by using the optimized parameters obtained by the preoperative simulation and parameter adjustment in the same way as the preoperative simulation. 
     The prediction of dynamics and/or functions of a postoperative biological organ is derived from the result of a postoperative simulation in accordance with the following method. 
     The prediction of dynamics and/or functions of a postoperative biological organ is derived from the result of a postoperative simulation in accordance with the same method as the method of the derivation of the prediction of dynamics and/or functions in the preoperative simulation. 
       FIGS. 18A and 18B  illustrate examples of visualization of parts of simulation results obtained before and after an operation.  FIG. 18A  illustrates visualization of a simulation result of a preoperative heart, and  FIG. 18B  illustrates visualization of a simulation result of a postoperative heart. In the example in  FIGS. 18A and 18B , since the preoperative heart has a hole in its atrial septum, blood in the right atrium and blood in the left atrium are mixed. Since a wall has been formed between the right atrium and the left atrium of the postoperative heart, the blood in the right atrium and the blood in the left atrium are not mixed. 
     To assist a doctor in selecting the most suitable operative procedure, an assumable postoperative finite element mesh model may be created from each of a plurality of different techniques. In this way, prediction of dynamics and/or functions of an individual postoperative biological organ can be derived from the corresponding postoperative simulation result, and the results can be compared with each other. 
     Examples of the prediction of dynamics and/or functions of a postoperative biological organ obtained by visualization or other derivation methods include prediction relating to formation of a new shape such as an autogenous, homogeneous, or heterogeneous biological material or an artificial object on a part of a biological organ structure, maintenance on malfunction of a biological organ by using the above biological material or artificial object, and replacement of a function by deforming a part of a biological organ structure. For example, a doctor can predict conditions of a heart that has undergone an operation for congenital heart disease, conditions of a cardiovascular system that has undergone percutaneous coronary intervention or an aortocoronary bypass operation, conditions of a cardiovascular system that has undergone a heart valve replacement operation, conditions of a cardiovascular system that has undergone cardiac valvuloplasty, conditions of a cardiovascular system that has undergone cardiac valve annuloplasty, conditions of a cardiovascular system that has undergone pacemaker treatment including cardiac resynchronization treatment, conditions of a cardiovascular system that has undergone treatment for aortic disease, conditions of a cardiovascular system that has undergone treatment for pulmonary arteriopathy, conditions of a cardiovascular system that has undergone installation of a circulatory assist device, and conditions of a cardiovascular system that has undergone other cardiovascular treatment. Examples of the operation for congenital heart disease include an open-heart operation for congenital heart disease, a catheterization operation for congenital heart disease, and an extracardiac operation for congenital heart disease. Examples of the catheterization operation for congenital heart disease include an operation using a defect closure device and an operation using a balloon catheter. Examples of the extracardiac operation for congenital heart disease include the Blalock-Taussig shunt, pulmonary artery banding, the Glenn operation, and TCPC. The heart valve replacement operation and the cardiac valvuloplasty include a catheterization operation. Examples of the circulatory assist device include medical devices generally used for circulatory assistance, such as an intra-aortic balloon pump, a percutaneous cardiopulmonary support device, a left ventricular assist device, and a right ventricular assist device. The cardiovascular treatment includes medical treatment. 
     In addition, regarding the prediction of dynamics and/or functions of a postoperative biological organ derived from a postoperative simulation result after parameter adjustment on virtual biological data, if the biological organ is a heart, for example, when the treatment content including medication, management of artificial respiration, or the like is changed after an operation on the heart, part or all of the change may be reflected on the prediction of dynamics and/or functions of the postoperative biological organ. As a result, more clinically accurate and practicable prediction of dynamics and/or functions is achieved. 
     The derivation results of the prediction of postoperative dynamics and/or functions indicate biological conditions to such an extent that the derivation results can practically be used for clinical purposes. A doctor or the like can compare the results about the pump performance of the postoperative heart, the hemodynamics, the load on the heart and the lungs, etc. Namely, a doctor or the like can determine the most suitable treatment in view of the results. 
     According to one aspect, a highly accurate fluid-structure interaction simulation on a biological organ having a deformable site that has an interface difficult to track is enabled. 
     According to another aspect, by displaying a fluid-structure interaction simulation result, prediction of dynamics or functions of a biological organ derived from the fluid-structure interaction simulation result or prediction of dynamics or functions after an actual operation performed by a doctor can effectively be used for clinical applications by doctors, for example. 
     All examples and conditional language provided herein are intended for the pedagogical purposes of aiding the reader in understanding the invention and the concepts contributed by the inventor to further the art, and are not to be construed as limitations to such specifically recited examples and conditions, nor does the organization of such examples in the specification relate to a showing of the superiority and inferiority of the invention. Although one or more embodiments of the present invention have been described in detail, it should be understood that various changes, substitutions, and alterations could be made hereto without departing from the spirit and scope of the invention.